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Springer Handbook of Lasers and Optics

Springer Handbooks provide a concise compilation of approved key information on methods of research, general principles, and functional relationships in physical sciences and engineering. The world’s leading experts in the fields of physics and engineering will be assigned by one or several renowned editors to write the chapters comprising each volume. The content is selected by these experts from Springer sources (books, journals, online content) and other systematic and approved recent publications of physical and technical information. The volumes are designed to be useful as readable desk reference books to give a fast and comprehensive overview and easy retrieval of essential reliable key information, including tables, graphs, and bibliographies. References to extensive sources are provided.

Springer

Handbook of Lasers and Optics Frank Träger (Ed.) With CD-ROM, 978 Figures and 136 Tables

123

Editor: Prof. Dr. Frank Träger University of Kassel Department of Physics Heinrich-Plett-Str. 40 34132 Kassel Germany

Library of Congress Control Number:

ISBN-10: 0-387-95579-8 ISBN-13: 978-0-387-95579-7

2007920818

e-ISBN: 0-387-30420-7 Printed on acid free paper

c 2007, Springer Science+Business Media, LLC New York All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+ Business Media, LLC New York, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. The use of designations, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Product liability: The publisher cannot guarantee the accuracy of any information about dosage and application contained in this book. In every individual case the user must check such information by consulting the relevant literature. Production and typesetting: LE-TeX GbR, Leipzig Handbook Manager: Dr. W. Skolaut, Heidelberg Typography and layout: schreiberVIS, Seeheim Illustrations: schreiberVIS, Seeheim & Hippmann GbR, Schwarzenbruck Cover design: eStudio Calamar Steinen, Barcelona Cover production: WMXDesign GmbH, Heidelberg Printing and binding: Stürtz AG, Würzburg SPIN 10892328

100/3100/YL 5 4 3 2 1 0

Foreword

After working for more than four decades in the field of laser science, I am delighted that Springer-Verlag has devoted one of the first volumes of the new Springer Handbook series to lasers and optics. The exhilarating pace of technological advances in our field is still accelerating, and lasers and optical techniques are becoming ever more indispensible as enabling tools in almost any field of science or technology. Since no single physicist, engineer, or graduate student can be an expert in all the important subfields of optical science, a concise, balanced, and timely compilation of basic principles, key applications, and recent advances, written by leading experts, will make a most valuable desk reference. The chosen readable style and attractive, well-illustrated layout is even inviting to casual studying and browsing. I know that I will keep my Springer Handbook of Lasers and Optics close at hand, despite the infinite amount of information (and misinformation) that is readily accessible via the Internet. Munich, February 2007

Theodor W. Hänsch

Prof. Theodor W. Hänsch Nobel Laureate in Physics 2005 Department of Physics Ludwig-MaximiliansUniversität (LMU) Munich, Germany

VII

Preface

It is often said that the 21st century is the century of the photon. In fact, optical methods, materials, and components have reached an advanced state of sophistication hitherto unknown. Optical techniques, particularly those based on lasers, not only find applications in the classical fields of physics and engineering but have expanded into many other disciplines such as medicine, the life sciences, chemistry and environmental research, to mention only a few examples. Nevertheless, progress in optics, photonic materials and coherent light sources continues at a rapid pace: new laser materials are being developed; novel concepts such as optics far beyond the diffraction limit, or nanooptics, are being explored; and coherent light sources generate wavelengths in ranges not previously accessible. In view of the pronounced interdisciplinary nature of optics, the Springer Handbook of Lasers and Optics is designed as a readable desk reference book to provide fast, upto-date, comprehensive, and authoritative coverage of the field. The handbook chapters are grouped into four parts covering basic principles and materials; fabrication and properties of optical components; coherent and incoherent light sources; and, finally, selected applications and special fields such as terahertz photonics, X-ray optics and holography. I hope that all readers will find this Springer Handbook useful and will enjoy using it. Kassel, February 2007

Frank Träger

Prof. Frank Träger Universität Kassel Experimentalphysik I Germany

List of Authors Andreas Assion Femtolasers Produktions GmbH Fernkorngasse 10 Vienna, 1100, Austria e-mail: [emailprotected] Thomas E. Bauer JENOPTIK Polymer Systems GmbH Coating Department Am Sandberg 2 07819 Triptis, Germany e-mail: [emailprotected] Thomas Baumert Universität Kassel Institut für Physik Heinrich-Plett-Str. 40 34132 Kassel, Germany e-mail: [emailprotected] Dietrich Bertram Philips Lighting Philipsstr. 8 52068 Aachen, Germany e-mail: [emailprotected]

Hans Brand Friedrich-Alexander-University of Erlangen-Nürnberg LHFT Department of Electrical, Electronic and Communication Engineering Cauerstr. 9 91058 Erlangen, Germany e-mail: [emailprotected] Robert P. Breault Breault Research Organization, Inc. 6400 East Grant Road, Suite 350 Tucson, AZ 85715, USA e-mail: [emailprotected] Matthias Brinkmann University of Applied Sciences Darmstadt Mathematics and Natural Sciences Schoefferstr. 3 64295 Darmstadt, Germany e-mail: [emailprotected] Uwe Brinkmann Bergfried 16 37120 Bovenden, Germany e-mail: [emailprotected]

Klaus Bonrad Schott Spezialglas AG Division Luminescence Technology Hattenbergstr. 10 55014 Mainz, Germany e-mail: [emailprotected]

Robert Brunner Carl Zeiss AG Central Research and Technology Carl-Zeiss-Promenade 10 07745 Jena, Germany e-mail: [emailprotected]

Matthias Born Philips Research Laboratories Aachen Weisshausstr. 2 52066 Aachen, Germany e-mail: [emailprotected]

Geoffrey W. Burr IBM Almaden Research Center 650 Harry Road San Jose, CA 95120 e-mail: [emailprotected]

Annette Borsutzky Technische Universität Kaiserslautern Fachbereich Physik Erwin-Schrödinger-Str. 49 67663 Kaiserslautern, Germany e-mail: [emailprotected]

Karsten Buse University of Bonn Institute of Physics Wegelerstr. 8 53115 Bonn, Germany e-mail: [emailprotected]

List of Authors

Carol Click Schott North America Regional Research and Development 400 York Avenue Duryea, PA 18642, USA e-mail: [emailprotected] Hans Coufal† IBM Research Division San Jose, CA, USA Mark J. Davis Schott North America Regional Research and Development 400 York Avenue Duryea, PA 18642, USA e-mail: [emailprotected] Wolfgang Demtröder TU Kaiserslautern Department of Physics Am Harzhübel 80 67663 Kaiserslautern, Germany e-mail: [emailprotected] Henrik Ehlers Laser Zentrum Hannover e.V. Department of Thin Film Technology Hollerithallee 8 30419 Hannover, Germany e-mail: [emailprotected] Rainer Engelbrecht Friedrich-Alexander-University of Erlangen-Nürnberg Department of Electrical, Electronic and Communication Engineering Cauerstr. 9 91058 Erlangen, Germany e-mail: [emailprotected] Martin Fally University of Vienna Faculty of Physics, Department for Experimental Physics Boltzmanngasse 5 Vienna, 1090, Austria e-mail: [emailprotected]

Yun-Hsing Fan University of Central Florida College of Optics and Photonics 4000 Central Florida Blvd. Orlando, 32816, USA e-mail: [emailprotected] Enrico Geißler Carl Zeiss AG Central Research and Technology Carl Zeiss Promenade 10 07745 Jena, Germany e-mail: [emailprotected] Ajoy Ghatak Indian Institute of Technology Delhi Physics Department Hauz Khas New Delhi, 110016, India e-mail: [emailprotected] Alexander Goushcha SEMICOA 333 McCormick Avenue Costa Mesa, CA 92626, USA e-mail: [emailprotected] Daniel Grischkowsky Oklahoma State University Electrical and Computer Engineering Engineering South 202 Stillwater, OK 74078, USA e-mail: [emailprotected] Richard Haglund Vanderbilt University Department of Physics and Astronomy 6301 Stevenson Center Lane Nashville, TN 37235-1807, USA e-mail: [emailprotected] Stefan Hansmann Al Technologies GmbH Deutsche-Telekom-Allee 3 64295 Darmstadt, Germany e-mail: [emailprotected]

List of Authors

Joseph Hayden Schott North America Regional Research and Development 400 York Avenue Duryea, PA 18642, USA e-mail: [emailprotected] Joachim Hein Friedrich-Schiller University Jena Institute for Optics and Quantum Electronics Max-Wien-Platz 1 07743 Jena, Germany e-mail: [emailprotected] Stefan W. Hell Max Planck Institute for Biophysical Chemistry Am Fassberg 11 37077 Göttingen, Germany e-mail: [emailprotected] Jürgen Helmcke Physikalisch-Technische Bundesanstalt (PTB) Braunschweig Former Head Quantum Optics and Length Unit (retired) Bundesallee 100 38116 Braunschweig, Germany e-mail: [emailprotected] Hartmut Hillmer University of Kassel Institute of Nanostructure Technologies and Analytics (INA) Heinrich-Plett-Str. 40 34128 Kassel, Germany e-mail: [emailprotected] Günter Huber Universität Hamburg Institut für Laser-Physik Department Physik Luruper Chaussee 149 22761 Hamburg, Germany e-mail: [emailprotected]

Mirco Imlau University of Osnabrück Department of Physics Barbarastr. 7 49069 Osnabrück, Germany e-mail: [emailprotected] Kuon Inoue Chitose Institute of Science and Technology Department of Photonics 758-65 Bibi 066-8655 Chitose, Japan e-mail: [emailprotected] Thomas Jüstel University of Applied Sciences Münster Stegerwaldstr. 39 48565 Steinfurt, Germany e-mail: [emailprotected] Jeffrey L. Kaiser Spectra-Physics Division of Newport Corporation 1335 Terra Bella Avenue Mountain View, CA 94043, USA e-mail: [emailprotected] Ferenc Krausz Max-Planck-Institut für Quantenoptik Hans-Kopfermann-Str. 1 85748 Garching, Germany e-mail: [emailprotected] Eckhard Krätzig University of Osnabrück Physics Department Barbarastr. 7 49069 Osnabrück, Germany e-mail: [emailprotected] Stefan Kück Physikalisch-Technische Bundesanstalt Optics Division Bundesallee 100 38116 Braunschweig, Germany e-mail: [emailprotected]

XII

List of Authors

Anne L’Huillier University of Lund Department of Physics P.O. Box 118 22100 Lund, Sweden e-mail: [emailprotected]

Ralf Malz LASOS Lasertechnik GmbH Research and Development Carl-Zeiss-Promenade 10 07745 Jena, Germany e-mail: [emailprotected]

Bruno Lengeler Aachen University (RWTH) II. Physikalisches Institut Templergraben 55 52056 Aachen, Germany e-mail: [emailprotected]

Wolfgang Mannstadt Schott AG Research and Technology Development Hattenbergstr. 10 55122 Mainz, Germany e-mail: [emailprotected]

Martin Letz Schott Glas Materials Science, Central Research Hattenbergstr. 1 55014 Mainz, Germany e-mail: [emailprotected]

Gerd Marowsky Laser-Laboratorium Göttingen e.V. Hans-Adolf-Krebs-Weg 1 37077 Göttingen, Germany e-mail: [emailprotected]

Gerd Leuchs University of Erlangen-Nuremberg Institute of Optics, Information and Photonics Guenther-Scharowsky-Str. 1 91058 Erlangen, Germany e-mail: [emailprotected] Norbert Lindlein Friedrich-Alexander University of Erlangen-Nürnberg Max-Planck Research Group Institute of Optics Information and Photonics Staudtstr. 7/B2 91058 Erlangen, Germany e-mail: [emailprotected]

Dietrich Martin Carl Zeiss AG Corporate Research and Technology Microstructured Optics Research Carl-Zeiss-Promenade 10 07745 Jena, Germany e-mail: [emailprotected] Bernhard Messerschmidt GRINTECH GmbH Research and Development, Management Schillerstr. 1 07745 Jena, Germany e-mail: [emailprotected]

Dennis Lo† The Chinese University of Hong Kong Hong Kong, P. R. China

Katsumi Midorikawa RIKEN Laser Technology Laboratory Hirosawa 2-1, Wako 351-0198 Saitama, Japan e-mail: [emailprotected]

Stefano Longhi University of Politecnico di Milano Department of Physics Piazza Leonardo da Vinci 32 20133 Milano, Italy e-mail: [emailprotected]

Gerard J. Milburn The University of Queensland Center for Quantum Computer Technology School of Physical Sciences St. Lucia, QLD 4072, Australia e-mail: [emailprotected]

List of Authors

Kazuo Ohtaka Chiba University Center for Frontier Science Photonic Crystals 1-33 Yayoi 263-8522 Chiba, Japan e-mail: [emailprotected]

Steffen Reichel SCHOTT Glas Service Division Research and Technology Development Hattenbergstr. 10 55014 Mainz, Germany e-mail: [emailprotected]

Motoichi Ohtsu Department of Electronics Engineering The University of Tokyo 2-11-16 Yayoi, Bunkyo-ku 113-8656 Tokyo, Japan e-mail: [emailprotected]

Hans-Dieter Reidenbach University of Applied Sciences Cologne Institute of Communications Engineering Institute of Applied Optics and Electronics Betzdorfer Str. 2 50679 Cologne, Germany e-mail: [emailprotected]

Roger A. Paquin Advanced Materials Consultant 1842 E. Pole Star Place Oro Valley, AZ 85737, USA e-mail: [emailprotected] Alan B. Peterson Spectra-Physics Division of Newport Corporation 1335 Terra Bella Avenue Mountain View, CA 94043, USA e-mail: [emailprotected] Klaus Pfeilsticker Universität Heidelberg Institut für Umweltphysik Fakultät für Physik und Astronomie Im Neuenheimer Feld 366 69121 Heidelberg, Germany e-mail: [emailprotected] Ulrich Platt Universität Heidelberg Institut für Umweltphysik Fakultät für Physik und Astronomie Im Neuenheimer Feld 366 69121 Heidelberg, Germany e-mail: [emailprotected] Markus Pollnau University of Twente MESA+ Institute for Nanotechnology P.O. Box 217 7500 AE, Enschede, The Netherlands e-mail: [emailprotected]

Hongwen Ren University of Central Florida College of Optics and Photonics Central Florida Blvd. 162700 Orlando, FL 32816, USA e-mail: [emailprotected] Detlev Ristau Laser Zentrum Hannover e.V. Department of Thin Film Technology Hollerithallee 8 30419 Hannover, Germany e-mail: [emailprotected] Simone Ritter Schott AG Division Research and Technology Development Material Development Hattenbergstr. 10 55122 Mainz, Germany e-mail: [emailprotected] Evgeny Saldin Deutsches Elektronen Synchrotron (DESY) Notkestr. 85 22607 Hamburg, Germany e-mail: [emailprotected] Roland Sauerbrey Forschungszentrum Dresden-Rossendorf e.V. Bautzner Landstr. 128 01328 Dresden, Germany e-mail: [emailprotected]

XIII

XIV

List of Authors

Evgeny Schneidmiller Deutsches Elektronen Synchrotron (DESY) Notkestr. 85 22607 Hamburg, Germany e-mail: [emailprotected]

Sune Svanberg Lund University Division of Atomic Physics P.O. Box 118 22100 Lund, Sweden e-mail: [emailprotected]

Bianca Schreder Schott Glas Division Research and Technology Development Material Development Hattenbergstr. 10 55122 Mainz, Germany e-mail: [emailprotected]

Orazio Svelto Politecnico di Milano Department of Physics Piazza Leonardo da Vinci 32 20133 Milan, Italy e-mail: [emailprotected]

Christian G. Schroer Dresden University of Technology Institute of Structural Physics Zellescher Weg 16 01062 Dresden, Germany e-mail: [emailprotected]

Bernd Tabbert Semicoa Engineering Department 333 McCormick Avenue Costa Mesa, CA 92626, USA e-mail: [emailprotected]

Markus W. Sigrist ETH Zurich, Institute of Quantum Electronics Department of Physics Schafmattstr. 16 8093 Zurich, Switzerland e-mail: [emailprotected]

K. Thyagarajan Indian Institute of Technology Delhi Physics Department Hauz Khas New Delhi, 110016, India e-mail: [emailprotected]

Glenn T. Sincerbox University of Arizona Optical Sciences 1630 East University Boulevard Tucson, AZ 85721, USA e-mail: [emailprotected]

Mary G. Turner Engineering Synthesis Design, Inc. 310 S. Williams Blvd. Tucson, AZ 85711, USA e-mail: [emailprotected]

Elisabeth Soergel University of Bonn Institute of Physics Wegelerstr. 8 53115 Bonn, Germany e-mail: [emailprotected] Steffen Steinberg LASOS Lasertechnik GmbH Carl-Zeiss-Promenade 10 07745 Jena, Germany e-mail: [emailprotected]

Giuseppe Della Valle Polytechnic Institute of Milan Department of Physics Piazza Leonardo da Vinci 32 20133 Milan, Italy e-mail: [emailprotected] Michael Vollmer University of Applied Sciences Brandenburg Department of Physics Magdeburger Str. 50 14770 Brandenburg, Germany e-mail: [emailprotected]

List of Authors

Silke Wolff SCHOTT Spezialglas AG Department of Research & Technology Development, Material Development Optical Glasses Hattenbergstr. 10 55122 Mainz, Germany e-mail: [emailprotected] Matthias Wollenhaupt Universität Kassel Institut für Physik Heinrich-Plett-Str. 40 34132 Kassel, Germany e-mail: [emailprotected] Shin-Tson Wu University of Central Florida College of Optics and Photonics Central Florida Blvd. 162700 Orlando, FL 32816, USA e-mail: [emailprotected]

Helen Wächter ETH Zurich, Institute of Quantum Electronics Department of Physics Schafmattstr. 16 8093 Zurich, Switzerland e-mail: [emailprotected] Mikhail Yurkov Deutsches Elektronen Synchrotron (DESY) Notkestr. 85 22607 Hamburg, Germany e-mail: [emailprotected] Aleksei Zheltikov M.V. Lomonosov Moscow State University Physics Department Vorobyevy gory Moscow, 119992, Russia e-mail: [emailprotected]

XVII

Contents

List of Abbreviations .................................................................................

XXIII

Part A Basic Principles and Materials 1 The Properties of Light Richard Haglund ..................................................................................... 1.1 Introduction and Historical Sketch ................................................. 1.2 Parameterization of Light .............................................................. 1.3 Physical Models of Light ................................................................ 1.4 Thermal and Nonthermal Light Sources .......................................... 1.5 Physical Properties of Light ............................................................ 1.6 Statistical Properties of Light.......................................................... 1.7 Characteristics and Applications of Nonclassical Light ...................... 1.8 Summary ...................................................................................... References ..............................................................................................

3 4 6 9 14 17 24 27 29 29

2 Geometrical Optics Norbert Lindlein, Gerd Leuchs ................................................................... 2.1 The Basics and Limitations of Geometrical Optics ............................ 2.2 Paraxial Geometrical Optics............................................................ 2.3 Stops and Pupils ........................................................................... 2.4 Ray Tracing ................................................................................... 2.5 Aberrations ................................................................................... 2.6 Some Important Optical Instruments .............................................. References ..............................................................................................

33 34 39 60 61 67 72 84

3 Wave Optics Norbert Lindlein, Gerd Leuchs ................................................................... 3.1 Maxwell’s Equations and the Wave Equation .................................. 3.2 Polarization .................................................................................. 3.3 Interference .................................................................................. 3.4 Diffraction .................................................................................... 3.5 Gaussian Beams ............................................................................ References ..............................................................................................

87 88 102 108 123 143 154

4 Nonlinear Optics Aleksei Zheltikov, Anne L’Huillier, Ferenc Krausz ........................................ 4.1 Nonlinear Polarization and Nonlinear Susceptibilities ..................... 4.2 Wave Aspects of Nonlinear Optics ................................................... 4.3 Second-Order Nonlinear Processes ................................................. 4.4 Third-Order Nonlinear Processes ....................................................

157 159 160 161 164

XVIII

Contents

4.5

Ultrashort Light Pulses in a Resonant Two-Level Medium: Self-Induced Transparency and the Pulse Area Theorem.................. 4.6 Let There be White Light: Supercontinuum Generation .................... 4.7 Nonlinear Raman Spectroscopy ...................................................... 4.8 Waveguide Coherent Anti-Stokes Raman Scattering ........................ 4.9 Nonlinear Spectroscopy with Photonic-Crystal-Fiber Sources ........... 4.10 Surface Nonlinear Optics, Spectroscopy, and Imaging ...................... 4.11 High-Order Harmonic Generation .................................................. 4.12 Attosecond Pulses: Measurement and Application........................... References ..............................................................................................

178 185 193 202 209 216 219 227 236

5 Optical Materials and Their Properties Matthias Brinkmann, Joseph Hayden, Martin Letz, Steffen Reichel, Carol Click, Wolfgang Mannstadt, Bianca Schreder, Silke Wolff, Simone Ritter, Mark J. Davis, Thomas E. Bauer, Hongwen Ren, Yun-Hsing Fan, Shin-Tson Wu, Klaus Bonrad, Eckhard Krätzig, Karsten Buse, Roger A. Paquin ................................................................. 5.1 Interaction of Light with Optical Materials ...................................... 5.2 Optical Glass ................................................................................. 5.3 Colored Glasses ............................................................................. 5.4 Laser Glass .................................................................................... 5.5 Glass–Ceramics for Optical Applications .......................................... 5.6 Nonlinear Materials ....................................................................... 5.7 Plastic Optics ................................................................................. 5.8 Crystalline Optical Materials ........................................................... 5.9 Special Optical Materials ................................................................ 5.10 Selected Data ................................................................................ References ..............................................................................................

249 250 282 290 293 300 307 317 323 327 354 360

6 Thin Film Optical Coatings Detlev Ristau, Henrik Ehlers ...................................................................... 6.1 Theory of Optical Coatings .............................................................. 6.2 Production of Optical Coatings ....................................................... 6.3 Quality Parameters of Optical Coatings............................................ 6.4 Summary and Outlook ................................................................... References ..............................................................................................

373 374 378 388 391 393

Part B Fabrication and Properties of Optical Components 7 Optical Design and Stray Light Concepts and Principles Mary G. Turner, Robert P. Breault ............................................................. 7.1 The Design Process ........................................................................ 7.2 Design Parameters ........................................................................ 7.3 Stray Light Design Analysis ............................................................. 7.4 The Basic Equation of Radiation Transfer ........................................

399 399 402 410 412

Contents

7.5 Conclusion .................................................................................... References ..............................................................................................

416 416

8 Advanced Optical Components Robert Brunner, Enrico Geißler, Bernhard Messerschmidt, Dietrich Martin, Elisabeth Soergel, Kuon Inoue, Kazuo Ohtaka, Ajoy Ghatak, K. Thyagarajan ........................................................................................ 8.1 Diffractive Optical Elements ........................................................... 8.2 Electro-Optic Modulators ............................................................... 8.3 Acoustooptic Modulator ................................................................. 8.4 Gradient Index Optical Components ............................................... 8.5 Variable Optical Components ......................................................... 8.6 Periodically Poled Nonlinear Optical Components............................ 8.7 Photonic Crystals ........................................................................... 8.8 Optical Fibers ................................................................................ References ..............................................................................................

419 419 434 438 440 449 459 463 471 494

9 Optical Detectors Alexander Goushcha, Bernd Tabbert ......................................................... 9.1 Photodetector Types, Detection Regimes, and General Figures of Merit .......................................................... 9.2 Semiconductor Photoconductors .................................................... 9.3 Semiconductor Photodiodes .......................................................... 9.4 QWIP Photodetectors ..................................................................... 9.5 QDIP Photodetectors ...................................................................... 9.6 Metal–Semiconductor (Schottky Barrier) and Metal–Semiconductor–Metal Photodiodes...................................... 9.7 Detectors with Intrinsic Amplification: Avalanche Photodiodes (APDs) ....................................................... 9.8 Detectors with Intrinsic Amplification: Phototransistors .................. 9.9 Charge Transfer Detectors............................................................... 9.10 Photoemissive Detectors ................................................................ 9.11 Thermal Detectors ......................................................................... 9.12 Imaging Systems ........................................................................... 9.13 Photography ................................................................................. References ..............................................................................................

503 505 510 512 527 529 530 532 537 539 546 549 553 555 560

Part C Coherent and Incoherent Light Sources 10 Incoherent Light Sources Dietrich Bertram, Matthias Born, Thomas Jüstel ........................................ 10.1 Incandescent Lamps ...................................................................... 10.2 Gas Discharge Lamps ..................................................................... 10.3 Solid-State Light Sources ............................................................... 10.4 General Light-Source Survey .......................................................... References ..............................................................................................

565 565 566 574 581 581

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Contents

11 Lasers and Coherent Light Sources Orazio Svelto, Stefano Longhi, Giuseppe Della Valle, Stefan Kück, Günter Huber, Markus Pollnau, Hartmut Hillmer, Stefan Hansmann, Rainer Engelbrecht, Hans Brand, Jeffrey Kaiser, Alan B. Peterson, Ralf Malz, Steffen Steinberg, Gerd Marowsky, Uwe Brinkmann, Dennis Lo† , Annette Borsutzky, Helen Wächter, Markus W. Sigrist, Evgeny Saldin, Evgeny Schneidmiller, Mikhail Yurkov, Katsumi Midorikawa, Joachim Hein, Roland Sauerbrey, Jürgen Helmcke ..................................... 11.1 Principles of Lasers ........................................................................ 11.2 Solid-State Lasers.......................................................................... 11.3 Semiconductor Lasers .................................................................... 11.4 The CO2 Laser................................................................................. 11.5 Ion Lasers ..................................................................................... 11.6 The HeNe Laser .............................................................................. 11.7 Ultraviolet Lasers: Excimers, Fluorine (F2 ), Nitrogen (N2 ) .................. 11.8 Dye Lasers ..................................................................................... 11.9 Optical Parametric Oscillators ......................................................... 11.10 Generation of Coherent Mid-Infrared Radiation by Difference-Frequency Mixing .................................................... 11.11 Free-Electron Lasers ...................................................................... 11.12 X-ray and EUV Sources ................................................................... 11.13 Generation of Ultrahigh Light Intensities and Relativistic Laser–Matter Interaction........................................ 11.14 Frequency Stabilization of Lasers ................................................... References ..............................................................................................

583 584 614 695 726 746 756 764 777 785 801 814 819 827 841 864

12 Femtosecond Laser Pulses: Linear Properties, Manipulation,

Generation and Measurement Matthias Wollenhaupt, Andreas Assion, Thomas Baumert .......................... 12.1 Linear Properties of Ultrashort Light Pulses ..................................... 12.2 Generation of Femtosecond Laser Pulses via Mode Locking .............. 12.3 Measurement Techniques for Femtosecond Laser Pulses .................. References ..............................................................................................

937 938 959 962 979

Part D Selected Applications and Special Fields 13 Optical and Spectroscopic Techniques Wolfgang Demtröder, Sune Svanberg........................................................ 987 13.1 Stationary Methods ....................................................................... 987 13.2 Time-Resolved Methods ................................................................ 1012 13.3 LIDAR ............................................................................................ 1031 References .............................................................................................. 1048 14 Quantum Optics Gerard Milburn ........................................................................................ 1053 14.1 Quantum Fields ............................................................................. 1053

Contents

14.2 States of Light ............................................................................... 14.3 Measurement ................................................................................ 14.4 Dissipation and Noise .................................................................... 14.5 Ion Traps....................................................................................... 14.6 Quantum Communication and Computation ................................... References ..............................................................................................

1055 1058 1061 1066 1070 1077

15 Nanooptics Motoichi Ohtsu ........................................................................................ 15.1 Basics ........................................................................................... 15.2 Nanophotonics Principles .............................................................. 15.3 Nanophotonic Devices ................................................................... 15.4 Nanophotonic Fabrications ............................................................ 15.5 Extension to Related Science and Technology ................................. 15.6 Summary ...................................................................................... References ..............................................................................................

1079 1079 1080 1082 1085 1088 1088 1089

16 Optics far Beyond the Diffraction Limit:

Stimulated Emission Depletion Microscopy Stefan W. Hell .......................................................................................... 1091 16.1 Principles of STED Microscopy ......................................................... 1092 16.2 Nanoscale Imaging with STED ......................................................... 1094 References .............................................................................................. 1097 17 Ultrafast THz Photonics and Applications Daniel Grischkowsky ................................................................................ 17.1 Guided-Wave THz Photonics .......................................................... 17.2 Freely Propagating Wave THz Photonics .......................................... References ..............................................................................................

1099 1101 1116 1145

18 X-Ray Optics Christian G. Schroer, Bruno Lengeler ......................................................... 18.1 Interaction of X-Rays with Matter .................................................. 18.2 X-Ray Optical Components ............................................................. References ..............................................................................................

1153 1154 1156 1162

19 Radiation and Optics in the Atmosphere Ulrich Platt, Klaus Pfeilsticker, Michael Vollmer ......................................... 19.1 Radiation Transport in the Earth’s Atmosphere ............................... 19.2 The Radiation Transport Equation .................................................. 19.3 Aerosols and Clouds....................................................................... 19.4 Radiation and Climate ................................................................... 19.5 Applied Radiation Transport: Remote Sensing of Atmospheric Properties ..................................... 19.6 Optical Phenomena in the Atmosphere .......................................... References ..............................................................................................

1165 1166 1169 1172 1174 1176 1182 1197

XXI

XXII

Contents

20 Holography and Optical Storage Mirco Imlau, Martin Fally, Hans Coufal† , Geoffrey W. Burr, Glenn T. Sincerbox ................................................................................... 20.1 Introduction and History ............................................................... 20.2 Principles of Holography ................................................................ 20.3 Applications of Holography ............................................................ 20.4 Summary and Outlook ................................................................... 20.5 Optical Data Storage ...................................................................... 20.6 Approaches to Increased Areal Density ........................................... 20.7 Volumetric Optical Recording ......................................................... 20.8 Conclusion .................................................................................... References ..............................................................................................

1205 1206 1207 1217 1222 1223 1225 1227 1239 1239

21 Laser Safety Hans-Dieter Reidenbach .......................................................................... 21.1 Historical Remarks......................................................................... 21.2 Biological Interactions and Effects.................................................. 21.3 Maximum Permissible Exposure ..................................................... 21.4 International Standards and Regulations ....................................... 21.5 Laser Hazard Categories and Laser Classes ....................................... 21.6 Protective Measures....................................................................... 21.7 Special Recommendations ............................................................. References ..............................................................................................

1251 1252 1253 1260 1267 1268 1270 1273 1275

Acknowledgements ................................................................................... About the Authors ..................................................................................... Detailed Contents...................................................................................... Subject Index.............................................................................................

1277 1279 1295 1313

XXIII

List of Abbreviations

A AEL AFM ANL AOM AOPDF APCVD AR ARS ASE AWG

accessible emission limit atomic force microscope Argonne National Laboratory acoustooptic modulator acoustooptic programmable dispersive filter atmospheric pressure chemical vapor deposition antireflection angle-resolved scattering amplified spontaneous emission arrayed waveguide

B BBO BH BLIP BZ

β-Barium-Borate buried heterostructure background-limited infrared photodetector Brillouin zone

C C–D CALIPSO CARS CAT CCD CCIS CCRF CGH CIPM CMOS COC COP CPA CRDS CRI CTIS CVD CW

Cole–Davidson fractional exponent β Cloud-aerosol lidar and infrared pathfinder satellite observations coherent anti-Stokes Raman scattering coplanar air transmission charge-coupled device charge-coupled image sensor capacitively coupled RF computer generated hologram Comité International des Poids et Mesures complementary metal–oxide–semiconductor detector cyclic olefin copolymer cyclic olefin polymer chirped-pulse amplification cavity-ring-down spectroscopy color rendering index charge transfer image sensor chemical vapor deposition continuous wave

D DARPA

United States Defense Advanced Research Projects Agency

DBF DBR DCF DEPFET DESY DEZn DFB DFG DFWM DGD DIAL DLA DOE DOM DOS DRO DRS DWDM

distributed feedback distributed Bragg reflector dispersion-compensating fiber depleted field effect transistor structure Deutsches Elektronen-Synchrotron diethylzinc distributed feedback difference-frequency generation degenerate four-wave mixing differential group delay differential absorption LIDAR direct laser acceleration diffractive optical element dissolved organic matter density of states doubly resonant OPO configurations double Rayleigh scattering dense wavelength division multiplexed

E ECDL EDFA EEDF EFDA EFS EL ELA EM EMC EMT EO EOM erf ESA ESRF EUV EWOD ErIG

extended-cavity diode laser erbium-doped fiber amplifier electron energy distribution function Er-doped fiber amplifiers equi-frequency surface electroluminescence excimer laser annealing electromagnetic electromagnetic compatibility effective-medium theory electrooptic electrooptic modulator error function excited-state absorption European Synchrotron Radiation Facility extreme ultraviolet electrowetting on dielectrics erbium iron garnet

F FBG FDPM FDTD FEL FET FIFO FLASH FOM FOV

fiber Bragg grating frequency-domain phase measurement finite-difference time domain free-electron laser field effect transistor first-in first-out free-electron-laser Hamburg figure of merit field of view

XXIV

List of Abbreviations

FP FR FROG FTS FWHM FWM FZP FoM

Fabry–Pérot Faraday rotator frequency-resolved optical gating Fourier-transform spectroscopy full width at half-maximum four-wave mixing Fresnel zone plate figure of merit

G GAC GASMAS GC GCF GDD GLAS GLS GRIN GSA GTI GVD

grating assisted coupler gas in scattering media absorption spectroscopy gain-coupled geometrical configuration factor group delay dispersion geoscience laser altimeter system sulfide glasses GaLaS gradient index ground-state absorption Gires–Tournois interferometer group velocity dispersion

H HDPE HDSS HHG HMO HOMO HR HVPE

high-density polyethylene holographic data storage system high-order-harmonic generation heavy metal oxide highest occupied molecular orbital highly reflecting hydride-vapor-phase epitaxy

I IAD IBAD IBS ICLAS ICP IL ILRC IR ITO

ion-assisted deposition ion-beam-assisted deposition ion-beam sputtering International Coordination Group for Laser Atmospheric Studies inductively coupled plasma interference lithography International Laser Radar Conferences infrared indium–tin oxide

K KB

Kirkpatrick–Baez

L lcp LC–SLM

left-circularly polarized light liquid-crystal spatial light modulator

LCVD LCoS LD LEAF LH LIBS LIDAR LIDT LIF LMJ LOQC LPCVD LPE LSHB LSO LT LT-GaAs LTG-GaAs LUMO LWFA

laser(-induced) chemical vapor deposition liquid crystal on silicon laser diode large effective area left-handed laser-induced breakdown spectroscopy light detecting and ranging laser-induced damage threshold laser-induced fluorescence laser megajoule linear optics quantum computing low-pressure CVD liquid-phase epitaxy longitudinal spatial hole burning laser safety officer low-temperature low-temperature GaAs low-temperature-grown GaAs lowest unoccupied molecular orbital laser wakefield acceleration

M MCP MCVD MFD MI MIS MMA MOCVD MOPA MOS MOT MPC MPE MPMMA MQW MSR MTF

microchannel plate modified chemical vapor deposition multilayer fluorescent disk modulation instability metal–insulator–semiconductor methyl methacrylate metalorganic chemical vapour epitaxy master-oscillator power-amplifier metal–oxide–semiconductor magnetooptical trap metallic photonic crystal maximum permissible exposure modified poly(methyl methacrylate) multiquantum well magnetic super-resolution modulation transfer function

N NA NCPM NEP NFL NGL NIF NIM NLSE NLSG NLTL NOHD NRI NSIC

numerical aperture noncritical phase matching noise equivalent power nanofocusing lenses next-generation lithography National Ignition Facility nearly index-matched nonlinear Schrödinger equation nonlinear signal generator nonlinear transmission line nominal ocular hazard distance nonresonant intrinsic National Storage Industry Consortium

List of Abbreviations

O OCT OFA OFHC OFI OLED OP OPA OPCPA OPD OPG OPL OPO OPS ORMOSIL OSNR OTF OVD

optical coherence tomography optical fibre amplifier oxygen-free high conductivity optical-field ionization organic light-emitting device oriented-patterned optical parametric amplifier optical parametric chirped pulse amplification optical path difference optical parametric generation optical path length optical parametric oscillator optically pumped semiconductor laser organically modified silicates optical signal-to-noise ratio optical transfer function outside vapor deposition

P PA PB PBG PBS PBS PC PCB PCF PD PDH PDLC PDMS PECVD PEDT/PSS PESRO PG PIC PICVD PL PLD PMD PMMA PMT POLLIWOG PPE PPKTP PPLN PPV PQ

PS PSA PSF PTV PVD PWM PZT

polystyrene projected solid angle point spread function peak-to-valley physical vapor deposition pulse width modulator piezoelectric transducer

Q QC QCL QD QDIP QND QPM QW QWIP QWOT QWP

quasicrystals quantum cascade laser quantum dot quantum-dot infrared photodetector quantum nondemolition quasi-phase matching quantum well quantum well infrared photodetector quarter-wave optical thickness quarter-wave plate

R photon avalanche photonic band photonic band gap photonic band structure polarizing beam splitter photonic crystal printed circuit board photonic-crystal fibers photodetector Pound-Drever-Hall technique polymer-dispersed liquid crystal polydimethylsiloxane plasma-enhanced chemical vapor deposition polyethylenedioxythiophene/ polystyrylsulfonat pump-enhanced SRO polarization gate particle-in-cell plasma impulse CVD photoluminescence pulsed-laser deposition polarization mode dispersion polymethylmethacrylate photomultiplier tube polarization-labeled interference versus wavelength for only a glint personal protective equipment periodically poled potassium titanyl phosphate periodically poled lithium niobate poly-para-phenylenevinylene phenanthraquinone

RAM rcp RCWA RDE RDS RE RESOLFT RF RFA RGB RIE RIKE RLVIP RMS RPE R/W

residual amplitude modulation right-circularly polarized light rigorous coupled wave analysis rotating disc electrode relative dispersion slope rare-earth reversible saturable optical fluorescence transition radio frequency Raman fiber amplifier red, green and blue reactive-ion etching Raman-induced Kerr effect reactive low-voltage ion plating root-mean-square retinal pigment epithelium rewritable

S SAR SASE SBS SC SCP SEM SFG SG SHG si SI SIL SLAR

synthetic-aperture radar self-amplified spontaneous emission stimulated Brillouin scattering supercontinuum stretcher–compressor pair scanning electron microscope sum-frequency generation sampled grating second-harmonic generation semiinsulating Système International solid-immersion lens side-looking airborne radar

XXV

XXVI

List of Abbreviations

SLM SM-LWFA SMF SMSR SNR SOA SOI SOS SPDC SPIDER SPM SRO SRS SS SSDL SSFS SSG SSI STED STP STPA STRUT SVEA

spatial light modulator self-modulated laser wakefield acceleration single-mode fiber side-mode suppression ratio signal-to-noise ratio semiconductor optical amplifier silicon-on-insulator silicon-on-sapphire spontaneous parametric down conversion spectral phase interferometry for direct electric field reconstruction self-phase modulation singly resonant OPO stimulated Raman scattering stainless-steel solid-state dye laser soliton self-frequency shift superstructure grating spatial–spectral interference stimulated emission depletion standard temperature and pressure sequential two-photon absorption spectrally temporally resolved upconversion technique slowly varying envelope approximation

T TADPOLE TCE TDSE TEF TEM TF THG THz-TDS TIP TIR TNSA TOD

temporal analysis by dispersing a pair of light E-fields transient collisional excitation time-dependent Schrödinger equation trap enhanced field transverse electric magnetic thin film third-harmonic generation THz time-domain spectroscopy truncated inverted pyramid total internal reflection target normal sheath acceleration third-order dispersion

TRO TS TTF TTG TV

triply resonant OPO total scattering TESLA test facility tunable twin guide television

U UV

ultraviolet

V VC VCSEL VLSI VPE

vertical cavity vertical-cavity surface-emitting laser very large scale integration vapor-phase epitaxy

W WDM WG WGP WORM

wavelength division multiplexing waveguide wire-grid polarizer write-once, read-many times

X XFEL XPM XUV

X-ray FELS cross-phase modulation extreme ultraviolet (soft X-ray)

Y YAG YAP YLF YSZ YVO

yttrium aluminium garnet yttrium aluminium perovskite yttrium lithium fluoride yttria-stabilized zirconia yttrium vanadate

Z ZAP zero additional phase ZAP-SPIDER zero-additional-phase SPIDER

1. The Properties of Light

The mystery of light has formed the core of creation stories in every culture, and attracted the earnest attentions of philosophers since at least the fifth century BCE. Their questions have ranged from how and what we see, to the interaction of light with material bodies, and finally to the nature of light itself. This chapter begins with a brief intellectual history of light from ancient Greece to the end of the 19th century. After introducing the physical parameterization of light in terms of standard units, three concepts of light are introduced: light as a wave, light as a quantum particle, and light as a quantum field. After highlighting the distinctive characteristics of light beams from various sources – thermal radiation, luminescence from atoms and molecules, and synchrotron light sources – the distinctive physical characteristics of light beams are examined in some detail. The chapter concludes with a survey of the statistical and quantum-mechanical properties of light beams. In the appropriate limits, this treatment not only recovers the classical description of light waves and the semiclassical view of light as a stream of quanta, but also forms a consistent description of quantum phenomena – such as interference phenomena generated by single photons – that have no classical analogs.

1.1

1.2

Introduction and Historical Sketch ......... 1.1.1 From the Greeks and Romans to Johannes Kepler....................... 1.1.2 From Descartes to Newton ............. 1.1.3 Newton and Huygens ................... 1.1.4 The 19th Century: The Triumph of the Wave Picture .... Parameterization of Light ..................... 1.2.1 Spectral Regions and Their Classification .................

1.2.2 Radiometric Units......................... 1.2.3 Photometric Units ........................ 1.2.4 Photon and Spectral Units .............

7 7 8

Physical Models of Light ........................ 1.3.1 The Electromagnetic Wave Picture .. 1.3.2 The Semiclassical Picture: Light Quanta................................ 1.3.3 Light as a Quantum Field ..............

9 9 12 13

1.4

Thermal and Nonthermal Light Sources .. 1.4.1 Thermal Light .............................. 1.4.2 Luminescence Light ...................... 1.4.3 Light from Synchrotron Radiation...

14 15 16 17

1.5

Physical Properties of Light ................... 1.5.1 Intensity ..................................... 1.5.2 Velocity of Propagation ................. 1.5.3 Polarization................................. 1.5.4 Energy and Power Transport .......... 1.5.5 Momentum Transport: The Poynting Theorem and Light Pressure .......... 1.5.6 Spectral Line Shape ...................... 1.5.7 Optical Coherence ........................

17 17 18 18 20

1.3

1.6

4 4 4 5

1.7

Statistical Properties of Light................. 1.6.1 Probability Density as a Function of Intensity.............. 1.6.2 Statistical Correlation Functions ..... 1.6.3 Number Distribution Functions of Light Sources ...........................

21 21 23 24 24 25 26

Characteristics and Applications of Nonclassical Light ............................. 1.7.1 Bunched Light ............................. 1.7.2 Squeezed Light ............................ 1.7.3 Entangled Light ...........................

27 27 27 28

1.8

Summary .............................................

References ..................................................

Part A 1

The Propertie

Part A

Basic Principles and Materials

Part A 1.1

1.1 Introduction and Historical Sketch 1.1.1 From the Greeks and Romans to Johannes Kepler The history of optics from the fifth century BCE until the early 17th century CE can be read as a single, prolonged attempt to elucidate, first qualitatively and then quantitatively, the nature of light as it is revealed through the phenomena of propagation, reflection and refraction. The earliest known theories about the nature of light originated with Empedocles of Agrigentum (fifth century BCE) and his contemporary, Leucippus. To the latter is attributed the notion that external objects are enveloped by eidola, “a kind of shadow or some material simulacrum which envelopes the bodies, quivers on the surface and can detach itself from them” in order to convey to the soul “the shape, the colors and all the other qualities of the bodies from which they emanate” [1.1]. A century later, Plato and his academy characterized light as a variant of elemental fire and theorized that seeing results from a conjunction of a ray emitted by the object seen and a “visual ray,” emitted by the seeing eye [1.2]. This picture was contentious from the start: Plato’s pupil Aristotle fumed that “to say, as the Ancients did, that colors are emissions and that this is how we see, is absurd” [1.1]. Nevertheless, the emission theory was debated well into the 16th century. Another of Plato’s pupils, the mathematician Euclid, wrote treatises on optics and catoptrics that were still being translated seven centuries later. Euclid’s work is distinguished from that of his predecessors by conclusions deduced from postulates; in the Optics, he adduces a model of ray optics that can be translated into recognizable principles of geometrical optics including the law of reflection from a plane surface; the concept of a near point for the eye; and the focusing of light by concave surfaces [1.3]. The Roman philosopher Lucretius (early first century BCE) gave to the world in his De Rerum Natura the most detailed ancient understanding of not only the geometry of light propagation, but also the effects of intensity on the observer. Two other ancient texts – by Hero (first century CE) and Ptolemy (second century CE), both of Alexandria – are important historically. Hero postulated the law of reflection in a form strikingly similar to that which emerged much later as Fermat’s principle of least time. Heros’s countryman Ptolemy produced a text on optics distinguished by its use of axiomatics coupled to experimental studies of reflection from curved surfaces and an attempt at developing a law of refraction. The data on

refraction are remarkably accurate, [1.4] and his attempt to provide a mathematical model, though unsuccessful, nevertheless stamps the work as modern. Building on the philosophical foundation laid by Aristotle, medieval opticians focused primarily on the phenomenon of refraction and made important predictions about the nature of light [1.1]. The ninth-century Baghdad philosopher Abu Hsuf Yaqub Ibn Is-haz (Alkindi) improved on the concept of the visual ray by requiring that it should have a physiological effect on the eye. In De Aspectibus, he mounted the first serious attack, supported by observations, on the theory of light as a stream of simulacra. Abu Ali al-Hasan ibn al-Haitham – known widely by his Latin name, Alhazen – published The Book of Optics (De aspectibus, or On Vision) in the 11th century CE. This text was translated into Latin and used until the early 17th century. His diagrams of the human visual apparatus correct some, though not all, the errors made by Galen, who worked only from dissections of animals. Because Alhazen understood how the eye lens refracted incoming rays of light, he was able to show that every point on the surface of an object in the visual field of the eye maps onto a point on the optic nerve to make a faithful, small-scale image of the object. By the beginning of the 12th century, western European scholars had in their possession both the works of the Greeks and those of the Muslim scholars. These centuries see a working out of the contradictions inherent in these competing views by late-medieval thinkers in England, France and Italy, [1.5] including Robert Grosseteste and Roger Bacon who were unwilling to accept the dogmatism of the Scholastics. In particular, they saw the phenomenology of the rainbow as a key to the understanding of refraction and reflection. The origin of the rainbow was correctly explained for the first time by Theodoric of Freiburg in the 15th century.

1.1.2 From Descartes to Newton By the time of Johannes Kepler’s death in the mid-17th century, the concept of light as a geometrical ray emanating from an object and collected by the eye was firmly established, and the emphasis shifted to theoretical questions about the mechanisms of refraction and reflection that could only be answered by understanding the properties of light. Moreover, there was increasing emphasis from the mid-17th century onward on carefully controlled experimentation, not simply ob-

The Properties of Light

optical instruments such as telescopes, microscopes and eventually spectrometers [1.10].

1.1.3 Newton and Huygens The early part of the 18th century saw the rise of the two competing theories about the nature of light that were to dominate the next century and a half. These are embodied in the lives and work of the two principals: Isaac Newton (1642–1728) and Christiaan Huygens (1629– 1695). The dispersion of light in a prism was known well before the young Isaac Newton “procured . . . a prism with which to try the celebrated phenomenon of colors.” Newton’s experimentum crucis was designed to show that white light could be decomposed into constituent colors that were dispersed according to a corpuscular model [1.11]. However, Newton’s Opticks, when published in 1710, was a curious admixture of projectile or corpuscular ideas and crude wave theories. Newton believed in the ether as a required medium to support the projectiles, and expected that the ether would undulate as light corpuscles passed through it. However, he was convinced on the basis of the corpuscular model that light traveled faster in material media, an assumption that would not be conclusively disproved until Foucault’s experiments in 1850. Challenges to Newton’s corpuscular theory came from kinematical theories that viewed light as one or another kind of vibrational motion: a vibratory motion supported by an ether (Hooke, 1665); or a propagating pulse-like disturbance in the ether (Huygens, 1690) [1.12]. Leonhard Euler explained refraction at an interface based on the vibrational theory, arguing that dispersion resulted from a variation of vibrational motion with color [1.13]. In Germany at least, Euler was seen as the originator of a wave model that could replace Newton’s corpuscular theory. In France, Huygens developed a geometrical construction of secondary wavelets to trace the propagation of a wave in time, laying a conceptual foundation for early 19th-century experiments in interference and diffraction that ultimately undermined the corpuscular hypothesis.

1.1.4 The 19th Century: The Triumph of the Wave Picture By the last quarter of the 18th century, it was clear that Newton’s corpuscular theory could not match the experimentally measured velocity of light in materials; moreover, experiments by Malus and Arago had shown

Part A 1.1

servation. Harmonized with mathematical models, this experimental philosophy proved to be the way to establish scientific knowledge of light on the strongest foundation [1.6]. René Descartes and the Cartesian thinkers who followed his lead, built a science of light and optics as part of a more general mathematical theory of physics, with his Dioptrics and the Discourses [1.7]. The Cartesian theory is distinguished by the concept of light as a vibration in a diaphanous medium that transmits the undulations from object to eye, a tendency to motion in particles of the embedding medium. Robert Hooke, Thomas Hobbes and Christiaan Huygens were likewise committed to vibrational theories of light. The first experimental evidence of what would eventually be convincing evidence for the wave theory of light came in 1658 with the publication of Grimaldi’s first memoir on diffraction. Pierre Fermat (1601–1665) solved one of the problems that the Dioptrics had treated badly, and did so in a way that was characteristic of what Newton would later call “mathematical philosophy.” Fermat’s simple idea was based on the rectilinear propagation of light, and the postulate that light travels less rapidly in a dense, material medium than in air. From this, he hypothesized that a light ray always follows that path that allows it to travel a given trajectory in the shortest time. It is possible to derive Snell’s law of refraction from this principle of least time [1.8]. Fermat based his theory on the assumption that the speed of light was finite, and that it was slower in material bodies than in air or vacuum – clearly contradicting Descartes, who believed the speed of light to be infinite. The Cartesian postulate was disproved when Cassini (in 1675) and Ole Römer (a year later), measured the time it took light to pass across the earth’s orbit based on observation of the transit time of Jupiter: about 11 minutes. The surveyors – Cassini in Paris and Jean Richer in Guyana – were measuring the Earth’s orbit. Christiaan Huygens, court astronomer to Louis XIV, proposed a figure of 12, 000 earth diameters for the orbital diameter, and thereby arrived in his Treatise on Light at an estimate of 2.3 × 108 m/s, within 20% of the currently accepted value and very close to the value calculated by Newton [1.9]. Grimaldi (1658) had discovered the phenomenon of diffraction, the explanation of which led in time to the ascendance of the wave theory. Progress in the science of light during this period was also aided immensely by the development of the differential and integral calculus and by the invention of high-quality clear glass for lenses, prisms and

1.1 Introduction and Historical Sketch

Part A

Basic Principles and Materials

Part A 1.2

that light has a new property, which came to be called polarization, that does not fit within the corpuscular picture at all. The earliest systematic studies of polarization phenomena associated with the propagation of light ´ waves are due to Etienne Malus in 1808, in response to a prize competition offered by the Paris Academy for a mathematical description of the phenomenon of double refraction in Iceland spar (calcite). Malus’s discoveries led to the recognition that light is a transverse electromagnetic wave, in which the electric and magnetic fields are perpendicular to each other and to the direction of propagation. Malus, using his ingenious refractometer, demonstrated in 1807 that the phenomenon of double refraction could be explained mathematically by Huygens’ construction. Fresnel, a dozen years later, was to win the prize competition for his theory of diffraction, even anticipating the objection of Poisson that light diffracted around a tiny opaque object would produce a bright spot in the middle of the geometrical shadow – to be known afterwards as Poisson’s spot [1.4]. Moreover, increasingly powerful mathematical descriptions [1.14] were applied to the phenomena of interference and diffraction studied experimentally by Thomas Young, the London polymath, and Fresnel. It was at last becoming clear that light constituted a qualitatively new kind of wave in which the vibrations were transverse to the direction of propagation of the light [1.15]. Indeed, the transverse character of the vibrations was first suggested by Young in a letter to Arago in 1812, thus hinting that Young was already reinterpreting his interference experiments in a way that differed sharply from previous thinking based on analogies with acoustic waves [1.16]. At virtually the same time, Biot and Savart, Ampère and Faraday were generating the experimental underpinnings for the eventual unification of optics and

electromagnetism. Galvani’s experiments on the stimulus of what was then called animal electricity had shifted attention from electrostatics, the major preoccupation of the eighteenth century, to time-dependent phenomena associated with electricity. However, it was Alessandro Volta who successfully showed that this phenomenon was not due to some vital magnetic force, but that it was no different from ordinary magnetism. While electrophysiology continued to be of major interest to biologists and students of medicine, it was thereafter studied by physicists primarily in relation to other electromagnetic phenomena. Oersted, by showing the deflection of a compass placed next to a current-carrying wire, demonstrated the interconnection of electrical and magnetic phenomena. And Faraday, in 1845, showed that the polarization of light could be rotated by applying a strong magnetic field to a medium through which the light was propagating. Thus the stage was set for the grand synthesis of classical electromagnetic theory. The first step was the publication of James Clerk Maxwell’s theory of electromagnetism in 1869. Maxwell’s theoretical prediction of electromagnetic radiation was verified experimentally by Heinrich Hertz in 1888 with the discovery of Hertzian waves in what now would be called the radiofrequency range of the spectrum. The classical theory of the electron developed by H. A. Lorentz would be the next step in the creation of a 19th-century theory of everything. The only clouds on the horizon were the unsolved problems of black-body radiation and the photoelectric effect, problems whose solutions would lead to the development of quantum physics and the evolution of a new view of light based on its dual character as wave and particle, and later of its accommodation into a fully quantum-mechanical field theory.

1.2 Parameterization of Light The properties of light are parameterized in similar ways in both the classical (wave) and semiclassical (photon) pictures of light. The fundamental physical properties of an electromagnetic wave are its wavelength λ, frequency ν and polarization state; alternatively, the first two of these properties may be stated in the form of a wave number k = 2πλ and angular frequency ω = 2πν. The photon model associates with individual light quanta a particle-like photon energy E photon = ω and momentum pphoton = k, where h = 2π is Planck’s constant. Photons are also associated with a helicity (photon spin) of ±1 that can be related to wave polarization.

The properties of light have been defined by international commissions in four kinds of units now in general use, depending on what properties are to be emphasized: radiometric units, based on the physical units, such as energy, and power, are used to describe the properties of electromagnetic waves or photons; photometric units, which refer to the properties of light as discerned by the human eye; photon units analogous to radiometric units that are normalized to photon energy; and spectral units that parameterize light in terms of its properties at specific frequencies or wavelength.

The Properties of Light

Photon energy hv (eV) e

The electromagnetic spectrum extends over an enormous range of frequencies and wavelengths, from low-frequency radio wavelength vibrations to extremely high-energy, short-wavelength nuclear gamma radiation. Figure 1.1 shows a typical classification scheme, relating wavelengths, frequencies, wave numbers and photon energies to the common designations of spectral regions of interest in optics, extending from the vacuum ultraviolet through the far-infrared. Some of the units employed match the Système International (SI) convention, others are habitually used in specialized science or technology communities.

Wave number v 1 (cm ) c0 10

0.1

Frequency v (Hz)

Wavelength c ë0 = 0 (µm) v 0.1

Vacuum ultraviolet Visible

Near-infrared Mid-infrared

10 10

kBT at 300 K 100

Far-infrared 100

Fig. 1.1 Chart showing the wavelengths, frequencies, wave numbers

1.2.2 Radiometric Units Radiometric units measure the properties of light in terms of physical units of energy and power, without reference to wavelength, and are therefore the most fundamental of the parameters used to describe light [1.19, 20]. The fundamental radiometric units are: radiance, a vector L whose magnitude is the power passing through a surface of unit area into a unit solid angle about the normal to the surface; irradiance, again a vector E, defined as the total power per unit spectral interval passing through a surface of unit area. As shown in Fig. 1.2, the magnitude of the radiance and irradiance depends on the shape of the surface over which one integrates, that is, over the projected area A⊥ as well as the solid angle dΩ into which light is emitted and the perpendicular area of the detector. The definition of spectral interval is not uniform; depending on the resolution or the parameterization desired, it might be given in Å, nm, cm−1 (not the same as 1/cm), or Hz, as here. To convert any radiometric unit X to the corresponding spectral radiometric unit X ν , recall that X = X ν dν for values of the frequency lying between ν and ν + dν.

and photon energies of electromagnetic radiation of interest in optics. (After [1.17])

terminology and symbols as the radiometric units, but with a subscript V for visual. The four fundamental photometric quantities, listed in Table 1.2, are: luminous intensity, the amount of light emitted by a source; luminous flux, the quantity of light transmitted in a given direction; illuminance, the measure of light falling on a surface; and luminance, which measures the brightness of a surface Direction of viewing Radiating surface A

è è

Normal r

Projected surface A cos è

1.2.3 Photometric Units Photometry refers to the measurement of light as it is perceived by the human eye; thus these units pertain principally to light with wavelengths of 380–760 nm. In astronomy, photometry also refers to the measurement of apparent magnitudes of celestial objects. Since these quantities depend on the spectral amplitude of light, it is not possible to convert photometric values directly into energy values. The photometric units use the same

Fig. 1.2 Geometry used to define radiometric units of radiance and irradiance in terms of emitting area, detecting area and solid angle of emission. The projected surface area in a given angular direction Θ is A⊥ = A cos Θ, while the solid angle in radiometric units is determined by the projected detector area A p perpendicular to the viewing direction, dΩ = Ap /r 2 . (After [1.18])

Part A 1.2

1.2.1 Spectral Regions and Their Classification

1.2 Parameterization of Light

Part A

Basic Principles and Materials

Part A 1.2

Table 1.1 Radiometric units Radiant energy Radiant energy density Radiant flux (power) Radiant exitance Irradiance Radiant intensity Radiance

Symbol

SI unit

Definition

Qe we Φe Me Ee Ie Le

J=Ws J/m−3 W W m−2 W m−2 W sr−1 W m −2 sr−1

– we = dQ e / dV Φe = dQ e / dt Me = dΦe / dA E e = dQ e / dt Ie = dΦe / dΩ L e = Ie /∆A ≡ d2 Φe / dΩ · dA

Table 1.2 Photometric units Luminous energy Luminous energy density Luminous intensity Luminous power Luminous exitance Illuminance Luminance (Apostilb)

Symbol

SI unit

Photometric unit

Definition

QV WV IV ΦV MV EV LV

J=W s J / m3 W sr−1 W W m−2 Wsr−1 W m−2 sr−1

lm s (talbot) lm s/m3 lm sr−1 = candela (cd) lm (lumen) lm m−2 lux (lx) = lm m−2 asb = 1/π cd/m2

— wV = dQ V / dV IV = dΦV / dΩ ΦV = dQ V / dt MV = dΦV / dA E V = dΦV / dA L V = d2 ΦV / dA · dΩ

considered as a light source. The standard source, or international standard candle, is defined as the intensity of a black-body radiator with an area of 1/60 cm2 Luminous flux Öv (lm)

Luminous efficiency V (ë)

700

600

500 400 510 nm

610 nm

300

200 Violet

100

400

450

Blue

Green

500

Yellow Orange

550

600

Red

650

700

685 lm

1.0

616 lm

0.9

448 lm

0.8

480 lm

0.7

411 lm

0.6

342 lm

0.5

274 lm

0.4

205 lm

0.3

137 lm

0.2

68 lm

0.1

750

Fig. 1.3 The standard CIE luminous efficacy curve for the human

eye, used as the basis for converting between photometric and radiometric units

heated to the melting point of platinum. Two auxiliary quantities, luminous energy and luminous energy density, correspond to the analogous radiometric units. The photometric units carry a subscript V for visual, to distinguish them from their radiometric counterparts; the overbar in the table below signifies an averaged quantity. The Commission Internationale de l’Eclairage (CIE) has developed a standard luminous efficacy curve for the human eye, with respect to which the photometric units are referred (Fig. 1.3). The lumen is defined such that the peak of the photopic (light-adapted) vision spectrum of an average eye has a luminous efficacy of 683 lm/W.

1.2.4 Photon and Spectral Units In the photon picture, there is a different set of descriptive quantities normalized to photon energy or photon number, as shown in Table 1.3. The overbarred quantities denote an average over photon wavelengths as well as over area and solid angle. In some cases – for example, when discussing the spectral brightness of laser or synchrotron sources – it is useful to distinguish physical quantities by their frequency ν. For example, in most cases involving spectroscopy or materials processing with lasers, the

The Properties of Light

1.3 Physical Models of Light

Photon number Photon density Photon flux (power) Photon irradiance Photon intensity Photon radiance

Symbol

SI unit

Definition

n¯ wn Φn En In Ln

Number Number m−3 Number s−1 Number s−1 m−2 Number s−1 sr−1 Number s−1 m−2 sr−1

n¯ = Q v / ω¯ wn = dn/ ¯ dV Φn = dn/ ¯ dt E n = dΦn / dA In = dΦn / dΩ L n = d2 Φn / dA · dΩ

Symbol

SI unit

Definition

Qν wν Φν Mν Eν Iν Lν

J Hz−1 = W s Hz−1 J Hz−1 m −3 W Hz−1 W m−2 Hz−1 W m−2 Hz−1 W sr−1 Hz−1 W m−2 sr−1 Hz−1

– wν = dQ ν / dV Φν = dQ ν / dt Mν = dΦν dA E ν = dQ ν / dt Iν = dΦν / dΩ L ν = d2 Φν / dΩ · dA

Table 1.4 Spectral radiometric units Spectral radiant energy Spectral radiant energy density Spectral radiant flux (power) Spectral radiant exitance Spectral irradiance Spectral radiant intensity Spectral radiance

quantity of interest is not simply the intensity or radiance, but the intensity or radiance available within a certain spectral bandwidth that defines the effective region of laser-materials interaction, in other words, the spectral brilliance. In Table 1.4, the overbarred quan-

tities indicate averages over time, space or solid angle, but not over frequency. To convert any radiometric unit X to the corresponding spectral radiometric unit X ν , recall that X = X ν dν for values of the frequency lying between ν and ν + dν.

1.3 Physical Models of Light By the end of the 19th century, it was generally accepted that the battle between the corpuscular (or emission) and wave theories of light had been resolved in favor of the latter. However, experimental developments – and the inability of classical electrodynamics and statistical mechanics to account for either the photoelectric effect or the shape of the black-body spectrum – drove Planck, Einstein and de Broglie to develop a semiclassical theory of the light quantum (eventually christened the “photon” by G. N. Lewis [1.21]). The picture of light as a quantum field emerged in the second half of the 20th century as it became experimentally possible to investigate phenomena associated both with small numbers of photons, statistically distinct ensembles of photons, and correlations between photons emitted in atomic cascades. We now consider the basic concepts underlying the classical, semiclassical and quantum models of light.

1.3.1 The Electromagnetic Wave Picture Maxwell’s classical theory of electromagnetism [1.22] is based on: Gauss’s law, a mathematical relationship governing the spatial properties of vector fields with or without sources; Faraday’s law of magnetic induction; and Ampère’s law linking currents with spatial variations in magnetic field. Maxwell’s recognition of the displacement current – the induced current due to time-varying electromagnetic fields – made it possible finally to establish the static and dynamical relationships between the electric field E (the force per unit charge) and magnetic induction B (the force per unit current). Together with the constitutive relations for the electric displacement and the magnetic field, Maxwell’s four equations, given below in differential and integral form, govern all classical

Part A 1.3

Table 1.3 Photon units

Part A

Basic Principles and Materials

Part A 1.3

electromagnetic phenomena including electromagnetic waves.

volume

∇·B=0, ∇ · BdV = 0 ,

E α (r, t) =

loop

1 Bα − (Mα + . . .) , µ0

∞

(1.1)

where ρ is the charge density and Q the enclosed charge. The constitutive relations electric displacement vector D and for magnetic field H are ∂ (2) Q αβ + . . . , Dα = ε0 E α + Pα − ∂xβ Hα =

(1.3)

The solutions to this equation include near-field or staticfield terms that decrease as 1/r 3 , an intermediate-range oscillatory field that exists in the so-called induction zone, and the propagating-wave terms whose amplitude decreases as 1/r. A time-dependent scalar field satisfying the wave equation in a source-free region may be decomposed into Fourier components at angular frequencies ωi , i = 0, 1, 2, 3, . . . that satisfy a Helmholtz wave equation, as follows:

volume

surface

n2 ∂2 E =0, c2 ∂t 2 2 2 n ∂ B ∇2 B − 2 2 = 0 , c ∂t c2 = (ε0 µ0 )−1 .

∇2 E −

∇ · D ≡ ∇ · (ε0 E + 4πε0 P) = ρ , Q ∇ · D dV = , ε0

∂B , ∇×E=− ∂t ∂ B · ds , (∇ × E) · dS = − ∂t open surface loop ∂D , ∇×H = J+ ∂t ∂D J+ · ds (∇ × H) · dS = ∂t

magnetic susceptibility µ0

(1.2)

where the terms in parentheses represent the contributions of dielectric and magnetic materials to the electric field (e.g., linear polarization Pα , quadrupole contri(2) butions Q αβ , and so on) and the magnetic induction (e.g., the bulk magnetization Mα ). The dielectric function and the magnetic susceptibilities are properties of materials, and vanish in vacuum. Further information on the constitutive relations, including their relativistic forms, may be found in textbooks [1.23, 24] The second term in Ampère’s law – the last of the four Maxwell equations – was deduced by James Clerk Maxwell from the asymmetry between the electric and magnetic fields that would exist if this displacement current were omitted. The last two of these equations can be combined using the vector identity ∇ × (∇ × V) = ∇(∇ · V) − ∇ 2 V to yield a wave equation that describes the propagation of transverse electromagnetic waves in a medium with a dielectric function ε = n 2 and

E α (r, ωi ) e−iωi t dωi ,

−∞

⇒ ∇ 2 + ki2 E α (r, ωi ) = 0 , ki2 =

ωi2 . c2

(1.4)

The Maxwell equations are generally presented in terms of the force fields E and B. However, it is often more convenient to represent electromagnetic waves in terms of vector and scalar potentials; this is particularly true for making the transition between the classical and quantum field pictures. Given the two homogeneous Maxwell equations, we can define a vector potential that is related to the electric field, the electric potential and the magnetic induction. Furthermore, because the magnetic induction remains unchanged by the addition of the gradient of a scalar function, we have an additional degree of freedom, the choice of gauge, which, in this case, is conveniently chosen to be the so-called Lorentz gauge: B=∇×A, ∂A E+ = −∇Φ , ∂t 1 ∂Φ ∇ · A+ 2 =0. c ∂t

(1.5)

With this choice of gauge, the inhomogeneous Maxwell equations can be decoupled to form a pair of inhomogeneous wave equations in the potentials, in which the sources of the wave fields are the charge and current

The Properties of Light

1 ∂2Φ ρ =− 2 2 ε0 c ∂t 2A 1 ∂ ∇ 2 A − 2 2 = −µ0 J . c ∂t ∇ 2Φ −

(1.6)

The other common gauge is the Coulomb or transverse gauge, which is so named because the source term is only the transverse component of the current density J. The radiation fields are determined by the vector potential alone; the Coulomb potential contributes only to the near field. However, although it will not be discussed further here, the optical near field has assumed significant importance as a probe of nanoscale phenomena now that there are optical techniques for sampling it and coupling it to far-field radiation that can be transported and observed [1.25, 26]. The ratio of electric to magnetic field amplitudes derivable from the Maxwell equations for a plane wave is |E(r, t)| = c|B(r, t)|; in fact, it turns out that this is a general property of electromagnetic waves. This means that for almost all practical purposes in describing the properties of electromagnetic waves, it is safe to focus on the electric field alone. For most purposes, three traveling-wave solutions of the scalar wave equation are sufficient to encompass the most common phenomena of wave optics: the spherical wave typical of light emanating from a point source; the plane wave that is the asymptotic form of the wavefront of a spherical wave at large distances from the source; and the Gaussian beam that describes light emitted by a laser source that is constrained in two spatial dimensions by the laser resonator. Spherical Wave The scalar solution for electromagnetic radiation emitted from a point source, in spherical coordinates, takes the form of a spherical wave (1) (1) (2) (2) Alm h l (kr) + Alm h l (kr) E (r, ω) = l,m

× Ylm (ϑ, ϕ) eiωt , (i)

(1.7)

where h l (kr), i = 1, 2 are the Hankel functions of the first and second kind, Ylm (ϑ, ϕ) are the spherical (i) harmonics, and the coefficients Alm , i = 1, 2 are determined by the boundary conditions. Since the electric field in general is a vector, the general solution is significantly more complex but systematic presentations may be readily found elsewhere [1.24].

Plane Wave The simplest possible solution to Maxwell’s equations is the plane wave, in which the phase fronts are infinite planar surfaces perpendicular to the direction of propagation. In Cartesian coordinates, assuming that the wave propagates in the z-direction, the plane wave for the Fourier component with frequency ω is described by the equation

E (r, t) = E0 (x, y) exp [i (kz z − ωt)] .

(1.8)

The plane wave is a convenient approximation to a small segment of an electromagnetic wave far from a point source where the spherical wavefronts have a very large radius of curvature. Gaussian Beam The beam of light emitted by a laser has properties that are determined by the geometry of the optical resonator for the laser, and for a cylindrically symmetric beam propagating in the z-direction is described by a function of the form [1.23]: r2 , E (r) = E 0 (r, z) exp − 2 w (z)

2 λz w2 (z) = w20 1 + , πw20

√ m 2r 1 2r 2 E 0 (r, z) = L lm w (z) w (z) w2 (z) kr 2 imϕ , × e exp i Φ (z) + 2R (z) (1.9) −1

Φ (z) = − (2l + m + 1) tan z 2R R (z) = z 1 + 2 , z

z − z0 zR

πw20 . (1.10) λ Here the functions L lm (r, z) are the generalized Laguerre polynomials, a complete orthogonal set of functions. The indices {l, m} represent different spatial modes of the Gaussian beam; for most laser experiments, one tries very hard to have a beam in the fundamental Gaussian mode, in which {l = 0, m = 0}. The quantity w0 is frequently called the beam waist, and represents the minimum focal spot size that will be reached at some point along the propagation axis. The intensity falls off zR =

Part A 1.3

densities [1.24]:

1.3 Physical Models of Light

Part A

Basic Principles and Materials

Part A 1.3

exponentially from the central maximum, and the rate at which that happens is governed by the beam waist at the given z coordinate. At that point, the wavefront is planar and perpendicular to the axis of propagation. The behavior of a Gaussian beam is often specified by the Rayleigh range z R (or the confocal beam parameter b = 2z R ) and the divergence angle Θ, shown schematically in Fig. 1.4; these are given respectively by 2πw20 , λ 2λ Θ= πw0 b=

(1.11)

Wave Packets Although all of these solutions to the Maxwell equations are monochromatic waves, in fact light beams often are mixtures of waves of differing frequencies. This comes about because of the spread in wavelengths of whitelight sources, the natural line width of even spectrally pure atomic sources, the broadening mechanisms (e.g., Doppler broadening) typical of many atomic or molecular light sources, and because of mode hopping from one frequency to another in laser sources. Mathematically, of course, this fact causes no difficulties because of the superposition principle for harmonic functions. The term wave packet refers to such a superposition of waves of many different modes. This should not be confused with the so-called wave-packet model of the photon found in many elementary texts; this model, usually presented as a way of thinking about wave– particle duality, presents a number of philosophical and pedagogical problems [1.27]. For coherent sources, there is an additional constraint on pulse duration and bandwidth, rather like the indeterminacy principle in quantum mechanics and

w (z)

b Ö2w0

w0 È

E0 /e

Fig. 1.4 Schematic of the spatial profile of a Gaussian beam

near a beam waist, showing the Rayleigh range or confocal beam parameter and the divergence angle Θ. The envelope of the beam represents the point at which the field has decreased to 1/ e of its maximum value on the beam axis

sometimes derived from it. This constraint says that, for a source with duration ∆τ, the spectral bandwidth satisfies the condition ∆τ · ∆ν ≥ 1, where ν is the frequency. Hence, light that comes from continuous-wave coherent sources can have extremely narrow bandwidths (i. e., high spectral purity). The best achieved so far is a bandwidth of order 1 kHz. On the other hand, the shortest laser pulses made to date, with durations of 4 fs, have spectral bandwidths of 200 nm or more.

1.3.2 The Semiclassical Picture: Light Quanta The theory of light quanta developed by Planck and Einstein explained the salient characteristics of blackbody radiation spectra in a semiclassical picture that quantized the allowable frequencies of radiation, but otherwise treated electromagnetic radiation in terms of classical fields [1.28]. This picture was incorporated into the quantum theory developed by Schrödinger, Heisenberg and especially Dirac; importantly, it avoided the troublesome mathematical divergences that occurred in a quantum-mechanical treatment of electromagnetic radiation. It eventually turned out, as noted by Heitler in a classic text, that these difficulties were primarily of a formal mathematical character [1.29] that was resolved in the modern version of quantum electrodynamics by Dyson, Feynman, Schwinger and Tomonaga [1.30] The spectrum of radiation from a body at thermal equilibrium at a temperature T , when modeled on the basis of classical theory, produced a catastrophic divergence of the predicted radiated intensity at ultraviolet wavelengths. Planck solved this problem by treating the radiation field as a collection of simple harmonic oscillators in the cavity (Hohlraumstrahlung), and then taking a new approach to calculating the density of modes based on Boltzmann statistics [1.31]. Since the radiation field must satisfy Maxwell’s wave equation, the three components of the electric field must satisfy the equations E x (r, t) = E x (t) cos(k x x) sin(k y y) sin(kz z) , k x = πνx L , νx = 0, 1, 2, 3, . . . E y (r, t) = E x (t) sin(k x x) cos(k y y) sin(kz z) , k y = πν y L , ν y = 0, 1, 2, 3, . . . E z (r, t) = E x (t) sin(k x x) sin(k y y) cos(kz z) , kz = πνz L , νz = 0, 1, 2, 3, . . . . (1.12) Any possible radiation field must be expressible as a sum of these cavity modes; the spatial quantization here is

The Properties of Light

k2 dk π2 ω2 dω ⇒ ρω dω = 2 3 . π c ρk dk =

(1.13)

For a system of harmonic oscillators in thermal equilibrium, the probability that an oscillator is excited into the nth mode is given by the Boltzmann probability exp(−E n /kB T ) Pn = exp(−E m /kB T ) m

exp[−(n + 1/2) ω/kB T ] m exp[−(m + 1/2) ω/kB T ] m=0

exp(−n ω/kB T ) , m exp(−m ω/kB T )

(1.14)

m=0

where is Planck’s constant. From this, with the substitution U ≡ exp(− ω/kB T ) the average occupation number of a cavity radiation mode for one polarization direction is calculated to be n = m Pm = (1 − U) mU m m

∂ m = (1 − U) U U ∂U m =

1 . exp ( ω/kB T )

(1.15)

This distribution function correctly reproduced the extant spectral measurements on black-body radiators, as well as the empirically derived Wien displacement law, which gave the shift in the measured maximum intensity as a function of wavelength. The energy density per unit angular frequency in a Hohlraum or cavity radiator is directly proportional to the irradiance (or intensity) of the radiation, and is found by multiplying the density of radiation modes (1.13) by the energy per quantum times the average occupation number: U(ω) dω = ωρω n dω =

ω3 1 dω . π 2 c3 exp ( ω/kB T ) − 1

(1.16)

1.3.3 Light as a Quantum Field Neither Planck’s explanation of black-body radiation nor Einstein’s theory of the photoelectric effect required one to think of light as a quantum object. In their semiclassical approach, light was treated as a classical electromagnetic wave and only the interaction with matter (e.g., in absorption or emission) was described in terms of quanta. However, working from experimental data of Kocher and Commins, [1.32], Clauser showed that polarization correlations of photons emitted in cascaded atomic transitions could not be accounted for by a semiclassical theory [1.33]. This turned out to be only one instance of what now is understood to be generally true: higher-order photon correlations, singlephoton experiments, photon entanglement and photon squeezing cannot be properly described without a fully quantum-mechanical theory of the radiation field. Such a theory requires, first of all, a prescription for converting the field variables E and B of classical radiation theory into a quantum mechanical operators [1.34]. This is accomplished by introducing the vector potential A in the Coulomb gauge, satisfying the conditions B=∇×A, E = −∇φ , ∇·A=0.

(1.17)

In free space, the vector potential satisfies the same wave equation that is satisfied by the electric and magnetic fields. For traveling waves subject to periodic boundary conditions, the vector potential can be expanded in a Fourier series. The vector potentials associated with the kth frequency mode of the radiation field can be expressed in terms of generalized position and momentum variables, Q k and Pk , associated with the kth oscillatory mode, as follows 1 1 Ak = (ωk Q k + iPk ) εk 2 4πε0 ωk V 1 1 A∗k = (ωk Q k − iPk ) εk ⇒ 2 4πε0 ωk V 1 ε0 Ek2 + µ−1 Bk2 dV E¯ k = 0 2 cavity

1 2 Pk + ω2k Q 2k , = (1.18) 2 where the overbar indicates spatial and temporal averages. Thus the quantum-mechanical description of light associates with each mode of the radiation field an oscillator with a frequency ωk , and likewise associates

Part A 1.3

the consequence of the boundary conditions at the cavity walls applied to the Maxwell equations. The density of modes is readily calculated to be

1.3 Physical Models of Light

Part A

Basic Principles and Materials

Part A 1.4

the quantum canonical coordinates with the electric and magnetic fields. The quantized radiation field is obtained by converting the dynamical variables {Q k , Pk } into corresponding operators {qˆk , pˆk }, and inserting these operators into the appropriate Hamiltonian for the field. The Hamiltonian operator for a harmonic oscillator of mass m=1 (in some natural units) is Hˆ = ( pˆ2 + ω2 qˆ 2 ); the operators obey the commutation relation [q, ˆ p] ˆ =i . It turns out to be useful to work not with these artificial coordinates and momenta, but with the so-called creation and annihilation operators for the oscillators, which are defined, together with their commutator, as follows: 1 1/2 aˆ = ωqˆ + i pˆ , 2ω 1 1/2 † aˆ = ωqˆ − i pˆ , 2ω a, (1.19) ˆ aˆ† = 1 . With these definitions, the harmonic-oscillator Hamiltonian for the radiation field turns out to be 1 2 pˆ + ω2 qˆ 2 Hˆ = 2 2 ω 1 2 aˆ − aˆ† i = 2 2 2 2 † aˆ − aˆ +ω 2ω (1.20) = ω aˆ† aˆ + 1/2 . A particularly important combination of these operators is the number operator given by nˆ = aˆ† a. ˆ Rewriting the Hamiltonian and the corresponding eigenvalue equation for the radiation field in terms of the annihilation and creation operators, we have 1 Hˆ = ω aˆ† aˆ + 2 1 ⇒ ≡ ω nˆ + 2

1 ˆ |n |n . |n = ω n+ H = En 2

(1.21)

As expected, the vacuum – the state with no quanta (n = 0) – still has an associated zero-point energy. The field operators corresponding to the number states are found by substituting the annihilation and creation operators into the expressions for the vector potentials and the electromagnetic fields; the operator for the electric field, for example, is found to be [1.34] ωk 1/2 Eˆ k = i 2ε0 V × εˆ k aˆk exp (−iωk t + ik · r) − aˆ† exp (iωk t − ik · r) , (1.22) where εˆ k is a unit polarization vector corresponding to the wave-vector mode k and the distance vector r has its usual meaning. This provides the required correspondence between the classical electromagnetic vector fields and the field operators needed for quantum field theory. In this quantum-mechanical model, the photon is not treated as a classical particle with definite energy, momentum and helicity, but as a quantum excitation associated with the normal modes of the electromagnetic field, specified by a wavevector k and the polarization ε. Localization of the photon – in essence, defining a wave packet comprising photons – is achieved by introducing a linear superposition of one-photon Fock states |1, k, ε (states with a well-defined number of photons), rather like any other quantum-mechanical superposition of states |ψ = C exp − (k − k0 )2 /2σ 2 k

× exp [−ik · r0 ] |1, k, ε ,

(1.23)

where C is a normalization constant. This wave function may be regarded as the quantum-mechanical analog of the classical wave packet described earlier.

1.4 Thermal and Nonthermal Light Sources The early papers on light quanta by Planck and Einstein implicitly assumed that it was possible to treat light as a classical electromagnetic wave, while treating matter quantum mechanically. This approach was successful in

providing a theoretical understanding of many important phenomena, including the spectrum of black-body radiation, spontaneous emission, stimulated absorption and stimulated emission, resonance fluorescence, the photo-

The Properties of Light

T = 5500 K

800

where m e is the mass of the electron. This classical lifetime turns out to be of the order of nanoseconds, typical for the lifetimes of many atomic transitions in the visible as calculated from quantum theory and measured using modern spectroscopic techniques.

T = 5000 K

400 T = 4500 K

200

Intensity

T = 4000 K T = 3500 K

500

7500

1000

1500

2000 ë (nm)

Fig. 1.5 Calculated spectrum of black-body radiators at

several different temperatures based on (1.16)

F = 13.1 J/cm2

7000

t = 40 ns

6500 6000 5500

electric effect, the Lamb shift and vacuum polarization. In this context, it is useful to note the different characteristics of light from thermal, nonthermal (or luminescent) and particle-beam sources. Both thermal and nonthermal light originate from atomic or molecular transitions from higher- to lowerenergy states, conventionally labeled as a transition from state 2 to state 1. In the semiclassical picture, these transitions are the outcome of a quantum process that may either be spontaneous or stimulated; the transition rates are described by the Einstein A and B coefficients: 1 A2→1 = , τ π 2 c3 π 2 c3 , A2→1 = B2→1 = 3 ω ω3 τ

E 02 (ω − ω0 )2 + γ 2

, γ ≡ 1/τ .

5000 4500 4000

t = 120 ns

3500

t = 160 ns

3000 2500

t = 180 ns

2000

t = 200 ns

1500

(1.24)

where τ is the mean lifetime of a given species. Because of the quantum-mechanical indeterminacy in the mean lifetime at which an atom or molecule will decay, there is a corresponding uncertainty in the distribution of frequencies that is described by the Lorentzian line-shape function of (1.45) |E (ω)|2 =

t = 80 ns

(1.25)

The Lorentzian line shape is also predicted for a driven classical harmonic oscillator with natural frequency ω0 . However, in this case the lifetime is not determined by

1000

t = 240 ns

500

t = 280 ns

260

265

270

275

280

285

290

295

t = 300 ns

Wavelength (nm)

Fig. 1.6 Time-lapse spectra of a laser-produced plasma generated

by Nd : YAG laser irradiation of silicon at a fluence of 13.1 J/cm2 . At early times, one observes only the black-body radiation of the large volume of material ablated from the surface; note that the peak of the black-body radiation curve shifts to the red with time, indicating a decrease in temperature. At later times, the spectra exhibit the luminescence lines from atomic transitions in silicon superimposed on a gradually disappearing black-body background. (After [1.35])

Part A 1.4

the properties of individual atoms, as it would be in quantum theory; instead, it is given by a classical radiative lifetime function defined by τ = 6πε0 m e c2 /e2 ω2 , (1.26)

u (ë) (kJ/nm)

600

1.4 Thermal and Nonthermal Light Sources

Part A

Basic Principles and Materials

Part A 1.4

1.4.1 Thermal Light Thermal or chaotic light sources are represented by the idealized black-body radiator, a large ensemble of emitters in thermal equilibrium with each other and with their Flux (Photons /s mrad into 0.1 % bandwidth) 10+14

ALS Berkeley SRS Daresbury

10+13

NSLS VUV NSLS X-Ray ESRF, Grenoble, France

10+12

APS, Argonne, USA Spring8, Nishi-Harima, Japan

10+11

Spectra normalized to electron beam current of 250 mA

10+10

10+9

10+8 104

102

10+2

10+4

Photon energy (keV)

Brightness (Photons/s mm2 mrad2 into 0.1 % bandwidth) 10+20

ALS Berkeley SRS Daresbury

10+19 10+18 10

NSLS VUV NSLS X-Ray

+17

ESRF, Grenoble, France

10+16 10

APS, Argonne, USA

+15

Spring8, Nishi-Harima, Japan

10+14 10

+13

+12

Spectra normalized to electron beam current of 250 mA

2.8978 × 10−6 , (1.27) T where T is the absolute temperature in Kelvin. This law can be derived by substituting in (1.16) and finding its extreme value in the usual way. The areal power density radiated through a small aperture in an enclosure containing radiation in thermodynamic equilibrium at a temperature T is given by the Stefan-Boltzmann law. This law was discovered empirically by Josef Stefan in 1879, and postulated on the basis of thermodynamics by Ludwig Boltzmann in 1884. This law states that the radiated power per unit area is λmax =

P(T ) ≡ E(T ) = σ T 4 , A π 2 kB4 = 5.67 × 10−8 W/(m2 K4 ) . (1.28) σ= 60 h 3 c2 This relationship requires that one integrate the radiated power for an ensemble of atoms over all wavelengths, again beginning with the Planck distribution. This integral can be carried out with the aid of (1.25) by a change of variables x = ω/kB T to yield the required result: c E(T ) = 4

∞ 0

ω3 dω π 2 c3 exp ( ω/kB T ) − 1

kB T

4 ∞

4π 2 c2 0

10+11 10+10

x 4 dx ∝ T4 . ex − 1

(1.29)

1.4.2 Luminescence Light

10+9 10+8 104

surroundings. The intensity spectrum of a black body calculated from (1.16) is shown in Fig. 1.5 for several different temperature. The Planck distribution formula in (1.16) yields two important empirical relations between the spectral and thermodynamic properties of blackbody radiation laws. The peak wavelength of black-body spectra is described by Wien’s displacement law, which shows a shift in the wavelength λmax at which the distribution function is a maximum proportional to the inverse temperature

102

10+2

10+4

Photon energy (keV)

Fig. 1.7a,b Spectral distribution of light from various synchrotron light sources identified in the caption. (a) Measured photon flux from the synchrotron light sources. (b) Measured spectral brilliance of several synchrotron sources. (After [1.36])

Electromagnetic radiation emitted by excited atoms and molecules not in thermal equilibrium with their surroundings typically exhibits much narrower spectra than black-body radiation, and is called luminescence. The excitation could come from any number of energy sources: energetic electrons (cathodoluminescence), light (photoluminescence), applied electric

The Properties of Light

1.4.3 Light from Synchrotron Radiation In recent years, accelerator-driven light sources have assumed an important place in both science and technology. Sources such as synchrotron storage rings [1.36,37]

and free-electron lasers [1.38, 39] generate light by bending relativistic electron beams either with dipole magnets or in insertion devices called undulators or wigglers. From such sources, extremely broad spectra can be generated that range from X-rays to the infrared region of the spectrum, with spectral brightnesses far exceeding those of any thermal source. The temporal and spatial characteristics of such radiation are not related to atomic or molecular properties, but instead to the particular characteristics of the associated electron accelerators or bending magnets. Bending magnets produce relatively narrow-band but incoherent radiation, and wavelength selectivity is achieved by the use of monochromators. Undulators and wigglers produce coherent beams of extremely high spectral brightness. In free-electron lasers, a quasi-monochromatic spectrum is produced by placing the wiggler inside an optical resonator. The spectral profile of light from synchrotron sources follows a universal curve derived from the classical theory of radiation by electrons moving on the arc of a circle. This distribution is quite unlike that of blackbody radiation. This is illustrated in Fig. 1.7a, where the measured photon flux from a number of synchrotron sources is compared for constant electron current. The spectral brilliance of some of the same sources is shown in Fig. 1.7b, illustrating the point that this quantity can be highly variable even for sources that have similar photon flux. Besides its broad spectral distribution, synchrotron light produced by wigglers and undulators (insertion devices) can have a high degree of spectral coherence.

1.5 Physical Properties of Light This section outlines the measurable physical properties of light associated with both the wave and photon pictures. These include intensity or irradiance, velocity of propagation, polarization, energy, power and momentum transport. These properties underlie the radiometric, photometric and spectral characterizations of light presented in Sect. 1.2.

1.5.1 Intensity Almost all measurements of the effects of light interacting with matter depend either on energy incident per unit volume, energy incident per unit area (fluence), power per unit area (intensity or irradiance), or a combination of these quantities. The time-averaged intensity or irradiance of an harmonic electromagnetic wave described

by an electric field vector E(r, t) ≡ E0 (r) e±iωt is T 1 I (r) = cε0 |E (r, t)|2 dt T 0 cε 0 |E 0 (r)|2 . (1.30) = cε0 |E (r, t)|2 = 2 In the photon picture, the analog of the intensity is the areal number density of photons per unit time, or photon flux; it is related to the energy density by I(ω) = cU(ω)/4, thus cU (ω) 4 ω3 1 . = 4π 2 c2 exp ( ω/kB T ) − 1

I (ω) =

(1.31)

Part A 1.5

fields (electroluminescence), sound waves (sonoluminescence), or chemical reactions (chemiluminescence). Light produced in this way exhibits a spectrum that is indicative both of its natural frequency and line width, and also characteristic of its environment. For example, impurity atoms in a solid, or excited atoms in a gas-discharge tube at moderate pressure, will exhibit line widths that convolute the natural (Lorentzian) line width with the Gaussian profile due to homogeneous or inhomogeneous broadening by the local electronic environment to produce the Voigt profile of (1.52). It is also possible for light sources to produce thermal and nonthermal light simultaneously from coupled equilibrium and nonequilibrium processes. Light produced by high-intensity laser irradiation of materials, for example, often has this characteristic. Figure 1.6 shows the spectrum of light emitted by a laser-produced plasma that at later times shows narrow luminescence lines riding on a broader, gradually decreasing blackbody background. In this case, the luminescence has the characteristics of radiation from individual atoms (with a line shape that convolutes the natural line width, Doppler and pressure broadening) and black-body radiation (due to hot ablated material in local thermodynamic equilibrium).

1.5 Physical Properties of Light

Part A

Basic Principles and Materials

Part A 1.5

The fluence is the integral of the intensity over a suitable time interval; in the case of laser beams, this interval is often simply the duration of the laser pulse.

1.5.2 Velocity of Propagation Two different concepts of velocity are associated with the idea of electromagnetic radiation. Phase velocity refers to the propagation of points with the same phase on a wavefront, while group velocity generally refers to the propagation velocity of energy or information in a wave packet. The phase velocity for a wave of frequency ν and vacuum wavelength λ in a medium with an index of refraction n is λ ω c ⇒ ω = vp kn . vp = = ν = (1.32) n n kn While this velocity represents the movement of the phase fronts, it is not the speed at which energy or information propagates; that requires some interaction with the oscillations that make up the wave. The group velocity vg is conventionally defined in terms of these quantities and the index of refraction of the medium in which the wave propagates as c k dn dω = 1− dk n n dk k dn = vp 1 − . (1.33) n dk Evidently in a medium with constant index of refraction, the phase velocity and group velocities are equal, while in a medium which exhibits normal dispersion ( dn/ dk > 0) the group velocity is at most equal to the phase velocity. However, in the case of anomalous dispersion ( dn/ dk < 0), the group velocity may actually exceed the speed of light in vacuum c; in such cases, it is always possible to demonstrate that the group velocity is not the speed at which information or energy propagates. This is the case, for example, in hollow waveguides witha cut-off frequency ω0 , in which the dispersion is k = ω2 − ω20 c, and also on the blue side, of many atomic resonances.

1.5.3 Polarization The polarization of a light wave is widely used as a marker and diagnostic in various spectroscopic techniques, [1.40] especially in laser spectroscopy, [1.41] and is conventionally described by three formalisms: the Mueller calculus, the Jones calculus, and the Poincaré sphere.

The Mueller calculus is based on a scheme first given by G. G. Stokes for measuring the polarization observables of a light beam; it uses four-component vectors to describe the measured intensities (all real numbers), and 4 × 4 matrices to describe the interaction of the light with various polarizing elements or materials. The Jones calculus is somewhat simpler computationally because it uses only two-component vectors and 2 × 2 matrices to calculate the polarization in a light beam; however, the parameters in the Jones vectors are complex. The Stokes vectors do not, however, constitute the basis of a linear vector space, while the Jones vectors do. The Poincaré sphere method is based, not on a numerical parameterization of polarized light, but on a mathematical way of relating different polarization forms to points on a sphere; transformations from one polarization form to another are embodied by rotations of the sphere. It is a useful qualitative construct, particularly in making experimental judgments where phase retardation effects rather than changes of intensity are involved. Various states of polarization are illustrated schematically in Fig. 1.8. The matrices that represent the operation of material polarizers such as wave-plates and phase retarders on Jones or Mueller vectors can be found in most optics textbooks. The Jones Calculus Consider a light wave propagating in the z-direction, so that the electric field has components only in the x–y-plane. In the previous notation, the electric field is then

E = E 0x exp [i (kz − ωt + φx )] xˆ + E 0y exp i kz − ωt + φ y yˆ . y

y á

(1.34)

y x

Ex = Ey

öx = öy

öx = öy + ð/2

öx > öy

Fig. 1.8 Schematic representation of linear, right-hand circular and elliptical states of polarization. The propagation direction is taken to be the z-axis of a Cartesian coordinate system. The various polarization states are characterized by differing relative values of electric-field amplitudes E x and E y and of the phase shifts Φx and Φ y

The Properties of Light

≡ exp [i (kz − ωt)] J .

(1.35)

The column vector, omitting the z and t dependence is called the full Jones vector J (or in some circles, the Maxwell column) of the light wave. The use of the same symbol as the electric current density rarely causes problems in optics, where electrical currents are usually negligible. Because the Jones vectors are the bases of a vector space, it is easy to calculate the action of a polarizing device on light having a given polarization state by simple matrix multiplication. For example, consider linearly polarized light with the polarization making equal 45◦ angles with the x- and y-axes of a quarterwave plate (QWP). The action of the QWP, which introduces a quarter-wave phase delay between x- and y-components of the electric field, is represented by the matrix (M)QWP . As shown below, the outgoing light beam is right-circularly polarized. Eout = (M)QWP (J)in

= exp [i (kz − ωt)] ≡ exp [i (kz − ωt)]

1 0 0 i E 0x eiπ/2 E 0y

ferential ∆ satisfies −π ≤ ∆ = φ y − φx ≤ π; and rcp/lcp refers to right-/left-circularly polarized light. The derivation of the relationships in the second column vector above is complicated, but may be found in a variety of sources [1.40]. An important advantage of the Mueller calculus is the fact that this formalism is able to describe unpolarized or partially polarized light, by taking time averages in time over the Stokes parameters [1.43]. For unpolarized light, only I is nonvanishing, and for partially polarized light, the remaining Stokes parameters must satisfy 0 < (Q 2 +U 2 + V 2 ) < I. On the other hand, for a completely polarized beam, (Q 2 + U 2 + V 2 ) = I, so that the points (Q,U,V ) lying on a unit sphere represent well-defined polarization states. The Poincaré Sphere In general, the polarization of a light wave is described by an ellipse, which can be characterized by the angle α describing the inclination of the ellipse with respect to a polar axis and by its ellipticity ε. These quantities can be related to the Stokes parameters by

S1 ≡ Q/I = cos (2ε) cos (2α) , S2 ≡ U/I = cos (2ε) sin (2α) , S3 ≡ V/I = sin (2ε) .

With these definitions, any desired polarization state can be represented as a point or collection of points on a sphere, as shown in Fig. 1.9 [1.44]. States of constant ellipticity are represented by the locus of points

E 0x E 0y (1.36)

Mueller Matrices The Mueller calculus for describing the polarization of a light beam is based on a measurement scheme for polarization developed by Stokes [1.42] The columnvector representation of the Stokes parameterization is ⎛ ⎞ ⎛ ⎞ I I ⎜ ⎟ ⎜ ⎟ ⎜ Q ⎟ ⎜ I ◦ − I90◦ ⎟ S=⎜ ⎟=⎜ 0 ⎟ ⎝ U ⎠ ⎝ I+45◦ − I−45◦ ⎠

⎛

Left-handed circular

Q M P'

P O

2á

ð/2

P (C0)

(1.37)

2E 0x E 0y sin ∆ The Stokes parameter I is the total intensity; the quantity Iφ is the intensity measured at the angle φ; the phase dif-

Constant azimuth

Constant ellipticity

2å

Ircp − Ilcp ⎞ 2

2 +E E 0x 0y ⎜ ⎟ 2 − E2 ⎜ E 0x ⎟ 0y =⎜ ⎟. ⎝ 2E 0x E 0y cos ∆ ⎠

(1.38)

Linear states

ç S

Right-handed circular

Fig. 1.9 Schematic representation of polarization states on

the surface of the Poincaré sphere

Part A 1.5

Factoring out the time and z dependence, and writing the x- and y-components of the electric field as a column vector, we have

E 0x eiφx E = exp [i (kz − ωt)] E 0y eiφ y

1.5 Physical Properties of Light

Part A

Basic Principles and Materials

Part A 1.5

on circles parallel to the equatorial plane; the equatorial plane itself is the locus of points representing linear polarization. States of circularly polarized light are represented by the poles. Unpolarized light is represented by the points constituting the surface of the sphere, while partially polarized light would be represented by a subset of points on the surface clustered around a particular point representing the highest degree of polarization. Photon Spin and Polarization In classical electromagnetic theory, the polarization of a light wave is determined by the transverse character of vibrations in the electromagnetic field. In the quantum theory of the photon, it becomes necessary to endow the photon with a property of angular momentum just as one does for the electron, but it is also necessary that this concept should be consistent with the wave description of polarization. The photon is a boson and has unit spin; since the photon is massless, it does not have orbital angular momentum, but only its intrinsic spin s = 1. A spin-1 particle can have possible spin projections +1, 0 and −1 with respect to some given axis; however, since light waves are transverse, the 0-component is ruled out. From quantum electrodynamics, it is known that photons with helicities of ±1 are related to left- and right-handed circular polarization states. The basis helicity states of the photon can be represented as column vectors assuming that the quantization axis is the direction of propagation: 0 1 |s = 1 ≡ |+ = . , |s = −1 ≡ |− = 1 0 (1.39)

Taking the quantization axis to be the z-axis, as well as the direction of light-wave propagation, we construct linearly polarized light in the x- and y-directions from these basis states as follows: 1 |ex = √ (|+ + |−) , 2 1 e y = √ (|+ − |−) . (1.40) 2 Once this correspondence is established, the overall polarization of an ensemble of photons, as in a light beam, can be treated in the same way that other ensembles of quantum particles are treated in the density-matrix formalism [1.45]. Suppose we have a beam of photons that is a mixture of two beams, prepared independently in polarization states characterized by the state vectors and ea , and eb with intensities Ia and Ib , respectively. If we

use the vectors of (1.39) as the basis states, then the state vectors characterizing the polarization are, respectively, (a) |ea = c(a) + |+ + c− |− , (b) |eb = c(b) + |+ + c− |− .

(1.41)

This seemingly quite abstract expression corresponds directly to the Stokes parameters defined for a classical light beam, using the standard projection-operator formalism of quantum mechanics. The normalized density matrix for this system of two beams in two possible states is ρ = Ia |ea ea | + Ib |eb eb | +1|+1 +1|−1 = −1|+1 −1|−1 ρ+1,+1 ρ+1,−1 . ≡ ρ−1,+1 ρ−1,−1

(1.42)

Now consider, for example, the Stokes parameter V , which is defined to be the difference of the right- and left-circular polarized intensities Ircp − Ilcp . From the definition of the photon helicity, it is clear that V and the density matrix elements are related by IV = −(ρ1,−1 +ρ−1,1 ). Following the same approach for the other Stokes parameters, it is found that I 1 + U −V + iQ ρ= , (1.43) 2 −V − iQ 1 − U thus clearly exhibiting the link between the classical and quantum descriptions of the polarization of the light wave and/or photons. Not surprisingly, it is the Stokes parameters – the classical observables – that also turn up in the quantum-mechanical treatment.

1.5.4 Energy and Power Transport In general, the energy density in a region of free space occupied by electric and magnetic fields E and B is given by 1 B2 u= + ε0 E 2 2 µ0 2 ε0 B 2 +E = 2 ε0 µ0 ε0 2 2 c B + E2 . = (1.44) 2 In material media, this energy density would be modified by replacing the vacuum values of the dielectric permittivity and magnetic permeability by their values in the

The Properties of Light

E y = E 0y cos (kz − ωt + φ) , Bx = −B0x cos (kz − ωt + φ) .

(1.45)

The time-dependent volumetric energy density associated with this wave is therefore given by ε0 2 2 2 cos2 (kz − ωt + φ) . (1.46) c B0x + E 0y u= 2 This equation exhibits explicitly the wave-like transport of energy associated with the existence of the electromagnetic wave. The total energy and power associated with the radiation field can be calculated by introducing the hypothesis that the energy in the field is related to the classical average field energy by 1 Ufield = ε0 |Eω (r, t)|2 dV 2 cavity

= (n + 1/2) ω .

(1.47)

1.5.5 Momentum Transport: The Poynting Theorem and Light Pressure The momentum and radiation pressure exerted by a light beam can be developed from the definition of energy contained in the beam. The total energy U delivered by a light beam in a time ∆t is related to the change in radiative momentum transferred and the radiation pressure prad by: U = u Ac∆t U ⇒ ∆ prad = = u A∆t c ∆ prad = uA ⇒ Frad = ∆t Frad =u. (1.48) ⇒ Prad = A The flow of energy and momentum carried by electromagnetic waves is described by Poynting’s theorem for a vector S = E × B/µ0 with units of energy/(time × area). With this definition, the energy density of the electromagnetic field U and the momentum density g = ε0 E × B satisfy ∂U = −∇ · S ∂t ∂g ← →(M) = −ρE − T · nˆ (1.49) ∂t

← → where ρE is the Lorentz force per unit volume, T (M) is the Maxwell stress tensor and the unit vector nˆ is directed along the outward normal to the surface bounding the relevant volume. In vacuum, the energy and momentum densities are equal apart from a factor 1 c; in material media the situation is much more complicated [1.23]. Planck’s quantum hypothesis asserts that the energy of a light quantum is related to the angular frequency of an electromagnetic wave ω by E = ω, where ≡ h 2π, with h being Planck’s constant 6.67 × of a light quantum 10−34 J s. Similarly, the momentum can be expressed as p = h λ = k. A similar relationship for the momentum of a material particle, of course, was famously postulated by de Broglie and has been verified in experiments with electrons and other quantum particles.

1.5.6 Spectral Line Shape One of the most important physical characteristics of a stream of light is its spectral line shape, which gives the probability density of intensity or irradiance as a function of wavelength or frequency. The line shape is also a unique signature of the source from which the light emanates. The line shape of a thermal or chaotic source is that of the black-body spectrum described by the Planck irradiance function, (1.16). The position of the peak in that spectrum is characteristic of the temperature, while its overall form is governed by the Bose–Einstein distribution, the second factor in (1.16). The shape of this function is illustrated in Fig. 1.7a for several temperatures. The shifting position of the peak wavelength at each temperature follows Wien’s displacement law (Sect. 1.4.1). Light from electronic transitions between energy levels E 2 and E 1 in an ensemble of non-interacting atoms (as, for example, in a dilute gas) has a resonance or Lorentzian line shape having the normalized form γ (1.50) L (ω; ω0 , γ ) = , π (ω − ω0 )2 + γ 2 where F0 is an appropriately normalized field amplitude, ω is the angular frequency of the light, ω0 = (E 2 − E 1 )/ is the frequency at the centroid of the line, and γ is called the natural line width. The natural line width γ is related to the intrinsic lifetime of the atom τ0 by γ = 1/2τ0 . Should those same atoms be in an environment in which the spectrum of light from the atoms is broadened because of, say, collisions or the Doppler effect, the line shape typically has a Gaussian intensity distribution. Its

Part A 1.5

material, namely, ε and µ. If we now consider a plane electromagnetic wave traveling in the z-direction for the sake of definiteness, the electric and magnetic fields are given respectively by

1.5 Physical Properties of Light

Part A

Basic Principles and Materials

Part A 1.5

a) 0.3

c) Intensity (arb. units)

0.25 ó = 1.53 ó = 1.3 ó=1 ó=0

0.2

ã=0 ã = 0.5 ã=1 ã = 1.8

A/cm2 2000 5000 7500 10 000

0.15 0.1 0.05 0 10

b) Transmission (%) 100

0.83

0.835

0.84

0.845

0.85

0.855 ë (µm)

Fig. 1.10a–c Spectral characteristics of light from various sources. (a) Comparison of spectral line shapes from

isolated atoms or molecules (Lorentzian), collisionally broadened sources (Gaussian) and from sources exhibiting both characteristics (Voigt profile). (b) Vibrationalrotational spectrum of HBr molecules, illustrating the band-like structure of radiation from even quite simple molecules. (After [1.46]). (c) Spectrum of light from a GaAs semiconductor laser as a function of driving current, showing the narrowing of the spectrum as the lasing threshold is exceeded. (After [1.47])

20 0

2700

v (cm1)

2600

2500

2400

2300

normalized form is

1 (ω − ω0 )2 , (1.51) G (ω; ω0 , σ) = √ exp − 2σ 2 σ π

where the angular frequencies have the same meaning as before and σ is the variance of the line-shape. For Doppler broadening of an atom with!mass m 0 at a temperature T , the variance is σ = ω0 kB T/m 0 c2 , where kB is Boltzmann’s constant. When the atomic emitters all experience the same broadening mechanism, the line shape is said to be homogeneously broadened; when the broadening mechanism differs within the ensemble, such as when the emitting atoms occupy different kinds of lattice sites in a solid, the emission line is said to be inhomogeneously broadened. In most realistic situations, of course, the broadening of atomic or molecular lines arises from a combination of Lorentzian and Gaussian mechanisms. In this case, the line shape assumes the so-called Voigt profile, which is the convolution of the Gaussian and Lorentzian

functions: V (ω, ω0 ; σ, γ ) ∞ G ω , ω0 ; σ L ω , ω0 ; γ dω = −∞

1 ω − ω0 + iγ , = √ Re erf √ σ 2π σ 2

(1.52)

where erf is the complex error function. The contrasting line shapes are illustrated in Fig. 1.10a; notice the greater amplitude of the Gaussian function far from the line center frequency ω0 . Light emission from molecules may arise from electronic, vibrational and/or rotational transitions, and this produces complex spectra in which the individual spectral lines are similar to those of atoms, but which appear in groups or bands of lines with rather regular spacings.

The Properties of Light

(1.53)

where B is the rotational constant of the molecule, J is the rotational quantum number, and ωv is the angular frequency of the vibrational transition. In practice, molecular spectra are complicated by anharmonicities, changes in rotational constants induced by stretching and so on; these issues are treated in a multitude of specialized monographs [1.48]. Coherent light from laser sources exhibits another characteristic line-shape phenomenon, the narrowing of the emission line as laser action begins. This line narrowing, illustrated in Fig. 1.10c, is due to the higher emission probability – and hence preferential amplification – of wavelengths near the line center. However, depending on the mechanism of line broadening, the narrowing effect may actually be reversed as amplification enters the saturation regime.

1.5.7 Optical Coherence The coherence phenomena associated with fundamental properties of light can be divided into spatial and temporal coherence [1.49]. The former refers to the effects on electromagnetic radiation emitted by sources of finite size, the latter to radiation emitted by sources with finite bandwidth. A beam of light is said to be spatially coherent when the phase difference between points on the wavefront remains constant in time, even if the phase fluctuates randomly at any given point. Thus an extended source comprising an ensemble of randomly fluctuating point sources can produce spatially coherent light if the interference fringes from nearby point sources accidentally overlap. For example, starlight of wavelength λ is coherent over a spot of diameter dcoh given by dcoh = 0.16

λ λR = 0.16 , ρ ϑ

(1.54)

where the stellar diameter is ρ, the distance to the point of observation is R, and ϑ is the angle that the star subtends at the point of observation. The coherence area associated with this diameter is Acoh = π(dcoh /2)2 . A light beam in which phase differences between points on the wavefront remain constant in time is said to be temporally coherent. Given the Fourier-transform relationship between time and frequency, this also implies that a temporally coherent beam has a high degree of spectral purity. If a light source emits a beam with frequencies ranging from ν to ν + δν, hence with a spectral bandwidth of δν, the extreme frequencies in the beam will lose temporal synchronization in a comparatively short time, called the coherence time, given by τcoh =

1 . 2πδν

(1.55)

The coherence length associated with this coherence time is simply lcoh = cτcoh . The coherence volume for a source with given coherence properties is then Vcoh = Acoh lcoh . The physical process giving rise to light emission largely determines the coherence of a source. In a thermal source, such as a gas-discharge lamp or an incandescent bulb, light is produced by microscopic or even atomic sources emitting spontaneously, hence at random times relative to one another. Wave packets from different emission events are essentially uncorrelated, and the degree of temporal and spatial coherence is low, though not zero [1.50]. In a laser source, on the other hand, the light is produced by stimulated emission and the degree of coherence is high; however, since temporally and spatially independent transverse and longitudinal modes can coexist simultaneously, only single-mode lasers can achieve the highest degree of temporal coherence. The coherence of a light source can be quantitatively characterized by interferometry [1.51]. For example, in a two-slit interference experiment (Young’s experiment), slits separated by a distance d and illuminated by monochromatic light of wavelength λ will produce intensity maxima and minima on a screen at a distance L, separated by a distance ∆y = λL/d. The contrast between maxima and minima can be characterized by the fringe visibility defined in terms of the measured intensities: V=

Imax − Imin . Imax + Imin

(1.56)

An important development in the last half century has been the recognition that coherence and polarization

Part A 1.5

An example is shown in Fig. 1.10b for the diatomic molecule HBr. The spacing of the lines is determined by the selection rules for transitions between rotational or vibrational energy levels. In this example, the spacing is approximately given by ⎧ ⎪ ωv − 2BJ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ J = 1, 2, 3, . . . (P branch) ∆E (v, J) = , ⎪ ⎪ ⎪ ωv + 2B(J + 1) , ⎪ ⎪ ⎪ ⎩ J = 0, 1, 2, . . . (R branch)

1.5 Physical Properties of Light

Part A

Basic Principles and Materials

Part A 1.6

are inextricably intertwined [1.52]. This convergence has made it possible to develop deep and detailed connections between the classical and quantum theories of light. It has also resulted in the discovery that single-point descriptions of polarization – such as those provided by the Jones or Mueller calculus – are inadequate to explain experimental observations of changes in polarization that occur during propagation, even in empty space [1.53, 54]. These discoveries have led to the development of a generalized coherence matrix in a vector electromagnetic theory that correctly describes these interrelated coherence and polariza-

tion phenomena. The form of this hermitian coherence matrix is similar to the matrices found in the Jones calculus:

& & E x (r, t) E ∗x (r, t) E x (r, t) E ∗y (r, t) J= & ∗ , & E x (r, t) E (r, t) E y (r, t) E ∗y (r, t) ¯ tr(J) = I.

(1.57)

However, in this case, the elements of the coherence matrix are not simply complex scalars, as in the Jones matrices, but components of the generalized coherence tensors.

1.6 Statistical Properties of Light The foregoing discussions are based on models either involving light rays or single photons with well-defined properties of wavelength and frequency, polarization, momentum and energy. However, all real light sources fluctuate in frequency and polarization, and a full treatment of the properties of light requires an accounting for this statistical character through probability density functions, correlation functions and such standard statistical measures as the variance. The next paragraphs present a statistical characterization of thermal (sometimes called chaotic) or black-body sources, coherent sources (such as lasers) and nonclassical light sources. More detailed treatments are available in many recent textbooks and review articles.

1.6.1 Probability Density as a Function of Intensity Consider a beam of unpolarized light from a thermal source, in a coordinate system defined such that the beam propagates in the z-direction. Such a beam comprises an equal mixture of x- and y-polarized components, with random amplitude and phase, each of which can be shown to obey a Rayleigh probability distribution given by [1.55]. 2 Ix , p (I x ) = exp −2 I¯ I¯ 2 Iy p I y = exp −2 , (1.58) I¯ I¯ ¯ where the average intensity √ I is related to the standard ¯ 2. The joint density function deviation σI by σI = I/ of the two independent random variables I x and I y is

found from a standard theorem of probability theory to be equal to 2 p (I) = I¯

I 0

I −ξ ξ exp −2 dξ exp −2 I¯ I¯

2 2 I = . I exp −2 I¯ I¯

(1.59)

A plot of this function is shown in Fig. 1.11; the physical interpretation of this is that the probability has a maximum value of 0.5, while the integral under the curves, as with all probability functions has the value 1. Applying the same general analysis, the probability distribution for light with varying degrees of polarization P from a thermal source is given by: 1 2I exp − pI (I) = P I¯ (1 + P) I¯ 2I . − exp (1.60) (1 − P) I¯ It is instructive to compare this distribution function to that of light from a laser source. The laser is proverbially a source of highly organized light, but its statistical character changes dramatically depending on whether the laser is operated below, near or well above the threshold for laser oscillation [1.56]. For a laser in the steady state, Mandel and Wolf show that the probability density p (I) as a function of intensity can be written as [1.57]

1 p (I) = C exp − (I − a)2 , (1.61) 4

The Properties of Light

0.3

p=1 p = 0.9 p = 0.5 p=0

0.6

a = 2 below threshold a = 0 at threshold a = 2 above threshold a = 10 far above threshold

0.2 0.4 0.1

0.2

0.5

1.5

2 2.5 3 Normalized intensity

Fig. 1.11 Probability density p(I ) for light with varying degrees of polarization from a thermal or chaotic source, calculated from (1.58) and (1.59)

where a is a pump parameter and C is a normalization constant. Below threshold, a < 0 and the thermal or chaotic output of the laser follows Bose–Einstein statistics; the probability density falls off exponentially. At threshold, a = 0 and the probability density has the form of a half-Gaussian cut off at I = 0. Above threshold, a > 0, and the probability density more nearly resembles the Gaussian distribution characteristic of coherent oscillators; far above threshold it is the closest we can make to a completely coherent source. The probability densities for these several cases are illustrated in Fig. 1.12.

1.6.2 Statistical Correlation Functions A particularly useful set of statistical measures are the correlation functions between either classical field variables or quantum field operators. The first-order classical and quantum-mechanical correlation functions are identical in form, and are given by E ∗ (r1 t1 ) E (r2 t2 ) g(1) (r1 t1 , r2 t2 ) ≡ & 1/2 , |E (r1 t1 )|2 |E (r2 t2 )|2

Part A 1.6

Normalized probability density 0.4

Normalized probability density 1

0.8

1.6 Statistical Properties of Light

20 15 Normalized intensity

Fig. 1.12 Probability densities p(I ) for a laser as a function of various pump parameters, as described above

the field operators defined in (1.62) replacing the classical field variables. The significance of the function g(1) (r1 t1 , r2 t2 ) is that it falls in the interval between 0 and 1, with 1 representing coherent light, 0 incoherent or chaotic light, and intermediate values representing varying degrees of partially coherent light. The results that follow from the first-order coherence properties of light, such as the results of Young’s experiment, are identical regardless of whether one chooses the classical or quantum-mechanical description. This follows from the fact that 0 ≤ |g(1) | ≤ 1. Experimentally this is tantamount to saying that the same kinds of interference phenomena are observed classically as well as quantum mechanically. On the other hand, the second-order coherence function leads to quite different and unanticipated differences between the quantum and classical cases. The secondorder coherence for quantum fields is defined by analogy with the classical quantities, and is g(2) (r1 t1 , r2 t2 ; r2 t2 , r1 t1 ) & − E (r1 t1 ) E − (r2 t2 ) E + (r2 t2 ) E + (r1 t1 ) & . ≡ & − E (r1 t1 ) E + (r2 t2 ) E − (r2 t2 ) E + (r1 t1 )

(1.62)

(1.63)

where {r1 t1 , r2 t2 } are the space–time variables that define the two electromagnetic fields. The quantummechanical expression is constructed analogously, with

The classical version of this equation is generated by replacing operators with the corresponding classical fields, whereupon one finds that the

Part A

Basic Principles and Materials

Part A 1.6

classical second-order correlation function satisfies 1 ≤ g(2) (0) ≤ ∞, g(2) (τ) ≤ g(2) (0). To see what this implies, consider a simple model for a thermal source such as an atomic discharge lamp, with Doppler-broadened lines and a Gaussian line shape. Its second-order correlation function turns out to be (1.64) g(2) (τ) = 1 + exp −(τ/τcoll )2 , where τcoll is the mean time between collisions. For perfectly coherent light, it is also easy to show that (1.54) implies g(2) (τ) = 1 for all values of τ. Figure 1.13 illustrates the differences between the two correlation functions; an experimental verification of this comparison for a discharge lamp and a laser was first published in 1966 [1.58]. In addition to these rather straightforward examples, however, there is also a range of values for which 0 ≤ g(2) (0) < 1, g(2) (0) < g(2) (τ) that corresponds to quantum-mechanical second-order coherence phenomena that have no equivalent in the classical realm, most commonly for photon streams with small, welldefined photon number [1.59].

1.6.3 Number Distribution Functions of Light Sources Another way of classifying the statistical properties of light beams is by comparing the number distribution functions of photons impinging on a detector, and comparing the variance in those distributions for different light sources. The statistical distribution of photon numbers arriving at a detector depends on the source of the light. We can generally distinguish thermal light, in which the process producing the photons is random, and coherent light, when it is not. A coherent light field can be viewed as one which is the closest thing to a perfectly classical state attainable in the quantum realm; such states are readily achieved in lasers, as seen in the previous section [1.60]. If the arrival and detection of photons in a stream are independent events, the probability distribution p(n) for n photons is given by the Poissonian distribution, with mean n¯ and standard deviation σn2 is given by n¯ n e−n¯ 2 , σn = n¯ . (1.65) n! In other words, for a Poisson distribution, the mean photon number equals the variance. Chaotic light, on the other hand, has a distribution function that can be described as super-Poissonian, p (n) =

g(2) (ô) 3 2.5 2

g(2) for chaotic source g(2) for coherent source

1.5 1 0.5 0

4 ô/ôcoll

Fig. 1.13 Second-order correlation functions g(2) (τ/τcoll )

for perfectly coherent light and chaotic light from a Doppler-broadened atomic discharge lamp

which is derivable from the Bose–Einstein or Planck distribution of photons in thermal equilibrium with their surroundings. That distribution function is readily shown Probability density 0.14 0.12 0.1 Sub-Poisson 0.08 Poisson 0.06

Super-Poisson

0.04 0.02 0

40 50 Photon number

Fig. 1.14 Probability density as a function of mean pho-

ton number n¯ for Poissonian (coherent), sub-Poissonian (nonclassical) and super-Poissonian (chaotic) light sources

The Properties of Light

1 p (n) = n¯ + 1

n¯ n¯ + 1

∝ e−

ω/kB T

1 . (1.66) e ω/kB T − 1 The variance for the Bose–Einstein distribution is σn2 = n¯ + n¯ 2 , which is always greater than it is for a Poissonian distribution. Interestingly, another characteristic of this thermal light is that the signal-to-noise ratio is always less than 1, so that thermal light cannot be used to transmit digital information. The difference between the two cases is illustrated in Fig. 1.14. In the case of nonclassical light, on the other hand, there are states in which σn2 < n¯ as shown in Figure 1.14; squeezed states of light (discussed in more detail below) are an example. n¯ =

A single-parameter way of characterizing the statistical characteristics of the various light sources is the Mandel Q-parameter, defined by & 2 2 nˆ nˆ − nˆ & = & −1 . Q≡ (1.67) nˆ nˆ Evidently, the case of chaotic or thermal light corresponds to Q > 0, coherent light to Q = 0, and sub-Poissonian light when 0 > Q > −1. Nonclassical light can be created in a number of different ways; some examples are fluorescence emitted by a single ion in an electromagnetic trap, [1.61] fluorescence from trapped atoms [1.62] and semiconductor quantum dots, [1.63, 64] and essentially arbitrary photon streams generated through cavity quantum electrodynamics [1.65–67].

1.7 Characteristics and Applications of Nonclassical Light Although the application of classical light – whether viewed in the electromagnetic wave or photon (semiclassical) models – are now ubiquitous, and the stuff of undergraduate textbooks, the study of nonclassical light, in the quantum-field model, is still an intensely active field of research. In this section, we describe in greater detail some of the characteristics of nonclassical light and consider one of its many interesting applications.

1.7.1 Bunched Light A particularly interesting statistical property of light is the arrival-time distribution of photons coming from the various sources described previously [1.68, 69]. It was experimentally observed a long time ago that light coming from thermal sources tends to be bunched, that is, the probability of a second hit on a detector immediately after one photon is detected is enhanced over the chances of a random detection event [1.70]. In contrast, the arrival-time distribution of photons from a coherent source is random, in keeping with the Poissonian nature of the emission process from a laser. Finally, however, there are light sources that generate arrival-time distributions exhibiting anti-bunching, with photons being detected at regular intervals. This anti-bunching phenomenon was first detected in fluorescence from sodium atoms [1.71], and has since been observed in quantum dots [1.72], atom [1.62] and ion traps, and even in atom lasers [1.73]. The transition between bunching and anti-

bunching behavior has also been observed [1.74, 75]. However, there is apparently no connection between sub-Poissonian statistics and anti-bunching, [1.76] so that the underlying physics of the two statistical properties may well likewise be different.

1.7.2 Squeezed Light In a classical wave, the phase and amplitude are independent variables. In quantum mechanics, however, these quantities are coupled through the uncertainty principle applied to the operators that define the quantum fields. Squeezed states are states of minimum uncertainty in which one or the other variable – number density or phase – is reduced below the Poissonian level [1.77–79] The origin of these states can be understood quantitatively by rewriting (1.22) as follows: ωk 1/2 Eˆ k = i 2ε0 V × εˆ k qˆk cos (k · r − ωk t) − pˆk sin (k · r − ωk t) , (1.68)

where the operators or canonical variables {qˆk , pˆk } are the quadrature amplitudes into which the electromagnetic field operator can be decomposed. These operators satisfy the standard indeterminacy relationship for the canonical variables in quantum mechanics. It follows, therefore, that while the product of their fluctuations

Part A 1.7

to be [1.17]

1.7 Characteristics and Applications of Nonclassical Light

Part A

Basic Principles and Materials

Part A 1.7

ÄX2

ÄÖ

uncertainties in the two field quadratures can in principle be equal

Än

ÄX2

∆X 1 = ∆X 2 =

ÄX1

Fig. 1.15a–c Relationships between the various quadrature states for light that is (a) quadrature squeezed in the vacuum state; (b) number-squeezed; and (c) phase-squeezed

must satisfy (∆qˆk )2 (∆ pˆk )2 ≥ 1, it is possible to generate coherent light in which the fluctuations in one variable are decreased at the expense of the other. It is conventional to redefine the complex electric field operator in (1.60) to make so-called dimensionless field quadratures X 1 and X 2 , as follows: ε0 V 1/2 Xˆ 1 (t) ≡ Eˆ k0 sin (ωt) 4 ω 1/2 qˆk (t) , = 2ω ε0 V 1/2 Eˆ k0 cos (ωt) Xˆ 2 (t) ≡ 4 ω 1/2 pˆk (t) . (1.69) = 2ω In this formalism, it is evident that the field quadratures obey the same indeterminacy relations as the quantum harmonic oscillator. Both the vacuum state and coherent states have minimum uncertainties, and moreover, the Detectors

Polarizer

+ Interference filters Logic circuits

Electrooptic modulator

From source Amplifier 0° 1'

/0 +/1 /0

Time tags

0° 0' Random number generator

Fig. 1.16 Schematic of an experiment demonstrating photon entan-

glement by correlations at a distance ([1.80]). The details of the experiment are described in the text

1 . 2

(1.70)

However, the indeterminacy relation requires only that the product of the uncertainties be a minimum. Hence, it is possible to imagine and to construct minimumuncertainty states in which the uncertainty in one field quadrature is reduced at the cost of increasing the uncertainty in the other quadrature [1.81]. Coherent light with such properties is said to be quadrature squeezed; the properties of such states are explored in detail in [1.57]. Another kind of coherent light is formed from the so-called number states of the electromagnetic field. These states, first used to describe optical correlations by Glauber [1.82–84] and Sudarshan [1.85], can in fact exhibit the properties of light that is number- or amplitude-squeezed. Additionally, of course, it is possible to squeeze the conjugate phase variable of the number states. Figure 1.15 illustrates the different relationships of the quadratures in the various kinds of squeezed light.

1.7.3 Entangled Light In 1935, Einstein, Podolsky and Rosen (EPR) published their observation that for entangled states (Schrödinger’s terminology [1.86]), quantum mechanics predicted that measurements on objects far apart could show a correlation. How then, argued Einstein, was it possible for quantum mechanics to be a complete, causal description of reality if the local results of a measurement could be influenced by quantum correlations at a large distance? Three decades later, J. S. Bell transformed this philosophical question into an experimentally testable issue by demonstrating that the quantum-mechanical predictions for a pair of interacting spin-1/2 particles were not consistent with the predictions of EPR for a system based on the idea of strict local causality [1.87, 88] In the event, it turned out to be easier to test these predictions using spin-1 photons instead of spin-1/2 particles, following procedures suggested by Clauser and Shimony [1.89]. Experiments by Aspect [1.90] and later by Zeilinger et al. [1.80] showed conclusively that photons emitted from a single source violated Bell’s inequality, and were thus entangled by correlations at a distance of precisely the kind that EPR argued were impossible.

The Properties of Light

400 m, a distance that precludes semiclassical correlations by virtue of the use of ultrashort light pulses and random switching of the preferred orientation of the polarization analyzers. The results appear to clearly demonstrate that quantum correlations at large distance are one of the novel realities of quantum physics. They also have applications in quantum cryptography [1.91, 92] and quantum teleportation [1.93, 94].

1.8 Summary The properties of light at this point in the history of optical science are as varied as the multiple ways in which light can now be generated and detected. The kinds of sources that can generate light range from classical thermal (chaotic) sources and coherent sources such as lasers, wigglers and undulators – and most recently to nonclassical sources generated by exquisite control of light-emission processes at the single-quantum level. While the statistical properties of light governed by firstorder correlations are virtually indistinguishable for both classical and quantum light sources, the second-order correlations reveal distinctive differences, particularly in the statistical properties of the light generated by these different sources. The wide-ranging impact of these studies, underlined by the recent award of a Nobel prize for the theory of quantum coherence, suggests not only that the field has reached the stature of a major part of physics, but that it has grown measurably in importance and is likely to continue to do so in the future. From a review written for the World Year of Physics in 2005 [1.95], one finds that two of the three celebrated papers in Einstein’s annus mirabilis had deep intellec-

tual connections with questions about the properties of light. The paper on special relativity was derived from a profoundly insightful critique of the classical theory of electromagnetic radiation [1.96]. In the second paper, Einstein literally solidified the concept of the light quantum (later christened the photon), which Planck had regarded as a mere heuristic, a mathematical device for saving appearances [1.97]. Einstein did so by showing from the Planck distribution that the entropy of cavity radiation has the same volumetric dependence as the entropy of an ideal gas – and therefore, he inferred, it was appropriate to treat light quanta as particles. But Einstein was also profoundly critical of the quantum physics of his time, even though, or perhaps because, his own interpretation of quantum phenomena – as in the EPR paradox – turned out not to be commensurate with experimental reality [1.98]. Nevertheless, and notwithstanding his objections, perhaps Einstein would also find it amusing that contemporary studies of both classical and quantum optics continue to generate informed criticism and new, exciting interpretations of the quantum physics whose foundations he helped to lay.

References 1.1 1.2 1.3

1.4

V. Ronchi: The Nature of Light: An Historical Survey (Heinemann, London 1970) Plato: Timaeus (Bobbs-Merrill, Indianapolis 1959) p. 117 D. C. Lindberg: The Beginnings of Western Science: The European Scientific Tradition in Philosphical, Religious, and Institutional Context, 600 B.C. to A.D. 1450 (Univ. Chicago Press, Chicago 1992) p. 455 D. Park: The Fire Within the Eye: A Historical Essay on the Nature and Meaning of Light (Princeton Univ. Press, Princeton 1997)

1.5 1.6 1.7 1.8

1.9

D. C. Lindberg: Studies in the History of Medieval Optics (Variorum Reprints, London 1983) A. R. Hall: The Revolution in Science 1500–1750, 2 edn. (Longman, London 1984) A. I. Sabra: Theories of Light from Descartes to Newton (Cambridge Univ. Press, Cambridge 1981) L. Davidovich: Sub-Poissonian processes in quantum optics, Rev. Mod. Phys. 68(1), 127–173 (1996) E. Hecht: Optics, 3 edn. (Addison–Wesley, Reading 1998)

Part A 1

The schematic of the experiment is shown in Fig. 1.16. Two photons are emitted from a common source, and pass through polarization-measuring devices that measure the photons emitted with orientations a and b perpendicular to the z-axis. At each terminus of the experiment, a computer monitors the orientation of the polarization as measured and the precise timing signal provided by an atomic clock. In the Zeilinger experiments, the polarization measurements are separated by

References

Part A

Basic Principles and Materials

Part A 1

1.10

1.11 1.12

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The Properties of Light

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Part A 1

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1.96 1.97

A. Einstein: Zur Elektrodynamik bewegter Körper, Ann. Phys. 17, 891–921 (1905) A. Einstein: Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heutis-

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Geometrical O 2. Geometrical Optics

2.1

The Basics and Limitations of Geometrical Optics ............................ 2.1.1 The Eikonal Equation .................... 2.1.2 The Orthogonality Condition of Geometrical Optics....................

34 34 35

2.2

2.1.3 The Ray Equation ......................... 2.1.4 Limitations of the Eikonal Equation 2.1.5 Energy Conservation in Geometrical Optics.................... 2.1.6 Law of Refraction ......................... 2.1.7 Law of Reflection .........................

35 36

Paraxial Geometrical Optics ................... 2.2.1 Paraxial Rays in Homogeneous Materials ............ 2.2.2 Refraction in the Paraxial Case....... 2.2.3 The Cardinal Points of an Optical System ..................... 2.2.4 The Imaging Equations of Geometrical Optics.................... 2.2.5 The Thin Lens............................... 2.2.6 The Thick Lens ............................. 2.2.7 Reflecting Optical Surfaces............. 2.2.8 Extension of the Paraxial Matrix Theory to 3 × 3 Matrices .................

38 38 39

39 42 44 49 51 52 55 56

2.3

Stops and Pupils................................... 2.3.1 The Aperture Stop......................... 2.3.2 The Field Stop ..............................

60 60 61

2.4

Ray Tracing .......................................... 2.4.1 Principle ..................................... 2.4.2 Mathematical Description of a Ray ...................................... 2.4.3 Determination of the Point of Intersection with a Surface ........ 2.4.4 Calculation of the Optical Path Length ............. 2.4.5 Determination of the Surface Normal ................... 2.4.6 Law of Refraction ......................... 2.4.7 Law of Reflection ......................... 2.4.8 Non-Sequential Ray Tracing and Other Types of Ray Tracing ......

61 61

Aberrations .......................................... 2.5.1 Calculation of the Wave Aberrations................ 2.5.2 Ray Aberrations and the Spot Diagram ................... 2.5.3 The Seidel Terms and the Zernike Polynomials ......... 2.5.4 Chromatic Aberrations ..................

2.5

62 63 65 65 65 66 67

68 68 69 71

Part A 2

This chapter shall discuss the basics and the applications of geometrical optical methods in modern optics. Geometrical optics has a long tradition and some ideas are many centuries old. Nevertheless, the invention of modern personal computers which can perform several million floating-point operations in a second also revolutionized the methods of geometrical optics and so several analytical methods lost importance whereas numerical methods such as ray tracing became very important. Therefore, the emphasis in this chapter is also on modern numerical methods such as ray tracing and some other systematic methods such as the paraxial matrix theory. We will start with a section showing the transition from wave optics to geometrical optics and the resulting limitations of the validity of geometrical optics. Then, the paraxial matrix theory will be used to introduce the traditional parameters such as the focal length and the principal points of an imaging optical system. Also, an extension of the paraxial matrix theory to optical systems with non-centered elements will be briefly discussed. After a section about stops and pupils the next section will treat ray tracing and several extensions to analyze imaging and non-imaging optical systems. A section about aberrations of optical systems will give a more vivid insight into this matter than a systematic treatment. At the end a section about the most important optical instruments generally described with geometrical optics will be given. These are the achromatic lens, the camera, the human eye, the telescope, and the microscope. For more information about the basics of geometrical optics we refer to text books such as [2.1–8].

Part A

Basic Principles and Materials

2.6

Some Important Optical Instruments ...... 2.6.1 The Achromatic Lens ..................... 2.6.2 The Camera ................................. 2.6.3 The Human Eye ............................

72 72 74 77

2.6.4 The Telescope .............................. 2.6.5 The Microscope ............................

78 82

References ..................................................

Part A 2.1

2.1 The Basics and Limitations of Geometrical Optics 2.1.1 The Eikonal Equation Geometrical optics is normally deﬁned to be the limiting case of wave optics for very small wavelengths λ → 0. In fact it is well-known that electromagnetic waves with a large wavelength λ such as radio waves cannot generally be treated with geometrical optical methods. X-rays and gamma radiation on the other hand propagate nearly like rays. They can generally be described quite well with geometrical optical methods provided the size of the optical elements (especially stops) is at least several hundred wavelengths. The accuracy of a geometrical optical calculation increases if the size of the optical element increases compared to the wavelength of the light used. The basic equations of geometrical optics [2.1, 7] are derived directly from the Maxwell equations. The restriction here is that only linear and isotropic materials are considered. Additionally, the electric charge density ρ is assumed to be zero. In this case the four Maxwell equations are (see Sects. 1.3, 3.1.1): ∂ B(r, t) ∇ × E(r, t) = − , ∂t ∂ D (r, t) + j(r, t) , ∇ × H(r, t) = ∂t ∇ · B(r, t) = 0 , ∇ · D(r, t) = 0 ,

(2.1) (2.2) (2.3) (2.4)

where the following quantities of the electromagnetic ﬁeld are involved: electric vector E, magnetic vector H, electric displacement D, magnetic induction B and electric current density j. The arguments illustrate that all quantities are in the general case functions of the spatial coordinates x, y, z with position vector r = (x, y, z) and of the time t. The so called nabla operator ⎛∂⎞ ∇

⎜ ∂x ∂ ⎟ = ⎝ ∂y ⎠ ∂ ∂z

(2.5)

is used and the symbol “×” indicates the vector product of two vectors whereas “·” indicates the scalar product of two vectors. The material equations in the case of linear and isotropic materials link the electromagnetic quantities with each other: D(r, t) = (r)0 E(r, t) , B(r, t) = µ(r)µ0 H(r, t) , j(r, t) = σ(r)E(r, t) .

(2.6) (2.7) (2.8)

The function is the dielectric function, µ is the magnetic permeability and σ is the speciﬁc conductivity. The constants 0 and µ0 are the dielectric constant of vacuum and the magnetic permeability of vacuum, respectively. A quite general approach for stationary monochromatic waves is used to describe the electric and the magnetic ﬁeld E(r, t) = e(r) eik0 L(r) e−iωt ,

(2.9)

H(r, t) = h(r) eik0 L(r) e−iωt .

(2.10)

The real function L is the optical path length and the vectors e and h are in the general case complex-valued to be able to represent all polarization states. The surfaces with constant optical path length L are the wave fronts, and the term Φ(r) = k0 L(r) is the phase of the wave; e and h are slowly varying functions of the position r whereas the term exp(ik0 L) varies rapidly because the constant k0 is deﬁned as k0 = 2π/λ with the vacuum wavelength λ. The angular frequency ω of the wave is linked to λ by ω = 2πc/λ = ck0 , where c is the speed of light in vacuum. By applying these equations to the Maxwell equations the so called time-independent Maxwell equations result: ∇ × e(r) eik0 L(r) = iωµ(r)µ0 h(r)

∇ × h(r) e

ik0 L(r)

· eik0 L(r) ,

(2.11)

= [−iω(r)0 + σ(r)] · e(r) eik0 L(r) ,

(2.12)

Geometrical Optics

∇ · µ(r)µ0 h(r) eik0 L(r) = 0 , ∇ · (r)0 e(r) eik0 L(r) = 0 .

(2.14)

= [ik0 ∇L(r) × e(r) + ∇ × e(r)] eik0 L(r) , ∇ × h(r) eik0 L(r)

(2.16)

= [ik0 ∇L(r) × h(r) + ∇ × h(r)] eik0 L(r) .

(2.17)

So, (2.11) and (2.12) give: ∇L(r) × e(r) − cµ(r)µ0 h(r) i = ∇ × e(r) , (2.18) k0 ∇L(r) × h(r) + c(r)0 e(r) i = (2.19) [∇ × h(r) − σ(r)e(r)] . k0 For the limiting case λ → 0, i. e. k0 → ∞, the right sides of both equations become zero (2.20) (2.21)

Now, (2.20) is inserted into (2.21) and the calculus for a double vector product is applied 1 ∇L(r) × [∇L(r) × e(r)] + c(r)0 e(r) = 0 cµ(r)µ0 ⇒ [∇L(r) · e(r)] ∇L(r) = 1/c2

(2.22)

Here, µ0 0 and µ are used, where n is the refractive index of the material. Equation (2.21) shows that the scalar product ∇L · e is zero and the ﬁnal result is the well-known eikonal equation [∇L(r)]2 = n 2 (r) .

= n2

(2.23)

L= const.

Fig. 2.1 Optical ray: the trajectory which is perpendicular to the surfaces of equal optical path length L

This is the basic equation of geometrical optics which provides e.g. the basis for the concept of optical rays. A ray is deﬁned as that trajectory which is always perpendicular to the wave fronts, which are the surfaces of equal optical path length L (Fig. 2.1). Therefore, a ray points in the direction of ∇L. Equation (2.23) has the name eikonal equation because the optical path length L is for historical reasons sometimes called the eikonal [2.1].

2.1.2 The Orthogonality Condition of Geometrical Optics Equations (2.20) and (2.21) can be solved for e and h: 1 ∇L(r) × e(r) , cµ(r)µ0 1 e(r) = − ∇L(r) × h(r) . c(r)0

h(r) =

(2.24) (2.25)

This shows on the one hand that h is perpendicular to e as well as ∇L and on the other hand that e is perpendicular to h as well as ∇L. Therefore, in the limiting case λ → 0 ∇L, e and h have to form an orthogonal triad of vectors. This conﬁrms the well-known fact that electromagnetic waves are transversal waves. At the end of the last section a light ray has been deﬁned as being parallel to ∇L and in Sect. 2.4 the important method of ray tracing will be explained. An extended method of ray tracing is polarization ray tracing where the polarization state of a ray which locally represents a wave is transported along with each ray [2.10, 11]. Using the results of this section it is clear that the vector e, which indicates the polarization (and amplitude) of the ray, has to be perpendicular to the ray direction ∇L.

2.1.3 The Ray Equation A surface with constant values L is a surface of equal optical path length. Now, a ray is deﬁned as that trajectory starting from a certain point in space which is

Part A 2.1

Using the rules of the nabla calculus the left-hand sides of (2.11) and (2.12) can be transformed to ∇ × e(r) eik0 L(r)

= ∇ eik0 L(r) × e(r) + eik0 L(r) ∇ × e(r)

− [∇L(r)]2 e(r) + n 2 (r)e(r) = 0 .

(2.13)

Equation (2.13) is not independent of (2.11) because it is well known that the quantity ∇ · (∇ × f ) of an arbitrary vector function f is always zero [2.9]. Therefore, if (2.11) is fulﬁlled, (2.13) will also be fulﬁlled. In the case of nonconducting materials, i. e. σ = 0, the same is valid for the relation between (2.12) and (2.14). In the more general case σ = 0, (2.12) and (2.14) require that ∇ · σ(r)e(r) eik0 L(r) = 0 . (2.15)

∇L(r) × e(r) − cµ(r)µ0 h(r) = 0 , ∇L(r) × h(r) + c(r)0 e(r) = 0 .

2.1 The Basics and Limitations of Geometrical Optics

Part A

Basic Principles and Materials

Part A 2.1

perpendicular to the surfaces of equal optical path length. Therefore, ∇L points in the direction of the ray. We use the arc length s along the curve that is deﬁned by the ray (Fig. 2.2). Then, if r describes now the position vector of a point on the ray, dr/ds is a unit vector which is tangential to the ray curve and the eikonal equation (2.23) delivers: ∇L = n

dr ds

(2.26)

Here and in the following L and n are not explicitly indicated as functions of the position to tighten the notation. From (2.26) a differential equation for the ray can be derived by again using (2.23) and the deﬁnition of d∇L/ds as being the directional derivative of ∇L along dr/ ds:

⇒

dr d dr d n = ∇L = · ∇(∇L) ds ds ds ds 1 1 = ∇L · ∇(∇L) = ∇(∇L)2 n 2n 1 ∇n 2 = ∇n , = 2n dr d n = ∇n . (2.27) ds ds

This is the differential equation for a ray in a general inhomogeneous isotropic and linear substance. Such materials where the refractive index is a function of the position are often called graded index (GRIN) materials. In this case the solution of the differential equation may be a quite complex curve. However, the most important case is that n is independent of the position, i. e. the ray propagates in

dr/ds

r(s)

Fig. 2.2 A curved light ray in a general (inhomogeneous) material. r(s) is the position vector of a point on the ray, where s is the arc length along the curve and O is the origin of the coordinate system. Then, the vector dr/ ds is a unit vector tangential to the ray

a homogeneous material. Then, a simple differential equation for the ray is obtained: d2r =0. ds2

(2.28)

The solution of this equation is a straight line. So, the ray equation in homogeneous materials is r = sa + p

(2.29)

with constant vectors a and p. This means that light rays propagate rectilinearly in a homogeneous and isotropic material if the eikonal equation is valid. Moreover, a has to be a unit vector, i. e. |a| = 1, because s is the geometrical path length along the ray. So, for the distance between two points P1 and P2 with position vectors r1 and r2 we have (s2 > s1 ): |r2 − r1 | = (s2 − s1 ) |a| = s2 − s1 ⇒ |a| = 1 . The limitations of the validity of the eikonal equation will be investigated in the next section.

2.1.4 Limitations of the Eikonal Equation Besides using directly the Maxwell equations the eikonal equation can also be derived from the wave equation and, in the case of a monochromatic wave, from the Helmholtz equation. This will be done in the following for a homogeneous, isotropic and linear dielectric material, i. e. where n is constant and σ = 0. Moreover, it is assumed that the scalar case is valid, i. e. that polarization effects can be neglected and only one component u(r) of the electric or magnetic vector has to be considered. In this limiting case it is easier to start directly with the scalar Helmholtz equation (Sect. 3.1.5) than to start as in Sect. 2.1.1 with the Maxwell equations and then to make the transition to the scalar case. The scalar Helmholtz equation is: ∇ · ∇ + (nk0 )2 u(r) = 0 . (2.30) Analogously to (2.9) or (2.10) the following approach for u is used u(r) = A(r) eik0 L(r) ,

(2.31)

where the amplitude A and the optical path length L are both real functions of the position and A varies only slowly with the position.

Geometrical Optics

∇

(2.32)

∇

Since A, L, k0 and n are all real quantities the real and the imaginary part of (2.32) can be simply separated and both have to be zero. To obtain the eikonal equation only the real part is considered: A − k02 (∇L)2 + n 2 k02 = 0 , A 1 A ⇒ (∇L)2 = n 2 + 2 . (2.33) k0 A =: γ ∇

∇

In the limiting case λ → 0 ⇒ k0 → ∞ the term γ can be neglected and again the eikonal equation (2.23) is obtained (∇L)2 = n 2 . But (2.33) shows that the eikonal equation can also be fulﬁlled for a ﬁnite value of λ with good approximation as long as the term γ is much smaller than 1, because the order of magnitude of n 2 is typically between 1 (vacuum) and 12 (silicon for infrared light). Therefore, the condition is: ∇

γ 1 ⇒

A λ2 1. 2 4π A

(2.34)

A = u0 ⇒

A=0 ⇒ γ =0,

L = na · r ⇒ ∇L = na ⇒ (∇L)2 = n 2 . Of course, a plane wave is also a solution of the Maxwell equations. A second example is a spherical wave, which is a solution of the scalar Helmholtz equation but not of the Maxwell equations themselves because the orthogonality conditions (2.24) and (2.25) cannot be fulﬁlled for a spherical wave in the whole of space. Nevertheless, a spherical wave u(r) = u 0 exp(ink0 r)/r with r = |r| is a very important approximation in many cases and a dipole radiation behaves in the far ﬁeld in a plane perpendicular to the dipole axis like a spherical wave. For the spherical wave we obtain: r

u0 A= ⇒ A = u0∇ · − 3 r r 3u 0 3u 0 r · r =− 3 + =0 ⇒ γ =0, r r5 r L = nr ⇒ ∇L = n ⇒ (∇L)2 = n 2 . r Here, the coordinate system has been chosen in such a way that the center of curvature of the spherical wave ∇

∇

A ∇ A · ∇L − k02 (∇L)2 + n 2 k02 + 2ik0 A A + ik0 L = 0 .

∇

Here, := ∇ · ∇ is the Laplace operator or Laplacian. So, by inserting the expression for u into the Helmholtz equation and dividing it by u the result is

This is fulﬁlled with good approximation if A is a slowly varying function of the position, i. e. if the relative curvature of A over the distance of a wavelength is very small. If the term γ is not very small the right-hand side of (2.33) depends on the position (because A depends generally on r) even though n is constant. Formally this is equivalent to an eikonal equation with positiondependent refractive index n so that light rays would formally be bent in regions of a rapidly changing amplitude, as e.g. in the focus. Therefore, the results of ray-tracing calculations (Sect. 2.4), which assume rectilinear rays in a homogeneous material, are not correct in the neighborhood of the focus where the amplitude changes rapidly. If aberrations are present the variation of the amplitude in the focal region will be less severe and the accuracy of geometrical optical calculations improves with increasing aberrations. In practice, a rule of thumb is that the focal region of an aberrated wave calculated with ray tracing approximates the actual focus very well if the result of the ray-tracing calculation gives a focus that has several times the size of the corresponding diffraction limited focus (Airy disc), which can easily be estimated (Sect. 3.4, PSF (point spread function)). There are also scalar waves which fulﬁll exactly the eikonal equation so that the term γ is exactly zero. One example is a plane wave with u(r) = u 0 exp(ink0 a · r). a is a constant unit vector in the direction of propagation and u 0 is also a constant. So, we have

Part A 2.1

Then, by omitting the arguments of the functions we can write

∇u = ∇ A eik0 L = eik0 L ∇ A + ik0 A eik0 L ∇L ∇A + ik0 ∇L u , = A

∇A + ik0 ∇L u u = ∇· A 2 ∇A + ik0 ∇L u = A A (∇ A)2 − + + ik0 L u A A2

A − k02 (∇L)2 = A ∇ A · ∇L + ik0 L u . +2ik0 A

2.1 The Basics and Limitations of Geometrical Optics

Basic Principles and Materials

Part A 2.1

is at the origin. Of course, it is quite straightforward to formulate the spherical wave with an arbitrary center of curvature r0 by replacing r with |r − r0 |. So, plane waves and spherical waves, which are very important in geometrical optics, fulﬁll both the eikonal equation (2.23) not only in the limiting case λ → 0 but also for ﬁnite wavelengths λ.

2.1.5 Energy Conservation in Geometrical Optics The imaginary part of (2.32) gives information about the intensity of the amount of transported light: ∇A =0. (2.35) A Since the intensity I of a light wave is proportional to the square of the amplitude A2 the following equality holds: ∇ A2 2A∇ A ∇A ∇I = . = =2 I A A2 A2 Therefore, (2.35) delivers L + 2∇L ·

I L + ∇L · ∇I = 0 or ∇ · (I∇L) = 0 .

(2.36)

Now, the integral theorem of Gauss can be applied ∇ · (I∇L) dV = I∇L · dS = 0 , (2.37) V

the mantle surface. Therefore, on the mantle surface the vectors ∇L (ray direction) and dS (surface normal) are perpendicular to each other and therefore ∇L · dS = 0. At the two face surfaces of the light tube (refractive index n) with surface vectors dS1 and dS2 , which are assumed to have an inﬁnitesimally small size, the electromagnetic power ﬂux P1 and P2 are given by Pj =

∇

Part A

∇

where the left integral symbolizes a volume integral over a volume V and the right integral a surface integral over the closed surface S which conﬁnes the volume V . A light tube (Fig. 2.3) is a tube-like entity (simple forms are e.g. a cylinder or a cone) where light rays form

Ij |∇L j · dS j | ; n

Using (2.37) the result is I∇L · dS = 0 = I1 ∇L 1 · dS1 + I2 ∇L 2 · dS2 . (2.38)

Hereby, the surface normals dS1 or dS2 always point out of the closed surface S and therefore ∇L 1 · dS1 and ∇L 2 · dS2 have opposite algebraic signs. In total, the power ﬂux P1 = −I1 ∇L 1 · dS1 /n which enters the light tube at the left is equal to the power ﬂux P2 = I2 ∇L 2 · dS2 /n which leaves the light tube at the right: P1 = P2 . This means that the energy is conserved and we can formulate the following lemma: In the scope of geometrical optics the electromagnetic power (energy) is transported along the light rays and the total light power is conserved in a light tube if no light is absorbed.

2.1.6 Law of Refraction Let us consider the interface between two materials with refractive index n 1 on one side and n 2 on the other. This interface is assumed to be replaced by a very thin layer in which the refractive index varies quite rapidly but continuously from n 1 to n 2 . An inﬁnitesimally small rectangular closed loop C is then constructed at the interface in such a way that two of the edges of the loop are parallel to the interface and the other two edges are parallel to the surface normal N (|N| = 1) of the interface

dS2

n2 t

Fig. 2.3 Scheme of a light tube which is bounded by a bundle of rays

N θ1

dS1

j ∈ {1, 2} .

θ2

Fig. 2.4 The parameters used in deriving Snell’s law

Geometrical Optics

where the left integral is a surface integral over the inﬁnitesimally small rectangular surface S which is bounded by the closed loop C. The right integral is a line integral over the closed loop C. If now the length of the side lines of the loop C parallel to N tends to zero the line integral is 0 = lt · (n 2 a2 − n 1 a1 ) (2.40) with l being the length of a side line of the loop parallel to the interface and t being a unit vector parallel to the interface. Another unit vector b is deﬁned as being perpendicular to both N and t and therefore also perpendicular to the surface S. This means that N, t and b form an orthogonal triad of unit vectors with t = b × N and therefore it holds that (b × N) · (n 2 a2 − n 1 a1 ) = 0 , ⇒ [N × (n 2 a2 − n 1 a1 )] · b = 0 . But the rectangular integration area can be rotated about N serving as axis. Therefore, the direction of b can be chosen arbitrarily as long as it is perpendicular to N. By fulﬁlling the upper equation for an arbitrary vector b we obtain the following equation, which is the vectorial formulation of the law of refraction, N × (n 2 a2 − n 1 a1 ) = 0 . (2.41) This means that n 2 a2 − n 1 a1 is parallel to N (or n 2 a2 − n 1 a1 = 0 what is only possible for the trivial

case n 1 = n 2 ) and therefore all three vectors a1 , a2 and N have to lie in the same plane. This means particularly that the refracted ray with direction vector a2 lies in the plane of incidence formed by N and a1 . By deﬁning the acute angles θ j between the rays a j ( j ∈ {1, 2}) and the surface normal N (Fig. 2.4) the modulus of (2.41) results in: n 1 sin θ1 = n 2 sin θ2 . (2.42) This is the well-known Snell’s law. If n 2 is bigger than n 1 there is always a solution θ2 for a given angle θ1 . But, if n 2 is smaller than n 1 there is the so-called critical angle of total internal reﬂection θ1,critical for which the refracted ray grazes parallel to the interface, i. e. θ2 = π/2: n 1 sin θ1,critical = n 2 sin θ2 = n 2 n2 ⇒ θ1,critical = arcsin . (2.43) n1 For angles θ1 > θ1,critical there exists no refracted ray because the sine function sin θ2 cannot be larger than 1. Then, all light is reﬂected at the interface and only a reﬂected ray exists.

2.1.7 Law of Reflection If a plane wave enters the interface between two materials there is, as well as a refracted wave, a reﬂected wave and in the case of total internal reﬂection there is only a reﬂected wave. A ray represents locally a plane wave and the law of reﬂection is formally obtained from (2.41) by setting n 2 = n 1 . Of course, the algebraic signs of the scalar products N · a1 and N · a2 have to be different in order to obtain a reﬂected ray whereas they have to be identical to obtain a refracted ray. This will be discussed later in Sect. 2.4 by ﬁnding explicit solutions of (2.41). After having now discussed the basics of geometrical optics the next section will treat the paraxial ray tracing through an optical system by using a matrix theory [2.12–15].

2.2 Paraxial Geometrical Optics 2.2.1 Paraxial Rays in Homogeneous Materials Some Basic Definitions In a homogeneous material the refractive index n is constant and therefore, according to the ray equation (2.29),

a ray propagates rectilinear. This means that a ray is deﬁnitely described by the position vector p of one point on the ray and the ray direction vector a. So, six scalar parameters (each vector with three components) are necessary. In principle one component of a is redundant (apart from the algebraic sign of this com-

Part A 2.2

where N points from material 2 to material 1 (Fig. 2.4). Since the direction vectors a of the light rays can be expressed as the gradient of a scalar function (2.26) the following identity is valid: dr = ∇ × ∇L = 0 . (2.39) ∇× n ds The ray direction vector is written in the following as a = dr/ ds. Using the integral theorem of Stokes equation (2.39) delivers ∇ × (na) · dS = na · dr = 0 ,

2.2 Paraxial Geometrical Optics

Part A

Basic Principles and Materials

Part A 2.2

ponent) because a is a unit vector (a2x + a2y + az2 = 1). Another component can be saved if a reference plane is deﬁned, e.g. the x–y plane at z = 0. Then, the xand y-components of the point of intersection of the ray with this plane are sufﬁcient. However, in the case of non-paraxial ray tracing (Sect. 2.4) all components of the two vectors p and a are stored and used since not all rays start in the same plane and the algebraic sign of each component of a is needed. Moreover, it is often more efﬁcient to store a redundant parameter instead of calculating it from other parameters. Most convenient optical systems consist of a sequence of rotationally symmetric centered refractive and reﬂective components. The rotation axis is called the optical axis of the system. For a simple lens with two spherical surfaces the optical axis is deﬁned by the two centers of curvature C1 and C2 of the spherical surfaces (Fig. 2.5). Using (2.41) it has been shown that a refracted ray (and also a reﬂected ray) remains in the plane of incidence. Therefore, it is useful to deﬁne the meridional plane, which is a plane containing an object point P and the optical axis (Fig. 2.6). All rays which come from the object point P and lie in the meridional plane are called meridional rays. A plane perpendicular to the meridional plane which contains a special reference ray, mostly the chief ray (Sect. 2.3), is called the sagittal plane and rays lying in it are called sagittal rays. In this section only meridional rays are discussed and moreover only socalled paraxial rays are considered. Paraxial rays are rays which fulﬁll the following conditions:

Optical axis C1

Fig. 2.5 The optical axis of a lens

Object point P

Meridional plane

Optical axis

Fig. 2.6 The meridional plane of an optical system

• •

The distance x of the ray from the optical axis is small compared to the focal length of each optical element of the system. The angle ϕ between the optical axis and the ray is small, i. e. ϕ 1. The same has to be valid for other angles, e.g. for the refraction angles at a lens.

For the angles this means that the following approximations have to be valid: sin ϕ ≈ tan ϕ ≈ ϕ ,

cos ϕ ≈ 1 .

The most important optical systems consist of optical elements (refractive, reﬂective or diffractive elements) which are embedded into piecewise homogeneous materials. Therefore, ray tracing (in the paraxial as well as non-paraxial case) through an optical system consists of the alternating sequence of propagation in a homogeneous material and refraction (or reﬂection or diffraction) at an element. Optical Imaging At this point some words on the term optical imaging have to be said. An object point which is either illuminated by external light or self-illuminating emits a ray fan, i. e. in geometrical optics an object point is the source of a ray fan. On the other side an image point is the drain of a ray fan and in the ideal case all rays of the fan should intersect each other in the image point (Fig. 2.7a). Therefore, the image point can be deﬁned in the ideal case by the point of intersection of only two rays. However, this is only useful in the case of paraxial ray tracing where all aberrations of the optical system are neglected. If the aberrations of an optical system also have to be taken into account, non-paraxial ray tracing (Sect. 2.4), also simply called ray tracing, has to be used. Then, there are several deﬁnitions of an image point because there is in general no longer a single point of intersection of all rays of the ray fan coming from the object point (Fig. 2.7b). The lateral deviation of the actual point of intersection of a ray with the image plane from the ideal image point is called ray aberration. A more advanced deﬁnition of optical imaging has of course to take into account interference effects between the different rays coming from the object point since the image point is a multiple-beam interference phenomenon. A typical example where the simple ray-based model fails would be an ideal spherical wave where a half-wave plate is introduced in half of the aperture (Fig. 2.7c). Then, the ray directions are unchanged and an ideal point of intersection of all rays exists, i. e. there are no ray aberrations. But, the image point would be

Geometrical Optics

2.2 Paraxial Geometrical Optics

Ideal lens

x' Object point φ

Fig. 2.8 Scheme showing the parameters of paraxial ray

Real lens

tracing for the transfer between two parallel planes with distance d

Object point

Aberrated image point (ray aberrations)

Ideal lens λ/2 plate

Object point

Aberrated image point (wave aberrations)

Fig. 2.7a–c Schematic display of three different situations in optical imaging: (a) ideal imaging, (b) image point show-

ing ray aberrations (and of course also wave aberrations), (c) image point showing no ray aberrations but nevertheless

wave aberrations

massively disturbed because there is destructive interference in the center of the image point due to the different optical path lengths of the rays. Therefore, a more advanced ray-based model additionally calculates the optical path length along each ray. The deviation in the optical path length of a ray from the ideal optical path length is called wave aberration. However, in this section we will treat the very simple model of paraxial ray tracing, which takes into account neither ray aberrations nor wave aberrations. Aberrations will be taken into account in Sect. 2.4 when we consider non-paraxial ray tracing.

A Note on the Validity of the Paraxial Approximation The approximation of sin ϕ by ϕ means that the next term of the Taylor series −ϕ3 /6 and all higher order terms are neglected. In the case of tan ϕ the next term of the Taylor series which is neglected is +ϕ3 /3. So, the equivalence of sin ϕ and tan ϕ is only valid if the difference of both third-order terms ϕ3 /2 is so small that it can be neglected. This is the case if the alternation of the optical path length from the object point to the image point by neglecting this term is smaller than the Rayleigh criterion of λ/4, where λ is again the wavelength. In the case of two rays with an optical path difference of λ/4 the phase difference is ∆Φ = π/2, i. e. the rays are in phase quadrature and the intensities have to be added because the interference term cos(∆Φ) is then zero. If the optical path difference is λ/2 the phase difference is ∆Φ = π and the amplitudes of both rays cancel each other (if the amplitudes have equal modulus). Then, the image point is strongly aberrated. So, the validity of the Rayleigh criterion is useful to deﬁne the limitations of the paraxial approximation. In practice, the paraxial theory is quite important because it allows the deﬁnition of such important parameters as the focus, the focal length or the principal points of a lens or optical system. An optical designer [2.16–19] will always ﬁrst design an optical system by using the paraxial matrix theory (or another paraxial method) so that the paraxial parameters are right. Afterwards he will try to optimize the non-paraxial parameters using ray tracing in order to correct aberrations of the system. Definition of a Paraxial Ray In the paraxial theory only rays in the meridional plane, which is here deﬁned as the x–z-plane, are regarded. Then, the y-component of the ray direction

Part A 2.2

Ideal image point

Part A

Basic Principles and Materials

Part A 2.2

vector a and the y-component of the starting point p of the ray are both zero: a y = 0 and p y = 0. We deﬁne for the x-component of the ray direction vector ax = sin ϕ ≈ tan ϕ ≈ ϕ. The z-component of the ray direction vector in the paraxial approximation is then az = cos ϕ ≈ 1. Therefore, a meridional paraxial ray at a certain z-position z can be described by the angle ϕ with the optical axis and the ray height x, which is indeed the x-component px of the starting point p of the ray. The z-component pz of a ray is noted in the paraxial matrix theory externally because in many cases several rays starting at the same z-position z = pz but having different values x and ϕ are considered. So, in total a paraxial ray is described by x and ϕ. Since matrix methods play an important role in optics [2.12, 13] these two parameters are noted as the components of a vector x ϕ so that the optical operations which we will discuss now can be represented as 2 × 2 matrices. Transfer Equation The paraxial ray tracing between two planes with a separation d and which are perpendicular to the optical axis is one of the basic operations. Here, only the lines of intersection of these two planes with the meridional plane are regarded (Fig. 2.8), even though later we will use the term planes slightly incorrectly. The ray parameters in the ﬁrst plane shall be x and ϕ and those in the second plane x and ϕ . Then, the transfer from the ﬁrst plane to the second plane with distance d is done by (Fig. 2.8): x x + ϕd . = (2.44) ϕ ϕ

This means that ray directions are not changed during the propagation of paraxial rays in a homogeneous material. Equation (2.44) can also be written by using a two times

two matrix [2.12–15]: 1 d x x x = = MT . ϕ 0 1 ϕ ϕ

(2.45)

The matrix MT is called the paraxial transfer matrix in a homogeneous material.

2.2.2 Refraction in the Paraxial Case Paraxial Law of Refraction The law of refraction connects the angle i between the incident ray and the surface normal with the angle i between the refracted ray and the surface normal (Fig. 2.9). The law of refraction (2.42) in the paraxial formulation is

ni = n i

(2.46)

where n and are the refractive indices of the two homogeneous materials in front of the surface and behind the surface, respectively. Refraction at a Plane Surface A paraxial ray with parameters x and ϕ hits a plane surface which is perpendicular to the optical axis (Fig. 2.9). The refractive index is n in front of the surface and n behind the surface. Then, the ray height x remains unchanged and only the ray parameter ϕ changes according to the paraxial law of refraction (2.46): 1 0 x x x = MR (2.47) = 0 nn ϕ ϕ ϕ

The matrix MR is the paraxial matrix for refraction at a plane surface. Refraction at a Plane Parallel Plate The plane parallel plate is the simplest case for a sequence of several surfaces and can be used to demonstrate the principle of tracing paraxial rays through an optical system by using the paraxial matrix theory. It is well known that the order of two matrices A and B is very important if two matrices have to be multiplied, i. e.

AB = BA . i' i

Fig. 2.9 Paraxial refraction at a (locally) plane surface

Therefore, the matrix for the ﬁrst operation has to be positioned immediately to the left of the vector (x, ϕ) of the paraxial ray that has to be traced through the system. The matrix of the next operation then has to be multiplied from the left side and so on for all other matrices. Using the notations of Fig. 2.10 the parameters

Geometrical Optics

Here, n is the refractive index left of the plane parallel plate and n is the refractive index right to the plane parallel plate. In total the parameters x and ϕ of a paraxial ray immediately behind the plane parallel plate are obtained from the parameters x and ϕ of the incident ray immediately in front of the plane parallel plate by multiplying them by the matrix MP of a plane parallel plate 1 d nnP x x x = MP . = (2.48) n ϕ ϕ ϕ 0 n The most important practical case is a plane parallel plate in air (n = n = 1). Then it holds that 1 ndP x x . = (2.49) ϕ ϕ 0 1 This means that the matrix of a plane parallel plate is identical to the transfer matrix in a homogeneous material on substitution of the transfer distance d in the

φ> 0 x> 0 x n > 1 and R > 0 (convex surface) a positive ray height x of the incident ray results in a negative ray angle ϕ of the refracted ray. This means that the convex spherical surface with lower refractive index on the left side has a positive refractive power and focuses a plane wave. For ϕ = 0, n > n > 1 and R < 0 (concave surface) the angle ϕ of the refracted ray is positive if the ray height x of the incident ray is also positive. This means that two rays would only intersect virtually in front of the lens. Therefore, a concave spherical surface with a lower refractive index on the left-hand side has a negative focal power.

2.2.3 The Cardinal Points of an Optical System

xO F

U φ

xI f

Fig. 2.13 Cardinal points of an optical system: F and F are the foci in the object space and the image space, respectively. N and N are the nodal points in the object and image space and U and U are the unit or principal points in the object and image space

An optical imaging system has several cardinal points [2.1, 3], and by knowing these values the paraxial properties of the optical system are determined deﬁnitely. The cardinal points are the principal points, the nodal points and the focal points and they are all situated on the optical axis. In order to deﬁne them some additional deﬁnitions have to be made. The cardinal points will be calculated in this section for a general optical system using the paraxial matrix theory [2.6]. At the end of this section the cardinal points of the simplest optical system, a refracting spherical surface, will be calculated explicitly to demonstrate the method. Assume a general optical imaging system such as that symbolized in Fig. 2.13. An object point PO with

Geometrical Optics

According to our sign convention the lateral magniﬁcation in Fig. 2.13 is negative since xO is positive and xI negative. The Principal Points The principal plane U or unit plane in the object space is the plane perpendicular to the optical axis that has the property that an object point in this principal plane is imaged to a point in the principal plane U of the image space with a lateral magniﬁcation β = +1. The points of intersection of the principal planes in the object and image space with the optical axis are called the principal or unit points U and U , respectively. So, U is the image of U. An important practical property of the principal planes following from this deﬁnition is that a ray which intersects the principal plane U in the object space at a height x is transferred to the principal plane U of the image space at the same height (Fig. 2.13). This property is e.g. used to construct the path of a paraxial ray graphically. The Nodal Points The second cardinal points of an optical system are the nodal points N (in the object space) and N (in the image space). A ray in the object space which intersects the optical axis in the nodal point N at an angle ϕ intersects the optical axis in the image space in the nodal point N at the same angle ϕ = ϕ. Therefore, the angular magniﬁcation γ deﬁned as

γ :=

ϕ ϕ

(2.53)

is γ = 1 for rays going through the nodal points. Additionally, since this has to be valid for arbitrary angles ϕ, the nodal point N is the image of the nodal point N. The Focal Points The focal points F (in the object space) and F (in the image space), also called principal foci or foci for short,

have the following properties. A ray starting from the focus F in the object space is transformed into a ray parallel to the optical axis in the image space. Vice versa, a ray which is parallel to the optical axis in the object space intersects the focus F in the image space. The planes perpendicular to the optical axis which intersect the optical axis in the focal points are called the focal planes. The distance between the principal point U and the focus F is called the focal length f in the object space and the distance between the principal point U and the focal point F is called the focal length f in the image space. In geometrical optics the sign convention for the focal length is usually that it is positive if the focus is right of the principal point. In Fig. 2.13 this means e.g. that the focal length f in the object space is negative whereas the focal length f in the image space is positive. A more general property of the focal planes is that rays starting from a point with object height xO in the focal plane of the object space form in the image space a bundle of parallel rays making an angle ϕ = −xO / f with the optical axis. The relation for ϕ can be easily understood by the fact that a ray starting from the object point parallel to the optical axis is transferred at the principal planes from U to U with the same height xO and then passes, after a distance f , the focal point F in the image space. The negative sign has to be taken due to the sign convention. Calculation of the Cardinal Points of a General Optical System Assume that we have a general optical system that is formed by an arbitrary combination of refracting spherical and plane surfaces which are all situated on a common optical axis (Fig. 2.14). Then, the system can be described by a 2 × 2 matrix M which is the product of a sequence of matrices MT , MR and MS (or further matrices for other optical elements). The matrix A B = MS,m · MT,m−1 · MS,m−1 · . . . M= C D

· MT,2 · MS,2 · MT,1 · MS,1

(2.54)

describes the propagation of a ray from a plane immediately in front of the vertex of the ﬁrst surface (surface 1) to a plane immediately behind the vertex of the last surface (surface m). Here, only matrices MS of refractive spherical surfaces are taken because a plane surface with matrix MR can be represented as a spherical surface with radius of curvature R = ∞. Additionally, behind each surface (apart from the last surface) the transfer to the

Part A 2.2

a lateral distance xO from the optical axis, which is called the object height, is imaged by the optical system to an image point PI with lateral distance xI , called the image height. The refractive indices are n in the object space and n in the image space. The lateral magniﬁcation β of an imaging system is deﬁned as the ratio of the image height xI to the object height xO : xI . (2.52) β := xO

2.2 Paraxial Geometrical Optics

Part A

Basic Principles and Materials

relation MS,1

n= n1

MS,2

n'1 =n2

Part A 2.2

N dN

n'2

MS,m – 1 MS,m

n'm– 1 n'm – 1 =nm

n i = n i−1

n' =n'm

F' dN'

dF' dU

dU'

Fig. 2.14 Distances between the cardinal points in the object space

and the vertex of the ﬁrst surface (quantities without apostrophe) and the vertex of the last surface and the cardinal points in the image space (quantities marked by an apostrophe) of a general optical are negative system consisting of refractive surfaces. dU and dU are again in the scheme, dF and dF are positive and dN and dN negative. n and n are the refractive indices in front of and behind the whole system, whereas n i and n i are the refractive indices in front of and behind the single refracting surface number i

next surface is described by using a matrix MT,i . In the special case of a thin lens (which does not exist in reality but which is an important idealization in geometrical optics) the propagation distance can just be set to zero so that the transfer matrix is identical to the unit matrix. The restriction to spherical surfaces is not stringent because in paraxial optics an aspheric surface is identical to a spherical surface if the radius of curvature of the aspheric surface at the vertex is identical to the radius of curvature of the spherical surface. From a mathematical point of view the determination of the radius of curvature in the paraxial regime just means that in both cases the parabolic terms are taken. Moreover, also cylindrical surfaces can be calculated with this method if the radius of curvature in the selected x–z-plane is taken. For a plane which contains the cylinder axis the cylindrical surface behaves like a plane surface whereas the cylindrical surface behaves like a spherical surface if the cylinder axis is perpendicular to the considered x–z-plane. A ray starts in front of the optical system in a material with refractive index n := n 1 and ends behind the system in a material with refractive index n := n m (Fig. 2.14). n i and n i with i ∈ {1, 2, . . . , m} are the refractive indices in front of and behind each refracting surface, which is described by the matrix MS,i . Of course, there is the

for i ∈ {2, 3, . . . , m} .

(2.55)

Now, a matrix M is calculated which describes the propagation of a ray from a plane P through the optical system to a plane P . The plane P is a distance d from the vertex of the ﬁrst surface of the optical system, whereas the vertex of the last surface is a distance d from the plane P . Using the paraxial sign conventions d is positive if P is in front of (i. e. left of) the ﬁrst surface. Similar d is positive if P is behind (i. e. right of) the last surface. It is very important to remember that d is measured from the plane P to the vertex of the ﬁrst surface whereas d is measured from the vertex of the last surface to the plane P . For these quantities the usual sign conventions are valid, i. e. they are positive if the propagation is from left to right and negative if the propagation is in the opposite direction. By using equation (2.45) the matrix M is: A B M = = MT MMT C D 1 d A B 1 d = 0 1 C D 0 1 A + Cd Ad + B + Cdd + Dd = . (2.56) C Cd + D Principal Points. To calculate the principal planes U

and U of the system the deﬁnition is used. If P is identical to the principal plane U and P is identical to U an object point in P has to be imaged to P with lateral magniﬁcation β = +1. Imaging means that all rays with arbitrary ray angles ϕ starting from the object point with height x have the same height x in P , independent of ϕ. Since the relation A x + B ϕ x A B x = = (2.57) ϕ C x + D ϕ ϕ C D holds, this means that the matrix element B has to be zero in order to have imaging between the planes P and P . Therefore, we have as a ﬁrst condition B = Ad + B + Cdd + Dd = 0 .

(2.58)

The second condition β = 1 means, by using B = 0, x = A x + B ϕ = A + Cd x = x , 1− A ⇒ A + Cd = 1 ⇒ dU . (2.59) = C

Geometrical Optics

Nodal Points. If P contains the nodal point N and P

contains the nodal point N the conditions for the ray parameters are x = x = 0 and ϕ = ϕ. Using (2.57) this gives:

0=x = A x+B ϕ= B ϕ ⇒ B = Ad + B + Cdd + Dd = 0 , ϕ = C x + D ϕ = D ϕ = ϕ ⇒ D = Cd + D = 1 .

ϕ = C x + D ϕ = D ϕ = 0 D . C

f = dU − dF = =

AD − BC C

D AD D (A − 1) − B + = −B. C C C (2.66)

The focus F in the image space can be calculated analogously. There, rays parallel to the optical axis (i. e. ϕ = 0) in front of the optical system in an arbitrary plane P , e.g. at d = 0, have to focus in the image space in the focus F at x = 0. If F is in the plane P , the distance dF := d between the vertex of the last surface of the optical system and the focus F is, by using (2.57), x = A x + B ϕ = A x = 0 ⇒ A = A + Cd = 0 ⇒ dF = −

(2.65)

A . C

(2.67)

Analogously, the focal length f , which is positive if F is to the right of U , can be calculated by 1 1 A 1− A − =− ⇒C=− . C C C f (2.68)

(2.62)

Focal Points and Focal Lengths. For the calculation of the focus F in the object space it is assumed that F is in the plane P . Then, all rays starting from the height x = 0 have to be in the image space rays parallel to the optical axis, i. e. ϕ = 0. Since this has to be valid in all planes in the image space the distance d in (2.56) is set to zero. So, the condition for the distance dF := d between the focus F and the vertex of the ﬁrst surface of the optical system is

⇒ D = Cd + D = 0 ⇒ dF = −

The focal length f is deﬁned as the distance between the principal point U and the focus F, where the sign convention in geometrical optics is that f is positive if F is right of U. Therefore, by using the sign conventions for dU and dF it is:

f = dF − dU =−

Then, the distances dN := d between the nodal point N and the vertex of the ﬁrst surface of the optical system := d between the vertex of the on the one hand and dN last surface of the optical system and the nodal point N on the other hand are: 1− D dN = , (2.63) C A (D − 1) − B . dN (2.64) = C

Now, the concrete meaning of the matrix coefﬁcient C as the negative reciprocal value of the focal length f in the image space becomes clear. By summarizing (2.59), (2.60), (2.63–2.65) and (2.67) the distances dU , dN and dF between the cardinal points in the object space and the vertex of the ﬁrst surface of the optical system as well as the distances , d and d between the vertex of the last surface of dU N F the optical system and the cardinal points in the image space are: D (A − 1) − B , C 1− D , dN = C D dF = − , C 1− A dU = , C A (D − 1) − B , dN = C A dF = − . C dU =

(2.69)

Part A 2.2

:= d (Fig. 2.14) to indicate Here, we use the name dU that it is the distance from the vertex of the last surface of the optical system to the principal point U . From the ﬁrst condition we then obtain the distance dU := d between the principal point U and the vertex of the ﬁrst surface 1− A B = Ad + B + (1 − A)d + D =0, C D (2.60) ⇒ dU = (A − 1) − B . C It has to be mentioned that, in the case of optical imaging, the coefﬁcient A of the matrix M has a concrete meaning. It is: x x = A x ⇒ β = = A (2.61) x Therefore, the coefﬁcient A is identical to the lateral magniﬁcation β deﬁned by (2.52).

2.2 Paraxial Geometrical Optics

Part A

Basic Principles and Materials

Also, the focal lengths can now be expressed as functions of the coefﬁcients A, B, C and D of the matrix M by summarizing (2.66) and (2.68): det(M) AD − BC = C C 1 f =− C f=

Part A 2.2

(2.70)

Relation Between the Focal Lengths in the Object and Image Spaces There is a very interesting relation between the focal length f in the object space and the focal length f in the image space. To derive it the ratio f / f is calculated by using (2.66) and (2.68):

−1/C 1 1 f = =− =− f (AD − BC)/C AD − BC det(M) (2.71)

Here, the determinant det(M) of the matrix M, deﬁned by (2.54), has been used. According to the calculus of linear algebra the determinant of the product of several matrices is equal to the product of the determinants of these matrices. Therefore, it holds that det(M) = det(MS,m ) · det(MT,m−1 ) · det(MS,m−1 ) · . . . · det(MT,1 ) · det(MS,1 ) .

(2.72)

1 di ⇒ det(MT,i ) = 1 , (2.73) MT,i = 0 1 1 0 ni ⇒ det(MS,i ) = . (2.74) MS,i = n i −n i n i − n R n ni Again, n i and n i

Here, relation (2.55) for the refractive indices of neighboring surfaces has been used. Therefore, the ratio of the focal length f and f is, according to (2.71), 1 n f =− =− f det(M ) n

f f =− . n n

are the refractive indices in front of and behind the respective surface; di is the distance between surface i and i + 1 (i ∈ {1, 2, . . . , m − 1}); and Ri is the radius of curvature of surface i. Now, we deﬁne again the refractive index in front of the ﬁrst surface as n := n 1 and the refractive index behind the last surface of the optical system as n := n m . Since the determinants of the transfer matrices MT,i are

(2.76)

The Cardinal Points of an Optical System with Identical Surrounding Refractive Indices An interesting special case is when the refractive indices n in front of the ﬁrst surface of the optical system and n behind the last surface of the optical system are identical: n = n . Then the determinant of the matrix M is, according to equation (2.75), det(M ) = 1. Therefore, the focal lengths in the object and image spaces have, due to (2.76), equal absolute value but different signs (due to the sign conventions of geometrical optics):

f =−f .

So, we have ﬁrst to calculate the determinants of the two elementary matrices of (2.45) and (2.51):

one the determinant of M is: m m n m+1−i det(M ) = det(M S,m+1−i ) = n i=1 i=1 m+1−i n m n m−1 n2 n1 = · ·...· · n m n m−1 n2 n1 n m−1 n m−2 n n n = · · . . . · 1 · = . (2.75) n n m−1 n2 n1 n

(2.77)

A second quite interesting property of an optical system with identical refractive indices in front of the ﬁrst surface and behind the last surface is that the principal points and the nodal points coincide. This can easily be derived from (2.59), (2.60), (2.63) and (2.64) by using det(M ) = AD − BC = 1: AD − D − BC D (A − 1) − B = C C 1− D = dN = C

dU =

(2.78)

and AD − A − BC A (D − 1) − B = C C 1− A = dU . = C

dN =

(2.79)

The Cardinal Points of a Spherical Refracting Surface The simplest optical imaging system is a single spherical refracting surface. As an application of (2.69) and (2.70)

Geometrical Optics

Then, according to (2.69), the result is: D (A − 1) − B = 0 , C 1− D dN = = −R , C nR D , dF = − = C n −n 1− A dU =0, = C A (D − 1) − B = R , dN = C n R A . dF = − = C n −n dU =

positions, i. e. F is right of the vertex of the surface and F is left of it. Similar the focal lengths are calculated using (2.70): det(M ) nR AD − BC = =− f= C C n −n n R 1 . f =− = (2.82) C n −n Since the principal points coincide with the vertex of the surface, the focal length f is of course identical to −dF and the focal length f is f = dF . The general equation (2.76) f /n = − f/n is of course also valid.

2.2.4 The Imaging Equations of Geometrical Optics

(2.81)

This means (Fig. 2.15): 1. Both principal points coincide with the vertex of the = 0). spherical surface (dU = dU 2. Both nodal points coincide with the center of cur = R). To vature of the spherical surface (−dN = dN understand this, the sign conventions have to be noticed: dN is positive if the vertex of the surface is is positive if the right to the nodal point N, but dN vertex of the surface is left to the nodal point N . 3. For a convex surface (R > 0) and n > n the surface has a positive optical power and the focus F is in front of the surface and F behind the surface. For a concave surface (R < 0) but still n > n the surface has a negative optical power and the foci change their

The Lens Equation On page 46 it has already been shown what imaging means. A point PO lying in the plane P a distance d in front of the vertex of the ﬁrst surface of an optical system with matrix M (2.54) is imaged onto a point PI in a plane P a distance d from the vertex of the last surface of the system. This is only the case if the matrix n

Optical system dO

PO d

d' d'F'

xO F

U dU

d'U'

xI f

dI F

U, U'

N,N'

R f

Fig. 2.15 Cardinal points and parameters of a spherical re-

fracting surface. (The position of F and F is drawn for the example n = 1 and n = 1.8)

Part A 2.2

the cardinal points of a spherical refracting surface shall be determined. In this special case the matrix M is, according to (2.51), 1 0 A B := MS = M= . (2.80) n − nn−n C D R n

2.2 Paraxial Geometrical Optics

Fig. 2.16 Parameters for explaining the imaging of an object point PO to an image point PI by a general optical system. The optical system is characterized by the vertices of the ﬁrst and last surface and its cardinal points (without nodal points). The sign conventions mentioned in the text mean for the classical geometrical optical parameters: xO > 0, xI < 0, dO < 0, dI > 0, Z < 0, Z > 0, f < 0, f > 0. But for the other parameters, which are only used in the paraxial matrix theory, we have: d > 0, d > 0, dF > 0, dF > 0, 0 and R2 < 0, plane–convex: R1 > 0 and R2 = ∞ (or R1 = ∞, R2 < 0) convex–concave (meniscus lens): R1 > 0 and R2 > 0 (or both negative) plane–concave: R1 < 0 and R2 = ∞ (or R1 = ∞, R2 > 0) biconcave: R1 < 0 and R2 > 0.

These lenses have different focal powers. For the case n L > n (e.g. for a lens made of glass which is used in air) biconvex and plane–convex lenses have generally positive focal lengths, i. e. they are positive lenses. On the other side, biconcave and plane–concave lenses have negative focal lengths, i. e. they are negative lenses. Meniscus lenses can be either positive (if the convex surface has the smaller radius of curvature) or negative (if the convex surface has the larger radius of curvature). Pay attention to the fact that in the case n L < n (which can be realized e.g. by a hollow lens made of thin plastic which is ﬁlled with air and used in water) the properties of the different types of lenses are reversed. In this case a biconvex lens has e.g. a negative focal length.

2.2.6 The Thick Lens

x = x , ϕ = ϕ −

Plane-convex

•

1 0 ⇒ ML0 = − f1 1

Biconvex

(2.99)

x . f

(2.100)

For a lens with a positive focal power the focal length f is also positive and the rays intersect behind the lens in a real focus. For a lens with a negative focal power

In the case of a thick lens the ray transfer through the thickness d := d1 between the two spherical surfaces is taken into account (see Fig. 2.20). Of course, the radii of curvature of the two spherical surfaces are still assumed to be so large that the point of intersection of a paraxial ray with the surface is in the same plane as the vertex of

Geometrical Optics

n=n1

2.2 Paraxial Geometrical Optics

nL: =n2 =n'1 n' =n'2 dUU'

R2 R1

f d:=d1

the surface. The matrix MLd of a thick lens is the product of three single matrices: matrix MS,1 for refraction at the ﬁrst spherical surface with radius of curvature R1 , matrix MT,1 for the transfer between the two surfaces by the distance d and matrix MS,2 for the refraction at the second spherical surface with the radius of curvature R2 . The refractive indices in front of, in, and behind the lens are n = n 1 , n L := n 1 = n 2 and n = n 2 , respectively. Then the matrix MLd of a thick lens is:

= =

1 d

n L nL − nn−n − nnLL−n 0 1 R n R1 n L 2 n 1 0 1 − nnLL−n R1 d n L d

L − nn−n R 2

− nnL −n R1

− nnLL−n R1

nL n

−

n nL

(n −n L )(n L −n) n n L n R1 R2 d n

−

n nL d n(n −n L ) n L n R2 d

(2.101)

In the most important case of identical external materials n = n, (2.101) reduces to MLd ⎛

1 − nnLL−n R1 d ⎜ ···

⎜ nL − n nL − n d 1 1 =⎜ + − ⎜− n R1 R2 n L R1 R2 ⎝ n nL d

···

⎞

⎟ ⎟ ⎟ 1 + n L −n d ⎠

Fig. 2.21 The principal planes of a thick lens. Notice that dU and dU

are both negative in the ﬁgure due to our sign convention

is the focal length of the thick lens in the image space f =

nn L R1 R2 . (2.103) (n L − n) [n L (R2 − R1 ) + (n L − n)d]

Because n = n , the focal length f in the object space is f = − f and the nodal points and the principal points coincide. So, it is now necessary to calculate the positions of the principal points U and U’ (Fig. 2.21). By using (2.69) one obtains AD − D − BC 1− D D (A − 1) − B = = C C C − nnLL−n d R2

= n L −n n L −n d 1 1 − n R1 − R2 + n L R1 R2

dU =

1 − nnLL−n R1 d

n −n L n R2

d'U'

Fig. 2.20 Parameters of a thick lens

MLd

=−1/ f

= dU =

ndR1 , n L (R2 − R1 ) + (n L − n) d

n L −n 1− A n L R1 d = C − n Ln−n R11 − R12 + n Ln−n L

=−

d R1 R2

ndR2 . n L (R2 − R1 ) + (n L − n) d

(2.104)

The distance dUU between the two principal planes, which is positive if U is right of U, is: dUU = d + dU + dU n(R2 − R1 ) . = d 1− n L (R2 − R1 ) + (n L − n)d

(2.105) (2.102)

n L R2

The matrix element C in the ﬁrst column of the second row is, according to (2.70), deﬁned as −1/ f , where f

A Thick Lens in Air Since the special case of a thick lens in air (n = 1) is the most important in practice, (2.102) for 1/ f , (2.104) for and (2.105) for d shall be repeated for dU and dU UU

Part A 2.2

Part A

Basic Principles and Materials

this case:

1 1 1 nL − 1 d = (n − 1) − + , L f R1 R2 n L R1 R2

dR1 , (2.107) n L (R2 − R1 ) + (n L − 1) d dR2 , =− (2.108) n (R − R1 ) + (n L − 1) d L 2 R2 − R1 . = d 1− n L (R2 − R1 ) + (n L − 1)d

(2.106)

dU =

Part A 2.2

dUU

(2.109)

In the following, three important cases of thick lenses in air will be described to illustrate the optical parameters of lenses. Special Cases of Thick Lenses in Air Ball Lens. For a ball lens with radius of curvature

R > 0 and refractive index n L the lens parameters are (Fig. 2.22) R1 = R ,

R2 = −R ,

d = 2R .

This means, according to (2.106–2.109), for the parameters in air 1 2(n L − 1) nL R , = ⇒ f = f Rn L 2(n L − 1) dU = −R , dU = −R , dUU = 0 . (2.110) This means that the principal points coincide and are at the center of curvature of the ball lens. For the special case n L = 2 the focal length would be equal to the radius of curvature f = R so that the focus in the image space would be on the backside of the sphere. For n L < 2 (e.g. nearly all glasses) the focus is outside of the sphere, whereas for n L > 2 (e.g. a silicon ball lens illuminated with infrared light) the focus would be inside the sphere.

Fig. 2.23 The meniscus of Hoegh

The Meniscus Lens of Hoegh. For the meniscus lens of

Hoegh (Fig. 2.23) with refractive index n L and thickness d the two radii of curvature are identical, i. e. R1 = R2 = R. Then, (2.106–2.109) deliver: (n L − 1)2 d 1 = , f n L R2 dU =

R , nL − 1

dU =−

R , nL − 1

dUU = d .

(2.111)

A thin meniscus with identical radii of curvature would have no optical effect. Contrary to this the thick meniscus of Hoegh has a positive optical power. At least one of the principal points is always outside of the lens and their separation is identical to the thickness of the lens (Fig. 2.23). Plane–Convex or Plane–Concave Lenses. We assume

now that the ﬁrst surface of the thick lens with refractive index n L and thickness d is curved (either convex, i. e.

d d

Fig. 2.22 Parameters of a ball lens

Fig. 2.24 Principal planes of a plane–convex or plane– concave lens

Geometrical Optics

R1 > 0, or concave) and the second is plane (R2 = ∞). Equations (2.106–2.109) are in this case (Fig. 2.24): nL − 1 1 = , f R1

d , nL (n L − 1)d 1 = = d 1− . nL nL

dU =−

dUU

(2.112)

This means that the ﬁrst principal point coincides with the vertex of the curved surface. Moreover, the focal length of a lens with one plane surface is calculated like the focal length of a thin lens. This is not astonishing since the plane–convex/plane–concave lens can be interpreted as a combination of a thin lens with focal length f and a plane parallel plate with thickness d and refractive index n L . This can easily be shown by calculating the matrix M = MP ML0 and comparing it with MLd of (2.102) for R2 = ∞.

2.2.7 Reflecting Optical Surfaces Up to now only refracting surfaces have been treated, which form lenses and complete objectives. But there are of course also reﬂecting surfaces which are e.g. very important in astronomical telescopes [2.21] or which will be very important in the near future for optical lithography systems [2.22] in the extreme ultraviolet (EUV) at a wavelength of 13 nm. However, a reﬂecting surface can easily be included in a paraxial design by calculating its paraxial 2 × 2 matrix and including it instead of the surface of a refracting surface in (2.54). We will see that the determinant of the matrix of a reﬂecting surface is one so that our general discussions

concerning the relation between the focal lengths f and f are valid. A Plane Reflecting Surface The reﬂection at a plane surface, which is perpendicular to the optical axis, is shown in Fig. 2.25. The law of reﬂection means that the angle i of the reﬂected ray with the surface normal is identical to the angle i of the incident ray, i. e. i = i . In the paraxial theory it is common practice not to take the reﬂected ray since then the light would travel from right to left. Instead, the unfolded ray path is taken, which is obtained by mirroring the reﬂected ray at the reﬂecting surface. By doing this the dashed ray in Fig. 2.25 is obtained and there is no change of the paraxial ray parameters x and ϕ. So, the paraxial ray matrix MRP of a reﬂecting plane surface is just the unit matrix: 1 0 . (2.113) MRP = 0 1

Its determinant is of course one. A Spherical Reflecting Surface The reﬂection at a spherical surface is treated analogously to the case of a plane surface and is shown for a convex mirror in Fig. 2.26. The ray that is reﬂected at the spherical surface is mirrored at a plane which goes through the vertex of the surface and is perpendicular to the optical axis. So, the dashed ray in Fig. 2.26 results. All angles in Fig. 2.26 are positive so that the following

Real reflected ray

Ray of the mirrored, unfolded ray path

i' i φ

φ'

x α R

φ'

i' i x =x' φ

Mirror plane

Fig. 2.25 Reﬂection at a plane surface

Fig. 2.26 Reﬂection at a spherical surface

Part A 2.2

dU = 0 ,

2.2 Paraxial Geometrical Optics

Part A

Basic Principles and Materials

relations are valid: i= α + i = i= α=

⎫ ϕ+α ⎪ ⎪ ⎪ ⎬ x ϕ ⇒ ϕ = ϕ + 2α = ϕ + 2 . ⎪ R i ⎪ ⎪ ⎭ x/R

Part A 2.2

(2.114)

Since the ray height x remains constant during reﬂection the paraxial ray matrix MRS is 1 0 . MRS = 2 (2.115) R 1 Again the determinant is one. The matrix (2.115) is also valid for a concave mirror. There, the radius of curvature R is negative so that the angle ϕ is smaller than the angle ϕ for a positive ray height x. This is just the effect of a concave mirror with a positive optical power. As an exercise the cardinal points of a spherical mirror shall be calculated by using (2.69) and (2.70): D dU = (A − 1) − B = 0 , C 1− D dN = =0, C Ray of mirrored path

R D =− , C 2 1− A dU = =0, C A (D − 1) − B = 0 , dN = C R A dF = − = − , C 2 det(M ) R AD − BC = = , f= C C 2 R 1 f =− =− . (2.116) C 2 So, the principal points U, U and the nodal points N, N all coincide with the vertex of the spherical mirror (Fig. 2.27). The focus F in the object space is at half the distance between the center of curvature of the spherical surface and the vertex. On the other side, the focus F in the image space would coincide with F for the real reﬂected ray. But since the unfolded ray path with the mirrored ray is taken the focus Fmirrored is also mirrored at the principal plane through the vertex that is perpendicular to the optical axis. The focal length is of course half the radius of curvature and a convex mirror has a negative optical power whereas a concave mirror has a positive optical power. If an optical system containing refractive and reﬂective surfaces has to be analyzed so that the same lens is e.g. passed twice or more it is necessary, on the way back, to change the order of surfaces and refractive indices and also the signs of the radii of curvature. dF = −

2.2.8 Extension of the Paraxial Matrix Theory to 3 × 3 Matrices U, U' F'mirrored R/2

F, F'

N, N' R/2

R Mirror plane = Principal plane

Fig. 2.27 Cardinal points of a convex mirror. A ray coming

from the left parallel to the optical axis has to go into the image space (virtually) through the focus Fmirrored . Fmirrored is the focus of the unfolded ray path, which is mirrored at the vertex plane. The real reﬂected ray would virtually go through the focus F’ which then coincides with F

The paraxial 2 × 2 matrix theory can only be used as long as all elements are centered around the optical axis and symmetric with respect to the optical axis. A tilted refractive plane surface or a diffraction grating, which both introduce a global tilt of all rays, can e.g. not be included in the 2 × 2 matrix theory. But there is an extension of this method by using 3 × 3 matrices [2.15]. This will be described in the following. Paraxial Ray Tracing at a Diffraction Grating A ray representing a plane wave with wavelength λ which hits a diffraction grating with a period Λ is diffracted according to the well-known grating equation [2.1] (Fig. 2.28): λ (2.117) sin ϕ = sin ϕ + m . Λ

Geometrical Optics

form

m=1 m=0 Λ

φ' m=–1

amplitude grating with period Λ. Three different diffraction orders m are shown

Here, the integer m is the diffraction order of the grating and, depending on the type of the grating, there may be only one efﬁcient order (e.g. for blazed gratings or volume holograms) or many orders with non-vanishing efﬁciency (e.g. for binary phase elements or amplitude gratings) [2.23–25]. In the case of many diffraction orders each order has to be calculated separately. The angles ϕ and ϕ are the angles of the incident and diffracted ray, respectively. In the paraxial approximation the sine of the angles is replaced by the angle itself so that the grating equation is ϕ = ϕ + m

λ . Λ

(2.118)

Together with the equation for the ray height x (x = x), which does not change in the case of diffraction at a grating, there are two equations relating the ray parameters before and after diffraction at the grating. However, it is no longer possible to write these two equations in a pure 2 × 2 matrix notation since it would be 0 x x 1 0 . + = (2.119) mλ/Λ ϕ 0 1 ϕ So, a constant additive vector at the end would be necessary and the calculation of one 2 × 2 matrix for a complete optical system containing one or more diffraction gratings and several other optical elements would be impossible. However, there is a possibility to change this by using 3 × 3 matrices instead of 2 × 2 matrices and a paraxial ray vector with three components instead of two, where the third component is always 1. The 3 × 3 matrices and the paraxial ray vectors are of the

⎛

where M is the normal paraxial 2 × 2 matrix with coefﬁcients A, B, C and D. The coefﬁcients ∆x and ∆ϕ are constant values which symbolize a lateral shift or a tilt which is exerted on the incident paraxial ray by the element. To obtain the 3 × 3 matrix appendant to a pure paraxial 2 × 2 matrix the coefﬁcients ∆x and ∆ϕ just have to be set to zero. The solution of our original example to deﬁne the paraxial 3 × 3 matrix MG,3×3 of a (non-tilted) diffraction grating is now quite easy: ⎛ ⎞ 1 0 0 ⎜ λ⎟ . (2.121) MG,3×3 = ⎝0 1 m Λ ⎠ 0 0

Tilted Refractive Plane Surface A refractive plane surface shall have a normal vector that is tilted by a small angle α with respect to the optical axis. The surface with refractive indices n in front of and n behind the surface is hit by a paraxial ray with ray parameters x and ϕ (Fig. 2.29). Since, the tilt angle α has to be small and the ray heights x also, the variation of the z-coordinates at the points of intersection of the tilted surface and rays with different heights x can be n

i α φ'

i φ

Fig. 2.29 The refraction at a tilted plane surface with tilt

angle α

Part A 2.2

Fig. 2.28 The diffraction at a grating symbolized here as an

⎞ ⎛ ⎞ ⎛ ⎞ B ∆x x x ⎟ ⎜ ⎟ ⎜ ⎟ D ∆ϕ⎠ ⇒ ⎝ϕ ⎠ = M3×3 ⎝ϕ⎠ 0 1 1 1 ⎞ x ∆x + ⎜M ⎟ =⎝ (2.120) ϕ ∆ϕ ⎠ , 1

A ⎜ M3×3 = ⎝C 0 ⎛

Grating

2.2 Paraxial Geometrical Optics

Part A

Basic Principles and Materials

neglected: ∆z = x tan α ≈ xα ≈ 0 ,

(2.122)

Part A 2.2

i. e. it is of second order and only ﬁrst-order terms are taken into account in the paraxial approximation. Additionally, the ray height x remains constant for refraction. The ray angles, tilt angles and refraction angles depend on each other through the following equations: ⎫ ϕ = i + α ⎪ ⎬ n n ⇒ ϕ = i + α = (ϕ − α) + α ϕ = i +α ⎪ n n ⎭ ni = n i n − n n α . (2.123) = ϕ+ n n So, the 3 × 3 matrix MR,α,3×3 for refraction at a tilted plane surface is ⎛ ⎞ 1 0 0 ⎜ ⎟ (2.124) MR,α,3×3 = ⎝0 nn n n−n α⎠ . 0 0

As an application and to see how the matrix of a complete system is determined the matrix of a thin prism will be calculated in the next paragraph. Thin Prism A thin prism consists of two tilted refractive surfaces and we assume that the prism is made of a material with refractive index n and the refractive index outside of the prism is n at both sides. Since the prism is assumed to be thin the propagation between the two refractive surfaces is neglected and the total matrix MPrism,3×3 of the prism is obtained by just multiplying the 3 × 3 matrices of the

single surfaces. The tilt angles of the two surfaces are α1 and α2 so that we have MPrism,3×3 = MR,α2 ,3×3 MR,α1 ,3×3 ⎛ ⎞⎛ 1 0 0 1 0 ⎜ ⎟⎜ n = ⎝0 nn n−n n α2 ⎠ ⎝0 n 0 0 1 0 0 ⎞ ⎛ 1 0 0 ⎟ ⎜ = ⎝0 1 n n−n (α1 − α2 )⎠ . 0 0 1

⎞

⎟ n −n n α1 ⎠ 1 (2.125)

By deﬁning the prism angle γ := α1 − α2 the total deﬂection angle δ of a thin prism with prism angle γ is δ = (n − n)γ /n. For the most important case of a thin prism in air (n = 1) it is δ = (n − 1)γ . The Transformation Matrices The matrix of a tilted plane surface or other tilted and also laterally shifted surfaces can be calculated quite formally by introducing the paraxial transformation matrix between two coordinate systems. The ﬁrst coordinate system with axes x and z will be named the global coordinate system. The second coordinate system with axes x and z is called the local coordinate system because, in this coordinate system, the surface will have a simple

x' Paraxial ray x

φ' φ z'

x' ∆φ γ

z' x

δ ∆x

z n

Fig. 2.30 The refraction at a thin prism with prism angle γ . The incident ray is deﬂected by an angle δ

Fig. 2.31 Transformation between two relatively tilted and shifted coordinate systems. The local x –z -coordinate system is laterally shifted with respect to the global x– z-coordinate system by a distance ∆x and tilted by an angle ∆ϕ

Geometrical Optics

x = (x − ∆x) cos(∆ϕ) ≈ x − ∆x , (2.126) z = (x − ∆x) sin(∆ϕ) ≈ (x − ∆x) ∆ϕ ≈ 0 , (2.127)

ϕ = ϕ − ∆ϕ .

(2.128)

Here, the paraxial approximations are used and since ∆x, ∆ϕ, x and ϕ are all paraxial (i. e. small) quantities, only terms of ﬁrst order are taken into account whereas terms of second order such as (x − ∆x)∆ϕ are set to zero. So, the z -coordinate remains zero if the ray has a global coordinate z = 0, what is always the case by choosing the global coordinate system accordingly. Therefore, the matrix MG→L,3×3 for the transformation of a paraxial ray from the global coordinate system to a local coordinate system is ⎛ ⎞ 1 0 −∆x ⎜ ⎟ MG→L,3×3 = ⎝0 1 −∆ϕ⎠ . (2.129) 0 0

Vice versa, the matrix ML→G,3×3 for the transformation of a paraxial ray from the local coordinate system to the global coordinate system is the inverse matrix to MG→L,3×3 : ⎛ ⎞ 1 0 ∆x ⎜ ⎟ −1 = ⎝0 1 ∆ϕ⎠ . (2.130) ML→G,3×3 = MG→L,3×3 0 0 1 It is also important to notice that in the paraxial approximation with small shifts ∆x and small angles ∆ϕ the order of shifting and tilting is arbitrary whereas this is not the case for ﬁnite quantities. Mathematically, this can be proved by calculating that the two matrices for a pure shift (i. e. ∆ϕ = 0) and for a pure tilt (i. e. ∆x = 0) permute: ⎛ ⎞⎛ ⎞ ⎛ ⎞ 1 0 ∆x 1 0 0 1 0 ∆x ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝0 1 0 ⎠ ⎝0 1 ∆ϕ⎠ = ⎝0 1 ∆ϕ⎠ 0 0 1 0 0 1 0 0 1 ⎛ ⎞⎛ ⎞ 1 0 0 1 0 ∆x ⎜ ⎟⎜ ⎟ (2.131) = ⎝0 1 ∆ϕ⎠ ⎝0 1 0 ⎠ . 0 0 1 0 0 1

This means, that in the paraxial approximation it is identical if the coordinate system is ﬁrst tilted and afterwards shifted or if it is ﬁrst shifted and then tilted. So, we can use one matrix for the whole transformation without taking care of the order of the single transformations. As an application of the transformation matrices the 3 × 3 matrix for refraction at a tilted and laterally shifted spherical surface with a radius of curvature R shall be calculated. The refractive indices are again n in front of the surface and n behind it. The vertex of the spherical surface is laterally shifted by a distance ∆x with respect to the optical axis (global coordinate system) and the surface is rotated around an axis perpendicular to the meridional plane by an angle ∆ϕ. The local coordinate system is of course that system in which the surface is neither tilted nor rotated. Then, a ray in the local coordinate system can be calculated by multiplying the incident ray (in the global coordinate system) by the transformation matrix MG→L,3×3 . In the local coordinate system, the ray is multiplied by the matrix of a normal non-tilted and non-shifted spherical surface MS,3×3 . Afterwards, the ray in the local coordinate system is transformed back into the global system by multiplying it with ML→G,3×3 . So, the matrix MS,∆x,∆ϕ,3×3 for refraction at a tilted and shifted spherical surface in the global coordinate system is just the product of the three matrices: MS,∆x,∆ϕ,3×3 = ML→G,3×3 MS,3×3 MG→L,3×3 ⎛ ⎞⎛ ⎞⎛ ⎞ 1 0 0 1 0 ∆x 1 0 −∆x ⎜ ⎟⎜ ⎟⎜ ⎟ n = ⎝0 1 ∆ϕ⎠ ⎝− nn−n R n 0⎠ ⎝0 1 −∆ϕ⎠ 0 0 1 0 0 1 0 0 1 ⎞ ⎛ ⎞⎛ 1 0 −∆x 1 0 ∆x ⎟ ⎜ ⎟⎜ n n −n n = ⎝0 1 ∆ϕ⎠ ⎝− nn−n R n n R ∆x − n ∆ϕ⎠ 0 0 1 0 0 1 ⎞ ⎛ 1 0 0 ⎟ ⎜ n n −n n −n (2.132) . = ⎝− nn−n ∆x + R n n R n ∆ϕ⎠ 0

The result shows that the ray height x remains, as expected, unchanged by refraction at the surface (x = x) and that there is for the ray angle ϕ , besides the usual term of a spherical surface, an additional term which does not depend on the angle of incidence but on the shift ∆x and the tilt ∆ϕ. It can also be seen that this additional term is zero if the condition ∆x/R = −∆ϕ is fulﬁlled. This is the well-known fact that a lateral shift of a spherical surface can be compensated by tilting it.

Part A 2.2

form, i. e. it is non-tilted and non-shifted in this local coordinate system. The local coordinate system is obtained from the global one by shifting a copy laterally in the x-direction by the small distance ∆x and rotating it by an angle ∆ϕ (Fig. 2.31). So, a paraxial ray with ray parameters (x, ϕ) in the global coordinate system has ray parameters (x , ϕ ) in the local coordinate system and the following relations are valid

2.2 Paraxial Geometrical Optics

Part A

Basic Principles and Materials

A special case is R → ∞ so that the spherical surface becomes a plane surface. In this case the matrix of (2.132) becomes ⎛ ⎞ 1 0 0 R→∞ ⎜ ⎟ MS,∆x,∆ϕ,3×3 ⇒ ⎝0 nn n n−n ∆ϕ⎠ = MR,∆ϕ,3×3 .

Part A 2.3

0 0

This is of course the same result for the matrix MR,α,3×3 for refraction at a tilted plane surface with ∆ϕ = α as that obtained in (2.124), which we derived directly from Fig. 2.29.

1 (2.133)

2.3 Stops and Pupils In the preceding section about paraxial optics only rays and object points in the neighborhood of the optical axis were considered. So, in the paraxial calculations stops have no inﬂuence. But this changes dramatically for the case of non-paraxial optics. There, stops are quite important optical elements which determine the light-gathering power of an optical system, its resolution, the amount of aberrations, its ﬁeld and so on. In the following only some elementary deﬁnitions about stops and pupils can be given. For more information we refer to the literature [2.1, 3, 8, 20]. There are two especially important stops, the aperture stop and the ﬁeld stop.

2.3.1 The Aperture Stop Assume ﬁrst of all a light-emitting object point which radiates in all directions. Then, the aperture stop is that physical stop which limits the cross-section of the image-forming pencil of rays. To determine the aperture stop the size and position of the images of all stops (e.g. lens apertures or real stops) in the system by that part of the system which precedes the respective stop have to be calculated. To do this the paraxial matrix theory of the last section can, for example, be used. If the distance of the image of stop i from the object point is li and the diameter of the stop image is di , then the aperture angle ϕi which can pass that stop is tan ϕi =

di . 2li

(2.134)

The aperture stop is now that stop number i that provides the minimum value ϕO of ϕi . The image of the aperture stop made by that part of the optical system which precedes the aperture stop is called the entrance pupil and the image of the aperture stop made by that part of the optical system which follows the aperture stop is called the exit pupil. The full aperture angle 2ϕO is called the

angular aperture on the object side and the corresponding quantity 2ϕI on the image side is called the angular aperture on the image side. ϕI can be determined by calculating the diameter dI of the exit pupil and the distance lI between the exit pupil and the image point and using again an equation like (2.134), replacing di with dI and li with lI . If the aperture stop is in front of the optical system the aperture stop and the entrance pupil will be identical. On the contrary, if the aperture stop is behind the whole optical system the aperture stop and the exit pupil are identical. In the general case, where the aperture stop is somewhere in the optical system the entrance pupil and the exit pupil can also be somewhere and they can be real or virtual images of the aperture stop. If an optical system consists of only one (thin) single lens the aperture stop, entrance pupil and exit pupil are of course all identical to the aperture of the lens itself. Another interesting case is e.g. an optical system where the aperture stop is in the back focal plane of the preceding part of the optical system. Then the entrance pupil is at inﬁnity and the system is called telecentric on the object side. In this case all chief rays on the object side (see later in this section) are parallel to the optical axis. Similarly, if the aperture stop is in the front focal plane of the part of the optical system which follows the aperture stop the exit pupil will be at inﬁnity and the system is called telecentric on the image side. Optical systems which are telecentric on both sides are quite important in optical metrology because in this case object points in different object planes have the same lateral magniﬁcation because the chief rays in the object and image spaces are both parallel to the optical axis. Therefore, the measured size of the object will be correct in a given image plane even though the object may be out of the object plane that is imaged sharply. A quite important quantity to characterize an optical system is the numerical aperture NA. The numerical

Geometrical Optics

aperture NAO on the object side is deﬁned as NAO = n O sin ϕO

(2.135)

and the numerical aperture NAI on the image side is NAI = n I sin ϕI ,

(2.136)

xI n I sin ϕI = xO n O sin ϕO

(2.138)

which is the usual formulation of the sine condition [2.1]. For the paraxial case this invariant reduces to the Smith– Helmholtz invariant: xI n I ϕI = xO n O ϕO

(2.139)

The numerical aperture determines how much light the optical system can gather from the object. It also determines (in the case of no aberrations) the resolution of the system due to diffraction. We will see in Sect. 2.5.3 that many aberrations depend on the numerical aperture. The position of the aperture stop in an optical system also inﬂuences the aberrations [2.26]. Another quite important deﬁnition of geometrical optics is the so-called chief ray or principal ray. This is that ray coming from the object point (which can of course be off-axis) which passes through the center of the aperture stop. Since the entrance pupil and the exit

pupil are both images of the aperture stop the chief ray also passes through the centers of the entrance pupil and exit pupil. If there are strong aberrations in the system this may not be exactly the case for object points which are strongly off-axis.

2.3.2 The Field Stop The second quite important stop is the ﬁeld stop, which limits the diameter of the object ﬁeld which can be imaged by an optical system. To ﬁnd the ﬁeld stop we calculate again the images of all stops by that part of the optical system which precedes the respective stop. Let us assume that the image of stop number i has again a diameter di and that the distance between the image of the stop and the entrance pupil of the system is L i . The ﬁeld stop is then that stop which has the smallest value φO of all values φi with tan φi =

di 2L i

(2.140)

The value 2φO is called the ﬁeld angle. The image of the ﬁeld stop by that part of the optical system which precedes the ﬁeld stop is called the entrance window and the image by that part of the optical system which follows the ﬁeld stop is called the exit window. If the line connecting an (off-axis) object point and the center of the entrance pupil is blocked by the entrance window the chief ray cannot pass the ﬁeld stop and so this object point cannot be imaged in most cases. However, there are cases where other rays coming from the object point can pass anyway and then there is no sharp border of the object ﬁeld but the outer parts of the object ﬁeld are imaged with lower intensity. This effect is called vignetting.

2.4 Ray Tracing It has been shown in Sect. 2.1.3 that light can be described by rays as long as the approximation of geometrical optics is valid. The propagation of such rays through an optical system is a very important tool to develop optical systems and to calculate their expected quality. The propagation of light rays through an optical system is called ray tracing [2.27, 28] and it is the basic tool of optical design, i. e. the design and optimization of optical systems concerning their imaging quality or other properties (e.g. tolerance of a system against misalignments or fabrication errors of components). In this

section the principle of ray tracing and some applications will be described. There is of course no room to discuss the basics of optical design itself. For this we refer to the literature [2.16–19].

2.4.1 Principle According to (2.29) a light ray propagates rectilinearly in a homogeneous and isotropic material. At an interface to another material the ray is partially refracted and partially reﬂected depending on the properties of the in-

Part A 2.4

where n O and n I are the refractive indices in the object and image space, respectively. It is an elementary property of optical imaging systems that NAO and NAI are connected by the lateral magniﬁcation β (2.52) of the optical system if the sine condition is fulﬁlled NAO NAI = . (2.137) β In fact, by replacing β by the ratio xI /xO of the image size and the object size this equation can be written as

2.4 Ray Tracing

Part A

Basic Principles and Materials

is the analysis of illumination systems with incoherent light. This will be discussed in Sect. 2.4.8 about non-sequential ray tracing. A precondition for ray tracing is that the optical system is known very well. It is not sufﬁcient to know some paraxial parameters but it is necessary to know the following data of the surfaces as well as the materials:

•

x-axis (mm)

Part A 2.4

• –5

• 0

z-axis (mm)

Fig. 2.32 Propagation of some light rays in a typical mi-

croscopic objective (NA = 0.4, magniﬁcation 20×, focal length f = 11.5 mm) calculated with our internal software RAYTRACE. In this case the microscopic objective is used in the reverse order, i. e. to focus light

terface. If a material is inhomogeneous (e.g. in GRIN lenses or in air ﬁlms with different temperatures) the light ray is curved during the propagation and the path of the ray has to be calculated by solving (2.27), in most cases numerically [2.29, 30]. However, in this section it is assumed that the optical system consists of different homogeneous materials which are separated by refracting or reﬂecting interfaces. Ray tracing means that the path of a bundle of rays, which are e.g. emitted by an object point or form a plane wave (i. e. object point with inﬁnite distance), is determined in an optical system (see Fig. 2.32 for tracing rays through an microscopic objective). In the approximation of geometrical optics the calculation is exact and no other approximations, such as e.g. paraxial approximations, are made. Since ray tracing can be easily automated with the help of computers it is nowadays the most important tool for designing lenses, telescopes and complete optical systems [2.16–19]. For complex optical systems it is even today with the help of modern computers not possible to replace ray tracing by pure wave-optical methods. Moreover, for most macroscopic optical systems, ray tracing in combination with wave-optical evaluation methods like the calculation of the point spread function [2.18], assuming that only the exit pupil of the system introduces diffraction, is a sufﬁciently accurate method to analyze imaging systems. Another quite modern application of ray tracing

• •

Type of the surface, e.g. plane, spherical, parabolic, cylindrical, toric or other aspheric surface. Characteristic data of the surface itself, e.g. the radius of curvature in the case of a spherical surface or the aspheric coefﬁcients in the case of an aspheric surface. Shape and size of the boundary of the surface, e.g. circular with a certain radius, rectangular with two side lengths or annular with an interior and an outer radius. Position and orientation of the surface in all three directions of space. Refractive indices of all materials and their dependence on wavelength.

The tracing of a given ray through an optical system has the following structure: 1. Determine the point of intersection of the ray with the following optical surface. If there is no point of intersection or if the hit surface is absorbing mark the ray as invalid and ﬁnish the tracing of this ray. Depending on the type of ray tracing it may also be necessary in this case to leave the ray unchanged and to go to step 4. If there is a point of intersection go to step 2. 2. Calculate the surface normal in the point of intersection. 3. Apply the law of refraction or reﬂection (or another law e.g. in the case of diffractive optical elements [2.31, 32]). Then, the new direction of the deﬂected ray is known and the point of intersection with the surface is the new starting point of the ray. 4. If there is another surface in the optical system go back to step 1 or, if not, then ﬁnish the tracing of this ray. In the case of step 1 the following surface can either be the physically next surface of the optical system which will really be hit by the ray (i. e. non-sequential ray tracing) or just the next surface in the computer list of surfaces where the order of the surfaces has been determined by the user (i. e. sequential ray tracing). In the next sections the mathematical realization of the different steps of ray tracing will be described.

Geometrical Optics

2.4.2 Mathematical Description of a Ray

r = p + sa .

(2.141)

The scalar parameter s is the arc length of the ray, i. e. in this case of rectilinear rays it is just the distance between r and p. The virtual part of the ray is described by s < 0 whereas that part where there is really light has s ≥ 0. In practice, there is also a maximum value smax if the ray hits a surface where it is deﬂected.

2.4.3 Determination of the Point of Intersection with a Surface

(2.142)

Concrete examples will be given later. By combining (2.141) and (2.142) the determination of the point of intersection is mathematically equivalent to the determination of the roots of a function G with the variable s G(s0 ) := F( p + s0 a) = 0 .

Plane Surface A plane surface can be described by the position vector C of a point on the surface (typically this point is in the center of the plane surface) and by the surface normal nz (see Fig. 2.34). Then, each point r of the surface fulﬁlls

F(r) = (r − C) · nz = 0 .

The determination of the point of intersection of a light ray described by (2.141) with a surface requires of course a mathematical description of a surface. It is well known from mathematics that a surface can be described in an implicit form with a function F fulﬁlling F(r) = 0 .

In many cases there can be several roots of G and it is also necessary to check whether the point of intersection is in the valid part of the surface, which is in practice limited by a boundary. Then, that root with the smallest positive value of s0 lying in the valid part of the surface has to be taken. These queries can be quite complex in a computer program. For general aspheric surfaces the solution of (2.143) will only be possible numerically, but for some simple cases the analytic solutions will be given in the following.

(2.143)

After having determined the value s0 at the root of G the position vector r0 of the point of intersection itself is obtained by applying s0 to (2.141). s

(2.144)

The solution of (2.143) in this case is ( p − C) · nz + s0 a · nz = 0 (C − p) · nz ⇒ s0 = . a · nz

(2.145)

In the case a · nz = 0 there is no deﬁnite point of intersection with the surface. Of course, (2.144) describes an unlimited surface whereas the surfaces of an optical system are limited. Therefore, it has to be checked whether the point of intersection is in the valid area of the surface. For a circular surface with radius R and center C this means e.g. that the point of intersection r0 has to fulﬁll the condition |r0 − C| ≤ R. For a rectangular surface a second vector nx (with nx · nz = 0 and |nx | = 1) along one of the sides (the vector n y along the second side is then just

r p r

C O

Fig. 2.33 Mathematical representation of a light ray as

a straight line. O symbolizes the origin of the coordinate system

Fig. 2.34 Mathematical description of a plane surface

Part A 2.4

A light ray (in a homogeneous material) can be described mathematically as a straight line with a starting point p and a direction vector a parallel to the ray (see Fig. 2.33). Here, a is a unit vector, i. e. |a| = 1. According to (2.29) an arbitrary point on the ray with position vector r is described by

2.4 Ray Tracing

Part A

Basic Principles and Materials

n y := nz × nx ) and the side lengths l x and l y of the rectangle have to be deﬁned, additionally. Then, it has to be checked whether the conditions |(r0 − C) · nx | ≤ l x /2 and (r0 − C) · n y ≤ l y /2 are fulﬁlled.

Part A 2.4

Spherical Surface A sphere whose center of curvature has position vector C and whose radius is R is described by

F(r) = |r − C| − R = 0 . 2

(2.146)

Therefore, (2.143) results in a quadratic equation for s0 : s02 + 2s0 ( p − C) · a + | p − C|2 − R2 = 0 . The two solutions are s01,2 = (C − p) · a ! ± [(C − p) · a]2 − |C − p|2 + R2 , (2.147) where the superscript 1, 2 is an index marking the two solutions. Depending on the argument of the square root there exist no (if the argument is negative), one (if the argument is zero) or two (if the argument is positive) solutions. After having determined the points of intersection with the full sphere it has to be checked whether the points of intersection are in the valid part of the spherical surface. To do this an additional vector nz (|nz | = 1) along the local optical axis and the lateral diameter D of the surface have to be deﬁned (Fig. 2.35). The radius of curvature R is positive if the vector nz points from the vertex to the center of curvature. In Fig. 2.35 R is, for example, positive. By using some trigonometric relations

R r

it is easy to see that the condition " D2 (C − r0 ) · nz ≥ 1− 2 R 4R has to be fulﬁlled by the point of intersection r0 if it lies on the valid spherical surface. General Surface z = f (x‚y) There are many important surfaces in optics, e.g. aspheric surfaces, which are described by a function f and the equation z = f (x, y). The implicit formulation with the function F is then

F(r) = z − f (x, y) = 0 ,

(2.148)

with r = (x, y, z). For a general function f the points of intersection of such a surface with a ray cannot be calculated analytically. But there are numerical methods such as Newton’s method combined with bracketing [2.33] to determine the roots of (2.143) where F of (2.148) is used. An important case is e.g. the description of rotationally symmetric aspheric surfaces with their axis of rotation along z by using the function [2.18]: z = f (x, y) = f (h) = +

i max $

ai h i

ch 2 # 1 + 1 − (K + 1)c2 h 2 (2.149)

i=1

# with h = x 2 + y2 ; c = 1/R is the curvature of the conical part of the surface with the conic constant K (K < −1 for a hyperboloid, K = −1 for a paraboloid and K > −1 for ellipsoids, with the special case K = 0 for a sphere). ai are aspheric coefﬁcients describing a polynomial of h. In most cases, only coefﬁcients with even integers i ≥ 4 are used and i max is in most cases less than or equal to ten. But in modern aspheric surfaces there may also be odd terms of i and i max > 10.

D C

Fig. 2.35 Mathematical representation of a spherical sur-

face (solid line) which is part of a full sphere (dashed plus solid line). O symbolizes the origin of the coordinate system

Coordinate Transformation In many cases there is a quite simple description of a surface in a local coordinate system [e.g. the description of a rotationally symmetric aspheric surface by using (2.149)] and it would not be useful to ﬁnd the implicit function F in the global coordinate system if the surface is e.g. tilted. In these cases it is more useful to transform the ray parameters p and a from the global coordinate system to the local system. Then, ﬁnding the point of intersection with the surface and the refraction or reﬂection (or “diffraction” if the element is a diffractive

Geometrical Optics

ny nx

2.4.4 Calculation of the Optical Path Length

Fig. 2.36 Parameters for transforming a vector p deﬁned in a global coordinate system to a vector p deﬁned in a local coordinate system. O symbolizes the origin of the global coordinate system

optical element) are done in the local coordinate system. Afterwards, the new ray is transformed back into the global coordinate system. Assume that the origin of the local coordinate system has the position vector C in the global coordinate system and the three unit vectors along the coordinate axes of the local system are nx , n y and nz in the global system (Fig. 2.36). To transform between the position vector p = ( px , p y , pz ) in the global system and p = ( px , py , pz ) in the local system one has the equations: p = C + px nx + py n y + pz nz

(2.150)

and px = ( p − C) · nx , py = ( p − C) · n y , pz = ( p − C) · nz .

L = L 0 + ns0 .

(2.154)

If the optical path length on another point r = p + sa on the ray has to be calculated this is done by just replacing s0 in equation (2.154) by s.

2.4.5 Determination of the Surface Normal If the function F of the implicit representation of the surface is known the surface normal N at the point of intersection is deﬁned as the normalized gradient of F at the point of intersection r0 ∇F . (2.155) |∇F| Some examples of surface normals are given in the following. N=

Plane Surface According to (2.144), one has (2.151)

For the ray direction vector with coordinates a = (ax , ay , az ) in the local system and a = (ax , a y , az ) in the global system the analogous equations are valid (but with C = 0 because direction vectors are measured from the origin of the respective coordinate system and can be shifted arbitrarily): a = ax nx + ay n y + az nz

The optical path length L along a ray at the point of intersection with the next surface is calculated by adding to the original optical path length L 0 at the starting point p of the ray the distance s0 between the starting point of the ray and the point of intersection with the next surface multiplied by the refractive index n of the material in which the ray propagates. Therefore, the optical path length is

F(r) = (r − C) · nz = 0 ⇒ N = nz .

(2.156)

Spherical Surface A spherical surface is described by (2.146):

F(r) = |r − C|2 − R2 = 0 , r0 − C . ⇒N= |r0 − C|

(2.157)

(2.152)

General Surface z = f (x,y)

and ax = a · nx , ay = a · n y , az = a · nz .

(2.153)

F(r) = z − f (x, y) = 0 , (− f x , − f y , 1) , ⇒N= ! 1 + f x2 + f y2

(2.158)

Part A 2.4

Of course it would also be possible to write the coordinate transformation using 3 × 3 matrices with the vectors nx , n y and nz as column or row vectors, but we prefer the vector notation here.

nz p'

2.4 Ray Tracing

Part A

Basic Principles and Materials

where f x := ∂ f/∂x and f y := ∂ f/∂y are the partial derivatives of f at the point of intersection r0 with the surface.

2.4.6 Law of Refraction Part A 2.4

For ray tracing a vectorial formulation of the law of refraction is necessary. In (2.41) an implicit formulation of the law of refraction (and also of the law of reﬂection) was given N × (n 2 a2 − n 1 a1 ) = 0 , where n 1 and n 2 are the refractive indices of the two materials and a1 and a2 are the unit direction vectors of the incident and refracted ray, respectively (Fig. 2.37). N is the local surface normal at the point of intersection of the incident ray with the surface. A solution of this equation can be found by the following steps: n1 a2 − a1 × N = 0 . n2 This means that the term in round brackets has to be parallel to N or itself zero. The later case is only possible for n 1 = n 2 so that, for n 1 = n 2 , we have n1 a2 = a1 + γ N n2 with a real value γ . Taking the square of both sides yields (as a1 , a2 and N are all unit vectors, i. e. |a1 | = |a2 | = |N| = 1) 2 n1 n1 + γ 2 + 2γ a1 · N 1= n2 n2 and therefore " 2 & n1 n1 % γ1,2 = − a1 · N ± 1 − 1 − (a1 · N)2 . n2 n2 n1

In total the result is: n1 n1 a2 = a1 − (a1 · N) N n2 n2 " 2 & n1 % ± 1− 1 − (a1 · N)2 N . n2

(2.159)

The vector term in front of the square root is parallel to the surface (the scalar product with N is zero). This means that the sign in front of the square root decides whether the component of a2 along N is parallel or antiparallel to N. Since the ray is refracted the sign of the component of a1 along N has to be equal to the sign of the component of a2 along N: sign (a1 · N) = sign (a2 · N) ,

(2.160)

where the sign function is +1 for a positive argument and −1 for a negative argument. Therefore, (2.159) can be written independently of the relative direction of N with respect to a1 as n1 n1 a2 = a1 − (a1 · N) N + sign (a1 · N) n2 n2 " 2 & n1 % 1 − (a1 · N)2 N (2.161) × 1− n2 So, this equation allows the calculation of the direction vector a2 of the refracted ray if the incident ray (direction vector a1 ), the local surface normal N and the two refractive indices n 1 and n 2 are known.

2.4.7 Law of Reflection The law of reﬂection is also formally described by (2.41) and therefore also by (2.159). But, in the case of reﬂection, ﬁrst of all the refractive indices are identical for the incident and the reﬂected ray, i. e. n 1 = n 2 , and sec-

a2 N

a1 a1

Fig. 2.37 Parameters for refraction of a ray at a surface

Fig. 2.38 Parameters for the reﬂection of a ray at a surface

Geometrical Optics

2.4.8 Non-Sequential Ray Tracing and Other Types of Ray Tracing The normal mode in most ray-tracing computer programs is so-called sequential ray tracing, i. e. the user deﬁnes the order in which the different surfaces of the optical system are passed by a ray. But this method is e.g. not useful for the analysis of illumination systems where the path of a ray and the order of surfaces can be different for each ray. The stability analysis of laser resonators [2.14, 15] is also quite exhausting with the sequential mode because the user knows the order of the surfaces but not how many times they will be hit by a ray. Of course, a stable resonator will be crossed by a light ray with an inﬁnite number of cycles. But for unstable resonators there is a ﬁnite number of cycles before the ray leaves the resonator. Therefore, non-sequential ray tracing is used in these cases. In this, the computer automatically calculates the

next surface that is physically hit by each ray. This is done, for example, by calculating the points of intersection of the ray with all surfaces and taking the surface with the smallest positive distance s0 . If there is no point of intersection with s0 > 0 the ray does not hit any surface of the system. Of course, non-sequential ray tracing is quite expensive in terms of computing time and therefore is normally only used if really necessary. Another speciality of non-sequential ray tracing is that a ray can be split at a surface into a refracted and a reﬂected ray (and in the case of diffractive optical elements also into more than two rays representing the different diffraction orders). Each ray is then recursively traced through the optical system. Some interesting modern optical systems such as Shack–Hartmann wavefront sensors [2.36] or beam homogenizers [2.37] use microlens arrays in combination with macroscopic optics. These array systems can also be analyzed with sequential or non-sequential ray tracing to obtain a ﬁrst insight [2.38, 39]. Of course, one has to be careful in these cases because diffraction and interference effects (for coherent or partially coherent illumination) cannot be neglected in several cases [2.40]. Sophisticated modern computer programs for sequential or non-sequential ray tracing implement in addition polarization ray tracing [2.10, 11]. In this, the local polarization state of each ray is taken into account and, for example, the split of the local power transported by each ray to the refracted and reﬂected ray for refraction/reﬂection at a surface is calculated according to the Fresnel equations [2.1]. A third type of ray tracing is so-called differential ray tracing or generalized ray tracing [2.28, 35, 41]. In this case, each ray is assumed to represent a local wave front with two principal curvatures and two principal directions. These parameters are then traced in addition to the normal ray parameters for each ray during propagation through the optical system. This allows, for example, the calculation of the local astigmatism of the wave front belonging to the ray by just tracing one ray. It also allows the calculation of the change of local intensity of the wave during the propagation.

2.5 Aberrations Whereas in the paraxial case the imaging quality of an optical system is ideal there are in practice aberrations of an optical system which deteriorate its imaging quality [2.1,6,8,26,42]. To explain the nature of aberrations,

see Fig. 2.39. At the exit pupil of an optical system there is a real wave front (solid line), i. e. the surface of equal optical path length, which intersects the exit pupil on the optical axis and which has its paraxial focus at the

Part A 2.5

ond the component of a2 along N has the opposite sign to the component of a1 along N (Fig. 2.38). This means that the other sign in front of the square root has to be taken and (2.159) results in a2 = a1 − (a1 · N) N − sign (a1 · N) ! × (a1 · N)2 N = a1 − 2 (a1 · N) N . (2.162) It is easy to prove that this equation correctly describes the reﬂection of a ray at a surface because all three vectors lie in a common plane (linearly dependent vectors) and the angle of incidence is equal to the angle of the reﬂected ray. The later can be seen by calculating the modulus of the cross product of (2.162) with N. Third, a2 really describes a reﬂected ray since twice the component of a1 along N is subtracted from a1 to obtain a2 . Besides refraction and reﬂection there is also a third quite important law for deﬂecting a ray at a surface, the vectorial local grating equation, which is used for ray tracing on holographic and more general diffractive optical elements. But for this equation and its solution we refer to the literature [2.31, 32, 34, 35].

2.5 Aberrations

Part A

Basic Principles and Materials

Exit pupil Real wave

Aberrations (wavelength)

Ideal spherical wave

0.1 W(x',y')

Part A 2.5

0.05 ∆x,∆y

0 –4

4 2

–2 x-axis (mm)

0 2

–2

y-axis (mm)

4 –4 Focal plane

Fig. 2.39 Explanation of the wave aberrations W and the ray aberrations ∆x, ∆y. The solid curve is the real wave front and the dashed curve the ideal spherical wave front. The solid rays are rays starting from the real wave front whereas the dashed rays are rays starting from the ideal spherical wave front. P is the (paraxial) focus of the wave front

point P, which lies in the focal plane of the system. However, there are deviations between an ideal spherical wave front (dashed line) with its center of curvature at P and the real wave front in the non-paraxial region. So, a ray starting at the point (x , y ) of the exit pupil has an optical path length difference between the real wave front and the ideal spherical wave front, which is called the wave aberration W(x , y ). Additionally, a ray with aberrations does not intersect the focal plane at the focus P but at a point with lateral displacements ∆x and ∆y in the x- and y-directions. These lateral deviations from the paraxial focus P are called the ray aberrations. Of course, the wave aberrations and ray aberrations are not independent of each other [2.43] and with good approximation the ray aberrations are proportional to the partial derivatives of the wave aberrations with respect to x and y .

2.5.1 Calculation of the Wave Aberrations The wave aberrations can be calculated by ray tracing. To do this a sphere which intersects the exit pupil on the optical axis and which has the (paraxial) focus P as its center of curvature is deﬁned. Then, the optical path lengths L(x , y ) of the points of intersection of this

Fig. 2.40 Wave aberrations for the on-axis point of the mi-

croscopic objective of Fig. 2.32 (NA = 0.4, magniﬁcation 20×, focal length f = 11.5 mm). The reference sphere is around the best focus of the wave aberrations. The focused spot is diffraction-limited since the peak-to-valley value is just 0.1 wavelengths (for a wavelength of 587.6 nm)

reference sphere with rays starting at the exit pupil at the points (x , y ) are calculated by using (2.147) and (2.154). The optical path length L(0, 0) of the chief ray is subtracted from the optical path length values of all other rays, resulting in the wave aberrations W(x , y ): W(x , y ) = L(x , y ) − L(0, 0) .

(2.163)

So, the wave aberrations are known for a grid of rays, i. e. the points (x , y ) in the exit pupil. In some cases it is useful not to take the paraxial focus for P but the so-called best focus. This is the point where either the wave aberrations or the ray aberrations have their smallest mean value (Fig. 2.40). So, in fact there are two different deﬁnitions of the best focus. If there is, for example, ﬁeld curvature the best focus will not be in the focal plane but on a sphere which intersects the optical axis in the focal plane.

2.5.2 Ray Aberrations and the Spot Diagram Ray aberrations can also be calculated by ray tracing. These aberrations are just the lateral deviations ∆x and ∆y between the focus P itself (which can be the paraxial focus or the best focus) and the points of intersection of the rays with a plane through the focus P. The surface normal of this plane is assumed to be nz (in most cases nz will be parallel to the optical axis) and the focus P has the position vector P. Additionally, the two unit vectors nx and n y lying in this plane and deﬁning the local x- and y-axis are known (nx , n y and nz form an orthogonal triad

Geometrical Optics

0.4

–0.2

–0.4 –0.2

0.2

0.4 x-axis (µm)

Fig. 2.41 Spot diagram for the on-axis point of the micro-

scopic objective of Fig. 2.32 with NA = 0.4, magniﬁcation 20×, focal length f = 11.5 mm. Since the numerical aperture of the lens used to focus the light is NA = 0.4 on the image side and the wavelength is λ = 587.6 nm the diffraction-limited Airy disc would have a diameter of 1.22λ/NA = 1.8 µm, i. e. larger than the ray aberrations. So, as in Fig. 2.40, it can also be seen from the ray aberrations that the on-axis spot of this lens is diffraction-limited

of unit vectors). Then, a ray number i with starting point pi and direction vector ai has its point of intersection ri with the plane according to (2.141) and (2.145) at (P − pi ) · nz ai . ai · nz

(2.164)

The ray aberrations are then deﬁned as: ∆x = (ri − P) · nx , ∆y = (ri − P) · n y .

(2.165) (2.166)

A quite demonstrative representation of the ray aberrations is a spot diagram. There, the points of intersection of the rays with a plane are graphically displayed by just drawing them as points (Fig. 2.41). This means that the spot diagram is a graphical representation of the ray aberrations (∆x, ∆y). Sometimes it is useful to determine the spot diagram not only in a plane through the focus but also in other planes to track the focusing of the rays.

In classical aberration theory [2.1,3] the primary aberration terms of Seidel (fourth-order wave aberration terms or third-order ray aberration terms) play an important role. The different terms are: spherical aberration, coma, astigmatism, curvature of ﬁeld and distortion. Whereas the ﬁrst three terms are point aberrations, i. e. aberrations that generate a blurred image point, the last two terms just cause a shift of the image point relative to the ideal paraxial image point but the image point itself would be sharp. There is not time in this chapter to go into details and to give a mathematical derivation so that only some facts will be stated for the different aberration terms. The distance of the object point from the optical axis will be called the object height rO in the following whereas the distance of a ray from the optical axis in the exit pupil will be called rA (from aperture height). For lenses with a small numerical aperture the maximum value rA is proportional to the numerical aperture NA of the image forming pencil of rays. Therefore, in the following the numerical aperture NA and the object height rO will be used to describe the functionality of the different Seidel terms. Spherical Aberration Spherical aberration is the only classical aberration that also occurs for object points on the optical axis of a rotationally symmetric optical system, i. e. for rO = 0. Spherical aberration of a normal single lens causes rays with a large height rA in the exit pupil of the lens to be refracted more strongly, so that they intersect the optical axis in front of the paraxial focus. In general optical systems the off-axis rays can also intersect the optical axis behind the paraxial focus. A typical property of spherical aberration is that it increases with the fourth power of the numerical aperture NA of the ray pencil forming the image point:

spherical aberration ∝ (NA)4

(2.167)

As already mentioned above the spherical aberration is independent of the object height rO . Coma Coma is an aberration which occurs only for off-axis points (of a rotationally symmetric optical system), i. e. rO = 0. The name coma is caused by the deformation of the image point, which looks like the coma of a comet. The coma depends on the third power of the numerical

Part A 2.5

0.2

ri = pi +

2.5.3 The Seidel Terms and the Zernike Polynomials

y-axis (µm)

–0.4

2.5 Aberrations

Part A

Basic Principles and Materials

aperture and linearly on the image height coma ∝ rO (NA)3 .

(2.168)

Part A 2.5

This is the reason why coma especially occurs for large numerical apertures whereas astigmatism dominates for small numerical apertures and large object heights (see the next paragraph). Coma can, for example, be generated in the microscopic objective of Fig. 2.32 by a lateral shift of the ﬁrst lens. A shift of 0.1 mm results in the aberrations shown in Fig. 2.42, which are dominated by coma although the spherical aberration of the original lens is still present. Astigmatism Astigmatism means that rays of the meridional plane and of the sagittal plane focus in different planes perpendicular to the optical axis. So, the geometrical shape of the image point is in general an ellipse. In two special planes, called the meridional and sagittal focal planes, the ellipses degenerate into two focal lines. The focal lines are perpendicular to each other. Between the meridional and the sagittal focal plane there is another plane where the shape of the image point is a circle, but of course this circle is extended whereas an ideal image point in geometrical optics would be a mathematical point. The astigmatism of an optical system is proportional to the square of the numerical aperture and the square of the object height:

astigmatism ∝ rO2 (NA)2 .

(2.169)

Aberrations (wavelength)

0.5 2 0 –4

–2

0 x-axis (mm)

–4

0 y-axis (mm)

Fig. 2.42 Wave aberrations for the on-axis object point of the misadjusted microscopic objective of Fig. 2.32 (NA = 0.4, magniﬁcation 20×, focal length f = 11.5 mm) by shifting the ﬁrst lens laterally by 0.1 mm. The image point which is then no more on-axis shows mainly coma but of course mixed with the spherical aberration of Fig. 2.40

Aberrations (wavelength)

0.5 2 0 –4

–2

0 x-axis (mm)

–4

0 – 2 y-axis (mm)

Fig. 2.43 Wave aberrations for an off-axis object point of the microscopic objective of Fig. 2.32 (NA = 0.4, magniﬁcation 20×, focal length f = 11.5 mm). The image point shows in this case mainly astigmatism but of course mixed with the spherical aberration of Fig. 2.40. There is nearly no coma because the well-adjusted microscopic objective fulﬁlls the sine condition

As mentioned above, this functionality is the reason that astigmatism also occurs for quite narrow pencils of rays. If there are cylindrical or toric surfaces in an optical system, astigmatism also occurs on the optical axis, whereas in the usual case of rotationally symmetric optical systems astigmatism occurs only for off-axis points. If we again take the microscopic objective of Fig. 2.32 but now with an off-axis object point (object height 15 mm, resulting image height 0.74 mm because of curvature of the ﬁeld) the resulting aberrations are mainly astigmatism, showing the typical saddle shape. Of course, the spherical aberration which is present onaxis remains, so that in fact the resulting aberrations represented in Fig. 2.43 are a mixture of astigmatism with a peak-to-valley value of about one wavelength and spherical aberration of 0.1 wavelength. Nearly no coma appears for off-axis points because a microscopic objective fulﬁlls the sine condition (2.138), which guarantees that object points in the neighborhood of the optical axis are imaged without coma. Curvature of Field As mentioned above the curvature of ﬁeld is not a point aberration but a ﬁeld aberration, i. e. the image point can be sharp but the position of the image point is shifted relative to the ideal paraxial value. In the case of the curvature of the ﬁeld the image points are situated on a spherical surface and, in connection with astigmatism, there are even two different spheres

Geometrical Optics

x-axis (mm)

2.5 Aberrations

27.6

27.7

27.8 z-axis (mm)

Fig. 2.44 Curvature of ﬁeld: ray-tracing picture of the im-

age points of different ﬁeld points of the microscopic objective of Fig. 2.32 (NA = 0.4, magniﬁcation 20×, focal length f = 11.5 mm) which is used in the reverse order, i. e. to build an image with reduced size. It can be seen that the image points are situated on a curve (dotted curve) which is in fact part of a sphere. Note that the scaling is quite different for the x- and z-axes to see the effect of the curvature of the ﬁeld

for rays in the meridional plane and in the sagittal plane. Figure 2.44 shows the curvature of the ﬁeld in the image plane of the microscopic objective of Fig. 2.32. For the off-axis points the best focus of the image points is behind the focal plane (light is coming, as usual, from the left). Of course, the off-axis points also show astigmatism so that the image points are blurred. Distortion The last Seidel term is distortion, which is also a ﬁeld aberration and not a point aberration. Distortion means that the lateral magniﬁcation for imaging is not a constant for all off-axis points but depends to some extent on the object height rO . The result is that each straight line in the object plane that does not pass through the optical axis is curved in the image plane. A regular grid like that in Fig. 2.45b is either pincushion-distorted (Fig. 2.45a) or barrel-distorted (Fig. 2.45c). The Zernike Polynomials A quite important method to calculate the different terms of the wave aberrations of an optical system

is to ﬁt the so-called Zernike polynomials [2.1, 44, 45] to it. The wave aberration data for this procedure can either be theoretically determined, e.g. by ray tracing, or experimentally determined, e.g. by interferometry. The condition for using Zernike polynomials is that the aperture of the optical system is circular because the Zernike polynomials are only orthogonal on the unit circle. There, they build a complete set of orthogonal polynomials and some of the terms correspond to the classical Seidel terms for the point aberrations, i. e. spherical aberration, coma and astigmatism. Besides this there are other terms such as trefoil or tetrafoil that result, for example, if the optical elements are stressed by ﬁxing them at three or four points. It has to be emphasized that there are no terms corresponding to the Seidel terms of curvature of ﬁeld and distortion since the Zernike polynomials can only represent point aberrations and not ﬁeld aberrations.

2.5.4 Chromatic Aberrations Up to now it was implicitly assumed that only light of one wavelength is considered and the presented aberrations were all monochromatic aberrations. Besides this, there are so-called chromatic aberrations, which are a result of the dispersion of a material, i. e. the dependence of the refractive index of a material on the wavelength (or, if there are diffractive optical elements in the system, the dispersion results from the strong dependence of the grating equation on the wavelength). Dispersion changes the paraxial parameters such as the focal length of a lens. For a thin lens with refractive index n in air we have for example (2.98): 1 1 1 . = − 1) − (2.170) (n f R1 R2

Part A 2.5

Fig. 2.45a–c Effect of distortion. The regular grid of (b) in the object plane is either distorted in the image plane to a pincushion shape (a) or a barrel shape (c). In (a) the lateral magniﬁcation increases with increasing object height whereas it decreases in (c)

–0.5

Part A

Basic Principles and Materials

If now n depends on the wavelength λ we have d dλ

1 f

=−

d f / dλ

f 2 dn/ dλ 1 . = n −1 f

dn dλ

1 1 − R1 R2

nd − 1 nF − nC

n(λd = 587.6 nm) . n(λF = 486.1 nm) − n(λC = 656.3 nm) (2.173)

(2.171)

Part A 2.6

By replacing the differentials by ﬁnite differences we can write, to a good approximation, ∆f ∆n . =− f n −1

Vd =

(2.172)

To characterize the dispersion of a material the so-called Abbe number Vd is used; it is deﬁned as

So, we have to a good approximation: 1 f (λC = 656.3 nm) − f (λF = 486.1 nm) = . f (λd = 587.6 nm) Vd (2.174)

For glasses with normal dispersion the Abbe number is a positive constant that has a small value for materials with high dispersion (e.g. materials like SF10) and a large value for materials with small dispersion (like e.g. BK7). The positive sign indicates that the focal length of a lens increases with increasing wavelength.

2.6 Some Important Optical Instruments In this section some important optical instruments such as the achromatic lens, the camera, the human eye, the telescope and the microscope will be discussed. However, this will be done in some cases quite brieﬂy because there are many text books on geometrical optics which treat these subjects quite amply [2.3, 5, 8].

2.6.1 The Achromatic Lens In paragraph 2.5.4 the chromatic aberrations of a single lens, i. e. the dependence of the focal length on the wavelength, were treated. An achromatic lens should have, in the ideal case, no chromatic aberrations. However, in practice the most important achromatic lens is an achromatic doublet consisting of two cemented lenses, for which the focal length can be identical for only two different wavelengths. So, in technical optics the term achromatic lens normally means a lens where the focal length is identical for two different wavelengths. For applications in the visible range these two wavelengths are commonly λF = 486.1 nm (blue line of atomic hydrogen) and λC = 656.3 nm (red line of atomic hydrogen), which are near the edge of the visible range. A lens where the focal length is identical for three wavelengths is called an apochromatic lens. To understand the principle of the achromatic correction of a lens doublet the paraxial matrix M of a combination of two thin lenses (paraxial matrices M1 and M2 ) situated in air with a zero separation is calculated. Of course, this is a simpliﬁcation because in

practice no thin lens really exists and, if the principal points of thick lenses are taken as reference elements, the distance between two lenses will normally be different from zero. But nevertheless, the calculation with two zero-distant thin lenses explains the principle and, according to equation (2.97), the result is 1 0 1 0 M = M2 M1 = − f1 1 − f1 1 2 1 1 0 , = (2.175) − f1 − f1 1 1

where the focal lengths of the two thin lenses are f 1 and f 2 . Therefore, the focal length f of the combination of these two lenses is 1 1 1 = + ; f f1 f2

(2.176)

that is, the optical powers of the single lenses are just added. The refractive index of the ﬁrst lens is n 1 and that of the second lens is n 2 , and both are situated in air. The optical powers 1/ f i (i ∈ {1, 2}) of refractive thin lenses are then, according to (2.98), 1 1 1 = [n i (λ) − 1] − fi (λ) Ri,1 Ri,2 =: [n i (λ) − 1] Ci , (2.177)

Geometrical Optics

1 1 1 1 + = + f 1 (λF ) f 2 (λF ) f 1 (λC ) f 2 (λC ) ⇒ [n 1 (λF ) − 1] C1 + [n 2 (λF ) − 1] C2 = [n 1 (λC ) − 1] C1 + [n 2 (λC ) − 1] C2 , ⇒ [n 1 (λF ) − n 1 (λC )] C1 = − [n 2 (λF ) − n 2 (λC )] C2 . By using (2.177) again the terms C1 and C2 can be expressed using the refractive indices at an intermediate wavelength between λF and λC , in our case λd = 587.6 nm (the yellow line of helium), and the focal lengths at this wavelength, and the result is: n 2 (λF ) − n 2 (λC ) n 1 (λF ) − n 1 (λC ) =− [n 1 (λd ) − 1] f 1 (λd ) [n 2 (λd ) − 1] f 2 (λd ) ⇒ V1,d f 1 (λd ) = −V2,d f 2 (λd ) . (2.178) Here, the Abbe numbers Vi,d (i ∈ {1, 2}) of the materials with refractive indices n i are deﬁned by (2.173). Since the Abbe number of a refractive material is always positive one of the refractive thin lenses has to be a negative lens and one a positive lens to fulﬁll (2.178). However, if one of the two thin lenses is not a refractive but a diffractive lens it formally has a constant negative Abbe number Vd = −3.453 ([2.23], Chap. 10). So, in the case of a so-called hybrid achromatic lens, which consists of a refractive and a diffractive lens, both lenses will have the same sign of optical powers, and so a positive hybrid achromatic lens consists of two positive single lenses: a refractive lens with a high optical power and a high Abbe number and a diffractive lens with a small optical power and a negative Abbe number with small modulus. However, a positive purely refractive achromatic lens consists of a positive lens with high optical power and high Abbe number (made of a crown glass such as BK7) and a negative lens with smaller optical power and smaller Abbe number (made of a highly dispersive ﬂint glass such as SF10), so that in total a positive optical power results. In our mathematical description only the case of two thin lenses with zero separation is treated. But it is no

problem to use the paraxial matrix theory to calculate the matrix of a real achromatic doublet consisting of two cemented lenses, i. e. three refractive spherical surfaces with ﬁnite distances embedding two different materials. However, in this case not only the focal length will depend on the wavelength but also to some degree the position of the principal planes. So, the position of the focus itself can vary a little, although the focal length is identical for the two selected wavelengths λF and λC . In practice, refractive achromatic doublets that can be bought do not only correct the chromatic errors but also fulﬁll the sine condition (2.138). This is possible because there are three surfaces with different radii of curvature, whereas to fulﬁll the paraxial properties only two of these three parameters are determined. Examples of Designing Achromatic Doublets In this paragraph the paraxial properties of different achromatic doublets will be calculated and compared with those of single refractive lenses. It will be assumed as above that the two lenses of the achromatic doublet are thin lenses with zero distance between the two lenses. This is of course a simpliﬁcation, but nevertheless it is a good approximation for most cases. Due to (2.178) the focal lengths f 1 (λd ) and f 2 (λd ) of the two lenses of the achromatic doublet at the wavelength λd = 587.6 nm have to fulﬁll the condition:

V1,d f 1 (λd ) = −V2,d f 2 (λd ) , f (λd ) or f 1 (λd ) = − V2,d 1,d 2 V

⇒

f 2 (λd ) = − V1,d f (λd ) . 2,d 1 V

Here, V1,d and V2,d are the Abbe numbers of the materials of the two lenses. Additionally, the focal length f of the achromatic doublet can be calculated according to (2.176) by 1 1 1 = + . f f1 f2 By combining both equations the focal lengths of the two single lenses can be expressed as functions of the focal length of the achromatic doublet: V1,d − V2,d f (λd ) , V1,d V2,d − V1,d f 2 (λd ) = f (λd ) . V2,d f 1 (λd ) =

(2.179)

A refractive achromatic doublet made of BK7 and SF10 has, for example, Abbe numbers of V1,d = 64.17 (BK7) and V2,d = 28.41 (SF10). Therefore, the focal lengths of

Part A 2.6

where the term Ci depends only on the two radii of curvature Ri,1 and Ri,2 of the thin lenses and is independent of the wavelength λ, whereas the refractive index n i depends on the wavelength. For an achromatic lens the optical powers at the two wavelengths λF and λC (or for two other wavelengths depending on the application) have to be identical. By using (2.176) and (2.177) this means:

2.6 Some Important Optical Instruments

Part A

Basic Principles and Materials

the two single lenses are in this case, due to (2.179), Lens made of BK7 : Lens made of SF10 :

f 1 (λd ) = 0.557 f (λd ) , f 2 (λd ) = −1.259 f (λd ) .

Part A 2.6

So, the second lens, made of the highly dispersive material SF10, is a negative lens if the achromatic doublet itself is a positive lens. As mentioned previously a diffractive lens (DOE) can be described by a negative and material-independent Abbe number Vd = −3.453. Therefore, in the following the focal lengths of the two single lenses of a hybrid achromatic doublet made of one refractive lens and one DOE shall be considered. First, the refractive lens with focal length f 1 is made of BK7 and the second lens with focal length f 2 is a DOE. According to equations (2.179) the focal lengths are: Lens made of BK7 : DOE :

f 1 (λd ) = 1.054 f (λd ) , f 2 (λd ) = 19.588 f (λd ) .

So, as mentioned previously, both lenses are positive lenses if the achromatic doublet has a positive optical power. Of course, most of the optical power is delivered by the refractive lens. A second hybrid achromatic doublet can be made, for example, by taking a refractive lens made of SF10 and a DOE: Lens made of SF10 : DOE :

1.001

f 1 (λd ) = 1.122 f (λd ) , f 2 (λd ) = 9.230 f (λd ) .

The remaining chromatic aberrations of an achromatic doublet, i. e. the variation of the focal length with the wavelength of the illuminating light, can be calculated by using (2.176). In this equation the optical power of a refractive lens as function of the wavelength is calculated by (2.177) and the optical power of a diffractive lens as a function of the wavelength is [2.23] 1 λ =: Cλ . = f DOE (λ) λd f DOE (λd )

Here, C = 1/[λd f DOE (λd )] is a constant value which depends on the focal length f DOE (λd ) of the DOE at the wavelength λd = 587.6 nm. So, the optical power of the DOE increases linearly with wavelength. This is easy to explain because, due to the paraxial grating equation (2.118), the angle of the diffracted light also increases linearly with the wavelength. The chromatic aberrations for the different types of achromatic doublets are shown in Fig. 2.46. The chromatic aberrations of a single refractive lens (made of either BK7 or SF10) compared to that of a refractive achromatic doublet (made of BK7 and SF10) are shown in Fig. 2.47. The result is that the best correction of the chromatic aberrations is made by the refractive achromatic doublet. However, the hybrid achromatic doublet made of a BK7 lens and a DOE also has quite small chromatic aberrations. Nevertheless, all types of achromatic doublets (purely refractive or hybrid) have lower chromatic aberrations than a single refractive lens and of course much lower chromatic aberrations than a single DOE, which is not shown in the ﬁgures.

f '/f 'd Refractive achromat (BK7 + SF10)

f '/f 'd Lens (SF10)

1.01

Lens (BK7)

0.999

Hybrid achromat (BK7 + DOE)

0.998

Achromatic doublet (BK7+ SF10)

0.99 0.997 0.996

Hybrid achromat (SF10 + DOE)

500

550

Fig. 2.46 The focal length

600 650 Wavelength λ (nm)

of different achromatic doublets (a refractive achromatic doublet made of BK7 and SF10, a hybrid achromatic doublet made of BK7 and a diffractive lens (DOE) and a hybrid achromatic doublet made of SF10 and a DOE) as a function of the wavelength λ normalized by the focal length f d at λd = 587.6 nm

0.98 500

550

600 650 Wavelength λ (nm)

Fig. 2.47 The focal length f of two single lenses (one made of BK7 and the other made of SF10) and a refractive achromatic doublet (made of BK7 and SF10) as a function of the wavelength λ normalized by the focal length f d at λd = 587.6 nm

Geometrical Optics

2.6.2 The Camera

x ≈ϕ f .

(2.180)

The moon has, for example, an angular extension of about half a degree if observed from the earth so that its image on a standard camera would be just x = 0.44 mm. This is the reason why the moon on a photo made with a miniature camera with a ﬁlm size of 24 mm × 36 mm is really small and details cannot be detected. However, this can be changed by using a telescope (Sect. 2.6.4) in front of the camera, which changes the angular extension ϕ of the object. In astronomical cameras the eyepiece of the telescope is commonly omitted and the detector is positioned directly in the focal plane of the objective lens Photosensitive device Diaphragm

or mirror, which has a large focal length f and which serves as the camera lens. Nevertheless, such a device is still called an astronomical telescope. The Depth of Field In geometrical optics an ideal camera lens (without aberrations) images one object plane very sharply onto the photosensitive image plane. But in reality each image point is imperfect for two reasons: ﬁrstly, due to the wave nature of light, it is not an ideal mathematical point but an airy disc, and secondly the resolution of the detector is in many cases smaller than the maximum possible resolution given by the wave nature of light. Object points in other planes than the ideal object plane are imaged to planes in front of or behind the detector plane (Fig. 2.49). Therefore, in the detector plane they will form small image discs, and if the diameter of the image discs is smaller than the pixel size p of the detector these other planes will also be imaged without loss of resolution onto the detector, which limits the resolution. The ideal object plane, which is imaged very sharply onto the detector, has object distance dO while the detector plane has image distance dI (where dO < 0 and dI > 0 for a real image in a camera). An object plane that is nearer the camera lens than |dO | and where the light rays of the object points form small discs in the detector plane with a diameter of exactly p is the nearest object plane which is imaged onto the detector with the maximum resolution, given by the pixel distance p. Its object distance is called dO,N (index “N” for “near”) and its image distance is called dI,N (Fig. 2.49). Similarly, that object plane a larger distance than |dO | from the lens where the rays coming from the object points also form discs in the detector plane with a diameter p is the farthest object plane which is imaged with the maximum resolution given by the detector. It has object

Lens Ideal object plane

Detector plane dO,N

dI,N

D dO

Fig. 2.48 Principle of a camera. The distance from the object to the lens of the camera compared to the focal length of the lens is in the presented case so large that the object can be assumed to be at inﬁnite distance

Part A 2.6

One of the simplest optical instruments is the camera [2.1, 8]. Of course, modern cameras are highly sophisticated technical instruments with complex wideangle or zoom lenses. However, the basic principle of each camera (see Fig. 2.48) is that a lens forms a real inverted picture of an object on a photosensitive surface, which can be a photographic ﬁlm or an electronic detector such as a charge-coupled device (CCD) chip. Additionally, each camera has a diaphragm near the lens. The standard camera objective of a miniature camera has a focal length of f = 50 mm so that each object at a distance of several meters can be assumed to be at an inﬁnite distance and the object distance dO in the lens equation (2.89) can be assumed to be dO → −∞. Then, the image is practically formed in the focal plane of the lens, i. e. dI ≈ f . So, the size x of the image of an object is determined by the angular extension ϕ of the object by

2.6 Some Important Optical Instruments

dO,F

p dI

dI,F

Fig. 2.49 Calculation of the depth of ﬁeld for the case of a camera with a thin lens

Part A

Basic Principles and Materials

Part A 2.6

distance dO,F and corresponding image distance dI,F (the index F indicates “far”). The depth of ﬁeld is now deﬁned as the axial extension of the object space between the near object plane and the far object plane, which are both just imaged with the maximum resolution of the detector. The depth of ﬁeld depends of course on the diameter D of the aperture stop and on the resolution of the detector, i. e. the pixel distance p. We assume in the following that we have a thin ideal lens with a focal length f on the image side and that the aperture stop is directly in the plane of the lens. Then, the aperture stop is also the entrance pupil and the exit pupil. An important quantity is the so called F number, f #, of the lens which is deﬁned as the ratio of the focal length f of the lens and the diameter D of the entrance pupil f#=

f . D

(2.181)

If the image is formed in the focal plane, as is nearly the case for a camera imaging a distant object, and if the diameter D is small compared to the focal length f the F number and the numerical aperture on the image side deﬁned by (2.136) are connected to a good approximation by: NAI = n I sin ϕI ≈ n I

D 1 . = nI 2f 2f#

(2.182)

Here, ϕI is the half aperture angle of the light cone on the image side and n I is the refractive index on the image side. In most cases there will be air on the image side, i. e. n I = 1. But for some camera-like systems such as the human eye n I will be different from 1 (Sect. 2.6.3). According to the lens equation (2.89), where n O and n I are the refractive indices on the object and image side, respectively, we have three equations for the different object and image distances:

Additionally, according to the theorem on intersecting lines, we have two additional equations (Fig. 2.49): D p p = ⇒ dI,N − dI = dI,N , dI,N dI,N − dI D p p D = ⇒ dI − dI,F = dI,F . dI,F dI − dI,F D

(2.186) (2.187)

By putting (2.183) and (2.184) in (2.186) and solving for dO,N the result is dO,N = =

n O f dO n O f − Dp (n O f + n I dO ) dO

. p 1 − D 1 + nnOI dfO

(2.188)

In the same way by combining (2.183), (2.185) and (2.187) the result for dO,F is dO,F = =

n O f dO n O f + Dp (n O f + n I dO ) dO

. p 1 + D 1 + nnOI dfO

(2.189)

It is common practice in photography to use the lateral magniﬁcation β, which was deﬁned by (2.52) as the ratio of the image height xI and the object height xO . For a lens which fulﬁlls the sine condition (2.138) the principal planes are in reality principal spheres which are centered around the object and the image point, respectively. The same is valid for the entrance and exit pupil [2.3]. Then, using (2.138), the lateral magniﬁcation can be expressed as xI n O sin ϕO n O D/(2dO ) n O dI = β= = = . xO n I sin ϕI n I D/(2dI ) n I dO (2.190)

By multiplying (2.183) with dO /n O it holds: n I dO n I dO 1 n I dO −1 = ⇒ = 1+ . n O dI nO f β nO f

(2.191)

So, (2.188) and (2.189) can be expressed by nI nO nI − = ⇒ dI = dI dO f

n I f dO n O f + n I dO

, (2.183)

nO nI n I f dO,N nI − = ⇒ dI,N = , dI,N dO,N f n O f + n I dO,N (2.184)

nO nI nI − = ⇒ dI,F = dI,F dO,F f

n I f dO,F n O f + n I dO,F

. (2.185)

dO dO , p = 1 − Dβ 1 − pf fβ# dO dO dO,F = . p = 1 + Dβ 1 + p f #

dO,N =

(2.192)

(2.193)

f β

In the last step, the F number, f #, deﬁned by (2.181) is used. In the case of a camera the focal length f is positive and the lateral magniﬁcation β is always negative

Geometrical Optics

since a real image is formed, i. e. β < 0. So, there is the interesting special case that the denominator of (2.193) can be zero:

space. By using (2.192) and (2.193) the result is ∆d = dO,N − dO,F = 2dO nO =2 pf# nI

(2.194)

On the right-hand side, (2.191) was used and solved for dO . So, if the camera is focused to the critical object distance dO,C given by (2.194) it holds that |dO,F | → ∞ and all objects which are farther from the camera lens than |dO,N | = |dO,C |/2 (this follows from (2.192)) will be imaged onto the detector with the maximum resolution, i. e. the image will look sharp. Of course, if the modulus |dO | of the actual object distance is larger than the modulus of the critical value |dO,C | given by (2.194), dO,F will formally be positive. This means that a virtual object behind the lens with distance dO,F , which can be produced by some auxiliary optics, can also be imaged sharp onto the detector. In fact, this still means that all real objects farther from the lens than |dO,N | will be imaged sharp onto the detector. If we have for example a camera with f = 50 mm, a minimum F number f # = 2.8, n O = n I = 1, and a pixel distance p = 11 µm (typical for a CCD chip) the critical object distance dO,C of (2.194) is dO,C = − 81.2 m. Therefore, all objects at a distance of more than |dO,N | = |dO,C |/2 = 40.6 m from the camera will be imaged sharply if the camera is focused to dO,C . If the F number is f # = 16 all objects with a distance of more than 7.1 m will be imaged sharp for a focusing distance of |dO,C | = 14.3 m. However, for larger F numbers the wave nature of light begins to limit the resolution because the radius rdiff of a diffraction limited spot will be rdiff = 0.61λ/NA≈ 1.22λ f # = 10.7 µm ≈ p for a wavelength λ = 550 nm and f # = 16. Of course, a larger F number means that the light intensity on the detector decreases because the light intensity on the detector is proportional to the effective area π D2 /4 of the light gathering lens and therefore proportional to 1/ f #2 = D2 / f 2 . So, a larger F number means that the exposure time has to be increased proportional to f #2 . All these facts are well-known from photography. If dO,F has a ﬁnite value, e.g. if the camera is focused on a near object, it is useful to calculate the axial extension ∆d = dO,N − dO,F of the sharply imaged object

1 β

1−

pf# f β

1−

− 1 β1

2 .

pf# f β

(2.195)

pf# f β

In the last step (2.191) has been used to express the object distance dO by the lateral magniﬁcation β because these two quantities are of course coupled to each other. Again, we see in (2.195) the limiting case that the denominator can approach zero [if (2.194) is fulﬁlled] and that therefore the depth of ﬁeld has an inﬁnite range. But, for near objects (for example |dO | ≤ 1 m) we normally have the case that f |β| p f #. Then, ∆d has ﬁrst of all a ﬁnite positive value and second there is a quite good approximation which is often used for the photography of near objects [2.8]

1 1 β −1 β nO ∆d = 2 pf#

2 nI 1 − pf fβ# nO nO 1−β 1 1 −1 =2 ≈2 pf# pf# 2 . nI β β nI β (2.196)

As an example we take again a common electronic camera with f = 50 mm, p = 11 µm and n O = n I = 1. The F number is assumed to be f # = 10 and the object is at dO = −1 m. Then, the lateral magniﬁcation is, according to (2.191), β = −0.05263. The extension of the depth of ﬁeld ∆d is according to the exact (2.195) ∆d = 83.75 mm and according to the approximate (2.196) ∆d = 83.60 mm. So, the error of the approximate equation is about 0.2% and the depth of ﬁeld has an extension of about 8.4 cm, i. e. objects with an axial extension in this range (for a medium object distance |dO | = 1 m) will be imaged without loss of resolution onto the detector.

2.6.3 The Human Eye The human eye is based on the principle of a camera that builds an inverted real image of the surroundings on the retina [2.1,8]. However, the actual structure and performance of the human eye is quite complex [2.46–48] so that only the most important features of the normal emmetropic eye can be discussed in this section.

Part A 2.6

p p 1+ =0⇒β=− Dβ D D nO f 1+ ⇒ dO,C = − nI p f nO f 1+ . =− nI pf#

2.6 Some Important Optical Instruments

Part A

Basic Principles and Materials

Part A 2.6

The optical power of the eye is delivered by the cornea and the deformable crystalline lens (Fig. 2.50). The main part of the optical power is delivered by the cornea with about 43 diopters because at the ﬁrst surface the difference between the refractive indices of air and the cornea (1.376) is quite high. The crystalline lens, with a refractive index between 1.386 in the outer parts and 1.406 in the core, is immersed on one side in the aqueous humour and on the other side in the vitreous body, both of which have a refractive index of 1.336. Therefore, the lens has about 19 diopters in the case of distant vision. The resulting total optical power of the eye is, due to the ﬁnite distance between the cornea and the crystalline lens, about 59 diopters for distant vision. The accommodation of the eye for near objects, which is performed by the crystalline lens, can vary between about 14 diopters in young age and nearly 0 diopters above 50 years of age because the crystalline lens loses its ﬂexibility with increasing age. Since the normal distance for reading is about 25–30 cm an accommodation of less than 3–4 diopters has to be corrected by wearing eyeglasses for reading. The photosensitive surface of the eye is the curved retina and the diaphragm of the eye is the iris, which can change its diameter between about 2 mm and 8 mm to control the irradiance on the retina depending on the intensity of the illuminating light. The effective focal length of the eye, which, as mentioned above, is an immersion system, is f /n ≈ 1/(59 diopters) ≈ 17 mm (n = 1.336 is the refractive index of the vitreous body between the eye lens and the retina). The so-called least distance of distinct vision of a normal adult eye is about 25 cm, which requires an accommodation of 4 diopters. The angular resolution ∆ϕ of a normal eye is about 1’ (one arc min) and, under optimal conditions, it can achieve 30”. The later corresponds to a distance of ∆x = ∆ϕ f /n = 2.5 µm on the retina. So, the light-sensitive cells (cones) in the fovea (about 200 µm in diameter), which is the central part of the retina, have to be about 2.5 µm or less in diameter and distance. In the fovea there are mainly color-sensitive cones whereas in the outer parts the rods, which are more sensitive to light but which cannot distinct between different colors, dominate. It is interesting to note that a human eye with normal vision is a nearly diffraction-limited optical system for a diameter of the pupil of up to 3 mm (the diameter for sharpest vision). This can be seen because in this case the radius r of the airy disc which limits the resolution according to the Rayleigh criterion (see Sect. 3.4.2) is about r = 0.61λ/NA = 3.8 µm for a wavelength of λ = 0.55 µm and a numerical aperture

x-axis (mm)

Iris

Crystalline lens

–5

Cornea 0

Retina 10

–5

20 z-axis (mm)

Fig. 2.50 Ray tracing of an eye model as used in ophthalmology. Here, the crystalline lens consists of a nucleus with higher refractive index (1.406) and a cortex with lower refractive index (1.386). The ﬁrst surface of the cornea and all surfaces of the crystalline lens are aspheric surfaces. It can be seen that the foci show aberrations for this large pupil (iris) diameter. The drawn off-axis point with quite high aberrations is of course far out of the fovea so that it is only used in the eye to detect motions and not to image an object

NA ≈ 1.5 mm/17 mm ≈ 0.088. So, the aforementioned value ∆x = 2.5 µm for the smallest resolvable distance on the retina is even slightly smaller than the distance r given by the Rayleigh criterion due to diffraction, which assumes that a drop of 26 percent in irradiance can be detected. The reason is that the Rayleigh criterion is slightly arbitrary and, under optimal conditions, the eye can also detect smaller drops in irradiance between two adjacent points. For larger diameters of the pupil than 3–4 mm spherical aberration and chromatic aberrations of the eye reduce the resolution. Therefore, at night or in badly illuminated rooms the resolution of the eye is reduced and all tasks which need a high resolution, for example reading, are more difﬁcult or impossible if the irradiance on the retina is too low even for the largest pupil diameter.

2.6.4 The Telescope One of the most important optical instruments is the telescope [2.1, 3, 8]. It has well-known applications in terrestrial and astronomical observations [2.21]. But there are at least as important applications in optics to

Geometrical Optics

d = f 1 − f 2 = f 1 + f 2 .

So, the coefﬁcient C of the ABCD-matrix M, which is according to (2.68) the negative value of the optical power, is zero and therefore the focal length of the telescope is inﬁnity. Such a system with zero optical power is called an afocal system. So, a telescope can also be deﬁned to be an afocal optical system where the trivial case that all lenses themselves have zero optical power, i. e. 1/ f 1 = 1/ f 2 = 0, is excluded. Telescope as a Beam Expander and Imaging System for Far-Distant Objects An important property of an afocal system is that it transforms a collimated bundle of rays into another collimated bundle of rays. The application as a beam expander for a collimated beam or as an imaging system for far-distant objects can easily be seen from (2.198) by taking two parallel rays with paraxial ray parameters (x1 , ϕ1 ) and (x2 , ϕ2 ) (ϕ2 = ϕ1 ) in front of the telescope. The paraxial ray parameters (x1 , ϕ1 ) and (x2 , ϕ2 ) of the

–f2 = f2' F1' = F2 f1'

F1' = F2 f2

(2.197)

The paraxial matrix M of the telescope from the objectsided principal plane U1 of the ﬁrst lens to the imagesided principal plane U2 of the second lens is 1 0 1 0 1 d M= − f1 1 − f1 1 0 1 2 1 ⎛ ⎞ 1 − fd d 1 ⎠ =⎝ 1 − f − f1 + f df 1 − fd 2 1 2 2 ⎛ f1 ⎞ 2 − f f1 + f2 ⎠. =⎝ 1 (2.198) f 0 − f 1

Part A 2.6

expand or compress a collimated (laser) beam, to ﬁlter spatial frequencies in an optical system, to transport an intermediate optical image to another plane, and so on. A telescope consists in principle of two lenses or two other focusing optical elements such as spherical or aspheric mirrors. Here, to demonstrate the principle we assume that it consists of two lenses with focal lengths f 1 and f 2 and a distance d between the two lenses. In order to have a telescope the image-sided focus F1 of the ﬁrst lens and the object-sided focus F2 of the second lens have to coincide (Fig. 2.51). Additionally, we assume that the two lenses are situated in air, so that we have f 2 = − f 2 for the image-and object-sided focal length. So, by taking into account the sign conventions for the focal lengths the condition for the distance between the two lenses of a telescope is

2.6 Some Important Optical Instruments

f1'

Fig. 2.51a,b Principle set ups of an astronomical telescope (a) and a Galilean telescope (b)

rays behind the telescope are then: ⎞ ⎛ f2 − f f 1 + f 2 xi ⎠ xi =⎝ 1 f 1 ϕi ϕi 0 − f 2 ⎞ ⎛ f − f2 xi + ( f 1 + f 2 )ϕi ⎠ =⎝ 1 f − f1 ϕi

(2.199)

with i ∈ {1, 2}. The angular magniﬁcation γ , deﬁned by equation (2.53), i. e. the ratio of the angle ϕ := ϕ1 = ϕ2 between the bundle of rays and the optical axis behind the afocal system to the angle ϕi := ϕ1 = ϕ2 in front of the system, in the paraxial case is γ=

f ϕ = − 1 . ϕ f2

(2.200)

So, the angular magniﬁcation, which determines the size of the image of a far-distant object, only depends on the ratio of the focal lengths of the two lenses. The beam-expanding property can be seen by calculating the distance ∆x between two parallel rays (ϕ2 = ϕ1 ) in front of the telescope and the distance ∆x behind the telescope: ∆x = x2 − x1 = −

f 2 f (x2 − x1 ) = − 2 ∆x (2.201) f1 f1

Part A

Basic Principles and Materials

So, the beam expanding ratio ∆x /∆x is the reciprocal of the angular magniﬁcation.

Part A 2.6

Imaging Property of a Telescope for Finite Distant Objects Although a telescope has zero optical power it images an object from one plane to another plane. This can be seen by calculating the paraxial matrix M from an object plane with a distance d1 to the principal plane U1 of the ﬁrst lens (keep in mind that, contrary to the normal sign conventions of paraxial optics, in the paraxial matrix theory d1 is positive if the object plane is in front of U1 and negative if it is behind U1 ) to an image plane at a distance d2 behind the principal plane U2 of the second lens (d2 is positive if the image plane is real and behind U2 and negative if it is a virtual image plane in front of U2 ). Figure 2.52 shows the parameters to calculate M : ⎞ ⎛ f 2 f + f − 2 1 d2 ⎝ f1 1 1 d 1 ⎠ M = f 0 1 0 1 0 − f 1 2 ⎞ ⎛ f f f − f 2 f 1 + f 2 − d1 f 2 − d2 f1 1 2⎠ . (2.202) =⎝ 1 f 0 − f1 2

In the case of imaging, the parameter B of the matrix has to be zero. So, the condition for the distances d1 and d2 is f f f 1 + f 2 − d1 2 − d2 1 = 0 , f1 f2 ⇒ d2 = f 2 +

( f 2 )2 ( f 2 )2 − d . 1 f 1 ( f 1 )2

(2.203)

As mentioned before, the image is real if d2 is positive and it is virtual for a negative value d2 . So, a real image 1

–f2 = f2' F1' = F2

Object plane

f1'

Image plane

Fig. 2.52 Parameters to calculate the paraxial matrix M for imaging

of an object point to an image point with the help of a telescope

of a real object point (i. e. d1 ≥ 0) means d2 ≥ 0 ⇒ f 1 +

f 1 2 1 1 ≥ d1 ≥ 0 ⇒ + ≥ 0 . f2 f1 f2 (2.204)

It can easily be seen that the Galilean telescope (see next paragraph) cannot deliver a real image of a real object point whereas the astronomical telescope delivers real images as long as 0 ≤ d1 ≤ f 1 + f 1 2 / f 2 . A quite interesting property of a telescopic imaging system is the lateral magniﬁcation β [see (2.52)]. It is, according to equation (2.202) for the imaging case, i. e. matrix element B = 0, equal to the matrix element A: β=

f x = − 2 . x f1

(2.205)

So, the lateral magniﬁcation of the telescopic system depends only on the focal lengths of the two lenses and is independent of the axial position of the object point. If we additionally place the aperture stop into the focal plane of the ﬁrst lens (only for the astronomical telescope) the telescopic imaging system is telecentric (see also page 60). A very important system is the so-called 4 f -system with f := f 1 = f 2 > 0. Then we have, with the help of (2.203), d2 = f + f − d1

⇒

d1 + d2 = 2 f .

(2.206)

This means that the sum of the two distances d1 and d2 is always 2 f and in total the distance from the object plane to the image plane is 4 f (in the case of thin lenses where the thickness of the lenses can be neglected compared to 4 f ) because the length of the telescope has to be added. This also means that, for a 4 f -system, the shift of the image plane is equal to the shift of the object plane and, therefore, the telescope itself can, for example, be shifted relative to the object and image plane without changing the imaging situation. Of course, in the non-paraxial realm aberrations will change the imaging quality if the telescope of a 4 f -system is moved because the aberrations depend on the actual position of the telescope relative to the object and image plane. The Astronomical and the Galilean Telescope There are two different types of telescopes (Fig. 2.51). The astronomical telescope (also called a Kepler telescope) and the Galilean telescope (also called a Dutch telescope).

Geometrical Optics

Astronomical Telescope. The astronomical telescope

Aperture stop

Field stop

Part A 2.6

(Fig. 2.51a or Fig. 2.53) consists of two positive lenses so that the ﬁrst lens (called the objective) forms a real image of a far-distant object near the focal plane (or exactly in the focal plane for an object with inﬁnite distance). Then, the second lens (called the eyepiece) also forms an inﬁnite-distant image but with an increased or decreased angular magniﬁcation. Since the focal lengths f 1 and f 2 are both positive the angular magniﬁcation γ is, according to (2.200), γ = − f 1 / f 2 < 0. Therefore, the image is upside down so that an astronomical telescope without additional optics to reverse the image is not practical for terrestrial inspections. However, for astronomical purposes or for image transfers in optical systems this is no disadvantage. Additionally, the advantage of the astronomical telescope is that the entrance pupil coincides with the objective in the case of imaging inﬁnite-distant objects. This means that the exit pupil, which is the image of the objective formed by the eyepiece, is typically near the focal plane of the eyepiece if f 1 f 2 , which is necessary to obtain an image with angular magniﬁcation |γ | 1. Therefore, the pupil of the eye can be positioned at the exit pupil of the telescope and all light rays with the same off-axis angle (i. e. from the same inﬁnite-distant object point) entering the telescope contribute to the image on the retina of the eye. Another advantage of the astronomical telescope is, as mentioned previously, that it can deliver a telecentric real image of objects with a ﬁnite distance. It is quite interesting to think a little bit more about the position of the aperture stop and the ﬁeld stop for the two cases of imaging inﬁnite-distant objects or ﬁnite-distant objects (Fig. 2.53). As mentioned, the aperture stop for the imaging of inﬁnite-distant objects (Fig. 2.53a) is the aperture of the ﬁrst lens. The ﬁeld stop lies in this case in the back focal plane of the ﬁrst lens. For the case of imaging of ﬁnite-distant objects (Fig. 2.53b) the situation is different and it is useful to put the aperture stop in the back focal plane of the ﬁrst lens to have a well-deﬁned numerical aperture for all object points which are not too far away from the optical axis. Then, the aperture of the ﬁrst lens can act as the ﬁeld stop. Of course, in this case the ﬁeld stop has no sharp rim because parts of the light cone of points with a similar distance from the optical axis than the radius of the ﬁrst lens can pass the system if the aperture stop is large enough. In this case a kind of vignetting occurs. So, an additional stop directly in the object or image plane serving as a ﬁeld stop would be desirable.

2.6 Some Important Optical Instruments

Vignetted image point

Aperture stop Field stop

Fig. 2.53a,b The aperture stop and the ﬁeld stop for the imaging of (a) inﬁnite-distant objects or (b) ﬁnite-distant objects (here shown

for the case that the object plane is in the front focal plane of the ﬁrst lens) with the help of an astronomical telescope

For astronomical observations most modern telescopes use mirrors as focusing elements instead of lenses [2.21]. There are telescopes with a primary mirror with a diameter of D = 8 m. From wave optics it is well known that two (inﬁnite-distant) object points can just be resolved by an aberration-free telescope with a diameter D of the primary mirror and a wavelength λ of the observed light if their angular separation is larger or equal to ∆ϕ with ∆ϕ = k

λ , D

(2.207)

where k is a constant of approximately k = 1 (for a full circular aperture it is k = 1.22). The exact value of k depends on the actual design of the instrument because a reﬂective telescope has in many cases an annular or more complicated aperture because the secondary mirror and the mounting shadow the central and other parts of the primary mirror. So, a telescope with a large-diameter primary mirror has of course an increased light-gathering power and an increased angular resolution.

Part A

Basic Principles and Materials

Part A 2.6

Galilean Telescope. The Galilean telescope (Fig. 2.51b) consists of a positive lens (the objective) with the focal length f 1 > 0 and a negative lens (the eyepiece) with focal length f 2 < 0 or f 2 = − f 2 > 0 and | f 2 | < | f 1 |. Of course, the telescope can also be rotated by 180◦ so that it reduces the angular magniﬁcation. But, in the following we assume that f 1 > 0 and f 2 < 0. The total length of the Galilean telescope is only | f 1 | − | f 2 | (for thin lenses) compared to | f 1 | + | f 2 | for the astronomical telescope (we use here the absolute values of the focal lengths although f 1 is always positive and only f 2 has a different sign for an astronomical and a Galilean telescope). Another advantage of the Galilean telescope is that the angular magniﬁcation γ is, according to (2.200) positive, γ = − f 1 / f 2 > 0. Therefore, the image is upright and can be directly used for terrestrial inspections. However, a disadvantage of the Galilean telescope is that the image of the objective made by the second lens is between the two lenses. Therefore, the exit pupil of the Galilean telescope is not accessible for the eye and the pupil of the eye works itself as the aperture stop of the complete system whereas the diameter of the objective limits the ﬁeld. So, Galilean telescopes have a limited ﬁeld of view and only small magniﬁcations of two to ﬁve are useful. Another disadvantage is that the Galilean telescope cannot deliver a real image of a real object. So, a Galilean telescope cannot be used to transport a real intermediate image to another plane in an optical system. However, the compact overall length and the positive angular magniﬁcation provide applications for a Galilean telescope as a beam expander or terrestrial telescope like a lorgnette.

to obtain at the distance |dI | = dS a magniﬁed virtual image of an object that has itself a smaller separation |dO | from the eye than the standard distance dS . The image at a distance dI from the image-sided principal plane of the lens with focal length f has to fulﬁll the imaging equation (2.89) whereby the refractive index on the image side has to be n = 1 because the human eye is normally used in air and only in this case delivers a sharp image. Then, the imaging equation is n 1 1 − = . dI dO f

(2.208)

Here, n is the refractive index on the object side, which is often 1 (object in air) but sometimes also larger than 1 if the object is in immersion (for example in water or oil). Due to the sign conventions of geometrical optics dO is negative since the object is in front of the lens. The image distance dI is also negative for a virtual image. Then, the lateral magniﬁcation β of the image is, according to (2.52), (2.208) and Fig. 2.54, β=

xI ϕI dI dI dI dS = =n = 1− = 1+ . xO ϕO dO dO f f (2.209)

Here, we use that, in the paraxial case, the angles ϕI and ϕO have to fulﬁll the condition n ϕI = nϕO , where n = 1 is valid in our case. Additionally, it is used that the virtual image is at the standard distance for distinct The Magnifier

2.6.5 The Microscope

dI xI

The last important optical instrument that will be discussed here is the microscope [2.1, 3, 8]. Whereas, the telescope, especially an astronomical telescope, is used to achieve an angular magniﬁcation of distant objects, a microscope is used to obtain a magniﬁed image of a very small near object. The Magnifier If somebody wants to see details of a small object he brings the object as close to the eye as possible since then the image of the object on the retina of the eye is as large as possible. However, a typical human eye can only form a sharp image of an object at a smallest distance of about dS = 25 cm, which is the standard distance for distinct vision. So, it is obvious that a positive lens, called a magniﬁer, directly in front of the eye can be used

φI = φO

dO xO

Fig. 2.54 The principle of a magniﬁer. A thin lens is used and it is assumed that the refractive indices in the object and image space are both equal so that ϕI = ϕO

Geometrical Optics

The Two-Stage Microscope There is of course a limitation to the lateral magniﬁcation achievable by using a magniﬁer because the object has to be very close to the magniﬁer and therefore also to the eye to achieve large lateral magniﬁcations. Therefore, microscopes have been invented that make a magniﬁcation of the object in two stages (Fig. 2.55). First, a magniﬁed real image of the object with magniﬁcation βobjective is formed by using a lens with a small focal length, called the objective. This real image is of course inverted. Then, a magniﬁer, called the eyepiece, with a (mostly) larger focal length is used to form a magniﬁed virtual image of the intermediate real image, where the virtual image is at the standard distance of distinct vision of the eye. The lateral magniﬁcation for this second operation is βeyepiece . This means, that the lateral magniﬁcations of both operations are multiplied and the total lateral magniﬁcation of the microscope βmicroscope is

βmicroscope = βobjective βeyepiece .

(2.210)

In practice, the objective of a microscope is a quite complex lens consisting of many single lenses to correct the aberrations (especially spherical aberration, coma and chromatic aberrations) of the objective and to guarantee a large ﬁeld of view [2.8]. Moreover, modern microscope objectives are corrected for inﬁnity. This means that their aberrations are only corrected if the object is exactly in the object-sided focal plane. Therefore, the image would be at inﬁnite distance and an additional lens (called the tubus lens) with a ﬁxed focal length (the so-called tubus length, which is often 160 mm) must be used to get the real image with the magniﬁcation imprinted on the objective. For biological investigations, where the object is often covered by a thin coverslip, the spherical aberrations, which result from a high-NA spherical wave passing through a plane-parallel plate, also have to be corrected. Another very important parameter of the objective is its numerical aperture NA (2.135). It determines on the one hand the light-gathering power of the objective and

Eyepiece Objective Object

F '1

Real intermediate image

Part A 2.6

vision so that the image distance dI , which is negative, is replaced by −dS , where dS is the absolute value of the standard distance for distinct vision. If the lens has, for example, a focal length of f = 5 cm a lateral magniﬁcation of β = 1 + 25/5 = 6 is obtained. In order to have a large ﬁeld of view without aberrations and especially without chromatic aberrations the magniﬁer itself is in practice not a single lens but an achromatic combination of different single lenses.

2.6 Some Important Optical Instruments

Virtual image

Fig. 2.55 The principle of a microscope illustrated by using thin lenses. The objective forms a real magniﬁed intermediate image of the object, which is then transformed by the eyepiece into a virtual further-magniﬁed image. The distance of this virtual image from the eyepiece and the eye, which is directly behind the eyepiece, has to be the standard distance of distinct vision

on the other the resolution which is possible. From wave optics we know that the smallest distance ∆x between two points which can be resolved by a microscope is ∆x = k

λ , NA

(2.211)

where λ is the wavelength of the used light and k is a constant (typically about 0.5) that depends on the illumination conditions (coherence) and the exact aperture shape of the objective (mostly circular). If the image of a microscope has to be on a camera chip (for example a CCD chip) a real image has to be made on the camera chip. Therefore, the eyepiece, which produces a virtual image, cannot be used and indeed it is sufﬁcient just to bring the CCD chip to the position of the real image of the objective (plus the tubus lens). A typical magniﬁcation of a high-NA objective in air with, for example, |β| = 50 is sufﬁcient if a CCD chip with a typical pixel size of 11 µm is used. This would mean that a structure size of 0.22 µm on the object is magniﬁed to the size of a pixel of the CCD chip. However, due to (2.211), 0.22 µm is approximately the resolution of an objective with NA< 1 and a wavelength in the visible spectral range. By bringing an immersion oil between the object and the objective, which has to be a special immersion objective, the NA can be increased up to about 1.4. So, the resolution can be increased accordingly. Another possibility is of course to reduce the wavelength. Modern microscopes for the inspection of integrated circuits use ultraviolet light with a wavelength of 248 nm.

Part A

Basic Principles and Materials

References 2.1 2.2 2.3

Part A 2

2.4 2.5 2.6

2.7 2.8 2.9

2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26

M. Born, E. Wolf: Principles of Optics, 6th edn. (Cambridge Univ. Press, Cambridge 1997) R. Ditteon: Modern Geometrical Optics (Wiley, New York 1998) H. Haferkorn: Optik, 4th edn. (Wiley–VCH, Weinheim 2003) E. Hecht: Optics, 3rd edn. (Addison–Wesley, Reading 1998) R. S. Longhurst: Geometrical and Physical Optics, 3rd edn. (Longman, New York 1973) V. N. Mahajan: Optical Imaging and Aberrations, Part I: Ray Geometrical Optics (SPIE, Bellingham 1998) D. Marcuse: Light Transmission Optics, 2nd edn. (Van Nostrand, New York 1982) H. Naumann, G. Schröder: Bauelemente der Optik, 5th edn. (Hanser, München 1987) I. N. Bronstein, K. A. Semendjajew: Taschenbuch der Mathematik, 23rd edn. (Thun, Frankfurt/Main 1987) R. A. Chipman: Mechanics of polarization ray tracing, Opt. Eng. 34, 1636–1645 (1995) E. Waluschka: Polarization ray trace, Opt. Eng. 28, 86–89 (1989) W. Brouwer: Matrix Methods in Optical Instrument Design (Benjamin, New York 1964) A. Gerrard, J. M. Burch: Introduction to Matrix Methods in Optics (Wiley, London 1975) H. Kogelnik, T. Li: Laser beams and resonators, Appl. Opt. 5, 1550–1567 (1966) A. E. Siegman: Lasers (Univ. Science Books, Mill Valley 1986) R. Kingslake: Lens Design Fundamentals (Academic, San Diego 1978) R. Kingslake: Optical System Design (Academic, New York 1983) D. Malacara, Z. Malacara: Handbook of Lens Design (Dekker, New York 1994) D. C. O’Shea: Elements of Modern Optical Design (Wiley, New York 1985) M. V. Klein, Th. E. Furtak: Optics, 2nd edn. (Wiley, New York 1986) R. Riekher: Fernrohre und ihre Meister, 2nd edn. (Verlag Technik, Berlin 1990) H. J. Levinson: Principles of Lithography (SPIE, Bellingham 2001) H. P. Herzig: Micro-Optics (Taylor Francis, London 1997) B. Kress, P. Meyrueis: Digital Diffractive Optics (Wiley, Chicester 2000) S. Sinzinger, J. Jahns: Microoptics (Wiley–VCH, Weinheim 1999) B. Dörband: Abbildungsfehler und optische Systeme. In: Technische Optik in der Praxis, ed. by G. Litfin

2.27 2.28 2.29

2.30

2.31

2.32

2.33

2.34 2.35 2.36

2.37

2.38

2.39

2.40

2.41

2.42 2.43 2.44

2.45

(Springer, Berlin Heidelberg New York 1997) pp. 73– 101 G. H. Spencer, M. V. R. K. Murty: General ray tracing procedure, J. Opt. Soc. Am. 52, 672–678 (1962) O. N. Stavroudis: The Optics of Rays, Wavefronts, and Caustics (Academic, New York 1972) A. Sharma, D. V. Kumar, A. K. Ghatak: Tracing rays through graded-index media: a new method, Appl. Opt. 21, 984–987 (1982) A. Sharma: Computing optical path length in gradient-index media: a fast and accurate method, Appl. Opt. 24, 4367–4370 (1985) R. W. Smith: A note on practical formulae for finite ray tracing through holograms and diffractive optical elements, Opt. Commun. 55, 11–12 (1985) W. T. Welford: A vector raytracing equation for hologram lenses of arbitrary shape, Opt. Commun. 14, 322–323 (1975) W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling: Root Finding and Nonlinear Sets of Equations, Numerical Recipes in C (Cambridge Univ. Press, Cambridge 1988) pp. 255–289 J. N. Latta: Computer-based analysis of holography using ray tracing, Appl. Opt. 10, 2698–2710 (1971) N. Lindlein, J. Schwider: Local wave fronts at diffractive elements, J. Opt. Soc. Am. A 10, 2563–2572 (1993) I. Ghozeil: Hartmann and other screen tests. In: Optical Shop Testing, ed. by D. Malacara (Wiley, New York 1978) pp. 323–349 Z. Chen, W. Yu, R. Ma, X. Deng, X. Liang: Uniform illumination of large targets using a lens array, Appl. Opt. 25, 377 (1986) N. Lindlein, F. Simon, J. Schwider: Simulation of micro-optical array systems with RAYTRACE, Opt. Eng. 37, 1809–1816 (1998) N. Lindlein: Simulation of micro-optical systems including microlens arrays, J. Opt. A: Pure Appl. Opt. 4, S1–S9 (2002) A. Büttner, U. D. Zeitner: Wave optical analysis of light-emitting diode beam shaping using microlens array, Opt. Eng. 41, 2393–2401 (2002) J. A. Kneisly: Local curvature of wavefronts in an optical system, J. Opt. Soc. Am. 54, 229–235 (1964) W. T. Welford: Aberrations of Optical Systems (Hilger, Bristol 1986) J. L. Rayces: Exact relation between wave aberration and ray aberration, Opt. Acta 11, 85–88 (1964) C.-J. Kim, R. R. Shannon: Catalog of Zernike Polynomials, Applied Optics and Optical Engineering, Vol. X, ed. by R. R. Shannon, J. C. Wyant (Academic, San Diego 1987) pp. 193–221 F. M. Küchel, Th. Schmieder, H. J. Tiziani: Beitrag zur Verwendung von Zernike–Polynomen bei der

Geometrical Optics

2.46

automatischen Interferenzstreifenauswertung, Optik 65, 123–142 (1983) W. N. Charman: Optics of the Eye, Handbook of Optics, Vol. I, 2nd edn., ed. by M. Bass (McGraw-Hill, New York 1995) pp. 24.3–24.54

2.47

2.48

References

W. S. Geisler, M. S. Banks: Visual Performance, Handbook of Optics, Vol. I, 2nd edn., ed. by M. Bass (McGraw-Hill, New York 1995) pp. 25.1–25.55 P. L. Kaufman, A. Alm: Adler’s Physiology of the Eye, 10th edn. (Mosby, St. Louis 2002)

Part A 2

Wave Optics

3. Wave Optics

can be performed very easily if some approximations of paraxial optics are valid. The formulae for this are treated in the last section of this chapter.

3.1

3.2

3.3

3.4

3.5

Maxwell’s Equations and the Wave Equation......................... 3.1.1 The Maxwell Equations ................. 3.1.2 The Complex Representation of Time-Harmonic Waves .............. 3.1.3 Material Equations ....................... 3.1.4 The Wave Equations ..................... 3.1.5 The Helmholtz Equations............... Polarization ......................................... 3.2.1 Different States of Polarization ...... 3.2.2 The Poincaré Sphere ..................... 3.2.3 Complex Representation of a Polarized Wave ...................... 3.2.4 Simple Polarizing Optical Elements and the Jones Calculus..................

88 88 94 95 98 99 102 105 105 106 106

Interference ......................................... 3.3.1 Interference of Two Plane Waves .... 3.3.2 Interference Effects for Plane Waves with Different Polarization States ... 3.3.3 Interference of Arbitrary Scalar Waves................................ 3.3.4 Some Basic Ideas of Interferometry

108 108

Diffraction ........................................... 3.4.1 The Angular Spectrum of Plane Waves ............................ 3.4.2 The Equivalence of the Rayleigh–Sommerfeld Diffraction Formula and the Angular Spectrum of Plane Waves ............................ 3.4.3 The Fresnel and the Fraunhofer Diffraction Integral ....................... 3.4.4 Numerical Implementation of the Different Diffraction Methods......... 3.4.5 The Influence of Polarization Effects to the Intensity Distribution Near the Focus .............................

123

111 115 119

123

125 126 135

138

Gaussian Beams ................................... 143 3.5.1 Derivation of the Basic Equations ... 143 3.5.2 The Fresnel Diffraction Integral and the Paraxial Helmholtz Equation .... 145

Part A 3

The quest to understand the nature of light is centuries old and today there can be at least three answers to the single question of what light is depending on the experiment used to investigate light’s nature: (i) light consists of rays that propagate, e.g., rectilinear in homogeneous media, (ii) light is an electromagnetic wave, (iii) light consists of small portions of energy, or so-called photons. The first property will be treated in the chapter about geometrical optics, which can be interpreted as a special case of wave optics for very small wavelengths. On the other hand, the interpretation as photons is unexplainable with wave optics and, above all, contradictory to wave optics. Only the theory of quantum mechanics and quantum field theory can simultaneously explain light as photons and electromagnetic waves. The field of optics treating this subject is generally called quantum optics. In this chapter about wave optics the electromagnetic property of light is treated and the basic equations describing all relevant electromagnetic phenomena are Maxwell’s equations. Starting with the Maxwell equations, the wave equation and the Helmholtz equation will be derived. Here, we will try to make a tradeoff between theoretical exactness and a practical approach. For an exact analysis see, e.g., [3.1]. After this, some basic properties of light waves like polarization, interference, and diffraction will be described. The propagation of coherent scalar waves is especially important in optics. Therefore, the section about diffraction will treat several propagation methods like the method of the angular spectrum of plane waves, which can be easily implemented on a computer, or the well-known diffraction integrals of Fresnel–Kirchhoff as well as Fresnel and Fraunhofer. In modern physics and engineering, lasers are very important and therefore the propagation of a coherent laser beam is of special interest. A good approximation for a laser beam is a Hermite–Gaussian mode and the propagation of a fundamental Gaussian beam

Part A

Basic Principles and Materials

3.5.3 Propagation of a Gaussian Beam.... 146 3.5.4 Higher-Order Modes of Gaussian Beams ....................... 147 3.5.5 Transformation of a Fundamental Gaussian Beam at a Lens .............. 151

3.5.6 ABCD Matrix Law for Gaussian Beams ...................... 152 3.5.7 Some Examples of the Propagation of Gaussian Beams ....................... 153 References .................................................. 154

3.1 Maxwell’s Equations and the Wave Equation The sources of the electric displacement D are the electric charges with density ρ.

3.1.1 The Maxwell Equations Part A 3.1

The well known Maxwell equations regarding electrodynamics [3.1] are the basis for our considerations and will be presented here in international SI units. The following physical quantities are used:

• • • • • •

E: electric (field) vector, unit [E] = 1 V/m D: electric displacement, unit [D] = 1 A s/m2 H: magnetic (field) vector, unit [H] = 1 A/m B: magnetic induction, unit [B] = 1 V s/m2 = 1 T j: electric current density, unit [ j] = 1 A/m2 ρ: electric charge density, unit [ρ] = 1 A s/m3

All quantities can be functions of the spatial coordinates with position vector r = (x, y, z) and time t. In the following, this explicit functionality is mostly omitted in the equations if it is clear from the context. The Maxwell equations are formulated in differential form by using the so-called nabla operator ⎛ ⎞ ∂ ⎜ ∂x ⎟ ⎜ ⎟ ⎜∂ ⎟ ⎟ (3.1) ∇ := ⎜ ⎜ ∂y ⎟ . ⎜ ⎟ ⎝∂ ⎠ ∂z The four Maxwell equations and the physical interpretation are ∂ B (r, t) ∇ × E (r, t) = − . (3.2) ∂t The vortices of the electric field E are caused by temporal variations of the magnetic induction B. ∂ D (r, t) + j (r, t) . (3.3) ∇ × H (r, t) = ∂t The vortices of the magnetic field H are either caused by an electric current with density j or by temporal variations of the electric displacement D. The quantity ∂ D/∂t is called the electric displacement current. ∇ · D (r, t) = ρ (r, t)

(3.4)

∇ · B (r, t) = 0

(3.5)

The magnetic field (induction) is solenoidal, i. e., there exist no “magnetic charges”. The Continuity Equation From (3.3) and (3.4) the conservation of the electric charge can be obtained by using the mathematical identity ∇ · (∇ × H) = 0,

∂ρ +∇ · j = 0 . ∂t

(3.6)

This equation is called the continuity equation of electrodynamics because it is analogous to the continuity equation of hydrodynamics. By integrating over a volume V with a closed surface A and applying Gauss’ theorem the following equation is obtained: ∂ρ dV = − ∇ · j dV = − j · dA . (3.7) ∂t V

Note that the integral V f dV always indicates a volume integral of a scalar

function f over the volume V , whereas the symbol A a · dA always indicates a surface integral of the vector function a over the closed surface A, which borders the volume V . The vector dA always points outwards from the closed surface. Therefore, the left side of (3.7) is the temporal variation of the total electric charge Q in the volume V and the right-hand side of (3.7) is the net electric current Inet (i. e., the current of positive charges flowing out of the surface plus the current of negative charges flowing in the surface minus the current of positive charges flowing in the surface minus the current of negative charges flowing out of the surface), which flows through the closed surface A ∂Q = −Inet . ∂t

(3.8)

Wave Optics

If the net current Inet is positive the total charge in the volume decreases during time, i. e., it becomes “more negative.” Energy Conservation in Electrodynamics From (3.2) and (3.3) a law of energy conservation of electrodynamics can be deduced by calculating the scalar product of E with (3.3) minus the scalar product of H with (3.2):

∇ · S = ∇ · (E × H) = − [E · (∇ × H) − H · (∇ × E)] . The quantity S S= E× H

(3.9)

is called the Poynting vector and has the physical unit of an intensity: [S] = 1 V A/m2 = 1 W/m2 , i. e., power per surface area. The Poynting vector arises due to the property of a cross product of two vectors perpendicular to both the electric and magnetic vector. Its absolute value describes the flow of energy per unit area and unit time through a surface perpendicular to the Poynting vector. It therefore describes the energy transport of the electromagnetic field. The sources of S are connected with temporal variations of the electric displacement, or of the magnetic induction, or with explicit electric currents. ∂B ∂D + E· j + H· . ∇ · S= − E· (3.10) ∂t ∂t In the next section it will be shown for the special case of an isotropic dielectric material that this equation can be interpreted as an equation of energy conservation.

where ε is the dielectric function of the material and µ the magnetic permeability. Both are functions of the position r. The dielectric constant of the vacuum ε0 = 8.8542 × 10−12 A s V−1 m−1 and the magnetic permeability of the vacuum µ0 = 4π × 10−7 V s A−1 m−1 are related to the speed of light in a vacuum c by: 1 c= √ ε0 µ0

(3.13)

with c = 2.99792458 × 108 m/s. In fact, the light speed in a vacuum is defined in the SI system as a fundamental constant of nature to exactly this value so that the basic unit of length (1 m) can be connected with the basic unit of time (1 s). The magnetic permeability of the vacuum is also defined in order to connect the basic SI unit of the electric current (1 A) with the mechanical basic SI units of mass (1 kg), length (1 m) and time (1 s). So, only the dielectric constant of the vacuum has to be determined by experiment, whereas c and µ0 are defined constants in the SI system. In dielectrics, by using (3.11) and (3.12), (3.10) is reduced to the following equation ∂E ∂H ∇ · S = − ε0 εE · + µ0 µH · ∂t ∂t 1∂ =− (3.14) (ε0 εE · E + µ0 µH · H) . 2 ∂t By integrating both sides of (3.14) over a volume V , which is bounded by a closed surface A (see Fig. 3.1), dA

Surface area A

Volume V

Energy Conservation in the Special Case of Isotropic Dielectric Materials In Sect. 3.1.3 we will see that isotropic dielectric materials are described with the following equations. The charge density and the electric currents are both zero

ρ = 0,

j =0.

(3.11)

Additionally, there are the following linear interrelations between the electric and magnetic quantities D (r, t) = ε0 ε (r) E (r, t) , B (r, t) = µ0 µ (r) H (r, t) ,

(3.12)

Fig. 3.1 Illustration of the quantities used for applying Gauss’ theorem. The surface A need not be the surface of a sphere with volume V , but can be an arbitrary closed surface. The small dotted vector symbolizes the infinitesimal surface vector dA, whereas the other vectors represent the vector field of the local Poynting vectors S at a fixed time

Part A 3.1

E · (∇ × H) − H · (∇ × E) ∂D ∂B = E· + E· j + H· . ∂t ∂t According to the rules of calculating using a nabla operator, the following equation is obtained:

3.1 Maxwell’s Equations and the Wave Equation

Part A

Basic Principles and Materials

Gauss’ theorem can be applied: Pnet := S· dA = ∇ · SdV A V 1 ∂ − = (ε0 εE · E + µ0 µH · H) dV 2 ∂t V ∂ 1 1 =− ε0 εE · E + µ0 µH · H dV ∂t 2 2 V ∂ w dV . (3.15) =− ∂t

Part A 3.1

The integral V ∇ · S dV symbolizes the volume integral

of ∇ · S over the volume V , whereas the integral A S· dA symbolizes the surface integral of the Poynting vector over the closed surface A. The vector dA in the integral always points outwards in the case of a closed surface. Therefore, Pnet is equal to the net amount of the electromagnetic power (difference between the power flowing out of the closed surface and the power flowing in the closed surface), which flows through the closed surface. So, a positive value of Pnet indicates that more energy flows out of the surface than into the system. Since the right-hand side of (3.15) must therefore also have the physical unit of power (unit 1 W = 1 J/s), it is clear that the quantity w :=

1 (ε0 εE · E + µ0 µH · H) = we + wm 2

(3.16)

is the energy density of the electromagnetic field in isotropic dielectric materials having the unit 1 J/m3 . The first term, 1 we = ε0 εE · E 2

(3.17)

is the electric energy density and the second term, 1 wm = µ0 µH · H 2

(3.18)

is the magnetic energy density. The negative sign on the right-hand side of (3.15) just indicates that the amount of energy in the volume decreases over time if the net amount of power Pnet flowing through the surface is positive, because this means that in total energy flows out of the closed surface. If the net electromagnetic power flow through the surface is zero, i. e., Pnet = 0, the total amount of electromagnetic energy V w dV in the volume is constant. This again shows that it is useful to interpret w as an energy density.

The Wave Equation in Homogeneous Dielectrics In this section the behavior of light in homogeneous dielectric materials will be discussed. In homogeneous materials the dielectric function ε and the magnetic permeability µ are both constants. A special case is the vacuum, where both constants are equal to 1 (ε = 1, µ = 1). The conclusion that electromagnetic waves can also exist in a vacuum without any matter was one of the most important discoveries in physics in the 19th century. In homogeneous dielectrics the Maxwell equations (3.2–3.5) can be simplified by using (3.11) and (3.12) with ε and µ being constant

∂ H (r, t) , ∂t ∂ E (r, t) , ∇ × H (r, t) = ε0 ε ∂t ∇ · E (r, t) = 0 , ∇ · H (r, t) = 0 . ∇ × E (r, t) = −µ0 µ

(3.19) (3.20) (3.21) (3.22)

These equations are completely symmetrical to a simultaneous replacement of E with H and ε0 ε with −µ0 µ. In the following, the vector identity ∇ × (∇ × E) = ∇ (∇ · E) − (∇ · ∇) E = ∇ (∇ · E) − ∆E

(3.23)

must be used. Thereby, the Laplacian operator ∆ = ∂ 2 /∂x 2 + ∂ 2 /∂y2 + ∂ 2 /∂z 2 has to be applied to each component of E! Equation (3.19) together with (3.21) then results in ∇ × (∇ × E) = −∆E = −µ0 µ∇ ×

∂H ∂t

∂ (3.24) (∇ × H) . ∂t Using (3.20) the wave equation for the electric vector in a homogeneous dielectric is obtained ∂ ∂E ε0 ε −∆E = − µ0 µ ∂t ∂t ∂2 E (3.25) ⇒ ∆E − ε0 µ0 εµ 2 = 0 . ∂t By using (3.13) this is usually written as = −µ0 µ

n2 ∂2 E =0. (3.26) c2 ∂t 2 The refractive index n of a homogeneous dielectric is defined as √ n = εµ . (3.27) ∆E −

Wave Optics

Because of the symmetry in E and H of the Maxwell equations (3.19–3.22) in homogeneous dielectrics, the same equation also holds for the magnetic vector ∆H −

n2 ∂2 H =0. c2 ∂t 2

3.1 Maxwell’s Equations and the Wave Equation

t = t0 +∆t

t = t0

(3.28) v∆ t e

Plane Waves in Homogeneous Dielectrics By defining the so-called phase velocity c v= (3.29) n a solution of (3.26) or (3.28) is r

(3.30) (3.31) O

This can be seen by using u := e · r ∓ vt = ex x + e y y + ez z ∓ vt e2x + e2y + e2z

so that it holds ∆E =

∂2 ∂x 2

∂2 ∂y2

∂2 ∂z 2

⇒

e · r = vt .

e · r = vt0

Fig. 3.2 The plane surfaces of constant optical path in the (3.32)

⎛

⎞ Ex ⎜ ⎟ ⎝ Ey ⎠ Ez

∂ 2 f (u) ∂ 2 f (u) = e2x + e2y + e2z = , ∂u 2 ∂u 2 ∂2 E ∂ 2 f (u) = v2 , 2 ∂t ∂u 2 and thus the same is valid for H in (3.31). The quantity nu has the physical unit of a path and the first term ne · r is called the optical path difference OPDbecause it is the product of the geometrical path and the refractive index n. A solution of type (3.30) or (3.31) is called a plane wave because for the following reason. The value of E remains constant for a constant argument u = u 0 defined by (3.32). The same is valid for H. Now, if we consider e.g., u = u 0 = 0 and take the negative sign in (3.32) we obtain: u=0

Part A 3.1

E(r, t) = f (e · r ∓ vt) , H(r, t) = g(e · r ∓ vt) .

with

case of a plane wave. O is the origin of the coordinate system and the dashed lines indicate the planes at two different times for the fixed value u = 0

unit vector e is perpendicular to the planes and points in the direction of propagation if the negative sign is used in (3.32) (as done here and in the following) and points in the opposite direction if the positive sign is used. The Orthogonality Condition for Plane Waves in Homogeneous Dielectrics The Maxwell equations (3.19–3.22) do not allow all orientations of the electric and magnetic vector relative to the propagation direction e of a plane wave. Equations (3.30) and (3.32) deliver

∂f ∂E = ∓v ∂t ∂u and ∂ fy ∂ fz ∂E z ∂E y − = ey − ez ∂y ∂z ∂u ∂u ∂f ∂f . = e× ⇒ ∇ × E = e× ∂u x ∂u

(∇ × E)x =

(3.33)

The geometrical path at time t = 0 must then also be zero and this means that the plane then passes through the origin of the coordinate system. For a fixed value of t0 , (3.33) describes a plane surface in space (see Fig. 3.2) at a distance vt0 from the origin. At the time t0 + ∆t it again describes a plane surface parallel to the plane at t = 0 but with a distance v(t0 + ∆t) from the origin. The

An analogous expression is valid for H. Therefore, the Maxwell equations (3.19) and (3.20) deliver ∂f ∂g = ±µ0 µv , ∂u ∂u ∂g ∂f = ∓ε0 εv . e× ∂u ∂u

e×

(3.34) (3.35)

Part A

Basic Principles and Materials

These equations can be integrated with respect to the variable u and by setting the integration constant to zero and using (3.13), (3.27) and (3.29) the result is µ0 µ E= f =∓ e× H , (3.36) ε0 ε ε0 ε e× E . H=g=± (3.37) µ0 µ These two equations show that E is perpendicular to e and H and that H is perpendicular to e and E. This can only be the case if e, E and H form an orthogonal triad of vectors. Therefore, a plane wave in a homogeneous dielectric is always a transversal wave.

Part A 3.1

The Poynting Vector of a Plane Wave In this section, the physical interpretation of the Poynting vector will be illustrated for plane waves. The Poynting vector defined in (3.9) is parallel to e µ0 µ ε0 ε e× H × ± e× E S= E× H = ∓ ε0 ε µ0 µ = − [(e × H) · E] e + [(e × H) · e] E . (3.38)

The second scalar product is zero so that only the first term remains. By using (3.36) this finally results in ε0 ε S = − [(e × H) · E] e = ± (E · E) e . (3.39) µ0 µ This means that the energy transport is along the direction of propagation of the plane wave and that the absolute value of the Poynting vector, i. e., the intensity of the light wave, is proportional to |E|2 . By using the vector identity (a × b) · c = − (a × c) · b and (3.37) we can also write S = − [(e × H) · E] e = [(e × E) · H] e µ0 µ =± (3.40) (H · H) e . ε0 ε This means that the absolute value of the Poynting vector is also proportional to |H|2 and that the following equality holds: µ0 µ ε0 ε |H|2 = |E|2 ε0 ε µ0 µ ⇒

µ0 µ |H| = ε0 ε |E| . 2

(3.41)

Comparing this with (3.16), (3.17) and (3.18) for the energy density of the electromagnetic field we have for a plane wave in a homogeneous dielectric 1 we = wm = w 2 (3.42) ⇒ w = ε0 ε |E|2 = µ0 µ |H|2 .

By again using (3.13), (3.27) and (3.29) the equations (3.39) and (3.40) can be transformed to S = ±vµ0 µ |H|2 e = ±vε0 ε |E|2 e = ±vwe . (3.43)

This means that the absolute value of the Poynting vector is, in a homogeneous dielectric, the product of the energy density (energy per volume) of the electromagnetic field and the phase velocity of light. This confirms the interpretation of the Poynting vector as being the vector of the electromagnetic wave transporting the energy of the electromagnetic field with the phase velocity of light. Figure 3.3 illustrates this. In the infinitesimal time interval dt the light covers the infinitesimal distance dz = v dt. We assume that the distance dz is so small that the local energy density w of the electromagnetic field is constant in the volume dV = A dz, whereby A is the area of a small surface perpendicular to the Poynting vector. Therefore, all the energy dW = w dV that is contained in the infinitesimal volume dV passes the surface area A in the time dt and we have, for the intensity I (electromagnetic power per area), I=

dW w dV wA v dt = = = wv = |S| . (3.44) A dt A dt A dt

This is exactly the absolute value of the Poynting vector S. The light intensity on a surface, which is not perpendicular to the direction of the Poynting vector is calculated by the equation I = S· N ,

(3.45)

where N is a local unit vector perpendicular to the surface.

A S

Fig. 3.3 Illustration of the Poynting vector S as transporting the energy of the electromagnetic field. A is the area of the circular surface

Wave Optics

Here, we have introduced the value λ, which has the physical unit of a length so that the argument of the cosine function has no physical unit. Its meaning will soon become clear. The characteristic property of a time-harmonic wave is that it has, for a fixed point r, periodically the same value after a certain time interval. The smallest time interval for which this is the case is called period T : 2πn E (r, t + T ) = E0 cos [e · r − v (t + T )] λ

2πn = E0 cos (e · r − vt) = E (r, t) λ 2πn ⇒ vT = 2π λ λ or cT = λ . ⇒ vT = (3.48) n Therefore, λ/n is the distance the light covers in the material in the period T and is called the wavelength of the harmonic wave in the material. The quantity λ itself is the wavelength in a vacuum. The reciprocal of T is called the frequency ν of the wave and the term 2πν = 2π/T is called the angular frequency ω of the

wave. Therefore the two following equations are valid cT = λ ⇒ c = λν , 2π 2π c= = 2πν = ω . λ T

(3.49) (3.50)

Additionally, we introduce the wave vector k, which is defined by k=

2πn e. λ

(3.51)

Then (3.46) and (3.47) for E and H can be written as E (r, t) = E0 cos (k · r − ωt) , H (r, t) = H0 cos (k · r − ωt) .

(3.52) (3.53)

Because of the orthogonality condition k (or e, which is parallel to k), E0 and H0 have to form an orthogonal triad. This can be explicitly seen in this case by using Maxwell’s first equation (3.19) in a homogeneous dielectric and the mathematical rules for the nabla operator ∇ × E = ∇ × [E0 cos (k · r − ωt)] = [∇ cos (k · r − ωt)] × E0 = −k × E0 sin (k · r − ωt) , ∂H = −ωµ0 µH0 sin (k · r − ωt) ; ∂t ωµ0 µH0 = k × E0 1 λ k × E0 = k × E0 H0 = ωµ0 µ 2πcµ0 µ ε0 ε e × E0 . (3.54) = µ0 µ

−µ0 µ ⇒ ⇒

In the last step (3.50), (3.13) and (3.27) are used and the geometrical interpretation of the result is that H0 is perpendicular to both e and E0 . The third Maxwell equation (3.21) delivers ∇ · E = ∇ · [E0 cos (k · r − ωt)] = E0 · [∇ cos (k · r − ωt)] = −E0 · k sin (k · r − ωt) = 0 ; ⇒ E0 · k = 0 .

(3.55)

This also means that k (or e) and E0 are perpendicular to each other. The two other Maxwell equations (3.20) and (3.22) are automatically fulfilled because of the symmetry in E and H.

Part A 3.1

A Time-Harmonic Plane Wave Up to now a plane wave was defined with (3.30) and (3.31) to be E(r, t) = f (u) and H(r, t) = g(u). The argument u is defined by (3.32) to be u = e · r − vt. This means that all points with position vector r at a fixed point t in time lie on a plane surface for a constant value u. Additionally, we saw that e, E and H have to form an orthogonal triad of vectors (3.36, 3.37). But the concrete form of the functions f and g can be quite arbitrary to fulfill these conditions. A wave which is very important in optics because of its simple form is a time-harmonic wave. Furthermore, it should be linearly polarized, i. e., the direction of the electric and magnetic vector are both constant. A linearly polarized time-harmonic plane wave is represented by the equations: 2πn E (r, t) = E0 cos u λ

2πn (3.46) = E0 cos (e · r − vt) , λ 2πn u H (r, t) = H0 cos λ

2πn (3.47) = H0 cos (e · r − vt) . λ

3.1 Maxwell’s Equations and the Wave Equation

Part A

Basic Principles and Materials

3.1.2 The Complex Representation of Time-Harmonic Waves

Part A 3.1

In paragraph “A Time harmonic Plane Wave” of Sect. 3.1.1 a linearly polarized time-harmonic plane wave is expressed with real cosine functions for the electric and the magnetic vector. Because E and H are observable physical quantities, they must, of course, be expressed by real functions. The fact that usual detectors are not fast enough to detect the electric and magnetic vector of light waves directly does not matter here. Nevertheless, the calculation with a complex exponential function is more convenient than the calculation with real cosine or sine functions. Now, the Maxwell equations (3.2–3.5) are linear. Therefore, if the functions E1 , D1 , H1 , B1 , j1 and ρ1 on the one hand and E2 , D2 , H2 , B2 , j2 and ρ2 on the other are both solutions of the Maxwell equations, then a linear combination of these functions is also a solution: ∂ B1 ∇ × E1 = − , ∂t ∂ D1 + j1 , ∇ × H1 = ∂t ∇ · D1 = ρ1 , ∇ · B1 = 0 , ∂ B2 , ∇ × E2 = − ∂t ∂ D2 + j2 , ∇ × H2 = ∂t ∇ · D2 = ρ2 , ∇ · B2 = 0 ; ⇒

z 2 (x) = a2 (x) + ib2 (x) Re (z 1 + z 2 ) = Re (z 1 ) + Re (z 2 )

⇒

(3.58) (3.59)

where α and β are arbitrary real or complex constants. The Euler equation delivers:

Re (z 1 z 1 ) = a12 − b21 = a12 + b21 = |z 1 |2 .

∗ eix + eix eix + e−ix = . 2 2

(3.60)

(3.61)

(3.63)

So, if the Poynting vector or products of the electric or magnetic vectors have to be calculated it is not allowed to just take the complex functions. Nevertheless, there are some useful applications of the complex notation. As we mentioned before, the frequency of a light wave is so high that no usual detector can directly measure the vibrations. For a typical wavelength of visible light of 500 nm the frequency of a wave in vacuum is determined according to (3.49) ν=

cos x =

(3.62)

Re (z 1 z 2 ) = a1 a2 − b1 b2 = a1 a2 = Re (z 1 ) Re (z 2 ) , a1 a2 + b1 b2 a1 Re (z 1 ) z1 = , = = Re 2 2 z2 a Re (z 2 ) a2 + b2 2

(3.57)

Re (z 1 − z 2 ) = Re (z 1 ) − Re (z 2 ) d dz 1 Re = Re (z 1 ) dx dx Re z 1 dx = Re (z 1 ) dx

Here, f is an arbitrary real function or constant. Only if two complex functions have to be multiplied or divided, or the absolute value has to be built, do we have to be careful:

∂ (αB1 + β B2 ) , (3.56) ∂t ∂ (αD1 + β D2 ) + α j1 + β j2 , ∇ × (αH1 + β H2 ) = ∂t

eix = cos x + i sin x

z 1 (x) = a1 (x) + ib1 (x)

Re ( fz 1 ) = f Re (z 1 )

∇ × (αE1 + β E2 ) = −

∇ · (αD1 + β D2 ) = αρ1 + βρ2 , ∇ · (αB1 + β B2 ) = 0 ,

Due to the linearity of the Maxwell equations it is obvious that if a function containing a cosine function is a solution of the Maxwell equations the replacement of the cosine function by a complex exponential function will then also be a solution. Therefore, it is quite normal that waves are expressed by using a complex function although only the real part of this function represents the real physical quantity. The addition, subtraction, integration and differentiation of such a complex function is also a linear operation, so that we can in the end build the real part and have the real solution:

⇒

c 2.998 × 108 m s−1 = = 5.996 × 1014 s−1 , λ 5.0 × 10−7 m T=

1 = 1.668 × 10−15 s = 1.668 fs . ν

Wave Optics

A x , A y , A z , Φx , Φ y and Φz are all real functions that depend only on the position r. Additionally, A x , A y and A z , which are called the components of the amplitude, are slowly varying functions of the position. On the other hand, the exponential terms with the components of the phase Φx , Φ y and Φz are rapidly varying functions of ˆ the position. The components of the complex vector E(r) are often called the complex amplitudes of the electric vector of the wave. Equation (3.39) gives the relation between the Poynting vector and the electric vector in a homogeneous dielectric ε0 ε S= (E · E) e . µ0 µ Now, the time average S¯ of the absolute value of the Poynting vector, i. e., the intensity, which is really measured by a common detector, will be calculated for the general time-harmonic wave. Therefore, we have to integrate the absolute value S of the Poynting vector over one period T and divide it by T S¯ (r) :=

1 T

T |S (r, t)| dt

ε0 ε 1 µ0 µ T

T E (r, t) · E (r, t) dt . 0

Using (3.64) for a general time-harmonic wave and (3.13) we obtain S¯ =

ε0 c T

ε µ

T 0

A2x cos2 (Φx − ωt) + A2y cos2 (Φ y − ωt)

+ A2z cos2 (Φz − ωt) dt ε ε0 c 2 A x + A2y + A2z . = µ 2

(3.65)

(3.66)

But, if we calculate directly the square of the absolute ˆ we also obtain value of the time-independent vector E ! !2 !ˆ! ˆ ·E ˆ ∗ = A2x + A2y + A2z . (3.67) ! E! = E By combining equations (3.66) and (3.67) we finally obtain !2 ε ε c " #∗ ε ε0 c !! 0 ˆ ˆ (r)!! = ˆ (r) . E (r) · E S¯ (r) = !E µ 2 µ 2 (3.68)

Therefore, the complex representation of time-harmonic waves allows a fast calculation of the time average of the Poynting vector, i. e., of the intensity of the light wave.

3.1.3 Material Equations In the last two sections we often concentrated on the electromagnetic field in an isotropic and homogeneous dielectric material where the Maxwell equations are simplified to (3.19–3.22). In other materials the general Maxwell equations (3.2–3.5) have to be used and more complex interrelations between the electric displacement and the electric vector on one hand and the magnetic induction and the magnetic vector on the other have to be found. Since the atomic distances are small compared to the wavelength of light, a macroscopic description with smooth functions is possible. To calculate the influence of the material, first of all the interrelations between D and E as well as B and H are considered in vacuum. These equations in a vacuum are obtained from (3.12) for the case µ = ε = 1. Then, additional terms are added to the equations in a vacuum. The electric polarization P and the magnetization M are introduced by: D (r, t) = ε0 E (r, t) + P (r, t) , B (r, t) = µ0 H (r, t) + M (r, t) .

(3.69) (3.70)

The atomic theory goes far beyond our scope. But in a macroscopic theory the effect of the atoms (i. e., mainly

Part A 3.1

So, the period is just a little bit more than a femtosecond. This means that in most cases only the time average of the light intensity over many periods is measured. A general time-harmonic wave with the angular frequency ω has the representation ⎞ ⎛ A x (r) cos [Φx (r) − ωt] ⎜ ⎟ ⎟ E (r, t) = ⎜ ⎝ A y (r) cos Φ y (r) − ωt ⎠ Az (r) cos [Φz (r) − ωt] ⎧⎛ ⎞⎫ iΦx (r) − iωt ⎪ ⎪ A e (r) x ⎪ ⎪ ⎨⎜ ⎟⎬ − iωt iΦ (r) ⎜ ⎟ y = Re ⎝ A y (r) e ⎠⎪ ⎪ ⎪ ⎪ ⎩ ⎭ A z (r) eiΦz (r) − iωt ⎧ ⎛ ⎞⎫ iΦx (r) ⎪ ⎪ A e (r) x ⎪ ⎪ ⎨ ⎜ ⎟⎬ −iωt iΦ (r) ⎜ ⎟ y = Re e ⎝ A y (r) e ⎠⎪ ⎪ ⎪ ⎪ ⎩ ⎭ A z (r) eiΦz (r) ˆ (r) . (3.64) =: Re e−iωt E

3.1 Maxwell’s Equations and the Wave Equation

Part A

Basic Principles and Materials

the electrons of the atoms) on the electric polarization is that it is a function of the electric vector. In the same way the magnetization is a function of the magnetic vector. The most general equations are

Pi (r, t) = P0 (r, t) + ε0

3 $

(1)

ηij (r, t) E j (r, t)

j=1 3 3 $ $

(2)

ηijk (r, t) E j (r, t) E k (r, t)

j=1 k=1

Part A 3.1

3 $ 3 3 $ $ (3) ηijkl (r, t)E j (r, t)E k(r, t)El(r, t)+. . . , + j=1 k=1 l=1

(3.71)

Mi (r, t) = M0 (r, t) + µ0

3 $

(1)

χij (r, t) H j (r, t)

j=1 3 3 $ $

(2)

χijk (r, t) H j (r, t) Hk (r, t)

j=1 k=1 3 $ 3 3 $ $ (3) χijkl (r, t)H j (r, t)Hk (r, t)Hl (r, t)+. . . . + j=1 k=1 l=1

(3.72)

Here, the lower indices running from 1 to 3 indicate the components of the respective electromagnetic vectors, e.g., E 1 := E x , E 2 := E y and E 3 := E z . The tensor func(1) (2) tions ηij , ηijk and so on describe the influence of the electric vector on the electric polarization and the same (1) (2) is valid for the tensor functions χij , χijk and so on in the magnetic case. The tensor functions are defined here (1) (1) in such a way that ηij and χij have no physical unit and are pure numbers. Nevertheless, the tensor functions of higher degree have different physical units. The ten(1) (2) sors ηij , ηijk and so on are called the tensors of dielectric (1) (2) susceptibility. The tensors χij , χijk and so on are called the tensors of magnetic susceptibility. In (3.71) a bias P0 for the polarization is also assumed and the same is made for the magnetization. In our general material equations, the different terms can depend on the position r as well as on the time t. But in most cases the material functions will not depend explicitly on the time t. Additionally, there have to be equations for the current density j and the charge density ρ. In optics the most important materials are either dielectrics or metals (e.g., for mirrors). In both cases we can assume ρ = 0. For the current density we can in most cases take the

equation: j = σE .

(3.73)

The conductivity σ indicates how good an electric current is conducted in a material and has the physical unit [σ] = 1 A V−1 m−1 . For ideal dielectric materials σ is zero so that we obtain j = 0. In this case the material does not absorb light. For metals, σ is, of course, not zero and for an ideal conductor it would become infinity, so that all light would be absorbed or reflected at once. There are also anisotropic absorbing materials like special crystals where σ is not a scalar but a tensor [3.1]. But this is out of our present scope. Discussion of the General Equations %3 Material (1) Polarization. The term j=1 ηij E j in (3.71) is re-

sponsible for linear responses of the electric polarization on the electric vector and is the most important effect. The following terms and the bias term are responsible for so-called nonlinear effects and are subject to nonlinear optics [3.2] (e.g., second-harmonic generation or self-focusing effects). In the following, the bias and all tensors with upper index (2) and more of the dielectric (2) (3) susceptibility ηijk , ηijkl , . . . will be set to zero because only linear optics will be treated in this chapter. In “normal” materials like different glass types the linear case is the normal case. Only if the absolute value of the electric vector of the electromagnetic field is in the range of the atomic electric field nonlinear effects occur in these materials. An estimation of the electric fields in atoms and in a light wave is helpful. In a typical atom the electric field on an outer electron can be estimated by applying Coulomb’s law and assuming an effective charge of the nucleus of one elementary charge and a distance of the electron of r = 10−10 m e E = |E| = , 4πε0r 2 r = 10−10 m, e = 1.6022 × 10−19 A s , ε0 = 8.8542 × 10−12 As V−1 m−1 ⇒

E ≈ 1.4 × 1011 V m−1 .

(3.74)

The electric field oscillates very rapidly in a light wave. Therefore, to estimate the electric field in a light ˆ of the wave (here in a vacuum) the amplitude | E| modulus of the time-independent complex-valued electric field is calculated. This can be done by using (3.68) for the relation between the time average S¯ of the modulus of the Poynting vector and the modulus of the time-independent complex-valued electric

Wave Optics

field. Here, the values are calculated in a vacuum (ε = µ = 1): & ! ! 2 S¯ cε0 !! !!2 ! ! ⇒ ! Eˆ ! = . (3.75) S¯ = ! Eˆ ! 2 cε0 The result for the electric field of the light on a sunny day is: ! ! ! ! S¯ ≈ 1 kW/m2 ⇒ ! Eˆ ! ≈ 868 V/m . In the focused spot of a medium-power continuous wave (cw) laser beam we have e.g.,

This shows that in normal materials and with “normal” light intensities the electric field of a light wave is quite small compared to the electric field of the atoms. Therefore, the electrons are only moved a little bit and this results normally in a linear response of the dielectric function to the exciting electric field of the light wave. Of course, there are also so-called nonlinear materials which show for smaller electric fields nonlinear effects. In addition, ultrashort pulsed lasers, e.g., so-called femtosecond lasers, can achieve much higher intensities in their focus so that electric fields, which are comparable to or higher than the electric field in atoms, result. Then the response is, of course, nonlinear. Magnetization. In practice, there are nearly no materials relevant to optics that show nonlinear magnetic (2) effects, so that χijk and all higher order tensors are zero. In fact, most optically interesting materials are not magnetic at all, so that the remaining tensor of the magnetic (1) susceptibility χij is also zero. In some materials the (1) magnetic susceptibility χij is not zero but it can be written as a scalar constant χ multiplied by a 3 × 3 unit matrix. χ is a negative constant for diamagnetic materials or a positive constant for paramagnetic materials. The magnetic permeability µ of the material, which is a pure real number without a physical unit, is then defined as

µ := 1 + χ .

(3.76)

Then we have, due to (3.70) and (3.72) B = µµ0 H .

(3.77)

This equation, which is also used in (3.12) will be used in the rest of this chapter and in many cases µ will really be a constant that does not depend on the position r.

Specialization to the Equations of Linear and Nonmagnetic Materials For linear materials, only the tensor of lowest degree of (1) the dielectric susceptibility ηij is different from zero. It can be expressed as a matrix

⎛ (1) (1) (1) ⎞ η η η ⎜ 11 12 13 ⎟ ⎜ (1) (1) (1) ⎟ ⎜η21 η22 η23 ⎟ . ⎝ ⎠ (1) (1) (1) η31 η32 η33

(3.78)

The dielectric tensor is defined as ⎛ ⎞ ⎛ ⎞ (1) (1) (1) ε11 ε12 ε13 η13 1 + η11 η12 ⎜ ⎟ ⎜ (1) (1) ⎟ ⎝ ε21 ε22 ε23 ⎠ := ⎝ η(1) 1 + η22 η23 ⎠ . 21 (1) (1) (1) ε31 ε32 ε33 η31 η32 1 + η33 (3.79)

Using (3.69) and (3.71) the relation between the dielectric displacement and the electric vector is ⎞ ⎛ ⎞⎛ ⎞ ε11 ε12 ε13 Ex Dx ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎝ D y ⎠ = ε0 ⎝ ε21 ε22 ε23 ⎠ ⎝ E y ⎠ . Dz ε31 ε32 ε33 Ez ⎛

(3.80)

In anisotropic materials like noncubic crystals or originally isotropic materials that are subject to mechanical stresses, the dielectric tensor has this general matrix form and the effects that occur are e.g., birefringence [3.1,3]. It can be shown that the dielectric tensor is symmetric, i. e., εij = ε ji . But the treatment of anisotropic materials is out of the scope of this chapter so that we will have in the following only isotropic materials. Then the dielectric tensor reduces to a scalar ε times a unit matrix whereby ε is in general a function of the position r. Material Equations for Linear and Isotropic Materials If the material is isotropic the dielectric tensor and all other material quantities are scalars times a unit matrix. Due to (3.80) and (3.77) we have, in this case, the wellknown relations between the electric displacement and the electric vector on one hand and between the magnetic induction and the magnetic vector on the other, which we also used in (3.12):

D (r, t) = ε0 ε (r) E (r, t) , B (r, t) = µ0 µ (r) H (r, t) ,

(3.81)

Part A 3.1

S¯ = 1 W/µm2 = 1012 W/m2 ! ! ! ! ⇒ ! Eˆ ! ≈ 2.74 × 107 V/m .

3.1 Maxwell’s Equations and the Wave Equation

Part A

Basic Principles and Materials

where ε(r), µ(r) means that the material functions will in general depend on the position. An explicit dependence on time is most often not the case so it is omitted here. Additionally, we assume that the charge density is zero and (3.73) is valid ρ=0, j (r, t) = σ (r) E (r, t) .

(3.82) (3.83)

Part A 3.1

If ε, µ and σ are constant, i. e., independent of the position, the material is called homogeneous. Due to the dispersion theory, which will not be treated here, the material functions will, in general, depend on the frequency of the stimulating electric or magnetic fields. Therefore, the Fourier components of the electric and magnetic field have to be calculated and treated separately. The electric vector and the electric displacement are written as Fourier integrals, i. e., as a superposition of time-harmonic waves with the angular frequency ω: 1 E (r, t) = √ 2π 1 D (r, t) = √ 2π

+∞ −∞ +∞

˜ (r, ω) e−iωt dω , E ˜ (r, ω) e−iωt dω . D

(3.84)

−∞

The magnetic vector and the magnetic induction are treated in the same way so that we can omit this. If the ˜ is calculated by function E is given E 1 ˜ (r, ω) = √ E 2π

+∞

E (r, t) eiωt dt .

(3.85)

−∞

Since E is a real function the complex Fourier components have to fulfill the condition ˜ (r, −ω) = E ˜ ∗ (r, ω) . E

3.1.4 The Wave Equations The Maxwell equations (3.2–3.5) contain the five vector fields E, D, H, B and j and the scalar field ρ. These quantities are related to each other by the material equations. Here, only the case of isotropic, linear and uncharged (ρ = 0) materials will be treated. Additionally, all material parameters like ε, µ and σ will be independent of the time t, but functions of the position r (and the frequency or wavelength of the light). In the following the explicit dependence of the functions on r and t will be omitted, but there are the following functionalities: E (r, t), H (r, t), µ (r), ε (r), σ (r). Using (3.81) and (3.83) for this case results in the following specialized Maxwell equations ∂H , ∂t ∂E +σE , ∇ × H = ε0 ε ∂t ∇ · (εE) = 0 , ∇ · (µH) = 0 . ∇ × E = −µ0 µ

(3.88) (3.89) (3.90) (3.91)

These equations contain, for given material functions ε, µ and σ, only the electric and the magnetic vector. To eliminate the magnetic vector the cross product of the nabla operator with (3.88) is taken ∂H ∇ × (∇ × E) = −µ0 ∇ × µ (3.92) ∂t ∂ ∂H . = −µ0 µ (∇ × H) − µ0 (∇µ) × ∂t ∂t By using (3.88), (3.89) and the nabla operator calculus for a double cross product, this results in

(3.86)

The same is valid for the electric displacement, the magnetic vector and the magnetic induction. In total, the material equations can be written for isotropic and linear materials with the Fourier components of the four electromagnetic vector quantities: ˜ (r, ω) = ε0 ε (r, ω) E ˜ (r, ω) D ˜ (r, ω) . ˜ (r, ω) = µ0 µ (r, ω) H B

is equivalent to just deal with time-harmonic waves of a certain angular frequency ω, where the quantities ε and µ are functions of ω.

(3.87)

In the following the tilde on the different quantities will mostly be omitted to simplify the notation. This

∂2 E ∂E − µ0 µσ ∂t ∂t 2 + [∇ (ln µ)] × (∇ × E) . (3.93)

∇ (∇ · E) − ∆E = − ε0 µ0 εµ

Here, ∆ = ∇ · ∇ is the Laplacian operator, which has to be applied on each component of E. Equation (3.90) can be used to eliminate the term ∇ · E from (3.93) ∇ · (εE) = E · ∇ε + ε∇ · E = 0 ; 1 ⇒ ∇ · E = − E · ∇ε = −E · ∇ (ln ε) . ε

Wave Optics

So, (3.93) gives the final so-called wave equation for the electric vector E εµ ∂ 2 E ∂E − µ0 µσ ∂t c2 ∂t 2 (3.94) + [∇ (ln µ)] × (∇ × E) = 0 .

∆E + ∇ [E · ∇ (ln ε)] −

Additionally, (3.13) was used to replace ε0 µ0 by 1/c2 . An analogue equation for the magnetic vector can be found using (3.88), (3.89) and (3.91): ∇ × (∇ × H) = ∇ (∇ · H) −∆H ' () *

3.1 Maxwell’s Equations and the Wave Equation

Wave Equations for Pure Dielectrics If the material is a pure dielectric the conductivity σ, which is responsible for absorption is zero. Then (3.94) and (3.96) reduce to:

εµ ∂ 2 E c2 ∂t 2 + [∇ (ln µ)] × (∇ × E) = 0 ,

(3.97)

εµ ∂ 2 H ∆H+∇ [H · ∇ (ln µ)] − 2 c ∂t 2 + [∇ (ln ε)] × (∇ × H) = 0 .

(3.98)

∆E+∇ [E · ∇ (ln ε)] −

Here, the equations are really symmetrical for a replacement of E with H and ε with µ. In practice, there are, of course, no materials that are completely transparent to light. But in the visible or infrared region most glasses can be assumed with a good approximation to be dielectrics.

∂2 H + ε0 (∇ε) ∂t 2 ∂H ∂E − µ0 µσ + (∇σ) × E . × ∂t ∂t

= −ε0 µ0 εµ

(3.95)

Using (3.89) again, this equation can be resolved with respect to ∂ E/∂t: ∂E +σE ∂t 1 ∂E = (∇ × H − σ E) . ∂t ε0 ε

∇ × H = ε0 ε ⇒

Then, ∂ E/∂t can be eliminated in (3.95) resulting in: ∆H + ∇ [H · ∇ (ln µ)] −

εµ ∂ 2 H c2 ∂t 2

∂H + [∇ (ln ε)] × (∇ × H) ∂t + [∇σ − σ∇ (ln ε)] × E = 0 .

− µ0 µσ

Wave Equations for Homogeneous Materials The second interesting special case is for homogeneous materials. Here, ε, µ and σ are constants that do not depend on r and their gradients are zero. In this case (3.94) and (3.96) reduce to

εµ ∂ 2 E ∂E =0, − µ0 µσ 2 2 ∂t c ∂t εµ ∂ 2 H ∂H =0. ∆H − 2 − µ0 µσ 2 ∂t c ∂t ∆E −

(3.99) (3.100)

These equations are symmetrical to a replacement of E with H. In practice, homogeneous materials are the most important case because all conventional lenses (with the exception of graded index lenses, or GRIN lenses) are made of homogeneous glasses or at least of glasses with a very small inhomogeneity.

3.1.5 The Helmholtz Equations (3.96)

Unfortunately, it is not possible to completely eliminate the electric vector from this wave equation of the magnetic field. Equations (3.94) and (3.96) are nearly symmetrical for a replacement of E with H and ε with µ. Only the terms containing the conductivity σ are not symmetrical. Nevertheless, there are two important special cases that provide symmetries of the wave equations of the electric and magnetic vector.

Assume a time-harmonic wave with angular frequency ω represented in its complex notation (3.64): ˆ (r) e−iωt , E (r, t) = E ˆ (r) e−iωt . H (r, t) = H

(3.101) (3.102)

As long as linear operations are made using this complex representation is permitted. The partial derivatives with respect to t can be calculated directly where again the

Part A 3.1

=−H·∇(ln µ)

= −∇ [H · ∇ (ln µ)] − ∆H ∂E + ∇ × (σ E) = ε0 ∇ × ε ∂t ∂E ∂E + ε0 (∇ε) × = ε0 ε∇ × ∂t ∂t + σ∇ × E + (∇σ) × E

100

Part A

Basic Principles and Materials

functionalities are omitted in our notation: ∂E ∂t ∂H ∂t ∂2 E ∂t 2 ∂2 H ∂t 2

simple form εµ ˆ + iωµ0 µσ E ˆ =0, E c2 εµ ˆ + ω2 ˆ =0. ˆ + iωµ0 µσ H ∆H H c2 ˆ + ω2 ∆E

ˆ −iωt , = −iω Ee ˆ −iωt , = −iω He ˆ −iωt , = −ω2 Ee ˆ −iωt . = −ω2 He

Part A 3.1

These equations can be used in the wave equations (3.94) and (3.96) for linear and isotropic materials. The result is " # εµ ˆ +∇ E ˆ ˆ + iωµ0 µσ E ˆ · ∇ (ln ε) + ω2 ∆E E c2 ˆ =0, + [∇ (ln µ)] × ∇ × E (3.103) " # εµ ˆ +∇ H ˆ ˆ + iωµ0 µσ H ˆ · ∇ (ln µ) + ω2 ∆H H c2 ˆ + [∇σ − σ∇ (ln ε)] × E ˆ =0. +[∇ (ln ε)] × ∇ × H (3.104)

These two time-independent equations are called the Helmholtz equations for the electric and the magnetic vector. Since only the position-dependent complex elecˆ and H ˆ are used, only tric and magnetic vectors E time-averaged quantities can be calculated using the Helmholtz equations. Again two special cases are of interest. Helmholtz Equations for Pure Dielectrics For pure dielectric materials the conductivity is zero (σ = 0). In this case (3.103) and (3.104) result in " # εµ ˆ +∇ E ˆ ˆ · ∇ (ln ε) + ω2 ∆E E c2 ˆ =0, + [∇ (ln µ)] × ∇ × E (3.105) " # εµ ˆ ˆ +∇ H ˆ · ∇ (ln µ) + ω2 H ∆H c2 ˆ =0. + [∇ (ln ε)] × ∇ × H (3.106)

Again, both equations are symmetric with regards to ˆ with H ˆ and ε with µ. a replacement of E Helmholtz Equations for Homogeneous Materials For homogeneous materials the gradients of ε, µ and σ are zero. In this case (3.103) and (3.104) obtain a quite

(3.107) (3.108)

ˆ and H. ˆ Both equations are completely symmetric in E The angular frequency ω is defined as 2πν and the frequency ν and the wavelength in vacuum λ are connected by (3.49): νλ = c. Therefore, (3.107) and (3.108) can be written as ˆ =0, ∆ + kˆ 2 E (3.109) ˆ =0, ∆ + kˆ 2 H (3.110) with εµ kˆ 2 = ω2 2 + iωµ0 µσ c 2 λ 2π nˆ 2 2π εµ + i µσ = . = λ 2πcε0 λ (3.111)

Here, the refractive index nˆ is defined as nˆ 2 = εµ + i

λ µσ . 2πcε0

(3.112)

This means that nˆ is a complex number if the conductivity σ is not zero. Therefore, the real part n and the imaginary part n I of nˆ can be calculated: nˆ = n + in I ⇒ ⇒ ⇒

⇒

nˆ 2 = n 2 − n 2I + 2inn I ; (3.113) λ n 2 − n 2I = εµ and 2nn I = µσ , 2πcε0 2 λ µσ , n 2I = 4πcε0 n 2 λ µσ = 0 . n 4 − εµn 2 − 4πcε0 + , 2 2 2 , , εµ + ε2 µ2 + λ µ2 2σ 2 4π c ε0 n= ; (3.114) 2 + , λ 2 , nI = µσ , . 2 2 2 4πcε0 εµ + ε2 µ2 + λ µ2 2σ 2 4π c ε0

(3.115)

Only the positive solution of the two solutions of the quadratic equation with the variable n 2 is taken since n should be a real number and additionally only the

Wave Optics

positive square root of n 2 is taken since n should be a positive real number. For a pure dielectric (σ = 0) the imaginary part vanishes and the refractive index is a real number like it was defined in (3.27): √ σ = 0 ⇒ nˆ = n = εµ (3.116) A Simple Solution of the Helmholtz Equation in a Homogeneous Material A simple solution of (3.109) is, e.g., a linearly polarized plane wave propagating in the z direction (3.117)

ˆ 0 is a constant vector and its modulus repreHere, E sents the amplitude of the electric vector at z = 0. If nˆ is complex the effective position-dependent amplitude decreases exponentially nˆ = n + in I

⇒

ˆ =E ˆ 0 e−2πn I z/λ ei2πnz/λ . E (3.118)

So, the extinction of a wave can be formally included in the notation of a wave using complex exponential terms by just assuming a complex refractive index. The real part of this complex refractive index is responsible for the “normal” refractive properties and the imaginary part is responsible for absorption. For metals, n I can be larger than one so that the wave can enter the metal for only a fraction of a wavelength before the electric (and magnetic) vector vanishes. Instead of using the imaginary part n I of the refractive index the so-called absorption coefficient α is often used. It is defined by α :=

4π nI λ

⇒

ˆ =E ˆ 0 e−αz/2 ei2πnz/λ . E (3.119)

After having propagated a distance z = 1/α the electric energy density of the wave, which is proportional to ˆ 2 , decreases to 1/e of its starting value. | E| Whereas α is, according to our definition, always positive in lossy materials there are also active gain media, e.g., in lasers, which have a negative coefficient α. Then α is not an absorption but an amplification or gain coefficient. Inhomogeneous Plane Waves The solution of the Helmholtz equation (3.109) in a homogeneous but lossy material defined by (3.118) is the simplest form of a so-called inhomogeneous plane

101

wave [3.1, 4]. The general inhomogeneous plane wave is obtained from (3.109) by the approach ˆ =E ˆ 0 eikˆ · r = E ˆ 0 e−g · r eik · r E ⇒

−kˆ · kˆ + kˆ 2 = 0 ,

(3.120)

where kˆ = k + ig is a constant but complex wave vector with the real part k and the imaginary part g. In the ˆ 0 is also a constant but complex electric general case, E vector so that all polarization states can be represented (Sect. 3.2.3). By using the (3.111), (3.113) and (3.119) the complex quantity kˆ is defined as kˆ =

2π α 2π 2πn nˆ = +i (3.121) (n + in I ) = λ λ λ 2 2 2 2 2πnα 4π n α +i . ⇒ kˆ 2 = − 4 λ λ2

This means that the two vectors k and g have to fulfill the conditions kˆ · kˆ = (k + ig) · (k + ig) = |k|2 − |g|2 + 2ig · k = kˆ 2 =

2πnα 4π 2 n 2 α2 +i . − 4 λ λ2

(3.122)

A separation of the real and imaginary part gives 4π 2 n 2 α2 − , 4 λ2 πnα . g·k = λ

|k|2 − |g|2 =

(3.123) (3.124)

So, the projection of the vector g onto the vector k has to fulfill the second equation. An important and interesting case is that of a lossless material, i. e., α = 0. Then g and k have to be perpendicular to each other meaning that the planes of constant phase, which are perpendicular to k, and the planes of constant amplitude, which are perpendicular to g, are also perpendicular to each other. Inhomogeneous plane waves do not exist in the whole space because the amplitude decreases exponentially along the direction of g, but on the other side it increases exponentially for the direction antiparallel to g and would tend to infinity. Therefore, only the halfspace with the exponentially decreasing part can exist in the real world, whereas in the other direction there

Part A 3.1

ˆ ˆ 0 ei2π nz/λ ˆ =E ˆ 0 eikˆ z = E . E

3.1 Maxwell’s Equations and the Wave Equation

102

Part A

Basic Principles and Materials

has to be a limit. An example of an inhomogeneous plane wave is an evanescent wave in the case of total internal reflection at an interface between two dielectric materials with different refractive indices. There, a plane wave propagating in the material with higher refractive index with an angle of incidence at the interface of more than the critical angle of total internal reflection is reflected. Besides the reflected wave, however, there exists an evanescent wave in the material with the

lower refractive index. Its vector k is parallel to the interface between the two materials, while its amplitude decreases exponentially with increasing distance from the interface. The evanescent wave transports no energy into the material with the lower refractive index, but the total energy is in the reflected wave. In the following sections some basic properties of light waves will be described. For more information see textbooks on optics such as e.g., [3.1, 5–11].

3.2 Polarization Part A 3.2

In Sect. 3.1.1 it is shown that a so-called linearly polarized plane wave is a solution of Maxwell’s equations. There, the electric vector has a well-defined direction which remains constant during the propagation of the wave. There are other solutions of Maxwell’s equations where the direction of the electric vector does not remain constant during the propagation, but nevertheless, it has at a certain point and at a certain time a well-defined direction. All these solutions are called polarized light. Contrary to this, light that is emitted by an electric bulb is unpolarized. This means that there are many light waves with stochastically distributed phase relations to each other, i. e., incoherent light, and where the polarization varies in time. So, these light waves are added incoherently and there is no preferred direction of the electric vector. In practice, light is often partially polarized, i. e., some of the light is unpolarized and the other is polarized. Natural sun light on the earth is, e.g., partially polarized because of the influence of the atmosphere to the originally unpolarized light of the sun. Here, only the case of a fully polarized plane wave in a homogeneous dielectric material will be investigated. In the paragraph “The Orthogonality Condition for Plane Waves in Homogeneous Dielectrics” it is shown that the electric vector E and the magnetic vector H of a plane wave in a homogeneous and isotropic linear material are always perpendicular to each other and both are perpendicular to the direction of propagation e of the plane wave. Therefore, for a given direction of propagation e it is sufficient to consider only the electric vector. The magnetic vector is then automatically defined by (3.37) ε0 ε H=± e× E . µ0 µ The electric vector E has to fulfill the wave equation (3.99) with σ = 0 n2 ∂2 E ∆E − 2 2 = 0 . c ∂t

Without loss of generality, the direction of propagation will be parallel to the z-axis, i. e., e = (0, 0, 1). Because of the orthogonality relation E can then only have a x and a y component. A quite general plane wave solution of the wave equation is in this case ⎛

⎞ E x (z, t) ⎜ ⎟ E (z, t) = ⎝ E y (z, t) ⎠ E z (z, t) ⎛ ⎞ A x cos (kz − ωt + δx ) ⎟ ⎜ = ⎝ A y cos kz − ωt + δ y ⎠ . 0

(3.125)

Here, k = 2πn/λ = ωn/c (3.50) is the modulus of the wave vector k = 2πne/λ. It holds A x ≥ 0 and A y ≥ 0. By applying a trigonometric theorem, introducing the parameter α := kz − ωt + δx and the phase difference δ := δ y − δx this equation can be written as .

E x (α) E y (α)

. = . =

A x cos α A y cos (α + δ)

A x cos α A y cos α cos δ − A y sin α sin δ

/ .

(3.126)

This equation is the parametric representation of an ellipse, which is formed by the apex of the twodimensional vector (E x , E y ) in the xy-plane for different values of the parameter α. Unfortunately, in general, the principal axes of this ellipse will be rotated with respect to the x-axis and y-axis. Therefore, a transformation has to be done to calculate the principal axes of this ellipse with lengths 2a and 2b. To do this, the following quantity is calculated where the argument α of E x and E y is

Wave Optics

omitted in the notation 2 2 Ex E y Ey Ex + −2 cos δ Ax Ay Ax A y − 2 cos α cos δ (cos α cos δ − sin α sin δ) = cos2 α + (cos α cos δ − sin α sin δ)

So, the equation of the rotated ellipse in the coordinate system (x, y) indexed by terms in E 2x , E 2y and E x E y is

× (− cos α cos δ − sin α sin δ) = cos2 α 1 − cos2 δ + sin2 α sin2 δ = sin2 δ;

−2

Ex A x sin δ

Ey A y sin δ

Ex E y cos δ = 1 . A x A y sin2 δ

E 2x

(3.127)

This is the implicit representation of an ellipse that is rotated with respect to the x- and y-axis (see Fig. 3.4). On the other side an ellipse with the half-axes a and b parallel to the coordinate axes x and y and the coordinates of the electric vector E x and E y are written as 2 . /2 Ey Ex + =1. (3.128) a b This equation is transformed into a coordinate system with axes x and y whereby the system (x , y ) is rotated by an angle ϕ relative to the system (x, y) by applying

Ay b

In the last step the trigonometric equality sin(2ϕ) = 2 sin ϕ cos ϕ is used. By comparing this equation with (3.127) the coefficients of the terms in E 2x , E 2y and E x E y have to be equal to calculate the rotation angle ϕ and the principal axes. This results in three equations cos2 ϕ sin2 ϕ 1 + 2 = 2 2 , a2 b A x sin δ sin2 ϕ cos2 ϕ 1 + = 2 2 , a2 b2 A y sin δ cos δ 1 1 sin (2ϕ) 2 − 2 = −2 . a b A x A y sin2 δ

cos2 ϕ sin2 ϕ + a2 b2 2 sin ϕ cos2 ϕ + + E 2y a2 b2 1 1 + E x E y sin (2ϕ) 2 − 2 = 1 . a b

Part A 3.2

⇒

103

the well-known rotation matrix to the coordinates / . /. / . cos ϕ sin ϕ E x Ex = E y Ey − sin ϕ cos ϕ / . E x cos ϕ + E y sin ϕ . = −E x sin ϕ + E y cos ϕ

= cos2 α + (cos α cos δ − sin α sin δ)2

3.2 Polarization

By (1) adding the second equation to the first equation and (2) subtracting the second equation from the first equation and applying the trigonometric equation cos(2ϕ) = cos2 ϕ − sin2 ϕ two new equations are obtained and written together with the old third equation 1 1 1 + = a2 b2 sin2 δ

Fig. 3.4 The polarization ellipse on which the apex of the

electric vector moves if the time t or the coordinate z changes

1 1 + 2 A2x Ay

/ ,

. / 1 1 1 1 1 cos (2ϕ) 2 − 2 = − 2 , a b sin2 δ A2x Ay cos δ 1 1 sin (2ϕ) 2 − 2 = −2 . a b A x A y sin2 δ

104

Part A

Basic Principles and Materials

Division of the third equation by the second equation gives the tangent of twice the rotation angle ϕ of the ellipse tan (2ϕ) =

−2 cos δ A x A y A12 − x

= 1 A2y

−2A x A y cos δ . A2y − A2x

Using a trigonometric relation and (3.129) it holds that + , 2 , A2x A2y cos2 δ 1 = 1 + tan2 (2ϕ) = , 1+4 2 cos (2ϕ) A2y − A2x 2

(3.129)

Part A 3.2

So, the rotation angle ϕ can be calculated from the known variables A x , A y and δ. There is a null of the denominator for the case A x = A y . Then the argument of the tangent is 2ϕ = ±π/2, i. e., ϕ = ±π/4. Only if the numerator is also zero, i. e., cos δ = 0 and therefore δ = (2n + 1)π/2, the angle ϕ is not defined. In this case, the ellipse is degenerated into a circle as we will see later and then the rotation angle is, of course, not defined. It should also be mentioned that the rotation angle ϕ is only defined between −π/4 and π/4 and therefore 2ϕ between −π/2 and π/2 so that the arctangent function is well defined. This is sufficient because first, a rotation of an ellipse by π (= 180 degrees) does not change anything. And second, the principal axes of the ellipse can be chosen in such a way that either a is the large axis or b. So, the rotation angle ϕ has to be defined only in an interval of the length π/2, i. e., [−π/4; π/4]. After having calculated ϕ, the half-axes a and b of the polarization ellipse can also be calculated using the first two of the above equations (i)

(ii)

/ 1 1 + 2 , A2x Ay . / 1 1 1 1 1 − = − 2 ; a2 b2 cos (2ϕ) sin2 δ A2x Ay

1 1 1 + = a2 b2 sin2 δ

(i) + (ii) ⇒ 1 = 2 sin2 δ

1 a2 1 1 1 + 2+ 2 Ax A y cos (2ϕ)

1 1 − 2 2 Ax Ay

1 = 2 sin2 δ

1 b2 1 1 1 + 2− A2x A y cos (2ϕ)

1 1 − 2 A2x Ay

Remember that 2ϕ is only defined between −π/2 and π/2 so that cos(2ϕ) ≥ 0. Defining the value s by 3 s=

+1 if A2y − A2x ≥ 0 , −1 if A2y − A2x < 0

(3.132)

the reciprocal values of the squares of the half-axes can be explicitly written as 1 1 = a2 2A2x A2y sin2 δ 0 1 2 2 2 2 2 2 2 2 × A y + A x + s A y − A x + 4A x A y cos δ , (3.133)

1 1 = 2 2 b 2A x A2y sin2 δ 0 1 2 2 2 2 2 2 2 2 × A y + A x − s A y − A x + 4A x A y cos δ . (3.134)

Now it is easy to calculate the ratio of the squares of the half-axes, whereby the equality cos2 δ − 1 = − sin2 δ is used

/1 ,

(3.130)

(i) − (ii) ⇒

A2y − A2x + 4A2x A2y cos2 δ ! ! . ! 2 ! !A y − A2x !

/1 .

(3.131)

b2 a2

A2y +

A2x + s

A2y + A2x

− 4A2x A2y sin2 δ

2 A2y + A2x − s A2y + A2x − 4A2x A2y sin2 δ & A2 A2 1 + s 1 − 4 sin2 δ x y 2

& 1 − s 1 − 4 sin2 δ

A2y +A2x A2x A2y A2y +A2x

. 2

(3.135)

Wave Optics

So, for given parameters A x , A y and δ the half-axes a and b of the ellipse can be calculated by (3.133) and (3.134). The rotation angle can be calculated by (3.129) and the ratio of the half-axes by (3.135). There are several interesting special cases of polarization states and these will be discussed in the following.

3.2.1 Different States of Polarization

or A x = 0 or A y = 0 .

(3.136)

The two cases A x = 0 or A y = 0 are obvious because in this case there is only an x or a y component of the electric vector. If both components are different from zero there is nevertheless linear polarization if the phase difference δ between the two components of the electric vector is δ = 0 or δ = π. Circular Polarization If the apex of the electric vector moves on a circle, the polarization state is called circular polarization. This means that both half-axes have to be equal: a = b. Using (3.135) this requires

Ax A y b2 = 1 ⇒ 2 sin δ 2 = ±1 . a2 A y + A2x Since | sin δ| ≤ 1 this requires (A x and A y are both positive)

(3.138)

So, the two different signs of the phase difference δ correspond to different directions of rotation of the apex of the electric vector. These two cases are called right-handed circular polarization (δ = π/2) and left-handed circular polarization (δ = −π/2). The definition of right-handed and left-handed is not always identical in textbooks so that we use the definition of [3.1]. Elliptic Polarization The general polarization state is, of course, the so-called elliptic polarization. In this case the apex of the electric vector moves on an ellipse if the time t or the position z is changed. This state is the case if neither δ = 0 or δ = π nor δ = ±π/2. Also if δ = ±π/2 the light is elliptically polarized if A x = A y . There, we have to again distinguish between right-handed and left-handed elliptic polarization.

3.2.2 The Poincaré Sphere A method to visualize the different states of polarization is the so-called Poincaré sphere, which was introduced by H. Poincaré in 1892. By using (3.126) as the definition of the electric field, the so-called Stokes parameters of a plane monochromatic wave can be defined [3.1] s0 := A2x + A2y , s2 := 2A x A y cos δ , s3 := 2A x A y sin δ .

This condition can only be fulfilled for the case A x = A y and then there is the additional condition sin δ = ±1. Finally, the conditions for circular polarization are ±π . Ax = A y ∧ δ = 2

Using the original (3.125) this means for the electric vector / / . . A x cos (kz − ωt + δx ) E x (z, t) = E y (z, t) A x cos kz − ωt + δx ± π2 / . A x cos (kz − ωt + δx ) . = ∓A x sin (kz − ωt + δx )

s1 := A2x − A2y ,

Ax A y 1 ≥ ⇒ 2A x A y ≥ A2x + A2y 2 2 2 A y + Ax 2 ⇒ 0 ≥ Ax − A y .

(3.139)

It is obvious that the four quantities are connected by the relation s12 + s22 + s33 = A4x + A4y − 2A2x A2y + 4A2x A2y

(3.137)

105

= A4x + A4y + 2A2x A2y = s02 .

(3.140)

Part A 3.2

Linear Polarization An important and quite simple polarization state is the case of linear polarization. In this case the polarization ellipse degenerates to a line and the apex of the electric vector just oscillates on a line. This is the case if either the numerator or the denominator of (3.135) is zero so that a or b is zero. This means + , , A2x A2y 2δ 1−, 1 − 4 sin 2 = 0 ⇒ sin δ = 0 A2y + A2x

3.2 Polarization

106

Part A

Basic Principles and Materials

Right-handed circular polarization

3.2.3 Complex Representation of a Polarized Wave

Part A 3.2

Linear polarization S1 Left-handed circular polarization

Fig. 3.5 The Poincaré sphere and the visualization of the

different states of polarization

So, only three of the parameters are independent and the parameter s0 is proportional to the intensity of the wave. If s1 , s2 and s3 are now used as the cartesian coordinates of a point in space, all allowed combinations will be situated according to (3.140) on a sphere with radius s0 . The radius s0 is proportional to the intensity of the wave and the sphere is called the Poincaré sphere (Fig. 3.5). The different polarization states correspond to different positions on the Poincaré sphere. For linearly polarized light it is, e.g., either A x = 0 or A y = 0 or δ = 0 or δ = π. In all four cases the parameter s3 will be zero. This means that points lying in the equatorial plane of the Poincaré sphere represent linear polarization. Another interesting case is circular polarization. In this case the conditions are according to (3.137): A x = A y and δ = ±π/2. Therefore, s1 and s2 are both zero and circular polarization corresponds to the poles of the Poincaré sphere. At the north pole (s1 = s2 = 0 and s3 = s0 ) the light is right-handed circularly polarized (δ = π/2). At the south pole (s1 = s2 = 0 and s3 = −s0 ) the light is left-handed circularly polarized (δ = −π/2). All other states of polarization (elliptic polarization) correspond to points somewhere else on the Poincaré sphere. In the upper hemisphere of the Poincaré sphere (s3 > 0) the light is always right-handed polarized and in the lower hemisphere (s3 < 0) the light is left-handed polarized.

In (3.125) the electric vector is expressed as a real quantity. As we have seen in other sections it is in many cases useful to take a complex notation, where only the real part has a physical meaning ⎞ ⎛ E x (z, t) ⎟ ⎜ E (z, t) = ⎝ E y (z, t) ⎠ E z (z, t) ⎛ ⎞ A x cos (kz − ωt + δx ) ⎟ ⎜ = ⎝ A y cos kz − ωt + δ y ⎠ 0 ⎞ ⎤ ⎡⎛ A x eiδx ⎟ ⎥ ⎢⎜ = Re ⎣⎝ A y eiδ y ⎠ eikz e−iωt ⎦ 0 ˆ ikz e−iωt . = Re Ae (3.141) ˆ is a constant but comIn this case the vector A plex vector in order to represent all possible states of polarization. With a real vector only a linear polarization state could be represented. The complex notation has the advantage that it is quite easy to calculate the time average of the intensity. According to (3.68) the time average of the intensity I is proportional to the square of the absolute value of the time-independent complex electric vecˆ tor E: ˆ (z) = Ae ˆ ikz ⇒ I ∝ E ˆ ·E ˆ ∗ = A2x + A2y . (3.142) E Therefore, it is clear that a detector sensitive only to the intensity of a light wave cannot distinguish between different polarization states.

3.2.4 Simple Polarizing Optical Elements and the Jones Calculus There are, of course, optical elements that influence the polarization state like polarizers, quarter-wave plates, half-wave plates and many others. Here, only the basic idea of their effects can be discussed. For more information about the treatment of polarizing optical elements see, e.g., [3.12–14]. In this section only fully polarized light is treated. This can be produced from natural unpolarized light with the help of a polarizer. In the following the word polarizer is always used for a po-

Wave Optics

This vector is called the Jones vector. Now each polarizing optical element can be represented by a 2 × 2 matrix, the Jones matrix. The resulting Jones vector of the light that has passed such a polarizing optical element is calculated by multiplying the Jones vector of the incident light with the Jones matrix. Several polarizing optical elements can be passed by simply multiplying the Jones matrices of these elements. Polarizer A polarizer is, e.g., a device that produces linearly polarized light from an arbitrary polarization state. If the polarizer passes only light polarized in x direction its Jones matrix is / . 1 0 . Px = (3.144) 0 0

Similarly Jones matrices of polarizers in other directions (y direction, 45◦ or −45◦ ) can be represented by / / . . 1 1 1 0 0 ; ; P45◦ = Py = 2 1 1 0 1 / . 1 1 −1 P−45◦ = . (3.145) 2 −1 1 As an example consider a plane wave linearly polarized in the x direction, which first passes a polarizer in the x direction and then in the direction 45◦ . The resulting vector is: /. /. / . 1 1 1 E0 1 0 Jfinal = 2 1 1 0 0 0 /. / . . / 1 1 1 1 E0 E0 = = . 2 1 1 2 E0 0

107

Of course, the first polarizer in the x direction has no effect and the second polarizer selects the component of the electric vector in the 45◦ direction. The resulting intensity of the wave is then proportional to |Jfinal |2 = E 02 /2, i. e., half of the intensity is absorbed. Another well-known effect is the combination of two crossed polarizers (e.g., in the x and y directions). Their matrix is, of course, . /. / . / 0 0 1 0 0 0 Pcrossed = Py Px = = . 0 1 0 0 0 0 This means that no light passes this combination. Only if other polarizing elements are between the two crossed polarizers can light pass. If, e.g., a polarizer with direction 45◦ is inserted between the two crossed polarizers light can pass this combination if it originally has a component in the x direction. The resulting light will, of course, only have a y component: Pcrossed+45◦ = Py P45◦ Px / /. / . . 1 0 0 0 1 1 1 = 0 0 0 1 2 1 1 . / 1 0 0 = . 2 1 0 Though this is a well-known example it is nevertheless a typical example that polarizing optical elements can produce quite astonishing results by inserting additional elements. Quarter-Wave Plate Another elementary polarizing optical element is a quarter-wave plate (λ/4-plate), which consists of a birefringent material. If the axes of the material are correctly oriented the refractive index for light polarized in x direction is, e.g., different from the refractive index for polarized light in y direction. The resulting effect is a phase difference between the two components of the Jones vector of π/2. The Jones matrix for a higher phase velocity in the y direction is, e.g., / . 1 0 iπ/4 . Pλ/4 = e (3.146) 0 i

This means that linearly polarized light with components in only the x or y direction remains linearly polarized

Part A 3.2

larization filter whereas other polarizing elements are simply called polarizing elements or polarizing optical elements. The case of partially polarized light will not be treated. A quite useful algorithm for the treatment of fully polarized light is the so-called Jones calculus, which was invented by Jones [3.15]. If we again have a plane wave propagating in the z direction the polarization state can be described by a two-dimensional vector J in the x and y direction containing the x and y components of the ˆ of (3.141): vector A / . / . A x eiδx Jx = . (3.143) J= Jy A y eiδ y

3.2 Polarization

108

Part A

Basic Principles and Materials

and the intensity is unchanged (in practice some light is, of course, absorbed). But for light that is linearly polarized and has equal components in the x and y direction (i. e., linearly polarized with a direction of 45◦ ) the resulting polarization state is circularly polarized light / . . / 1 1 0 E0 iπ/4 Jfinal = e √ 0 i 2 E0 . / 1 E0 = √ eiπ/4 . iE 2 0

Part A 3.3

Again, the intensity of the light is unchanged, only the polarization state has changed. Linearly polarized incident light with other directions of polarization will result in elliptically polarized light. Half-Wave Plate A third interesting case is when the circularly polarized light passes an identical quarter-wave plate a second

time. Then the Jones vector is 1 iπ/4 E 0 1 0 iπ/4 Jfinal = e √ e 0 i iE 0 2 1 E0 . = √ eiπ/2 −E 0 2 The result is again linearly polarized light, but with a rotation of the direction of polarization of 90◦ . The effect of two identical quarter-wave plates is, of course, the effect of a half-wave plate (λ/2-plate). So, a half-wave plate rotates the direction of polarization of linearly polarized light by 90◦ if the incident light is polarized in the 45◦ direction. If the incident light is polarized in the x or y direction nothing happens. The matrix of a half-wave plate is 1 0 1 0 iπ/4 iπ/4 e Pλ/2 = e 0 i 0 i (3.147) = eiπ/2 1 0 . 0 −1

3.3 Interference Interference is the property of all types of waves to form characteristic stationary variations of the intensity by the superposition of two or more waves. Of course, in the case of light some conditions have to be fulfilled because with natural light from the sun or light from a bulb it is quite difficult to get interference effects. On the other hand it is no problem to obtain interference effects with the help of a laser. In fact, the condition is that the light has to be coherent or at least partially coherent[3.16]. There are complete books about interference effects and the application of these optical effects in the field of interferometry [3.17–20]. So, in this section only the basic ideas can be treated.

3.3.1 Interference of Two Plane Waves First, the interference of two monochromatic plane waves in a homogeneous and isotropic material will be treated. The two plane waves with the angular frequency ω are propagating in the direction of their wave vectors k1 and k2 . The corresponding unit vectors in the direction of propagation are e1 = k1 /|k1 | and e2 = k2 /|k2 |. Their polarization state shall be arbitrary. In Sect. 3.2 we investigated the different polarization states and used the fact that for one plane wave the coordinate system can be chosen so conveniently that

the direction of propagation is in the z direction and the electric vector can only have x and y components because of the orthogonality condition. For two plane waves, which are not propagating parallel, a more generalized description has to be found. Therefore, the unit vector e⊥ , which is perpendicular to the plane formed by the two propagation vectors e1 and e2 is defined as e⊥ :=

e1 × e2 . |e1 × e2 |

(3.148)

Only for the case that e1 and e2 are parallel or antiparallel, i. e., e2 = ±e1 , e⊥ is not defined and we define in this case e1 := (0, 0, 1) and e⊥ = (0, 1, 0). But in the following, it is sufficient to assume that e⊥ is well defined via (3.148) or otherwise. So, for each wave we can define a unit vector e,1 or e,2 , which lies in the plane defined by the directions of propagation of the two waves but perpendicular to the respective propagation vector e,1 := e⊥ × e1 , e,2 := e⊥ × e2 .

(3.149) (3.150)

Now, each plane wave will only have components along e⊥ and the respective vector e,1 or e,2 . The components along e⊥ are called transversal electric (TE) components. The components along e,1 or e,2

Wave Optics

are called transversal magnetic (TM) components, because in this case the corresponding component of the magnetic vector is perpendicular to the plane of propagation. Therefore, using a generalization of (3.141) the electric vectors of both plane waves can be represented as: " E1 (r, t) = Re A,1 eiδ,1 e,1 + A⊥,1 eiδ⊥,1 e⊥ # × eik1 · r e−iωt " # ˆ ,1 + A ˆ ⊥,1 eik1 · r e−iωt , = Re A

(3.152)

The quantities with a hat are complex, the others are real. Using this representation the orthogonality condition for electromagnetic waves is automatically fulfilled. The magnetic vector is not explicitly noted here because it is automatically defined by (3.37). Moreover, the interaction of an electromagnetic wave with matter is normally due to an electric field. Therefore, the electric vector is used in our calculation. The interference of these two plane waves just means that the electric vectors have to be added. Since this is a linear operation and we are, in the end, only interested in the time average of the intensity it is sufficient to add ˆ 1 and the time-independent complex electric vectors E ˆ 2 . The resulting electric vector E ˆ 1+2 is E ˆ 1+2 = E ˆ1+E ˆ2 = A ˆ ,1 + A ˆ ⊥,1 eik1 · r (3.153) E ˆ ⊥,2 eik2 · r . ˆ ,2 + A + A The intensity of a plane wave measured on a surface perpendicular to the direction of propagation is, according ˆ 2 to (3.68), √ proportional to | E| . The proportionality factor is ε/µ ε0 c/2. The intensity of the plane wave on a plane surface that is not perpendicular to the direction of the energy flow is decreased by the cosine of the angle of incidence. In the following, the plane on which we define the intensity of our interference pattern is perpendicular to the effective direction of the energy flow, i. e., perpendicular to k1 + k2 if the two waves are not antiparallel or perpendicular to k1 if k1 = −k2 . The co-

sine factors are then identical for both waves and the intensity I1+2 and the square of the modulus of the reˆ 1+2 |2 are really proportional to sulting electric vector | E each other with a constant of proportionality a. So, for the interference pattern the following holds true: ! !2 !ˆ ! ∗ ˆ 1+2 · E ˆ 1+2 ! E1+2 ! = E " ˆ ,1 + A ˆ ⊥,1 eik1 · r = A # ˆ ,2 + A ˆ ⊥,2 eik2 · r + A " ˆ ∗,1 + A ˆ ∗⊥,1 e−ik1 · r × A # ˆ ∗,2 + A ˆ ∗⊥,2 e−ik2 · r + A ˆ ,1 · A ˆ ∗,1 + A ˆ ⊥,1 · A ˆ ∗⊥,1 =A ˆ ,2 · A ˆ ∗,2 + A ˆ ⊥,2 · A ˆ ∗⊥,2 +A ˆ ∗,2 + A ˆ ⊥,1 · A ˆ ∗⊥,2 ˆ ,1 · A + A × ei (k1 − k2 ) · r ˆ ,2 + A ˆ ∗⊥,1 · A ˆ ⊥,2 ˆ ∗,1 · A + A × e−i (k1 − k2 ) · r .

(3.154)

All other terms vanish because of the orthogonality of the respective vectors. To evaluate this equation further, the scalar product of e,1 and e,2 has to be calculated. It is e,1 · e,2 = (e⊥ × e1 ) · (e⊥ × e2 ) = [e1 × (e⊥ × e2 )] · e⊥ = [(e1 · e2 ) e⊥ − (e1 · e⊥ ) e2 ] · e⊥ = e1 · e2 .

(3.155)

This relation is, of course, obvious and we can use it to evaluate the interference pattern. To abbreviate the notation the phase differences between the two waves are defined as δ := δ,1 − δ,2 and δ⊥ := δ⊥,1 − δ⊥,2 ! !2 !ˆ ! ! E1+2 ! = A2,1 + A2⊥,1 + A2,2 + A2⊥,2 " + A,1 A,2 eiδ (e1 · e2 ) # + A⊥,1 A⊥,2 eiδ⊥ ei (k1 − k2 ) · r " + A,1 A,2 e−iδ (e1 · e2 ) # + A⊥,1 A⊥,2 e−iδ⊥ e−i (k1 − k2 ) · r

109

Part A 3.3

(3.151)

" E2 (r, t) = Re A,2 eiδ,2 e,2 + A⊥,2 eiδ⊥,2 e⊥ # × eik2 · r e−iωt # " ˆ ⊥,2 eik2 · r e−iωt . ˆ ,2 + A = Re A

3.3 Interference

110

Part A

Basic Principles and Materials

= A2,1 + A2,2 + 2A,1 A,2 (e1 · e2 ) × cos (k1 − k2 ) · r + δ

Λ G = k1 – k2

+ A2⊥,1 + A2⊥,2 + 2A⊥,1 A⊥,2 × cos [(k1 − k2 ) · r + δ⊥ ] .

k1 (3.156)

Part A 3.3

The two terms that depend on the parameters of both waves and are functions of the position are called interference terms. These interference terms distinguish the superposition of coherent waves and incoherent waves. In the case of incoherent waves the interference terms vanish and the resulting intensity is just the sum of the single intensities of both waves. It can be seen that the interference pattern resolves into terms that depend only on TM components and those that depend only on TE components. Since both are perpendicular to each other, the intensities of both waves can also be divided into the sum of a “TM intensity” and a “TE intensity” I1 = a A2,1 + A2⊥,1 = I,1 + I⊥,1 , (3.157) I2 = a A2,2 + A2⊥,2 = I,2 + I⊥,2 . Here, the constant of proportionality a is used, which was explained above. Then, the intensity of the interference pattern is I1+2 = I,1 + I,2 : + 2 I,1 I,2 (e1 · e2 ) cos (k1 − k2 ) · r + δ + I⊥,1 + I⊥,2 : + 2 I⊥,1 I⊥,2 cos [(k1 − k2 ) · r + δ⊥ ] . (3.158) The Grating Period and the Fringe Period Equation (3.158) shows that the surfaces of constant intensity are plane surfaces with

(k1 − k2 ) · r = constant .

(3.159)

The planes are perpendicular to the so-called grating vector G (Fig. 3.6) with G = k1 − k2 .

(3.160)

Since the cosine function is periodic the distance between the two neighboring planes of equal intensity is called the grating period Λ of the interference pattern. It can be calculated by taking a point r1 on the first plane and a point r2 on the neighboring second plane, so that the vector ∆r := r2 − r1 is parallel to G and

ϕ/ 2

λ /n

Fig. 3.6 Interference of two plane waves. The solid lines indicate the planes of constant phase of the two plane waves at a fixed time having the distance λ/n, respectively, whereas the dashed lines indicate the interference planes with constant intensity having a distance Λ. The planes themselves are perpendicular to the drawing plane

simultaneously perpendicular to the planes. Its modulus is the grating period cos [(k1 − k2 ) · r2 ] = cos [(k1 − k2 ) · r1 ] ; ⇒ (k1 − k2 ) · r2 = (k1 − k2 ) · r1 + 2π ⇒ G · ∆r = |G| |∆r| = 2π 2π λ 2π ⇒ Λ = |∆r| = = = 2πn . |G| |e n − e2 | |e | − e 1 1 2 λ (3.161)

Here, n is the refractive index of the material in which the waves propagate and λ = 2πc/ω would be the wavelength in vacuum. So, the grating period is infinity if both waves are propagating parallel, i. e., e1 = e2 , and the smallest grating period can be obtained if both waves are propagating antiparallel, i. e., e1 = −e2 . Then the grating period Λmin is λ Λmin = . 2n In the general case the grating period can be expressed by using the angle ϕ between the two wave vectors (Fig. 3.6) : |e1 − e2 | = (e1 − e2 ) · (e1 − e2 ) : = 2 − 2e1 · e2 λ . ⇒Λ= √ (3.162) n 2 (1 − cos ϕ) This can√ also be seen in Fig. 3.6 using the trigonometric identity 2(1 − cos ϕ) = 2 sin(ϕ/2).

Wave Optics

(3.163)

In analogy to (3.161) the result is 2π 2π p= ! ! = , !G ! |G − (G · N) N|

(3.164)

where N is a unit vector perpendicular to the detector plane. By defining the two angles of incidence β1 and β2 of the plane waves onto the detector plane as cos β1 := e1 · N;

cos β2 := e2 · N;

the fringe period p can be written as: 2π 2π p= : =: 2 2 |G| − (G · N)2 [G − (G · N) N] 2π =: |k1 − k2 |2 − [(k1 − k2 ) · N]2 λ = 2 n 2 (1 − cos ϕ) − (cos β1 − cos β2 )2 or 2π Λ p= : =√ 2 2 1 − cos2 α |G| − (G · N) λ Λ = √ = . sin α n 2 (1 − cos ϕ) sin α (3.165)

In the last equation the angle α between the grating vector G and the surface normal N, i. e., the angle α between the grating planes and the detector plane, is defined as cos α = (G · N)/|G|. From (3.165) it can be seen that the fringe period is infinity if the grating planes are parallel to the detector plane and that the fringe period is minimal if the grating planes are perpendicular

111

to the detector plane, i. e., if G is parallel to the detector plane. Then the fringe period p is equal to the grating period Λ.

3.3.2 Interference Effects for Plane Waves with Different Polarization States In the calculation of the grating period and the fringe period we assumed that the interference terms in (3.158) are different from zero so that interference occurs. But, this is not always the case as we will now discuss. A quite interesting quantity in interferometry is the visibility V of the interference fringes. It is defined as V :=

Imax − Imin , Imax + Imin

(3.166)

where Imax is the maximum intensity and Imin the minimum intensity at a point in the interference pattern when the phase (i. e., the argument of the cosine function) of the interference terms is varied in a range of 2π. For the interference of plane waves the maximum and minimum intensity can also be taken at different points because the intensity of the two single waves is then independent of the position. The visibility can vary between 0 for Imin = Imax , i. e., no interference occurs, and 1 for Imin = 0. Linearly Polarized Plane Waves For the case of linearly polarized plane waves the phase constants of each wave δ,1 and δ⊥,1 on the one hand and δ,2 and δ⊥,2 on the other are equal or differ only by π. There are, in fact, effectively two different cases:

δ,1 = δ⊥,1 ∧ δ,2 = δ⊥,2 δ,1 = δ⊥,1 + π ∧ δ,2 = δ⊥,2 + π

;

δ⊥ = δ⊥,1 − δ⊥,2 ⇒ = δ,1 − δ,2 = δ =: δ and s := +1 ; δ,1 = δ⊥,1 + π ∧ δ,2 = δ⊥,2 δ,1 = δ⊥,1 ∧ δ,2 = δ⊥,2 + π δ⊥ = δ⊥,1 − δ⊥,2 ⇒ δ,1 − δ,2 ∓ π = δ ∓ π =: δ ∓ π and s := −1

(3.167)

(3.168)

The parameter s characterizes the different cases and is either +1 or −1. Then the intensity of the interference

Part A 3.3

What is really observed are the lines of intersection of the planes of constant intensity with a detector plane. The resulting lines are called the interference fringes. In the case of the interference of two plane waves the interference fringes are straight, parallel and equidistant lines with the distance p, called the period of the fringes. Only in the case when the grating vector G is parallel to the detector plane is the fringe period p equal to the grating period Λ. In the general case, only the component of the grating vector parallel to the detector plane has to be used to calculate the fringe period. This relation can be easily seen by taking the plane with z = 0 as detector plane. Then the fringes in the xy-plane are described in analogy to (3.159) by kx,1 − kx,2 x+ ky,1 − ky,2 y = G · r = constant .

3.3 Interference

112

Part A

Basic Principles and Materials

pattern (see (3.158)) can be expressed as

I1+2 = I,1 + I,2 : + 2 I,1 I,2 cos ϕ cos [(k1 − k2 ) · r + δ] + I⊥,1 + I⊥,2 : + 2s I⊥,1 I⊥,2 cos [(k1 − k2 ) · r + δ] . (3.169) Here, the angle ϕ between the directions of propagation of the two waves is used. So, there are several interesting special cases of (3.169):

• Part A 3.3

Both waves have only TE components, i. e., I,1 = I,2 = 0, I⊥,1 = 0 and I⊥,2 = 0. Then, we obtain ITE,TE = I⊥,1 + I⊥,2 : + 2s I⊥,1 I⊥,2 cos [(k1 − k2 ) · r + δ] .

•

If the intensities of both waves are equal the visibility is 1. One wave has only a TE component, i. e., I,1 = 0 and I⊥,1 = 0, and the other wave has only a TM component, i. e., I,2 = 0 and I⊥,2 = 0. Then the interference terms vanish and the intensity is constant ITE,TM = I⊥,1 + I,2 = constant .

•

(3.172)

This just means that orthogonally polarized waves cannot interfere. Both waves have only TM components, i. e., I⊥,1 = I⊥,2 = 0, I,1 = 0 and I,2 = 0. Then, we obtain ITM,TM = I,1 + I,2 : +2 I,1 I,2 cos ϕ cos [(k1 − k2 ) · r +δ] . (3.173)

The visibility VTM,TM is in this case: : 2 I,1 I,2 cos ϕ . VTM,TM = I,1 + I,2

e⊥

Fig. 3.7a,b The different possibilities for the interference of linearly polarized waves. (a) s = +1, i. e., the two electric vectors are oscillating in the same quadrant; (b) s = −1,

i. e. the two electric vectors are oscillating in different quadrants. The dashed line indicates the plane in which the wave vectors of both waves are situated

(3.170)

This is the well-known interference equation that is also used for scalar waves where an arbitrary component of the electric vector, but for a small angle ϕ between the two wave vectors of the waves, is regarded. In this case the visibility VTE,TE (defined with (3.166)) of the interference pattern is : 2 I⊥,1 I⊥,2 VTE,TE = . (3.171) I⊥,1 + I⊥,2

•

So, the visibility is always smaller than 1 and the interference term vanishes if both waves are propagating perpendicular to each other, i. e., e1 · e2 = cos ϕ = 0. For the case I,1 = I,2 , where the visibility is 1 in the TE polarized case, the visibility in the TM polarized case is VTM,TM = cos ϕ. Only for small angles between the directions of propagation of the two plane waves the visibility is high. Of course, for small angles there is in fact no real difference between TE- and TM-polarized light and for ϕ = 0 or ϕ = π the difference between TE and TM exists no longer. TE and TM components of both waves are present. Then the value of the constant s is important. For s = +1 the interference terms have identical signs and add. But for s = −1 the interference terms have different signs and cancel each other out if :

I,1 I,2 cos ϕ =

I⊥,1 I⊥,2 .

(3.175)

The meaning of this is that for s = −1 the electric vectors of both waves are oscillating in different quadrants. If the above equation is fulfilled they are again perpendicular to each other and cannot interfere. This is explained in Fig. 3.7. Circularly Polarized Plane Waves Circularly polarized plane waves exist according to (3.137) only if

1 I1 ; 2 1 I,2 = I⊥,2 = I2 ; 2 I,1 = I⊥,1 =

(3.174)

π , 2 π δ,2 − δ⊥,2 = ± . 2 δ,1 − δ⊥,1 = ±

Wave Optics

Then, the intensity of the interference pattern is determined according to (3.158) : 1 I1 + I2 + 2 I1 I2 I1+2 = 2 × cos ϕ cos (k1 − k2 ) · r + δ : 1 + I1 + I2 + 2 I1 I2 2

+ cos [(k1 − k2 ) · r + δ⊥ ] .

(3.176)

and (3.176) reduces to : I↑↓ = I1 + I2 + I1 I2

× (cos ϕ − 1) cos (k1 − k2 ) · r + δ . (3.179)

The visibility (3.166) is √ I1 I2 (1 − cos ϕ) . (3.180) V↑↓ = I1 + I2 So, for different chirality the behavior is reverse to that of equal chirality. Waves that are propagating parallel (ϕ = 0) and have different chirality cannot interfere, whereas waves that are propagating antiparallel (ϕ = π) and have different chirality interfere very well.

Now, we have to differentiate between several cases:

•

Both waves have the same chirality, i. e., either δ,1 − δ⊥,1 = π/2 and δ,2 − δ⊥,2 = π/2 or δ,1 − δ⊥,1 = −π/2 and δ,2 − δ⊥,2 = −π/2. Then, the phase differences are δ = δ,1 − δ,2 ;

δ⊥ = δ⊥,1 − δ⊥,2 = δ =: δ

and (3.176) reduces to : I↑↑ = I1 + I2 + I1 I2

× (cos ϕ + 1) cos (k1 − k2 ) · r + δ . (3.177)

•

The visibility (3.166) is √ I1 I2 (cos ϕ + 1) V↑↑ = . (3.178) I1 + I2 For small angles ϕ between the directions of propagation of the two waves the intensity is identical to the interference pattern of two linearly TE-polarized waves and the visibility can reach 1 for equal intensities in both waves. If both waves are propagating perpendicular to each other the interference term has only half the size and the visibility is for equal intensities only 1/2. If the angle ϕ is larger than π/2 and approaches π the interference term and the visibility vanish. This means that waves propagating antiparallel and with the same chirality cannot interfere. Both waves have different chirality, i. e., either δ,1 − δ⊥,1 = π/2 and δ,2 − δ⊥,2 = −π/2 or δ,1 − δ⊥,1 = −π/2 and δ,2 − δ⊥,2 = π/2. Then, the phase differences are δ = δ,1 − δ,2 =: δ; δ⊥ = δ⊥,1 − δ⊥,2 = δ ± π = δ ± π

113

The Application of Two-Beam Interference for an Electron Accelerator A quite interesting modern application of the interference of two waves is the laser-driven electron accelerator [3.21],[3.22]. This accelerator can in principle also be used for any other charged particle whereby the efficiency increases if the velocity of the particle is nearly identical to the speed of light. So, heavy particles need a lot of kinetic energy (at best a multiple of their rest energy m 0 c2 ) if they shall be accelerated by laser light with a good efficiency. Here, only the basic principle can be discussed. Figure 3.8 shows two interfering waves both with linear TM polarization, equal amplitudes and a phase E1

Ez = E1+ E2 E2

E1 k2 θ z

2θ

k1 E2

Fig. 3.8 The interference of two linearly polarized waves

with TM polarization and π phase difference results in a longitudinal component Ez of the electric field

Part A 3.3

× cos [(k1 − k2 ) · r + δ⊥ ] : = I1 + I2 + I1 I2 × cos ϕ cos (k1 − k2 ) · r + δ

3.3 Interference

114

Part A

Basic Principles and Materials

Part A 3.3

difference of π on points along the z-axis. First of all, we assume that the waves are plane. The z-axis bisects the angle 2θ between the wave vectors k1 and k2 of both waves so that θ is the angle between the z-axis and one of the wave vectors. Then, the electric vectors E1 and E2 are oriented as indicated and the components perpendicular to the z-axis, i. e., parallel to the x-axis, cancel each other. The magnetic vectors of both waves, which are perpendicular to the drawing plane, i. e., parallel to the y-axis, also cancel each other out on points along the z-axis because they are antiparallel. However, due to the configuration of the two interfering waves there is a resulting longitudinal component Ez of the electric vector for points on the z-axis. From a mathematical point of view, we have for the electric vectors using the coordinate system of Fig. 3.8 E1 (x, z, t) " #⎞ ⎛ E 0 cos θ cos 2π λ (−x sin θ + z cos θ) − ωt ⎟ ⎜ ⎟ =⎜ 0 ⎝ " # ⎠, E 0 sin θ cos 2π λ (−x sin θ + z cos θ) − ωt E2 (x, z, t) " #⎞ ⎛ −E 0 cos θ cos 2π λ (x sin θ + z cos θ) − ωt ⎜ ⎟ ⎟ =⎜ 0 ⎝ " # ⎠; E 0 sin θ cos 2π λ (x sin θ + z cos θ) − ωt ⇒ Ez (x = 0, z, t) = E1 (x = 0, z, t) + E2 (x = 0, z, t) ⎞ ⎛ 0 ⎟ ⎜ 0 =⎝ ⎠. 2π 2E 0 sin θ cos λ z cos θ − ωt =: E z (0, z, t) (3.181)

Here, E 0 is the maximum amplitude of the electric vector of one of the two interfering waves (E 0 = max |E1 | = max |E2 |), λ is the wavelength of the waves and ω is their angular frequency. So, we see that along the z-axis with x = 0 there exists only a component of the electric field parallel to the z-axis and by using the relations λν = c and ω = 2πν = 2πc/λ (ν is the frequency) for our waves, which are situated in a vacuum, we obtain E z (x = 0, z, t)

cos θ c z −t . = 2E 0 sin θ cos 2π λ c

(3.182)

Now, a relativistic electron, i. e., the speed v of the electron should be nearly the speed of light c, travels along the z-axis from left to right. It should pass the regarded point at that time where E z = −2E 0 sin θ. Then, the electron will be maximally accelerated along the z-axis due to its negative electric charge. But, the interference maximum with E z = −2E 0 sin θ seems to propagate along the z-axis with the phase velocity c/cos θ, which is faster than the speed of light for θ > 0. Of course, this is in no contradiction to special relativity because no information nor energy is transported with this speed. So, for a small angle θ the relativistic electron with v ≈ c (but nevertheless v < c) travels a certain distance nearly in phase with the accelerating electric field before it comes out of phase because of c/cos θ > c > v. The distance after which the electron is out of phase can also be calculated very easily. The electron travels in the laboratory framework in the time interval t, a distance z = vt, i. e., t = z/v. The velocity v can be assumed to be constant during the acceleration process because it should be nearly the speed of light and therefore it does not change considerably although the electron may gain a lot of kinetic energy. By introducing this into (3.182) the argument Φ of the cosine function as a function of the position z on the z-axis is: c cos θ c cos θ 1 Φ(z) = 2π z − t = 2π z − . λ c λ c v (3.183)

If the phase Φ of the electric field, which affects the electron changes by ±π, the electric field that first accelerated the electron will now slow it down. So, the distance ∆z on the z-axis between being accelerated and decelerated by the electric field is c cos θ 1 ∆Φ = 2π ∆z − λ c v = ±π ⇒ ∆z = ±

λ . 2 (cos θ − c/v)

(3.184)

For v ≈ c the longest distance for being in phase would be achieved by θ = 0. But then, the accelerating electric field itself would be zero because of the factor sin θ in (3.182). So, in practice, a tradeoff has to be found between a big value sin θ and another big value cos θ. Additionally, the velocity of the electron (or other charged particle) should be nearly the speed of light. However, we see that for the interference of two infinitely extended plane waves the electron would be accelerated and slowed down periodically. But, we can

Wave Optics

3.3.3 Interference of Arbitrary Scalar Waves The interference phenomena are, of course, not restricted to plane waves but can occur for arbitrary waves. Since for arbitrary waves the polarization can change locally we will neglect the polarization in this section and concentrate on so-called scalar waves. Some Notes on Scalar Waves In the case of scalar waves only one Cartesian component of the electric (or magnetic) vector is regarded and the complete polarization state is neglected. Nevertheless, for two linearly polarized interfering waves, which are both TE polarized, the result of the scalar calculation is identical to the exact result. Since the orthogonality condition is neglected for scalar waves the result of the scalar wave equation is not automatically a solution of the Maxwell equations. A scalar wave, which is often used in optics, is a spherical wave with its center of curvature at the point

115

r0 with the complex amplitude u (r) = a

eik |r − r0 | |r − r0 |

(3.185)

and the modulus of the wave vector k = 2πn/λ, which is also called a wave number. a is a constant. It should be mentioned that a spherical wave is a solution of the scalar Helmholtz equation of homogeneous materials (Sect. 3.1.5), where only one component of the electric or magnetic vector is regarded. But, a spherical wave is not a solution of the Maxwell equations itself because this would violate the orthogonality conditions of an electromagnetic wave. But, in the far field and in the plane perpendicular to its dipole axis, a dipole radiation is a good approximation for a spherical wave. Here, the scalar complex amplitude u was introduced, which can stand, apart from a constant of proportionality, for one component of the electric or magnetic vector. Again, the intensity of this scalar wave is at least proportional to the square of the modulus of u and in the following we just define for scalar waves I := uu ∗ .

(3.186)

A general scalar wave can be described by u (r) = A (r) eiΦ (r) .

(3.187)

In this case, A is a real function, which changes only slowly with the position r, where Φ is also a real function but the complex exponential factor exp(iΦ) varies rapidly with the position r. The Interference Equation for Scalar Waves By using two general scalar waves (3.187)

u 1 (r) = A1 (r) eiΦ1 (r) ,

u 2 (r) = A2 (r) eiΦ2 (r)

instead of using plane waves in (3.158) or (3.170) the interference equation of scalar waves is obtained by : I1+2 = I1 + I2 + 2 I1 I2 cos Φ (3.188) with I1 = A21 , I2 = A22 and Φ = Φ1 − Φ2 . In some cases it is more convenient to write this equation as I1+2 = I0 (1 + V cos Φ) .

(3.189)

Here, I0 = I1 + I2 is defined as the resulting intensity for incoherent light, where only the intensities of the single waves have to be added. The visibility V is defined in (3.166) and is here √ √ Imax − Imin 2 I1 I2 2 I1 I2 V= = = . (3.190) Imax + Imin I1 + I2 I0

Part A 3.3

replace the plane waves by focused laser beams, i. e., Gaussian beams (Sect. 3.5), so that the region with a high electric field has a quite limited length smaller than ∆z. The beam waist of each laser beam should be at the crossing point of the two Gaussian beams on the z-axis to achieve a high amplitude of the electric vector. Then, the electric field amplitude E 0 is not constant along the z-axis but decreases outside of the beam waist like a Gaussian curve. So, it is possible to achieve a net acceleration of the electron if it is harmonized with the phase of the electric field of the laser beams when it crosses the beam waist. If the electron is not harmonized with the phase of the laser beams it can also be slowed down. The concrete calculation in the case of Gaussian beams is, of course, a little bit more complex than with plane waves because the wave vector and therefore the direction of the electric vector changes locally in the case of a Gaussian beam. By using ultrashort and focused laser pulses the resulting electric field in the beam waist, which accelerates the electron can be as high as 1 GV/m or higher. Of course, the acceleration distance is only as long as the beam waist, i. e., several µm for a strongly focused laser beam. So, some keV of kinetic energy can be gained by the electron. But, by repeating many acceleration devices in series (and, of course, harmonized in phase) the effective acceleration distance can be increased so that the laser-driven electron accelerator may become an alternative to conventional particle accelerators in the future.

3.3 Interference

116

Part A

Basic Principles and Materials

For general scalar waves the fringe period in the detector plane with coordinates (x, y) is not constant but will vary. But at a point (x, y) in the neighborhood of a fixed point (x0 , y0 ) the phase function Φ can be written as a Taylor expansion neglecting all terms of second and higher order: / . / . ∂Φ(x0 ,y0 ) x − x0 ∂x Φ (x, y) ≈ Φ (x0 , y0 ) + ∂Φ(x0 ,y0 ) · y − y0 ∂y = Φ (x0 , y0 ) + ∇⊥ Φ (x0 , y0 ) · ∆r . (3.191)

Part A 3.3

Here, the two-dimensional nabla operator ∇⊥ is introduced. The local fringe period p is defined as the distance between two fringes, i. e., the distance where the phase function increases or decreases by 2π taken along a path parallel to the local phase gradient. Therefore, we have for the vector ∆r pointing from one fringe to the next neighboring fringe at the position (x0 , y0 ) ∇⊥ Φ ⇒ ∇⊥ Φ · ∆r = p|∇⊥ Φ| = 2π; |∇⊥ Φ| 2π 2π = ⇒ p= . (3.192) |∇⊥ Φ| ∂Φ 2 ∂Φ 2 + ∂x ∂y

∆r = p

All quantities have to be calculated at the point (x0 , y0 ). By comparing this equation with (3.164) it is clear that the component G of the grating vector in the xy-plane is defined by ⎛ ∂Φ ⎞ ⎜ G = ⎜ ⎝

∂x ∂Φ ∂y

⎟ ⎟. ⎠

(3.193)

0 The grating vector G itself is defined as ⎛ ∂Φ ⎞ ⎜ G = ∇Φ = ⎜ ⎝

∂x ∂Φ ∂y ∂Φ ∂z

⎟ ⎟, ⎠

(3.194)

with Φ defined as a function of (x, y, z). The local fringe frequency ν is defined as the reciprocal of the local fringe period ∂Φ 2 ∂Φ 2 + ∂y ∂x 1 ν= = (3.195) p 2π and describes the number of fringes per length unit. If the fringe frequency is too high the interference pattern

cannot be resolved in practice since common detector arrays like a CCD camera only have a limited number of pixels per length unit and integrate the light intensity over the area of one pixel. Interference of Scalar Spherical and Plane Waves The interference of two plane waves is investigated in detail in the last section for general polarization states. Simple examples of the interference of two scalar waves are the interference of a spherical wave and a plane wave or the interference of two spherical waves. In principle, the general statements to the effects of different polarization states also hold for spherical waves as long as the numerical aperture is not too high. So, the investigation of scalar waves is not really a restriction. Two spherical waves with the wavelength λ and the corresponding wave number k = 2πn/λ having their center of curvature at the points r1 = (x1 , y1 , z 1 ) and r2 = (x2 , y2 , z 2 ) have the complex amplitude functions

eik |r − r1 | , |r − r1 | eik |r − r2 | u 2 (r) = a2 . |r − r2 | u 1 (r) = a1

In the following, only the interference pattern in the xy-plane at z = 0 in an area centered around the origin of the coordinate system at x = y = z = 0 is evaluated. Additionally, the distances |ri | (i = 1, 2) of the centers of curvature of both spherical waves from the origin of the coordinate system shall : be large compared to the maximum distance |r| = x 2 + y2 of the origin of the coordinate system from a point lying in the evaluated aperture of the interference pattern. Then the amplitude of the spherical waves can be assumed to be constant because of |ri | |r| ⇒ : |r − ri | = (r − ri ) · (r − ri ) 2 = |ri |2 + |r|2 − 2 |ri | |r| cos α ≈ |ri | . Here, α is the angle between the two vectors r and ri . Therefore, the two spherical waves are written as u (r) = A eik |r − r1 | , 1

u 2 (r) = A2 eik |r − r2 | with constant amplitudes A1 = a1 /|r1 | and A2 = a2 /|r2 |. The arguments of the complex exponential func-

Wave Optics

tions can, of course, not be replaced by the constant terms |ri | because these are very fast oscillating functions. Then, the intensity I1+2 of the interference pattern is, according to (3.188), : I1+2 (x, y) = I1 + I2 + 2 I1 I2 cos Φ (x, y) with I1 = A21 ; I2 = A22 and

(3.196)

The square roots √ can be developed into a Taylor series according to 1 + x ≈ 1 + x/2 − x 2 /8 for x 1. Since |ri | |r|, the most important terms are & x 2 + y2 − 2xxi − 2yyi |ri | 1 + |ri |2 ≈ |ri | + −

x 2 + y2 xxi + yyi − |ri | 2 |ri | 2 2 x + y2 − 2xxi − 2yyi

. (3.197) 8 |ri |3 The last term can be neglected for 2 2 x + y2 − 2xxi − 2yyi k 1 8 |ri |3 2 2 x + y2 − 2xxi − 2yyi λ . (3.198) ⇒ 2πn 8 |ri |3 By using spherical coordinates ri , ϑi , ϕi for ri , where ri is the distance from the origin, ϑi is the polar angle and ϕi is the azimuthal angle, xi = ri cos ϕi sin ϑi , yi = ri sin ϕi sin ϑi , z i = ri cos ϑi ,

(3.199) (3.200) (3.201)

the condition for neglecting the last term is 2 2 2 x + y2 x + y2 (x cos ϕi + y sin ϕi ) sin ϑi − ri2 4ri3 +

λ (x cos ϕi + y sin ϕi )2 sin2 ϑi . ri πn

(3.202)

117

If this condition is fulfilled for both spherical waves the phase function of the interference pattern can be written as πn 1 1 2 x + y2 Φ (x, y) ≈ δ + − λ r1 r2 2πn − (cos ϕ1 sin ϑ1 − cos ϕ2 sin ϑ2 ) x λ 2πn − (sin ϕ1 sin ϑ1 − sin ϕ2 sin ϑ2 ) y . λ (3.203)

The phase constant δ is defined as δ = 2πn(r1 − r2 )/λ. The term dependent on x 2 + y2 is called defocus and is proportional to the difference of the curvatures of both spherical waves. The linear terms in x and y are called tilts and are only present if the centers of curvature of the two spherical waves and the origin do not lie on a common line. In the interferometric testing of spherical surfaces or in the measurement of the wave aberrations of lenses there often appears defocus and tilts due to an axial (⇒ defocus) or lateral (⇒ tilts) misalignment of the test object. Then the coefficients of these terms are determined by a least-squares fit of the function Φmisalign = a + bx + cy + d x 2 + y2 (3.204) to the measured phase function Φmeasured . Afterwards, the phase function Φreduced , which is freed from misalignment aberrations is calculated by Φreduced = Φmeasured − Φmisalign

(3.205)

using the fitted coefficients a, b, c and d. The phase function Φreduced then only contains the desired wave aberrations or the desired surface deviations from the ideal surface plus systematic errors of the experimental setup. A special case of (3.203) is that one of the waves is a plane wave. Without loss of generality the second wave shall be plane. This means that the parameter r2 is infinity. Then (3.203) reduces to: πn 2 x + y2 Φ (x, y) ≈ δ + λr1 2πn cos ϕ1 sin ϑ1 − ex,2 x − λ 2πn − sin ϕ1 sin ϑ1 − e y,2 y , (3.206) λ where ex,2 := cos ϕ2 sin ϑ2 and e y,2 := sin ϕ2 sin ϑ2 are the x and y components of the unit vector e2 = k2 /|k2 | parallel to the wave vector k2 of the plane wave. Of course, the phase constant δ, which is only defined modulus 2π, is therefore not infinity, but has a certain value

Part A 3.3

Φ (x, y) = k (|r − r1 | − |r − r2 |) & . x 2 + y2 − 2xx1 − 2yy1 = k |r1 | 1 + |r1 |2 & / x 2 + y2 − 2xx2 − 2yy2 . − |r2 | 1 + |r2 |2

3.3 Interference

118

Part A

Basic Principles and Materials

that depends on r1 and the phase offset of the plane wave at (x = 0, y = 0, z = 0). In the case of the interference of a spherical and a plane wave the defocus term is directly proportional to the curvature of the spherical wave. The tilt terms again depend on both waves but for the case that either the spherical wave has its center of curvature at x = y = 0 (⇒ sin ϑ1 = 0) or the plane wave is perpendicular to the xy-plane (⇒ ex,2 = e y,2 = 0) the tilt terms depend only on the parameters of one wave.

Part A 3.3

Two Examples of Interference Patterns Assume that we have two interfering monochromatic waves with a wavelength λ = 0.5 µm. Additionally, we know that the first of the waves is a plane wave that propagates parallel to the optical axis, which is defined to be parallel to the surface normal of the detector and to intersect the detector in its center. So, the parameters of the first wave in (3.203) are r1 → ∞, ϑ1 = 0 and ϕ1 = 0 y(mm) 0.4

and the phase function Φ of the interference pattern depends only on the parameters of the second wave 2π π 2 x + y2 + Φ (x, y) ≈ δ − cos ϕ2 sin ϑ2 x λr2 λ 2π sin ϕ2 sin ϑ2 y . (3.207) + λ The refractive index n has been set to 1 because the measurements should be made in air. Let us now assume that we detect the interference pattern, which is displayed in Fig. 3.9. Such an interference pattern is often called an interferogram. In this case it has straight, parallel and equidistant fringes parallel to the y-axis. Therefore, the second wave also has to be a plane wave (r2 → ∞), which can be described by the two angles ϕ2 and ϑ2 . The interference pattern only changes along the x-axis and there the period p is 0.2 mm. Therefore, the phase function Φ has to be of the form Φ(x, y) = ax with a = 2π/ p = 10π/mm. Comparing this with (3.207) results in 2π 2π cos ϕ2 sin ϑ2 x + sin ϕ2 sin ϑ2 y = ax; λ λ aλ = 0.0025 . ⇒ δ = 0 ∧ ϕ2 = 0 ∧ sin ϑ2 = 2π

δ+

0.2 0

(3.208)

–0.2

By looking at the intensity pattern it can also be seen that the visibility has the maximum value of V = (Imax − Imin )/(Imax + Imin ) = (2 − 0)/(2 + 0) = 1. Therefore, the second plane wave has the same intensity as the first wave. In a second (simulated) measurement the interferogram of Fig. 3.10 is obtained. It can be estimated that the local fringe frequency increases linearly with the distance from the center and that we therefore have a defocus term with a quadratic phase function. No linear phase function is present and therefore the tilt angle ϑ2 of the second wave has to be zero. A more detailed evaluation of the intensity pattern using the interference equation (3.188) confirms the estimation that the phase function is Φ(x, y) = b(x 2 + y2 ) with b = 20π/mm2 . Additionally, it can be seen that the visibility is V = (1.28 − 0.72)/(1.28 + 0.72) = 0.28. Using (3.207) to calculate the radius of curvature r2 of the second wave results in π 2 x + y2 = b x 2 + y2 Φ(x, y) = − λr2 π = 100 mm . (3.209) ⇒ |r2 | = bλ

–0.4 –0.4

–0.2

Intensity (arb. units) 0.5

0.2 1

1.5

0.4 x(mm) 2

Intensity

1.5 1 0.5

–0.4

–0.2

0.2

0.4 x(mm)

Fig. 3.9 Example of an interference pattern (interfero-

gram) with straight, parallel and equidistant fringes. Top: (simulated) camera picture as also seen directly with the eye. Bottom: section through the intensity function

Wave Optics

y(mm) 0.4

0.2

–0.4 –0.4 –0.2 Intensity (arb.units) 0

0.2 0.5

0.4 x (mm) 1

Intensity

3.3.4 Some Basic Ideas of Interferometry

1.2 1 0.8 0.6 0.4 0.2 0

–0.4

This means that the intensity of the second wave is either about 50 times higher than that of the first wave or about 50 times smaller. Which one of these two values is valid can be determined by measuring first the intensity I1 of the first wave alone and then the intensity of the interference pattern and therefore also I1 + I2 . The example also shows that the visibility decreases quite slowly if the intensity difference factor α increases. This is the reason why, e.g., a scattered spherical wave of a dust particle, which has quite a small intensity compared to the intensity of the illuminating coherent wave, produces in many cases quite high-contrast fringes that disturb the measurement. These two examples are, of course, quite simple and can be evaluated manually. However, already in the second example it can be seen that it is not so easy to decide whether it is really a pure defocus term or mixed with some other terms. Therefore, in practice, an automated evaluation of the interference pattern has to be made [3.20]. One step to this is the phase shifting technique which will be discussed briefly on page 121.

–0.2

0.2

0.4 x (mm)

Fig. 3.10 Example of an interference pattern showing a de-

focus term with a low contrast. Top: (simulated) camera picture as also seen directly with the eye. Bottom: section through the intensity function

The sign of r2 cannot be detected in this case where only one interferogram without a carrier frequency is known. This is quite clear because the cosine function is an even function and so cos Φ = cos(−Φ). By using (3.190) and the approach I2 = αI1 the coefficient α can be calculated using the visibility V √ 2 α V= 1+α 2 ⇒ α = 2 −1± V

3 2 2 49 −1 −1 = . V2 0.02 (3.210)

The basic principle of an interferometer is that an incident wave is divided into two waves, which can then interfere with each other. In most interferometers, like, e.g., a Michelson or a Mach–Zehnder interferometer, there exists a so-called reference arm and an object arm. The object arm often contains an object to be tested that changes the object wave. Together with the unchanged wave of the reference arm the interference pattern is formed and carries information about the test object. Nevertheless, there are also interferometers, like, e.g., shearing interferometers, which do not have an object and a reference arm but two copies of an object wave that interfere. The applications of interferometers are the measurement of surface deviations or aberrations of optical elements and wavefront characterization. Another application is the high-precision length measurement. In the following, the principles of the most important two-beam interferometers with monochromatic light are described. Other types of interferometers, which are not treated here, are interferometers with two or more wavelengths and so-called white-light interferometers with a broad spectrum of wavelengths. Additionally, there are multiple beam interferometers where three or more light beams interfere, e.g., a Fabry–Perot interferometer. For more information about interferometry we refer to [3.17–20, 23–28].

119

Part A 3.3

–0.2

3.3 Interference

120

Part A

Basic Principles and Materials

Part A 3.3

Michelson Interferometer One of the simplest interferometers is the Michelson interferometer (Fig. 3.11). A plane wave is divided by a beam splitter into two plane waves. One of these plane waves hits the reference mirror and the other hits the object mirror. Both waves are then reflected back and each wave is again divided into two plane waves. Therefore, the Michelson interferometer has two exits, where one is identical to the entrance so that only the other can really be used. If the beam splitter is exactly oriented at 45◦ relative to the incoming plane wave and both mirrors are exactly perpendicular to the plane waves the wave vectors k1 and k2 of the two plane waves at the exit are parallel, i. e., k1 = k2 . Then, according to (3.170), which is also valid for scalar waves, the intensity in the whole space behind the exit, e.g., in the detector plane, which shall be perpendicular to the z-axis, depends only on the phase difference δ, which is constant over the whole exit pupil: : I1+2 = I1 + I2 + 2 I1 I2 cos [(k1 − k2 ) · r + δ] : (3.211) = I1 + I2 + 2 I1 I2 cos δ .

In this case one speaks of fluffed out fringes because no interference fringes are present. Depending on the optical path difference δ between object and reference arm the intensity can have a maximum or a minimum. If the object mirror is axially shifted the intensity on the detector changes periodically and one period corresponds to an axial shift ∆z of half a wavelength in the material with refractive index n, in which the light propagates (normally air), because of the double pass arrangement λ . (3.212) ∆z = 2n If the beam splitter or one of the mirrors are tilted, there are interference fringes on the detector. An axial

shift of the object mirror then causes a lateral movement of the fringes. Again, the fringes move by one period if the axial shift is λ/(2n). A typical application of a Michelson interferometer is the length measurement whereby the relative shift of the object mirror can be measured. There are several variations of the Michelson interferometer. One is a Twyman–Green interferometer, where a lens is placed in the object arm and instead of using a plane object mirror a spherical mirror is used. In the case of the ideal adjustment of the interferometer the focus of the lens and the center of curvature of the spherical mirror have to coincide. If the quality of either the lens or the spherical mirror (and of all other components of the interferometer) is known, the resulting interference pattern can be used to determine the errors of the other component. Mach–Zehnder Interferometer Another very important interferometer is a Mach– Zehnder interferometer (Fig. 3.12). There the light of an incoming plane wave is again divided by a beam splitter into two waves. Then the transmitted plane wave is reflected at the upper mirror and passes the second beam splitter or is reflected at it. The plane wave reflected at the first beam splitter is reflected at the lower mirror and can pass an optional transmissive object to be tested. At the second beam splitter this wave can be transmitted or reflected. So, the Mach–Zehnder interferometer has two exits, which can both be used. Similarly to the case of the Michelson interferometer there are fluffed out fringes if all mirrors and beam splitters are oriented by exactly 45◦ relative to the incoming plane wave. If one of the mirrors or beam splitters is tilted there are fringes on the detector. Beam splitter 1

Exit

Entrance

Reference arm

Mirror 2 (reference arm)

Entrance Beam splitter Mirror 1 (object arm)

Fig. 3.11 Basic principle of a Michelson interferometer

Exit 1 Object

Beam splitter 2 Exit 2

Fig. 3.12 Basic principle of a Mach–Zehnder interferometer

Wave Optics

Shearing Interferometer An interesting interferometer that needs no external reference arm is a shearing interferometer. There, by some means, a copy of the object wave is generated, which is either laterally or radially sheared [3.19, 29]. Here, only the case of lateral shearing will be discussed. The coordinate system is chosen in such a way that the shearing is along the x-axis by a distance ∆x. Then, the phases of the two copies are Φ1 (x, y) = Φo (x + ∆x, y) and Φ2 (x, y) = Φo (x, y), whereby Φo is the phase of the object wave itself. The phase difference Φ, which appears in the interference term of the interference (3.188), is then

Φ(x, y) = Φ1 (x, y) − Φ2 (x, y)

≈ ∆x

∂Φo (x, y) . ∂x

The approximation of taking the first partial derivative at the point (x, y) is valid for small shearing distances ∆x. The shearing interferometer can be similarly evaluated as other interferometers with phase shifting techniques and phase unwrapping, whereby a continuous function for Φ ≈ ∆x∂Φo /∂x results. To obtain the phase Φo of the wavefront itself a kind of integration has to be made [3.19], [3.30]. To obtain an unambiguous wavefront both partial derivatives of Φo in the x and y directions have to be determined before the integration can be performed. Phase-Shifting Interferometry A typical technique to extract the object phase from the measured intensity values is phase-shifting interferometry [3.25, 31]. There, the reference mirror (or the object) is axially shifted by a well-known small distance and at least three different intensity distributions with different but well-known reference phases have to be observed, whereby there are also other possibilities to shift the phase [3.32]. By shifting the reference phase by δφ, which is a well-defined integer multiple m of π/2 (m) the intensity of the interference pattern I1+2 changes according to (3.188) as : (m) I1+2 = I1 + I2 + 2 I1 I2 cos (Φ + δφ) : π . (3.214) = I1 + I2 + 2 I1 I2 cos Φ + m 2 Three measurements with different reference phases are in principle enough because (3.214) contains the three unknowns I1 , I2 and the desired phase Φ. It is quite simple to combine the three measured intensity distri(0) (1) (2) butions I1+2 , I1+2 and I1+2 with different values m in order to calculate Φ: : (0) I1+2 = I1 + I2 + 2 I1 I2 cos Φ , : π (1) I1+2 = I1 + I2 + 2 I1 I2 cos Φ + 2 : = I1 + I2 − 2 I1 I2 sin Φ , : (2) I1+2 = I1 + I2 + 2 I1 I2 cos (Φ + π) : = I1 + I2 − 2 I1 I2 cos Φ; (0)

(2)

⇒ I1+2 + I1+2 = 2 (I1 + I2 ) , : (0) (2) I1+2 − I1+2 = 4 I1 I2 cos Φ, : (0) (2) (1) I1+2 + I1+2 − 2I1+2 = 4 I1 I2 sin Φ, (0)

⇒ tan Φ =

= Φo (x + ∆x, y) − Φo (x, y) (3.213)

121

(2)

(1)

I1+2 + I1+2 − 2I1+2 (0)

(2)

I1+2 − I1+2

(3.215)

However, this simple phase shifting algorithm with only three measurements is quite sensitive to phase shifting

Part A 3.3

A Mach–Zehnder interferometer can be used to detect inhomogeneities of an optional object in the object arm. A special type is, e.g., an interferometer where the object is a tube filled with gas. By changing the pressure or the temperature in the tube the refractive index n is changed. Then the optical path difference between the object and the reference arm also changes and the fringes move or, in the case of fluffed out fringes, the overall intensity changes. So, the dependence of the refractive index of the gas on the pressure and temperature can be measured. In another type of Mach–Zehnder interferometer the object is a combination of a well-known lens and a lens to be tested, which together form a telescope. Using phase shifting interferometry (see later) by shifting one of the mirrors the errors of the lens to be tested can be determined. Of course, in practice there are always so-called adjustment errors that have to be eliminated from the measurement results and there are also systematic errors of the other components in the setup. Additionally, a very important fact is that the object to be tested has to be imaged onto the detector using auxiliary optics. Then, it is possible to say that the measured errors correspond to errors of the object at a certain point. This is, of course, also valid for the Twyman–Green interferometer and all other interferometers used for the measurement of an optical object with a limited depth. In an interferometer for the measurement of refractive index changes where the object is very long, it is not possible to image the complete object sharply onto the detector.

3.3 Interference

122

Part A

Basic Principles and Materials

Part A 3.3

errors, i. e., if the phase shifts are not integer multiples of π/2. If more measurements are made, the correction of phase shifting errors is possible [3.20, 26, 31]. A principal problem of two beam interferometry is that the phase values obtained with a phase shifting algorithm, like, e.g., with (3.215), are only defined modulo 2π. (Note: The arctangent function itself is only unambiguously defined between −π/2 and +π/2. But by evaluating the signs of the numerator and denominator of (3.215) the phase Φ can be unambiguously calculated between −π and +π.) Therefore, so-called phase unwrapping algorithms [3.20] have to be used in order to obtain a continuous phase profile of, e.g., the surface deviations or wave aberrations of a lens. Some Ideas on the Energy Conservation in Interferometers Here, some principal ideas to the conservation of energy in an interferometer shall be given because the laws of energy conservation have to be fulfilled everywhere in optics. Let us consider, e.g., the Mach–Zehnder interferometer of Fig. 3.12. We assume that each beam splitter has a splitting ratio of 1:1, i. e., half of the light power is transmitted and half is reflected. So, if the intensity of the incoming plane wave is I0 at the entrance, the intensities of the transmitted and reflected plane waves are each I0 /2. The two mirrors are assumed to reflect all light without losses and diffraction effects at apertures are also neglected because we use plane waves. So, at the second beam splitter each of the plane waves is again divided into two waves with equal intensity, i. e., each of the four waves now has the intensity I0 /4. At exit 1, two of these waves interfere and we assume that the phase difference Φ between these two waves shall be zero or an integer number times 2π. Then, the resulting intensity I1+2 is, according to (3.188),

: I1+2 = I1 + I2 + 2 I1 I2 cos Φ & I2 I0 I0 = + + 2 0 = I0 . 4 4 16

odd-numbered multiple of π & I2 I0 I0

I1+2 = + + 2 0 cos Φ

4 4 16 = 0 ⇒ Φ = (2m + 1) π

(3.217)

with an integer number m. So, if we take the basic solution Φ = π this requires that the phase shift between the transmitted and the reflected wave at each beam splitter is half of this value, i. e., π/2. Then, the law of energy conservation is fulfilled. To explain this further consider Fig. 3.12 regarded. The phase differences between the two waves interfering at exits 1 and 2, respectively, due to the geometrical path are identical for both exits. The same is valid for reflections at the mirrors because each wave is exactly reflected once at a mirror. So, there has to be a phase shift between a reflected and a transmitted wave at a beam splitter in order to fulfill the law of energy conservation. The two waves that interfere at exit 1 (symmetrical exit) are each reflected one time a a beam splitter and transmitted one time by a beam splitter. So, the assumed phase shifts of π/2 due to a reflection at a beam splitter cancel out each other because the phase difference between both interfering waves is taken. But, at the exit 2 (antisymmetric exit) the first wave is transmitted by both beam splitters and the other wave is reflected at both beam splitters. Therefore, the phase difference between the two interfering waves is in this case π Φ = Φ + 2 = Φ + π . (3.218) 2 This guarantees that the sum of the intensities I1+2 +

I1+2 at both exits is equal to the intensity at the entrance: & ⎞ ⎛ 2 I I I 0 0

I1+2 + I1+2 = ⎝ + + 2 0 cos Φ ⎠ 4 4 16 & ⎞ I02 I I 0 0

+⎝ + +2 cos Φ ⎠ 4 4 16 ⎛

(3.216)

This, however, means that all of the incoming light power

has to be at exit 1. Therefore, the intensity I1+2 at exit 2 has to automatically be zero. Since the intensities of the two single waves, which interfere at exit 2 are also I0 /4, this is only possible if the phase difference Φ is an

I0 I0 (1 + cos Φ) + (1 − cos Φ) = I0 . 2 2 (3.219)

So, the energy is conserved if the phase shift of a wave, which is reflected at a beam splitter, is π/2 compared to the phase of the transmitted wave.

Wave Optics

3.4 Diffraction

123

3.4 Diffraction dicular to the z-axis) to another parallel plane in the distance z 0 in a homogeneous and isotropic material with the refractive index n. The only approximation here is that u is assumed to be a scalar function. But, since a plane wave can easily be defined by taking into account polarization (Sect. 3.2) an extension of this formalism is possible, but will not be treated here. According to (3.52) a scalar plane wave u(r) = u 0 eik · r

(3.220)

fulfills the Helmholtz equation (3.109), which is written for scalar waves as ∇ 2 + k2 u (x, y, z) = 0 . (3.221) The condition for the modulus |k| = k of the wave vector is 2 2πn . (3.222) |k| = k2x + k2y + k2z = λ According to the linearity of the Helmholtz equation a sum of plane waves with different directions of propagation is also a solution of the Helmholtz equation and in the limit a continuous spectrum of plane waves is a solution (Fig. 3.13). There, the integration has to be done over two angles or more exactly over two components of the wave vector. The third component is then automatically defined by (3.222) as long as only plane waves propagating in the positive z direction are taken into account, which will be the case here. Since the complex amplitude is always regarded in a plane perpendicular to the z-axis, the two components of the wave

3.4.1 The Angular Spectrum of Plane Waves The knowledge of the angular spectrum of plane waves allows the exact propagation of a complex amplitude function u from one plane (which is chosen perpen-

Fig. 3.13 Arbitrary scalar wave as superposition of plane

waves

Part A 3.4

Up to now we have mostly investigated the propagation of plane waves and other waves that are not affected by any limiting apertures. A plane wave has, e.g., an infinite spatial extension and therefore it does not exist in the real world. Nevertheless, if the diameter of the limiting aperture is very large compared to the wavelength of the light, a plane wave can be a quite good approximation if the propagation distance is not very large. But also in this case there are disturbances at the rim of the wave which are called diffraction effects. In this section diffraction theory will mostly be treated for scalar waves and only at the end of this section will the influence of polarization effects on the electric energy density in the focal region of a lens be treated [3.33, 34]. In contrast to most textbooks on optics like [3.1, 35], we will not start historically with the Huygens–Fresnel principle or with the integral theorem of Helmholtz and Kirchhoff, but with the angular spectrum of plane waves. Only Kirchhoff’s boundary conditions will be used, i. e., a wave that is incident on an absorbing screen with a hole will be undisturbed in the area of the hole and completely absorbed in the other parts of the screen. Starting from the angular spectrum of plane waves the Fresnel– Kirchhoff diffraction integral will be derived and it will be shown that both formulations are nearly equivalent [3.36–38]. The approximations of Fresnel diffraction and Fraunhofer diffraction will be discussed afterwards. A quite interesting application of Fraunhofer diffraction is, e.g., the calculation of the intensity distribution in the focal region of a lens [3.1, 39]. Afterwards, some ideas to the numerical implementation of scalar diffraction formula are given [3.38]. At the end of this section we will briefly reflect on the combination of polarization and diffraction by using the superposition of plane waves taking into account their polarization states. This is used to calculate the influence of polarization effects to the electric energy density in the focal region of a lens. There are many modern applications of diffraction and interference effects in optical holography [3.40– 43] and computer-generated diffractive optics [3.44–50]. But, there is no space to cover these fields in this chapter and we refer therefore to the literature.

124

Part A

Basic Principles and Materials

Part A 3.4

vector used for the integration will be the x and y components. To obtain a symmetrical formulation the vector ν of the spatial frequencies is introduced by ⎛ ⎞ νx 1 ⎜ ⎟ ν= k = ⎝ ν y ⎠ with 2π νz 2 n (3.223) |ν| = νx2 + ν2y + νz2 = . λ The complex amplitude u(r) of a wave can then be written as a superposition of plane waves +∞ +∞ u˜ νx , ν y e2πiν · r dνx dν y . (3.224) u(r) = −∞ −∞

The integration takes into account arbitrary spatial frequencies νx and ν y so that later the mathematical formalism of the Fourier transformation [3.51, 52] can be used. To fulfill (3.222) the z component of the vector of the spatial frequencies is defined by the x and y components since we allow only waves propagating in the positive z direction & n2 − νx2 − ν2y . (3.225) νz = λ2 Nevertheless, the square root only delivers a real solution for a positive argument. Therefore, two cases have to be distinguished & n2 2πiz − νx2 − ν2y 2 n λ2 νx2 + ν2y ≤ 2 ⇒ e2πiνz z = e , λ &

to (3.224) it holds that u 0 (x, y, 0)

+∞ +∞ u˜ 0 νx , ν y , 0 e2πi(νx x + ν y y) dνx dν y , = −∞ −∞

where u˜ 0 (νx , ν y , 0) is the Fourier transform of u 0 in the plane at z = 0 and can be calculated using the Fourier relation u˜ 0 νx , ν y , 0 (3.229) =

+∞ +∞ u 0 (x, y, 0)e−2πi(νx x + ν y y) dx dy .

−∞ −∞

Since u˜ 0 is now known the complex amplitude u in another parallel plane at z = z 0 can be calculated with (3.224) and (3.225): u(x, y, z 0 ) =

+∞ +∞ u˜ 0 νx , ν y , 0 e2πi νx x + ν y y

−∞ −∞

× e2πiνz z 0 dνx dν y ; ⇒ u(x, y, z 0 )

(3.227)

where in both cases the result of the square root is a real number. The second case corresponds to an exponentially decreasing amplitude and so the waves with such high spatial frequencies propagate only along very small distances z of the range of some wavelengths and are called evanescent waves. If the complex amplitude u 0 of a wave is known in a plane and the coordinate system is chosen such that this plane is perpendicular to the z-axis at z = 0 according

(3.230)

+∞ +∞ = u˜ 0 νx , ν y , 0 −∞ −∞

(3.226)

n2 −2πz νx2 + ν2y − 2 2 n λ , νx2 + ν2y > 2 ⇒ e2πiνz z = e λ

(3.228)

& λ2 nz 0 1 − 2 νx2 + ν2y 2πi λ n ×e × e2πi νx x + ν y y dνx dν y .

(3.231)

So, this is an inverse Fourier transformation where the function u˜ νx , ν y , z 0 & nz 0 λ2 2 2 2πi λ 1 − n 2 νx + ν y = u˜ ν , ν , 0 e 0

(3.232)

has to be Fourier transformed. In total, by using a Fourier transformation (3.229), multiplying u˜ 0 with the propagation factor exp(2πiνz z 0 ), and applying an inverse Fourier transformation ((3.231) for both operations) the complex amplitude in a plane parallel to the original

Wave Optics

plane in the distance z 0 can be calculated. Hereby, it must be taken into account that according to (3.226) and (3.227) the propagation factor exp(2πiνz z 0 ) can either be a pure phase factor (for νx2 + ν2y ≤ n 2 /λ2 ) or an exponentially decreasing real term. The propagation factor is also known as the transfer function of free space H & nz 0 λ2 2πi 1 − 2 νx2 + ν2y λ n H ν ,ν ,z = e . x

u(x, y, z 0 ) =

+∞ +∞ u˜ 0 νx , ν y , 0

−∞ −∞

(3.233)

=−

1 2π

+∞ +∞ u˜ 0 νx , ν y , 0 H νx , ν y , z 0 u(x, y, z 0 ) =

(3.234)

3.4.2 The Equivalence of the Rayleigh– Sommerfeld Diffraction Formula and the Angular Spectrum of Plane Waves According to the convolution theorem of Fourier mathematics, (3.234) can be written as a convolution of two functions, whereby these two functions are the inverse Fourier transforms of u˜ 0 and H. The inverse Fourier transform of u˜ 0 is, due to (3.228), the complex amplitude distribution u 0 at z = 0. The other term is not so obvious. But in [3.36] and [3.37] it is shown that the following relation is valid & nz λ2 0 +∞ +∞ 2πi 1 − 2 νx2 + ν2y λ n e × e2πi νx x + ν y y dνx dν y ⎛ nr ⎞ . / 2πi ikr λ ⎟ e 1 ∂ 1 ∂ ⎜ e ⎜ ⎟=− =− ⎠ 2π ∂z 0 ⎝ r 2π ∂z 0 r

−∞ −∞

1 1 z 0 eikr =− ik − , 2π r r r 2 whereby r := x 2 + y2 + z 20 and k := 2πn/λ.

λ2 ikz 0 1 − 2 νx2 + ν2y n ×e × e2πi νx x + ν y y dνx dν y

+∞ +∞ 1 u 0 x , y , 0 ik − l

−∞ −∞

z 0 eikl

dx dy l l

(3.236)

with l :=

2 (x − x )2 + (y − y )2 + z 20 .

(3.237)

The right-hand side of (3.236) is known as the general Rayleigh–Sommerfeld diffraction formula. So, the complex amplitude u(x, y, z 0 ) can either be expressed as a superposition of plane waves or as a convolution of the original complex amplitude u 0 (x, y, 0) with a spherical Huygens wavelet h of the form h (x, y, z 0 ) = −

1 2π

1 z 0 eikr ik − , r r r

(3.238)

which is the inverse Fourier transform of the transfer function of free space H; h is also called the impulse response since it results if the stimulating complex amplitude u 0 (x, y, 0) has the form of a δ function. Equation (3.236) is a mathematical expression of the Huygens–Fresnel principle. The term z 0 /l is the cosine obliquity factor. By using (3.236) and (3.238) the complex amplitude can be written as u(x, y, z 0 )

(3.239)

+∞ +∞ = u 0 x , y , 0 h x − x , y − y , z 0 dx dy , −∞ −∞

(3.235)

i. e., as a convolution of the original complex amplitude u 0 in the first plane and the impulse response h. In most cases the term r is much larger than the wavelength in the medium λ/n. Then the relation k

Part A 3.4

Then, (3.231) can be written as

× e2πi νx x + ν y y dνx dν y .

125

So, in total (3.231) can be written as

−∞ −∞

3.4 Diffraction

126

Part A

Basic Principles and Materials

1/r is valid and the impulse response of (3.238) can be written as 1 1 z 0 eikr h (x, y, z 0 ) = − ik − 2π r r r n z 0 eikr . (3.240) λ r r Then (3.236) results in the more familiar but less general Rayleigh–Sommerfeld diffraction formula ≈ −i

u(x, y, z 0 )

Part A 3.4

≈ −i

n λ

+∞ +∞ z 0 eikl

dx dy . (3.241) u 0 x , y , 0 l l

−∞ −∞

In the case that the complex amplitude u 0 is different from zero inside an aperture A and zero outside of the aperture A (3.241) is also known as the Fresnel– Kirchhoff diffraction integral [3.1]. Then, the effective integration is not carried out from minus infinity to plus infinity but on the area of the aperture A. It has to be mentioned that (3.236) can also be written in a similar form that does not explicitly assume that the original complex amplitude, which is now named u 0 (r ) has to be known in a plane and that the complex amplitude u(r), which has to be calculated, is also defined in a plane. The generalization is 1 1 u (r) = − u 0 r ik − 2π l A

N · (r − r ) eikl dS . (3.242) l l Here, r = (x , y , z ) defines an arbitrary point on a curved surface (the aperture A) where the original complex amplitude u 0 is defined and r = (x, y, z) is an arbitrary point on the second curved surface on which the complex amplitude u has to be calculated. Additionally, N is a unit vector perpendicular to the aperture A at the point r . The integration is done over the aperture A and the integration element dS indicates a two-dimensional integration. Additionally, the distance l is then defined by: 2 (3.243) l := |r − r |2 . ×

Nevertheless, in the following we will always assume that both complex amplitudes u 0 and u are defined in parallel planes. This allows, as we will see in the next section, that approximate integrals like the Fresnel diffraction integral can be numerically calculated using the efficient fast Fourier transformation.

3.4.3 The Fresnel and the Fraunhofer Diffraction Integral In the following, we always assume that the plane aperture A on which the integration of the diffraction integral is carried out is limited and has a maximum diameter of D. There is no other restriction on the form of the aperture, which can be circular, rectangular or irregularly formed. The parameter D is therefore the diameter of a circle, which contains the aperture and which is centered around the z-axis. Again, we have the complex amplitude u 0 (x , y , 0) in a first plane at z = 0, which is zero outside of the aperture. Additionally, the distance z 0 of the second plane to the first parallel plane is much larger than the diameter D of the aperture A, i. e., D z0 . Then, the distance l (see (3.237)) of a point P =

(x , y , 0) in the first plane and a point P = (x, y, z 0 ) in the second plane can be written as (Fig. 3.14) 2 (x − x )2 + (y − y )2 + z 20 2 = x 2 + y2 + z 20 + x 2 + y 2 − 2xx − 2yy

& 2 x 2 + y 2 − 2xx − 2yy

2 2 2 = x + y + z0 1 + . x 2 + y2 + z 20

(3.244)

We define the term l0 as l0 :=

x 2 + y2 + z 20 D ⇒

D 1. l0

(3.245)

Because of x ≤ D/2, y ≤ D/2 and (3.245) all terms on the order of x 3/l03 , y 3/l03 and higher can be neglected. Then l can be approximated by the first terms of its y

P x'

l z

Fig. 3.14 Coordinate systems used in the calculation of the

different diffraction integrals

Wave Optics

Taylor expansion: & x 2 + y 2 − 2xx − 2yy

l = l0 1 + l02 ⎛

−

x 2 + y 2 − 2xx − 2yy

2 ⎞ ⎟ ⎠

8l04

π ⇒ n

8l03

x 2 + y 2

4λl03

The last two terms are on the order of x /l03 , y 3 /l03 or higher. Therefore, they can be neglected and the result is 2

xx + yy

x 2 + y 2 xx + yy

− − , l ≈ l0 + 2l0 l0 2l03 (3.247)

where l in the denominator of the Huygens wavelet of (3.241) can then be replaced with a good approximation by only the first term of the Taylor expansion, i. e., l0 , and the cosine obliquity factor z 0 /l can be replaced by the term z 0 /l0 . This means that the Fresnel–Kirchhoff diffraction integral (3.241) can be approximated by u(x, y, z 0 ) +∞ +∞ u 0 x , y , 0 eikl dx dy ,

The Fresnel Diffraction Integral In the paraxial regime, i. e., x 2 + y2 ≤ D2 , we can write

x 2 + y2 ≤ D2 z 20 & x 2 + y2 x 2 + y2 ⇒ l0 = z 0 1 + ≈ z0 + . (3.251) 2 2z 0 z0 The second term is only interesting in the rapidly oscillating exponential factor exp(ikl0 ). In the other cases we can write l0 ≈ z 0 and z 0 /l0 ≈ 1. Additionally, we have to approximate 1/l0 in some terms in the exponential factor . /−1/2 x 2 + y2 1 1 1 x 2 + y2 1+ = ≈ − . 2 l0 z0 z0 z0 2z 30 (3.252)

Since the term 1/l0 appears only in terms that are themselves on the order of x 2 , y 2 or xx , yy , the second term (x 2 + y2 )/(2z 30 ) can be neglected because it would lead to terms of higher order. In the paraxial regime the last term in (3.247) can also be neglected. Finally, we obtain from (3.248): u(x, y, z 0 )

−∞ −∞

(3.248)

where l is defined via (3.247). This equation is valid as long as the last two terms of (3.246) can be neglected in the exponential factor exp(ikl). Therefore, two conditions have to be valid: x 2 + y 2 xx + yy

π k 2l03 x 2 + y 2 xx + yy

1, (3.249) ⇒n λl03

n i e = −i λz 0

x 2 + y2 2πnz 0 iπn λz 0 λ e

iπn x , y , 0 e

x 2 + y 2 λz 0

−2πin ×e

xx + yy

λz 0 dx dy .

(3.253)

This is the Fresnel diffraction integral, where the integration is made over the aperture A. The condition for the

Part A 3.4

(3.246)

n z0 λl0 l0

There are two especially interesting approximations of the Fresnel–Kirchhoff diffraction integral. The first is the Fresnel diffraction integral where only points P in the neighborhood of the axis, i. e., in the paraxial regime, are considered and the second is the Fraunhofer diffraction integral where only points in the far field or in the focal plane of a lens are considered.

2 xx + yy

x 2 + y 2 xx + yy

− − = l0 + 2l0 l0 2l03 2 x 2 + y 2 xx + yy

x 2 + y 2 + − . 2l03 8l03

≈ −i

127

(3.250)

x 2 + y 2 xx + yy

⎜ ≈ l0 ⎝1 + − 2l02 l02

x 2 + y 2

3.4 Diffraction

128

Part A

Basic Principles and Materials

validity of the Fresnel diffraction integral is, according to (3.249) and (3.250), x 2 + y 2 xx + yy

n 1 λl03 2 x 2 + y 2 1 and n 4λl03 ⇒ Q Fresnel :=

n(D/2)4 λz 30

n D4 16λz 30

(3.254)

Part A 3.4

As an example, we take n = 1, λ = 0.5 µm, D = 10 mm and z 0 = 1 m. Then the term Q Fresnel has the value Q Fresnel = 0.00125 and the condition for the validity of the Fresnel diffraction integral is very well fulfilled. If the distance z 0 is only 0.1 m the term Q Fresnel is 1.25 and the Fresnel approximation is at the limit of its validity. This shows that the Fresnel diffraction integral is a good approximation in a distance between the near and the far field. In the near field (which ranges from z 0 = 0 up to a distance z 0 of several times D) the Fresnel– Kirchhoff diffraction integral or the angular spectrum of plane waves has to be used. In the far field there is another more simple approximation, the Fraunhofer diffraction formula, which will be discussed in the next section. But before doing this the Fresnel diffraction integral of (3.253) will be discussed a little further. Equation (3.253) shows that the integral itself is formally the Fourier transformation of the function f x , y

⎧ ⎪ u x , y , 0 exp iπn x 2 +y 2 ⎪ ⎨ 0 λz 0 = if (x , y ) ∈ A . (3.255) ⎪ ⎪ ⎩ 0 if (x , y ) ∈ /A This gives the quite efficient possibility of calculating the integral numerically by using a fast Fourier transformation (FFT) [3.52]. But (3.253) can also be written in a different form 2πnz 0 n i u(x, y, z 0 ) = −i e λ λz 0 2 2 x − x + y − y

iπn λz 0 u 0 x , y , 0 e dx dy . × A

(3.256)

So, this form shows the Fresnel diffraction integral as a convolution of the functions u 0 and exp[iπn(x 2 +

y 2 )/(λz 0 )]. According to the convolution theorem the Fresnel diffraction integral can then formally be written as u(x, y, z 0 )

(3.257)

2πnz 0 n i = −i e λ FT−1 λz 0 ⎧ ⎧ ⎫⎫ ⎪ ⎪ ⎪ x 2 + y 2 ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ < ⎨ iπn ⎬⎪ = λz 0 . × FT u 0 (x , y , 0) · FT e ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ⎩ ⎭⎪ Now, according to (3.229) we have u˜ 0 νx , ν y , 0 +∞ +∞

u 0 (x , y , 0)e−2πi(νx x + ν y y ) dx dy

= −∞ −∞

< = = FT u 0 (x , y , 0) .

(3.258)

The second Fourier pair is ⎧ ⎫ ⎪ x 2 + y 2 ⎪ ⎪ ⎪ ⎨ iπn ⎬ λz 0 FT e ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

2 +∞ +∞ iπn x + y

λz 0 e−2πi(νx x + ν y y ) dx dy

= e −∞ −∞

λz 0 2 νx + ν2y λz 0 −iπ n . e =i n

(3.259)

In total, (3.257) results in u(x, y, z 0 ) i =e

2πnz 0 +∞ +∞ λ u˜ 0 νx , ν y , 0 −∞ −∞

λz 0 2 νx + ν2y −iπ n ×e × e2πi(νx x + ν y y) dνx dν y . (3.260)

This is the Fresnel diffraction integral expressed in the Fourier domain. The same equation can also be obtained from the angular spectrum of plane waves (3.231) by

Wave Optics

expanding the square root of the transfer function of free space in a Taylor series & λ2 λ2 1 − 2 νx2 + ν2y ≈ 1 − 2 νx2 + ν2y ; n 2n & 2 nz 0 λ 2πi 1 − 2 νx2 + ν2y λ n ⇒e λz 0 2 nz 0 νx + ν2y −iπ 2πi n λ e . ≈e

(3.261)

2 2 nz 0 λ4 2 λ3 z 0 2 2 2 ν ν 2π + ν = π + ν π; x y x y λ 8n 4 4n 3 3 2 λ z0 2 νx + ν2y 1 . ⇒ Q Fresnel, Fourier := 3 4n (3.262)

To give an estimation of this term a spherical wave with the half-aperture angle ϕ is regarded. Then, the maximum spatial frequency of the spherical wave will be n sin ϕ/λ. The function u˜ 0 will be considerably different from zero only for spatial frequencies with νx2 + ν2y < n 2 sin2 ϕ/λ2 . Therefore, the condition (3.262) can be transformed to nz 0 sin4 ϕ 1 . (3.263) Q Fresnel, Fourier = 4λ It is obvious that the error term Q Fresnel, Fourier increases with increasing distance z 0 whereas the error term Q Fresnel of (3.254) decreases with increasing z 0 . So, the formulation of the Fresnel diffraction integral in the Fourier domain (3.260) is better used for the near field whereas the Fresnel diffraction integral of (3.253) is better used for the medium or far field. So, both formulations are contrary. The numerical evaluation of both diffraction integrals can be done using FFT’s whereas only one FFT is necessary in the formulation of (3.253) and two FFT’s (one for calculating u˜ 0 and one for the integral itself ) in the formulation of (3.260). The Fraunhofer Diffraction Formula An approximation of the Fresnel–Kirchhoff diffraction integral for the far field can be obtained from (3.247) and (3.248). First, we define the direction cosines α and β as

α :=

x ; l0

β :=

y l0

(3.264)

129

so that (3.247) can be written as

2 αx + β y

x 2 + y 2

− αx + β y − . l ≈ l0 + 2l0 2l0 (3.265)

With increasing l0 the second and fourth term decrease more and more and only the first and the third term remain. The condition that only these two terms have to be considered is that the contribution of the other terms does not remarkably vary the exponential factor in (3.248). This is fulfilled if πn

x 2 + y 2 n D2 π ⇒ Q Fraunhofer := 1. λl0 4λz 0 (3.266)

In the last step it is required that all points in the aperture A fulfill the condition x 2 + y 2 ≤ (D/2)2 , where D is again the maximum diameter of the aperture. We also used that according to (3.245) l0 ≥ z 0 . Then, (3.248) can be written as u(α, β, z 0 ) n z 0 ikl0 = −i e λl0 l0 n −2πi αx + β y

λ × u0 x , y , 0 e dx dy . A

(3.267)

This is the well-known Fraunhofer diffraction integral. It means that the complex amplitude in the far field is the Fourier transform of the complex amplitude at z = 0. The importance of (3.267) would be quite marginal if it is only valid for the far field. This can be seen by the following example. We assume n = 1, λ = 0.5 µm, D = 10 mm. Then (3.266) would require z0

n D2 = 50 m . 4λ

(3.268)

But, there is another quite important case: the complex amplitude in the focal plane of a lens. The Complex Amplitude in the Focal Plane of a Lens We assume a complex amplitude u 0 in the starting plane that is defined as different from zero in the aperture A and zero outside of the aperture. The influence of an ideal thin lens which is positioned in the starting plane would be that u 0 has to be multiplied by the transmission

Part A 3.4

This approximation is valid as long as the higher-order terms do not contribute to a considerable variation of the exponential factor. The condition for this is

3.4 Diffraction

130

Part A

Basic Principles and Materials

function tlens, ideal of the lens, which is an exponential phase factor of the form ⎛ & ⎞

2 + y 2 x −ik ⎝ f 1 + − f⎠ f2 tlens, ideal x , y = e =: e−ikllens .

(3.269)

Part A 3.4

Here, f is the focal length of the lens and a positive value f corresponds to a positive lens, whereas in our case we will only have a positive lens. Of course, an ideal lens does not exist in reality and a more adapted transmission function is t (3.270) x , y = e−ikllens + iW x , y ,

Then, similar to the case of the Fresnel diffraction integral the cosine obliquity factor z 0 /l is one and the distance l in the denominator of (3.272) can be replaced by z 0 . Only l in the exponential phase factor of (3.272) has to be considered carefully since the phase factor will rapidly oscillate if l varies by more than one wavelength λ/n. So, using (3.270), (3.272) and the approximations the intermediate result is u(x, y, z 0 ) = −i

n λz 0

lens

where W are the wave aberrations of the lens. However, in reality the wave aberrations of a lens will depend on the incident wavefront and a wave-optical simulation of a lens including aberrations is not so easy. But here, we assume that W is known for a given complex amplitude u 0 . For an ideal lens we just have to put W equal to zero. So, the new complex amplitude u 0 behind the lens is defined by u 0 x , y , 0 = u 0 x , y , 0 tlens x , y

(3.271) and the complex amplitude in a parallel plane at the distance z 0 is, according to the Fresnel–Kirchhoff diffraction integral (3.241), z 0 eikl

n dx dy , u 0 x , y , 0 u(x, y, z 0 ) = −i λ l l A

(3.272)

with l defined by (3.237) as 2 l = (x − x )2 + (y − y )2 + z 20 . Now, we are only interested in points in the neighborhood of the Gaussian focus of the lens at (0, 0, f ) and since the radius D/2 of the aperture shall be several times smaller than the focal length f of the lens we have the following conditions and approximations f = z 0 (1 + ε)

with λ D |ε| 1, f ; 2 n 2 D 2 2 z 20 , x + y ≤ 4 D/2 y D/2 x 1 and 1. z0 z0 z0 z0

u 0 x , y , 0 eiW x , y × eik (l − llens ) dx dy .

(3.273)

The term l − llens has to be evaluated: l − llens & = z0 1 + & −f

x 2 + y2 + x 2 + y 2 − 2 (xx + yy ) z 20

x 2 + y 2 +f f2

≈ z0 +

x 2 + y2 x 2 + y 2 xx + yy x 2 + y 2 + − − 2z 0 2z 0 z0 2f

= z0 +

x 2 + y2 ∆z x 2 + y 2 xx + yy

− + 2z 0 fz 0 2 z0 (3.274)

with ∆z := f − z 0 . As in the case of the Fresnel diffraction integral the terms of higher than second order in x, y, x or y have been neglected because of our conditions. Nevertheless, it should be pointed out that the restrictions on the sine of the half-aperture angle ϕ of the lens (sin ϕ ≈ D/(2 f )) are not so severe as in the case of the Fresnel diffraction integral since only points in the neighborhood of the Gaussian focus, i. e., when x, y and ∆z are small, are interesting. To make this clear the higher-order terms have to be estimated whereby only one section along x and x

is considered. But this is no restriction if the aperture is circular or if the section along x, x has a larger diameter than along y, y . The maximum value of x is D/2 and the maximum interesting value of x is just some wavelengths if we are near the focal plane of the lens. Additionally, we can replace the term D/(2z 0 ) with a good approximation by sin ϕ because of f ≈ z 0 . For the same reason

Wave Optics

Again, these terms have to be much smaller than π so that they can be neglected. A numerical example can illustrate this: A lens with a focal length of f = 100 mm and D/2 = 30 mm, i. e., sin ϕ = 0.29, is illuminated with light of the wavelength λ = 0.5 µm. The refractive index for the light propagation is n = 1. The propagation distance behind the lens is z 0 = 99.9 mm and the maximum value of x, which is interesting for us, is xmax = 10 µm. Later, in (3.294), we will see that the radius of the diffraction-limited Airy disc of a lens with sin ϕ = 0.29 and λ = 0.5 µm is ρ0 = 0.61 λ/NA = 0.61λ/(n sin ϕ) ≈ 1 µm. Therefore, an area with radius xmax = 10 µm contains all interesting structures of the intensity distribution of the focus. Using these values, the higher-order terms are 4 xmax = 5 × 10−12 π, 4λ f 3 3∆z 4 sin ϕ = 1.1 π , πn 4λ 3x 2 sin2 ϕ = 2.5 × 10−4 π, πn max 2λ f x 3 sin ϕ = 6 × 10−8 π, πn max 2 λf xmax sin3 ϕ = 0.49π . πn λ

πn

We see that the “defocus” term of higher order is 1.1 π and cannot be neglected. But if we go directly in the focal plane, i. e., z 0 = f , this term will completely vanish. The second term that cannot be neglected is the last term (0.49 π), which is proportional to xmax /λ and the third power of the sin ϕ. But in the direct neighborhood of

131

the Airy disc, which has in the diffraction-limited case a radius of about 1 µm this term will be a factor of 10 smaller. This means that for sin ϕ of 0.3 (and the given other parameters) the higher order terms can only be neglected in the direct neighborhood of the focus and the complex amplitude calculated outside may have some errors. But by decreasing the numerical aperture of the lens the accuracy of the calculation increases. So, finally we have the following result for the complex amplitude in the neighborhood of the Gaussian focus by neglecting higher-order terms u(x, y, z 0 ) x 2 + y2 n ikz 0 ik 2z 0 = −i e e λz 0 ×

u0 A

−ik ×e

2 ik ∆z x + y 2 x , y , 0 eiW x , y e z 0 f

xx + yy

z0 dx dy .

(3.276)

This integral is similar to the Debye integral. In [3.1] a slightly different but in fact nearly identical form of this integral is evaluated for an ideal lens, i. e., W = 0, using the Lommel functions (E. Lommel invented these functions when he was a professor of physics at the University of Erlangen in 1868–1886). For us it is especially interesting that (3.276) expresses the complex amplitude in the neighborhood of the Gaussian focus as a Fourier transformation of the pupil function G G x , y = ⎧ ⎨u (x , y , 0) exp iW(x , y ) exp ik ∆z x 2 +y 2 0 fz 0 2 ⎩0 if (x , y ) ∈ A (3.277) if (x , y ) ∈ / A. Here, u 0 is the complex amplitude of the incident wave, the term exp(iW) describes the influence of the wave aberrations of the lens and the third term is a defocus term. In the focal plane itself the defocus term vanishes since then z 0 = f . So, in fact we see that in the focal plane of an ideal lens (i. e., W = 0 and ∆z = 0) we again have a kind of Fraunhofer diffraction and the complex amplitude u is calculated by a Fourier transformation of u 0 . The intensity distribution I in the focal plane for an incident plane on-axis wave, i. e., u 0 (x , y , 0) =

Part A 3.4

we also have 1/z 30 − 1/ f 3 ≈ 3∆z/ f 4 . The fourth-order terms of k(l − llens ) are therefore of the following form ⎛ ⎞ 2 4

x 2 + x 2 − 2xx

x ⎟ ⎜ k⎝ − 3⎠ 3 8 f 8z 0 / . 0 1 1 x4 4 x

+ − = 2πn 8λz 30 8λz 30 8λ f 3 1 3x 2 x 2 x 3 x

xx 3 + − − 4λz 30 2λz 30 2λz 30 4 3∆z 4 3x 2 sin2 ϕ xmax sin ϕ + max + ≤ πn 3 4λ 2λ f 4λ f 3 sin ϕ xmax xmax sin3 ϕ . − − (3.275) λ λf2

3.4 Diffraction

132

Part A

Basic Principles and Materials

a = constant with I0 = a2 , is according to (3.276)

n2 (3.278) I(x, y, z 0 = f ) = I0 2 2 λ f ! !2

! ! −2πin xx + yy ! ! ! iW x , y

! λ f ×! e dx dy ! , e ! ! ! A !

Part A 3.4

where I0 is the intensity of the incident plane wave. The intensity distribution in the focal plane is also called the point spread function (PSF) of the lens. It is often usual to normalize the PSF by dividing it by the intensity IF of an nonaberrated lens of the same type at the Gaussian focus. Using (3.278) IF is obtained by setting W = 0 and (x, y) = (0, 0): ! !2 ! ! ! n 2 !! n2 dx dy !! = I0 2 2 S2 . IF (0, 0, f ) = I0 2 2 ! λ f ! λ f ! A

(3.279)

dx dy

Here, S = is the surface area of the aperA ture A. Then, the normalized point spread function PSF is I(x, y, f ) (3.280) PSF(x, y) = IF (0, 0, f ) ! !2

! ! −2πin xx + yy ! ! 1 ! iW x , y

! λ f = 2! e dx dy ! . e ! S ! ! A ! The dimensionless number σ = PSF(0, 0) of the aberrated lens at the Gaussian focus is called the Strehl ratio of the lens and is defined by ! !2 ! ! ! ! 1 eiW x , y dx dy !! . σ = PSF(0, 0) = 2 !! S ! ! A

(3.281)

In this section we calculated the PSF and the Strehl ratio only for a thin lens with the aperture in the plane of the lens. For general optical systems the same concept and the same equations are used but then the aperture A is the exit pupil of the optical system and f is the distance of the Gaussian focus from the exit pupil of the system. Two Examples of Fraunhofer Diffraction Two simple examples that can be solved analytically are the Fraunhofer diffraction at a rectangular aperture or at a circular aperture.

Fig. 3.15 Parameters of a rectangular transparent aperture

in an opaque screen Fraunhofer Diffraction at a Rectangular Aperture.

The intensity distribution in the focal plane of an ideal lens (focal length f ) with a rectangular aperture of the diameter 2a in x direction and 2b in y direction has to be calculated (Fig. 3.15). The wavelength of the light is λ and the refractive index of the material in which the wave propagates is n. The lens itself is illuminated with a uniform plane on-axis wave with the intensity I0. Then, according to (3.278) the intensity in the focal plane of the ideal lens (i. e., W = 0) is I(x, y, z 0 = f ) ! !2 ! ! xx + yy

! ! 2 −2πin n !

! λ f e dx dy ! = I0 2 2 ! ! λ f ! ! A ! ! !2 ! a ! xx

! ! −2πin !

! λ f = I0 2 2 ! e dx ! ! λ f ! !−a ! ! !2 ! b ! yy

! ! −2πin ! λ f dy !! . ×! e (3.282) ! ! !−b ! The first integral is ⎡ ⎤

xx x =a a −2πin xx −2πin −λ f λ f dx = ⎢ λf ⎥ e e ⎣ ⎦ 2πinx n2

−a

x =−a

xa 2λ f sin 2πn . = 2πnx λf

(3.283)

Wave Optics

sin2(π xˆ )/(π xˆ )2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 –5 –4 –3 –2

3.4 Diffraction

133

y' a

ρ' 0

φ' x'

–1

Fig. 3.16 Normalized intensity distribution along the x-axis

in the focal plane of a lens for a rectangular aperture. 2 Shown is the function sin(π x)/(π x) ˆ ˆ . The function 2 sin(π y)/(π y) ˆ ˆ along the y-axis is identical

The second integral is analogous and so the intensity in the focal plane is ⎞2 ⎛ nx 2a n 2a 2b 2 ⎝ sin π λ f ⎠ I(x, y, z 0 = f ) = I0 λf π nxλ 2a f ⎞2 sin π nλy 2b f ⎠ . ×⎝ π nλy 2b f ⎛

(3.284)

By introducing the variables xˆ = nx 2a/(λ f ) and yˆ = n y 2b/(λ f ), which are pure numbers without a physical unit, the normalized intensity distribution along one of the axes (along the x- or y-axis) can be easily calculated and is shown in Fig. 3.16. The minima of the intensity distribution along the x-axis are at λf xˆ = m with m = 1, 2, 3 . . . ⇒ x = m 2na λ . (3.285) ≈m 2NAx Here, the numerical aperture NAx := n sin ϕ ≈ na/ f with the half-aperture angle ϕ 1 of the lens in the x direction has been used. Fraunhofer Diffraction at a Circular Aperture. The in-

tensity distribution in the focal plane of an ideal lens (focal length f ) with a circular aperture of the radius a can also be calculated using (3.278) (Fig. 3.17). The wavelength of the light is again λ and the refractive

Fig. 3.17 Parameters of a circular transparent aperture in an

opaque screen

index of the material in which the wave propagates is n. The lens itself is again illuminated with a uniform plane on-axis wave with intensity I0 . Because of the circular symmetry it is useful to introduce polar coordinates x = ρ cos φ ; x = ρ cos φ y = ρ sin φ ; y = ρ sin φ = ρρ cos φ − φ .

; ⇒ xx + yy

(3.286)

Then, according to (3.278) the intensity in the focal plane of the ideal lens (i. e., W = 0) is written in polar coordinates I(ρ, φ, z 0 = f ) ! !2 ! ! xx + yy

! ! 2 −2πin n !

! λ f = I0 2 2 ! e dx dy ! ! λ f ! ! A ! ! !2 ! a 2π ! ρρ cos φ − φ ! ! −2πin ! n 2 !!

! λ f = I0 2 2 ! e ρ dρ dφ ! λ f ! ! !0 0 ! ! ! !a 2π !2 ρρ cos φ

! −2πin n 2 !! ! λf = I0 2 2 ! e ρ dρ dφ ! . ! λ f ! !0 0 ! (3.287)

Part A 3.4

5 xˆ

134

Part A

Basic Principles and Materials

To solve the double integral the well-known Bessel functions [3.51] of the first kind Jm (x) are introduced by the integral representation i−m 2π

Jm (x) =

2π

(3.288)

1 2π

0.7 0.6

For m = 0 we obtain J0 (x) =

0.9 0.8

eix cos α eimα dα .

2π

[2 J1(πρˆ )/(πρˆ )]2 1

0.5 0.4

eix cos α dα .

(3.289)

0.3 0.2

Part A 3.4

0.1

Therefore, the intensity is

0 –3

I(ρ, φ, z 0 = f ) ! !2 ! !a ! ρρ

2πn 2 !!

! −2πn ρ J dρ = I0 0 ! ! . λf λf ! !

–2

–1

3 ρˆ

Fig. 3.18 Normalized intensity distribution in the focal 2 plane of a lens. Shown is the function 2J1 (π ρ)/(π ρ) ˆ ˆ

(3.290)

There is another integral relation which connects the two Bessel functions J0 and J1 : x x J1 (x) =

x J0 (x ) dx .

(3.291)

By substituting x = −2πnρρ /(λ f ) −2πnρ/(λ f ) dρ we obtain I(ρ, φ, z 0 = f ) ! ! 2 ! 2 2 2πn !! λ f = I0 ! 4π 2 n 2 ρ2 λf ! !

I(ρ, ˆ z 0 = f ) = I0

nπa2 λf

/2 2 . J1 π ρˆ . 2 π ρˆ (3.293)

−2πn λρaf

and

dx =

!2 ! ! ! x J0 x dx !! ! !

2 λ fa ρa 2 − J1 −2πn 2πnρ λf ⎛ ⎞2 ρa 2 2πna2 ⎝ J1 2πn λ f ⎠ = I0 . (3.292) λf 2πn λρaf = I0

can be written as

2πn λf

Here, the symmetry of the Bessel function J1 (−x) = J1 (x) has been used. By defining the variable ρˆ := 2nρa/(λ f ), which is a pure number without a physical unit, the intensity in the focal plane of the ideal lens

2 The function 2J1 (π ρ)/(π ρ) is shown in Fig. 3.18. ˆ ˆ The first minimum is at the value ρˆ 0 = 1.22, i. e., at the radius ρ0 with λf λ λf = 0.61 = 0.61 . (3.294) 2na na NA Here, again the numerical aperture of the lens NA ≈ na/ f is used. The area inside the first minimum of the diffraction limited focus is called the Airy disc. It is also interesting to compare the maximum intensity I(ρˆ = 0, z 0 = f ) in the central peak of the focus with the intensity I0 of the incident plane wave. The ratio is 2 I(ρˆ = 0, z 0 = f ) nπa2 = . (3.295) I0 λf ρ0 = 1.22

For a lens with a focal lens of f = 10 mm, an aperture radius of a = 1 mm and a wavelength of λ = 0.5 µm and n = 1 we obtain, e.g., I(ρˆ = 0, z 0 = f )/I0 = (200π)2 ≈ 4 × 105 . The quantity na2 /(λ f ) is also known as the Fresnel number F of a lens, which is the number of Fresnel zones of the lens in the paraxial case. This can be easily seen by calculating the optical path difference OPDbetween

Wave Optics

a ray from the center of the lens to the focus and a ray from the rim of the lens to the focus. It is ⎛ & ⎞ 2 2 a OPD = n a2 + f 2 − f = n ⎝ f 1 + 2 − f ⎠ f na2 λ =F ; 2f 2 na2 ⇒ F= . λf

∆x

∆νx

∆y

∆νy

Dνy Ny Fourier transformation Dvx

3.4.4 Numerical Implementation of the Different Diffraction Methods Many of the proposed diffraction integrals can be solved by performing one or two Fourier transformations (Table 3.1). For a numerical implementation, a discrete Fourier transformation is necessary and to increase the speed of the calculation it makes sense to take a fast Fourier transformation (FFT) [3.52]. Of course, it has to be noted that the sampling theorem is fulfilled and that the size of the field is large enough. In practice, large field sizes of, e.g., more than 2048 × 2048 samples need a lot of computer memory and computing time. To use a FFT the field of the complex amplitude u 0 is in each spatial direction x and y uniformly sampled at N x × N y points, where N x and N y are powers of two. The diameters of the field in the spatial domain are called Dx in the x and D y in the y direction. In most cases the field will be quadratic and sampled with equal number of points, i. e., N x = N y and Dx = D y . Nevertheless, there are cases (e.g., systems with cylindrical or toric optical elements) where it is useful to have different sampling rate and number of sampling points along x and y. Then, the sampling interval ∆x and ∆y in the x and y direction between two neighboring sampling points in the spatial

Fig. 3.19 Discrete fields for solving diffraction integrals using a FFT

domain are (Fig. 3.19) Dx ∆x = , Nx Dy ∆y = . Ny

(3.297)

We call the conjugated variables in the Fourier domain νx and ν y , where these can be really the spatial frequencies defined in (3.223) or other variables like in the Fresnel diffraction integral, which just have the physical dimension of a spatial frequency. Table 3.1 shows the conjugated variables for the different diffraction integrals. The diameters of the fields in the Fourier domain are called Dνx and Dνy along the νx - and ν y direction. The associated sampling intervals between two neighboring sampling points in the Fourier domain are Dνx , ∆νx = Nx Dνy . (3.298) ∆ν y = Ny For a FFT the product of the respective diameters in the spatial domain and in the Fourier domain is equal to the number of sampling points. Therefore, the two relations are valid Dx Dνx = N x , D y Dνy = N y .

(3.299)

Table 3.1 Conjugated variables and number of FFTs for calculating the different diffraction integrals Diffraction method

Spectrum of plane waves

Fresnel (Fourier domain)

Fresnel (convolution)

Equation

(3.231)

(3.260)

(x, y) ↔ νx , ν y

(3.253) nx x , y ↔ λz ,

Conjugated variables

(x, y) ↔ νx , ν y

Fraunhofer, Debye integral

ny λz 0

(3.267), (3.276) nα nβ x ,y ↔ λ , λ , nx n y x , y ↔ λz , λz 0

Number of FFTs

Part A 3.4

This means that the intensity in the central peak of the focus of an ideal lens is proportional to the square of the Fresnel number of this lens.

135

Dy Ny

≈

(3.296)

3.4 Diffraction

136

Part A

Basic Principles and Materials

By using (3.297) and (3.298) it is clear that for the product of the sampling intervals in the spatial and in the Fourier domain the following equations apply 1 1 1 ∆x∆νx = ⇒ ∆νx = ; ∆x = , Nx Dx Dνx 1 1 1 ⇒ ∆ν y = ; ∆y = . ∆y∆ν y = Ny Dy Dνy (3.300)

Part A 3.4

The variables are in our case always symmetrical around the origin of the coordinate system. Therefore, the minimum and maximum values of the respective variables are Dx Dx xmin = − , xmax = ; 2 2 Dy Dy , ymax = ; ymin = − 2 2 Nx Nx νx,min = − ∆νx = − , 2 2Dx Nx Nx ; νx,max = + ∆νx = + 2 2Dx Ny Ny ν y,min = − ∆ν y = − , 2 2D y Ny Ny ν y,max = + ∆ν y = + . (3.301) 2 2D y Since a discrete Fourier transformation has periodic boundary conditions the function values at the left boundary will be equal to those at the right boundary, i. e., u(xmin , y) = u(xmax , y), u(x, ymin ) = u(x, ymax ), and so on. This fact is important because it generates aliasing effects if the field size and the sampling are not correct. In the following some special aspects of the different diffraction integrals, which can be calculated using one or two FFTs are presented. Numerical Implementation of the Angular Spectrum of Plane Waves or the Fresnel Diffraction in the Fourier Domain To solve the diffraction integrals (3.231) and (3.260) two FFTs are necessary. The first is to transform the complex amplitude u 0 into the Fourier domain with the spatial frequencies νx and ν y . In order to represent all propagating waves the maximum spatial frequencies have to fulfill the conditions n 2n Dx νx,max ≥ ⇒N x ≥ , λ λ 2n D y n . ν y,max ≥ ⇒N y ≥ (3.302) λ λ Here, (3.223) and (3.301) have been used. However, it is in general not necessary that all spatial frequencies belonging to propagating waves can be represented. Es-

pecially, in the case of the Fresnel diffraction integral (3.260) formulated in the Fourier domain high spatial frequencies will in most cases not be allowed because the equation is, depending on the propagation distance z 0 , only valid for small spatial frequencies (3.262). Let us assume that the angular spectrum of plane waves has a maximum tilt angle ϕ for which the Fourier transform u˜ 0 of u 0 has a function value that is noticeably different from zero. Then, it is sufficient that instead of (3.302) the following conditions are used n 2n Dx sin ϕ ⇒ N x ≥ sin ϕ, λ λ 2n D y n sin ϕ . ν y,max ≥ sin ϕ ⇒ N y ≥ λ λ νx,max ≥

(3.303)

If these conditions are not fulfilled aliasing effects occur and the numerical result is wrong. It is clear that only for microoptical elements the condition (3.302), i. e., sin ϕ is allowed to be 1, can be fulfilled. For a wavelength of λ = 0.5 µm, refractive index n = 1 and a field diameter in the x direction of Dx = 1 mm, we obtain, e.g., N x ≥ 2n Dx /λ = 4000. For a two-dimensional FFT a field size of 4096 × 4096 samples is at the upper limit of modern personal computers. Also in the case of the second (inverse) FFT, which calculates u using (3.231) or (3.260) aliasing effects can occur. Graphically, this means that parts of a diverging propagating wave leave the border of the field. Because of the periodic boundary conditions these parts of the waves will then enter the field at the opposite boundary. Figure 3.20 illustrates this effect for a diverging spherical wave that has a diameter D0 at the starting plane and would have a diameter Dz0 > Dx in the distance z 0 . If the half-aperture angle of the spherical wave is ϕ, where we assume that ϕ is so small that sin ϕ ≈ tan ϕ,

ϕ tan ϕ ≈sin ϕ D0

Dz0

Fig. 3.20 Illustration of aliasing effects. The parts of the wave that leave the field at a boundary enter the field at the opposite boundary and interfere with the other parts of the wave

Wave Optics

the diameter Dz0 will be Dz0 ≈ D0 + 2z 0 sin ϕ .

3.4 Diffraction

137

diameters Dx,z0 and D y,z0 in the second plane, where Dνx = n Dx,z0 /(λz 0 ) and Dνy = n D y,z0 /(λz 0 ) (3.304)

Numerical Implementation of the Fresnel (Convolution Formulation) and the Fraunhofer Diffraction Using the method of the angular spectrum of plane waves guarantees that the field sizes Dx and D y do not change during the propagation as long as no manipulations are made in the Fourier domain. The reason for this is that two FFTs are used, one “normal” and one inverse FFT. If the Fresnel diffraction integral of (3.253) or the Fraunhofer diffraction integrals (3.267) or (3.276) are used, only one FFT is made. Therefore, the field size changes because theconjugated variables are now, ac cording to Table 3.1, x , y and [nx/(λz 0 ), n y/(λz 0 )] or [(nα/λ, nβ/λ) , where α = x/z 0 and β = y/z 0 ]. This means that according to (3.299) the following relations are valid if we introduce the field diameters Dx,0 = Dx and D y,0 = D y in the first plane and the field

λz 0 , n Dx,0

D y Dνy = N y ⇒ D y,z0 = N y

λz 0 . n D y,0

(3.305)

Of course, the sampling densities ∆x z0 and ∆yz0 in the second plane are then ∆x z0 =

λz 0 , n Dx,0

∆yz0 =

λz 0 . n D y,0

(3.306)

Let us calculate the intensity distribution in the focal plane of a lens with focal length f (z 0 = f ) using (3.276). The aperture of the lens shall be quadratic with diameters 2a = 2b. Then, it is enough to just consider one dimension, e.g., the x direction. If the field size Dx,0 would now only be 2a the sampling interval would be ∆x z0 =

λ λf ≈ , n2a 2NA

(3.307)

where, the numerical aperture of the lens NA := n sin ϕ ≈ na/ f has been used. But by comparing this result with (3.285) it is clear that the sampling density is so low that the secondary maxima are not observed because the sampling is only in the minima. Therefore, it is necessary that the aperture of the lens is embedded into a field of zeros so that the effective field diameter Dx,0 is at least doubled Dx,0 ≥ 4a. So, by embedding the aperture of the lens by more and more zeros the effective sampling density in the focal plane is increased. Increasing the field size by a factor m and filling the new area with zeros reduces the sampling interval by a factor m to ∆x z0 = λ/(2mNA). In other words we can say: the total field size Dx,z0 in the focal plane is proportional to the number N x /m of samples with which the lens aperture is sampled whereas the sampling density is proportional to the factor m of the zero embedding. Of course, there is also another reason for embedding the lens aperture with zeros: the aliasing effects. A quadratic lens aperture with Dx,0 = 2a would, due to the periodic boundary conditions, mean that the aperture is repeated periodically without spaces in between and fills the whole space. Therefore, no diffraction at all would occur and the focus would be a delta peak as in geometrical optics.

Part A 3.4

If Dz0 > Dx aliasing effects will appear. This effect can be useful to numerically simulate the self-imaging Talbot effect for infinitely extended periodic structures [3.53]. However, in most cases aliasing effects are disturbing and have to be avoided. For converging waves, aliasing effects will not appear as long as the propagation distance is not so large that the wave passes the focus and becomes diverging. In practice, the limitations on the number of samples will limit the application of the propagation of a wave using the angular spectrum of plane waves. It is also very interesting to note that the field diameters Dx and D y normally do not change between the two planes at z = 0 and z = z 0 in the case of the propagation using the angular spectrum of plane waves or the Fresnel integral in the Fourier domain formulation. Only, if the field size in the Fourier domain is manipulated will the field size in the spatial domain change. This is only possible by taking each second sampling point and embedding the new field with so many zeros that the total number of sampling points remains unchanged. Then the effective diameters in the Fourier domain Dνx and Dνy are doubled and therefore, according to (3.299) the diameters Dx and D y in the spatial domain will be halved. But this manipulation reduces the sampling density in the Fourier domain and is therefore a kind of high-pass filtering operation suppressing the long-periodic spatial structures.

Dx Dνx = N x ⇒ Dx,z0 = N x

138

Part A

Basic Principles and Materials

3.4.5 The Influence of Polarization Effects to the Intensity Distribution Near the Focus

Part A 3.4

In the previous cases of diffraction we considered only the scalar case. But, in this section we want to discuss the influence of polarization effects to the intensity distribution in the focal region of a lens. However, we will only discuss a simple numerical simulation method, which is in fact identical to the semi-analytical vectorial Debye integral formulation discussed, e.g., in [3.54] or [3.34]. Richards and Wolf [3.34] were one of the first who calculated that the light distribution in the focus of a lens, which is illuminated by a linearly polarized plane wave, is asymmetrical for a high numerical aperture of the lens. Some Elementary Qualitative Explanations To illustrate the influence of polarization effects to the focus refer to Fig. 3.21. Assume a plane wave with linear polarization (electric vector in the y direction), which is focused by a lens. Then, there is one plane (xz-plane) where the electric vectors are perpendicular to the plane of refraction of the rays (Fig. 3.21a). There, the electric vectors in the focus add like scalars and a quite large transversal component is obtained. But, there is also the yz-plane, where the electric vectors change their direction after being refracted by the lens. Then, they add in the focus like real vectors and a smaller transversal component than in the xz-plane is obtained (Fig. 3.21b). Especially, for very steep rays corresponding to high numerical aperture rays the electric vectors nearly cancel each other in the focus. Due to the broken symmetry of a)

b) y z

Fig. 3.22a,b Local polarization vectors in the aperture in front of the lens for (a) linearly polarized light and (b) a ra-

dially polarized doughnut mode

this problem the intensity distribution in the focus will also be nonrotationally symmetric. Of course, this effect is only visible for very high numerical apertures, because otherwise the vector character of the electric field is not so obvious. It is especially visible for annular apertures and a high numerical aperture lens. But there is a rotationally symmetric polarization pattern, called the doughnut mode [3.33], where the direction of the electric vector varies locally, so that it always points away from the optical axis in radial direction. Figure 3.22 shows in (a) the electric vectors in the aperture of the lens at a certain time for linearly polarized light, i. e., they all point in the same direction. In (b) the electric vectors for the radially polarized doughnut mode are drawn at a certain time. They all point radially away from the optical axis. Of course, in the case of the doughnut mode there has to be the intensity zero on the optical axis due to the symmetry of the problem. So, the intensity distribution in the aperture of the lens is not homogeneous in the case of the radially symmetric doughnut mode but in fact it is the superposition of a Hermite–Gaussian TEM10 and a TEM01 mode (Fig. 3.32) with relative phase difference zero. Then, the time-independent complex-valued electric vector Erad before the lens has the value ⎛ ⎞ x 2 2 2 ⎜ ⎟ − x + y /w0 Erad (x, y, z = 0) = E 0 ⎝ y ⎠ e . 0

Fig. 3.21a,b Addition of the electric vectors in the focus of a lens for linearly polarized light. (a) The electric vectors are perpendicular to the xz-plane, i. e., they add arithmetically like scalars; (b) the electric vectors lie in the yz-plane and add vectorially to a transversal component

(3.308)

Here, w0 is the beam waist of the Gaussian function and E 0 is a constant. The : maximum of the √ amplitude of the electric vector is at x 2 + y2 = w0 / 2 as can be easily calculated. But now, for the radially polarized doughnut mode in all planes containing the optical axis, the electric vectors are oriented as in Fig. 3.23. Then, the electric vectors add

Wave Optics

3.4 Diffraction

139

the orthogonality condition of plane waves (Sect. 3.1.1) (see the paragraph on page 91). Examples of different polarization states are (see Sect. 3.2.4):

• • • Fig. 3.23 Orientation of the electric vectors in front of

and behind a lens for the radially symmetric doughnut mode. The electric vectors add to a longitudinal component parallel to the optical axis

The lens itself fulfills the sine condition so that we have for the refracted ray behind the lens : 2 x 2 + y2 2 2 , h = x + y = f sin ϑ ⇒ sin ϑ = f (3.309)

in the focus to a longitudinal component and this is the case for all planes containing the optical axis. Therefore, the focus is completely rotationally symmetric. Numerical Calculation Method To calculate the electric energy density in the focal region of a lens the vectorial Debye integral of [3.34] or the method of [3.55] can be used. Both say that the electric vector in the focal region can be written as a superposition of plane waves, which propagate along the direction of the rays running from the exit pupil of the lens to the geometrical focus. In this model, diffraction effects at the rim of the aperture are neglected, as is also done in the scalar formulation of the Debye integral. But this is a quite good approximation as long as the diameter of the aperture of the lens is large compared to the wavelength (2raperture λ) and as long as the numerical aperture of the lens is sufficiently high (which is always the case here because polarization effects are only interesting for high numerical apertures). For the numerical calculation we make a uniform sampling of rays in the planar entrance pupil of the lens, e.g., N × N rays in an orthogonal pattern, where the rays outside of the aperture (e.g., circular or annular aperture) have zero amplitude. The optical axis shall be the z-axis. Each ray number (i, j) at the coordinate (xi , y j , z 0 ) in the entrance pupil has a wave vector ki, j = (0, 0, k) = (0, 0, 2π/λ) and a generally complex polarization vector Pi, j = (Px,i, j , Py,i, j , 0), which is in fact proportional to the electric vector E at this point if the sampling is uniform. Of course, Pi, j is orthogonal to ki, j because of

where, f is the focal length of the lens and ϑ the polar angle (Fig. 3.24). By defining the azimuthal angle ϕ we have the following relations for the rays in front of the lens and behind the lens x = h cos ϕ y = h sin ϕ and

⇒

ϕ = arctan

y x

(3.310)

⎛

⎞ − cos ϕ sin ϑ ⎜ ⎟ k = k ⎝ − sin ϕ sin ϑ ⎠ . cos ϑ

Pi,j hi,j

ki,j

k'i,j

P'i,j

(3.311)

Fig. 3.24 Principal scheme of the distribution of rays used to calculate the electric energy density in the focus. The lens : has to fulfill the sine condition so that the ray height h = x 2 + y2 is h = f sin ϑ

Part A 3.4

A linearly polarized homogeneous plane wave (polarization in the y direction) with the polarization vector Pi, j = (0, P0 , 0) with a constant real value P0 . A circularly polarized homogeneous plane √ wave = (P / 2, iP0 / with the polarization vector P i, j 0 √ √ 2, 0), where the factor 1/ 2 is introduced to have |Pi, j |2 = P02 . The radially polarized doughnut mode with the polarization vectors according to (3.308).

140

Part A

Basic Principles and Materials

The polarization vector P of each ray has to be separated into a component lying in the plane of refraction and a component perpendicular to it. The unit vector e along the component in the plane of refraction in front of the lens is ⎛ ⎞ cos ϕ ⎜ ⎟ e = ⎝ sin ϕ ⎠ . (3.312) 0 Behind the lens the new unit vector e along the component in the is ⎛ plane of refraction ⎞ cos ϕ cos ϑ ⎜ ⎟ (3.313) e = ⎝ sin ϕ cos ϑ ⎠ .

Part A 3.4

sin ϑ So, the polarization vector P behind the lens can be calculated by keeping in mind that the component perpendicular to the plane of refraction remains unchanged and that the component in the plane of refraction keeps its amplitude but is now parallel to e . In total this means for P

" # 1 P = √ P − P · e e + P · e e cos ϑ ⎛ ⎞ cos ϕ ϑ) (1−cos ⎜ ⎟ P− Px cos ϕ+ Py sin ϕ ⎝ sin ϕ (1−cos ϑ) ⎠ −sin ϑ = . √ cos ϑ (3.314)

√ Here, the factor 1/ cos ϑ is necessary in order to conserve the energy of the tilted plane wave [3.56]. In other words, behind the lens the density of the numerical sampling by the polarization vectors P i, j depends on the angle ϑ whereas in front of the lens the sampling of Pi, j is selected to be uniform. So, the electric vector E behind the lens is no longer proportional to the polarization vector P alone, but to the ratio of P and the small surface element dA perpendicular to P which is assigned to each ray by the distance of the ray sampling, and which √ is itself depending on the angle ϑ. So, the factor 1/ cos ϑ corrects the non uniform numerical sampling of the polarization vectors behind the lens. If the lens does not fulfill the sine condition but another equation the correction factor will of course change. For an ideal lens the plane waves along the rays have to be all in phase at the focus. So, we just can set the focus to the coordinate r = (x = 0, y = 0, z = 0). To calculate the electric vector E at a point r near the focus the plane waves have to be summarized according to $ ik · r

E r ∝ Pi, j e i, j . (3.315) i, j

Small wave aberrations W(x, y) of the lens can also be taken into account by just adding them to the phase term of each plane wave, i. e., ki, j · r → ki, j · r + Wi, j with Wi, j := W(xi , y j ). Then we have: $ ik · r + iWi, j E r ∝ Pi, j e i, j . (3.316) i, j

It can be seen that in the focal plane z = 0 and for linearly polarized light with a small numerical aperture, this sum reduces to a discretized formulation of the scalar diffraction integral of (3.276) for calculating the light distribution near the focus of a lens. By just changing the polarization vectors of the field distribution in front of the lens the effects of a quite arbitrary state of polarization to the focus can be investigated. The squares of the x , y, or z components of the electric field E can also be calculated separately. Of course, the electric energy density we , which is the physical quantity normally detected by a light detector, can then be calculated by we r ∝ |E|2 = E · E∗ . (3.317) The proportionality factors are omitted because we are only interested in relative energy distributions and not in absolute values. Some Simulation Results Figures 3.25, 3.26 and 3.27 show the results of some simulations for the electric energy density in the focal plane (xy-plane) of an ideal lens. In all cases the numerical aperture of the lens is assumed to be 1.0 and the wavelength of the illuminating light is λ = 632.8 nm. The typical number of plane waves used to sample the aperture of the lens is N × N = 100 × 100 (or 200 × 200), whereby in the case of a circular aperture (normal case) and especially in the case of an annular aperture (see later) the effective number of plane waves is smaller (factor π/4 smaller for a circular aperture) because the sampling is made in an orthogonal and uniformly sampled xy-pattern and plane waves outside of the aperture just have zero amplitude. The number of sampling points in the focal plane is 200 × 200 and so a simulation on a modern 1 GHz Pentium PC takes less than two minutes. In Fig. 3.25 the lens is illuminated by a linearly polarized plane wave (polarization in y direction). The squares of the different components of the electric vector in the focal plane are displayed. The x component in Fig. 3.25a is quite small and vanishes in the center of the geometrical focus (pay attention to the scale). The biggest component is the y component in Fig. 3.25b, i. e., along the direction in which the light is polarized

Wave Optics

a) y(µm)

b) y(µm)

Energy density (arb. units)

0.8

0.5

0.6

–0.5

0.4

0.2

–0.5

0.5 x(µm)

–0.5

Energy density⏐Ex⏐2 (arb. units)

0.005

0.5 x(µm)

Energy density⏐Ey⏐2 (arb.units)

0.01

–0.5

0.5 x(µm)

141

Energy density (arb. units)

–0.5

0.5 y(µm)

0.2 0.4 0.6 0.8 1

Fig. 3.26a,b Sections along (a) the x-axis and (b) the y-axis of the total electric energy density in the focal plane of an ideal lens with NA = 1.0, which is illuminated by a plane linearly polarized (in the y direction) wave with λ = 632.8 nm (see also Fig. 3.25d)

d) y(µm)

0.5

–0.5

–0.5 –0.5

0.5 x(µm)

Energy density⏐Ez⏐ (arb. units) 2

–0.5

0.5 x(µm)

Total energy density (arb.units)

0.2 0.4 0.6 0.8 1

Fig. 3.25a–d Squares of the components of the electric

vector in the focal plane of an ideal lens with NA = 1.0, which is illuminated by a plane linearly polarized (in the y direction) wave with λ = 632.8 nm. (a) x component, (b) y component, (c) z component and (d) sum of all components, i. e., total electric energy density

in front of the lens. But, there is also a z component (Fig. 3.25c) for points besides the geometrical focus. This component is mainly responsible for the asymmetric shape of the total electric energy density, which is displayed in Fig. 3.25d. Figure 3.26 shows sections along the x- and y-axis of the total electric energy density. It can be seen that the diameter of the central maximum along the y-axis is increased whereas the diameter along the x-axis has nearly the value of the scalar calculation dfocus = 1.22λ/NA = 0.77 µm. All quantities are normalized in such a way that the total energy density has a maximum value of 1. In Fig. 3.27 the lens is illuminated by a radially polarized doughnut mode, whereby the beam waist w0 is at 95% of the lens aperture radius. Then, there are radial, i. e., transversal components of the electric field in Fig. 3.27a, which are of course rotationally symmetric due to the symmetry of the field. But, the strongest com-

ponent is the longitudinal z component in Fig. 3.27b, which is also rotationally symmetric and has a central maximum with a diameter slightly smaller than the value of the scalar calculation. The total electric energy density in Fig. 3.27c is also rotationally symmetric but the diameter of the central maximum is increased due to the transversal components of the electric field. However, the surface area S, which is covered in the focus by a total electric energy density of more than half the maximum value is S = 0.29λ2 in the linearly polarized case and only S = 0.22λ2 for the radially polarized doughnut mode. So, if the light spot is used to write into a nonlinear material, which is only sensitive to a total electric energy density beyond a certain threshold a tighter spot can be obtained using the radially polarized doughnut mode. Figure 3.28 shows for the same parameters as in Figs. 3.26 and 3.27 the total electric energy density near the focus in planes containing the optical axis (z-axis) and one of the lateral coordinates. In Figs. 3.28a,b the xz-plane and the yz-plane are shown for the case of an illumination of the lens with linearly polarized homogeneous light (polarization in the y direction). Figure 3.28c shows the same for the case of the radially polarized doughnut mode, where here the focal region is completely rotationally symmetric so that only one section has to be shown. It can again be seen that in the case of linear polarization the lateral diameter of the focus is broader in the yz-plane than in the xz-plane. The focus of the radially polarized doughnut mode has a lateral diameter nearly that of the small axis of the linearly polarized case focus. From the scalar theory it is well-known that the diameter of the central maximum of the focus is decreased if an annular aperture is used instead of a circular one. It is

Part A 3.4

c) y(µm)

0.05 0.1 0.15 0.2

3.4 Diffraction

142

Part A

Basic Principles and Materials

a) Energy density⏐E ⏐2 (arb. units) radial

b) Energy density⏐E ⏐2 (arb. units) z

0.15

0.1

Total energy density (arb. units) 1

0.8

0.6

0.4

0.2

0.05

Part A 3.4

–0.5

0.5

–0.5

r (µm)

0.5

–0.5

r(µm)

0.5

r(µm)

Fig. 3.27a–c Squares of the components of the electric vector in the focal plane of an ideal lens with NA = 1.0, which is illuminated by a radially polarized doughnut mode with λ = 632.8 nm. The beam waist w0 of the Gaussian function is w0 = 0.95raperture , whereby raperture is the illuminated aperture radius of the lens. (a) Radial components, i. e., |E x |2 + |E y |2 , (b) z component, (c) sum of all components, i. e., total electric energy density

x/ µm

y / µm

1.5

0.5

–0.5

–1

–1.5

–2 –2 –1.5 –1 –0.5

0.5 1

1.5 2 z/ µm

–2 –2 –1.5 –1 –0.5 0

Energy density (a.u.) 0

0.2

0.4

0.6

0.8

0.5 1

1.5 2 z / µm

r/ µm

–2 –2 –1.5 –1 –0.5 0

Energy density (a.u.) 1

0.2

0.4

0.6

0.8

0.5 1

1.5 2 z / µm

Energy density (a.u.) 1

0.2

0.4

0.6

0.8

Fig. 3.28a–c Simulation of the total electric energy density in the focal region by using a circular aperture (NA = 1.0, λ = 632.8 nm). (a) Linearly polarized homogeneous light (xz-plane), (b) linearly polarized homogeneous light (yz-plane), (c) radially polarized doughnut mode with w0 = 0.95raperture (xz-plane or yz-plane because of rotational symmetry)

also well-known from the scalar theory that in this case, the secondary maxima increases compared to a circular aperture. So, by using an annular aperture a smaller focus should be reached and this should especially be the case for the radially polarized doughnut mode. The reason is that in this case the transversal components in the focus of the doughnut mode decrease so that the total energy density is more and more dominated by the longitudinal

z component, which has a small diameter. So, by using a radially symmetric doughnut mode, an annular aperture (e.g., rannulus = 0.9raperture ), and a high numerical aperture lens (NA > 0.9) a very tight rotationally symmetric spot with a small surface area can be obtained. This effect can be used to achieve a higher resolution in optical data storage or in optical lithography where a mask is written spot by spot by a so-called laser pattern

Wave Optics

x/ µm

y / µm

1.5

0.5

–0.5

–1

–1.5

0.5 1

1.5 2 z/ µm

–2 –2 –1.5 –1 –0.5 0

Energy density (a.u.) 0

0.2

0.4

0.6

0.8

0.5 1

1.5 2 z / µm

r/ µm

–2 –2 –1.5 –1 –0.5 0

Energy density (a.u.) 1

0.2

0.4

0.6

0.8

0.5 1

1.5 2 z / µm

Energy density (a.u.) 1

0.2

0.4

0.6

0.8

Fig. 3.29a–c Simulation of the total electric energy density in the focal region by using an annular aperture with an inner radius of 90% of the full aperture (NA = 1.0, λ = 632.8 nm). (a) linearly polarized homogeneous light (xz-plane), (b) linearly polarized homogeneous light (yz-plane), (c) radially polarized doughnut mode with w0 = 0.95raperture (xz-plane or yz-plane because of rotational symmetry)

generator. Figure 3.29 shows the simulation results of the electric energy density in the focal region for an annular aperture with rannulus = 0.9raperture . As in the former cases, the numerical aperture itself is NA = 1.0 and the wavelength is λ = 632.8 nm. Figures 3.29a and b show the xz-plane and yz-plane, respectively, for the case of linear polarization and Fig. 3.29c shows the same for the doughnut mode (w0 = 0.95raperture ), where the result is rotationally symmetric so that only one section is shown. It can be seen that in the case of an annular aperture the lateral diameter of the focus is decreased but on the other side the depth of focus along the optical axis is increased compared to the full aperture case. Besides the central maximum there are also, as expected, some secondary maxima with increased height. Nevertheless, a comparison of the cases of linear polarization and the doughnut mode shows that the lateral diameter of the

focus is decreased for the doughnut mode, especially in the yz-plane. By calculating the surface area S again, which is covered in the focus by a total electric energy density of more than half the maximum value, the result is S = 0.29λ2 for the linearly polarized light and only S = 0.12λ2 for the radially polarized doughnut mode. Moreover, the height of the secondary maxima is not as high for the doughnut mode as for the linear polarization. In some applications these secondary maxima can be disturbing, but in applications where a certain threshold value of the energy density has to be reached in order to obtain an effect, these secondary maxima have no influence. So, the annular aperture, especially combined with the radially polarized doughnut mode, allows a high lateral resolution. Experiments verify the simulation results for linearly polarized light [3.57] and for the radially polarized doughnut mode [3.33, 54].

3.5 Gaussian Beams Gaussian beams are a paraxial solution of the scalar Helmholtz equation and are suitable to describe the propagation of coherent laser beams [3.10,11,58]. However, the influence of apertures on the laser beam is not considered in this description because apertures would generally disturb the Gaussian beam. The transformation of laser beams at a lens is, of course, also only

143

treated in a paraxial sense and aberrations of the lens are not taken into account.

3.5.1 Derivation of the Basic Equations The typical property of a collimated laser beam is that it propagates straight on in a homogeneous material.

Part A 3.5

–2 –2 –1.5 –1 –0.5

3.5 Gaussian Beams

144

Part A

Basic Principles and Materials

Part A 3.5

Nevertheless, because of diffraction effects, the laser beam diverges during the propagation. Depending on the diameter of the laser beam this effect will be quite small or large. However, the laser beam will behave along the direction of propagation (z-axis) nearly as a plane wave. But instead of having a constant amplitude, as for a plane wave, the amplitude will be a function of the transversal coordinates (x and y) and also a slowly varying function of the propagation distance along the zaxis. Mathematically, this means that the scalar complex amplitude u(x, y, z) of a laser beam can be described by the product of a (generally) complex function Ψ (x, y, z), which changes only slowly along the z-axis, and the complex amplitude of a plane wave propagating in the z direction exp(ikz) u(x, y, z) = Ψ (x, y, z)eikz .

(3.318)

The constant k is again defined as 2πn/λ. n is the refractive index of the homogeneous material in which the Gaussian beam propagates and λ is the wavelength in a vacuum. To simplify the notation the wavelength λn = λ/n in the material is used in the following. By using the scalar Helmholtz equation (3.221) ∇ 2 + k2 u(x, y, z) = 0

(3.319)

the following equation for u is obtained ∂Ψ ikz ∂u = e , ∂x ∂x ∂2u ∂ 2 Ψ ikz = e , ∂x 2 ∂x 2 ∂Ψ ikz ∂u = e + ikΨ eikz , ∂z ∂z ∂2u ∂ 2 Ψ ikz ∂Ψ ikz e − k2 Ψ eikz ; = e + 2ik ∂z ∂z 2 ∂z 2 2 ∂ Ψ ∂2Ψ eikz + ⇒ ∇ 2 + k2 u = ∂x 2 ∂y2 ∂2Ψ ∂Ψ ikz + 2 eikz + 2ik e =0. ∂z ∂z (3.320)

According to our assumption that Ψ changes only slowly along the z direction the term ∂ 2 Ψ /∂z 2 is assumed to be so small that it can be neglected. This is the case if the relative variation of ∂Ψ /∂z during the propagation by one wavelength is much smaller than one. In a mathematical

formulation this means ! ! ! ! 2 ! ! ! ∂ Ψ ! ! ∂Ψ ! 4π ! ∂Ψ ! != ! ! !2k ! ! ! ∂z 2 ! ! ∂z ! λ ! ∂z ! n ! |∆(∂Ψ /∂z)| !! ⇒ 4π . |∂Ψ /∂z| !∆z=λn

(3.321)

Using this simplification the following equation for Ψ is obtained ∂2Ψ ∂2Ψ ∂Ψ =0. + 2 + 2ik ∂z ∂x 2 ∂y

(3.322)

This equation is called the paraxial Helmholtz equation because it corresponds to the case of Fresnel diffraction (Sects. 3.4.3 and 3.5.2). To solve it we first use a quite simple approach for Ψ , which corresponds to a fundamental mode Gaussian beam 0 1 k x 2 + y2 i P(z) + 2q(z) Ψ (x, y, z) = Ψ0 exp (3.323) with the two complex functions P and q, which are both functions of z. Ψ0 is a constant that depends on the amplitude of the Gaussian beam and is determined by the boundary conditions. Using the notations P := dP/ dz and q := dq/ dz our approach gives . / ik x 2 + y2

∂Ψ

= iP − q Ψ, ∂z 2q 2 ∂Ψ kx = i Ψ, ∂x q k2 x 2 ∂2Ψ ik = Ψ− 2 Ψ. 2 q ∂x q Inserting these equations in (3.322) results in the following conditions for P and q k 2 x 2 + y2

2ik k2 x 2 + y2

− − 2k P + q =0. q q2 q2 (3.324)

This equation has to be fulfilled for arbitrary values of x and y. Therefore, the equation finally gives two equations i P = and q = 1 . (3.325) q By integration we obtain q(z) = q0 + z, z . P(z) = i ln 1 + q0

(3.326) (3.327)

Wave Optics

The integration constant of P has been put to zero because it would just introduce a constant phase factor in Ψ . Equation (3.323) has a similar form like a paraxial spherical wave, i. e., a parabolic wave, if q is interpreted as a kind of complex radius of curvature. Therefore, it is useful to split 1/q into a real and an imaginary part 1 λn 1 = +i . q(z) R(z) πw2 (z)

(3.328)

q0 = −i

πw20 . λn

(3.329)

The propagation constant w0 , which corresponds to the curvature 1/R0 = 0, is called the beam waist. Later it will be shown (3.337) that the beam waist is the smallest beam radius of a Gaussian beam during its propagation. In summary by using (3.323), (3.326), (3.327), (3.328), and (3.329) the function Ψ of a fundamental mode Gaussian beam can be written as:

Ψ (x, y, z) = Ψ0

1 1 + i λn z2

The Fresnel diffraction integral is determined according to (3.256), where the argument z 0 is substituted by z, because this equation is not only valid in a plane, and k is defined as usual as k = 2πn/λ ik ikz e (3.331) u(x, y, z) = − 2πz 2 2

x−x + y−y ik 2z dx dy

× u0 x , y , 0 e

x 2 + y2 k(x 2 + y2 ) − 2 i e w (z) e 2R(z) .

πw0

(3.330)

3.5.2 The Fresnel Diffraction Integral and the Paraxial Helmholtz Equation In Sect. 3.4.3 the Fresnel diffraction integral is derived as a paraxial solution of the Fresnel–Kirchhoff diffraction formula. Here, it will be shown that the Fresnel diffraction integral describes the propagation of waves with complex amplitudes that fulfill the paraxial Helmholtz equation (3.322). So, the name “paraxial Helmholtz equation” is appropriate.

= Ψ (x, y, z)eikz ; ⇒ Ψ (x, y, z) = − ×

ik 2πz

ik u 0 x , y , 0 e

x − x

2 + y − y

2z dx dy .

So, the function Ψ is defined in accordance with (3.318) and it has to be shown that this function Ψ is a solution of the paraxial Helmholtz equation (3.322). We have the following equations for the partial derivatives of Ψ k2 ∂Ψ = u 0 x , y , 0 x − x

(3.332) 2 ∂x 2πz A 2 2 x − x + y − y

ik 2z ×e dx dy , 2 k3 ∂2Ψ = i u 0 x , y , 0 x − x

2 3 ∂x 2πz A 2 2 x − x + y − y

ik 2z ×e dx dy

2 k + u 0 x , y , 0 2πz 2 A 2 2 x − x + y − y

ik 2z dx dy , ×e 2 k3 ∂2Ψ = i u 0 x , y , 0 y − y

∂y2 2πz 3 A 2 2 x − x + y − y

ik 2z ×e dx dy

2 k + u 0 x , y , 0 2πz 2 A 2 2 x − x + y − y

ik 2z dx dy , ×e

Part A 3.5

⇒

145

The real part is the curvature of the wave and R is the real radius of curvature. The selection of the imaginary part of 1/q becomes obvious by inserting (3.328) into (3.323). It:shows that the real function w describes the distance x 2 + y2 from the z-axis at which the amplitude decreases to 1/ e of the maximum value. Therefore, w is called the beam radius; w and R are both real functions of z. A further simplification can be made by choosing q0 = q(0) as an imaginary number. This means that the radius of curvature R is infinity at z = 0, i. e., the curvature of the wave is zero at z = 0 1 λn =i q0 πw20

3.5 Gaussian Beams

146

Part A

Basic Principles and Materials

ik ∂Ψ = ∂z 2πz 2

−

k2 4πz 3

u 0 x , y , 0

R(z) = z +

2 + y − y

ik 2z ×e dx dy

u0 x , y , 0 A

x − x

2 2 # x − x + y − y

2 2 x − x + y − y

ik 2z ×e dx dy . ×

Part A 3.5

So, it is clear that the function Ψ of (3.331) fulfills the paraxial Helmholtz equation ∂2Ψ ∂2Ψ ∂Ψ =0. + 2 + 2ik ∂z ∂x 2 ∂y Therefore, the Fresnel diffraction integral and the paraxial Helmholtz equation correspond to each other. The propagation of a Gaussian beam can be made either by calculating the Fresnel diffraction integral if the complex amplitude u 0 is given in a plane, or the propagation rules can be directly derived from the paraxial Helmholtz equation as has been done to obtain (3.326) and (3.327) [3.10].

3.5.3 Propagation of a Gaussian Beam The parameters w and R of a Gaussian beam change during the propagation of the beam along the z-axis. An explicit representation of w and R can be obtained by combining (3.326), (3.328), and (3.329) πw2 z + i λn 0

λn z 2 + z 2R λ2 z 2 = w20 + n 2 . (3.337) π zR π 2 w0

The last equation shows that the beam waist w0 is indeed the smallest value of the beam radius w and that it is obtained at z = 0. Simultaneously, the radius of curvature R is infinity at z = 0. The equation also shows that the √ beam radius of the Gaussian beam has the value w = 2w0 at the distance z = z R (Rayleigh length) from the beam waist. Another interesting limiting case is the far field, i. e., z → ±∞. Then we have R(z) = z, λn |z| w(z) = . πw0 The far field angle θ of a Gaussian beam is θ ≈ tan θ =

λn w(z) = . |z| πw0

(3.338) (3.339)

(3.340)

So, by measuring the far field angle θ and the wavelength λn of a laser diode its beam waist w0 can be calculated if we assume that the fundamental Gaussian beam is a good description for the wavefront of a laser diode. By using (3.337) the function Ψ (3.330) can be written in a more illustrating way λn z w0 w0 − i πw 1 w0 0 = = . λn z λ2 z 2 1 + i λn z2 w0 + i πw w20 + 2n 2 0 πw π w0

w0 − i

To simplify the notation the so-called Rayleigh length is defined as

with A=

(3.333)

So, by separating the real and the imaginary part, two equations are obtained z 1 = , R z 2 + z 2R zR λn = ; 2 2 πw z + z 2R

w2 (z) =

(3.336)

The term in parentheses of the numerator can be expressed as

λn 1 1 +i = = 2 . πw2 R πw2 πw20 z − i λn 0 2 z + λn

πw20 z R := . λn

⇒

π 2 w4 z 2R = z+ 2 0 , z λn z

and

& w20 +

λ2n z 2 = w(z) π 2 w20

w0 , w(z) λn z sin Φ = − ; πw0 w(z) λn z tan Φ = − . πw20

cos Φ =

(3.334) (3.335)

λn z = A eiΦ = A cos Φ + iA sin Φ πw0

⇒

Wave Optics

3.5 Gaussian Beams

147

In summary we have 1 1 + i λn z2 πw0

λn z w0 iΦ(z) e with tan Φ(z) = − . w(z) πw20 (3.341)

w z

The complex amplitude u of a Gaussian beam can then be expressed using (3.318), (3.330), and (3.341) Fig. 3.31 Scheme showing the propagation of a Gaussian beam along the z-axis. The Gaussian beam is laterally limited by the beam radius w and its wavefront has the local radius of curvature R

u(x, y, z)

(3.342)

This means that a Gaussian beam has a Gaussian profile for a constant value z (Fig. 3.30). The term w0 /w(z) ensures that the total power PG of the beam is conserved during the propagation along the z direction: +∞ +∞ |u(x, y, z)|2 dx dy PG (z) = −∞ −∞

x +y +∞ +∞ 2 −2 2 w w (z) dx dy Ψ02 2 0 e = w (z) 2

−∞ −∞ w2 = Ψ02 2 0 w (z)

πw20 πw2 (z) = Ψ02 = constant . 2 2 (3.343)

By interpreting the beam radius w as lateral extension of the Gaussian beam it can be graphically symbolized as in Fig. 3.31. At the beam waist the local curvature of the Gaussian beam is zero. In the far field, the radius of curvature R increases proportional to z like the radius of curvature of a spherical wave.

⏐u⏐

⏐u⏐max w

⏐u⏐max/e x

Fig. 3.30 Amplitude of a Gaussian beam at a constant

value z

3.5.4 Higher-Order Modes of Gaussian Beams In (3.323) a quite simple approach has been selected for the function Ψ , which mainly describes the lateral variation of the Gaussian beam. This approach is the fundamental mode in the case of rotational symmetry. In the following a more general approach is made. The beam can now have two different principal curvatures along the local x and y direction and also higher order modes are taken into account. The different principal curvatures are useful to describe the radiation of laser diodes (e.g., edge emitter), which often have different beam radii and radii of curvature in the x and y directions. Therefore, the following approach is taken for Ψ [3.58]: √ √ x y Ψ (x, y, z) =g h 2 2 (3.344) wx (z) w y (z)

ky2 kx 2 + i P(z) + 2qx (z) 2q y (z) . ×e The functions g and h have to describe the lateral variations of the amplitude of the different modes and therefore it is useful to take the normalized quantities x/wx and y/w y , which are pure numbers without a physical unit. Here, wx and w y are again the beam √radii in the x and y directions, respectively. The factor 2 in the arguments of g and h seems to be quite arbitrary at the moment. But it will be shown in the following that this leads to a well-known differential equation for g and h. This approach for Ψ has to fulfill (3.322). The functions are written without arguments in the following to simplify the notation. Additionally, the first or second derivative of a function f (η) with respect to its argument η is written as f or f

, respectively. However, it should be kept in mind that, e.g., g is in fact a function of x and z (since wx is a function of z) and therefore the

Part A 3.5

x 2 + y2 k(x 2 + y2 ) − i w0 2 e w (z) eiΦ(z) e 2R(z) eikz . = Ψ0 w(z)

148

Part A

Basic Principles and Materials

Part A 3.5

√ derivative g (η) := dg(η)/ dη with η := 2x/wx (z) remains a function of x and z. On the other hand, qx is, e.g., only a function of z so that the derivative qx (z) is just defined as dqx (z)/ dz. So, using these notations, we have: / 0 . ikx 2 iky2

∂Ψ

= gh iP − 2 qx − 2 q y ∂z 2qx 2q y /1 . √ x

√ y

+ −gh 2 2 w y − g h 2 2 wx wy wx 2 2 ky kx + i P+ 2q 2q x y ×e , 2 2 i P + kx + ky

ikx √ g ∂Ψ 2qx 2q y , = gh + 2 h e ∂x qx wx

√ ikx

k2 x 2 ik gh − 2 gh + 2 gh qx q x wx qx √ ikx

gh +2 2 h + 2 q x wx wx kx 2 ky2 + i P+ 2qx 2q y , ×e . √ iky k 2 y2 ∂2Ψ ik = gh − 2 gh + 2 gh

2 qy qy wy ∂y qy / √ iky h

+2 2 g + 2 gh qy wy wy kx 2 ky2 + i P+ 2qx 2q y . ×e ∂2Ψ = ∂x 2

By inserting these functions in (3.322) and dividing by gh the following equation is obtained ik k2 x 2 k 2 y2 ik + − 2k P + 2 (qx − 1) + 2 (q y − 1) qx q y qx qy

√ ikx g wx g w x − +2 2 −2 2 2 qx gwx wx g

√ wy h iky h = 0 . (3.345) w y − +2 2 −2 2 2 qy hw y wy h This equation also has to be fulfilled for x → ∞ and y → ∞. Then, the terms proportional to x 2 and y2 are very large compared to the other terms and, similarly to the case of the fundamental mode, the following two

conditions have to be fulfilled: qx = 1

⇒

qx = qx,0 + z

q y = 1

⇒

q y = q y,0 + z .

and (3.346)

Additionally, qx and q y are analogous to the fundamental mode split up into real and imaginary parts 1 1 λn = +i qx Rx πw2x

and

1 1 λn = +i . qy Ry πw2y (3.347)

Calculating the derivative with respect to z delivers for the first equation −

1 R

λn w x = − 2 = − 2x − 2i . 2 qx qx Rx πw3x

(3.348)

This equation is added to the square of (3.347). Then, the real and the imaginary part are split resulting in two equations R x = 1 −

λ2n R2x π 2 w4x

and w x =

wx . Rx

(3.349)

Analogous results are obtained for R y and w y . Inserting these results into (3.345) finally delivers ik ik + − 2k P

qx q y √ x g

+2 2 −4 2 3 gwx wx

√ y h +2 2 −4 2 3 hw y wy

g h

=0. h

(3.350)

Now, the terms in the first row depend only on z, whereas the terms in the second row depend on x and z and the terms in the third row on y and z. Therefore, a separation approach has to be made ik ik + − 2k P = − f x (z) − f y (z) , qx q y √ x g

2 2 − 4 2 3 = f x (z) , gwx wx g √ y h

2 2 − 4 2 3 = f y (z) , hw y wy h

(3.351) (3.352) (3.353)

where f x and f y are functions that only depend on z. The solution of the differential equation for g (and analogously for h) shall be described briefly because it is a quite general solution scheme, which is often applied in optics and physics. First, the differential equation

Wave Optics

is written by using η = α := f x w2x as

√ 2x/wx and the abbreviation

d2 g(η) dg(η) 1 − αg(η) = 0 . − 2η (3.354) 2 dη 2 dη The usual approach to solve such a differential equation is to write g as a polynomial ∞ $ am ηm , g(η) = dg(η) = dη d g(η) = dη2

m=1 ∞ $

mam ηm−1 , m(m − 1)am ηm−2 .

(3.355)

m=2

Inserting of this approach into the differential (3.354) and arranging for equal powers of η gives ∞ $ m(m − 1)am ηm−2 m=2

−2

∞ $ m=1

∞

1 $ mam η − α am ηm 2 m

m=0

1 − 2m + α am ηm = 0 . (3.356) 2 This equation can only be fulfilled for all possible values of η if each coefficient in front of ηm is zero, i. e., 1 (m + 2)(m + 1)am+2 − 2m + α am = 0 2 2m + 12 α am . ⇒ am+2 = (3.357) (m + 2)(m + 1) Now, if there would be no stop criterion for the progression of coefficients am this equation would tend towards very large values of m to 2 lim am+2 = am (3.358) m→∞ m because α has a finite value. But this is the same progression of coefficients as for exp(η2 ) ∞ 2 m ∞ $ $ 2 η η2m η = e = m! m! =

m=0 ∞ $

So, the progression of coefficients will in this case be: 1 bm = m 2 ! ⇒

bm+2 =

1 m+2 2

2 = bm . ! m +2

(3.360)

Therefore, for very large values m the progression of coefficients bm will have the same behavior as the coefficients am and the amplitude |Ψ | of the higher-order mode Gaussian beam would tend to infinity for large values η because the compensating term [(3.344), (3.347)] only has the form exp(−x 2 /w2x ) = exp(−η2 /2). But, for physical reasons, |Ψ | has to tend to zero for large values of η. Therefore, there has to be a stop criterion for the progression of coefficients, which just means that the variable α has to fulfill the following equation α = f x w2x = −4 j; 4j ⇒ fx = − 2 . wx

j = 0, 1, 2, . . . (3.361)

By inserting this into (3.354) the well-known differential equation for the Hermite polynomials H j is obtained

m=0

∞ $ (m + 2)(m + 1)am+2 =

m=0

∞ $ 1 m ηm = bm ηm . ! m=0,2,4,... 2 m=0,2,4,...

(3.359)

149

d2 g(η) dg(η) + 2 jg(η) = 0 . − 2η dη dη2

(3.362)

The progression of coefficients of the Hermite polynomials fulfill, according to (3.357) and (3.361), the condition 2m + 12 α am (m + 2)(m + 1) 2(m − j) am . = (m + 2)(m + 1)

am+2 =

(3.363)

But, we have two progressions of coefficient, one for odd numbers m and one for even numbers m. So, if j is odd, only the odd coefficient progression will stop and vice versa with even values of j. Therefore, one of the coefficients a0 or a1 , which are the two integration constants of our second-order differential equation, must additionally be zero. So, we now have the possibility to calculate the Hermite polynomials H j , which are the solutions of g and h. The Hermite polynomials are, in most textbooks, normalized but by using (3.363) it is only possible to calculate the unnormalized Hermite polynomials. But this is no problem since we do not need the normalized polynomials.

Part A 3.5

m=0 ∞ $

3.5 Gaussian Beams

150

Part A

Basic Principles and Materials

If we take a0 = 0 and a1 = 0 for the even Hermite polynomials H j and a0 = 0 and a1 = 0 for the odd Hermite polynomials we obtain up to the third order apart from the normalization constant H0 (η) = 1,

f y (taking n instead of j), in λn i 1 1 λn dP −m = + −n dz 2 qx q y πw2x πw2y i 1 1 1 1 = + − mIm − nIm . 2 qx q y qx qy

H1 (η) = η,

(3.365)

H2 (η) = −2η2 + 1,

Using (3.346) and

2 H3 (η) = − η3 + η . 3

(3.364)

Part A 3.5

Equation (3.351) for P results, together with (3.361) (taking m instead of j) and the analogous equation for

1 1 1 = = qx qx,0 + z Re(qx,0 ) + iIm(qx,0 ) + z =

z + Re(qx,0 ) − iIm(qx,0 ) z 2 + 2zRe(qx,0 ) + |qx,0 |2

(3.366)

finally gives –2

–2

–1

3 –3 –2 –1 0 1 TEM00 –3

3 –3 –2 –1

–2

–1

2 0 1 TEM11

3 –3 –2 –1

–3

–2

–1

3 –3 –2 –1 0 1 TEM21

P(z) = i ln

0 1 TEM10

0 1 TEM20

–3

–2

3 –3 –2 –1

z z 1+ 1+ qx,0 q y,0 z + Re(qx,0 ) − m arctan −Im(qx,0 ) z + Re(q y,0 ) . − n arctan −Im(q y,0 )

–3

3 –3 –2 –1

0 1 TEM22

Fig. 3.32 Simulation of the intensity distributions of some

Hermite–Gaussian modes using the normalized coordinates x/wx and y/w y

(3.367)

Note that −Im(qx,0 ) and −Im(q y,0 ) are used because these quantities are positive as will be seen in (3.369). In summary, the function Ψ of the higher-order mode Gaussian beams (Hermite–Gaussian modes) in the case of a cartesian coordinate system can be written by using (3.344)–(3.367) √ √ x y Hn 2 2 Ψ (x, y, z) = Hm wx (z) w y (z) 1 × z z 1 + qx,0 1 + q y,0

z + Re(qx,0 ) −i m arctan −Im(qx,0 ) ×e z + Re(q y,0 ) +n arctan −Im(q y,0 ) y2 x2 + iπ λn Rx (z) λn R y (z) ×e / . y2 x2 + − w2x (z) w2y (z) . (3.368) ×e

Wave Optics

3.5 Gaussian Beams

151

The functions wx and Rx are obtained by comparing the real and imaginary parts of (3.347) and (3.366) λn z 2 + 2zRe(qx,0 ) + |qx,0 |2 , π −Im(qx,0 ) z 2 + 2zRe(qx,0 ) + |qx,0 |2 . Rx (z) = z + Re(qx,0 )

w2x (z) =

/ . (3.371)

They are named TEMmn , where m is the index √ of the Hermite polynomial Hm with the argument 2x/wx and n is the index √ of the Hermite polynomial Hn with the argument 2y/w y . The number of zeros is equal to the mode number and the area covered by the modes increases with the mode number. So, the complete behavior of the higher order Gaussian beam is well defined if the complex quantities qx,0 and q y,0 at the plane z = 0 are known. This is the case if the beam radii wx,0 and w y,0 and the radii of curvature Rx,0 and R y,0 of the wavefront at the plane z = 0 are known. Hereby, the beam waists in x and y direction can be in different planes. If both beam waists are in the same plane the coordinate system can be chosen such that the beam waists are in the plane z = 0 and qx,0 = −iπw2x,0 /λn with the beam waist wx,0 in the x direction. Then a simplification similar to the case of the fundamental mode of a Gaussian beam can be made and (3.369) and (3.370) reduce to (3.336) and (3.337). Also, (3.368) can then be simplified.

3.5.5 Transformation of a Fundamental Gaussian Beam at a Lens The transformation of a fundamental Gaussian beam at a (thin) lens is performed using a paraxial approx-

Fig. 3.33 Transformation of a Gaussian beam at a thin ideal

lens

imation. This means that it is assumed that the beam radius immediately in front of the lens is identical to the beam radius immediately behind the lens. Additionally, the radius of curvature of the Gaussian beam changes in the same way as that of a spherical wave. The sign convention is that a positive lens has a positive focal length f > 0 and that a divergent spherical wave coming from the negative z direction (i. e., from “left” using the optical agreement) has a positive radius of curvature R > 0. R1 is the radius of curvature immediately in front of the lens and R2 is the radius of curvature immediately behind the lens. Then, a lens with focal length f transforms the radii of curvature according to the paraxial imaging equation of geometrical optics (Fig. 3.33) 1 1 1 = − . R2 R1 f

(3.372)

Since the beam radius remains constant, the complex beam parameters q1 immediately in front of the lens and q2 immediately behind the lens transform also with 1 1 1 = − . q2 q1 f

(3.373)

In the case of a thick lens or a lens system the two principal planes of the lens system have to be taken as reference planes for q1 and q2 according to the laws of paraxial geometrical optics. If the q parameters are different in x and y direction as in (3.347) both sets of parameters just have to be treated separately using (3.373). To calculate the relation between the q parameter q1 in the distance d1 in front of a lens with the focal length f and q2 in the distance d2 behind the lens, (3.326) and (3.373) have to be combined. We call qL the Gaussian beam parameter immediately in front of the lens and qR the beam parameter immediately behind the lens, where

Part A 3.5

x2 y2 −2 + 2 2 wx (z) w y (z) ×e

(3.370)

Analogous equations are, of course, valid for w y and R y , which are obtained by substituting the index x for y. Figure 3.32 shows the typical intensity distribution |Ψ |2 of some lower-order Hermite–Gaussian modes with 2

√ √ x y |Ψ (x, y, z)|2 = Hm Hn 2 2 wx (z) w y (z) !2 ! ! ! ! ! ! ! 1 ! × !! ! ! ! z z ! 1 + qx,0 1 + q y,0 ! .

(3.369)

152

Part A

Basic Principles and Materials

qL qR

Fig. 3.34 Scheme showing the complex beam parameters for the transformation of a Gaussian beam from a plane in the distance d1 in front of a lens (with focal length f ) to a plane in the distance d2 behind the lens

the parameters are illustrated in Fig. 3.34. Then we have

Part A 3.5

qL = q1 + d1 ,

Free Space Propagation The free space propagation in a homogeneous material with refractive index n is described by (3.326): q2 = q1 + z. On the other hand the paraxial matrix for free space propagation between two planes with a distance z is / / . . 1 z A B = 0 1 C D

⇒ q2 = q1 + z =

Aq1 + B 1 · q1 + z = . 0 × q1 + 1 Cq1 + D

(3.377)

So, the free space propagation fulfills the ABCD matrix law of Gaussian beams (3.376).

1 1 1 1 1 = − = − , qR qL f q1 + d1 f

Thin Lens For the transformation of a Gaussian beam at a thin lens, (3.373) is valid. The paraxial matrix of a thin lens with f (q1 + d1 ) focal length f is ; ⇒ qR = f − q1 − d1 . / . / 1 0 fq1 + fd1 + fd2 − d2 q1 − d1 d2 A B = , q2 = qR + d2 = − 1f 1 C D f − q1 − d1 Aq1 + B fq1 1 × q1 + 0 q1 1 − df2 + d1 + d2 − d1fd2 . (3.378) = = q1 ⇒ q2 = ⇒ q2 = . (3.374) f − q1 Cq1 + D − f +1 q1 d1 − f + 1− f So, the transformation of a Gaussian beam at a thin lens also fulfills the ABCD matrix law of (3.376).

3.5.6 ABCD Matrix Law for Gaussian Beams The propagation through an optical system can be described in the paraxial geometrical optics by an ABCD matrix (see the chapter about geometrical optics or [3.11,58–60]). We compare the terms of (3.374) with the paraxial ABCD matrix for the propagation from a plane with the distance d1 in front of a lens with focal length f to a plane with distance d2 behind the lens / . . / 1 − df2 d1 + d2 − d1fd2 A B = . (3.375) C D − 1f 1 − df1

A Sequence of Lenses and Free Space Propagation We assume that M1 and M2 are the paraxial matrices for two subsequent operations like free space propagation or transformation at a thin lens. The Gaussian beam has the q parameters q0 before the first operation, q1 after the first operation and q2 after the second operation. Both (3.377) and (3.378) fulfill (3.376). Therefore, we have the relations

q1 =

A1 q0 + B1 C1 q0 + D1

and q2 =

A2 q1 + B2 . C2 q1 + D2 (3.379)

We see that the Gaussian-beam parameter transforms as Aq1 + B . q2 = Cq1 + D

Substitution of q1 into q2 gives (3.376)

It can be shown that this ABCD matrix law is valid quite generally as long as the paraxial approximation holds. In the following it will be shown for a sequence of (thin) lenses and free space propagation.

q2 = =

+B1 A2 CA11qq00+D + B2 1

+B1 C2 CA11qq00+D + D2 1

(A2 A1 + B2 C1 )q0 + (A2 B1 + B2 D1 ) . (C2 A1 + D2 C1 )q0 + (C2 B1 + D2 D1 )

(3.380)

Wave Optics

However, the paraxial matrix M of both operations is . M=

A2 B2

= M2 M1 / .

C2 D2 .

d2 d2 =− =β, f d1 d1 d2 =0, d1 + d2 − f d1 d1 1 =− = . 1− f d2 β

/ A1 B1 C1 D1

C2 A1 + D2 C1 C2 B1 + D2 D1

/ .

(3.381)

This shows that the ABCD matrix law is valid for two subsequent operations of free space propagation or transformation at a thin lens. Therefore, it also has to be valid for an arbitrary number of those operations. Geometrical optics shows that a thick lens can be replaced by a thin lens and free space propagation. Therefore, the ABCD matrix law can also be applied for thick lenses or a system consisting of many lenses. We assume that the paraxial ABCD matrix of such a system is known for describing the propagation between two planes with the optical system in between. The transformation of the Gaussian beam parameter q1 at the first plane to the parameter q2 at the second plane is then described by (3.376). Of course, it is always assumed that no apertures are in the system and that the paraxial approximation is valid, i. e., the optical system is ideal and does not introduce any aberrations.

βq1 − qf1

1 β

⇒

(3.386)

1 1 1 + 2 . =− q2 β f β q1

(3.387)

Using (3.328) to split the complex q parameters into their real variables delivers 1 1 λn 1 λn + 2 +i +i =− ; 2 2 R2 β f β R1 πw2 β πw21 1 1 1 and w2 = |β|w1 . ⇒ = 2 − R2 β R1 β f (3.388)

The result is that the beam radius transforms from one plane to another plane with the lateral magnification β if the imaging equation is fulfilled for these two planes, i. e., if the first plane is imaged by the lens onto the second plane. Position and Size of the Beam Waist Behind a Lens It is assumed that a Gaussian beam has its beam waist w0 at the distance d1 in front of a lens with focal length f so that the beam parameter in the first plane is, according to (3.329), q1 = −iπw20 /λn . The position and size of the beam waist behind the lens has to be calculated. The desired parameters are the size w2 of the beam waist and its distance d2 from the lens (Fig. 3.35).

3.5.7 Some Examples of the Propagation of Gaussian Beams Transformation in the Case of Geometrical Imaging A Gaussian beam with beam parameter q1 at the distance d1 in front of the first principal plane of a lens (or a lens system) with focal length f is examined at the distance d2 behind the second principal plane of the lens. There, the Gaussian beam has the beam parameter q2 . Additionally, it is assumed that the distances d1 and d2 and the focal length of the lens fulfill the imaging equation of paraxial geometrical

(3.385)

f w2

Fig. 3.35 Scheme showing the parameters to calculate the

position and the size of the beam waist of a Gaussian beam behind a lens

Part A 3.5

q2 =

(3.382)

(3.384)

By using (3.374) we obtain

Summarizing, the relation between q2 and q0 is Aq0 + B q2 = . Cq0 + D

(3.383)

1−

A2 A1 + B2 C1 A2 B1 + B2 D1

153

optics 1 1 1 + = ⇒ d1 d2 f

C D .

A B

3.5 Gaussian Beams

154

Part A

Basic Principles and Materials

θ θ = λ /π w0

In the limiting case of geometrical optics, i. e., w0 → 0, this equation is equal to the paraxial imaging equation. The beam waist w2 can be determined using the imaginary part of (3.389) and replacing then the term d1 + d2 − d1 d2 / f with the help of the first part of (3.390) λn = πw22 2 1 − df2

Fig. 3.36 Special case of the transformation of a Gaussian

beam at a lens where the beam waist lies in the front focal plane of the lens

Part A 3

Using (3.328) and (3.374) results in 1 λn 1 = +i q2 R2 πw22 πw2 i λn f0 + 1 − df1 = 2 ; πw −i 1 − df2 λn 0 + d1 + d2 − d1fd2 ⇒

πw20 λn

⎛

π 2 w40 λ2n

⎝1 +

π 2 w4 0 λ2 f 2 n 2 d 1− f1

⎞ ; (3.391) ⎠

⎛ ⎞ π 2 w40 2 2 2 d2 ⎜ λn f ⎟ ⇒ w22 = w20 1 − ⎝1 + 2 ⎠ . f d1 1− f Equation (3.390) delivers 1−

1 λn +i R2 πw22 ⎧ ⎫ 2 4 πw2 π w ⎪ ⎨ i λ 0 − 1 − df2 λ2 f0 + ⎪ ⎬ n n ⎪ ⎩ 1 − d1 d1 + d2 − d1 d2 ⎪ ⎭ f f = 2 2 4 2 . π w0 d1 d2 1 − df2 + d + d − 1 2 f λ2

d2 = f

π 2 w40 λ2n f 2

1 − df1 2 + 1 − df1

(3.392)

so that the final result for w2 is w22 =

(3.389)

At the position of the beam waist the real part 1/R2 of this equation has to vanish. This gives a condition for calculating d2 : d2 π 2 w40 − 1− f λ2 f n d1 d1 d2 d1 + d2 − =0; + 1− f f π 2 w40 − d1 1 − df1 λ2n f ⇒ d2 = 2 4 (3.390) 2 . π w0 d1 + 1 − 2 2 f λ f n

π 2 w40 λ2n f 2

w20 2 . + 1 − df1

(3.393)

Of special interest is the case that the beam waist w0 in front of the lens lies in the front focal plane of the lens, i. e., d1 = f . Equations (3.390) and (3.393) reduce in this special case to d2 = f

and w2 =

λn f . πw0

(3.394)

So, if the beam waist of the incident Gaussian beam lies in the front focal plane of the lens, the beam waist of the transformed Gaussian beam lies in the back focal plane of the lens (Fig. 3.36). Additionally, its size w2 will be the product of the focal length f of the lens and the far field angle (3.340) of the incident Gaussian beam. This result shows that the transformation of Gaussian beams should not be confused with the transformation of paraxial spherical waves of geometrical optics.

References 3.1 3.2

M. Born, E. Wolf: Principles of Optics, 6th edn. (Cambridge University Press, Cambridge 1997) R. W. Boyd: Nonlinear Optics (Academic, San Diego 2003)

3.3 3.4

A. Yariv, P. Yeh: Optical Waves in Crystals (Wiley, New York 1984) A. E. Siegman: Propagating modes in gain-guided optical fibers, J. Opt. Soc. Am. A 20, 1617–1628 (2003)

Wave Optics

3.5 3.6 3.7 3.8 3.9

3.10 3.11

3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22

3.23

3.24

3.25

3.26

3.27

3.28

3.29

3.30

3.31

3.32

3.33

3.34

3.35 3.36 3.37

3.38

3.39 3.40 3.41 3.42

3.43 3.44 3.45

3.46 3.47 3.48

3.49

Techniques and Applications, ed. by P. K. Rastogi (Artech House, Norwood 1997) pp. 15–50 O. Bryngdahl: Applications of Shearing Interferometry, Prog. Opt., Vol. 4, ed. by E. Wolf (Elsevier, New York 1965) pp. 37–83 H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider: Characterization of microlenses using a phaseshifting shearing interferometer, Opt. Eng. 33, 2680–2686 (1994) K. Creath: Phase-Mesurement Interferometry Techniques, Prog. Opt., Vol. 26, ed. by E. Wolf (Elsevier, New York 1988) pp. 349–393 A. Hettwer, J. Kranz, J. Schwider: Three channel phase-shifting interferometer using polarizationoptics and a diffraction grating, Opt. Eng. 39, 960–966 (2000) S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs: Focusing light to a tighter spot, Opt. Commun. 179, 1–7 (2000) B. Richards, E. Wolf: Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system, Proc. R. Soc. A 253, 358–379 (1959) M. Françon: Diffraction (Pergamon, Oxford 1966) J. E. Harvey: Fourier treatment of near-field scalar diffraction theory, Am. J. Phys. 47, 974–980 (1979) E. Lalor: Conditions for the validity of the angular spectrum of plane waves, J. Opt. Soc. Am. 58, 1235– 1237 (1968) U. Vokinger: Propagation, modification, analysis of partially coherent light fields. Dissertation (University of Neuchatel (UFO), Allensbach 2000) J. J. Stamnes: Waves in Focal Regions (Hilger, Bristol 1986) H. J. Caulfield: Handbook of Optical Holography (Academic, New York 1979) P. Hariharan: Basics of Holography (Cambridge University Press, Cambridge 2002) E. N. Leith, J. Upatnieks: Recent Advances in Holography, Prog. Opt., Vol. 6, ed. by E. Wolf (Elsevier, New York 1967) pp. 1–52 G. Saxby: Practical Holography (Prentice Hall, New York 1988) B. R. Brown, A. W. Lohmann: Computer generated binary holograms, IBM J. 13, 160–168 (1969) O. Bryngdahl, F. Wyrowski: Digital holography – Computer-Generated Holograms, Prog. Opt., Vol. 28, ed. by E. Wolf (Elsevier, New York 1990) pp. 1–86 H. P. Herzig: Micro-Optics (Taylor & Francis, London 1997) B. Kress, P. Meyrueis: Digital Diffractive Optics (Wiley, Chichester 2000) W.-H. Lee: Computer-Generated Holograms: Techniques, Applications, Prog. Opt., Vol. 16, ed. by E. Wolf (Elsevier, New York 1978) pp. 119–232 D. Maystre: Rigorous Vector Theories of Diffraction Gratings, Prog. Opt., Vol. 11, ed. by E. Wolf (Elsevier, New York 1984) pp. 1–67

155

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J. W. Goodman: Introduction to Fourier Optics, 2nd edn. (McGraw-Hill, New York 1996) H. Haferkorn: Optik, 4th edn. (Wiley-VCH, Weinheim 2003) E. Hecht: Optics, 3rd edn. (Addison-Wesley, Reading 1998) M. V. Klein, Th. E. Furtak: Optics, 2nd edn. (Wiley, New York 1986) V. N. Mahajan: Optical Imaging and Aberrations, Part II: Wave Diffraction Optics (SPIE Press, Bellingham 2001) D. Marcuse: Light Transmission Optics, 2nd edn. (Van Nostrand, New York 1982) A. E. Siegman: Lasers (University Science Books, Mill Valley 1986) R. A. Chipman: Polarization analysis of optical systems, Opt. Eng. 28, 90–99 (1989) R. A. Chipman: Mechanics of polarization ray tracing, Opt. Eng. 34, 1636–1645 (1995) E. Waluschka: Polarization ray trace, Opt. Eng. 28, 86–89 (1989) R. C. Jones: A new calculus for the treatment of optical systems, J. Opt. Soc. Am. 31, 488–503 (1941) J. W. Goodman: Statistical Optics (Wiley, New York 1985) M. Françon: Optical Interferometry (Academic, New York 1984) P. Hariharan: Optical Interferometry (Academic, Sydney 1985) D. Malacara (Ed.): Optical Shop Testing, 2nd edn. (Wiley, New York 1991) D. W. Robinson, G. T. Reid (Eds.): Interferogram Analysis (IOP Publ., Bristol 1993) C. M. Haaland: Laser electron acceleration in vacuum, Opt. Commun. 114, 280–284 (1995) Y. C. Huang, R. L. Byer: A proposed high-gradient laser-driven electron accelerator using crossed cylindrical laser focusing, Appl. Phys. Lett. 69, 2175– 2177 (1996) R. Dändliker: Two-Reference-Beam Holographic Interferometry. In: Holographic Interferometry, ed. by P. K. Rastogi (Springer, Berlin, Heidelberg 1994) pp. 75–108 R. Dändliker: Heterodyne Holographic Interferometry, Prog. Opt., Vol. 17, ed. by E. Wolf (Elsevier, New York 1980) pp. 1–84 G. Schulz, J. Schwider: Interferometric Testing of Smooth Surfaces, Prog. Opt., Vol. 13, ed. by E. Wolf (Elsevier, New York 1976) pp. 93–167 J. Schwider: Advanced Evaluation Techniques in Interferometry, Prog. Opt., Vol. 28, ed. by E. Wolf (Elsevier, New York 1990) pp. 271–359 W. H. Steel: Two-Beam Interferometry, Prog. Opt., Vol. 5, ed. by E. Wolf (Elsevier, New York 1966) pp. 145–197 H. J. Tiziani: Optical metrology of engineering surfaces – scope and trends. In: Optical Measurement

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3.50 3.51

3.52

3.53

3.54

Part A 3

S. Sinzinger, J. Jahns: Microoptics (Wiley-VCH, Weinheim 1999) I. N. Bronstein, K. A. Semendjajew: Taschenbuch der Mathematik, 23rd edn. (Thun, Frankfurt 1987) W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling: Numerical Recipes in C (Cambridge University Press, Cambridge 1991) pp. 398–470 B. Besold, N. Lindlein: Fractional Talbot effect for periodic microlens arrays, Opt. Eng. 36, 1099–1105 (1997) S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs: The focus of light-theoretical calculation and experimental tomographic reconstruction, Appl. Phys. B 72, 109–113 (2001)

3.55

3.56

3.57

3.58 3.59 3.60

M. Mansuripur: Distribution of light at and near the focus of high-numerical-aperture objectives, J. Opt. Soc. Am. A 3, 2086–2093 (1986) M. Mansuripur: Distribution of light at and near the focus of high-numerical-aperture objectives: Erratum, J. Opt. Soc. Am. A 10, 382–383 (1993) R. Dorn, S. Quabis, G. Leuchs: The focus of light – linear polarization breaks the rotational symmetry of the focal spot, J. Mod. Opt. 50, 1917–1926 (2003) H. Kogelnik, T. Li: Laser beams and resonators, Appl. Opt. 5, 1550–1567 (1966) W. Brouwer: Matrix Methods in Optical Instrument Design (Benjamin, New York 1964) A. Gerrard, J. M. Burch: Introduction to Matrix Methods in Optics (Wiley, London 1975)

157

Nonlinear Opt 4. Nonlinear Optics

4.1

Nonlinear Polarization and Nonlinear Susceptibilities ............... 159

4.2

Wave Aspects of Nonlinear Optics ........... 160

4.3

Second-Order Nonlinear Processes ......... 161 4.3.1 Second-Harmonic Generation........ 161 4.3.2 Sum- and Difference-Frequency Generation and Parametric Amplification............................... 163

4.4 Third-Order Nonlinear Processes ............ 4.4.1 Self-Phase Modulation ................. 4.4.2 Temporal Solitons......................... 4.4.3 Cross-Phase Modulation ............... 4.4.4 Self-Focusing............................... 4.4.5 Four-Wave Mixing ........................

164 165 166 167 167 169

4.4.6 4.4.7 4.4.8 4.4.9

Optical Phase Conjugation ............. Optical Bistability and Switching .... Stimulated Raman Scattering......... Third-Harmonic Generation by Ultrashort Laser Pulses..............

4.5 Ultrashort Light Pulses in a Resonant Two-Level Medium: Self-Induced Transparency and the Pulse Area Theorem .................. 4.5.1 Interaction of Light with Two-Level Media .................. 4.5.2 The Maxwell and Schrödinger Equations for a Two-Level Medium 4.5.3 Pulse Area Theorem ...................... 4.5.4 Amplification of Ultrashort Light Pulses in a Two-Level Medium ................ 4.5.5 Few-Cycle Light Pulses in a Two-Level Medium ................ 4.6 Let There be White Light: Supercontinuum Generation.................. 4.6.1 Self-Phase Modulation, Four-Wave Mixing, and Modulation Instabilities in Supercontinuum-Generating Photonic-Crystal Fibers ................. 4.6.2 Cross-Phase-Modulation-Induced Instabilities ................................. 4.6.3 Solitonic Phenomena in Media with Retarded Optical Nonlinearity. 4.7

Nonlinear Raman Spectroscopy .............. 4.7.1 The Basic Principles ...................... 4.7.2 Methods of Nonlinear Raman Spectroscopy ............................... 4.7.3 Polarization Nonlinear Raman Techniques .................................. 4.7.4 Time-Resolved Coherent Anti-Stokes Raman Scattering........

169 170 172 173

178 178 178 180

181 183 185

185 187 189 193 194 196 199 201

4.8 Waveguide Coherent Anti-Stokes Raman Scattering ................................. 202 4.8.1 Enhancement of Waveguide CARS in Hollow Photonic-Crystal Fibers... 202

Part A 4

This chapter provides a brief introduction into the basic nonlinear-optical phenomena and discusses some of the most significant recent advances and breakthroughs in nonlinear optics, as well as novel applications of nonlinear-optical processes and devices. Nonlinear optics is the area of optics that studies the interaction of light with matter in the regime where the response of the material system to the applied electromagnetic field is nonlinear in the amplitude of this field. At low light intensities, typical of non-laser sources, the properties of materials remain independent of the intensity of illumination. The superposition principle holds true in this regime, and light waves can pass through materials or be reflected from boundaries and interfaces without interacting with each other. Laser sources, on the other hand, can provide sufficiently high light intensities to modify the optical properties of materials. Light waves can then interact with each other, exchanging momentum and energy, and the superposition principle is no longer valid. This interaction of light waves can result in the generation of optical fields at new frequencies, including optical harmonics of incident radiation or sum- or difference-frequency signals.

158

Part A

Basic Principles and Materials

4.8.2 Four-Wave Mixing and CARS in Hollow-Core Photonic-Crystal Fibers ......................................... 205 4.9 Nonlinear Spectroscopy with Photonic-Crystal-Fiber Sources....... 4.9.1 Wavelength-Tunable Sources and Progress in Nonlinear Spectroscopy 4.9.2 Photonic-Crystal Fiber Frequency Shifters ....................................... 4.9.3 Coherent Anti-Stokes Raman Scattering Spectroscopy with PCF Sources .......................... 4.9.4 Pump-Probe Nonlinear Absorption Spectroscopy using Chirped Frequency-Shifted Light Pulses from a Photonic-Crystal Fiber ........

209 209 210

211

213

4.11 High-Order Harmonic Generation .......... 4.11.1 Historical Background ................... 4.11.2 High-Order-Harmonic Generation in Gases ...................................... 4.11.3 Microscopic Physics ...................... 4.11.4 Macroscopic Physics...................... 4.12 Attosecond Pulses: Measurement and Application ............... 4.12.1 Attosecond Pulse Trains and Single Attosecond Pulses......... 4.12.2 Basic Concepts for XUV Pulse Measurement ........... 4.12.3 The Optical-Field-Driven XUV Streak Camera Technique ........................ 4.12.4 Applications of Sub-femtosecond XUV Pulses: Time-Resolved Spectroscopy of Atomic Processes ... 4.12.5 Some Recent Developments...........

219 219 220 222 225 227 227 227 230

234 236

Part A 4

4.10 Surface Nonlinear Optics, Spectroscopy, and Imaging ........................................ 216

References .................................................. 236

Although the observation of most nonlinear-optical phenomena requires laser radiation, some classes of nonlinear-optical effects were known long before the invention of the laser. The most prominent examples of such phenomena include Pockels and Kerr electrooptic effects [4.1], as well as light-induced resonant absorption saturation, described by Vavilov [4.2, 3]. It was, however, only with the advent of lasers that systematic studies of optical nonlinearities and the observation of a vast catalog of spectacular nonlinear-optical phenomena became possible. In the first nonlinear-optical experiment of the laser era, performed by Franken et al. in 1961 [4.4], a rubylaser radiation with a wavelength of 694.2 nm was used to generate the second harmonic in a quartz crystal at the wavelength of 347.1 nm. This seminal work was followed by the discovery of a rich diversity of nonlinear-optical effects, including sum-frequency generation, stimulated Raman scattering, self-focusing, optical rectification, four-wave mixing, and many others. While in the pioneering work by Franken the efficiency of second-harmonic generation (SHG) was on the order of 10−8 , optical frequency doublers created by early

1963 provided 20%–30% efficiency of frequency conversion [4.5, 6]. The early phases of the development and the basic principles of nonlinear optics have been reviewed in the most illuminating way in the classical books by Bloembergen [4.7] and Akhmanov and Khokhlov [4.8], published in the mid 1960s. Over the following four decades, the field of nonlinear optics has witnessed an enormous growth, leading to the observation of new physical phenomena and giving rise to novel concepts and applications. A systematic introduction into these effects along with a comprehensive overview of nonlinear-optical concepts and devices can be found in excellent textbooks by Shen [4.9], Boyd [4.1], Butcher and Cotter [4.10], Reintjes [4.11] and others. One of the most recent up-to-date reviews of the field of nonlinear optics with an in-depth discussion of the fundamental physics underlying nonlinear-optical interactions was provided by Flytzanis [4.12]. This chapter provides a brief introduction into the main nonlinear-optical phenomena and discusses some of the most significant recent advances in nonlinear optics, as well as novel applications of nonlinear-optical processes and devices.

Nonlinear Optics

4.1 Nonlinear Polarization and Nonlinear Susceptibilities

159

4.1 Nonlinear Polarization and Nonlinear Susceptibilities Nonlinear-optical effects belong to a broader class of electromagnetic phenomena described within the general framework of macroscopic Maxwell equations. The Maxwell equations not only serve to identify and classify nonlinear phenomena in terms of the relevant nonlinearoptical susceptibilities or, more generally, nonlinear terms in the induced polarization, but also govern the nonlinear-optical propagation effects. We assume the absence of extraneous charges and currents and write the set of Maxwell equations for the electric, E(r, t), and magnetic, H(r, t), fields in the form (4.1) (4.2) (4.3)

t J(ζ ) dζ ,

(4.5)

−∞

where J is the induced current density. Generally, the equation of motion for charges driven by the electromagnetic field has to be solved to define the relation between the induced current J and the electric and magnetic fields. For quantum systems, this task can be fulfilled by solving the Schrödinger equation. In Sect. 4.5 of this chapter, we provide an example of such a self-consistent analysis of nonlinear-optical phenomena in a model two-level system. Very often a phenomenological approach based on the introduction of field-independent or local-field-corrected nonlinear-optical susceptibilities can provide an adequate description of nonlinear-optical processes. Formally, the current density J can be represented as a series expansion in multipoles: ∂ J = (P − ∇ · Q) + c (∇ × M) , (4.6) ∂t where P and Q are the electric dipole and electric quadrupole polarizations, respectively. In the electric dipole approximation, we keep only the first term on the right-hand side of (4.6). In view of (4.5), this gives the following relation between the D, E, and P vectors: D = E + 4π P.

(4.7)

(4.8)

where PL is the part of the electric dipole polarization linear in the field amplitude and Pnl is the nonlinear part of this polarization. The linear polarization governs linear-optical phenomena, i. e., it corresponds to the regime where the optical properties of a medium are independent of the field intensity. The relation between PL and the electric field E is given by the standard formula of linear optics: PL = χ (1) (t − t )E(t ) dt , (4.9) where χ (1) (t) is the time-domain linear susceptibility tensor. Representing the field E and polarization PL in the form of elementary monochromatic plane waves,

(4.4)

Here, B = H + 4π M, where M is the magnetic dipole polarization, c is the speed of light, and D = E + 4π

P = PL + Pnl ,

and

E = E (ω) exp (ikr − ωt) + c.c.

(4.10)

PL = PL (ω) exp ikr − ωt + c.c. ,

(4.11)

we take the Fourier transform of (4.9) to find PL (ω) = χ (1) (ω)E(ω) , where

(4.12)

χ (1) (ω) =

χ (1) (t) exp(iωt) dt .

(4.13)

In the regime of weak fields, the nonlinear part of the polarization Pnl can be represented as a power-series expansion in the field E: Pnl = χ (2) (t − t1 , t − t2 ) : E(t1 )E(t2 ) dt1 dt2 + χ (3) (t − t1 , t − t2 , t − t3 ) .. .E(t1 )E(t2 )E(t3 ) dt1 dt2 dt3 + . . . ,

(4.14)

where χ (2) and χ (3) are the second- and third-order nonlinear susceptibilities. Representing the electric field in the form of a sum of plane monochromatic waves, Ei (ωi ) exp(iki r − ωi t) + c.c. , (4.15) E= i

we take the Fourier transform of (4.14) to arrive at Pnl (ω) = P (2) (ω) + P (3) (ω) + . . . ,

(4.16)

Part A 4.1

1 ∂B , c ∂t 1 ∂D , ∇×B= c ∂t ∇·D=0, ∇·B=0.

∇×E=−

We now represent the polarization P as a sum

160

Part A

Basic Principles and Materials

where (4.17) P (2) (ω) = χ (2) (ω; ωi , ω j ) : E(ωi )E(ω j ) , . P (3) (ω) = χ (3) (ω; ωi , ω j , ωk )..E(ωi )E(ω j )E(ωk ) ,

ω 1, k 1

χ (2)

(4.18)

χ (2) (ω; ωi , ω j ) = χ (2) (ω = ωi + ω j ) = χ (2) (t1 , t2 ) exp[i(ωi t1 + ω j t2 )] dt1 dt2 (4.19) is the second-order nonlinear-optical susceptibility and χ (3) (ω; ωi , ω j , ωk ) = χ (3) (ω = ωi + ω j + ωk ) = χ (3) (t1 , t2 , t3 ) exp[i(ωi t1 + ω j t2 + ωk t3 )] dt1 dt2 dt3

ω 3, k 3

ω 2, k 2

Fig. 4.1 Sum-frequency generation ω1 + ω2 = ω3 in a medium with a quadratic nonlinearity. The case of ω1 = ω2 corresponds to second-harmonic generation (2)

(4.20)

Part A 4.2

is the third-order nonlinear-optical susceptibility. The second-order nonlinear polarization defined by (4.17) gives rise to three-wave mixing processes, optical rectification and linear electrooptic effect. In particular, setting ωi = ω j = ω0 in (4.17) and (4.19), we arrive at ω = 2ω0 , which corresponds to second-harmonic generation, controlled by the nonlin(2) ear susceptibility χSHG = χ (2) (2ω0 ; ω0 , ω0 ). In a more general case of three-wave mixing process with ωi = ω1 = ω j = ω2 , the second-order polarization defined by (4.17) can describe sum-frequency generation (SFG) ωSF = ω1 + ω2 Fig. 4.1 or difference-frequency generation (DFG) ωDF = ω1 − ω2 , governed by the

nonlinear susceptibilities χSFG = χ (2) (ωSF ; ω1 , ω2 ) and (2) χDFG = χ (2) (ωDF ; ω1 , −ω2 ), respectively. The third-order nonlinear polarization defined by (4.18) is responsible for four-wave mixing (FWM), stimulated Raman scattering, two-photon absorption, and Kerr-effect-related phenomena, including selfphase modulation (SPM) and self-focusing. For the particular case of third-harmonic generation, we set ωi = ω j = ωk = ω0 in (4.18) and (4.20) to obtain ω = 3ω0 . This type of nonlinear-optical interaction, in accordance with (4.18) and (4.20), is controlled by (3) the cubic susceptibility χTHG = χ (3) (3ω0 ; ω0 , ω0 , ω0 ). A more general, frequency-nondegenerate case can correspond to a general type of an FWM process. These and other basic nonlinear-optical processes will be considered in greater details in the following sections.

4.2 Wave Aspects of Nonlinear Optics In the electric dipole approximation, the Maxwell equations (4.1–4.4) yield the following equation governing the propagation of light waves in a weakly nonlinear medium: 4π ∂ 2 Pnl 1 ∂ 2 E 4π ∂ 2 PL = 2 . ∇ × (∇ × E) − 2 2 − 2 2 c ∂t c ∂t c ∂t 2

axis, we represent the field E in (4.21) by E (r, t) = Re eA (z, t) exp (ikz − ωt)

(4.22)

and write the nonlinear polarization as Pnl (r, t) = Re ep Pnl (z, t) exp ikp z − ωt ,

(4.21)

(4.23)

The nonlinear polarization, appearing on the right-hand side of (4.21), plays the role of a driving source, inducing an electromagnetic wave with the same frequency ω as the nonlinear polarization wave Pnl (r, t). Dynamics of a nonlinear wave process can be then thought as a result of the interference of induced and driving (pump) waves, controlled by the dispersion of the medium. Assuming that the fields have the form of quasimonochromatic plane waves propagating along the z-

where k and A(z, t) are the wave vector and the envelope of the electric field, k p and Pnl (z, t) are the wave vector and the envelope of the polarization wave. If the envelope A(z, t) is a slowly varying function over the wavelength, |∂ 2 A/∂z 2 | |k∂A/∂z|, and ∂ 2 Pnl /∂t 2 ≈ −ω2 Pnl , (4.21) is reduced to [4.9] ∂A 1 ∂A 2πiω2 + = Pnl exp (i∆kz) , ∂z u ∂t kc2

(4.24)

Nonlinear Optics

where u = (∂k/∂ω)−1 is the group velocity and ∆k = kp − k is the wave-vector mismatch. In the following sections, this generic equation of slowly varying envelope approximation (SVEA)

4.3 Second-Order Nonlinear Processes

161

will be employed to analyze the wave aspects of the basic second- and third-order nonlinear-optical phenomena.

4.3 Second-Order Nonlinear Processes 4.3.1 Second-Harmonic Generation

∂A1 1 ∂A1 + = iγ1 A∗1 A2 exp (i∆kz) , ∂z u 1 ∂t 1 ∂A2 ∂A2 + = iγ2 A21 exp (−i∆kz) , ∂z u 2 ∂t

(4.25) (4.26)

where γ1 =

where L is the length of the nonlinear medium. The intensity of the second-harmonic field is then given by 2 sin ∆kL 2 2 2 L 2, (4.30) I2 (L) ∝ γ2 I10 ∆kL 2

where I10 is the intensity of the pump field. Second-harmonic intensity I2 , as can be seen from (4.30) oscillates as a function of L Fig. 4.2 with a period L c = π/|∆k| = λ1 (4|n 1 − n 2 |)−1 , where λ1 is the pump wavelength and n 1 and n 2 are the values of the refractive index at the frequencies of the pump field and its second harmonic, respectively. The parameter L c , defining the length of the nonlinear medium providing the maximum SHG efficiency, is referred to as the coherence length. Second-harmonic intensity (arb. units)

2πω21 (2) χ (ω; 2ω, −ω) k1 c2

(4.27)

and

Lc2 = 2Lc1

0.4

0.3

γ2 =

4πω21 (2) χ k2 c2 SHG

(4.28)

are the nonlinear coefficients, u 1 and u 2 are the group velocities of the pump and second-harmonic pulses, respectively, and ∆k = 2k1 − k2 is the wave-vector mismatch for the SHG process. If the difference between the group velocities of the pump and second-harmonic pulses can be neglected for a nonlinear medium with a given length and if the intensity of the pump field in the process of SHG remains much higher than the intensity of the second-harmonic field, we set u 1 = u 2 = u and |A1 |2 = |A10 |2 = const. in (4.25) and (4.26) to derive in the retarded frame of

0.2 Lc1

0.1

0.0

6 L /Lc

Fig. 4.2 Second-harmonic intensity as a function of the

length L of the nonlinear medium normalized to the coherence length L c for two values of L c : (dashed line) L c1 and (solid line) L c2 = 2L c1

Part A 4.3

In second-harmonic generation, a pump wave with a frequency of ω generates a signal at the frequency 2ω as it propagates through a medium with a quadratic nonlinearity (Fig. 4.1). Since all even-order nonlinear susceptibilities χ (n) vanish in centrosymmetric media, SHG can occur only in media with no inversion symmetry. Assuming that diffraction and second-order dispersion effects are negligible, we use (4.24) for a quadratically nonlinear medium with a nonlinear SHG (2) susceptibility χSHG = χ (2) (2ω; ω, ω) to write a pair of coupled equations for the slowly varying envelopes of the pump and second-harmonic fields A1 = A1 (z, t) and A2 = A2 (z, t):

reference with z = z and η = t − z/u ∆kL

i∆kL 2 sin 2 , (4.29) A2 (L) = iγ2 A10 ∆kL L exp 2 2

162

Part A

Basic Principles and Materials

Part A 4.3

Although the solution (4.29) describes the simplest regime of SHG, it is very instructive as it visualizes the significance of reducing the wave-vector mismatch ∆k for efficient SHG. Since the wave vectors k1 and k2 are associated with the momenta of the pump and second-harmonic fields, p1 = k1 and p2 = k2 , with being the Planck constant, the condition ∆k = 0, known as the phase-matching condition in nonlinear optics, in fact, represents momentum conservation for the SHG process, where two photons of the pump field are requited to generate a single photon of the second harmonic. Several strategies have been developed to solve the phase-matching problem for SHG. The most practically significant solutions include the use of birefringent nonlinear crystals [4.13, 14], quasi-phase-matching in periodically poled nonlinear materials [4.15, 16] and waveguide regimes of nonlinear interactions with the phase mismatch related to the material dispersion compensated for by waveguide dispersion [4.7]. Harmonic generation in the gas phase, as demonstrated by Miles and Harris [4.17], can be phase-matched through an optimization of the content of the gas mixture. Figure 4.3 illustrates phase matching in a birefringent crystal. The circle represents the cross section of the refractive-index sphere n 0 (ω) for an ordinary wave at the pump frequency ω. The ellipse is the cross section of the refractive-index ellipsoid n e (2ω) for an extraordinary wave at the frequency of the second harmonic 2ω. Phase matching is achieved in the direction where n 0 ω = n e (2ω), corresponding to an angle θpm with respect to the optical axis c of the crystal in Fig. 4.3. When the phase-matching condition ∆k = 0 is satisfied, (4.29) and (4.30) predict a quadratic growth of the

Pump amplitude (arb. units)

Second-harmonic amplitude (arb. units)

1.0

0.8

0.6

0.4

0.2

0.0

0.0 4 z/zsh

Fig. 4.4 The amplitudes of the pump and second-harmonic

fields as functions of the normalized propagation distance z/z sh with z sh = [γρ10 (0)]−1

second-harmonic intensity as a function of the length L of the nonlinear medium. This scaling law holds true, however, only as long as the second-harmonic intensity remains much less than the pump intensity. As |A2 | becomes comparable with |A1 |, depletion of the pump field has to be taken into consideration. To this end, we introduce the real amplitudes ρ j and phases ϕ j of the pump and second-harmonic fields, A j = ρ j exp iϕ j , with j = 1, 2. Then, assuming that u 1 = u 2 = u and γ1 = γ2 = γ , we derive from (4.25) and (4.26) ρ1 (η, z) = ρ10 (η) sech [γρ10 (η) z] ,

(4.31)

ρ2 (η, z) = ρ10 (η) tanh [γρ10 (η) z] .

(4.32)

ηo (ω)

ηe (2 ω)

Fig. 4.3 Phase-matching second-harmonic generation in

a birefringent crystal

The solutions (4.31) and (4.32) show that the entire energy of the pump field in the phasematching regime can be transferred to the second harmonic. As the pump field becomes depleted Fig. 4.4, the growth of the second-harmonic field saturates. Effects related to the group-velocity mismatch become significant when the length of the nonlinear −1 medium L exceeds the length L g = τ1 /|u −1 2 − u 1 |, where τ1 is the pulse width of the pump field. The length L g characterizes the walk-off between the pump and second-harmonic pulses caused by the group-velocity mismatch. In this nonstationary regime of SHG, the amplitude of the second harmonic in the constant-pump-

Nonlinear Optics

A2 (z, t) = iγ2

−1 A210 t − z/u 2 + ξ u −1 2 − u1

z A3 (z, t) = iγ3

× exp (−i∆kξ) dξ .

−1 A10 t − z/u 3 + ξ u −1 3 − u1

−1 × A20 t − z/u 3 + ξ u −1 3 − u2

(4.33)

Group-velocity mismatch may lead to a considerable increase in the pulse width of the second harmonic τ2 . For L L g , the second harmonic pulse width, −1 τ2 ≈ |u −1 2 − u 1 |z, scales linearly with the length of the nonlinear medium and is independent of the pump pulse width.

4.3.2 Sum- and Difference-Frequency Generation and Parametric Amplification

2πω21 (2) χ (ω1 ; ω3 , −ω2 ) , (4.37) k1 c2 2πω22 (2) γ2 = χ (ω2 ; ω3 , −ω1 ) , (4.38) k2 c2 2πω23 (2) γ3 = χ (4.39) k3 c2 SFG are the nonlinear coefficients, u 1 , u 2 , and u 3 and k1 , k2 , and k3 are the group velocities and the wave vectors of the fields with frequencies ω1 , ω2 , and ω3 , respectively, and ∆k = k1 + k2 − k3 is the wave-vector mismatch for the SFG process. As long as the intensity of the sum-frequency field remains much less than the intensities of the laser fields, the amplitudes of the laser fields can be can be assumed to be given functions of t, A1 (z, t) = A10 (t) and

× exp (−i∆kξ) dξ.

(4.40)

The efficiency of frequency conversion, as can be seen from (4.40) is controlled by the group delays ∆21 ≈ −1 −1 −1 −1 |u −1 2 − u 1 |L, ∆31 ≈ |u 3 − u 1 |L, and ∆32 ≈ |u 3 − −1 u 2 |L between the pulses involved in the SFG process. In particular, the laser fields cease to interact with each other when the group delay ∆21 starts to exceed the pulse width of the faster laser pulse. In difference-frequency generation (DFG), two input fields with frequencies ω1 and ω2 generate a nonlinear signal at the frequency ω3 = ω1 − ω2 . This process is of considerable practical significance as it can give rise to intense coherent radiation in the infrared range. In the limiting case of ω1 ≈ ω2 , this type of nonlinear-optical interaction corresponds to optical rectification, which has been intensely used over the past two decades for the generation of terahertz radiation. If the field at the frequency ω1 is strong and remains undepleted in the process of nonlinear-optical interaction, A1 (z, t) = A10 (t), the set of coupled equations governing the amplitudes of the remaining two fields in the stationary regime is written as 1 ∂A2 ∂A2 + = iγ2 A1 A∗3 exp (i∆kz) , ∂z u 2 ∂t 1 ∂A3 ∂A3 + = iγ3 A1 A∗2 exp (−i∆kz) , ∂z u 3 ∂t

(4.41) (4.42)

where, 2πω22 (2) χ (ω2 ; ω1 , −ω3 ) , k2 c2 2πω23 (2) γ3 = χ (ω3 ; ω1 , −ω2 ) k3 c2 γ2 =

(4.43) (4.44)

are the nonlinear coefficientsm and ∆k = k1 − k2 − k3 is the wave-vector mismatch for the DFG process. With no signal at ω3 applied at the input of the nonlinear medium, A3 (0, t) = 0, the solution to (4.41) and (4.42) in the stationary regime is given by [4.12]

∆k A2 (z) = A2 (0) cosh (κz) + i sinh (κz) , 2κ (4.45)

A3 (z) = iA2 (0) sinh (κz) ,

(4.46)

Part A 4.3

In sum-frequency generation Fig. 4.1, two laser fields with frequencies ω1 and ω2 generate a nonlinear signal at the frequency ω3 = ω1 + ω2 in a quadratically nonlinear medium with a nonlinear susceptibility (2) χSFG = χ (2) (ω3 ; ω1 , ω2 ). In the first order of dispersion theory, the coupled equations for slowly varying envelopes of the laser fields A1 = A1 (z, t) and A2 = A2 (z, t) and the nonlinear signal A3 = A3 (z, t) are written as ∂A1 1 ∂A1 + = iγ1 A3 A∗2 exp (i∆kz) , (4.34) ∂z u 1 ∂t 1 ∂A2 ∂A2 + = iγ2 A3 A∗1 exp (i∆kz) , (4.35) ∂z u 2 ∂t 1 ∂A3 ∂A3 + = iγ3 A1 A2 exp (−i∆kz) , (4.36) ∂z u 3 ∂t where γ1 =

163

A2 (z, t) = A20 (t), and the solution of (4.36) yields

field approximation is given by z

4.3 Second-Order Nonlinear Processes

164

Part A

Basic Principles and Materials

where κ

= 4γ2 γ3∗ |A1 |2 − (∆k)2 .

(4.47)

Away from the phase-matching condition, the amplification of a weak signal is achieved only when the intensity of the pump field exceeds a threshold, I1 > Ith =

n 1 n 2 n 3 c3 (∆k)2 , (2) 2 32π 3 χDFG ω2 ω3

(4.48)

where we took (2)

χ (2) (ω2 ; ω1 , −ω3 ) ≈ χ (2) (ω3 ; ω1 , −ω2 ) = χDFG .

Part A 4.4

Above, this threshold, the growth in the intensity I2 of a weak input signal is governed by γ γ ∗ |A10 |2 sin2 (κz) + 1 . (4.49) I2 (z) = I2 (0) 2 3 2 κ This type of three-wave mixing is often referred to as optical parametric amplification. A weak input field, referred to as the signal field (the field with the amplitude A2 in our case), becomes amplified in this type of process through a nonlinear interaction with a powerful pump field (the undepleted field with the amplitude A1 in the case considered here). In such a scheme of optical parametric amplification, the third field (the field with the amplitude A3 ) is called the idler field. We now consider the regime of optical parametric amplification ω1 = ω2 + ω3 where the pump, signal and idler pulses are matched in their wave vectors and group velocities. Introducing the real amplitudes ρ j and phases ϕ j of the pump, signal, and idler fields, A j = ρ j exp iϕ j , where j = 1, 2, 3, assuming that γ2 = γ3 = γ in (4.35) and (4.36), A1 (z, t) = A10 (t) and A3 (0, t) = 0, we write the solution for the amplitude of the signal field as [4.18] A2 (η, z) = A20 (η) cosh [γρ10 (η) z] .

(4.50)

The idler field then builds up in accordance with A3 (η, z) = A∗20 (η) exp [iϕ10 (η)] sinh [γρ10 (η) z] . (4.51)

As can be seen from (4.50), optical parametric amplification preserves the phase of the signal pulse. This property of optical parametric amplification lies at the heart of the principle of optical parametric chirped-pulse amplification [4.19], allowing ultrashort laser pulses to be amplified to relativistic intensities. It also suggests a method of efficient frequency conversion of few-cycle field waveforms without changing the phase offset between their carrier frequency and temporal envelope, making few-cycle laser pulses a powerful tool for the investigation of ultrafast electron dynamics in atomic and molecular systems. In the nonstationary regime of optical parametric amplification, when the pump, signal, and idler fields propagate with different group velocities, useful and important qualitative insights into the phase relations between the pump, signal, and idler pulses can be gained from energy and momentum conservation, ω1 = ω2 + ω3 and k1 = k2 + k3 . These equalities dictate the following relations between the frequency deviations δω j in the pump, signal, and idler fields ( j = 1, 2, 3): δω1 = δω2 + δω3

(4.52)

δω1 /u 1 = δω2 /u 2 + δω3 /u 3 .

(4.53)

and

In view of (4.52) and (4.53), we find δω2 = q2 δω1

(4.54)

δω3 = q3 δω1 ,

(4.55)

and

−1 −1 −1 where q2 = (u −1 1 − u 3 )/(u 2 − u 3 ), q3 = 1 − q2 . In the case of a linearly chirped pump, ϕ1 (t) = α1 t 2 /2, the phases of the signal and idler pulses are given by ϕm (t) = αm t 2 /2, where αm = qm α1 , m = 2, 3. With qm 1, the chirp of the signal and idler pulses can thus considerably exceed the chirp of the pump field.

4.4 Third-Order Nonlinear Processes Optical nonlinearity of the third order is a universal property, found in any material regardless of its spatial symmetry. This nonlinearity is the lowestorder nonvanishing nonlinearity for a broad class of centrosymmetric materials, where all the even-order nonlinear susceptibilities are identically equal to zero

for symmetry reasons. Third-order nonlinear processes include a vast variety of four-wave mixing processes, which are extensively used for frequency conversion of laser radiation and as powerful methods of nonlinear spectroscopy. Frequency-degenerate, Kerr-effect-type phenomena constitute another important class of third-

Nonlinear Optics

order nonlinear processes. Such effects lie at the heart of optical compressors, mode-locked femtosecond lasers, and numerous photonic devices, where one laser pulse is used to switch, modulate, or gate another laser pulse. In this section, we provide a brief overview of the main third-order nonlinear-optical phenomena and discuss some of their practical applications.

4.4.1 Self-Phase Modulation The third-order nonlinearity gives rise to an intensitydependent additive to the refractive index: n = n 0 + n 2 I (t) ,

(4.56)

∂I ω n2 L . (4.58) c ∂t The resulting spectral broadening of the pulse can be estimated in the following way: ∆ω (t) =

I0 ω n2 L , (4.59) c τ where I0 is the peak intensity of the light pulse and τ is the pulse duration. The first-order dispersion-theory equation for the slowly varying envelope A(t, z) of a laser pulse propagating in a medium with a Kerr-type nonlinearity is written as [4.9] ∆ω =

∂A 1 ∂A + = iγ˜ |A|2 A , (4.60) ∂z u ∂t where u is the group velocity of the laser pulse and 3πω γ˜ = 2 χ (3) (ω; ω, −ω, ω) . 2n 0 c is the nonlinear coefficient.

(4.61)

165

In the retarded frame of reference, z = z and η = t − z/u, the solution to (4.60) is written as

A (η, z) = A0 (η) exp iγ˜ |A0 (η)|2 z , (4.62) where A0 (η) is the initial field envelope. Since the group-velocity dispersion was not included in (4.60), the shape of the pulse envelope remains unchanged as the pulse propagates through the nonlinear medium. The intensity-dependent change in the refractive index gives rise to a nonlinear phase shift ϕnl (η, z) = γSPM I0 (η) z ,

(4.63)

where γSPM = 2πn 2 /λ and I0 (η) is the initial intensity envelope. The deviation of the instantaneous frequency of the pulse is given by δω (η, z) = −

∂ϕnl (η, z) ∂I0 (η) = −γSPM z . (4.64) ∂t ∂η

A quadratic approximation of the pulse envelope, η2 (4.65) I0 (η) ≈ I0 (0) 1 − 2 , τ0 where τ0 is the pulse width, which is valid around the maximum of the laser pulse, gives a linear chirp of the Spectral intensity (arb. units) 10

0.1

0.01 1.00

1.01

1.02

1.03

1.04

1.05 1.06 Wavelength (µm)

Fig. 4.5 Self-phase-modulation-induced spectral broadening of a laser pulse with a central wavelength of 1.03 µm (an input spectrum is shown by curve 1) in a fused-silica optical fiber with n 2 = 3.2 × 10−16 cm2 /W: γSPM I0 (0)z = 1.25 (curve 2), 2.50 (curve 3), and 6.25 (curve 4)

Part A 4.4

where n 0 is the refractive index of the medium in the absence of light field, n 2 = (2π/n 0 )2 χ (3) (ω; ω, ω, −ω) is the nonlinear refractive index, χ (3) (ω; ω, ω, −ω) is the third-order nonlinear-optical susceptibility, referred to as the Kerr-type nonlinear susceptibility, and I(t) is the intensity of laser radiation. Then, the nonlinear (intensity-dependent) phase shift of a pulse at a distance L is given by ω Φ (t) = n 2 I (t) L . (4.57) c Due to the time dependence of the radiation intensity within the light pulse, the nonlinear phase shift is also time-dependent, giving rise to a generally timedependent frequency deviation:

4.4 Third-Order Nonlinear Processes

166

Part A

Basic Principles and Materials

pulse δω (η, z) ≈ −2γSPM

I0 (0) ηz . τ02

(4.66)

The spectrum of a self-phase-modulated pulse is given by ∞ 2 S (ω) = I (η) exp [iωη + iϕnl (η)] dη . (4.67) 0

Part A 4.4

Figure 4.5 illustrates SPM-induced spectral broadening of a short laser pulse with a central wavelength of 1.03 µm, typical of ytterbium fiber lasers, in a fusedsilica optical fiber with nonlinear refractive index n 2 = 3.2 × 10−16 cm2 /W. Thus, self-phase modulation results in spectral broadening of a light pulse propagating through a hollow fiber. This effect allows compression of light pulses through the compensation of the phase shift acquired by the pulse in a hollow fiber. Compensation of a linear chirp, corresponding to a linear time dependence of the instantaneous frequency, is straightforward from the technical point of view. Such a chirp arises around the maximum of a light pulse, where the temporal pulse envelope can be approximated with a quadratic function of time [see (4.65, 4.66)]. It is physically instructive to consider the compression of chirped light pulses in the time domain. Since the frequency of a chirped pulse changes from its leading edge to its trailing edge, dispersion of our compressor should be designed in such a way as to slower down the leading edge of the pulse with respect to the trailing edge of the pulse. In other words, the group velocities for the frequencies propagating with the leading edge of the pulse should be lower than the group velocities for the frequencies propagating with the trailing edge of the pulse. This can be achieved by designing a dispersive element with the required sign of dispersion and appropriate dispersion relation. Systems of diffraction gratings and, recently, multilayer chirped mirrors [4.20] are now widely used for the purposes of pulse compression. In certain regimes of pulse propagation, self-phase modulation and pulse compression may take place in the same medium.

4.4.2 Temporal Solitons The nonlinear phase shift acquired by a laser pulse propagating through a medium with a Kerr nonlinearity can be balanced by group-velocity dispersion, giving rise

to pulses propagating through the nonlinear dispersive medium with an invariant or periodically varying shape: optical solitons. Optical solitons is a special class of solutions to the nonlinear Schrödinger equation (NLSE) i

∂q 1 ∂ 2 q + + |q|2 q = 0 . ∂ξ 2 ∂τ 2

(4.68)

The NLSE can describe the evolution of optical wave packets including the dispersion β(ω) of optical waves in a bulk material or in a waveguide structure through the power series expansion β (ω) ≈ β (ω0 ) +

1 (ω − ω0 ) u

1 + β2 (ω − ω0 )2 + . . . , (4.69) 2 where ω0 is the central frequency of the wave packet, u = (∂β/∂ω|ω=ω0 )−1 is the group velocity, and β2 = ∂ 2 β/∂ω2 |ω=ω0 . Thus, with the NLSE (4.68) projected on laser pulses propagating in a nonlinear medium, q is understood as the normalized pulse envelope, q = A/(P0 )1/2 , with ξ being the normalized propagation coordinate, ξ = z/L d , L d = τ02 /|β2 | being the dispersion length, P0 and τ0 defined as the pulse width and the pulse peak power, respectively, and τ = (t − z/u)/τ0 . The canonical form of the fundamental soliton solution to (4.68) is [20]

ξ . (4.70) q (ξ, τ) = sech (τ) exp i 2 The radiation peak power required to support such a soliton is given by P0 = |β2 | /(γτ02 ) .

(4.71)

Solitons retain their stable shape as long as their spectrum lies away from the spectrum of dispersive waves that can propagate in the medium. High-order dispersion perturbs solitons, inducing Cherenkov-type wave-matching resonances between solitons and dispersive waves [4.21, 22]. Under these conditions solitons tend to lose a part of their energy in the form of blueshifted dispersive-wave emission. For low pump-field powers, the generic wave-matching condition for such soliton–dispersive wave resonances is written [4.22] Ω = 1/2ε, where Ω is the frequency difference between the soliton and the resonant dispersive wave and ε is the parameter controlling the smallness of perturbation of the nonlinear Schrödinger equation, which can be represented as ε = |β3 /6β2 | for photonic-crystal fibers

Nonlinear Optics

(PCFs) with second-order dispersion β2 = ∂ 2 β/∂ω2 and third-order dispersion β3 = ∂ 3 β/∂ω3 . This dispersivewave emission of solitons is an important part of supercontinuum generation in nonlinear optical fibers, including photonic-crystal fibers.

4.4.3 Cross-Phase Modulation

where χ (3) (ωs ; ωs , ωp , −ωp ) is the third-order nonlinearoptical susceptibility of the medium; 1/σ = 11/u 1 − 1/u 2 ; u 1 and u 2 are the group velocities of the pump and probe pulses, respectively; and k2 is the wave number of the pump pulse. Taking the time derivative of the nonlinear phase shift, we arrive at the following expression for the frequency deviation of the probe pulse 3πω2 δωXPM (η, z) = − 2 2 χ (3) ωs ; ωs , ωp , −ωp c k2 2 z 2 (4.73) × σ Ap (η, 0) − Ap η − , 0 . σ Similarly to self-phase modulation, cross-phase modulation can be employed for pulse compression. The dependence of the chirp of the probe pulse on the pump pulse intensity can be used to control the parameters of ultrashort pulses [4.24]. Cross-phase modulation also opens the ways to study the dynamics of ultrafast nonlinear processes, including multiphoton ionization, and to characterize ultrashort light pulses through phase measurements on a short probe pulse [4.25].

4.4.4 Self-Focusing Self-focusing is a spatial counterpart of self-phase modulation. While SPM originates from the time-dependent

167

change in the refractive index induced by a laser pulse with an intensity envelope I(t) varying in time, selffocusing is related to a nonlinear lens induced by a laser beam with a spatially nonuniform intensity distribution I(r). Given a transverse intensity profile I(r), the nonlinear additive to the refractive index is written as n (r) = n 0 + n 2 I (r) .

(4.74)

If the field intensity peaks at the center of the beam at r = 0, the nonlinear change in the refractive index also reaches its maximum at r = 0, yielding a focusing or defocusing lens, depending on the sign of n 2 . The stationary regime of self-focusing is governed by [4.9] ∂A ∆n 2ik + ∆⊥ A = −2k2 A, (4.75) ∂z n0 where ∆n = n 2 I = n˜ 2 |E|2 , ∆⊥ is the transverse part of the Laplacian. We consider a Gaussian beam and assume that this beam retains its profile as it propagates through the nonlinear medium, A0 r2 A (r, z) = exp − 2 + iψ (z) , f (z) 2a0 f 2 (z) (4.76)

where a0 is the initial beam size, f (z) characterizes the evolution of the beam size along the propagation coordinate z [ f (0) = 1], and the function ψ(z) describes the spatial phase modulation of the field. In the paraxial approximation, r a0 f (z), (4.75) and (4.76) give [4.18] −2 L −2 df − L nl d2 f = , (4.77) dz 2 f 3 (z) where L df = 2πa02 /λ and L nl = a0 [2n 0 /(n 2 |A|2 )]1/2 are the characteristic diffraction and nonlinear lengths, respectively. Solving (4.77), we arrive at

P0 z 2 f 2 (z) = 1 + 1− , (4.78) L df Pcr where P0 is the total power of the laser beam and cλ2 (4.79) 16π 2 n 2 is the critical power of self-focusing. The focal length of the nonlinear lens is given by L df L sf = (4.80) 1/2 . P0 − 1 Pcr Pcr =

Part A 4.4

Cross-phase modulation (XPM) is a result of nonlinearoptical interaction of at least two physically distinguishable light pulses (i. e., pulses with different frequencies, polarizations, mode structures, etc.) related to the phase modulation of one of the pulses (a probe pulse) due to the change in the refractive index of the medium induced by another pulse (a pump pulse). The cross-action of a pump pulse with a frequency ω1 on a probe pulse with a frequency ω2 gives rise to a phase shift of the probe pulse, which can be written as [4.23]. 3πω2 ΦXPM (η, z) = 2 2 χ (3) ωs ; ωs , ωp , −ωp c k2

2 z ζ (4.72) × Ap η − , 0 dζ , σ

4.4 Third-Order Nonlinear Processes

168

Part A

Basic Principles and Materials

Part A 4.4

With P0 > Pcr , the nonlinear lens leads to a beam collapse. In reality, beam collapse can be arrested by the saturation of optical nonlinearity occurring at high field intensities. Beyond the paraxial approximation, the scenario of self-focusing is much more complicated. The beam does not collapse as a whole, as the focal length of the nonlinear lens for peripheral beams differs from the one for paraxial beams. In the quasistationary regime, i. e., when the pulse duration τ0 is much larger than the characteristic response time of optical nonlinearity τnl , the length of self-focusing is a function of time, giving rise to moving foci [4.26]. In the nonstationary regime, i. e., on time scales less than τnl , the leading edge of the pulse experiences no focusing, but induces a nonlinear lens that focuses the trailing edge of the pulse. As a result, the beam becomes distorted, evolving to a hornlike pattern [4.27]. The equation of self-focusing (4.75) allows a waveguide solution [4.28], which corresponds to the regime where the nonlinear lens exactly compensates for the diffraction of the laser beam. This solution is, however, unstable with respect to infinitely small fluctuations, which either give rise to a diffraction divergence or lead to a beam collapse. Such nonlinear waveguides can be stabilized, as shown by Fibich and Gaeta [4.29], by reflections of light from guiding boundaries in optical waveguides. To illustrate this regime of nonlinear beam dynamics, we consider a cylindrical gas-filled hollow waveguide and define nondimensional cylindrical coordinates r and z as r = R/R0 (R is the dimensional radial coordinate and R0 is the inner radius of the hollow fiber) and z = Z/L df (Z is the dimensional longitudinal coordinate). The radial profiles Q β (r) of light intensity distribution in the waveguide solutions ψ ∝ exp(iβz)Q β (r) to the NLSE governing self-focusing (i. e., waveguides induced through the Kerr nonlinearity along the z-axis on a bounded domain with a circular symmetry) are described [28, 29] by solutions to the ordinary differential equation ∆⊥ Q β − β Q β + Q 3β = 0 [∆⊥ = ∂ 2 /∂r 2 + (1/r)∂/∂r], subject to the boundary conditions dQ β (0)/ dr = 0, Q β (1) = 0. Although this model neglects the fields outside the fiber core (e.g., radiation modes), it provides useful physical insights into the spatial self-action in hollow PCFs. This differential equation has an infinite number of (n) solutions Q β , n = 0, 1, 2, . . .. The Hamiltonians for (n) all the guided modes Q β are positive, preventing the (n) blowup of the profiles Q β in the presence of small fluctuations, thus stabilizing the nonlinear waveguides (0) in a hollow fiber. The ground-state solution Q β is

a monotonically decreasing function of r, tending to a zeroth-order Bessel function in the case of low field (0) amplitudes Q β ∝ εJ0 (2.4r), where ε is a small parameter controlled by the field intensity and the nonlinearity of the gas filling the fiber. (n) Although the modes Q β are stable with respect to small perturbations, these solutions are centers, rather than attractors, in a conservative system [4.30]. However, mode solutions corresponding to Kerrnonlinearity-induced waveguides may become attractors in systems with dissipation, e.g., in hollow waveguides a)

ω 1, k 1

χ (3) FWM

ω 2, k 2

ω 4, k4

ω 3, k 3

b) χ (3) THG

ω, k

3ω, k TH

c) ω3

ω 1, k 1

χ (3) CARS

ω2 ω1

ω4

ω CARS , kCARS

ω 3, k 3 Ω ω 2, k 2

d) χ (3) DFWM

ω, k 1

ω, k 2

ω, k 3 ω, k 4

Fig. 4.6a–d Four-wave mixing: (a) general-type FWM, (b) third-harmonic generation, (c) coherent anti-Stokes Raman scattering, and (d) degenerate four-wave mixing

Nonlinear Optics

with losses. The circularly symmetric field distribution is then formed at the output of the hollow waveguide regardless of the initial beam profile [30]. As demonstrated by Moll et al. [4.31], collapsing light beams tend to form universal, Townesian profiles [4.28] while undergoing self-focusing on an infinite domain in bulk materials. Unlike Townesian beam profiles, which are known to be unstable in free space, the ground-state waveguide modes observed in hollow photonic-crystal fibers [4.32] remain stable with respect to small fluctuations, in agreement with the theory of self-focusing on bounded domains [4.29, 30], resulting in no blowup until the critical power of beam collapse is reached.

4.4 Third-Order Nonlinear Processes

169

The third-order nonlinear polarization responsible for the considered FWM process is (3)

PFWM = χ (3) (ωFWM ; ω1 , ω2 , ω3 ) .. .E (ω1 ) E (ω2 ) E (ω3 ) .

(4.83)

With no depletion of the pump fields, the SVEA equations give the following expression for the envelope of the i-th Cartesian component of the FWM field [4.9]: 2πω2FWM (3) χ (ωFWM ; ω1 , ω2 , ω3 ) kFWM c2

i∆kz × A1 j A2k A3l exp 2 ∆kz sin × exp (−αFWM z) ∆kz2 z ,

[AFWM (z)]i = −

4.4.5 Four-Wave Mixing

E j = A j exp i k j r − ω j t + c.c. ,

(4.81)

where j = 1, 2, 3 and k j = k j + iα j are the complex wave vectors of the pump fields. The FWM signal is written as EFWM = AFWM exp [i (kFWM r − ωFWM t)] + c.c. (4.82)

where kFWM = k FWM + iαFWM is the complex wave vector of the FWM fields.

(4.84)

where ∆k = k1 + k2 + k3 − kFWM + i (α1 + α2 + α3 − αFWM )

(4.85)

is the z-component of the wave-vector mismatch. Phase matching, as can be seen from (4.84), is the key requirement for high efficiency of frequency conversion in FWM. The fields involved in FWM can be phase matched by choosing appropriate angles between the wave vectors of the laser fields and the wave vector of the nonlinear signal or using the waveguide regime with the phase mismatch related to the material dispersion compensated by the waveguide dispersion component.

4.4.6 Optical Phase Conjugation Optical phase conjugation is generally understood as the generation of an optical field with a time-reversed wave front, or with a conjugate phase. This effect can be used to correct aberrations in certain types of optical problems and systems [4.33]. Suppose that a light beam with an initially plane wave front propagates through an aberrating medium, such as, for example a turbulent atmosphere or a material with inhomogeneities of the refractive index. The wave front of the light beam transmitted through such a medium is distorted. We now use an optical phase-conjugate process to generate the field with a wave front time-reversed with respect to the wave front of the beam transmitted through the aberrating system. As the phase-conjugate beam now propagates through the aberrating medium in the backward direction, its wave front becomes restored.

Part A 4.4

In general-type four-wave mixing (Fig. 4.6a), three laser fields with frequencies ω1 , ω2 , and ω3 generate the fourth field with a frequency ωFWM = ω1 ± ω2 ± ω3 . In the case when all three laser fields have the same frequency (for example, when all the three pump photons are taken from the same laser field), ωFWM = 3ω (Fig. 4.6b), and we deal with third-harmonic generation (THG), which is considered in greater detail for short-pulse interactions in Sect. 4.4.9. If the frequency difference ω1 − ω2 of two of the laser fields is tuned to a resonance with a Raman-active mode of the nonlinear medium (Fig. 4.6c), the FWM process ωFWM = ω1 − ω2 + ω3 = ωCARS is refereed to as coherent anti-Stokes Raman scattering (CARS). An FWM process involving four fields of the same frequency (Fig. 4.6d) with ωDFWM = ω = ω − ω + ω corresponds to degenerate four-wave mixing (DFWM). The FWM field in this nonlinear-optical process is the phaseconjugate of one of the laser fields, giving rise to another name of this type of FWM – optical phase conjugation. For the general-type FWM ωFWM = ω1 + ω2 + ω3 , we represent the pump fields as

170

Part A

Basic Principles and Materials

A phase-conjugate field can be produced through the degenerate four-wave mixing of light fields E j (r, t) = A j (r, t) exp i k j r − ωt + c.c. , (4.86)

Part A 4.4

where j = 1, 2, 3, 4. The phase-conjugate geometry of DFWM is shown in Fig. 6d. In this scheme, two strong counterpropagating pump fields E1 and E2 with the same frequency ω and wave vectors k1 and k2 = −k1 illuminate a medium (3) with a cubic nonlinearity χDFWM = χ (3) (ω; ω, −ω, ω). The DFWM interaction of these two pump fields with a weak signal of the same frequency ω and an arbitrary wave vector k3 gives rise to a field with a frequency ω that propagates in the opposite direction to the signal beam and that is phase-conjugate of the signal field. The phase-conjugate field generated through DFWM can be instructively thought of as a result of scattering of the forward pump field off the grating induced by the backward pump and the signal field or as a result of scattering of the backward pump from the grating induced by the forward pump and the signal field. The nonlinear polarization responsible for phase conjugation in DFWM is . (3) (3) PDFWM = 6χDFWM ..E1 E2 E3∗ . (4.87) If the depletion of the pump fields is negligible, the nonlinear propagation equation (4.24) for the considered DFWM process is reduced to the following two equations for the amplitude of the signal field and its phase-conjugate [4.1] dA3 = iκ3 A3 + iκ4 A∗4 , (4.88) dz dA4 = −iκ3 A4 − iκ4 A∗3 , (4.89) dz where 12πω (3) χDFWM |A1 |2 + |A2 |2 , κ3 = (4.90) cn 12πω (3) χ κ4 = A1 A2 . (4.91) cn DFWM By introducing A3 = B3 exp (iκ3 z) , A4 = B4 exp (−iκ3 z) ,

(4.92) (4.93)

we reduce the set of equations (4.88) and (4.89) to dB3 = iκ4 B4∗ , (4.94) dz dB4 = −iκ4 B3∗ . (4.95) dz

The solution can now be written as i |κ4 | sin (|κ4 | z) B4 (L) B3∗ (z) = − κ4 cos (|κ4 | L) cos [|κ4 | (z − L)] ∗ B3 (0) , + cos (|κ4 | L) cos (|κ4 | z) B4 (L) B4 (z) = cos (|κ4 | L) iκ4 sin [|κ4 | (z − L)] ∗ B3 (0) , − |κ4 | cos (|κ4 | L)

(4.96)

(4.97)

where B3 (0) and B4 (L) are the boundary conditions for the signal and DFWM fields. With B4 (L) = 0, (4.97) gives iκ4 B4 (0) = (4.98) tan (|κ4 | L) B3∗ (0) . |κ4 | This expression visualizes the structure of the phaseconjugate field generated through DFWM. The intensity reflection coefficient of the DFWM medium serving as a phase-conjugate mirror is given by RDFWM = tan2 (|κ4 | L) .

(4.99)

As can be seen from (4.99), the reflectivity of the DFWM-based phase-conjugate mirror can exceed 100%. This becomes possible due to the energy supplied by the strong pump fields.

4.4.7 Optical Bistability and Switching In optical bistability or multistability, an optical system has two or more than two stable states, most often represented by the intensity at the output of the system as a function of the input intensity. With the level of the output intensity determined by a certain operation on a light beam, a bi- or multistable system makes a decision in which state it will operate, acting as a switch for optical communications or optical data processing. As an example of an optical bistable system, we consider a Fabry–Perot cavity with a Kerr-nonlinear medium inside. We assume that the cavity mirrors are identical and have an amplitude reflectance r and transmittance t. Let A1 , A2 and A3 be the amplitudes of the incident, reflected, and transmitted fields, respectively. The amplitudes of forward and backward waves inside the cavity will be denoted as B1 and B2 , respectively. The amplitudes of the fields inside and outside the cavity are related by the expressions [4.1] B2 = rB1 exp (2ikL − αL) ,

(4.100)

B1 = tA1 + rB2 ,

(4.101)

Nonlinear Optics

where k is the wave number and α is the intensity absorption coefficient. Solving (4.100) and (4.101), we arrive at the relation tA1 B1 = , 1 − r 2 exp (2ikL − αL)

B1 =

tA1 , 1 − R exp (iδ)

I2 / I1

(4.102)

which is known as the Airy equation. When k or α is a strongly nonlinear function of the light intensity, (4.102) leads to a bistable behavior of the system in the transmitted field. Let us assume that the absorption is negligible and rewrite (4.102) as

3 1 2 1

(4.104)

where ω L c is the linear phase shift and ω δnl = 2n 2 I L c δ0 = ϕ = 2n 0

(4.105)

(4.106)

is the nonlinear phase shift, with I = [cn/(2π)](|B1 |2 + |B2 |2 ) ≈ [cn/(π)]|B1 |2 The ratio of the field intensity inside the cavity I2 = [cn/(2π)]|B1 |2 , to the intensity of the incident field I1 = [cn/(2π)]|A1 |2 is now given by I2 = F (I2 ) , I1

(4.107)

0 0.0

0.2

0 0.6 I2 / I0

0.4

Fig. 4.7 The function F(I2 ) and the ratio I2 /I1 , appearing on the right- and left-hand sides of (4.107), plotted as functions of the intensity I2 normalized to I0 = λ/(n 2 L) for different input field intensities I1 : (1) I1 = I , (2) I1 = I , and (3) I1 = I , I > I > I . The circles in the Airyfunction curve show the range of input intensities where the operation of the Fabry–Perot cavity is unstable

of the nonlinear Fabry–Perot cavity. The circles in the Airy-function curve show the range of input intensities where the operation is unstable. As the input intensity I1 is increased, the system displays a bistable behavior and hysteresis. More hysteresis loops and a multistable behavior are observed, as can be seen from Fig. 4.7, with a further increase in I1 . Control

where

n = n0 + n2l 2 T + 4R T sin

δ ,

(4.108)

Signal

BS1

where T is the intensity transmittance of the cavity mirrors, and 4n 2 ωL I2 . (4.109) δ = δ0 + c In Fig. 4.7, we present the function F(I2 ) and the ratio I2 /I1 , i. e., the left- and right-hand sides of (4.107), plotted as functions of the intensity I2 for different input field intensities I1 . The points where the straight lines representing the left-hand side of (4.107) cross the plot of the Airy-type function F(I2 ) define the operation points

M2 Φnl

BS2 Output 1

n = n0 M1

Φnl = 0 Output 2

Fig. 4.8 Mach–Zehnder interferometer with a nonlinear

medium in one of its arms, functioning as an all-optical switching device: BS1, BS2, beam splitters; M1, M2, mirrors

Part A 4.4

δ = δ0 + δnl ,

171

(4.103)

where R = ρ2 exp(−iϕ) is the intensity reflectance of the cavity mirrors and the phase shift δ, corresponding to a full round trip around the cavity, is given by

F (I2 ) =

4.4 Third-Order Nonlinear Processes

172

Part A

Basic Principles and Materials

Optical switching can now be implemented [4.1] by including a nonlinear medium into one of the arms of a Mach–Zehnder interferometer Fig. 4.8. The intensities at the output ports 1 and 2 are determined by the interference of light fields transmitted through the arms of the interferometer. If there is only one input beam applied to the system with symmetric beam splitters BS1 and BS2, output intensities display an oscillating behavior as functions of the nonlinear phase shift Φnl acquired by the light field in one of the arms of the interferometer. When Φnl = 0, the intensity at output port 1 is at its maximum, while the intensity at output port 2 is at its minimum. An opposite relation between the output intensities is achieved with Φnl = π. The requirement Φnl = π is typical of a broad class of all-optical switching devices.

4.4.8 Stimulated Raman Scattering Part A 4.4

Vibrations or rotations of molecules, electronic motions in atoms or collective excitations of matter can interact with light, shifting its frequency through an inelastic scattering process by the frequency Ω of Raman-active motions (as shown in Fig. 4.6c). This phenomenon was discovered by Raman and Krishnan [4.34] and almost simultaneously by Mandelstam and Landsberg [4.35] in 1928. In an intense laser field, pump laser photons and frequency-shifted photons act coherently to resonantly drive molecular motions, leading to the amplification of the Raman-shifted signal. This effect is called stimulated Raman scattering (SRS). In the SRS regime, Raman-active modes of a material function as optical modulators, forcing the driving laser field to oscillate at new frequencies. An intense laser field under these conditions not only gives rise to photons at new frequencies through interaction with Raman-active modes, but also amplifies the light consisting of those photons. In the continuous-wave regime, the interaction between the pump field and the frequency-shifted (Stokes) signal is governed by the following set of coupled equations for the intensities of the pump Ip and Stokes Is fields [4.23]: dIs = gR Ip Is − αs Is , dz ωp dIp = − gR Ip Is − αp Ip . dz ωs

(4.110) (4.111)

Here, αp and αs are the losses at the frequencies of the pump and Stokes fields, ωp and ωs are the frequencies of the pump and Stokes fields, and gR is the Raman gain, which is related to the imaginary part of the relevant

third-order susceptibility [4.9],

gR ∝ Im χ (3) ωs ; ωp , −ωp , ωs .

(4.112)

Neglecting the action of the Stokes field on the pump, i. e., omitting the first term on the right-hand side of (4.111), we arrive at the following solution for the intensity of the Stokes field at the output of a Raman-active medium with a length L: Is (L) = Is (0) exp (gR I0 L eff − αs L) ,

(4.113)

where I0 is the incident pump intensity at z = 0 and L eff =

1 1 − exp −αp L . αp

(4.114)

Thus, pump absorption restricts the nonlinear interaction length to L eff . In the absence of the Stokes field at the input of a Raman-active medium, Is (0) = 0, the Stokes field builds up from spontaneous Raman scattering inside the medium. The power of the Stokes signal at the output of a medium with a length L is then given by Ps (L) = P¯0 exp [gR (ωs ) I0 L eff − αs L] ,

(4.115)

where P¯0 = ωs Beff Beff = (2π)1/2 [|g2 (ωs )|I0 L eff ]−1/2 , and g2 (ωs ) = (∂ 2 gR /∂ω2 )|ω=ωs . With an assumption of a Lorentzian Raman gain band, the critical pump power corresponding to the threshold of SRS effect is given by an approximate formula Pcr ≈

16Seff , gR L eff

(4.116)

where Seff is the effective mode area of the pump field. In Sect. 4.6.3, we will consider in greater detail the influence of SRS on the propagation of ultrashort laser pulses in a nonlinear medium. It will be shown, in particular, that the Raman effect in fibers gives rise to a continuous frequency downshift of solitons propagating in optical fibers.

Nonlinear Optics

4.4.9 Third-Harmonic Generation by Ultrashort Laser Pulses

of standard fibers is replaced by a microstructure with an array of air holes running along the fiber parallel to its core Fig. 4.9 – have opened a new phase in nonlinear optics [4.38, 39]. Controlled dispersion of guided modes [4.40] and large interaction lengths provided by these fibers for light fields strongly confined in a small fiber core [4.41] result in a radical enhancement of nonlinear-optical frequency conversion and spectral transformation of laser radiation through selfand cross-phase modulation (SPM and XPM) [4.38], supercontinuum generation [4.42–44], four-wave mixing (FWM) [4.45], third-harmonic generation [4.42–52], modulation instabilities [4.53], and soliton frequency shifting [4.54]. Third-order nonlinear-optical processes enhanced in PCFs now offer a useful alternative to frequency-conversion schemes using χ (2) nonlinear crystals. Highly efficient THG has recently been observed in fused silica [4.46–50] and multicomponent glass PCFs [4.51], as well as in tapered fibers [4.52]. These b)

d) c)

Fig. 4.9a–d SEM images of photonic-crystal fibers: (a) periodic- and (b) double-cladding fused-silica PCFs, (c) highindex-step birefringent PCF, and (d) hollow-core PCF

173

Part A 4.4

Third-harmonic generation (THG) is one of the basic nonlinear-optical processes, which has been intensely studied and employed for numerous applications since the early days of nonlinear optics [4.7–9]. The seminal work by Miles and Harris [4.17] has demonstrated a tremendous potential of direct THG related to the cubic optical nonlinearity χ (3) of gases for efficient frequency conversion of laser radiation and for the diagnostics of the gas phase. Solid-state strategies of frequency conversion, on the other hand, mainly rely on the quadratic nonlinearity χ (2) of non-centrosymmetric crystals, with frequency tripling conventionally implemented through cascaded second-order nonlinear-optical processes, phase matched by crystal anisotropy [4.13, 14] or periodic poling of nonlinear materials [4.16]. Photonic-crystal fibers (PCFs) [4.36, 37] – optical fibers of a new type where the solid continuous cladding

4.4 Third-Order Nonlinear Processes

174

Part A

Basic Principles and Materials

Part A 4.4

experiments not only demonstrated the significance of THG for efficient, guided-wave frequency tripling of femtosecond laser pulses, but also revealed several new interesting nonlinear-optical phenomena. The thirdharmonic signal has been shown to display asymmetric spectral broadening [4.51, 52] or even a substantial frequency shift. We will demonstrate here that such a behavior is a universal intrinsic feature of multimode guided-wave THG. Based on the arguments of the slowly varying envelope approximation (SVEA), we will show that the sign and the absolute value of the third-harmonic frequency shift, observed in many PCF experiments, is controlled by the phase- and groupindex mismatch for the interacting pair of pump and third-harmonic modes. The possibility to tune the frequency of the main spectral peak in the spectrum of the third harmonic by varying the group-velocity mismatch is a unique property of THG-type processes, which is not typical of standard parametric FWM processes, where the first-order dispersion terms cancel out of the balance of the field momenta. New regimes of THG will be identified with no signal produced at the central frequency of the third harmonic 3ω0 and with the pump energy efficiently converted to spectrally isolated narrowband frequency components, which can be tuned within a spectral range of several tens of terahertz from the 3ω0 frequency. We start with qualitative arguments illustrating phase matching for third-harmonic generation generalized to include the phase and group-velocity mismatch of the pump and third-harmonic fields, as well as the Kerr effect, broadening the pump spectrum through SPM. We represent the wave numbers (or the propagation constants of guided modes in a waveguide regime) kp and kh at the frequencies of the pump field and the third harmonic as kp (ω) ≈ k (ω0 ) + vp−1 Ω/3 + κSPM P , kh (3ω) ≈ k (3ω0 ) + vh−1 Ω + 2κXPM P

(4.117) (4.118)

where ω0 is the central frequency of the pump field; vp,h = (∂k/∂ω)−1 ω0 ,3ω0 are the group velocities of the pump and its third harmonic; Ω = 3ω − 3ω0 ; κSPM = ω0 n 2 /cSeff and κXPM = 3ω0 n¯ 2 /cSeff are the SPM and XPM nonlinear coefficients (Seff is the effective beam, or mode, area and n 2 and n¯ 2 are the nonlinear refractive indices at ω0 and 3ω0 , respectively); and P is the power of the pump field. In writing (4.117) and (4.118), we neglect group-velocity dispersion and higher-order dispersion effects, as well as the SPM of the third-harmonic field. With n 2 ≈ n¯ 2 , the phase mismatch

is then given by ∆k = kh − 3kp ≈ ∆k0 + ξΩ + 3κSPM P ,

(4.119)

where ∆k0 = k(3ω0 ) − 3k(ω0 ) is the phase mismatch of the pump and third-harmonic wave numbers at the central frequencies of these fields and ξ = vh−1 − vp−1 is the group-velocity mismatch. As can be seen from (4.119), the group delay of the pump and third-harmonic pulses is an important factor in THG momentum conservation. In this respect, the balance of momenta for THG radically differs from standard phase-matching conditions for parametric FWM processes [4.23], where the first-order dispersion order terms cancel out, reducing the FWM momentum balance to group-velocity dispersion (GVD)-related issues. The phase-matching condition (4.119) suggests the possibility of substantially frequency shifting the maximum in the spectrum of the third harmonic. However, the amplitude of an Ω-shifted spectral component in the spectrum of the third harmonic and, hence, the efficiency of Ω-shifted peak generation is determined by the spectrum of the pump field. To specify this dependence, we proceed with an SVEA analysis of THG in the field of SPM-broadened pump field by writing SVEA coupled equations for the envelopes of the pump and third-harmonic fields, A(t, z) and B(t, z):

∂ 1 ∂ + A = iγ1 A |A|2 , (4.120) ∂t νp ∂z

1 ∂ ∂ + B = iβ (A)3 exp (−i∆k0 z) ∂t νh ∂z + 2iγ2 B |A|2 , (4.121) where vp and vh are the group velocities of the pump and third-harmonic pulses, respectively, and γ1 , γ2 and β are the nonlinear coefficients responsible for SPM, XPM, and THG, respectively; and ∆k = kh − 3kp is the phase mismatch (or the difference of propagation constants in the guided-wave regime) in the absence of the nonlinear phase shifts of the pump and third-harmonic fields. Solution of (4.120) and (4.121) yields [4.24, 55] A tp , z = A0 tp exp iϕSPM tp , z , (4.122) z B (th , z) = iσ dz A30 th + ξz 0 × exp −i∆β0 z + 3iϕSPM th + ξz , z (4.123) + iϕXPM th , z , z ) ,

where tl = (t − z/vl ) with l = p, h for the pump and the field, respectively; A0 (t) is the initial-condition envelope

Nonlinear Optics

of the pump pulse; 2 ϕSPM tp , z = γ1 A0 tp z

ϕXPM th , z , z = 2γ2

175

Third-harmonic intensity (arb. units)

(4.124)

0.4

is the SPM-induced phase shift of the pump field; and z

4.4 Third-Order Nonlinear Processes

A0 th + ξz 2 dz

Third-harmonic intensity (arb. units)

0.2

1.0

0.0 –1

1 0

2 Ωτ

(4.125)

−∞ −∞

2 × A Ω dΩ dΩ ,

(4.126)

where A(Ω) is the spectrum of the input pump field. Analysis of the regime of small nonlinear phase shifts is thus very instructive as it allows phase-matching effects to be decoupled from the influence of the spectrum of the pump field. While the phase matching is represented by the argument in the exponential in the first factor on the right-hand side of (4.126), the significance of the pump spectrum is clear from the convolution integral appearing in this expression. Depending on the signs of the phase and group-velocity mismatch, ∆k0 and ξ, the peak in the spectrum of the third harmonic can be either red- or blue-shifted with respect to the frequency 3ω0 . The spectral width of this peak, as can be seen from (4.126), is given by δ ≈ 2π(|ξ|z)−1 , decreasing as z −1 with the growth in the propagation coordinate z (see inset in Fig. 4.10). With low pump powers, the generalized phasematching condition (4.119), as can be seen also from the exponential factor in (4.126), defines the central frequency 3ω0 + Ωmax , Ωmax = −∆k0 /ξ, of the peak in the spectrum of the third harmonic. The amplitude of this peak, as shown by (4.126), is determined by the amplitudes of the pump field components with frequencies ω1 = ω0 + Ωmax − Ω , ω2 = ω0 + Ω − Ω and ω3 = ω0 + Ω , which can add up to transfer the energy to the 3ω0 + Ωmax component in the spectrum of the

3 0.5 1

2 0.0

–4

–2

Ωτ

Fig. 4.10 Spectra of the third harmonic generated in the

regime of weak self-phase modulation of the pump field with ∆k0 τ/ξ = −2 (curve 1), −3 (2) and 0 (3). The inset illustrates narrowing of the main peak in the spectrum of the third harmonic with ∆k0 τ/ξ = −1 and ξ L/τ = 2 (1), 4 (2), and 10 (3)

third harmonic through the ω1 + ω2 + ω3 = 3ω0 + Ωmax process. The spectrum of the pump field should therefore be broad enough to provide a high amplitude of this peak. In the regime when the SPM-induced broadening of the pump spectrum is small, the tunability range of the third harmonic (i. e., the range of frequency offsets Ω) is mainly limited by the bandwidth of the input pump field. This regime of THG is illustrated in Fig. 4.10. As the ratio ∆k0 τ/ξ changes from −2 (curve 1) to −3 (curve 2), the peak in the spectrum of the third harmonic is bound to shift from Ωmax = 2/τ to 3/τ. A similar spectral shift limited by a field-unperturbed pump spectrum has been earlier predicted by Akhmanov et al. for second-harmonic generation [4.56]. For large |∆k0 /ξ|, however, the pump power density at the wings of the spectrum is too low to produce a noticeable peak with Ωmax = −∆k0 /ξ in the spectrum of the third harmonic (cf. curves 2 and 3 in Fig. 4.10). In the general case of non-negligible nonlinear phase shifts, phase-matching effects cannot be decoupled from the influence of the pump spectrum, and we resort to approximate integration methods to identify the main features of the THG process. For a pump pulse with a Gaussian envelope, A0 (t p ) = A˜ exp[−tp2 /(2τ 2 )], where A˜ and τ are the amplitude and the initial duration

Part A 4.4

is the XPM-induced phase shift of the third-harmonic field. In the regime where the nonlinear phase shifts, given by (4.124) and (4.125), are small, the Fourier transform of (4.123) yields the following expression for the spectrum of the third-harmonic intensity: z 2 2 sin (∆k0 + Ωξ) 2 I (Ω, z) ∝ β (∆k0 + Ωξ)2 ∞ ∞ × A Ω − Ω A Ω − Ω

176

Part A

Basic Principles and Materials

of the pump pulse, the spectrum of the third harmonic, in view of (4.123), is given by ∞ z 3 dth dz B (Ω, z) = iβ A˜ −∞

2 2 3 th + ξz × exp − − i∆k0 z + 3iγ1 A˜ 2 2τ 2 th + ξz z × exp − τ2 ⎡ z 2 ⎣ ˜ × exp 2iγ2 A dz

Third-harmonic intensity (arb. units) 6.0 x 10 –4 4.0 x 10 –4

10 –2

2.0 x 10 –4

2 1

– 20

–10

0 th/τ

10 –3

10 –4

2 th + ζ z × exp − + iΩth τ2

10 –5 (4.127)

Part A 4.4

Changing the order of integration in (4.127), we apply the saddle-point method to estimate the integral in dth : z B(Ω, z) ∝ iβ A˜ 3 τ dz 0 ˜ 2 z × exp −i ∆k0 + Ωξ − 3γ1 | A| ξ τ ˜ 2 Φ (z − z ) , (4.128) +2iγ2 | A| ξ τ x where Φ(x) = exp(−x 2 ) dx. 0

With ζ z/τ 1 and Φ[(z − z )ζ ] constant, the phase matching is controlled by the factor

2 sin2 ∆k + Ωξ + 3γ1 A˜ 2z . F (Ω, z) = 2 2 ∆k + Ωξ + 3γ1 A˜ (4.129)

Thus, the saddle-point estimate of the SVEA integral for the third-harmonic field recovers the generalized phase-matching condition in the form of (4.119). The frequency offset Ωmax providing the maximum efficiency of THG is now determined by the dispersion of the material, its nonlinear properties, and the intensity of the pump field. Spectral broadening of the pump field due to SPM radically expands the tunability range of the third harmonic (Fig. 4.11). With γ I0 L = 1 (L is the interaction length and I0 is the pump-field peak intensity), the spectrum of the pump field is broad enough to produce a high-amplitude peak at Ωmax = −6/τ. The peak be-

–5

10 Ωτ

Fig. 4.11 Spectra of the third harmonic generated with no

SPM of the pump (curve 1) and with an SPM-broadened pump (curves 2, 3) with γ I0 L = 1 (1) and 2 (2); ∆k0 τ/ξ = −6, ξ L = 20. The insets show the temporal structure of the third harmonic for γ I0 L = 1 (1) and 2 (2) with ξ L/τ = 20

comes narrower as the length of the nonlinear medium increases (cf. curves 2 and 3 in Fig. 4.11), suggesting a convenient way for an efficient generation of shortwavelength radiation with a well-controlled spectrum for spectroscopic and metrological applications. In the time domain, the third harmonic tends to break up into two pulses, as shown in the inset to Fig. 4.11 for γ I0 L = 1 (curve 1) and 2 (curve 2) with ξ Lτ = 20. The first peak at th τ = −20 represents a third-harmonic pulse that propagates with the pump pulse and that is phasematched to the pump field in the sense of (4.119). The second pulse propagates with the group velocity of the third harmonic and is group-delayed in our case with respect to the pump field. Due to the phase matching, the ratio of the amplitudes of the first and second peaks increases with the growth in γ I0 L (cf. curves 1 and 2 in the inset to Fig. 4.11). Examples of frequency-shifted THG can be found in the recent literature on nonlinear optics of PCFs and tapered fibers [4.48–52]. Interesting collinear and Cherenkov-type intermode phase-matching options [4.50, 51] have been highlighted. The nature of the frequency shift has been identified in [4.57, 58]. Generation of an asymmetrically broadened and spectrally shifted third harmonic is the most frequently encountered situation, observed in nonlinear-optical ex-

Nonlinear Optics

∆k (µm–1); ξ (fs/µm); ∆k + ξΩ (µm–1) Third-harmonic intensity (arb. units) 1.2

Pump intensity (arb. units) 1

0.8

0.1

0.4

0.01

0.0 350

400

450 λ (nm)

10 – 3

1000 1100 1200

1300 λ (nm)

3 2

0 1

0.30

0.35

0.40

0.45 λ (µm)

Fig. 4.12 The phase mismatch ∆k0 (1), group-velocity mismatch ξ (2), and the effective phase mismatch ∆k0 + ξΩ (3) for third-harmonic generation in an air-clad fused-silica fiber with a core radius of 0.9 µm. The pump wavelength is 1.25 µm. The vertical dashed line shows the wavelength λ = 383 nm, where ∆k0 + ξΩ = 0. The insets show a typical experimental spectrum of blue visible emission (1) and the spectrum of the Cr:forsterite laser pump field (2) at the output of a photonic-crystal fiber with a length of 10 cm (after [4.57])

periments with PCFs [4.49–51] and tapered fibers [4.52]. The two most striking recent experimental findings include the observation of a 260 nm spectral component generated by Ti:sapphire laser pulses in high-delta PCFs [4.49] and the generation of a spectrally isolated frequency component at 380 nm by femtosecond pulses of 1.25 µm Cr:forsterite laser radiation [4.59]. In the former case, the central wavelength of the ultraviolet (UV) component remained stable [4.49] as the central pump wavelength was tuned from 770 to 830 nm [which seems to agree well with the phase-matching condition of (4.119)]. In the latter case, generation of the blue-shifted third harmonic allowed a highly efficient frequency conversion of pump radiation over the frequency interval exceeding 540 THz [4.59]. The spectral peak of visible blue emission was shifted from the frequency 3ω0 in these experiments by 34 nm (a typical

spectrum is presented in the inset to Fig. 4.12), with no signal produced at 3ω0 . To identify the main features of the dramatic frequency shift in the spectrum shown in the inset to Fig. 4.12, we consider the dispersion of guided modes in an air-clad fused-silica thread as a generic example of PCF dispersion. In Fig. 4.12, we plot the phase and group-velocity mismatch, ∆k0 and ξ, as well as the effective phase mismatch ∆keff = ∆k0 + ξΩ for intermodal THG in an air-clad fused silica fiber with a core radius of 0.9 µm. We assume that the pump field is coupled into the fundamental mode of the fiber, while the third harmonic is generated in one of high-order HE13 type modes. The effective phase mismatch ∆keff passes through zero at the wavelength of 383 nm, corresponding to a frequency shift of about 63 THz relative to 3ω0 , which agrees well with the typical spectrum of blue emission from a PCF [4.59] presented in inset 1 to Fig. 4.12. The main features and dominant tendencies of the third-harmonic spectrum are adequately described by (4.119) and (4.127). Comparison of the pump (inset 2 to Fig. 4.12) and third-harmonic spectra presented in Fig. 4.11 also suggests that a blue shifting of the main peak in the spectrum of the third harmonic by more than 60 THz becomes possible due to a strong broadening of the pump field, which is, of course, not only due to SPM in realistic conditions of PCF experiments. We have shown that third-harmonic generation in the field of spectrally broadened short pump pulses can display interesting and practically significant new features. A short-pulse pump field broadened due to selfphase modulation can generate its third harmonic within a broad spectral range. However, the phase-matching condition generalized to include the phase and groupvelocity mismatch of the pump and third-harmonic fields, as well as the Kerr-nonlinearity-induced spectral broadening of the pump field, tends to select a narrow spectral region of efficient THG. The possibility to tune the frequency of the main spectral peak in the spectrum of the third harmonic by varying the group-velocity mismatch is a unique property of THG-type processes, which is not typical of standard parametric FWM processes, where the first-order dispersion terms cancel out of the balance of the field momenta. For pump fields with large nonlinear frequency deviations, this spectral region may lie several tens of terahertz away from the central frequency of the third harmonic 3ω0 . This non−3ω THG process, leading to no signal at 3ω0 , is shown to result in interesting and practically significant spectral-transformation phenomena in photonic-crystal fibers.

177

Part A 4.4

–1

4.4 Third-Order Nonlinear Processes

178

Part A

Basic Principles and Materials

4.5 Ultrashort Light Pulses in a Resonant Two-Level Medium: Self-Induced Transparency and the Pulse Area Theorem 4.5.1 Interaction of Light with Two-Level Media

Part A 4.5

The phenomenological approach based on nonlinearoptical susceptibilities does not include nonstationary phenomena related to a dynamic modification of the nonlinear medium by the laser field. Interaction of a resonant laser field with a two-level system is a physically interesting and methodologically important regime where the equations governing the evolution of the laser field in a nonlinear medium can be solved self-consistently with the equations of motion for the quantum system interacting with the field. Such a self-consistent analysis reveals the existence of a remarkable regime of nonlinear-optical interactions. A resonance laser pulse whose amplitude and pulse width are carefully matched to the two-level system can propagate in a two-level medium with no absorption-induced attenuation of the pulse amplitude, a phenomenon known as self-induced transparency. The interaction of laser radiation with a two-level system is a classic problem of laser physics [4.60]. It has been extensively studied with the use of various approximate analytical approaches and numerical procedures over four decades, revealing important aspects of the interaction of laser radiation with a two-level system and, generally, laser–matter interactions. The standard approach based on the slowly varying envelope approximation (SVEA) and rotating-wave approximation gives Maxwell–Bloch equations in the case of a two-level atom [4.60, 61]. These equations provide an adequate description of laser radiation propagating in resonant media within a broad range of parameters and give the key to understanding such fundamental resonant optical phenomena as the formation of 2π solitons and self-induced transparency [4.60–63]. As shown in the classic works by McCall and Hahn [4.62, 63], light pulses with pulse areas multiple of π can propagate in a two-level medium with no changes in their shape, while pulses with other pulse areas tend to change their areas during the propagation in a two-level medium evolving to pulses with areas multiple of p. Much analytical and numerical work has been done over the past decade to extend this SVEA result to ultrashort pulses. Eberly [4.64] has rederived the area theorem for the case of short light pulses, modifying this theorem to include pulse chirping and homogeneous damping. Ziolkowski et al. [4.65]

applied the finite-difference time-domain (FDTD) technique [4.66] to solve the semiclassical Maxwell–Bloch equations numerically. This approach revealed several important features of short-pulse propagation in a twolevel medium and allowed a more detailed analysis of self-induced transparency effects. Hughes [4.67, 68] has employed the FDTD approach to demonstrate the possibility of generating sub-femtosecond transients in a two-level medium. Tarasishin et al. [4.69] have applied FDTD technique to integrate jointly the Maxwell and Schrödinger equations for an ultrashort light pulse propagating in a two-level medium. Below in this section, we explain the FDTD-based algorithm [4.69] solving the Maxwell and Schrödinger equations to model the interaction of ultrashort laser pulses with an ensemble of twolevel atoms. FDTD simulations presented below reveal interesting regimes of short-pulse propagation and amplification in two-level media, including the evolution of the pulse to a 2π soliton, amplification of a single-cycle pulse in a medium with a spatially modulated distribution of dipole moments of resonant transitions, and amplification of chirped light pulses.

4.5.2 The Maxwell and Schrödinger Equations for a Two-Level Medium We shall start with the extension of the standard FDTD procedure to the case of short pulses propagating in a two-level medium, when the Maxwell equation for the fields and the Schrödinger equation for the wave functions should be solved without any assumptions that are usually employed in the SVEA approach. In the onedimensional case, the FDTD algorithm involves step-bystep integration of two Maxwell curl equations ∂Dz (x, t) ∂Hy (x, t) = , ∂t ∂x ∂Hy (x, t) ∂E z (x, t) = . ∂t ∂x

(4.130) (4.131)

To perform this integration, we have to define the relation between the components of the electromagnetic induction and the electromagnetic field. This can be done through the equation for the polarization of the medium. In our case of a two-level medium, this involves the solution of the Schrödinger equation for the wave functions of the energy levels.

Nonlinear Optics

4.5 Ultrashort Light Pulses: Self-Induced Transparency and the Pulse Area Theorem

We will consider an ensemble of noninteracting twolevel atoms or molecules whose wave functions can be represented as superpositions of two basis states 1 and 2: ψ(t) = a(x, t)ψ1 + b(x, t)ψ2 ,

(4.132)

where ψ1 and ψ2 are the eigenfunctions of an unperturbed system corresponding to the states with energies E 1 and E 2 (we assume for definiteness that E 1 > E 2 ), respectively, and a(x, t) and b(x, t) are complex coefficients. Then, the Schrödinger equation for the wave function yields the following set of differential equations: da(x, t) = E 1 a(x, t) − µE z (x, t)b(x, t), dt db(x, t) = E 2 b(x, t) − µE z (x, t)a(x, t), i dt

(4.133) (4.134)

∗

r1 (x, t) = a(x, t)b (x, t) + a (x, t)b(x, t) , r2 (x, t) = i[a(x, t)b∗ (x, t) − a∗ (x, t)b(x, t)] , r3 (x, t) = a(x, t)a∗ (x, t) − b(x, t)b∗ (x, t) , r4 (x, t) = a(x, t)a∗ (x, t) + b(x, t)b∗ (x, t) .

(4.135) (4.136) (4.137) (4.138)

The physical content of the parameters defined by (4.135–4.138) is well known from classic textbooks on coherent optics [4.60, 61]. It can be easily verified using the set (4.133) and (4.134) that r4 (x, t) is independent of time and can be interpreted as the probability to find the system in either state 1 or state 2. The quantities r1 (x, t) and r3 (x, t) play an especially important role. Depending on the sign of r3 (x, t), resonant electromagnetic radiation can be either amplified or absorbed by a two-level system. The quantity r1 (x, t) controls the polarization of the medium in the case of a linearly polarized light field: Pz (x, t) = 4πµr1 (x, t)N ,

(4.141)

dr3 (x, t) = −2 (µ/ ) E z (x, t)r2 (x, t) , dt

(4.140)

(4.142)

where ω0 = (E 1 − E 2 )/ . Differentiating (4.141), using (4.140) for dr1 (x, t)/ dt, and taking into consideration that r1 (x, t) = Pz (x, t)/(µN) = [Dz (x, t) − E z (x, t)]/(4πµN), we find that d2 Dz (x, t) d2 E z (x, t) − + ω20 [Dz (x, t) − E z (x, t)] dt 2 dt 2 4πµ2 ω0 N E z (x, t)r3 (x, t) = 0 , (4.143) +2 2 dr3 (x, t) = E z (x, t) dt 4π ω0 N

dDz (x, t) dE z (x, t) − . (4.144) × dt dt The FDTD approach involves difference approximation [4.66] of time and spatial derivatives involved in (4.130) and (4.131): ∆t n+1/2 n+1/2 (H − Hy,i−1/2 ) , (4.145) ∆x y,i+1/2 ∆t n+1 n+3/2 n+1/2 n+1 (E Hy,i+1/2 = Hy,i+1/2 + − E z,i ) , (4.146) ∆x z,i+1 n Dn+1 z,i = Dz,i +

where i and n indicate the values of discrete spatial and temporal variables, respectively, x = i∆x, t = n∆t, ∆x and ∆t are the steps of discretization in spatial and temporal variables. This approach yields the following set of difference equations: n+1 n−1 E z,i = Dn+1 + Dn−1 z,i − E z,i z,i n − 4Dn + 4∆t 2 4πµ2 ω0 N E n−1 r n−1 4E z,i z,i z,i 3,i + , 2 + ∆t 2 ω20

(4.139)

where N is the volume density of two-level atoms or molecules. Thus, the quantity r1 (x, t) defines the sought relation between the components of the electromagnetic induction and the electromagnetic field. Using (4.133) and (4.134), we arrive at the following set of equations for the quantities defined by (4.135– 4.137): dr1 (x, t) = −ω0r2 (x, t) , dt

dr2 (x, t) = ω0r1 (x, t) + 2 (µ/ ) E z (x, t)r3 (x, t) , dt

n+1 r3,i

2 n+1 n E z,i + E z,i ω0 N n+1 n−1 n+1 n−1 . × Dz,i − Dz,i − E z,i + E z,i

(4.147)

n−1 = r3,i + 0.5 4π

(4.148)

Thus, we arrive at the following closed algorithm of numerical solution: 1. substitute Dn+1 z,i determined from (4.145) for the current value of the discrete-time variable into the set

Part A 4.5

where µ is the dipole moment of transition between the levels 1 and 2. Following Feynman et al. [4.68], we introduce real combinations of the complex quantities a(x, t) and b(x, t):

179

180

Part A

Basic Principles and Materials

of (4.147) and (4.148) and determine the values of n+1 n+1 r3,i and E z,i , 2. substitute the quantities thus determined into (4.146) and determine the values of the magnetic field n+3/2 Hy,i , 3. substitute these values of the magnetic field into (4.145) and repeat the procedure for the next value of the discrete-time variable.

∂θ(x) 2πωµN =− θ(x) sin θ(x), ∂x ncE 0

To test the FDTD-based procedure of simulations described in Sect. 4.5.2, we model the propagation of light pulses (4.149)

Part A 4.5

where A(x, t) is the pulse envelope, in a two-level medium and compare the results of these simulations with the predictions of the McCall–Hahn area theorem [4.62, 63]. This plan can be accomplished by keeping track of the pulse area ∞ Ω(x, τ) dτ , (4.150) θ(x) =

−∞

2πωµN ∂A(x, t) n ∂A(x, t) + =− sin[θ(x, t)] . ∂x c ∂t nc

(4.152)

The celebrated solution to (4.152) is a pulse with a hyperbolic-secant shape, which propagates in a resonant two-level medium with no changes in its envelope:

t − x/V A(x, t) = sech , (4.153) µτ τ where V is the pulse velocity in the medium,

−1 4πµ2 ωNτ 2 n + V= . (4.154) nc c

(4.156)

ωµN where α = 8πncE . 0 According to (4.156), pulses whose areas θ(x0 ) are multiples of π propagate in a two-level medium with no changes in their envelopes (the soliton propagation regime). However, solitons with pulse areas equal to π, 3π, 5π, . . . are unstable. Thus, a pulse with an arbitrary initial area changes its waveform propagating in a twolevel medium until its pulse area becomes multiple of 2π. The characteristic spatial scale of this process is estimated as α−1 . Numerical simulations [4.69] have been performed for hyperbolic-secant pulses: 2

−∞

where Ω(x, t) = 2µ A(x, t) is the real Rabi frequency. The SVEA equation for the pulse envelope A(x, t) can be represented as 2πω ∂A(x, t) n ∂A(x, t) + =i P(x, t) , (4.151) ∂x c ∂t nc where n is the refractive index of the medium, P(x, t) is the slowly varying amplitude of the polarization induced in the medium. As shown by McCall and Hahn [4.62, 63], on exact resonance, P(x, t) = iµN sin[θ(x, t)], with t θ(x, t) = Ω(x, τ) dτ. Hence, (4.151) yields

(4.155)

where E 0 is the initial pulse amplitude. With θ ≈ 2π, we arrive at α ∂θ(x) = − sin θ(x), ∂x 2

4.5.3 Pulse Area Theorem

E(x, t) = A(x, t) eiφ+ikz−iωt + c.c. ,

In the regime of weak absorption (when the pulse area only slightly deviates from its stable values), the evolution of the pulse area is governed by the equation

E(x0 , t) = E 0

2 cos [ω(t − t0 )] . exp[−(t − t0 )/T ] + exp[(t − t0 )/T ] (4.157)

The parameters of the medium and the pulse were chosen as follows: Ω = 2µE 0 /2 = 0.0565ω, 4πµNr3 (0) = −0.12E 0 , and ω = ω0 (exact resonance). In accordance with the area theorem, a pulse with a duration T = 5.631 2π/ω should propagate in the twolevel medium with these specified parameters values without any changes in its envelope. FDTD simulations [4.69] reveal no changes in the waveform and the amplitude of such a pulse within a distance of 100 λ with an accuracy better than 0.1%. Now, we examine how an arbitrary light pulse evolves to a 2π pulse in a two-level medium. FDTD simulations were performed for the propagation of a pulse (4.157) with a duration T = 7.0392π/ω in a two-level medium with these specified parameters values. While the amplitude of such a pulse increases by 15% as it propagates through the medium, the duration of the pulse decreases by a factor of about two. The pulse areas FDTD-simulated for these values of the propagation coordinate are equal to 2.5π, 2.33π, 2.24π, and 2.1π, while the pulse areas for the same values of x calculated from (4.156) are equal to 2.5π, 2.35π, 2.22π,

Nonlinear Optics

4.5 Ultrashort Light Pulses: Self-Induced Transparency and the Pulse Area Theorem

and 2.09π, respectively. Thus, the results of FDTD simulations agree very well with the predictions of the area theorem, indicating the adequacy of the numerical approach.

4.5.4 Amplification of Ultrashort Light Pulses in a Two-Level Medium

Modulation of the Spatial Distribution of the Dipole Moment First, let us explain why we expect that the modulation of the spatial distribution of the dipole moment may improve the efficiency of amplification of an ultrashort light pulse in a two-level medium. To do this, we multiply (4.152) by A(x, t) and integrate the resulting expression ∞in time. Then, introducing the pulse energy, Φ(x) = −∞ A2 (x, t) dt, and taking into consideration that the pulse energy remains constant if θ(x0 ) = 0, 2π, . . . , we arrive at the following equation relating the pulse energy and area: ∂Φ(x) 2πωN = (4.158) [cos θ(x) − 1] . ∂x nc

Thus, if the pulse area is kept equal to π, as the pulse propagates through an inverted two-level medium, then the energy of such a pulse would grow linearly as a function of the distance: Φ(x) = (1 + βx) Φ(0) ,

(4.159)

where β = 4πωN Φ(0)nc and Φ(0) is the initial energy of the pulse. By looking at (4.150) and (4.159) and taking into consideration that Φ(0) = 2τ E 02 = 2E 0 /µ(0) for a hyperbolic-secant pulse (4.157), we find that the pulse area can be kept equal to π by modulating the spatial distribution of the dipole moment in accordance with µ0 µ(x) = √ . (4.160) 1 + βx Such a modulation of the spatial distribution of dipole moments can be achieved, for example, by preliminarily orienting molecules in the medium. In accordance with the area theorem, the evolution of the amplitude of a π pulse propagating in an inverted two-level medium with a spatial profile of the dipole moment described by (4.160) is governed by E(x) = E 0 1 + βx . (4.161) This growth of the pulse amplitude is exactly compensated by the decrease in the dipole moment of transitions in the two-level medium. The net effect is that the pulse area remains constant and equal to π. Figures 4.13a–d present the results of FDTD simulations for the amplification of a light pulse with a duration T = 2π/ω corresponding to a single optical cycle in an inverted two-level medium with a uniform distribution of the dipole moment (Figs. 4.13a,b) and in an inverted twolevel medium where the spatial distribution of dipole moments of resonant transitions is modulated in accordance with (4.160) (Figs. 4.13c,d). Simulations were performed for the case when Ω = 2µ0 E 0 /2 = 0.159ω, 4πµNr3 (0) = 0.12E 0 , and ω = ω0 . The time in these plots is measured from the moment when the center of the pulse passes through the entrance boundary of the medium. Figures 4.13a,b display the evolution of a π pulse in an inverted uniform two-level medium and the population difference in this medium at the distances 0, 3β −1 and 6β −1 . These plots show that a two-level medium with a uniform distribution of the dipole moment of transitions cannot ensure an efficient amplification of a transform-limited resonant π pulse. Figures 4.13c,d present the results of FDTD simulation for a π pulse

Part A 4.5

Since the numerical algorithm based on the FDTD technique is intended to simulate the evolution of very short pulses, it enables one to explore many important aspects of short-pulse amplification in a two-level medium, providing a deeper understanding of the problems arising in the amplification of short pulses and the ways that can be employed to resolve these problems. Propagation of π pulses seems to provide optimal conditions for amplification in a two-level system, since such pulses transfer atoms (or molecules) in an initially inverted medium to the lower state. However, the Rabi frequency increases as a light pulse propagates through the medium and its amplitude increases due to the amplification. Because of this change in the Rabi frequency, the pulse now cannot transfer inverted atoms or molecules to the lower state, but leaves some excitation in a medium, which reduces the gain and leads to pulse lengthening due to the amplification of the pulse trailing edge in a medium with residual population inversion. Thus, some precautions have to be taken to keep the area of the pulse constant in the process of pulse amplification in the medium. Below, we use the FDTD technique to explore two possible ways to solve this problem: (i) modulation of the spatial distribution of the dipole moments of resonant transitions in a two-level system and (ii) amplification of a frequency-detuned chirped pulse.

181

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Basic Principles and Materials

propagating through a medium with a spatial evolution of the dipole moment described by (4.160). These simulations illustrate the possibility of efficient amplification of a single-cycle pulses in such a medium. Note that the pulse amplitude in Figs. 4.13c and 4.13d increases linX ( β –1)

a) E (arb. units)

early with the propagation coordinate x (the deviation from the linear dependence does not exceed 10−4 ). This result agrees very well with the predictions of the area theorem and can also be considered as another test of the reliability of the developed algorithm. X ( β –1)

b) r3

Part A 4.5

3 2 1 0 –1 –2 –3 –8

6 1.0

0.5 0.0 –0.5 –4

t (2π/ω)

8 X ( β –1)

c) E (arb. units)

–1.0 –8

–4

X ( β –1)

6 3 2 1 0 –1 –2 –3 –8

t (2π/ω)

d) r3

1.0

0.5 0.0 –0.5 –4

t (2π/ω)

e) E (arb. units)

–1.0 –8

–4

t (2π/ω)

f) E (arb. units)

–1

–2

–3

–4 –8 –7 –6 –5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 7 8 t (2π/ω)

–4 –1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 t (2π/ω)

Nonlinear Optics

4.5 Ultrashort Light Pulses: Self-Induced Transparency and the Pulse Area Theorem

Frequency-Detuned Chirped Pulses The second idea of keeping the area of the amplified pulse unchanged is to use chirped pulses whose frequency is detuned from the exact resonance. Such a pulse experiences compression as it propagates in the considered two-level medium. With an appropriate choice of pulse parameters, this decrease in pulse duration may serve to preserve the area of the light pulse undergoing amplification. The Rabi frequency in this case is written

1 2µ0 E 0 2 Ω= ∆2 + , (4.162) 2

with k = 1.4, propagating in a uniform two-level medium with the same parameters as in Figs. 4.13a,b. The distance covered by the pulse in the medium is 6β −1, and the frequency detuning is ∆ = ω − ω0 = −0.02ω0 . As can be seen from Figs. 4.13e,f, although higher-order dispersion effects noticeably distort the waveform of the pulse, the peak intensity of an initially chirped pulse at the output of an amplifying uniform medium is nearly twice as high as the output energy of a transform-limited pulse. Furthermore, comparison of Figs. 4.13a,b,e,f shows that, chirping a light pulse, one can avoid undesirable pulse-lengthening effects. Fig. 4.13 (a)–(d) The evolution of (a), (c) the pulse waveform and (b), (d) the population difference for a single-cycle

light pulse propagating in an inverted two-level medium with (a), (b) a uniform distribution of the dipole moment and (c), (d) a spatial distribution of dipole moments of resonant transitions modulated in accordance with (4.160) simulated with the use of the FDTD technique [4.69] for Ω = 2µ0 E 0 /2 = 0.159ω, 4πµNr3 (0) = 0.12E 0 , and ω = ω0 . The time in these plots is measured from the moment when the center of the pulse passes through the input boundary of the medium. (e), (f) Amplification of a singlecycle chirped laser pulse (4.163) in a two-level medium: (e) the input pulse and (f) the output pulse simulated with the use of the FDTD technique for Ω = 2µ0 E 0 /2 = 0.159ω, 4πµNr3 (0) = 0.12 × 100 , and ω = ω0

4.5.5 Few-Cycle Light Pulses in a Two-Level Medium The results of FDTD simulations, as shown by Tarasishin et al. [4.70], typically agree very well with the predictions of McCall and Hahn theory [4.62, 63] for light pulses propagating in a two-level medium until the pulse duration T becomes less than the duration T0 of a single optical cycle. In agreement with the general predictions of McCall and Hahn, a 2.9π pulse with T = T0 , 2µE 0 πT/ = 2.9π, µN/E 0 = 0.00116, for example, is transformed until its area becomes equal to 2π. The peak amplitude of the pulse increases by a factor of 1.31 under these conditions, while its duration decreases to 0.55T0 . Noticeable deviations from the McCall–Hahn regime were observed for pulses with durations shorter than the duration of a single field cycle. In particular, half-cycle 2π pulses become asymmetric as they propagate through a two-level medium (Fig. 4.14a), and the characteristic length corresponding to the phase shift (4.9) equal to π estimated on the basis of FDTD simulations for such pulses was equal to 9.4 λ, which appreciably differs from the SVEA estimate for the characteristic length corresponding to the π phase shift, L = ( nc)/(4ω2 µ2 T 2 N) = (E 02 nc)/(4ω2 N ). Quartercycle 2π pulses display noticeable distortions and lengthening in the process of propagation through a twolevel medium (Fig. 4.14b). Deviations observed in the behavior of very short 2π pulses from the McCall–Hahn scenario are due to the fact that, although, formally, such pulses have an area of 2π, the cycle of interaction between light and a two-level system remains incomplete in this case, as the pulses do not even contain a full cycle of the field (Figs. 4.14a,b). As a result, such pulses leave some excitation in a two-level medium (Fig. 4.14d) instead of switching excited-state population back to the ground state, as in the case of longer 2π pulses (Fig. 4.14c). The amplitude of the leading edge of the pulse becomes higher than the amplitude of its trailing edge, and the pulse waveform becomes noticeably asymmetric (Fig. 4.14a). The group velocity of such very short pulses increases due to this incompleteness of the light-twolevel-system interaction cycle, leading to a discrepancy between the SVEA estimate and the FDTD result for the characteristic length corresponding to the π phase shift. The residual population in the medium and the asymmetry of the pulse waveform increase with pulse shortening. The results of FDTD simulations presented in this section thus show that the general predictions of McCall and Hahn for the evolution of the amplitude and the

Part A 4.5

where ∆ = ω − ω0 is the detuning of the central frequency of the pulse from the resonance. Figures 4.13e,f present the results of FDTD simulations for a single-cycle pulse with a quadratic initial chirp 2 cos ωt + kt 2 /T 2 , (4.163) E(0, t) = E 0 exp (−t/T ) + exp (−t/T )

183

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Basic Principles and Materials

a) E (arb. units)

b) E (arb. units)

X ( L)

1.0

0.5

0.0

–0.5

– 0.5

–1.0 –5

c) Population

t (2π/ω)

– 1.0 –5

r1 r2

Part A 4.5

0.8

1 1.0

0.8

0.6

0.4

0.2

0.0 –5

t (2π/ω)

X ( L)

1 1.0

d) Population

X ( L)

r1 r2

t (2π/ω)

0.0 –5

t (2π/ω)

Fig. 4.14 (a) Evolution of a half-cycle 2π pulse in a two-level medium: T = 0.5T0 , 2µE 0 πT/ = π, and µN/E 0 = 0.0016, (b) Evolution of a quarter-cycle 2π pulse in a two-level medium: T = 0.25T0 , 2µE 0 πT/ = 2π, and µN/E 0 = 0.0032. (c), (d) Evolution of the excited- and ground-state populations in a two-level medium under the action of (c) a half-cycle 2π pulse and (d) a quarter-cycle 2π pulse: (dashed line) ground-state population r1 and (solid line) excited-state population r2 .

Simulations were performed with the use of the FDTD technique [4.70]

phase of short pulses in a two-level medium generally agree reasonably well with the results of numerical simulations until the pulse duration becomes less than the duration of a single optical cycle. Numerical analysis reveals several interesting physical features in the formation of 2π solitons produced as a result of splitting of single-cycle pulses propagating in a two-level medium. In particular, the resulting pulses may have different amplitudes, durations, and group velocities, allowing the formation of sub-femtosecond pulses and slowing down of the light in two-level media. Noticeable deviations from the McCall–Hahn regime can be observed

for pulses with durations shorter than the duration of a single field cycle. Half-cycle 2π pulses become asymmetric as they propagate through a two-level medium, while quarter-cycle 2π pulse display considerable distortions and lengthening in the process of propagation through a two-level medium. Deviations observed in the behavior of very short 2π pulses from the McCall–Hahn scenario are due to the fact that the cycle of interaction between light and a two-level system remains incomplete in this case, and light pulses leave some excitation in a two-level medium instead of switching excited-state population back to the ground state.

Nonlinear Optics

4.6 Let There be White Light: Supercontinuum Generation

185

4.6 Let There be White Light: Supercontinuum Generation Supercontinuum (SC) generation – a physical phenomenon leading to a dramatic spectral broadening of laser pulses propagating through a nonlinear medium – was first demonstrated in the early 1970s [4.71, 72] (see [4.73] for an overview of early experiments on supercontinuum generation). Presently, more than three decades after its discovery, supercontinuum generation is still one of the most exciting topics in laser physics and nonlinear optics [4.44], the area where high-field a)

4.6.1 Self-Phase Modulation, Four-Wave Mixing, and Modulation Instabilities in Supercontinuum-Generating Photonic-Crystal Fibers c)

Fig. 4.15a–c Spectral transformations of ultrashort pulses in photonic-crystal fibers: (a) supercontinuum generation, (b) third-harmonic generation, and (c) frequency shifting

Propagation of laser pulses in PCFs is always accompanied by SPM-induced spectral broadening. The basic features of SPM are discussed in Sect. 4.4.1. For very short laser pulses and broadband field waveforms, SPM can be thought of as a four-wave mixing ω p1 + ω p2 = ω3 + ω4 with frequency components ω p1 and ω p2 from the spectrum of the laser field serving as pump photons generating new frequency components ω3 and ω4 . In the case of a frequency-degenerate pump, ω p1 = ω p2 = ω p , the new frequency components generated through FWM appear as Stokes and anti-Stokes sidebands at frequencies ωs and ωa in the spectrum of the output field. Such FWM processes become especially efficient, as emphasized in Sect. 4.4.4, when phase matching is achieved for the fields involved in the nonlinear-optical interaction. Under certain condi-

Part A 4.6

science meets the physics of low-energy unamplified ultrashort pulses in the most amazing way. The advent of photonic-crystal fibers [4.36, 37], capable of generating supercontinuum emission with unamplified, nano- and even sub-nanojoule femtosecond pulses, has resulted in revolutionary changes in frequency metrology [4.74–77] opened new horizons in ultrafast science [4.78, 79] and allowed the creation of novel wavelength-tunable and broadband fiber-optic sources for spectroscopic [4.80] and biomedical [4.81] applications. The rainbow of colors produced by a laser beam Fig. 4.15 has become an optical instrument and a practical tool. As a physical phenomenon, supercontinuum generation involves the whole catalog of classical nonlinear-optical effects, such as self- and cross-phase modulation, four-wave mixing, stimulated Raman scattering, solitonic phenomena and many others, which add up to produce emission with an extremely broad spectra, sometimes spanning over a couple of octaves. Below, we discuss the basic physical processes contributing to supercontinuum generation in greater detail, with special emphasis made on self-phase modulation, fourwave mixing, and modulation instabilities (Sect. 4.6.1), cross-phase modulation (Sect. 4.6.2), as well as the solitonic phenomena and stimulated Raman scattering (Sect. 4.6.3).

186

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Basic Principles and Materials

tions, the pump fields modify the effective refractive indices of the guided modes involved in the FWM process, inducing the phase matching for the FWM and leading to a rapid growth of spectral sidebands representing the Stokes and anti-Stokes fields. This regime of four-wave mixing, referred to as modulation instability (MI), plays an especially important role in nonlinear-optical spectral transformations of ultrashort pulses in PCFs. In the simplest, scalar regime of modulation instability, two photons of the pump field with the frequency ω p generate Stokes and anti-Stokes sidebands with frequencies ωs = ω p − Ω and ωa = ω p + Ω. To illustrate this regime of MI, we represent the propagation constants of the Stokes and anti-Stokes sidebands as Taylor-series expansions around ω p ,

Part A 4.6

1 β ω p + Ω ≈ β0 ω p + Ω up 1 2 + β2 ω p Ω + 2γ P , (4.164) 2 1 β ω p − Ω ≈ β0 ω p − Ω up 1 2 + β2 ω p Ω + 2γ P , (4.165) 2 where P is the peak power of the pump field, β0 (ω p ) is the Kerr-effect-free propagation constant of the pump field mode (i. e., the propagation constant of the pump field corresponding to the regime with P = 0), u p = (∂β/∂ω|ω=ω p )−1 is the group velocity of the pump pulse, β2 (ω p ) = ∂ 2 β/∂ω2 |ω = ω p , γ = (n2 ω p )/(cS and eff ) is the nonlinear ∞ ∞ coefficient, ∞ ∞ Seff =[ −∞ −∞ |F(x, y)|2 dx dy]2 / −∞ −∞ |F(x, y)|4 dx dy is the effective area of a guided mode with the transverse field profile F(x, y). With the propagation constant of the pump field written as β ω p = β0 ω p + γ P , (4.166) the mismatch of the propagation constants of the fields involved in the FWM process is given by ∆βFWM = β ω p + Ω + β ω p + Ω − 2β ω p ≈ β2 ω p Ω 2 + 2γ P . (4.167) The phase matching can thus be achieved for this type of FWM at 1/2 2γ P Ω = ± (4.168) β2 ω p

only when the central frequency of the pump field lies in the range of anomalous group-velocity dispersion, where β2 (ω p ) < 0. Figure 4.16 illustrates typical features of the scalar MI in PCFs observed by Fedotov et al. [4.82]. In those experiments, unamplified 50 fs pulses of 790–810 nm Ti:sapphire laser radiation with a repetition rate of 10 MHz and an energy of 0.1–1.4 nJ were coupled into micro-waveguide channels off the central core of the PCF (the inset in Fig. 4.16). The zero group-velocity dispersion (GVD) wavelength for the micro-waveguide channel used to observe MI is estimated as λ0 ≈ 720 nm, providing an anomalous GVD for the pump field. With a special choice of the pump pulse power, it is then possible to use the SPM phase shift to induce phase matching for efficient four-wave mixing (Fig. 4.16). This process can be understood as SPM-induced modulation instability, resulting in an exponential growth of spectral sidebands phase-matched with the pump field. In the output spectrum presented in Fig. 4.16, the 795 nm pump field generates sidebands centered at 700 and 920 nm. To understand this result, we use the standard result of MI theory (4.168) for the frequency shift corresponding to the maximum MI gain. With γ ≈ 50 W−1 km−1 , D ≈ 30 ps/(nm · km), pulse energy E ≈ 0.5 nJ, initial pulse duration τ ≈ 50 fs, we find Ωmax /2π ≈ 50 THz, which agrees well with the frequency shifts of the sidebands observed in the output spectra presented Intensity (arb. units) 1.0

0.5

0.0 600

700

800

900

1000 λ (nm)

Fig. 4.16 Sideband generation through modulation insta-

bility in the spectrum of an ultrashort pulse transmitted through a photonic-crystal fiber (shown in the inset). The input energy of laser pulses is 0.5 nJ. A scanning electron microscopy (SEM) image of the photonic-crystal fiber is shown in the inset

Nonlinear Optics

in Fig. 4.16. SPM-induced broadening sufficient to seed the considered MI-type FWM process is achieved within a fiber length z ≈ (2L d L nl )1/2 , where L d = τ 2 /|β2 | and L nl = (γ P)−1 are the dispersion and nonlinear lengths, respectively. For the above-specified parameters of laser pulses and a PCF, we have L d ≈25 cm and L nl ≈ 0.2 cm. SPM-induced broadening can thus provide seeding for sideband generation at Ωmax with PCF lengths exceeding 3.2 cm. This condition was satisfied in our experiments, where the PCF length was equal to 8 cm. Our experimental results thus demonstrate efficient regimes of SPM-induced MI in microchannel waveguides of PCFs, which offer much promise for parametric frequency conversion and photon-pair generation.

4.6.2 Cross-Phase-Modulation-Induced Instabilities

GVD (ps/(nm × km))

νg (10 10cm/s)

200 2.04 0

2.02

–200

–400 2.00 –600

0.6

0.9

1.2

1.5

1.8 λ (µm)

Fig. 4.17 The group velocity (the solid curve and the righthand axis) and the group-velocity dispersion (the dashed curve and the left-hand axis) calculated as functions of the radiation wavelength λ for the fundamental mode of the PCF with a cross-sectional structure shown in the inset

Experiments [4.83] were performed with highindex-step fused-silica PCFs having a cross-sectional geometry shown in the inset to Fig. 4.17 with a core diameter of 4.3 µm. Figure 4.17 displays the group velocity and GVD calculated for this PCF using the polynomial expansion technique [4.84]. The standard theory of XPM-induced MI, as presented by Agrawal [4.85], was used to analyze the main features of this phenomenon for fundamental-wavelength and second-harmonic femtosecond pulses of a Cr:forsterite laser co-propagating in a PCF with the above-specified structure. This theory predicts that stationary solutions to slowly varying envelope approximation equations for the pump and probe fields including dispersion up to the second order become unstable with respect to a small harmonic perturbation with the wave vector K and the frequency Ω if K has a nonzero imaginary part. The domains of this instability can be found by analyzing the dispersion relation

(K − Ωδg/2)2 − h 1 (K + Ωδg/2)2 − h 2 = C 2 , (4.169)

where

h j = β22 j Ω 2 Ω 2 + 4γ j P j g/β2 j /4 , C = 2Ω 2 β21 β22 γ1 γ2 P1 P2 ,

(4.170) (4.171)

γ j = n 2 ω j /(cS j ) are the nonlinear coefficients, δ = (vg2 )−1 − (vg1 )−1 , β2 j = ( d2 β j / dω2 )ω=ω j , P j , ω j , vg j and β j are the peak powers, the central frequencies, the group velocities, and the propagation constants of the pump ( j = 1) and probe ( j = 2) fields, n 2 is the nonlinear refractive index, and S j are the effective mode areas for the pump and probe fields. The gain of instabilities with a wave number K is given by G(Ω) = 2 Im(K ). Analysis of the dispersion properties of the PCFs employed in our experiments (Fig. 4.17) yields β21 ≈ −500 fs2 /cm, β22 ≈ 400 fs2 /cm, and δ = 150 fs/cm. The dimensionless frequency shift of the probe field, f = Ω/Ωc [where Ωc = (4γ2 P2 /|β22 |)1/2 ] changes from approximately 3.3 up to 3.8 as the γ1 P1 /γ2 P2 ratio is varied from 0.3 to 2.5. As highlighted by Agrawal [4.85], such a weak dependence of the frequency shift of the probe field on the pump power is typical of XPM-induced MI in the regime of pump– probe group-velocity mismatch. With γ2 P2 ≈ 1.5 cm−1 , the frequency shift f ≈ 3.8 gives sidebands shifted by Ω/2π ≈ 74 THz with respect to the central frequency ω2 of the second harmonic (which corresponds to approximately 90 nm on the wavelength scale). As will

187

Part A 4.6

Self-induced MI in PCFs has been shown [4.53, 82] to provide a convenient means for broadband parametric amplification, permitting the creation of compact and convenient all-fiber optical parametric oscillators. Cross-phase-modulation-induced MI, on the other hand, does not require anomalous dispersion [4.23], suggesting convenient and practical control knobs for the frequency conversion of ultrashort pulses in PCFs, allowing the frequency shifts and amplitudes of sidebands in output spectra of the probe field to be tuned by varying the amplitude of the pump field.

4.6 Let There be White Light: Supercontinuum Generation

188

Part A

Basic Principles and Materials

Part A 4.6

be shown below, this prediction agrees well with our experimental results. The laser system used in experiments [4.83] consisted of a Cr4+ :forsterite master oscillator, a stretcher, an optical isolator, a regenerative amplifier, and a compressor. The master oscillator, pumped with a fiber ytterbium laser, generated 30–60 fs light pulses of radiation with a wavelength of 1.23–1.25 µm at a repetition rate of 120 MHz. These pulses were then transmitted through a stretcher and an isolator, to be amplified in a Nd:YLF-laser-pumped amplifier and recompressed to the 170 fs pulse duration with the maximum laser pulse energy up to 40 µJ at 1 kHz. A 1 mm-thick β barium borate (BBO) crystal was used to generate the second harmonic of amplified Cr:forsterite laser radiation. Fundamental-wavelength, 1235 nm radiation of a femtosecond Cr:forsterite laser and its second harmonic were used as pump and probe fields, respectively. As can be seen from Fig. 4.16, the pump wavelength falls within the area of anomalous dispersion for the fundamental mode of the PCF, while the second-harmonic probe lies in the range of normal dispersion. The faster pump pulse Fig. 4.17 was delayed in our experiments with respect to the slower probe pulse at the input of the PCF by a variable delay time of τ. Figure 4.18 presents the results of experimental measurements performed with 170 fs pump pulses (the fundamental radiation of the Cr:forsterite laser) with an energy ranging from 0.2 up to 20 nJ and 3 nJ, 180 fs probe pulses (the second-harmonic output of the Cr:forsterite laser) transmitted through a 5 cm PCF with the cross-sectional structure shown in the inset to Fig. 4.17. For delay times τ around zero, the slower probe pulse sees only the trailing edge of the faster moving pump pulse. In such a situation, XPM predominantly induces a blue shift of the probe field. For τ ≈ δL ≈ 750 fs, where L = 5 cm is the PCF length, the leading edge of the pump pulse catches up with the probe field closer to the output end of the fiber, which results in a predominant red shift of the probe. To symmetrize the interaction between the pump and probe fields with respect to the XPM-induced frequency shift, we choose the delay time τ = δL/2 ≈ 375 fs. In the regime of low peak pump powers (less than 3 kW), the output spectrum of the probe field displays only slight broadening due to self-phase modulation (Fig. 4.18a). Pump pulses with higher peak powers lead to radical changes in the output spectra of the probe field, splitting the central spectral component of the probe field and giving rise to intense symmetric sidebands around the central frequency ω2 (Figs. 4.18b–d).

The general tendencies in the behavior of the output spectrum of the probe field as a function of the pump power agree well with the prediction of the standard theory of XPM-induced MI. In view of the splitting and slight blue-shifting of the central spectral component of the probe field (Figs. 4.18b–d), we define the effective central wavelength of the pump-broadened probe spectrum as 605 nm. As the pump power changes from 5 kW up to 42 kW in our experiments, the shift of the short-wavelength sideband in the output spectrum of the second harmonic increases from 80 nm up to approximately 90 nm. The theory predicts the wavelength shifts of 76 nm and 90 nm, respectively, indicating the predominant role of XPM-induced MI in the observed spectral transformations of the probe field. The amplia) Intensity (arb. units) 1.5

b) Intensity (arb. units) 1

1.0 0.1 0.5 0.01 700 800 500 600 700 800 λ (nm) λ (nm) c) Intensity (arb. units) d) Intensity (arb. units) 0.0

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500

600

700 800 λ (nm)

e) Intensity (arb. units)

0.0

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700 800 λ (nm)

600

700 800 λ (nm)

f) Intensity (arb. units)

1.0

0.0

500

0.0

500

600

700 800 λ (nm)

Fig. 4.18a–f Output spectra of the second-harmonic field transmitted through a 5 cm PCF. The power of the pump pulses is (a) 3, (b) 7, (c) 30, (d) 42, (e) 70, and (f) 100 kW. The power of the probe pulse is 8 kW

Nonlinear Optics

4.6.3 Solitonic Phenomena in Media with Retarded Optical Nonlinearity Optical solitons propagating in media with noninstantaneous nonlinear response experience reshaping and continuous frequency down-shifting due to the Raman effect phenomenon, called soliton self-frequency shift (SSFS) [4.86, 87]. Photonic-crystal fibers substantially enhance this nonlinear-optical process due to strong field confinement in a small-size fiber core and the possibility to tailor dispersion of guided modes by varying the fiber structure. Liu et al. [4.54] have shown that 200 fs input pulses of 1.3 µm laser radiation can generate sub-100 fs soliton pulses with a central wavelength tunable down to 1.65 µm through the SSFS in a tapered PCF. Photoniccrystal fibers with the wavelength of zero group-velocity dispersion (GVD) shifted to shorter wavelengths have been used for the soliton frequency downshifting of 800–1050 nm laser pulses [4.88, 89]. Abedin and Kubota [4.90] have employed a PCF to demonstrate a 120 nm SSFS for 10 GHz-repetition-rate picosecond pulses. In recent experiments [4.91, 92], PCFs with a special dispersion profile have been shown to provide an efficient spectral transformation of chirped sub-6 fs Ti:sapphire laser pulses through SSFS, leading to the generation of a well-resolved solitonic spectral component cen-

189

tered at 1.06 µm. Red-shifted soliton signals formed by sub-6 fs laser pulses in PCFs have been demonstrated to allow to a synchronized seeding of a picosecond Nd:YAG pump laser, permitting a considerable simplification of a few-cycle optical parametric chirped-pulse amplification (OPCPA) scheme [4.79]. With many of the key tendencies in the evolution of ultrashort pulses in PCFs analyzed in the extensive literature, we focus here on the possibility of using the SSFS phenomenon for widely tunable frequency shifting of few-cycle laser pulses. Our theoretical analysis is based on the numerical solution of the generalized nonlinear Schrödinger equation [4.93]

6 ∂A i ∂ (i) k (k) ∂ k A =i β + iγ 1 + ∂z k! ∂τ k ω0 ∂τ k=2 ⎤ ⎡ ∞ R(η) |A (z, τ − η)|2 dη⎦ , × ⎣ A(z, τ) −∞

(4.172)

where A is the field amplitude, β (k) = ∂ k β/∂ωk are the coefficients in the Taylor-series expansion of the propagation constant β, ω0 is the carrier frequency, τ is the retarded time, γ = (n 2 ω0 )/(cSeff ) is the nonlinear coefficient, n 2 is the nonlinear refractive index of the PCF material, ∞ ∞ 2 |F (x, y)|2 dx dy Seff =

−∞ −∞ ∞ ∞

−∞ −∞

(4.173)

|F (x, y)|4 dx dy

is the effective mode area [F(x, y) is the transverse field profile in the PCF mode], and R(t) is the retarded nonlinear response function. For fused silica, we take n 2 ≈ 3.2 × 10−16 cm2 /W, and the R(t) function is represented in a standard form [4.93, 94]: R(t) = (1 − f R )δ(t) + f R Θ(t)

τ12 + τ22 τ1 τ22

− τt

t sin τ1

(4.174)

where f R = 0.18 is the fractional contribution of the Raman response; δ(t) and Θ(t) are the delta and the Heaviside step functions, respectively; and τ1 = 12.5 fs and τ2 = 32 fs are the characteristic times of the Raman response of fused silica. We now apply (4.172) and (4.174) to compute the evolution of ultrashort pulses in two types of PCFs (Figs.

Part A 4.6

tudes of sidebands generated by pump pulses with a peak power of about 40 kW, as can be seen from Fig. 4.18d, become comparable or may even exceed the amplitude of the spectral components at the central part of the probe spectrum. The maximum frequency shift of the probe-field sidebands achieved in our experiments with 45 kW pump pulses is estimated as 80 THz, which is substantially larger than typical frequency shifts resulting from XPM-induced MI in conventional fibers [4.23]. With pump powers higher than 50 kW, both the central spectral components of the probe field and its sidebands featured a considerable broadening (Figs. 4.18e,f) and tended to merge together, apparently due to the cross-phase modulation induced by the pump field. XPM-induced instabilities thus open an efficient channel of parametric FWM frequency conversion in photonic-crystal fibers. Fundamental-wavelength femtosecond pulses of a Cr:forsterite laser were used in our experiments as a pump field to generate intense sidebands around the central frequency of co-propagating second-harmonic pulses of the same laser through XPMinduced MI in a PCF. This effect leads to efficient pump-field-controlled sideband generation in output spectra of the second-harmonic probe field.

4.6 Let There be White Light: Supercontinuum Generation

190

Part A

Basic Principles and Materials

Part A 4.6

4.19, 4.20). PCFs of the first type consist of a fused silica core with a diameter of 1.6 µm, surrounded with two cycles of air holes (inset in Fig. 4.20a). To find the parameters β (k) for these fibers, we numerically solved the Maxwell equations for the transverse components of the electric field in the cross section of a PCF using a modification of the method of polynomial expansion in localized functions [4.84]. Polynomial approximation of the frequency dependence of the propagation constant β for the fundamental mode of the PCF computed with the use of this numerical procedure with an accuracy better than 0.1% within the range of wavelengths 580–1220 nm yields the following β (k) coefficients for the central wavelength of 800 nm: β (2) ≈ − 0.0293 ps2 /m, β (3) ≈ 9.316 × 10−5 ps3 /m, β (4) ≈ − 9.666 × 10−8 ps4 /m, β (5) ≈ 1.63 × 10−10 ps5 /m, (6) −13 6 β ≈ − 3.07 × 10 ps /m. For the fundamental mode of such PCFs, the GVD, defined as D = −2πcλ−2 β (2) , vanishes at λz ≈ 690 nm. Fibers of the second type are commercial NL-PM-750 PCFs (from Crystal Fibre). The core diameter for these PCFs was equal to 1.8 µm. The parameters β (k) for these PCFs were defined as polynomial expansion coefficients for the dispersion profile of the fundamental mode of these fibers provided by the manufacturer. The group-velocity dispersion for PCFs of this type vanishes at λz ≈ 750 nm. In the case studied here, the laser field at the input of a PCF has the form of a few-cycle pulse (the upper a) Intensity envelope (arb. units)

b) Spectral Intensity (arb. units) Input

1.0

Input

0.002

0.0

0.000 z = 2 cm

0.4

z = 2 cm 0.004 0.000

0.0 z = 4 cm

0.4

z = 4 cm

0.004

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0.000 z = 8 cm

0.4

z = 8 cm

0.004

0.0

0.000 0.004

z = 12 cm

0.4 0.0 0.4 0.0

panel in Fig. 4.19a) with a broad spectrum (the upper panel in Fig. 4.19b) and a complicated chirp [4.79, 92]. For both types of PCFs, the short-wavelength part of the spectrum lies in the range of normal dispersion, while the wavelengths above λz experience anomalous dispersion. A typical scenario of spectral and temporal evolution of a few-cycle laser pulse in PCFs of the considered types is illustrated by Figs. 4.19a,b. The initial stage of nonlinear-optical transformation of a few-cycle pulse involves self-phase modulation, which can be viewed as four-wave mixing of different frequency components belonging to the broad spectrum of radiation propagating through the fiber. Frequency components lying near the zero-GVD wavelength of the PCF then serve, as shown in the classical texts on nonlinear fiber optics [4.23], as a pump for phase-matched FWM. Such phase-matched FWM processes, which involve both frequency-degenerate and frequency-nondegenerate pump photons, deplete the spectrum of radiation around the zero-GVD wavelength and transfer the radiation energy to the region of anomalous dispersion (spectral components around 920 nm for z = 2 cm in Fig. 4.19b). A part of this frequencydownconverted radiation then couples into a soliton, which undergoes continuous frequency downshifting due to the Raman effect (Fig. 4.19b), known as soliton self-frequency shift [4.23,85,86]. In the time domain, the red-shifted solitonic part of the radiation field becomes

z = 12 cm

0.000 z = 24 cm 0

1000

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3000

4000

5000 τ (fs)

0.004 0.000

z = 24 cm 0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2 λ (µm)

Fig. 4.19 Temporal (a) and spectral (b) evolution of a laser pulse with an initial energy of 0.25 nJ and an input temporal

envelope and chirp shown in Fig. 4.1b propagating through the second-type PCF (shown in the inset)

Nonlinear Optics

delayed with respect to the rest of this field (Fig. 4.19a) because of the anomalous GVD of the fiber. As a result of these processes, the red-shifted soliton becomes increasingly isolated from the rest of the light field in both the time and frequency domains, which reduces, in particular, the interference between the solitonic and nonsolitonic part of radiation, seen in Fig. 4.19b. High-order fiber dispersion induces soliton instabilities, leading to Cherenkov-type emission of dispersive waves [4.21, 22] phase-matched with the soliton, as discussed in the extensive literature (see, e.g., [4.95, 96]). a) Intensity envelope (arb. units)

This resonant dispersive-wave emission gives rise to a spectral band centered around 540 nm in Fig. 4.19b. As a result of the above-described nonlinear-optical transformations, the spectrum of the radiation field for a PCF with a characteristic length of 20 cm typically features four isolated bands, representing the remainder of the FWM-converted pump field (the bands centered at 670 and 900 nm in Fig. 4.19b), the red-shifted solitonic part (reaching 1.06 µm for z = 24 cm in Fig. 4.19b), and the blue-shifted band related to the Cherenkov emission of dispersive waves in the visible. In the time domain, as

Input

1.0 0.0

0.0 z = 2 cm

0.5 0.0

z = 4 cm

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z = 30 cm

Part A 4.6

z = 3 cm

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191

b) Intensity envelope (arb. units) Input

1.0

4.6 Let There be White Light: Supercontinuum Generation

z = 5 cm

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1000

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d) Spectral intensity (arb. units)

0.006 0.003 0.000 0.006 0.003 0.000 0.006 0.003 0.000

0.006 0.003 0.000 0.006 0.003 0.000 0.006 0.003 0.000 0.006 0.003 0.000 0.006 0.003 0.000 0.006 0.003 0.000 0.4

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z = 4 cm z = 5 cm 0.5

0.6

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Fig. 4.20 Temporal (a), (b) and spectral (c), (d) evolution of laser pulses with an initial energy of (a), (c) 0.15 nJ and (b), (d) 0.5 nJ and an initial pulse width of 6 fs in the first-type PCF (shown in the inset). The input pulses are assumed to be

transform limited

192

Part A

Basic Principles and Materials

Part A 4.6

can be seen from Fig. 4.19a, only the solitonic part of the radiation field remains well localized in the form of a short light pulse, the remaining part of the field spreading out over a few picoseconds. In Figs. 4.20a–d, we illustrate tunable frequency shifting of few-cycle laser pulses through SSFS in PCFs by presenting the results of simulations performed for an idealistic input pulse with an initial pulse width of 6 fs and a Gaussian pulse shape. For the first-type PCF (shown in the inset to Fig. 4.20a), almost the entire spectrum of the input pulse falls within the range of anomalous dispersion, and the pulse tends to form solitons, observed as well-resolved prominent spikes in the time domain (Figs. 4.20a,b). In the frequency domain, the Raman effect leads to a continuous frequency downshifting of the soliton (Figs. 4.20c,d). The rate of this frequency shift dν/ dz, where ν is the carrier frequency and z is the propagation coordinate, rapidly grows with a decrease in the pulse duration τ0 . With a linear approximation of the Raman gain as a function of the frequency, the integration of the nonlinear Schrödinger equation, as shown by Gordon [4.97], yields dν/ dz ∝ τ0−4 . Although high-order dispersion and deviations of the Raman gain curve from the linear function generally make the relation between dν/ dz and τ in soliton dynamics much more complicated [4.98], the soliton pulse width remains one of the key parameters controlling the soliton frequency shift for a given Raman gain profile. In the case considered here, the short duration of 6 fs input pulses provides a high rate of soliton-frequency shifting at the initial stage of pulse propagation through the PCF. As the spectrum of the soliton is shifted toward the spectral range with larger values of GVD, the pulse width increases, which slows down the frequency shift. It is instructive to illustrate the main tendencies in the spectral and temporal evolution of few-cycle laser pulses in PCFs using the results of analysis of ideal solitons, i. e., solitons governed by the nonlinear Schrödinger equation [(4.68) in Sect. 4.4.2]. The NLSE (4.68) is recovered from (4.172) by setting β (k) = 0 for k ≥ 3, taking f R = 0, and keeping only the term representing the Kerr effect, i. e., the term proportional to iγ A |A|2 , in the nonlinear part of the equation. In normalized, soliton units, the energy carried by a soliton j is [4.98] E j = 4ξ j , where ξ j = W − j + 0.5 is the soliton eigenvalue, controlled by the input pulse energy 2W 2 . The soliton pulse width is given by τ j = τ0 /2ξ j , where τ0 is the input pulse width. The soliton pulse width can thus be reduced, leading to higher SSFS rates, by increasing the energy of the input pulse.

In the case of solitary waves evolving in fibers with high-order dispersion and retarded nonlinearity, the results of NLSE analysis for the soliton energy and the soliton pulse width are no longer valid. In particular, as the soliton spectrum is shifted toward larger values of GVD, the soliton pulse width is bound to increase, while the soliton amplitude decreases (Figs. 4.20a,b). These changes in the soliton pulse width and amplitude are dictated by the balance between the dispersion and the nonlinearity, necessary for the existence of the soliton. On the qualitative level, however, being applied to short sections of a fiber, these simple relations provide important clues for the physical understanding of the evolution of Raman-shifted solitons in a PCF. Indeed, as can be seen from the comparison of the results of simulations performed for input pulses with the same initial pulse width (6 fs), but different energies, the SSFS rate in the case of higher energy pulses can substantially exceed the frequency-shift rate of solitons produced by lower energy pulses. A pulse with an input energy of 0.15 nJ, as can be seen from Fig. 4.20c, is coupled into a soliton, which undergoes a permanent red-shift as it propagates through the fiber. At z = 30 cm, the spectrum of this soliton peaks at 1.06 µm. A similar input pulse that has an initial energy of 0.5 nJ forms a soliton that exhibits a much faster frequency downshift. The central wavelength of this soliton reaches 1.12 µm already at z = 5 cm. Intensity envelope (2) (arb. units) 7650

7700

Time (1) (fs) 7750 7800

Intensity envelope (1) (arb. units) 7850

7900 0.5

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0.2 1

0.1 0.0

700

750

800

850

0.1

900

950 Time (2) (fs)

0.0

Fig. 4.21 Temporal envelopes of red-shifted solitons

(close-up of the peaks labeled with boxes in Figs. 4.20a,b) generated by laser pulses with an initial pulse width of 6 fs and an initial energy of (1) 0.15 nJ and (2) 0.5 nJ in a PCF at z = 30 cm (1) and 3 cm (2)

Nonlinear Optics

Intensity (arb. units)

– 10

– 20

– 30 400

600

800

1000

1200

1400 λ (nm)

Fig. 4.22 The spectrum of supercontinuum emission produced by 820 nm pump pulses with an initial duration of 35 fs and an input power of 320 mW in a microstructure fiber with a length of 30 cm and the cross-section structure shown in the inset to Fig. 4.20a.

tinuum radiation generated in PCFs has been intensely employed through the past few years to measure and control the offset between the carrier and envelope phases of ultrashort laser pulses, as well as for the creation of novel broadband sources for nonlinear spectroscopy, biomedical applications, and photochemistry. Examples of applications of PCF light sources based on enhanced nonlinear-optical interactions of guided modes will be given in Sect. 4.9.

4.7 Nonlinear Raman Spectroscopy Nonlinear Raman spectroscopy is one of the most powerful techniques of nonlinear spectroscopy, which has found numerous applications in condensed- and gasphase analysis, plasma diagnostics, investigation of molecular relaxation processes, temperature and concentration measurements, condensed-phase studies, and femtochemistry. While many of modifications of nonlinear Raman spectroscopy have become a routine tool of modern optical experiments, giving rise to many successful engineering applications, some of nonlinear Raman experiments carried out in the last decade have shown that the potential of this technique for many topical and sometimes interdisciplinary problems of modern

193

physics, chemistry, and biology is far from being completely realized. Similar to frequency-tunable sources of coherent radiation, which revolutionized nonlinear optics in its early days, allowing many delicate spectroscopic experiments, including nonlinear spectroscopic studies, to be performed, the impressive progress of femtosecond lasers in the 1990s has resulted in the breakthrough of the nonlinear Raman spectroscopy to new unexplored areas, giving rise to several elegant new ideas and approaches, permitting more-complicated systems and problems to be attacked, and leading to the measurements of fundamental importance. This new phase of nonlinear Raman spectroscopy also promoted

Part A 4.7

To provide illustrative physical insights into the observed behavior of red-shifted solitons in PCFs as a function of the input pulse energy, we plot in Fig. 4.21 the snapshots of temporal envelopes of the solitonic part of the field corresponding to the input energies of 0.15 nJ (curve 1) and 0.5 nJ (curve 2). We take these snapshots of solitons, representing close-up views of intensity envelope sections labeled with boxes in Figs. 4.20a,b, for two different values of the propagation coordinate, z = 30 cm in the case of a 0.15 nJ input pulse and z = 3 cm for the 0.5 nJ input energy. With the spectra of red-shifted solitons centered around 1.06 µm in both cases (Figs. 4.20c,d), these values of the propagation coordinate allow a fair comparison of SSFS dynamics in terms of the dependence of the frequency shift rate on the soliton pulse width. As is seen from Fig. 4.21, the SSFS rate correlates well with the soliton pulse width. While the soliton produced by a pulse with an initial energy of 0.15 nJ has a pulse width of about 50 fs (curve 1 in Fig. 4.21), the pulse width of the soliton emerging from the 0.5 nJ laser pulse is about 20 fs. In qualitative agreement with predictions of Gordon [4.97] and Lucek and Blow [4.98], this shorter soliton in Figs. 4.20b,d displays a faster downshifting as compared with the longer soliton in Figs. 4.20a,c. In the following section, this dependence of the SSFS rate on the energy of the pulse launched into the fiber will be used for the experimental demonstration of widely tunable soliton frequency shift of 6 fs pulses produced by a Ti:sapphire oscillator. At higher input powers, the spectral features originating from FWM, SSFS, dispersive-wave emission of solitons experience broadening due to SPM and XPM effects, merging together and giving rise to a broadband white-light emission (Figs. 4.15, 4.22). This supercon-

4.7 Nonlinear Raman Spectroscopy

194

Part A

Basic Principles and Materials

Part A 4.7

the development of new spectroscopic concepts, including time-resolved schemes, broadband spectroscopy, polarization measurements, and CARS generalizations based on higher-order nonlinear processes. This conceptual and technical progress achieved in the last decade shows us some very important features of what nonlinear Raman spectroscopy is going to be in the nearest future, encouraging the application of new ideas, techniques, and methods in this area of spectroscopy. This section provides a brief introduction to the main principles of nonlinear Raman spectroscopy, giving a general idea of how the measurements are performed and the spectroscopic data are extracted from the results of these measurements. Following this plan, we will first give a brief introduction to the basic concepts of nonlinear Raman spectroscopy. Then, we will consider various modifications of coherent Raman four-wave mixing (FWM) spectroscopy, including the standard CARS scheme, stimulated Raman scattering, Raman-induced Kerr effect, degenerate four-wave mixing (DFWM), and coherent hyper-Raman scattering. We will also briefly describe polarization techniques for nonlinear Raman spectrometry and coherent ellipsometry, allowing selective investigation of multicomponent molecular and atomic systems and permitting the sensitivity of nonlinear Raman spectrometry to be radically improved. Finally, in the context of the growing interest in the applications of short-pulse spectroscopy for the investigation of ultrafast processes, we will provide an introduction to time-resolved nonlinear Raman spectroscopy.

4.7.1 The Basic Principles Nonlinear Raman spectroscopy is based on nonlinearoptical interactions in Raman-active media. The nonlinear character of light interaction with a medium implies that molecular, atomic, or ionic vibrations in a medium are no longer independent of the light field. Instead, pump light waves with frequencies ω1 and ω2 modulate Raman-active vibrations in a medium at the frequency Ω ≈ ω1 − ω2 , which can be then probed with another light beam, generally having a frequency ω3 (Fig. 4.6c). This wave-mixing process involving the inelastic scattering of the probe wave by molecular vibrations gives rise to coherent Stokes- and anti-Stokes-frequency-shifted signals, whose amplitude I, polarization (the ellipticity χ and the tilt angle ψ of the principal axis of the polarization ellipse), and phase ϕ carry the spectroscopic information concerning the medium under study. This is the general idea of nonlinear Raman spectrometry, illustrated by Fig. 4.6c.

In practical terms, to undertake a simple threecolor CARS experiment, one generally needs three laser sources generating radiation with the frequencies meeting the requirements specified above. In the most popular scheme of two-color CARS, where the anti-Stokes signal is generated through the frequencymixing scheme ωa = 2ω1 − ω2 , the number of lasers required is reduced to two. The light beams have to be brought into coincidence in space to excite Ramanactive transitions in a medium and to generate the anti-Stokes signal. Introducing some delay time between pumping and probing pulses, one can also perform time-resolved CARS measurements to keep track of the temporal dynamics of excitation in the system under investigation. While spontaneous Raman scattering often suffers from the low quantum yield, which eventually results in the loss of sensitivity, coherent Raman scattering allows much more intense signals to be generated, thus allowing very high sensitivities to be achieved. Coherent Raman scattering also offers several other very important advantages that stem from the coherent character of the signal, which is thus well collimated and generated in a precisely known direction. Generally, the analysis of the amplitude, phase, and polarization of the signal resulting from a nonlinear wave-mixing process in a Raman-resonant medium involves the calculation of the nonlinear response of the medium in terms of the relevant nonlinear susceptibilities and the solution of Maxwell equations for the field of the signal. Below, we restrict our brief introduction to the theory of nonlinear Raman processes to the description of the basic notions and terminology, which will be employed later to explain the main concepts of various nonlinear Raman methods, including frequencyand time-domain CARS, coherent ellipsometry, and the Raman-induced Kerr effect (RIKE). Excitation of Raman Modes Within the framework of a simple, but physically instructive, semiclassical model of a nonlinear medium consisting of noninteracting molecules with a Ramanactive vibration with frequency Ω, the interaction of light with molecules (atoms or ions) can be described in terms of the electronic polarizability of molecules α depending on the generalized normal coordinate Q (e.g., defined as the distance between the nuclei in a molecule) [4.99]:

∂α α (Q) = α0 + Q +... , (4.175) ∂Q 0

Nonlinear Optics

where α0 is the equilibrium polarizability of a molecule, (∂α/∂Q)0 is the derivative of the electronic polarizability in the normal coordinate taken for the equilibrium position of nuclei, and we restrict our consideration to terms in this expansion linear in Q. The term (∂α/∂Q)0 Q in (4.175) is responsible for the modulation of light by molecular vibrations, as it gives rise to new frequency components in the induced polarization of the system, whose frequency shift is determined by the frequency of molecular vibrations. This can be seen from the expression for the polarization of a medium: P = Np ,

(4.176)

where N is the number density of Raman-active molecules,

∂α p = α(Q)E = α0 E + QE +. . . (4.177) ∂Q 0

H = − pE = −α (Q) E 2 .

(4.178)

Thus, the light-induced force driving molecular vibrations is given by ∂α 2 ∂H = E . F=− (4.179) ∂Q ∂Q As can be seen from (4.179) the force acting on a molecule in a light field may result in a resonant excitation of Raman-active vibrations with a frequency Ω if the field involves frequency components ω1 and ω2 with ω1 − ω2 ≈ Ω. Wave Mixing in Raman-Active Media The propagation of light waves in a nonlinear medium is governed by the wave equation

n 2 ∂ 2 E 4π ∂ 2 Pnl = 2 , (4.180) c2 ∂t 2 c ∂t 2 where E is the field in the light wave, n is the refractive index, c is the speed of light, and Pnl is the nonlinear polarization of the medium. Consider for example the CARS process, as an example of a nonlinear Raman process, assuming that it involves plane and monochromatic waves, ∆E −

E (r, t) = E1 exp (−iω1 t + ik1 r) + E2 exp (−iω2 t + ik2r) + E exp (−iωt + ikr) + Ea exp (−iωat + ikar) + c.c. ,

(4.181)

195

where ω1 and ω2 are the frequencies of pump waves, ω is the frequency of the probe wave, ωa = ω + ω1 − ω2 , the field envelopes E, E1 , E2 , and Ea are slowly varying functions of coordinates r and time t, k, k1 , k2 , and ka are the wave vectors of light waves with frequencies ω, ω1 , ω2 , and ωa , respectively, and c.c. stands for a complex conjugate. Then, representing the nonlinear polarization in the medium as a superposition of plane waves, we can write the equation for the amplitude of the anti-Stokes wave as 2πωa n a ∂ Ea ∂ Ea + =i Pnl (ωa ) exp (ika z) , c ∂t ∂z cn a (4.182)

where Pnl (ωa ), n a , and ka are the amplitude of the nonlinear polarization, the refractive index, and the zcomponent of the wave vector at the frequency ωa . The pump wave amplitudes will be assumed to be constant. Now, the nonlinear polarization of a medium has to be found with the use of some model of the nonlinear medium. Raman-Resonant Nonlinear Polarization and Nonlinear Susceptibilities Within the framework of our model of a nonlinear medium, the third-order nonlinear polarization of the medium at the anti-Stokes frequency ωa is given by [4.99]

N ∂α 2 E E 1 E 2∗ , P (3) (ωa ) = 4MD (Ω, ω1 − ω2 ) ∂Q 0 (4.183)

where D (Ω, ω1 − ω2 ) = Ω 2 − (ω1 − ω2 )2 − 2iΓ (ω1 − ω2 ) ,

(4.184)

derivatives are taken in the equilibrium position, Γ is the phenomenologically introduced damping constant, M is the reduced mass of a molecule, N is the number density of molecules, an asterisk indicates a complex conjugate, Q is the amplitude of the Raman-active molecular vibration, which can be expressed in terms of the density matrix ρ of an ensemble of molecules, Q = Sp (ρq) = ρab qba ,

(4.185)

where Sp is the trace operator and q is the operator of the vibrational coordinate. Introducing the third-order nonlinear-optical susceptibility of the medium,

N ∂α 2 , (4.186) χ (3)R = 24MD (Ω, ω1 − ω2 ) ∂Q 0

Part A 4.7

is the dipole moment of a molecule and E is the electric field strength. The energy of a molecule in a light field is written as

4.7 Nonlinear Raman Spectroscopy

196

Part A

Basic Principles and Materials

and solving (4.182), we arrive at the following expression for the intensity of the CARS signal Ia : 2 Ia ∝ χ (3) (ωa ; ω, ω1 , −ω2 ) 2 ∆kl 2 sin 2 × II1 I2l , ∆kl

(4.187)

Part A 4.7

where I, I1 , I2 are the intensities of the pump and probe beams, l is the length of the nonlinear medium, and ∆k = |∆k| = |ka − k − k1 + k2 | is the wave-vector mismatch. As can be seen from (4.187), the anti-Stokes wave is generated especially efficiently in the direction of phase matching, where ∆k = 0. The cubic nonlinear-optical susceptibility of (3) a medium χijkl is a fourth-rank tensor. The knowledge of the form of this tensor is important, in particular, for understanding the polarization properties of the signal of nonlinear Raman scattering. The form of the (3) χijkl tensor is determined by the symmetry properties of a medium. For an isotropic medium, only 21 of 81 (3) tensor components of χijkl are nonvanishing. Only three of these components are independent of each other, as the relations (3)

(3)

χ1111 = χ2222 = χ3333 ,

(4.188) (3)

(3)

χ1122 = χ1133 = χ2211 = χ2233 = χ3311 = χ3322 , (4.189) (3)

(3)

χ1212 = χ2121 = χ1313 = χ3131 = χ2323 = χ3232 , (4.190) (3)

(3)

χ1221 = χ2112 = χ1331 = χ3113 = χ2332 = χ3223 , (4.191) (3)

(3)

χ1111 = χ1122 = χ1212 = χ1221

(4.192)

hold true for an isotropic medium [4.9, 99]. The form of (3) the χijkl tensor for all the crystallographic classes can be found in many textbooks on nonlinear optics [4.9, 10]. Phase Matching of Focused Beams Generally, in the case of focused beams, phase-matching effects in nonlinear Raman scattering are taken into account through the phase-matching integral. In particular, for Gaussian beams, the expression for the overall power of the two-color CARS signal occurring in accordance with the scheme ωa = 2ω1 − ω2 is written

as [4.11, 100, 101] 2 2 3π ωa (3) 2 Pa = χ c2 n a 2 ∞ 32 b 2 × P1 P2 3 2πr |J|2 dr . π w30

(4.193)

Here, P1 and P2 are the powers of the pump beams with frequencies ω1 and ω2 , respectively, w0 is the waist size of the focused pump beams, b is the confocal parameter, 2 r C2 exp − bH dξ , (4.194) J= (1 + iξ ) (k − ik ξ ) H −C1

where −C1 and C2 are the coordinates of the boundaries of the nonlinear medium, H=

1 + ξ 2 ξ − ζ − i , k − ik ξ k

(4.195)

k = 2k1 + k2 , k = 2k1 − k2 , ζ = 2(z − f )/b is the normalized coordinate along the z-axis ( f is the coordinate of the beam waist along the z-axis). Equations (4.193–4.195) show that the information on the nonlinear cubic susceptibility in the CARS signal may be distorted by phase-matching effects. This problem has been analyzed both theoretically and experimentally for different modifications of nonlinear Raman spectroscopy [4.102]. The influence of absorption at the wavelength of pump and probe waves and at the frequency of the FWM signal can be taken into account by including the imaginary parts of the relevant wave vectors. The integral in (4.194) can be calculated in an analytical form for several particular cases, giving a clear physical understanding of the role of phase-matching and absorption effects in coherent FWM spectroscopy and imaging [4.103].

4.7.2 Methods of Nonlinear Raman Spectroscopy In this section, we will briefly consider standard and widely used schemes for nonlinear Raman spectroscopy (Fig. 4.2), including coherent Raman scattering, stimulated Raman scattering, and the Raman-induced Kerr effect and provide a brief introduction into the vast area of DFWM. Stimulated Raman Scattering The idea of using stimulated Raman scattering (SRS) as a spectroscopic technique is based on the measurement

Nonlinear Optics

of the frequency dependence of the SRS small-signal gain, which is proportional to the imaginary part of the nonlinear cubic susceptibility of a Raman-active medium [4.9, 99]. The power of the pump wave in such measurements has to be chosen in such a way as to avoid uncontrollable instabilities and to obtain noticeable SRS gain. The SRS-based approach was successfully employed, in particular, for high-resolution spectroscopy of Raman transitions [4.104]. Limitations of SRS as a spectroscopic technique are due to instabilities arising for light intensities exceeding the threshold SRS intensity. These instabilities build up under conditions when several nonlinear processes, including self-focusing and self-phase modulation, compete with each other, often rendering the SRS method impractical for spectroscopic applications.

dynamics in molecular systems [4.113]. The nonlinear nature and the spectral selectivity of CARS make this method an ideal tool for nonlinear spectroscopy [4.114, 115]. The most recent advances in nonlinear Raman techniques include coherence-controlled CARS [4.116], enhancing the potential of CARS microscopy [4.117], and CARS in photonic-crystal fibers [4.118, 119]. The widely employed geometry of nonlinear Raman measurements implies the use of collinear focused laser beams. While focusing allows the intensity sufficient to ensure a reliable detection of the nonlinear Raman signal to be achieved, the collinear geometry of wave mixing increases the length of nonlinear interaction. However, such an approach is reasonable as long as the spatial resolution along the propagation coordinate is not important. The scheme of nonlinear Raman spectroscopy becomes resolvable along the propagation coordinate as soon as collinear beams are replaced by a noncollinear one (Fig. 4.6c). In CARS, this technique is called the boxcars geometry [4.120]. The interaction area is confined in this case to the region where the beams intersect each other, allowing a high spatial resolution to be achieved. In the broad-beam CARS geometry Fig. 4.23, focused laser beams are replaced with broad or sheetlike beams. This approach allows the nonlinear Raman signal to image the whole areas of a nonlinear medium a) ω2

ω1

k FWM

ω FWM

Object

ω1 k1

b) k FWM β

k0 k1 α

θ k2

Fig. 4.23a,b Broad-beam folded coherent-anti-Stokes Raman scattering: (a) beam arrangements and (b) wave-vector diagram

197

Part A 4.7

Coherent Anti-Stokes Raman Scattering Instead of measuring the gain of one of two waves, as is done in SRS, Maker and Terhune [4.105] have demonstrated a spectroscopic technique based on measuring the frequency dependence of the intensity of a new wave generated at the anti-Stokes frequency ωa = 2ω1 − ω2 in the presence of two light waves with frequencies ω1 and ω2 chosen in such a way as to meet the condition of a Raman resonance with a Raman-active transition in a medium: ω1 − ω2 ≈ Ω (Fig. 4.6c). This approach, called coherent anti-Stokes Raman scattering, has become one of the most widespread nonlinear Raman methods, allowing many urgent spectroscopic problems to be successfully solved and stimulating numerous engineering applications of nonlinear laser spectroscopy (so-called three-color CARS with ωa = ω1 − ω2 + ω3 is shown in Fig. 4.6c). Similar to the SRS process described in the previous section, CARS involves the stimulated scattering of light in a Raman-active medium. However, in contrast to the standard SRS scheme, where a Stokes wave is generated or amplified, the CARS process gives rise to the appearance of a new frequency component, suggesting a spectroscopic approach that is free of instabilities typically arising in SRS due to the competition of different nonlinear processes. Due to its high spatial, temporal, and spectral resolution, the possibilities of studying highly luminous objects, and a rich variety of polarization methods, the CARS technique has gained a wide acceptance for temperature and concentration measurements in excited gases, combustion, and flames [4.99, 106–108], gas-phase analysis [4.99, 109, 110], high-resolution molecular spectroscopy [4.111, 112]. Short-pulse CARS gives an access to ultrafast processes and wave-packet

4.7 Nonlinear Raman Spectroscopy

198

Part A

Basic Principles and Materials

Part A 4.7

on a charge-coupled device (CCD) camera. The idea of broad-beam CARS imaging, which was discussed by Regnier and Taran [4.121] in 1973, has later proved to be efficient for the solution of many problems of gas-phase and plasma diagnostics [4.102, 122, 123]. Significant progress in extracting the data concerning parameters of a gas medium was achieved, allowing CARS signals from molecules of different types to be simultaneously detected, with the development of a dual broadband CARS scheme [4.124, 125] and angularly resolved CARS [4.126]. Figure 4.23 illustrates the application of the broadbeam folded CARS geometry for the investigation of excited and ionized gases [4.122, 123]. In this scheme, a pair of cylindrically focused coplanar broad light beams with frequencies ω1 and ω2 and wave vectors k1 and k2 , forming a small angle θ, irradiate a thin plasma layer in a plane parallel to the plane of the target (Fig. 4.23a). A cylindrically focused or a collimated nonfocused laser beam with frequency ω3 and wave vector k3 , which makes an angle α with the plane of the k1 and k2 vectors (Fig. 4.23b), irradiates the laser-produced spark from above. The FWM signal is generated in the direction kFWM determined by phase-matching conditions, forming an angle β with the plane of the target (Figs. 4.23a,b). Imaging the one-dimensional FWM signal onto a CCD array, we were able to map the spatial distribution of resonant particles in the plasma line by line. The use of a collimated unfocused beam ω3 enables slice-by-slice plasma imaging [4.102]. Raman-Induced Kerr Effect The Raman-induced Kerr effect (RIKE) [4.9] is understood as an optical birefringence induced in an initially isotropic medium due to an anisotropic Ramanresonant third-order polarization of a medium. In this scheme, a nonlinear medium is irradiated with a pair of light beams with frequencies ω1 and ω2 whose difference is tuned, in accordance with the general idea of probing Raman-active vibrations, to a resonance with Raman-active transitions in a medium. Then, the polarization of a probe wave at the frequency ω1 becomes perturbed due to the anisotropic nonlinear polarization induced in the Raman-active medium, which can be detected with the use of a polarization analyzer and a detector. The RIKE technique provides us with a convenient method for measuring the frequency depen(3) dence of the cubic susceptibility χijkl around a Raman resonance. The transmission coefficient T of a polarization analyzer aligned in such a way as to block the probe beam in the absence of the pump beam is given

by [4.99]

(3) Tl (ω1 − ω2 ) ∝ sin2 2γ χ1122 (ω1 ; ω2 , ω1 , −ω2 ) 2 (3) + χ1221 (ω1 ; ω2 , ω1 , −ω2 ) I22 (4.196)

in the case of a linearly polarized pump (I2 is the intensity of the pump beam and γ is the angle between the polarization vectors of the pump and probe waves) and (3)R Tc (ω1 − ω2 ) ∝ χ1122 (ω1 ; ω2 , ω1 , −ω2 ) 2 (3)R − χ1221 (ω1 ; ω2 , ω1 , −ω2 ) I22 (4.197)

in the case of a circularly polarized pump. As can be seen from (4.197), the coherent background is completely suppressed in the case of a circularly polarized pump. Importantly, RIKE is one of the four-wave mixing processes where the phase-matching condition is satisfied automatically regardless of the arrangement of the wave vectors of pump and probe waves, since the phase-matching condition k1 = k2 + k1 − k2 becomes an identity in this case. Degenerate Four-Wave Mixing Although, rigorously speaking, degenerate four-wave mixing does not employ Raman transitions and the models used to describe DFWM may sometimes differ from the standard ways of CARS description [4.9, 10], it is reasonable to briefly introduce DFWM as a nonlinear technique here, as it very frequently offers a useful alternative to CARS, allowing valuable data on a medium to be obtained in a convenient and physically clear way. DFWM is closely related to CARS as both processes are associated with the third-order nonlinearity of a medium. The main difference between these methods is that CARS implies the use of a two-photon Raman-type resonance (Fig. 4.6c), while DFWM is a frequency-degenerate process (Fig. 4.6d), involving either four one-photon resonances or a pair of two-photon resonances. With modern lasers capable of generating very short pulses, having large spectral widths, the DFWM signal can be detected simultaneously with CSRS and CARS in the same experimental geometry with the same molecular system by simply tuning the detection wavelength [4.127]. The combination of these nonlinear-optical approaches allows a more elaborate study of molecular relaxation and photochemistry processes, providing a much deeper insight

Nonlinear Optics

into the ultrafast molecular and wave-packet dynamics [4.127, 128]. The main advantages of DFWM as a spectroscopic technique are associated with the technical simplicity of this approach, which requires only one laser source and allows phase-matching conditions to be automatically satisfied regardless of the dispersion of the medium under study. Broadband DFWM [4.129] makes it possible to measure the temperature of excited gases, including atomic gases [4.130], with a single laser pulse. Folded broad-beam DFWM schemes are employed in several convenient and elegant methods for two-dimensional imaging of spatial distributions of gas parameters [4.131, 132].

4.7.3 Polarization Nonlinear Raman Techniques

4.7 Nonlinear Raman Spectroscopy

ψ θ

µ Pε

Pnr

Part A 4.7

Polarization Properties of the Coherent FWM Signal When analyzing polarization properties of the FWM signal, one has to take into account the interference of resonant components of FWM related to various (molecular or atomic) transitions in the medium and the

x Pnr

Methods of polarization-sensitive four-photon spectroscopy provide an efficient tool for the solution of many problems arising in the investigation of Raman resonances. In particular, the polarization technique is a standard method to suppress the coherent background in CARS measurements [4.99, 106], which makes it possible to considerably improve the sensitivity of spectroscopic measurements [4.99] and improves the contrast in CARS microscopy [4.115]. Polarization techniques in FWM spectroscopy can separately measure the real and imaginary parts of the relevant third-order nonlinear-optical susceptibility [4.133, 134], resolve closely spaced lines in FWM spectra of molecules [4.99, 135] and atoms [4.102], and improve the contrast of cubic-susceptibility dispersion curves near Raman resonances [4.136, 137]. Polarization methods in nonlinear Raman spectroscopy [4.138–140] help to analyze the interference of vibrational Raman resonances with oneand two-photon electronic resonances in CARS spectra [4.141], and can be used to determine invariants of atomic and molecular Raman and hyper-Raman scattering tensors [4.102] and to perform conformational analysis for complex organic molecules [4.141]. A comprehensive review of polarization techniques employed for molecular spectroscopy was provided by Akhmanov and Koroteev [4.99].

199

θ Φ e1

Fig. 4.24a,b Polarization technique of nonlinear Raman spectrometry: (a) coherent ellipsometry and (b) polariza-

tion suppression of the nonresonant background in CARS spectroscopy

nonresonant coherent background. In particular, it is the interference of the resonant FWM component with the nonresonant coherent background that ensures the possibility to record complete spectral information concerning the resonance under study, including the data on the phase of resonant FWM. The polarization ellipse of a Raman-resonant FWM signal is characterized by its ellipticity χ (which is defined as χ = ±atan(b/a), where atan stands for the arctangent function, b and a are the small and principal semiaxes of the polarization ellipse, respectively) and the tilt angle ψ of its principal axis (Fig. 4.24a). These parameters are related to the Cartesian components of the third-order polarization of a medium Px and Py by the following expressions [4.99]: tan(2ψ) = tan(2β) cos(δ) , sin(2χ) = sin(2β) sin(δ) ,

(4.198) (4.199)

200

Part A

Basic Principles and Materials

where β and δ are defined as Py tan (β) = , Px δ = arg Py − arg (Px ) .

(4.200) (4.201)

The frequency dependencies of the FWM polarization ellipse parameters, as can be seen from (4.198) and (4.199), provide information concerning the phase of the resonant FWM component, allowing a broad class of phase measurements to be performed by means of nonlinear Raman spectroscopy.

Part A 4.7

Suppressing the Nonresonant Background Polarization suppression of the nonresonant background in CARS is one of the most useful, practical, and widely employed polarization techniques in nonlinear Raman spectroscopy. Physically, the possibility of suppressing the nonresonant background in coherent Raman spectroscopy is due to the fact that the resonant and nonresonant components of the nonlinear polarization induced in a Raman-active medium are generally polarized in different ways. Let us illustrate this technique for the CARS process ωa = 2ω1 − ω2 , where ωa is the frequency of the anti-Stokes signal and ω1 and ω2 are the frequencies of pump waves, in the case of an isotropic medium. The third-order polarization responsible for the generation of the signal with the frequency ωa is then written as

P (3) = [Pr + Pnr ] E 12 E 2∗ ,

(4.202)

where E 1 and E 2 are the amplitudes of the light fields, Pr and Pnr are the resonant and nonresonant components of the third-order polarization induced in the Raman-active medium. In the case of an isotropic medium, relations (4.188– 4.192) are satisfied for both resonant and nonresonant components of the nonlinear-optical susceptibility. However, only the nonresonant part of the cubic susceptibility satisfies the Kleinman relations [4.9, 99], (3)nr

(3)nr

χ1111 = 3χ1122 = 3χ1221 = 3χ1212 ,

(4.203)

while the resonant part of the cubic susceptibility is usually characterized by a considerable dispersion near a Raman resonance, which implies that the resonant cubic susceptibility tensor components are not invariant with respect to the permutation of their frequency arguments. Taking into account relations (4.203) for the nonresonant part of the cubic susceptibility, we arrive at the following expression for the nonresonant and

resonant components of the third-order polarization of a Raman-active medium: (3)nr (4.204) Pnr = χ1111 2e1 e1 e∗2 + e∗2 (e1 e1 ) , ∗ (3)r ∗ Pr = 3χ1111 1 − ρ¯ e1 e1 e2 + ρe ¯ 2 (e1 e1 ) , (4.205) (3)r

(3)r

where ρ¯ = χ1221 /χ1111 and e1 and e2 are the unit polarization vectors of the light fields with frequencies ω1 and ω2 , respectively (Fig. 4.24b). Suppose that the pump fields with frequencies ω1 and ω2 are linearly polarized and their polarization vectors e1 and e2 are oriented at an angle µ with respect to each other, as shown in Fig. 4.24b. The vectors Pnr and Pr , in accordance with (4.204) and (4.205), generally have different orientations in space making a nonzero angle θ with each other. Therefore, by orienting the polarization analyzer in such a way as to suppress the nonresonant component of the CARS signal (by setting the angle ε measured from the direction perpendicular to the vector Pnr equal to zero, see Fig. 4.24b), one can analyze background-free CARS spectra. In many situations, nonlinear Raman study of Raman-active media would be simply impossible without this technique. This is the case, for example, when the CARS signal from a resonant gas under investigation is too weak to be reliably detected against the nonresonant CARS signal from the windows of a gas cell. Another example is the CARS spectroscopy on low concentrations of complex biological molecules, when the coherent background due to solvent molecules may be so strong that it leaves no way to detect the CARS signal from the molecules being studied without polarization suppression of the nonresonant nonlinear Raman signal. Even small deviations of the orientation of the polarization analyzer from the background-suppression position may be crucial, leading to dramatic changes in the signal-to-noise ratio. Another important conclusion that can be made from (4.204) and (4.205) is that, measuring the ratio of the CARS signals for different polarization arrangements is a convenient way of determining the properties of the nonlinear susceptibility tensor and, thus, characterizing the symmetry properties of molecular transitions under investigation. Coherent Ellipsometry Coherent ellipsometry, i. e., the measurement of the parameters of the polarization ellipse corresponding to the FWM signal, is one of widely used modifications of polarization-sensitive four-photon spectroscopy. Below,

Nonlinear Optics

we consider the main physical principles and discuss the main ideas of coherent ellipsometry. The possibility of reconstructing the real and imaginary parts of the nonlinear-optical susceptibility of a medium as functions of frequency and time is due to the interference of the resonant FWM component with the nonresonant coherent background, which ensures the recording of the phase information for the resonance being studied. For a broad class of problems, the nonlinear polarization of a medium responsible for coherent FWM processes can be represented as a sum of the nonresonant and resonant components described by a real vector Pnr and a complex vector Pr , respectively. Choosing the x-axis along the vector Pnr (Fig. 4.24b), we can write the Cartesian components of the total polarization of a medium cubic in the external field as (4.206)

Py = Pr eiϕ sin(θ),

(4.207)

where ϕ is the phase of the resonant component of the nonlinear polarization and θ is the angle between the resonant and nonresonant components. Knowing the parameters of the polarization ellipse from ellipsometric measurements, we can reconstruct, with the use of (4.198), (4.199), (4.206), (4.207), the real and imaginary parts of the resonant nonlinear polarization as functions of frequency and time from the experimental data of coherent ellipsometry. In the important particular case when the resonant component of the FWM signal can be considered as a small correction to the nonresonant component, the general procedure of separating the real and imaginary parts of the nonlinear polarization of a medium becomes especially simple. One can easily verify that the relations [4.99] ψ = β cos(ϕ) ∝ Re(P) χ = β sin(ϕ) ∝ Im(P)

(4.208) (4.209)

are satisfied in this case, showing that the spectral or temporal dependencies of the parameters ψ and χ of the FWM polarization ellipse respectively reproduce the spectral or temporal dependencies of the real and imaginary parts of the resonant component of the nonlinear polarization of a medium. Thus, the data obtained by means of coherent ellipsometry enable one to extract complete information concerning the resonant component of the nonlinear polarization of a medium, including information on its phase [4.134]. Note that no assumptions regarding the

201

shape of the line observed in a nonlinear Raman spectrum was made in our consideration, which means that this approach can be applied to a broad class of spectral lines. This procedure can be also extended to the time domain, allowing not only the spectra but also the time dependencies of the real and imaginary parts of the nonlinear polarization of a medium to be reconstructed. Finally, close molecular and atomic lines unresolvable in amplitude nonlinear Raman spectra can be also resolved in certain cases with the use of the phase information stored in coherent nonlinear spectra.

4.7.4 Time-Resolved Coherent Anti-Stokes Raman Scattering The method of time-resolved FWM spectroscopy implies that information on the parameters of atomic or molecular systems is extracted from an impulse response of a coherently excited system rather than from the frequency dispersion of nonlinear susceptibilities, as is done in frequency-domain FWM spectroscopy. The original idea of time-resolved CARS is that a light pulse with a duration shorter than the characteristic transverse relaxation time T2 induces coherent molecular vibrations with amplitude Q(t), and the decay kinetics of these vibrations is analyzed with the use of another, probe light pulse, which is delayed in time with respect to the pump pulses. The complete set of equations governing the processes related to time-domain CARS includes the SVEA wave equation (4.182) and the equations for the amplitude of coherent molecular vibrations Q(t), defined in accordance with (4.185), and the normalized population difference between the levels involved in the Raman resonance, n = ρaa − ρaa : ∂2 Q 2 ∂Q 1 ∂α + Ω2 Q = n E2 , + (4.210) T2 ∂t 2M ∂Q ∂t 2 1 ∂α 2 ∂Q ∂n n − 1 + E , (4.211) = ∂t T1 2 Ω ∂Q ∂t where T1 is the population relaxation time. In many cases, (4.182), (4.210), and (4.211) can be simplified with the use of the slowly varying envelope approximation. In this approximation, the energy of the CARS signal as a function of the delay time τ of the probe pulse in the scheme of time-resolved CARS with short light pulses and ∆k = 0 is given by [4.99] ∞ Wa (τ) ∝ −∞

|Q(t)A (t − τ)|2 dt .

(4.212)

Part A 4.7

Px = Pnr + Pr eiϕ cos(θ),

4.7 Nonlinear Raman Spectroscopy

202

Part A

Basic Principles and Materials

Since the intensity of the CARS signal in such a scheme is determined by (4.212), the use of sufficiently short probe pulses makes it possible to measure the kinetics of Q(t). Experiments by Alfano and Shapiro [4.142] and von der Linde et al. [4.143] have demonstrated the possibility of using the time-domain CARS technique to directly measure the T2 time for Raman-active modes in crystals and organic liquids. Formulas (4.185–4.187) and (4.212) show the relation between the information that can be obtained by frequency- and time-domain CARS spectroscopy. In fact, this information is essentially the same, and frequency- and time-domain CARS methods can successfully complement each other in studies of complex inhomogeneously broadened spectral bands. For example, in frequency-domain CARS spectroscopy, the phase information on molecular resonances can be extracted

through polarization measurements and coherent ellipsometry (see the discussion above), and the level of coherent nonresonant background can be suppressed by means of the relevant polarization technique [see (4.204), (4.205) and Fig. 4.24b]. In time-domain CARS, on the other hand, the nonresonant background appears only at zero delay time between the pump and probe pulses, having no influence on the transient signal, while the impulse-response measurements may directly provide the information not only on the amplitude, but also on the phase of a molecular or atomic resonance. The development of femtosecond laser systems resulted in the impressive technical and conceptual progress of time-domain FWM spectroscopy, allowing photochemistry processes and molecular dynamics to be monitored in real time (see [4.109, 110, 113] for a review).

Part A 4.8

4.8 Waveguide Coherent Anti-Stokes Raman Scattering 4.8.1 Enhancement of Waveguide CARS in Hollow Photonic-Crystal Fibers The general idea of waveguide CARS [4.144–149] is to improve the efficiency of four-wave mixing by increasing the interaction length and increasing the intensity of pump waves with given pump powers by reducing the transverse size of the wave-guiding layer in planar waveguides or the core diameter in optical fibers. CARS spectroscopy of the gas phase leaves no alternative to hollow waveguides and fibers. Since the refractive index of the core in such waveguides is lower than the refractive index of the cladding, the modes guided in the hollow core are always characterized by nonzero losses. The magnitude of these losses scales [4.150,151] as λ2 /a3 preventing fibers with small inner diameters to be used in nonlinear-optical experiments, which limits the waveguide CARS enhancement factors attainable with such fibers. In hollow-core photonic-crystal fibers [4.36,152,153], which guide light due to photonic band gaps, optical losses can often be kept low even in the case of small core diameters. Hollow PCFs with an inner diameter of ≈15 µm demonstrated by Benabid et al. [4.154] have the magnitude of optical losses on the order of 1–3 dB/m. We will show below that this property of hollow-core microstructure fibers makes them very attractive for waveguide CARS spectroscopy. We start with the expression [4.11] for the power of the three-color CARS signal generated at the frequency

ωs = ω0 + ω1 − ω2 by pump fields with frequencies ω0 , ω1 and ω2 : PCARS = 1.755 × 10−5

ω4s k0 k1 k2 2 D c4 ks2 k

(3) 2 × χeff P0 P1 P2 F2 ,

(4.213)

where k0 , k1 , k2 , ks are the wave numbers of light fields with frequencies ω0 , ω1 , ω2 , ωs , respectively; P0 , P1 , P2 are the powers of the fields with frequencies ω0 , ω1 , (3) ω2 , respectively; χeff is the effective combination of cubic nonlinear-optical susceptibility tensor components corresponding to the chosen set of polarization vectors of pump and signal fields; D is the frequency degeneracy factor of the four-wave mixing process defined after Maker and Terhune [4.105]; F2 =

∞ 2k 2π R dR πb 0 2 ξ 2 exp ib∆kξ 2 R exp − × dξ bH (1 + iξ ) (k − ik ξ ) H −ζ (4.214)

is the phase-matching integral, ∆k = ks − (k0 + k1 − k2 ), k = k0 + k1 − k2 , k = k0 + k1 + k2 , ξ = 2(z − f )/b, ζ = 2 f /b, b = n j ω j w20 /c is the confocal parameter, w0 is the

Nonlinear Optics

beam waist diameter, 2 1 + ξ ξ − ξ −i . H = k (k − ik ξ )

(4.215)

In the limiting case of tight focusing, when the confocal parameter b is much less than the length of the nonlinear medium l, b l, no increase in the CARS power can be achieved by reducing the pump-beam waist radius because of the simultaneous decrease in the interaction length. Mathematically, this well-known result is a consequence of the tight-focusing limit existing for the phase-matching integral (4.214). For small phase mismatches, ∆kl π, the phase-matching integral can be written in this limiting case as 4π 2 F2 = . 2 1 + kk

(4.216)

F2 =

k 4l 2 . k b2

(4.217)

Since the latter regime is exactly the case of waveguide CARS, we can use (4.216) and (4.217) to estimate the enhancement of waveguide CARS with respect to the regime of tight focusing. Phase mismatches in waveguide CARS should be understood as difference of propagation constants of waveguide modes involved in the wave-mixing process, and the mode-overlapping integral should generally be included to allow for the contribution of waveguide effects, in particular, the influence of higher-order waveguide modes. Assuming that the beam waist radius of focused pump beams is matched to the inner radius of a hollow fiber, a, we find from (4.216) and (4.217) that the waveguide CARS enhancement factor scales as λ2l 2 /a4 . The length l can be made very large in the case of fibers, but the fundamental limitation of waveguide CARS in hollow fibers comes from optical losses, whose magnitude scales as λ2 /a3 . The influence of optical losses and phase-mismatch effects on the CARS process in the loose-focusing regime can be included through the factor M ∝ exp [− (∆α + α4 ) l] 2 ∆kl sinh2 ∆αl 2 + sin 2 × l2, ∆αl 2 ∆kl 2 + 2 2

(4.218)

203

where ∆α = (α1 + α2 + α3 − α4 )/2, α1 , α2 , α3 , α4 are the magnitudes of optical losses at frequencies ω0 , ω1 , ω2 , ωs , respectively. It is straightforward to see from (4.218) that the amplitude of the CARS signal in a lossy waveguide reaches CARS , which is its maximum at some optimal length lopt given by

α1 + α2 + α3 1 CARS ln lopt = (4.219) . ∆α α4 With α1 ≈ α2 ≈ α3 ≈ α4 = α, (4.219) yields ln 3 . (4.220) α Then, setting ∆k = 0 for phase matching and w0 = 0.73a for the best matching of input beams with the fiber mode radius, assuming that the refractive index of the gas filling the fiber core is approximately equal to unity, and taking into consideration that M = (31/2 − CARS = 3−1/2 )2 /(3 ln 3)2 ≈ 0.123 for ∆k = 0 and l = lopt ln 3/α, we arrive at the following expression for the waveguide CARS enhancement factor: 2 2 k + k λ . (4.221) µ = 1.3 × 10−3 k k α2 a4 We can now see from (4.221), that the waveguide CARS enhancement factor scales as λ2 /α2 a4 and is limited by fiber losses. We will show in the next section that, due to the physically different mechanism behind light guiding, hollow microstructure fibers allow CARS enhancement factors to be substantially increased with respect to standard, solid-cladding hollow fibers. We will examine also the CARS enhancement factors as functions of the core radius for the fibers of both types and investigate the influence of the phase mismatch. We start with the case of standard, solid-cladding hollow fibers. The magnitude of optical losses for E Hmn modes in such fibers is given by [4.150] u 2 λ2 n 2 + 1 mn α= (4.222) , √ 2π a3 n 2 − 1 where u mn is the eigenvalue of the characteristic equation for the relevant hollow-fiber mode (the mode parameter), n is the refractive index of the fiber cladding, and the refractive index of the gas filling the fiber core is set equal to unity. Plugging optical losses into the CARS enhancement factor by substituting (4.222) into (4.221) with u n = 2.4 for the limiting eigenvalue of the E H11 mode of a hollow fiber, we derive the following expression for the factor of CARS enhancement in a solid-cladding hollow fiber CARS = lopt

Part A 4.8

In the opposite limiting case of loosely focused pump beams, b l, weak absorption and negligible phase mismatches, the phase-matching integral is reduced to

4.8 Waveguide Coherent Anti-Stokes Raman Scattering

204

Part A

Basic Principles and Materials

relative to the tight-focusing regime in the case of exact phase matching: 2 2 a n2 − 1 −2 k + k ρ = 6.1 × 10 2 . (4.223) k k λ n2 + 1

Part A 4.8

Optical losses, which grow with decreasing inner radius a, limit the CARS enhancement, with the factor ρ rapidly lowering with decreasing a for small values of the fiber inner radius. The situation radically changes in the case of a microstructure fiber. The magnitude of optical losses for such fibers, as mentioned above, may be on the order of 1–3 dB/m in the case of fibers with a hollow core diameter of about 15 µm [4.154]. In the case of small inner radii, microstructure fibers provide much higher CARS enhancement factors than solid-core hollow fibers. The CARS enhancement factor in hollow microstructure fibers with the magnitude of optical losses equal to 0.1 and 0.01 cm−1 starts to exceed the CARS enhancement factor in a solid-cladding hollow fiber for core radii less than 20 and 45 µm, respectively. For hollow fibers with small core radii, the factor µ may be several orders of magnitude higher than the enhancement factor ρ. An additional source of radiation losses in hollow fibers is related to radiation energy transfer to higher-order waveguide modes. The efficiency of this nonlinear-optical mode cross-talk process depends on radiation intensity and the mismatch ∆kc of propagation constants of waveguide modes involved in energy exchange. Starting with the standard expression for the propagation constants of E Hmn modes in a hollow fiber, we arrive at the following formula for the coherence length lc = π(2|∆kc |)−1 of the mode cross-talk process: lc =

a2 2π 2 2 , λ u 2 − u 21

(4.224)

where u 2 and u 1 are the parameters of cross-talking fiber modes. The coherence length lc , as can be seen from (4.224), becomes very small for guided modes of high orders, making the efficiencies of energy transfer from the fundamental to very high order modes negligible. For the cross-talk between the lowest order E H11 and E H12 modes, with u 1 ≈ 2.4 and u 2 ≈ 5.5, the coherence length can be estimated as lc ≈ 0.8a2 /λ. The coherence length of such a cross-talk process is typically much smaller than the optimal length for the wave-mixing process (4.220). However, for highintensity pump beams, the efficiency of this cross-talk

process increases [4.155], and the energy lost from the fundamental mode due to the excitation of higherorder modes may become comparable with radiation energy leakage with the characteristic length governed by (4.222). Importantly, the scaling law of the waveguide CARS enhancement factor as a function of the magnitude of optical losses, fiber inner radius, and radiation wavelength differs from a similar scaling law of the waveguide SRS enhancement factor [4.23], η = λ/αa2 . Physically, this difference stems from differences in scattering mechanisms involved in SRS and CARS, with SRS and CARS signals building up in different fashions as functions of the interaction length and pump field amplitudes. The difference in waveguide enhancement factors for SRS and CARS suggests different strategies for optimizing fibers designed to enhance these processes. Phase mismatch, resulting from the difference in propagation constants of guided modes involved in the CARS process, is another important factor limiting the efficiency of CARS in a hollow fiber. In the case of nonzero phase mismatch ∆k, the optimal length for the CARS process can be found from a transcendental equation that immediately follows from (4.218): CARS CARS ∆α sinh ∆αlopt + ∆k sin ∆klopt + (∆α + α4 )

CARS CARS − cosh ∆αlopt =0. × cos ∆klopt (4.225)

Phase mismatch reduces the maximum waveguide CARS enhancement attainable with a hollow microstructure fiber, with the power of the CARS signal becoming an oscillating function of the fiber length. The characteristic period of these oscillations is determined by the coherence length. Oscillations become less pronounced and then completely flatten out as optical losses build up. No oscillations is observed when the attenuation length becomes less than the coherence length. An important option offered by hollow microstructure fibers is the possibility to compensate for the phase mismatch related to the gas dispersion with an appropriate choice of waveguide parameters due to the waveguide dispersion component, scaling as a−2 in the case of a hollow fiber. We have shown in this section that hollow microstructure fibers offer a unique opportunity of implementing nonlinear-optical interactions of waveguide modes with transverse sizes of several microns in a gas medium, opening the ways to improve the

Nonlinear Optics

4.8.2 Four-Wave Mixing and CARS in Hollow-Core Photonic-Crystal Fibers Hollow-core photonic-crystal fibers (PCFs) [4.36, 152, 153] offer new interesting options for high-field physics and nonlinear optics. Waveguide losses can be radically reduced in such fibers relative to standard, solid-cladding hollow fibers due to the high reflectivity of a periodically structured fiber cladding within photonic band gaps (PBGs) [4.152, 153, 156], allowing transmission of high-intensity laser pulses through a hollow fiber core in isolated guided modes with typical transverse sizes of 10–20 µm. Due to this unique property, hollow PCFs can substantially enhance nonlinear-optical processes [4.157], including stimulated Raman scattering [4.154,158,159], four-wave mixing (FWM) [4.160], coherent anti-Stokes Raman scattering (CARS) [4.118], and self-phase modulation [4.161]. Air-guided modes in hollow PCFs can support high-power optical solitons [4.162, 163], allow transportation of high-energy laser pulses for technological [4.68, 164] and biomedical [4.165] applications. In this section, we discuss phase-matched FWM of millijoule nanosecond pulses in hollow PCFs with a period of the photonic-crystal cladding of about 5 µm and a core diameter of approximately 50 µm. We will show that Raman-resonant FWM in large-core hollow

PCFs enhances the potential of waveguide CARS in hollow fibers, providing a convenient sensing tool for condensed-phase species adsorbed on the inner fiber walls and trace-gas detection. Large-core-area hollow PCFs employed in experiments [4.119, 166] were fabricated using a standard procedure, which involves stacking glass capillaries into a periodic array and drawing this preform at a fiberdrawing tower. Several capillaries have been omitted from the central part of the stack, to produce a hollow core of the fiber. While in standard hollow PCFs, the number of omitted capillaries is seven, PCFs used in our experiments had a hollow core in the form of a regular hexagon with each side corresponding to five cane diameters. The inset in Fig. 4.25 shows an image of a hollow PCF with a period of the cladding of approximately 5 mm and a core diameter of about 50 µm. The baking of capillaries forming the photonic-crystal structure, as shown in the image, allows a hollow waveguide with a nearly ideal 50 µm-diameter hexagonal core to be fabricated. It is still to be explored whether this technique can be scaled up to the fabrication of hollow PCFs with even larger core diameters. Transmission spectra Transmission (arb. units) 1.5

1.0

0.5

0.0

400

500

600

700

800

900 λ (nm)

Fig. 4.25 Transmission spectrum of the hollow-core PCF. The inset shows the cross-section view of the PCF with a period of the cladding structure of about 5 µm

205

Part A 4.8

efficiency of nonlinear-optical processes, including fourwave mixing and coherent anti-Stokes Raman scattering, and suggesting the principle for the creation of highly sensitive gas-phase sensors based on nonlinear spectroscopic techniques. Hollow-core microstructure fibers have been demonstrated to allow the waveguide CARS efficiency to be substantially increased as compared to standard, solid-cladding hollow fibers. The theorem predicting an l 2 /a4 enhancement for a waveguide CARS process in a hollow fiber with an inner radius a and length l has been extended to include new solutions offered by microstructure fibers. The maximum CARS enhancement in a hollow microstructure fiber was shown to scale as λ2 /α2 a4 with radiation wavelength λ, radiation losses α, and the inner fiber radius, allowing CARS efficiency to be substantially improved in such a fiber. This λ2 /α2 a4 CARS enhancement factor differs from the λ/αa2 ratio, characterizing waveguide SRS enhancement in a hollow microstructure fiber, which is related to the difference in the physical nature of SRS and CARS signals and suggests different strategies for optimizing fibers designed to enhance CARS and SRS processes.

4.8 Waveguide Coherent Anti-Stokes Raman Scattering

206

Part A

Basic Principles and Materials

Part A 4.8

of hollow PCFs employed in our experiments display well-pronounced passbands (Fig. 4.25), indicating the PBG guidance of radiation in air modes of the fiber. The laser system used in experiments [4.119, 166] consisted of a Q-switched Nd:YAG master oscillator, Nd:YAG amplifiers, frequency-doubling crystals, a dye laser, as well as a set of totally reflecting and dichroic mirrors and lenses adapted for the purposes of CARS experiments. The Q-switched Nd:YAG master oscillator generated 15 ns pulses of 1.064 µm radiation, which were then amplified up to about 30 mJ by Nd:YAG amplifiers. A potassium dihydrogen phosphate (KDP) crystal was used for the frequency doubling of the fundamental radiation. This secondharmonic radiation served as a pump for the dye laser, generating frequency-tunable radiation within the wavelengths ranges 540–560 and 630–670 nm, depending on the type of dye used as the active medium for this laser. All the three outputs of the laser system, viz., the fundamental radiation, the second harmonic, and frequency-tunable dye-laser radiation, were employed as pump fields in FWM, as described below. The frequency dependencies of the anti-Stokes signals produced through different FWM processes were measured point by point by scanning the frequency of dye-laser radiation. The energies of these pump fields were varied in our experiments from 0.5 up to 10 mJ at the fundamental wavelength, from 0.5 to 8 mJ in the second harmonic, and from 0.05 to 0.7 mJ for dye-laser radiation. To couple the laser fields into the fundamental mode of the PCF, we focused laser beams into spots with a diameter of 35 µm at the input end of the fiber. The PCF could withstand the energy of fundamental radiation up to 10 mJ, corresponding to a laser fluence of approximately 630 J/cm2 , without an irreversible degradation of fiber performance because of optical breakdown. Laser-induced breakdown on PCF walls was judged by a dramatic irreversible reduction in fiber transmission and intense sideward scattering of laser radiation, visible through the fiber cladding. While the achieved level of input energies was sufficient to produce reliably detectable FWM signals in our experiments, a further increase in the laser radiation energy coupled into the PCF is possible through a more careful optimization of the coupling geometry. FWM processes with the CARS-type frequencymixing scheme ωa = 2ω1 − ω2 (ω1 and ω2 are the frequencies of the pump fields and ωa is the frequency of the anti-Stokes signal produced through FWM) were studied in our experiments for two different sets of pump and signal frequencies. In the first FWM process, used

in our experiments to test phase matching and assess the influence of waveguide losses, two waves with the wavelength λ1 = 2πc/ω1 ranging from 630 to 665 nm, provided by the dye laser, are mixed with the fixedfrequency field of the fundamental radiation at λ2 = 1064 nm, to generate an anti-Stokes signal within the range of wavelengths λa from 445 to 485 nm. The second FWM process, designed to demonstrate the potential of CARS spectroscopy with hollow PCFs, is a standard Nd:YAG-laser CARS arrangement with λ1 = 532 nm and λ2 ranging from 645 to 670 nm. To assess the influence of phase matching and radiation losses on the intensity of the FWM signal generated in a hollow PCF, we use (4.213), (4.218), and (4.225) to write the power of the (3) anti-Stokes signal as Pa ∝ |χeff |2 P1 P22 M, where P1 and P2 are the powers of the fields with frequen(3) cies ω1 and ω2 , respectively; χeff is the effective combination of cubic nonlinear-optical susceptibility tensor components; and the factor M includes optical losses and phase-mismatch effects: M(∆αl, αa l, δβl) = exp[(∆α + αa )l][sinh2 (∆αl/2) + sin2 (δβl/2)][(∆αl/2)2 + (δβl/δβl/2)2 ]−1l 2 , where ∆α = (2α1 + α2 − αa )/2, α1 , α2 , and αa are the magnitudes of optical losses at frequencies ω1 , ω2 , and ωa , respectively, and δβ is the mismatch of the propagation constants of waveguide modes involved in the FWM process. In order to provide an order of magnitude estimate on typical coherence lengths lc = π/(2|δβ|) for FWM processes in hollow PCFs and to choose PCF lengths L meeting the phasematching requirement L ≤ lc for our experiments, we substitute the dispersion of a standard hollow fiber with a solid cladding for the dispersion of PCF modes in these calculations. As shown by earlier work on PBG waveguides [4.167], such an approximation can provide a reasonable accuracy for mode dispersion within the central part of PBGs, but fails closer to the passband edges. For the waveguide FWM process involving the fundamental modes of the pump fields with λ1 = 532 nm and λ2 = 660 nm, generating the fundamental mode of the anti-Stokes field in a hollow fiber with a core radius of 25 µm, the coherence length is estimated as lc ≈ 10 cm. Based on this estimate, we choose a fiber length of 8 cm for our FWM experiments. With such a choice of the PCF length, effects related to the phase mismatch can be neglected as compared with the influence of radiation losses. Phase matching for waveguide CARS in the PCF was experimentally tested by scanning the laser frequency difference ω1 − ω2 off all the Raman resonances (with λ1 ranging from 630 to 665 nm and λ2 = 1064 nm)

Nonlinear Optics

FWM intensity (arb. units) Transmission Signal wavelength (nm) 450 455 460 465 470 475 480 12

1.0 2

10 8 4

0.5 1

4 2 0

3 630

635

640

645

650 655 660 665 Dye-laser wavelength (nm)

0.0

Fig. 4.26 Intensity of the ωa = 2ω1 − ω2 four-wave mixing

stretching vibrations of water molecules typically fall within a broad frequency band of 3200–3700 cm−1 . The frequency dependence of the FWM signal from the PCF substantially deviates from the spectral profile of the factor M(∆αL, αa L, 0) (cf. lines 1 and 4 in Fig. 4.27), clearly indicating the contribution of Ramanactive species to the FWM signal. To discriminate between the CARS signal related to water molecules adsorbed on the PCF walls against the OH contamination of the PCF cladding, we measured the spectrum of the CARS signal from a PCF heated above a burner. Heating by 30 K during 30 min reduced the amplitude of the Raman resonance in the spectrum of the CARS signal by a factor of about seven. The high level of the CARS signal was then recovered within several days. This spectrum of the CARS signal from the dry PCF was subtracted from the CARS spectrum recorded at the output of the hollow PCF under normal conditions. The difference spectrum was normalized to the spectral profile of the factor M(∆αL, αa L, 0). The result of this normalization is shown by line 5 in Fig. 4.27. FWM intensity (arb. units) 452

and using the above expression for M(∆αl, αa l, δβl) with δβL ≈ 0 to fit the frequency dependence of the FWM signal. Dots with error bars (line 1) in Fig. 4.26 present the intensity of the anti-Stokes signal from hollow PCFs measured as a function of the frequency of the dye laser. Dashed lines 2 and 3 in this figure display the transmission of the PCF for dye-laser radiation and the anti-Stokes signal, respectively. Solid line 4 presents the calculated spectral profile of the factor M(∆αL, αa L, 0). Experimental frequency dependencies of the FWM signals, as can be seen from the comparison of lines 1 and 4 in Fig. 4.26, are fully controlled by the spectral contours of PCF passbands (lines 2 and 3), indicating that phase-mismatch effects are much less significant for the chosen PCF lengths than variations in radiation losses. The second series of experiments was intended to demonstrate the potential of waveguide CARS in a hollow PCF for the sensing of Raman-active species. For this purpose, the frequency difference of the secondharmonic and dye-laser pump fields was scanned through the Raman resonance, ω1 − ω2 = 2πcΩ, with O−H stretching vibrations of water molecules, adsorbed on the inner PCF walls. The frequencies Ω of O−H

Transmission

Signal wavelength (nm) 450 448 446 444

442 1.0

2 1 0.5

5 4

3 0 650

655

0.0 660 665 670 Dye-laser wavelength (nm)

Fig. 4.27 Intensity of the ωa = 2ω1 − ω2 four-wave mixing

signal from the hollow PCF with the length of 8 cm versus the wavelength of dye-laser radiation with λ1 = 532 nm and λ2 ranging from 645 to 670 nm: (1) the measured spectrum of the FWM signal, (2) fiber transmission for dye-laser radiation, (3) fiber transmission for the FWM signal, (4) the spectral profile of the factor M and (5) the spectrum of the FWM signal corrected for the factor M upon the subtraction of the spectrum of the CARS signal from the heated hollow PCF

207

Part A 4.8

signal from the hollow PCF with the length of 8 cm versus the wavelength of dye-laser radiation with λ1 ranging from 630 to 665 nm and λ2 = 1064 nm: (1) the measured spectrum of the FWM signal, (2) fiber transmission for dye-laser radiation, (3) fiber transmission for the FWM signal, and (4) the spectral profile of the factor M

4.8 Waveguide Coherent Anti-Stokes Raman Scattering

208

Part A

Basic Principles and Materials

Part A 4.8

Notably, the contrast of the experimental wavelength dependence of the FWM intensity (squares with error bars) in Fig. 4.26 is higher than the contrast of a similar dependence for the CARS signal in Fig. 4.27. This variation in the ratio of the maximum amplitude of the nonlinear signal correlates well with the behavior of transmission for dye-laser radiation and the nonlinear signal, shown by curves 2 and 3 in both figures. With the dye-laser radiation wavelength set around 650 nm, both the pump and nonlinear signal wavelengths λ1 and λa in Fig. 4.26 are close to the respective maxima of PCF transmission. The CARS signal, on the other hand, is detected away from the maximum transmission for the dye-laser radiation and the nonlinear signal (Fig. 4.27). It is therefore important to normalize the measured CARS spectrum to the wavelength dependence of the M factor, taking into account wavelength-dependent losses introduced by the PCF. This normalization procedure considerably improves the contrast of the CARS spectrum, as shown by curve 5 in Fig. 4.27. Experiments presented above demonstrate the potential of waveguide CARS in PCFs to detect trace concentrations of Raman-active species, suggesting PCF CARS as a convenient diagnostic technique. However, CARS signals detected in these experiments do not allow the origin of Raman-active species in the fiber to be reliably identified, as it is not always possible to discriminate between the contributions to the CARS signal provided by the hollow core and PCF walls. As an example of a more easily quantifiable Raman medium, permitting the CARS signal from the PCF core to be separated from the signal from PCF walls, we chose gas-phase molecular nitrogen from atmosphericpressure air filling the hollow core of a PCF. A two-color Raman-resonant pump field used in these experiments consisted of 15 ns second-harmonic pulses of Nd:YAG laser radiation with a wavelength of 532 nm (ω1 ) and dye-laser radiation (ω2 ) with a wavelength of 607 nm. The dye-laser frequency was chosen in such a way as to satisfy the condition of Raman resonance ω1 − ω2 = Ω with a Q-branch Raman-active transition of N2 at the central frequency Ω = 2331 cm−1 . Coherently excited Q-branch vibrations of N2 then scatter off the secondharmonic probe field, giving rise to a CARS signal at the frequency ωCARS = 2ω1 − ω2 (corresponding to a wavelength of 473 nm). The hollow PCF (shown in inset 1 to Fig. 4.28) was designed to simultaneously provide high transmission for the air-guided modes of the second harmonic, dye-laser radiation, and the CARS signal. With an appropriate fiber structure, as can be

CARS intensity (arb. units) Transmission (arb. units) 1

473 532 607nm

0.1

400

500

1.0

σβ (cm–1) 4

600

700

800 λ (nm)

2 0 –2 2320

2330

0.5

0.0

2324

2340 λ (nm)

2331

2338 Ω (cm–1)

Fig. 4.28 CARS spectrum of Q-branch Raman-active vibrations of N2 molecules in the atmospheric-pressure air filling the hollow core of the PCF. The insets show: (1) an image of the PCF cross section; (2) the transmission spectrum of the hollow PCF designed to simultaneously support air-guided modes of the pump, probe, and CARS signal fields (their wavelengths are shown by vertical lines) in the hollow core of the fiber; and (4.3) the mismatch of the propagation constants δβ = 2β1 − β2 − βa calculated for the ωa = 2ω1 − ωa 2 CARS process in fundamental air-guided modes of the hollow PCF with β1 , β2 , and βa being the propagation constants of the Nd:YAG-laser second-harmonic (ω1 ) and dye-laser (ω2 ) pump fields and the CARS signal (ωa ) in the hollow PCF

seen from inset 2 to Fig. 4.28, PCF transmission peaks can be centered around the carrier wavelengths of the input light fields and the CARS signal (shown by vertical lines in inset 2 to Fig. 4.28). Phase matching for CARS with the chosen set of wavelengths has been confirmed [4.168] (inset 3 in Fig. 4.28) by a numerical analysis of PCF dispersion based on a modification of the field-expansion technique developed by Poladian et al. [4.169].

Nonlinear Optics

The resonant CARS signal related to Q-branch vibrations of N2 in these experiments can be reliably separated from the nonresonant part of the CARS signal originating from the PCF walls. The spectra of the CARS signal at the output of the PCF Fig. 4.28 are identical to the N2 Q-branch CARS spectrum of the atmospheric air [4.99] measured in the tight-focusing regime. In view of this finding, the CARS signal can be completely attributed to the coherent Raman scattering in the gas filling the fiber core with no noticeable contribution from the nonlinearity of PCF walls. Results presented here show that large-core-area hollow PCFs bridge the gap between standard, solidcladding hollow fibers and hollow PCFs in terms of

4.9 Nonlinear Spectroscopy with Photonic-Crystal-Fiber Sources

209

effective guided-mode areas, allowing energy fluence scaling of phase-matched waveguide four-wave mixing of laser pulses. We used hollow PCFs with a core diameter of about 50 µm to demonstrate phase-matched FWM for millijoule nanosecond laser pulses. Intense CARS signal has been observed from stretching vibrations of water molecules inside the hollow fiber core, suggesting CARS in hollow PCFs as a convenient sensing technique for pollution monitoring and trace gas detection. Hollow PCFs have been shown to offer much promise as fiber-optic probes for biomedical Raman applications, suggesting the way to substantially reduce the background related to Raman scattering in the core of standard biomedical fiber probes [4.170].

4.9 Nonlinear Spectroscopy with Photonic-Crystal-Fiber Sources

The progress in wavelength-tunable light sources through the past decades has been giving a powerful momentum to the development of nonlinear laser spectroscopy. Nonlinear Raman spectroscopy, in particular, has benefited tremendously more than 30 years ago from the application of tunable laser sources, as optical parametric oscillators [4.171] and dye lasers [4.172] were demonstrated to greatly simplify measurements based on coherent anti-Stokes Raman scattering (CARS), making this technique much more informative, efficient, and convenient. Broadband laser sources later contributed to the technical and conceptual progress in nonlinear Raman spectroscopy [4.99, 106–110], allowing singleshot CARS measurements. In the era of femtosecond lasers, several parallel trends have been observed in the development of laser sources for nonlinear Raman spectroscopy [4.113]. One of these tendencies was to adapt broadband femtosecond pulses for spectroscopic purposes [4.113, 173, 174] and to use different spatial phase-matching geometries to simultaneously generate coherent Stokes and anti-Stokes, as well as degenerate four-wave mixing signals [4.113, 127, 128]. The rapid progress in nonlinear materials, on the other hand, resulted in the renaissance of optical parametric oscillators and amplifiers (OPOs and OPAs) for nonlinear spectroscopy [4.175]. Chirped pulses were used [4.176,177] to probe broad spectral regions and large ranges of delay times, suggesting efficient single-shot nonlinear spectroscopic approaches [4.178–180].

In this section, we focus on the potential of photoniccrystal fibers [4.36,37] as novel efficient sources for nonlinear spectroscopy. PCFs are unique waveguide structures allowing dispersion [4.40] and spatial field [4.181] profiles to be engineered by modifying the design of the fiber structure. Nonlinear-optical PCF components and novel PCF-based light sources have been intensely used through the past few years in frequency metrology [4.74, 77], biomedical optics [4.81], ultrafast photonics [4.78, 79], and photochemistry [4.182]. Coherent nonlinear spectroscopy and microscopy open up a vast area for applications of PCF light sources and frequency shifters. Efficient frequency conversion and supercontinuum generation in PCFs have been shown to enhance the capabilities of chirped-pulse CARS [4.183] and coherent inverse Raman spectroscopy [4.184]. Cross-correlation frequency-resolved optically gated CARS (XFROG CARS) has recently been demonstrated [4.80] using specially designed PCF frequency converters for ultrashort laser pulses. Novel light sources based on frequency shifting in PCFs provide a useful tool for the measurement of second-order optical nonlinearities in organic materials [4.185] and offer interesting new options in CARS microscopy [4.186]. Efficient spectral broadening of ultrashort pulses in PCFs with carefully engineered dispersion profiles [4.42–44] makes these fibers ideal light sources for pump–supercontinuum probe time- and frequency-resolved nonlinear-optical measurements [4.187]. Below in this section, we demonstrate applications of PCF light sources for chirped-pulse CARS

Part A 4.9

4.9.1 Wavelength-Tunable Sources and Progress in Nonlinear Spectroscopy

210

Part A

Basic Principles and Materials

and nonlinear absorption spectroscopy. We will show that PCFs can provide efficient nonlinear-optical transformations of femtosecond Cr:forsterite laser pulses, delivering linearly chirped frequency-shifted broadband light pulses with central wavelengths ranging from 400 to 900 nm. These pulses were cross-correlated in our experiments with the femtosecond second-harmonic output of the Cr:forsterite laser in toluene solution, used as a test object, in boxcars geometry to measure CARS spectra of toluene molecules (XFROG CARS). The blue-shifted chirped-pulse output of a photoniccrystal fiber with a spectrum stretching from 530 to 680 nm is shown to be ideally suited for the nonlinear absorption spectroscopy of one- and two-exciton bands of thiacarbocyanine J aggregates in a polymer film excited by femtosecond second-harmonic pulses of the Cr:forsterite laser.

b) PCF output (arb. units) 1

4 1 3

Part A 4.9

4.9.2 Photonic-Crystal Fiber Frequency Shifters Spectroscopic measurements were performed with multicomponent-glass PCFs [4.51,188], fabricated with the use of the standard PCF technology [4.36, 37]. In PCFs used in our experiments (Fig. 4.29a), the solid fiber core is surrounded by a single ring of thin-wall capillaries whose outer diameters are equal to the diameter of the PCF core. The outer part of the microstructure cladding consists of 11 rings of capillaries with outer diameters approximately three times larger than the diameter of the PCF core and a high air-filling fraction. Dispersion and nonlinearity of the PCFs were managed by scaling the geometric sizes of the PCF structure. Technologically, this was realized by using the same preform to fabricate PCFs with the same type of the structure, but with different magnifying factors. This procedure allowed to scale the sizes of PCF structure without changing its geometry. The laser system used in experiments consisted of a Cr4+ :forsterite master oscillator, a stretcher, an optical isolator, a regenerative amplifier, and a compressor [4.189]. The master oscillator, pumped with a fiber ytterbium laser, generated 30–60 fs light pulses of radiation with a wavelength of 1.23–1.25 µm at a repetition rate of 120 MHz. These pulses were then transmitted through a stretcher and an isolator, to be amplified in a Nd:YLF-laser-pumped amplifier and recompressed to the 200 fs pulse duration with the maximum laser pulse energy up to 20 µJ at 1 kHz. The energy of laser pulses used in experiments presented in this paper ranged from 0.5 up to 200 nJ.

2 0.1

0.01 400

500

600

700

900 800 Wavelength (nm)

Fig. 4.29 (a) An SEM image of soft-glass photonic-crystal fibers. (b) Intensity spectra of the dispersion-managed blue-

shifted output of soft-glass PCFs. The wavelength offset δ is 50 nm (1), 110 nm (2), 150 nm (3), 190 nm (4), 220 nm (5), and 300 nm (6)

For the PCFs used in experiments, the central wavelength of Cr:forsterite laser pulses falls within the range of anomalous dispersion. Such pulses can therefore form solitons in the fiber. High-order dispersion induces wave-matching resonances between solitons and dispersive waves [4.21, 22], giving rise to intense blueshifted emission, observed as prominent features in the PCF output spectra (Fig. 4.29b). The central wavelength of this blue-shifted emission correlates well with the wavelength where dispersive waves guided in the fundamental mode of the PCF are phase-matched with the soliton produced by Cr:forsterite laser pulses. The phase matching between the soliton produced by the input laser pulse and the dispersive wave, which defines the frequency of the dominant peaks in output

Nonlinear Optics

4.9.3 Coherent Anti-Stokes Raman Scattering Spectroscopy with PCF Sources In this section, we show that PCF frequency shifters can serve as convenient sources of chirped wavelengthtunable pulses for CARS spectroscopy. In CARS

Cr: forsterite Laser

211

Objective LBO

Delay line PCF MS fiber Lens

Filter Objective

Raman medium Spectrograph CARS Lens signal

Fig. 4.30 Diagram of femtosecond CARS spectroscopy with the use of ultrashort pulses frequency-upconverted and chirped in a photonic-crystal fiber. The inset shows the cross-sectional view of the central part of the photoniccrystal fiber

experiments, sub-microjoule Cr:forsterite-laser pulses with an initial duration of about 90 fs are launched into the central core of the PCF (Fig. 4.30), resulting in the efficient generation of a blue-shifted signal (Fig. 4.31), with a central wavelength dictated by phase matching for dispersive-wave emission and controlled by fiber dispersion. The wavelength of the blue-shifted signal can be finely tuned by changing the intensity of the pump pulse (Fig. 4.31) due to the nonlinear change in the refractive index of the fiber core and the spectral broadening of the pump pulse. Cross-correlation frequency-resolved optical gating (XFROG) [4.190, 191] was used to characterize the blue-shifted output of the PCF. An XFROG signal was generated by mixing the blue-shifted signal from the fiber E a with the 620 nm 90 fs secondharmonic output of the Cr:forsterite laser E SH in a BBO crystal. A two-dimensional XFROG spectrogram, ∞ S(ω, τ) ∝ | −∞ E a (t) E SH (t − τ) exp(−iωt) dt|2 , was then plotted by measuring the XFROG signal as a function of the delay time τ between the second-harmonic pulses and the blue-shifted output of the PCF and spectrally dispersing the XFROG signal. The XFROG spectrogram shown in inset 1 to Fig. 4.31 visualizes

Part A 4.9

spectra of PCFs, is controlled by the dispersion of the fiber. The central frequency of the blue-shifted signal in the output of PCFs can thus be tuned by modifying the dispersion profile of the fiber. The GVD profiles of waveguide modes were modified in our experiments by scaling the geometric sizes of the fiber without changing the type of the structure shown in Fig. 4.29a. The core diameter of PCFs fabricated for these experiments was varied from 0.9 to 3.8 µm. Figure 4.29b shows the spectra of the blue-shifted output of the fiber, tuned by changing the offset δ = λ0 − λz between the central wavelength of the input laser field, λ0 , and the zeroGVD wavelength λz . With the PCF length remaining unchanged (10 cm), larger blue shifts are achieved by increasing the offset δ (cf. curves 1–6 in Fig. 4.29b). The power of the input laser field needs to be increased for larger δ in these experiments to keep the amplitude of the blue-shifted signal constant. In these experiments, dispersion-managed soft-glass PCFs are shown to serve as frequency shifters of femtosecond Cr:forsterite laser pulses, providing an anti-Stokes output tunable across the range of wavelengths from 400 to 900 nm. Experiments presented in this section show that the structural dispersion and nonlinearity management of multicomponent-glass photonic-crystal fibers allows a wavelength-tunable frequency shifting and white-light spectral transformation of femtosecond Cr:forsterite laser pulses. We have explored the ways toward optimizing non-silica PCFs for frequency shifting and white-light spectral superbroadening of femtosecond Cr:forsterite laser pulses and identified important advantages of multicomponent-glass PCFs over silica microstructure fibers for the spectral transformation of laser pulses in the 1.2–1.3 µm spectral range. By coupling 200 fs pulses of 1.24 µm Cr:forsterite laser radiation into different types of multicomponent-glass PCFs, where the zero-GVD wavelength is tuned by scaling the sizes of the fiber structure, we have demonstrated spectrally tailored supercontinuum generation and frequency upshifting providing a blue-shifted output tunable across the range of wavelengths from 400 to 900 nm.

4.9 Nonlinear Spectroscopy with Photonic-Crystal-Fiber Sources

212

Part A

Basic Principles and Materials

a) ω

Intensity (arb. units) λ (nm)

λ (nm)

1 2

1.0

340

0.8 0.6

330

310 –10

580 576 –7 –4 –1

0.0

310 –10 –5 0

10 τ/100 (fs)

Phase (rad) 40

1.0 0.8

3 0.4 0.0 – 2000

c) ˜ | χ | 1000

584

100

– 20

2000 t (fs)

Part A 4.9

0.5

d) 588

2 5 τ/100 (fs)

5 10 15 τ/100 (fs) (3) 2

Intensity (arb. unit) 1.2

1.0 0.8 0.6 0.4 0.2 0.0

Ω 588 584

320

0.2

592

330

0.4 320

ω1

b) λCARS (nm)

ω2

580 576 –4

–1

2 τ/100 (fs)

0.1 950 1000 1050 1100 (ω1 – ω2) / 2πc (cm–1)

Fig. 4.32 (a) A diagram of femtosecond CARS spec0.0

600

700

800

900 λ (nm)

Fig. 4.31 The spectrum of the blue-shifted output of

a photonic-crystal fiber pumped by 1.24 µm 90 fs Cr:forsterite-laser pulses with an input energy of (solid line) 170 nJ, (dashed line) 220 nJ, and (dash–dotted line) 270 nJ. The insets show: (1) the intensity of the sum-frequency signal generated in a BBO crystal by the second-harmonic pulse from the Cr:forsterite laser and the blue-shifted PCF output as a function of the wavelength and the delay time τ between the second-harmonic and anti-Stokes pulses, (2) theoretical fit of the XFROG trace, and (3) the pulse envelope and the phase of the anti-Stokes pulse providing the best fit

the temporal envelope, the spectrum, and the chirp of the blue-shifted PCF output. A reasonable fit of the experimental XFROG trace was achieved (inset 2 in Fig. 4.31) with a blue-shifted pulse having a duration of about 1 ps and a linear positive chirp corresponding to the phase ϕ(t) = αt 2 with α = 110 ps−2 . The spectrum and the phase ϕ of the blue-shifted pulse reconstructed with this procedure are shown in inset 3 in Fig. 4.31.

troscopy with chirped pulses is shown on the left. The first pulse (frequency ω1 ) is transform limited. The second pulse (frequency ω2 ) is linearly chirped. The linear chirp of the second pulse maps the delay time between the pulses on the frequency, allowing the frequency difference ω1 − ω2 to be scanned through Raman resonances Ω by tuning the delay time τ. (b) The intensity of the CARS signal generated in a noncoplanar boxcars geometry from toluene solution as a function of the wavelength and the delay time τ between the second-harmonic and the blue-shifted PCF output pulses used as a biharmonic pump. (c) A model of the Raman spectrum, including a doublet of Lorentzian lines and a frequency-independent nonresonant background used in the theoretical fit. (d) Theoretical fit of the XFROG CARS spectrogram

The linear chirp defines a simple linear mapping between the instantaneous frequency of the blue-shifted PCF output and the delay time τ, allowing spectral measurements to be performed by varying the delay time between the pump pulses. Experiments were performed with 90 fs second-harmonic pulses of the Cr:forsterite laser (at the frequency ω1 ) and the linearly chirped pulses from the PCF (the frequency ω2 ) as a biharmonic pump for the CARS spectroscopy of toluene solution. The frequency difference ω1 − ω2 was scanned

Nonlinear Optics

213

Although the slope of the XFROG CARS trace and positions of Raman peaks can be adequately described with the use of this simple model, some of the spectroscopic features of the experimental XFROG CARS trace deviate from the theoretical fit. These deviations may originate from variations in the coherent background as a function of the frequency. For a quantitative spectroscopic analysis, these inhomogeneities in the frequency dependence of the coherent background should be carefully measured and included in the fit. On the other hand, the nonresonant contribution to CARS spectral profiles can be efficiently suppressed [4.99, 106] by using three input pulses in the CARS arrangement instead of two and by introducing the delay time between the third pulse (probe) and the two-color pump, tuned to a Raman resonance under study.

4.9.4 Pump-Probe Nonlinear Absorption Spectroscopy using Chirped Frequency-Shifted Light Pulses from a Photonic-Crystal Fiber Time-resolved nonlinear-optical spectroscopy of molecular dynamics and fast excitation transfer processes typically involves specifically designed sequences of pump and probe pulses with a variable delay time and a smoothly tunable frequency of the probe field. In a laboratory experiment, such pulse sequences can be generated by femtosecond optical parametric amplifiers (OPAs). Femtosecond OPAs, however, inevitably increase the cost of laser experiments and make the laser system more complicated, unwieldy, and difficult to align. An interesting alternative strategy of pump–probe spectroscopy employs a pump field in the form of a broadband radiation (supercontinuum) with a precisely characterized chirp. A time–frequency map, defined by the chirp of such a supercontinuum pulse, allows time- and frequency-resolved measurements to be performed by tuning the delay time between the transform-limited pump pulse and chirped supercontinuum pulse [4.176,193]. The supercontinuum probe pulse for pump–probe experiments is most often generated by focusing amplified femtosecond pulses into a silica or sapphire plate. The chirp of the probe pulse in such an arrangement is dictated by the regime of nonlinear-optical spectral transformation and dispersion of the nonlinear material, leaving little space for the phase tailoring of the probe field. In this section, we demonstrate the potential of PCFs as compact and cost-efficient fiber-optic sources

Part A 4.9

through the frequencies of Raman-active modes of toluene molecules by tuning the delay time between the pump pulses (Fig. 4.32a). The second-harmonic pulse also served as a probe in our CARS scheme, generating the CARS signal at the frequency ωCARS = 2ω1 − ω2 through the scattering from Raman-active vibrations coherently excited by the pump fields. The light beams with frequencies ω1 and ω2 were focused into a cell with toluene solution at a small angle with respect to each other (Fig. 4.30). The CARS signal generated in the area of beam interaction in this non-coplanar boxcars geometry had a form of a sharply directed light beam with a low, phase-matching-controlled angular divergence spatially separated (Fig. 4.30) from the pump beams. Figure 4.32b presents the map of CARS spectra from the toluene solution measured for different delay times τ between the biharmonic pump pulses. This procedure of measurements, in fact, implements the XFROG technique [4.190, 191]. However, while FROG-based techniques [4.192] are usually employed to characterize ultrashort pulses, our goal here is to probe Raman-active modes of toluene molecules, used as a test object, by means of CARS spectroscopy. In the case of a positively chirped pulse from the PCF (insets 2 and 3 in Fig. 4.31), small delay times τ correspond to the excitation of low-frequency Raman-active modes (τ ≈ −200 fs in Fig. 4.32b). In particular, the 1004 cm−1 Raman mode of toluene is well resolved in the presented XFROG CARS spectrogram. This mode is excited with the second harmonic of Cr:forsterite-laser radiation and the spectral slice around the wavelength of λ2 ≈ 661 nm picked with an appropriate τ (Fig. 4.32a) from the positively chirped blue-shifted PCF output, giving rise to a CARS signal with the wavelength λCARS ≈ 584 nm. Raman modes with higher frequencies are probed at larger delay times (τ ≈ 100–200 fs in Fig. 4.32b). As a simple model of the Raman spectrum of toluene molecules in the studied spectral range, we employ a doublet of Lorentzian lines interfering with the coherent nonresonant background (Fig. 4.32c). The frequencies of the peaks in these spectra are used as fitting parameters. The ratio of the peak value of the resonant part of the CARS susceptibility, χ¯ (3) , to the nonresonant susceptibility, χ (3)nr , was estimated by measuring the intensities of the CARS signal on and off the Raman resonances, yielding |χ (3)nr |/|χ¯ (3) | ≈ 0.05. The best fit (Fig. 4.32d) is achieved with the Raman peaks centered at 1004 and 1102 cm−1 , which agrees well with earlier CARS studies of toluene [4.99].

4.9 Nonlinear Spectroscopy with Photonic-Crystal-Fiber Sources

214

Part A

Basic Principles and Materials

Part A 4.9

of probe pulses with a tunable frequency and tailored phase for a time-resolved nonlinear spectroscopy of molecular aggregates. We will show that PCFs can provide efficient nonlinear-optical transformations of femtosecond Cr:forsterite laser pulses, delivering linearly chirped frequency-shifted broadband light pulses, optimized for pump–probe nonlinear absorption spectroscopy of molecular aggregates. The blue-shifted output of a photonic-crystal fiber with a spectrum stretching from 530 to 680 nm will be used to probe one- and two-exciton bands of thiacarbocyanine J aggregates in a polymer film excited by femtosecond second-harmonic pulses of the Cr:forsterite laser. Molecular aggregates are interesting and practically significant objects encountered in many physical, chemical and biological systems [4.194]. Interactions between the molecules forming aggregates give rise to collective electronic states, which can be delocalized over large chains of molecules, modifying the optical response of the system [4.195]. Specific types of molecular aggregation, known as J and H aggregation, give rise to a pronounced spectral shift and dramatic narrowing of absorption bands, indicating the cooperative nature of the optical response of molecular aggregates [4.196]. In natural systems, molecular aggregates are involved in the processes and functions of vital importance as they play the key role in light harvesting and primary charge separation in photosynthesis [4.197]. On optical resonances, the nonlinear susceptibility of molecular aggregates displays a collective enhancement [4.195], scaling as N 2 with the number of particles N forming an aggregate. Dramatic enhancement of optical nonlinearity and the available pathways for ultrafast relaxation of excited states [4.194] suggest a variety of interesting applications of molecular aggregates, such as terahertz demultiplexing of optical signals [4.198], spectral sensitization in optical data storage and photography [4.199], energy transfer from light-harvesting antennas and complexes in artificial photosynthesis [4.197], mode locking in laser cavities, and creation of novel devices for ultrafast photonics. Collective electronic eigenstates in aggregates of strongly coupled molecules are grouped into excitonic bands [4.194–196]. In the ground state of this band structure, all N molecules of the aggregate reside in the ground state. In the lowest excited band, the molecules coupled into an aggregate share one excitation. The eigenstates of this first excited band (one-excitons, or Frenkel excitons) are represented by linear combinations of basis states corresponding to one excited and N − 1 ground-state molecules. Eigenstates that can be

reached from the ground state via two optical transitions, making the molecules of an aggregate share two excitations, form a two-exciton band. The third-order susceptibility χ (3) , responsible for four-wave mixing (FWM) processes, can involve only one- and twoexcitons. Although three-exciton states can be reached Intensity (arb. units)

Optical density (arb. units)

1.0

0.5

0.5 2

0.0

500

600

700

800 λ (nm)

0.0

Two-exciton band

Chirped probe

One-exciton band

fs pump

Ground state

Fig. 4.33 (a) Absorption spectrum of the thiacarbocya-

nine film (1), spectrum of the second-harmonic output of the Cr:forsterite laser (2), and spectrum of the blueshifted PCF output (3). (b) Diagram of the pump–probe nonlinear-absorption spectroscopy of exciton bands in molecular aggregates using a femtosecond (fs) pump pulse and a broadband chirped probe field

Nonlinear Optics

measured by the pump–probe technique for thiacarbocyanine J aggregates in a polymer film with a delay time between the probe and pump pulses equal to (a) 100 fs, (b) 500 fs, and (c) 1100 fs

0.1

0.0

– 0.1

630

700

770

b) 0.04 0.02

– 0.02 – 0.04 550

600

650

700

750

550

600

650

700

750 λ (nm)

c) 0.03 0.02 0.01 0.00 – 0.01 – 0.02

by three transitions, they do not contribute to the thirdorder polarization as they do not have a transition dipole that would couple them to the ground state. Timeresolved nonlinear spectroscopic methods based on χ (3) processes, such as spectroscopy of nonlinear absorption, degenerate four-wave mixing, and third-harmonic generation, have proven to be convenient and infor-

mative techniques for the characterization of one- and two-exciton states of aggregates, providing the data on the strength of dipole–dipole interaction between molecules in aggregates, as well as on the disorder and typical relaxation and exciton annihilation times in aggregates [4.194]. Experiments [4.200] were performed with thin-film samples of J aggregates of thiacarbocyanine dye. Film samples of J aggregates were prepared by spin-coating the solution of thiacarbocyanine dye on a thin substrate. The concentration of thiacarbocyanine dye was 5 × 10−3 mol/l. The spinning speeds ranged from 1000 up to 3000 revolutions per minute. A mixture of acetonitrile, dichloroethane, and chloroform with a volume ratio of 2:2:1 was used as a solvent. The thickness of the dye layer applied to a substrate was estimated as 30 nm, and the total thickness of the sample was about 1 µm. Absorption spectra of J-aggregate films (curve 1 in Fig. 4.33a) display two pronounced peaks. The broader peak centered at 595 nm corresponds to thiacarbocyanine monomers, while the narrower peak at 660 nm represents the excitonic absorption of J aggregates. To provide a rough estimate of the delocalization length Nd of excitons in aggregates, we apply the formula [4.201] NdW ≈ (3π 2 |J|3π 2 |J|W)1/2 − 1, which expresses Nd through the half-width at half-maximum W of the aggregate peak in the absorption spectrum and the energy J of dipole–dipole interaction between nearest-neighbor molecules in the aggregate (J < 0 for J aggregates). With the J parameter estimated as J ≈ 900 cm−1 from the aggregation-induced red shift of the absorption peak in Fig. 4.33a, we find Nd ≈ 6. Experimental results presented below in this paper show that, with such a delocalization length of excitons in molecular aggregates, spectra of nonlinear absorption display well-pronounced nonoverlapping features indicating [4.194–196] bleaching through transitions between the ground and one-exciton states and induced absorption via transitions between one- and two-exciton states of molecular aggregates. The laser system used in experiments [4.200] was based on the Cr4+ :forsterite laser source with regenerative amplification, as described in Sect. 4.9.2. A 1 mm-thick BBO crystal was used to generate the second harmonic of amplified Cr:forsterite laser radi-

Part A 4.9

0.00

– 0.03

215

Fig. 4.34a–c Differential spectra of nonlinear absorption

a) Differential absorption (arb. units)

560

4.9 Nonlinear Spectroscopy with Photonic-Crystal-Fiber Sources

216

Part A

Basic Principles and Materials

Part A 4.10

ation. The spectrum of the second-harmonic output of the Cr:forsterite laser was centered at 618 nm (curve 2 in Fig. 4.33a). Second-harmonic pulses with a pulse width of about 120 fs and the energy ranging from 10 to 80 nJ were used as a pump field in our experiments on the nonlinear spectroscopy of molecular aggregates. Frequency-tunable upconversion of fundamentalwavelength Cr:forsterite-laser pulses was performed through the nonlinear-optical spectral transformation of these pulses in soft-glass PCFs with the crosssection structure shown in Fig. 4.29a. The properties of such fibers and the methods of frequency conversion of Cr:forsterite laser pulses in these PCFs have been discussed in Sect. 4.9.2 Intensity spectrum of the frequency-shifted output of the PCF best suited as a probe field for time-resolved nonlinear-absorption spectroscopy of J aggregates is presented by curve 3 in Fig. 4.33a. At the level of 20% of its maximum, the intensity spectrum of the blue-shifted PCF output stretches from 530 to 680 nm. Dispersion of the PCF frequency shifter used in our experiments provided a linear chirp of the output pulse Sect. 4.9.3 with the pulse chirp rate controlled by the fiber length. The spectrum of 120 fs second-harmonic pulses of the Cr:forsterite laser partially overlaps the absorption spectrum of molecular aggregates (Fig. 4.33a). These pulses were used in our experiments to excite the aggregates through the transitions from the ground state to the one-exciton band (Fig. 4.33b). The spectra of absorption modified by the pump field were measured by chirped blue-shifted pulses delivered by the PCF (Fig. 4.33b). Figures 4.34a–c present the results of experimental measurements upon the subtraction of absorption spectra measured in the absence of the pump pulse and normalization to the spectrum of the probe field. Nonlinear absorption spectra shown in Figs. 4.34a–c display well-pronounced minima at 665 nm and blue-shifted peaks at 640 nm. Such features are typical of nonlinear absorption spectra of J aggregates measured by the pump–probe technique (see [4.194] for a review).

The negative feature is indicative of bleaching through pump-induced transitions between the ground state and the one-exciton band, while the blue-shifted peak originates from induced absorption due to transitions between one- and two-exciton bands of molecular aggregates (Fig. 4.34a–c). For highly ordered aggregates, the spectrum of nonlinear absorption is dominated by transitions between the ground state and lowest one- and two-exciton states [4.195, 196]. The exciton delocalization length can be then estimated from the spectral shift ∆ of the induced-absorption peak relative to the bleaching minimum using the following formula [4.202]: Nd∆ ≈ (3π 2 |J|/∆)1/2 − 1. With the spectral shift estimated as ∆ ≈ 470 cm−1 , we find that Nd∆ ≈ 6, in perfect agreement with the value of NdW obtained from aggregate absorption spectra. The amplitudes of both positive and negative features in nonlinear absorption spectra decay on a sub-picosecond time scale with increasing delay time between the pump and probe pulses (Fig. 4.34a–c), indicating a sub-picosecond relaxation rate of the oneexciton state of molecular aggregates in our experiments. This finding suggests, in agreement with earlier studies of ultrafast excitation energy transfer processes in molecular aggregates, that the relaxation dynamics of aggregates in our experimental conditions is mainly controlled by the quenching of excited states of aggregates through exciton–exciton annihilation [4.194]. We have thus shown that photonic-crystal fibers with a specially designed dispersion offer the ways to create efficient sources of ultrashort pulses for coherent nonlinear spectroscopy. These fibers provide a high efficiency of frequency upconversion of femtosecond laser pulses, permitting the generation of sub-picosecond linearly chirped anti-Stokes pulses ideally suited for femtosecond coherent anti-Stokes Raman scattering spectroscopy. Experimental studies demonstrate that PCFs can deliver linearly chirped frequency-shifted broadband light pulses, optimized for pump–probe nonlinear absorption spectroscopy.

4.10 Surface Nonlinear Optics, Spectroscopy, and Imaging In this section, we will dwell upon the potential of nonlinear-optical methods for the investigation of surfaces and interfaces. The ability of second-order nonlinear-optical processes, such as SHG and SFG (Sect. 4.3), to probe surfaces and interfaces is most

clearly seen in the case of a centrosymmetric material. In this situation, the electric-dipole SHG and SFG response from the bulk of the material vanishes. At a surface or an interface, the bulk symmetry is broken, and electric-dipole second-order nonlinear-optical

Nonlinear Optics

effects are allowed. Such surface-specific SHG and SFG processes enable a highly sensitive nondestructive local optical diagnostics of surfaces and interfaces (Fig. 4.35a–b). Illuminating and physically insightful discussion of this technique can be found in the classical texts on nonlinear optics [4.9, 203]. The χ (2) signal from a surface or an interface is, however, not entirely background-free, as the second-order a)

Reflected second harmonic

Incident beam 2ω

Medium 1 Surface / Interface Medium 2

2ω

ω1 ω2

ω SF ω2

ω SF

ν2

ω1

Forward CARS detection

ν1 Lens Filter

Objective

CARS

Sample

ω1 ω2

Objective

ω1

ω CARS

nonlinear optical processes are not strictly forbidden even in a centrosymmetric medium. Beyond the electric-dipole approximation, the second-order nonlinear signal, as can be seen from (4.6–4.8) and (4.14), can be generated through the electric-quadrupole and magnetic-dipole effects. It is generally very difficult, often impossible, to completely distinguish between the surface and bulk contributions to the nonlinear signal. Luckily enough, the electric-quadrupole and magneticdipole components in the nonlinear-optical response are typically ka times less significant than the dipoleallowed part [4.9], with k = 2π/λ and a being a typical size of an atom or a unit cell in a crystal. The ra(2) tio of the surface dipole-allowed susceptibility χs to (2) the bulk susceptibility γb can be then estimated as (2) (2) |χs |/|γb | ∼ d/(ka), where d is the thickness of the surface layer. In reflection SHG (Fig. 4.35a), the bulk contribution is typically generated in a subsurface layer with a thickness of d ∼ λ/(2π). The ratio of the surface part of the total reflected SHG signal to the bulk contributions in this case is on the order of d 2 /a2 , which can be easily made much larger than unity. This ratio can be substantially enhanced on frequency resonances or with an appropriate choice of polarization arrangement. A combination of the high spatial and temporal resolution with a spectral selectivity makes χ (2) techniques a powerful tool for time-resolved species-selective studies of surfaces and buried interfaces (see, e.g., [4.204] for a comprehensive review of recent results), detection and size and shape analysis of adsorbed species, nanoparticles, and clusters on surfaces [4.205, 206], as well as imaging and microscopy of biological species [4.207]. The sensitivity and selectivity of the χ (2) technique are enhanced when the frequency of one of the laser fields (ω1 in the inset to Fig. 4.35b) is tuned to a resonance with a frequency of one of the vibrations typical of species on a surface or an interface (Fig. 4.35b). This method of surface analysis is referred to as sum-frequency surface vibrational spectroscopy. The capabilities of this technique have been impressively

ν2 ω1 ω2

Dichroic mirror CARS

Filter Lens Epl-CARS detection

ν1

Fig. 4.35 Nonlinear optics, spectroscopy, and imaging of surfaces and buried structures. (a) Nonlinear-optical probing of surfaces and interfaces using second-harmonic generation. (b) Surface vibrational spectroscopy using sumfrequency generation. Diagram of vibrational transitions probed by SFG is shown on the right. (c) Nonlinear microscopy based on coherent anti-Stokes Raman scattering. Diagram of Raman-active transitions selectively addressed through CARS is shown on the left

217

Part A 4.10

Transmitted second harmonic

4.10 Surface Nonlinear Optics, Spectroscopy, and Imaging

218

Part A

Basic Principles and Materials

demonstrated for vapor–liquid and liquid–solid interfaces [4.205]. In sum-frequency surface vibrational spectroscopy [4.207] an infrared laser pulse E1 with a frequency ω1 overlaps on the surface of a sample with the second laser pulse E2 , which typically has a frequency ω2 in the visible, to induce a second-order polarization at the sum frequency ωSF = ω1 + ω2 : (2)

PSF (ωSF ) = χ (2) (ωSF ; ω1 , ω2 ) : E1 E2 .

(4.226)

The intensity of the optical signal at the sum frequency is given by (2) 2 ISF ∝ χeff I1 I2 . (4.227) Here, I1 and I2 are the intensities of the laser beams and (2) χeff = L SF · eSF ·χ (2) s : L 2 · e2 L 1 · e1 , (4.228)

Part A 4.10

where L 1 , L 2 and L 3 are the Fresnel factors at the frequencies ω1 , ω2 , and ωSF , respectively, e1 , e2 , and eSF are the unit polarization vectors of the laser and sum-frequency fields, and the surface quadratic suscep(2) tibility χ s is written as (2)nr + χ (2) s = χ

aq , ω1 − ωq + iΓq

(4.229)

with χ (2)nr being the nonresonant quadratic susceptibility andaq , ωq , and Γq being the strength, the frequency, and the damping constant for the q-th vibrational mode, respectively. When the infrared field is scanned over the frequency of the q-th vibrational mode, the SFG signal is resonantly enhance, and its spectrum gives the spectrum of the vibrational mode. Tunable dye lasers [4.204], optical parametric oscillators and amplifiers [4.205], or PCF frequency shifters [4.185, 208] are employed as sources of frequency tunable radiation, allowing selective probing of vibrational (as well as electronic and excitonic) transitions in molecules and molecular aggregates. Through the past few years, nonlinear-optical methods of surface spectroscopy have been extensively involved in the rapid growth of nonlinear microscopic techniques based on χ (2) and χ (3) processes. In particular, SHG and THG processes have proven to be convenient techniques for high-resolution three-dimensional microscopy of biological objects [4.207,209], as well as laser-produced plasmas and micro-explosions [4.210]. In early experimental demonstrations of SHG microscopy, grain structures and defects on the surface

of thin films were visualized using SHG in transmission [4.211] and surface monolayers were imaged by reflection-geometry SHG microscopy [4.212]. In recent years, progress in laser technologies and the advent of new-generation imaging and scanning systems made it possible to extend nonlinear microscopy techniques to three-dimensional structures, buried objects, and biological tissues [4.207,209]. In this modification of nonlinear microscopy, the nonlinear signal is generated in the focal region of the laser beam in the bulk of a sample, originating from optical micro-inhomogeneities, which break the point-group symmetry of the medium or change phase-matching conditions. In two transverse dimensions, the high spatial resolution of nonlinear microscopy techniques is controlled by the nonlinear nature of the process, tightly confining the area where the signal is generated to the focal region. Resolution in the direction of probing is achieved either due to symmetry breaking, similar to nonlinear-optical surface-imaging techniques, or through phase-matching modification. In CARS microscopy [4.207,209,213,214], the nonlinear signal is resonantly enhanced when a frequency difference between two laser fields is tuned to a Ramanactive mode of molecules under study, as described in Sect. 4.4.8 (see also Fig. 4.35c). This makes microscopy also species-selective as Raman resonances serve as fingerprints of a certain type of molecules or molecular aggregates. Forward CARS and backward CARS (also called epi-CARS) geometries have been developed (Fig. 4.35c). In the forward-CARS microscopy, the mismatch |∆k| of the wave vectors involved in wave mixing (Sect. 4.7) is typically much smaller than 2π/λCARS , where λCARS is the wavelength of the CARS signal. For epi-CARS, |∆k| is larger than 2π/λCARS . The intensity of the epi-CARS signal can thus be comparable with the forward-CARS signal intensity only for a thin sample with a sample thickness d meeting the inequality |∆k|d < π. CARS microscopy has been one of the most successful recent developments in the field of nonlinear-optical microscopy. In addition to the high spatial resolution, this technique has a number of other important advantages over, for example, microscopy based on spontaneous Raman scattering. In particular, in CARS microscopy, laser beams with moderate intensities can be used, which reduces the risk of damaging biological tissues. The anti-Stokes signal in CARS microscopy is spectrally separated from the laser beams and from fluorescence, as the anti-Stokes wavelength is shorter than the wavelengths of the laser beams. In transparent materials, CARS microscopy can be used to image tiny buried objects inside the sample in three

Nonlinear Optics

dimensions. Due to the coherent nature of CARS, the capabilities of CARS microscopy can be enhanced by

4.11 High-Order Harmonic Generation

219

means of coherence control, as recently demonstrated by Dudovich et al. [4.117].

4.11 High-Order Harmonic Generation 4.11.1 Historical Background

Part A 4.11

The invention of the laser in 1960 opened many new fields of research. One of them is nonlinear optics triggered by Franken [4.215] with the first demonstration of frequency doubling in a crystal (1961) and pioneered by Bloombergen and coworkers. Third-harmonic generation in gases was observed for the first time by New and Ward in 1967 [4.216], fifth and higher-order harmonic generation a few years later by Reintjes and others [4.217]. The main objectives of this research were to increase the conversion efficiency, to cover a large (continuous) spectral range with, for example, frequency mixing processes and to reach very short wavelengths. The most natural route for the latter goal was to use fundamental fields with the shortest possible wavelength, to produce short-wavelength radiation through a low-order nonlinear-optical process. Another research area, which started in the late sixties, is the study of atoms in strong laser fields. The objective of this fundamental research was simply to understand the behavior of atoms and molecules exposed to intense electromagnetic fields. This field evolved in parallel with the development of pulsed lasers towards increasing peak powers, increasing repetition rates and decreasing pulse durations. The character of the laser–atom interaction also evolved from being essentially perturbative for laser intensities below 1013 W/cm2 to strongly nonperturbative for higher intensities. For many years, this regime of nonlinear optics was studied only by looking at ionization processes. The number and charge of the produced ions, as well as the energy and angular emission of the electrons, were experimentally detected and compared with theoretical predictions. At the end of the 1980s, it was realized that looking at the emitted photons would bring complementary information on the physical processes taking place. Indeed, efficient photon emission in the extreme ultraviolet (XUV) range, in the form of high-order harmonics of the fundamental laser field, was observed for the first time in 1987, in Saclay [4.218] (33rd harmonic of a Nd:YAG laser) and in Chicago [4.219] (17th harmonic of a KrF laser). The harmonic spectra were characterized by a decrease in efficiency for the first harmonics,

followed by a broad plateau of nearly constant conversion efficiency, ending up by an abrupt cutoff. The existence of such a plateau was clearly a nonperturbative signature of the laser-atom interaction. Most of the early work concentrated on the extension of the plateau, i. e. the generation of harmonics of higher frequency and shorter wavelength going progressively from ≈ 20 nm at the end of the 1980s to ≈ 7 nm by the middle of the 1990s [4.220–224]. It was soon realized that, in contrast to the ideas promoted in the nonlinear-optics community, the shortest wavelengths were obtained with long-wavelength lasers. Today, harmonic spectra produced with short and intense laser pulses extend to the water window (below the carbon K-edge at 4.4 nm) [4.225, 226]. A breakthrough in the theoretical understanding of high-order harmonic generation processes was initiated by Krause and coworkers [4.227] who showed that the cutoff position in the harmonic spectrum follows the universal law Ip + 3Up , where Ip is the ionization potential, whereas Up = e2 E 2 /4mω2 , is the ponderomotive potential, i. e. the mean kinetic energy acquired by an electron oscillating in the laser field. Here e is the electron charge, m is its mass, and E and ω are the laser electric field and its frequency, respectively. An explanation of this universal fact in the framework of a simple semiclassical theory was found shortly afterwards [4.228, 229], and confirmed by quantum-mechanical calculations [4.230, 231]. Progress in experimental techniques and theoretical understanding stimulated numerous studies of harmonic generation. The influence of the laser polarization was investigated in great detail [4.232–239] (see also [4.240–242] for the theory). The nonlinear conversion process was optimized with respect to the laser parameters [4.243, 244], and to the generating medium [4.245–250]. Finally, the spatial [4.251–256] and temporal [4.257–263] properties of the radiation were characterized and optimized. The specifications of the harmonic emission (ultrashort pulse duration, high brightness, good coherence) make it a unique source of XUV radiation, used in a growing number of applications ranging from atomic [4.264, 265] and molecular [4.266–268] spectroscopy to solid-state [4.269–

220

Part A

Basic Principles and Materials

271] and plasma [4.272–274] physics. Finally, it has recently been demonstrated that the low-order harmonics are intense enough to induce nonlinear optical processes in the XUV range [4.275–277]. Almost immediately after the first observation of the harmonic plateau, it was realized that, if the harmonics were emitted in phase, i. e. phase-locked, the temporal structure of the radiation emitted from the medium would consist of a “train” of attosecond pulses separated by half the laser period [4.278, 279]. There is a clear analogy here with mode-locked lasers, where axLens

Gasjet

5ω 3ω

Laser pulse

.. .

Part A 4.11 Photons (arb. units) 10 H37 8

ial modes oscillating in a laser cavity are locked in phase, leading to the production of trains of short pulses. Attosecond pulses have remained, however, a theoretical prediction until recently [4.280–282]. A first possible indication of harmonic radiation containing an attosecond sub-structure was in a high-order autocorrelation of the driving laser pulse [4.283]. This was followed by a beautiful experiment showing evidence for phaselocking between five consecutive harmonics generated in argon, thus indicating that trains of 250 as pulses were formed [4.284]. In a series of experiments performed in Vienna, single pulses of duration of a few hundred attoseconds were demonstrated by using ultrashort (5 fs) laser pulses and spectrally filtering a few harmonics in the cutoff spectral region [4.285, 286]. These experimental results are the beginning of a new field of research attophysics, where processes in atoms and molecules can be studied at an unprecedented time scale. The purpose of this section is to present to the nonexpert reader a simple description of high-order-harmonic generation, and its application to attosecond metrology. In Sect. 4.11.2, we describe the most important aspects of high-order-harmonic generation processes. We begin with a short description of the experimental set up needed to obtain high-order harmonics. Then we discuss the microscopic and macroscopic physics underlying the generation of high-order harmonics. We focus on the physics of importance for the generation of attosecond pulse trains and single attosecond pulses. We refer the reader to several review articles for a more complete overview of this research topic [4.245, 287, 288]. In Sect. 4.12, the emerging field of attosecond science is reviewed. The different measurement techniques are described and the first application of attosecond pulses is presented.

4.11.2 High-Order-Harmonic Generation in Gases

H43

H31

Experimental Method Generating high-order harmonics is experimentally simple. A typical set up is shown in Fig. 4.36. A laser pulse

H49

6 4

Fig. 4.36 Schematic representation of a typical experimen-

H53

0 50

80 Energy (eV)

tal setup for high-order-harmonic generation. An intense short pulse laser is focused into a vacuum chamber containing a gas medium. Harmonics are emitted along the laser propagation axis. A photograph of a gas target is shown below. A typical spectrum obtained in neon is shown at the bottom. It shows a plateau ending with a sharp cutoff

Nonlinear Optics

Intensity

10 –6 10 –7

Dephasing limited

4.003

10 –8

20.18

Oxygen K-edge

10 –9

39.95

10 –10

Kr 83.80

10 –11

Xe 131.3

221

Harmonic conversion efficiency

Ionisation energy Ip He

4.11 High-Order Harmonic Generation

Order

Carbon K - edge

Absorption limited

10 –12 100

300 Harmonic order

Fig. 4.37 Illustration of the influence of the gas species on the generation efficiency (left). The result presented at the right has been obtained with 5 fs 800 nm laser pulses (after [4.302])

laser pulse. Rare gases are the favored species, for obvious technical reasons. In addition, some work has been done with alkali atoms [4.296], ions [4.220, 304], molecules [4.305–308] and clusters [4.309–311]. Photons are separated in energy and detected by an XUV spectrometer, including a grating, sometimes a refocusing mirror, and a detector (electron multiplier, microchannel plates, etc.). A typical experimental spectrum is shown in Fig. 4.36. This result has been obtained in a gas of neon using a 100 fs 800 nm Ti:sapphire laser [4.312]. It shows odd harmonic peaks up to the 53rd order, with a rapid decrease beyond the 49th harmonic, characteristic of the cutoff region. The spectral range of the harmonic emission as well as the conversion efficiency depends strongly on the gas medium. As schematically illustrated in Fig. 4.37 (left), the efficiency is highest in the heavy atoms Ar, Xe, Kr, but the highest photon energies are obtained in He and Ne [4.222, 245, 302]. Figure 4.37 (right) presents Laser

Atom Harmonic emissiom

Laser

Medium Coherent superposition

Fig. 4.38 Illustration of the two aspects of high-orderharmonic generation. (Top) harmonic emission from a single atom. (Bottom) phase matching in the nonlinear medium

Part A 4.11

is focused into a vacuum chamber containing a rather small gas target with an atomic pressure of at least a few mbar. The harmonics (only of odd order, owing to inversion symmetry) are emitted along the laser propagation axis. Many types of lasers have been used, excimers, Nd:YAG, Nd:glass, Ti:sapphire, dye lasers, etc. [4.218, 219, 222, 223, 289, 290]. In addition, the second harmonic of these lasers [4.291–293] as well as the radiation from sum- or difference-frequency mixing processes [4.294, 295] for example to get into the mid-infrared range [4.296], have also been employed. Typical energies are between a fraction to a few tens of mJ, typical pulse durations are from a few femtoseconds to a few tens of picoseconds. In the last five years, the favorite tool has become the Ti:sapphire laser, providing very short pulse lengths, high laser intensities at high repetition rates. The shortest laser pulses used today to study atoms in strong laser fields are about 5 fs (two cycles) long [4.285]. The advantage of using short pulses is that atoms get exposed to higher laser intensity before they ionize, leading to higher-order harmonics. The different parameters of the laser pulses, such as the polarization, the focusing characteristics, the spatial and temporal profiles are often varied and optimized in the experiments. Recent studies do not simply vary a given parameter, but attempt to shape a laser pulse (by varying, for example, its phase [4.244], its degree of ellipticity [4.297–299], or its spatial properties [4.300, 301]) to tailor the harmonics for different applications. The gas medium is provided by a gas jet, hollow fiber [4.246–248] or a (small) gas cell [4.249, 250, 303]. Figure 4.36 shows a photograph of such a gas target containing argon atoms irradiated by an intense

222

Part A

Basic Principles and Materials

a comparison between the generation efficiency for three rare gases Ar, Ne, He, when ultrashort laser pulses (of duration 5 fs) are used for the excitation [4.245, 302]. As illustrated in Fig. 4.38, two conditions are necessary to observe efficient harmonic emission. First, each individual atom must generate light at these frequencies, requiring a highly nonlinear response to the radiation field. Second, the harmonic field results from the coherent superposition of all the emitting atoms in the medium. Harmonic generation will be efficient only if phase matching is achieved, requiring the generated field to be in phase with the nonlinear polarization creating it over the medium’s length. We discuss these two aspects in more details in the following sections.

V(r)

5ω

V(r)

4.11.3 Microscopic Physics r

Part A 4.11

Electrons in an atom in the presence of a time-dependent radiation field oscillate. This is described by a dipole moment d(t) = Φ(t)|er|Φ(t), where |Φ(t) is the timedependent electronic wavefunction, the solution of the Schrödinger equation. When the radiation field is weak, there is mainly one oscillation frequency, that of the field. In a strong radiation field, the oscillatory motion becomes distorted and the dipole moment now includes a series of higher-order frequencies, odd harmonics of the fundamental one. The harmonic emission from a single atom can thus be calculated by taking the Fourier transform of the dipole moment. Theorists often use a single-active-electron approximation, assuming that the interaction with the field involves essentially one active electron, to describe the response of the atom to a strong laser field. A number of methods have been proposed to solve this problem and it is beyond the scope of this paper to review all of these approaches. The most realistic approach is probably the numerical solution of the time-dependent Schrödinger equation, pioneered by Kulander at the end of the 1980s [4.313]. Many important results, such as the determination of the cutoff law for high-order harmonics in 1992 [4.227] or proposals for single attosecond pulse generation using few-cycle laser pulses at the end of the 1990s [4.245,314,315] were obtained directly from numerical calculations. The semiclassical strong field approximation, originating from a seminal paper of Keldysh in 1964 [4.316], applied by Lewenstein in the 1990s to high-order-harmonic generation [4.230, 231] allows to explore a larger parameter space as well as to gain intuitive insight. This model and a related one approximating the atomic potential by a δ(r) potential [4.240] have been extensively used to interpret experimental results.

Fig. 4.39 Schematic picture of different regimes for harmonic generation. The multiphoton, perturbative regime is illustrated on the top, while the semiclassical model is presented on the bottom

Many insights in the physical understanding of the interaction between atoms and strong laser fields have been provided by a simple semiclassical model, proposed first by Van der Linden, van der Heuvell, and Muller in the context of above-threshold ionization and extended by Corkum and others [4.228, 229] to multiple ionization and high-order-harmonic generation. According to this model, illustrated in Fig. 4.39 (bottom), the electron tunnels through the Coulomb energy barrier modified by the presence of the relatively slowly varying linearly polarized electric field of the laser. It then undergoes (classical) oscillations in the field, during which the influence of the Coulomb force from the nucleus is practically negligible. If the electron comes back to the vicinity of the nucleus, it may be rescattered one or several times by the nucleus, thus acquiring a high kinetic energy, and in some cases, kicking out a second or third electron. It may also recombine back to the ground state, thus producing a photon with energy Ip , the ionization potential, plus the kinetic energy acquired during the oscillatory motion. We also show in Fig. 4.39 (top) for comparison the more traditional harmonic-generation process based on

Nonlinear Optics

multiphoton absorption in the (barely perturbed) atomic potential. An intuitive understanding of some of the properties of harmonic generation can be gained by elementary classical calculations of the electron motion outside the binding potential. Assuming the electron to have zero velocity immediately after it has tunneled through the potential barrier at time t = t0 , and the laser field to be simply described by E = E 0 sin(ωt), we obtain: v = −v0 cos(ωt) + v0 cos(ωt0 ) , (4.230) v0 x = − sin(ωt) ω v0 + sin(ωt0 ) + (t − t0 )v0 cos(ωt0 ) , (4.231) ω where v0 = qE 0 /mω. Depending on the time at which the electron is released into the continuum (t0 ), it will follow different trajectories, as illustrated in Fig. 4.40. Only those electrons released between T/4 and T/2

4.11 High-Order Harmonic Generation

(where T is the laser period) are of interest for harmonic generation. When the laser field changes its sign, they come back towards the core (at x = 0) with a certain kinetic energy. This energy, which determines the emitted harmonic order, is proportional to the square of the slope of the trajectory as it crosses the time axis [open circles in Fig. 4.40 (top)]. Except for the trajectory starting at approximately 0.3T , giving the highest kinetic energy (cutoff), there are two (main) trajectories leading the same kinetic energy. This is illustrated in Fig. 4.40 (bottom), showing the kinetic energy when the electron returns to the core (solid line), as well as the time spent in the continuum (dotdashed line), as a function of release time. As shown in the figure, for each energy (dashed line), and hence Dipole strength (arb. units) 10–10

Dipole phase (rad) 0

–40

10–14

–60

10–16

–80 T/4

Time

T/2

10–18 1

–100 5 4 Intensity (1014 W/cm2)

Intensity (1014 W/cm2) Electric field

Return energy (Up)

5 Time in the continuum (optical cycle)

4.5 4

3.17 3

1.0 Return energy

3.5

τ2 3

2 τ1

0 0.2

Time in the continuum

0.5

2.5 2

0.3

0.0

0.4 0.5 Release time (optical cycle)

Fig. 4.40 (Top) Electron trajectories in the continuum corresponding to different release times. The laser electric field is represented in dotted line. (Bottom) kinetic energy (solid line) and time in the continuum (dot-dashed line)as a function of the release time

1.5 –20 –10

50 60 70 α (10 –14 cm2/W)

Fig. 4.41 (Top) Single atom response within the strong-

field approximation. Intensity (dark line) and phase (light line) of the 35-th harmonic in neon as a function of the laser intensity. (Bottom) Quantum path analysis of the same harmonic

Part A 4.11

–20

10–12 Distance from the nucleus (arb. units)

223

224

Part A

Basic Principles and Materials

for each harmonic order, there are mainly two trajectories, a short and a long one, contributing to the harmonic emission. The periodicity (for a pulse several cycles long) of the process implies that the light emission is not continuous but at discrete (odd-harmonic) frequencies. The influence of the complex electron dynamics inherent to the harmonic-generation process is clearly visible on the intensity dependence of the harmonic components of the quantum-mechanical dipole moment. Figure 4.41 shows for example the 35-th harmonic Detector

a) Phase-locked generating beams

Part A 4.11

c) Coherence time (fs) 60 50 40 30 20 10 17

31 27 29 Harmonic order

Fig. 4.42a–c Spatially-resolved temporal coherence mea-

surements of high-order harmonics. The principle of the measurement is shown in (a). Two images obtained on the 13-th harmonic in xenon obtained for two different time delays (0 and 25 fs) between the two pulses are shown in (b). A summary of the measured coherence times in the two spatial regions for high harmonics in xenon is presented in (c)

generated in neon, calculated within the strong field approximation. The intensity and phase are represented respectively in solid and dashed line as a function of the laser intensity. The sharp intensity and phase variation at low intensity corresponds to the cutoff region. In the plateau, oscillations are clearly visible in the variation of both the intensity and the phase. They originate from interference effects between the contributions from the multiple trajectories. This fascinating conclusion stimulated the development of analysis techniques to extract the contributing electron trajectories (or rather the relevant quantum paths) from quantum-mechanical approaches [4.317, 318]. The result of such an analysis for the 35-th harmonic in neon is shown at the bottom in Fig. 4.41. The contributions to the dipole phase varying linearly with the intensity φdip (I) = α j I, corresponding to a quantum path ( j), are identified as vertical lines in Fig. 4.41. In this case, the dominant trajectory is the “second” one, with a coefficient α2 = 24 × 10−14 cm2 W. At low intensity, in the cutoff region, there is only one quantum path. The microscopic physics (the quantum orbits) behind high-order-harmonic generation was shown in a series of experiments investigating the temporal coherence of the harmonics [4.319–321]. We present in Fig. 4.42 the principle as well as some results obtained in xenon [4.320]. Two phase-locked spatially separated harmonic sources are created by splitting the laser into two replicas in a Michelson interferometer and by slightly misaligning one of the arms. A variable time delay can be introduced between the two pulses. The generated harmonics are separated by a grating and interfere in the far field. The variation of the contrast of the fringes as a function of time delay gives the coherence time. Figure 4.42b shows images obtained for the 13-th harmonic in xenon. (For experimental reasons, the images are not symmetrical.) These images present two spatial regions with different coherence times. The central region exhibits a long coherence time, whereas the outer region a much shorter one. Figure 4.42c summarizes measurements on the harmonics generated in xenon. These results can be interpreted in a simple way by recalling that the harmonic field is a sum of the contributions from each quantum path j E q (r, t) = A j (r, t) e−[iqωt+α j I (r,t)] (4.232) j

where A j is an envelope function, representing the strength of each path. The temporal variation of the laser intensity I(t) will produce a chirp in the emitted field,

Nonlinear Optics

and consequently a spectral broadening (or a reduced coherence time). The radial variation of the laser intensity I(r) will affect the curvature of the phase front of the harmonics, and therefore their divergence. The contribution from the quantum paths with a long excursion time in the continuum (Fig. 4.40), resulting in a large α j , will have a short coherence time and a pronounced curvature in the far field, whereas that from the quantum paths with a short excursion time will have a longer coherence time and be more collimated [4.322]. Similar evidence for the existence of (at least) two quantum paths has been obtained by analyzing the harmonic spectra in combination with measuring the harmonic pulse duration [4.321]. Contributions of different quantum paths can be selected macroscopically, either by spatial or spectral filtering.

4.11.4 Macroscopic Physics

pol

δφq = φq − φq ≈ kq z − qk1 z + q arctan(2z/b) − φdip . (4.233) In this equation, the first two terms characterize the difference in dispersion at the fundamental and q-th harmonic frequencies, mainly due to the effect of the free electrons in the medium. The third one is a geometrical term originating from the Gouy phase shift of a Gaussian beam with confocal parameter b across the focus. The fourth term is the dipole phase described above. According to the previous discussion, it is more physically correct to consider separately the contributions from the different quantum paths, before making a (coherent) sum. The phase difference to minimize depends on the trajectory and is given by pol

δφq, j = φq − φq ≈ kq z − qk1 z + q arctan(2z/b) − α j I(r, z, t) .

δΦq (rad) 120 100 80 60 40 20 0 –20 –40 –10 r (µm) 60 40 20 0 –20 –40 –60 –10

–5

10 z (mm)

L coh (mm) 1 0.8 0.6 0.4 0.2 0 10 z (mm)

Fig. 4.43 (Top) phase difference between the harmonic

wave and the polarization for the 35-th harmonic generated in neon on the propagation axis. In dashed line, we show the term coming from the geometrical phase shift of a Gaussian beam and in dotted line, the term coming from the dipole phase. The laser intensity is 6 × 1014 W/cm2 . The confocal parameter is chosen to be b = 1 cm. (Bottom) corresponding phase matching map (after [4.317])

a term decreasing with z, usually quite small. In contrast, the contribution of the dipole phase, here due mainly to the second trajectory, with a transition to the cutoff for |z| ≥ 7.5 mm, first increases, then decreases with z. In the particular case shown in Fig. 4.43, phase matching on the propagation axis is best realized for z ≥ 7 mm, requiring the medium to be located after the focus. In general, the situation can be rather complicated. Methods have been developed to visualize where (in the nonlinear medium) and when (during the laser pulse) phase matching was best realized [4.317], using in particular three-dimensional maps representing the local coherence length, defined as

(4.234)

Figure 4.43 illustrates the variation of δφq along the propagation axis (z). Most contributions are monotonic. Focusing as well as the free electron dispersion (not shown in Fig. 4.43) lead to a contribution to δφq increasing with z, while the (normal) dispersion leads to

225

Lq = π

∂δφq ∂z

−1

(4.235)

An example of such a map for the situation corresponding to Fig. 4.43 (top) is shown at the bottom of the same

Part A 4.11

We now turn to the second aspect of harmonic generation, namely the response of the whole medium. To achieve phase matching, i. e. to ensure efficient conversion, the wave vector difference (traditionally called the phase mismatch) between the harmonic wave and the polarization must be minimized, so that the phase difference varies as little as possible over the medium’s length. For an incident Gaussian beam, this phase difference on the propagation axis (z) is given by

4.11 High-Order Harmonic Generation

226

Part A

Basic Principles and Materials

Part A 4.11

figure. The white areas indicate where phase-matching is best realized. The two parts of the figure correspond to the transition between cutoff and plateau. The brief analysis presented above can easily be generalized to include propagation in waveguides [4.246– 248], which modifies somewhat the phase-matching conditions, since the geometrical and dipole phase effects are much reduced (owing to a constant intensity in the waveguide). Absorption in the nonlinear medium starts to play a role for high conversion efficiencies and long medium lengths and/or high pressures. The optimization of phase matching of high-order-harmonic generation has stimulated a great deal of efforts during the last five years. The so-called absorption limit, where the limitation on the conversion efficiency is due to absorption, and not to the coherence length, has been reached in different wavelength ranges [4.249,250,302]. Recently, the use of extremely long focusing geometries (which minimize both the effect of the dipole phase and that of the Gouy phase shift) has led to conversion efficiencies as high as a few times 10−5 and energies in the µJ range [4.303,323]. Phase matching of high-orderharmonic generation is by no means a solved issue, since it is a complicated three-dimensional problem, involving a number of parameters (laser focusing, pressure, medium length and geometry, laser intensity). In addition, the problem is quite different for the low-order harmonics with energy around a few tens of eV, where most of the work has been done, and for the highorder harmonics at 100 eV or more. An interesting idea, similar to quasi-phase-matching has recently been investigated [4.324]. A modulated waveguide, used for the generation of high-order harmonics, induces a periodic variation of the degree of ionization leading to enhanced conversion efficiency, especially for the high harmonics. Finally, a conclusion of importance for the generation of attosecond pulse trains is that the different quantum paths contributing to harmonic generation discussed in the previous section are not phase matched in the same conditions (4.234). The axial variation of the laser intensity I(z) leads to different phase matching conditions for the contributions from the different quantum paths. In other words, depending on the geometry, ionization, pressure conditions, phase matching will enhance one of the contributing trajectories to the detriment of the others. This conclusion is extremely important for the generation of attosecond pulse trains. As illustrated in Fig. 4.44 (top), the electron trajectories contributing to harmonic generation in the single-atom response lead to bursts of light at different times during

τ1

τ2

Esinω0t |eq | 2 τ2

τ1

0 |Eq | 2

1 Time (2π/ω0)

0.5 τ1

300 as

0.5

1 Time (2π/ω0)

Fig. 4.44 Illustration of the temporal structure of a few harmonics in the single atom response (top, middle). In some conditions, phase matching selects only one burst of light per half cycle, leading to phase locking (bottom)

the laser half-cycle. Phase locking between consecutive harmonics is not realized. In some conditions, however, phase matching results in efficient generation of only one of these contributions [Fig. 4.44 (bottom)] [4.325, 326], leading to a train of attosecond pulses. Another possibility, that makes use of the different spatial properties of the contributing trajectories, is to select the shortest trajectory by spatially filtering the harmonic beam with an aperture. Finally, a spectral filter can be used to select only the harmonics in the cutoff region, where the electron dynamics is much simpler, with only one electron trajectory in a single-atom response.

Nonlinear Optics

4.12 Attosecond Pulses: Measurement and Application

227

4.12 Attosecond Pulses: Measurement and Application 4.12.1 Attosecond Pulse Trains and Single Attosecond Pulses

E L (t) = E a (t) cos(ωL t + ϕ).

(4.236)

Only pulses with ϕ ≈ 0 have the potential for producing a single burst. For such a fundamental pulse with a “cosine” waveform, the peak intensity of the laser pulse can be adjusted to give rise to a single burst within a preselected frequency band (beige band in Fig. 4.46). Recently, carrier-envelope-phase-controlled few-cycle pulses opened the door to the reproducible generation of single sub-femtosecond XUV pulses [4.330]. The feasibility of a single sub-femtosecond pulse produced by few-cycle-driven high-order harmonic generation was corroborated by numerical studies based on a computer code [4.331] solving Maxwell’s wave equations in three dimensions and calculating the radiation of the strongly driven atomic dipoles using the quantum theory of Lewenstein et al. [4.231]. These numerical calculations predict the feasibility of isolated near-bandwidth-limited XUV pulse generation with durations down to the 100 as range.

4.12.2 Basic Concepts for XUV Pulse Measurement The most direct information about the duration of short flashes of electromagnetic radiation can be gained from time-domain measurements. To this end, the burst to

Part A 4.12

The first experimental indication of harmonic radiation containing a substructure was observed in a high-order autocorrelation of the driving laser pulse [4.283]. The experimental set up was similar to that described in Fig. 4.42 (top), except that the two harmonics sources were spatially overlapped and no spectral selection with a grating was done. The signal was studied as a function of time delay, and showed substructures. More-recent experiments [4.284] showed conclusive evidence for a group of harmonics beating together to form a sequence of XUV bursts. In perfect phase-locking conditions, a combination of N = 5 harmonics generated by a 800 nm laser can form a train of pulses separated by half the laser period T0 /2 = 1.35 fs, with a duration as short as T0 /2N. In the experiment described in [4.284], the signal used to study the phase locking of the harmonics were the sidebands in the photoelectron spectra, which are due to the coherent superposition of two processes, combined absorption of a harmonic and infrared (IR) laser photon, and absorption of the next harmonic and emission of a laser photon [Fig. 4.45 (top)]. The sideband intensity depends on the phase difference between the two harmonics involved, as well as on the time delay between the XUV and IR pulses. Monitoring the intensity as a function of time delay for several sideband peaks allows one to study the variation of the phase difference between consecutive harmonics [Fig. 4.45 (bottom)]. The relative phases of the harmonics were found to be fairly constant, yielding a train of 250 as pulses spaced by half the oscillation period of the laser field [Fig. 4.45 (middle)]. The high repetition rate (twice the laser frequency) of this train of pulses may pose problems when it comes to spectroscopic applications, because the interpretation of pump–probe experiments tends to be ambiguous. Using such a train for either triggering or probing a dynamic process that does not completely come to a standstill within T0 /2(≈ 1.3 fs for 800 nm driver laser light from a Ti:sapphire laser) gives rise to multiple pumping or probing, complicating the interpretation of measured data. This recognition led to the proposal that a single pulse could possibly be selected from the attosecond train by a driving laser pulse with a polarization state rapidly varying in time to confine pure linear polarization (which is a prerequisite for harmonic emission) to a single laser cycle at the pulse peak [4.280, 281, 327].

This polarization gating technique has been experimentally implemented using different methods to achieve the modulation of the degree of ellipticity of the fundamental laser pulse. A significant confinement of the harmonic emission (however not in the attosecond regime) has been shown using both spectral and temporal diagnostics [4.297,298,328,329]. This technique could be a way to isolate single as pulses in the plateau energy region. A more straightforward approach to confining highharmonic emission to a single laser cycle became feasible with the availability of intense few-cycle (sub10 fs) laser pulses. When driven with a laser pulse consisting of only a few oscillation cycles, the harmonic emission at the highest photon energies (near cutoff region) can be confined to one half of the oscillation period near the pulse peak. By selecting a well-defined photon energy band near the cutoff region, which requires the highest driving field intensity, it is expected to be possible to isolate one single sub-femtosecond burst, as illustrated in Fig. 4.46. This diagram also shows the sensitivity of isolated pulse generation to the absolute (or carrier-envelope) phase ϕ of the fundamental laser pulse

228

Part A

Basic Principles and Materials

Electron energy

q q+2 I(t) 10

1.35 fs

Part A 4.12

250 as

5 Time (fs)

Sideband-peak area normalized to total signal 0.30

0.15 0.30

0.15 0.08 0.04 0.030

0.015 –4

–3

–2

–1

1 Delay τ (fs)

Fig. 4.45 Experimental evidence for phase-locking of high-order harmonics; principle of measurement (top), experimental results (bottom) and reconstructed temporal intensity profile (middle). Reprinted from [4.284] with permission from the authors

be characterized must be probed by a sampling pulse that is comparable to or shorter than the duration of the burst. In the optical regime this concept has been implemented by nonlinear autocorrelation techniques, where the burst to characterize is probed by itself. The extension of these techniques to the XUV spectral range is not trivial, because of the unfavorable scaling of the nonlinear polarizability with wavelength and because of the low power of the XUV pulse to be measured. Only a few results have been obtained so far for relatively low-order harmonics. The nonlinear process used for the autocorrelation is two-photon ionization of a gas target. Experiments have demonstrated two-photon ionization in Xe, Kr [4.277, 332] by low-order harmonics and recently two-photon ionization in He with a combination of harmonics from the 7-th to the 13-th [4.333]. In the first autocorrelation experiments [4.275, 334], harmonics generated by a 1 TW 30 fs laser beam were used to ionize He atoms. The number of He+ ions was measured as a function of the delay between two identical ninth-harmonic pulses (14.1 eV) generated by two replicas of the fundamental pulse. A pulse width of 27 fs was obtained by assuming a sech2 pulse shape (Fig. 4.47). In a more recent experiment a two-photon-absorption-based auto-correlation was implemented for a frequency-resolved optical gating (FROG) measurement [4.335]. Here the two replicas of the fifth harmonic of a 7 mJ Ti:sapphire laser pulse were focused into a Xe gas jet. From the recorded FROG trace the intensity profile and chirp of a high-harmonic pulse could be retrieved. In another recent experiment [4.336], an autocorrelation of a bunch of harmonics (from the 7-th to the 11-th) gives evidence for attosecond bunching. The extension of auto-correlation techniques to higher-order harmonics, with energies in the XUV/softX-ray region, remains, however, a formidable challenge. The atomic cross section of the absorption process σ scales with λ6 , implying a dramatic decrease of the two-photon transition probability with decreasing wavelength. As a consequence auto-correlation schemes require photon fluences orders of magnitude higher than those available from existing harmonic sources in the XUV regime. In order to circumvent the difficulties resulting from low two-photon transition cross sections

Nonlinear Optics

EL (t) = Ea (t) cos ωLt Harmonic photon energy

4.12 Attosecond Pulses: Measurement and Application

229

EL (t) = Ea (t) sin ωLt Harmonic photon energy

Multicycle driven pulse

Time

Harmonic photon energy

Time

Fewcycle driven pulse

Time

Fig. 4.46 Comparison of the temporal structure of harmonics of multi-cycle or few-cycle pulses for different values of the carrier-envelope phase. The peaks sketch the temporal structure (horizontal axis) of emitted XUV radiation as a function of XUV photon energy (vertical axis)

at short wavelengths, the XUV pulse can be correlated with the strong laser pulse previously used for its generation. The cross-correlation can be implemented by XUV photoionization in the presence of the laser pulse. XUV pulses have been cross-correlated with the femtosecond Number of He+ ions 4000

3000

2000 FWHM = 27 fs 1000

infrared (IR) laser pulses in a number of experiments over the past decade [4.257–260,262,337,338]. All these experiments had in common that the temporal intensity envelope of the laser pulse served as a sampling function for measuring the XUV pulse shape and duration, which limits the temporal resolution to several femtoseconds or more. Here we review a novel implementation of XUV/IR cross-correlation technique that correlates the XUV pulse envelope with the oscillating laser field rather than its envelope, providing a sub-T0 /4 probe (T0 is the laser cycle) for measuring the XUV pulse duration. The method exploits the dependence of the final kinetic energy of the XUV-pulse-generated photoelectron on the phase (and strength) of the laser field at the instant of photoionization. The width and position of the resultant photoelectron spectra are measured as a function of the relative delay between XUV and laser pulse and yield a convolution of the oscillating laser light field with the XUV pulse envelope. Careful deconvolution allows one

0 –80

–40

80 40 Relative delay (fs)

Fig. 4.47 Auto-correlation trace of the ninth harmonic by two-photon ionization of He (after [4.334])

Part A 4.12

Time

230

Part A

Basic Principles and Materials

to determine the duration of the XUV pulse and its timing jitter with respect to the light field with as resolution [4.261, 285]. The concept has been termed attosecond streak camera because it is the streaking of the electron’s kinetic energy by the rapidly varying laser field that yields direct time-domain information of the time structure of emitted electron wave packet, which mimics that of the XUV pulse [4.339–341].

4.12.3 The Optical-Field-Driven XUV Streak Camera Technique

Part A 4.12

XUV photoionization in the presence of a strong laser field can be accounted for by a simple quasi-classical model, similar to that presented above, treating the interaction as a two-step process [4.261]. According to this model, the photoelectron is first ejected by a short XUV pulse with a distribution of initial momenta known from conventional photoionization studies [4.342]. Subsequently, it is accelerated (or decelerated) by the light field. For XUV pulse durations τx very short compared to T0 /2, the model predicts that, depending on the oscillation phase of the light field at the instant of “birth” of the electron, a momentum component is added to the initial momentum of the electron. The concept of laser-field-assisted XUV photoionization can be implemented in various detection geometries. The two most important ones are shown in ∆p pi

∆t > 0 ∆p

pf pi

∆t > 0

Fig. 4.48 Two-color (XUV/IR) photoionization in differ-

ent geometries: the strong electric field of the infrared laser pulse can be polarized along the direction of detection of the photoelectrons or orthogonally to it, giving rise to substantial differences in the photoelectron’s motion after its liberation. The polarization of the XUV pulse has little influence on the electron motion

Fig. 4.48. The photoelectrons are detected within a cone aligned parallel and orthogonally to the electric field vector of the linearly polarized laser field (henceforth referred to as the parallel and orthogonal detection geometry, respectively). Here we focus on the orthogonal geometry because the first sub-femtosecond pulse measurement used this configuration [4.285]. In the next section we shall demonstrate that the parallel geometry can also be employed for efficient as diagnostics. Figure 4.49 depicts the final momentum distribution of the photoelectron for different instants of release in the strong laser field. In general, a laser-induced momentum component is added to the initial momentum vector of the photoelectron, resulting in a shift of the photoelectron angular distribution along the laser polarization in momentum space. The momentum transferred from the IR light field is largest if the electric field of the light wave is zero, i. e. its slope is maximum, at the instant of birth of the photoelectron. The width of the XUV photoelectron energy spectrum ∆W, which is equal to the bandwidth of the XUV pulse spectrum in the absence of the light field, increases with increasing momentum shift, within a finite detection cone aligned orthogonally to the laser polarization. Scanning the instant of birth of the photoelectron through the light field oscillations by changing the relative delay td between the light pulse and XUV pulse results in a modulation of the center of gravity as well as the width ∆W(td ) of the XUV photoelectron spectrum with a period equal to one half of the light oscillation period T0 . In the first time-resolved sub-femtosecond XUV pulse measurement [4.285] the modulation of the fieldinduced spectral broadening ∆W(td ) was recorded. The detection of such a modulation relies on an XUV pulse short compared to half the laser period, i. e. τx T0 /2, implying also a fast (attosecond) ionization of the bound electrons by the XUV photons, and a timing firmly fixed (with as precision) of the XUV pulse with respect to the laser field. A finite XUV pulse duration or a finite timing jitter of the XUV pulse (relative to the phase of the light field) of any origin results in a reduced contrast of the predicted variation of ∆W(t d ) versus time delay (illustrated in Fig. 4.49). An XUV pulse duration or a timing jitter exceeding T0 /2 (≈ 1.25 fs in the case of nearinfrared laser light from the Ti:sapphire laser) would smear out the modulation completely. Hence, the modulation depth of ∆W(td ) provides a sub-femtosecond probe for setting a reliable upper limit on the XUV pulse duration and timing jitter. Figure 4.50 displays the experimental set up for the generation of sub-femtosecond XUV pulses by har-

Nonlinear Optics

4.12 Attosecond Pulses: Measurement and Application

EL(t)

231

EL(t)

Time

Wkin

EL(t)

Time

Wkin

Wkin WD

EL(t)

EL(t) Total scan

Time

Wkin

τx « T0

Fig. 4.49 Principle of measuring XUV-intensity/IR-light-field cross-correlation with attosecond resolution. The photo-

electrons created initially with isotropic momentum distribution by the absorption of XUV photons pick up momentum from the strong laser light field. The light-induced momentum change ∆ px deforms the final photoelectron momentum distribution at instants t1 , t2 , t3 = t2 + T0 /4, t4 = t2 + T0 /2, and t5 = t2 + 3T0 /4 as shown, where E L (t3 ) = E L (t5 ) = 0. Photoelectrons detected within a cone aligned orthogonally to the direction of the light field vector (x-direction) display a kinetic energy spread at t3 and t5 whilst remaining unaffected by the light field at t1 , t2 and t4 . The full scan shows schematically the expected modulation of the spectra if τx < T0 /2

monic generation and their time-resolved measurement by the attosecond streak-camera technique described in the previous section. A 200 nm-thick zirconium foil with an aperture of 2 mm is placed 150 cm downstream from the XUV pulse source. The aperture is matched to the beam diameter of the harmonic beam, which has a low divergence of ≈ 0.7 mrad. This filter is placed in the beam to block the laser and low-order harmonics across the XUV beam, transmitting only photons

with an energy of higher than 70 eV. It is mounted on a nitrocellulose membrane of 5 µm thickness to cover a 2 mm-diameter hole in the membrane. This device is virtually dispersion-free and produces an annular laser beam confining the XUV beam in its center. The energy in the laser beam is adjusted by a motorized iris and measured by a photodiode. Both the laser and the collinearly propagating XUV beam are focused with a two-component focusing mirror directing the XUV

Part A 4.12

232

Part A

Basic Principles and Materials

TOF Kr

Ionization detector V–

Iris

Zr filter on pellicle PZT Ne

Part A 4.12

Ag-mirror Pump laser pulse Duration≈ 7 fs Energy ≈ 0.5 mJ Repetition rate = 1 kHz

Two-component Mo/Si multilayer mirror Reflectivity > 60 % at 90 eV Bandwith ≈ 5 eV

Fig. 4.50 Experimental set up for the attosecond pulse measurement. The focused 7 fs laser beam interacts with neon atoms to produce high-harmonic radiation. The laser and the highly collimated XUV beam co-propagate collinearly through a 2 m beam line towards the measurement. In the beaml ine the laser and harmonic beams pass through a 200 nmthick 3 mm-diameter zirconium foil placed on a 5 µm-thick nitrocellulose pellicle to cover a hole of 2 mm diameter. The energy transported by the resulting annular beam can be adjusted with a motorized iris between a fraction of a microjoule and a few tens of microjoules. The Mo/Si multilayer consists of an annular part having an outer diameter of 10 mm with a concentric hole of 3 mm diameter hosting a miniature mirror of slightly smaller diameter. Both parts originate from the same substrate, ensuring identical radii of curvature (R = 70 mm). The miniature central mirror is mounted on a wide-range nanometer-precision piezo-driven stage, allowing alignment and translation with respect to the external part

and IR beam with an adjustable time delay into a common focus of the two focusing mirrors, where a nozzle supplying the target atoms is placed. The focusing mirror is coated with a Mo/Si multilayer stack designed to reflect photons with 90 eV energy within a 5 eV bandwidth. This bandwidth is large enough to support XUV pulses as short as 0.4 fs. The contour plot in Fig. 4.51 depicts the variation of the energy distribution of photoelectrons knocked out

Photoelectron energy (eV) 85 80 75 70 65 60 –15

–10

–5

10 15 Delay td(fs)

Fig. 4.51 Energy distributions of the Kr-4p photoelectrons (binding energy ≈15 eV) as a function of the delay td between the ionizing sub-femtosecond 90 eV XUV pulse and the dressing few-cycle laser wave over some tens of femtoseconds

Nonlinear Optics

Counts per 1eV bin 120

td = – 0.45 fs

td = 0 fs

0 60

85 75 80 Photoelectron energy (eV)

Fig. 4.52 Kr-4p photoelectron spectra extracted from the contour plot in Fig. 4.51 at delays indicated by the arrows. The dots represent spectra normalized to the same number of electron counts in the time-of-flight electron spectrometer. The lines show asymmetric Gaussian fits to the data

νinst (PHz)

233

λinst (nm)

0.55

550

0.50

600 650

0.45

700 750 800

0.40 –6

–4

–2

0 2 τrise < 1 fs

4 6 Delay td(fs)

Fig. 4.53 Calculated (line) and measured blue shift of the fundamental pulse, probed by the XUV pulse. The steep rising edge indicates the presence of single XUV pulses without significant satellites

can be evaluated from the modulation in ∆W(td ) by fitting a sinusoidal half-oscillation of adjustable period to ∆W(td ) at different values of td . The sampling interval was scanned through the range of −8 fs ≤ td ≤ 8 fs. The dots in Fig. 4.53 show the carrier-frequency sweep evaluated in this manner, revealing a dynamic blue shift from a carrier wavelength of ≈750 to ≈550 nm. This strong dynamic frequency shift carried by the few-cycle laser pulse has been found to originate from the interaction of the pulse with the ionizing gas source of the XUV pulses. In fact, the line in Fig. 4.53 is obtained from propagating a bandwidth-limited 7 fs laser pulse through the volume of neon gas emitting the high harmonics in a numerical experiment [4.331]. The measured ≈30% dynamic frequency blue shift at the center of the pulse is larger than predicted by the simulations but reflects qualitatively the predicted behavior. The qualitative agreement suggests that the observed blue shift can be attributed to ionization-induced self-phase modulation in the high-harmonic-generation process. This direct probing of the field oscillations of a light wave can be regarded as the first application of sub-femtosecond XUV pulses. The observed sub-femtosecond rise time of the ionization-induced dynamic blue shift provides conclusive evidence for the isolated nature of the sub-femtosecond XUV burst. Any satellite of substantial energy spaced by some T0 /2 ≈ 1.30 fs from the main burst would broaden this rise time to more than 1 fs. The agreement of the measured XUV pulse duration and that obtained from simulations [4.285] within the experimental error suggests that the timing jitter of

Part A 4.12

from the 4 p shell of krypton by 90 eV XUV pulses (y-axis) as a function of delay td (x-axis). The XUV pulses were filtered out from the high-harmonic emission spectrum of Ne atoms driven by few-cycle (7 fs, 750 nm) laser pulses by the Mo/Si mirror within a 5 eV bandwidth at 90 eV near the cutoff of the emission spectrum. The XUV pulse were delayed by 150 as steps with respect to the few-cycle laser pulse to record the laser-affected XUV photoelectron spectra versus delay. The data clearly bring to light a quasi-periodic evolution of the photoelectron energy spectrum with a period of ≈ T0 /2. From the depth of this modulation an XUV pulse duration of τXUV = 650 as ± 150 as was evaluated [4.285]. Figure 4.52 shows two representative spectra sliced out from the contour plot in Fig. 4.51 at delays of td = −450 as and td = 0, as indicated by the arrows. The marked difference between the two spectra recorded at delays just 450 as apart provides a clear evidence for the photoelectron emission time (and hence the XUV pulse) being temporally confined to a fraction of a femtosecond. In the analysis presented in the previous subsections, the laser field was used to probe the XUV pulse duration. With its duration known, the sub-femtosecond XUV pulse can take over the role of the probe as it is scanned across the laser pulse (in time) and measure the frequency of light for the first time in a time-resolved experiment. A possible sweep of instantaneous frequency νinst (or wavelength λinst ) of the few-cycle laser pulse

4.12 Attosecond Pulses: Measurement and Application

234

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Basic Principles and Materials

the XUV pulse with respect to the phase of the IR light wave must be small compared to 1 fs, indicating that the sub-femtosecond XUV pulse is locked to the carrier wave of its generating few-cycle light pulse with as precision. The attosecond timing stability of the subfemtosecond XUV pulse to a few-cycle IR light wave makes these pulses a unique tool for investigating the dynamic behavior of matter on an attosecond time scale. The light-field-controlled photoemission experiment already demonstrates this capability. In the measured energy distribution ∆W(td ) it is implicit, that in the investigated spectral range near 90 eV bound–free electronic transitions from the 4p state in krypton respond to XUV excitation within less than 500 as, constituting I x (t)

Part A 4.12

E L (t)

Detector

the first time-resolved measurement on an atomic time scale. Another implementation of the attosecond streakcamera technique is shown in Fig. 4.54 [4.286]. Here the photoelectrons are detected in the parallel geometry as sketched in the top of Fig. 4.48. If the photoelectron is ejected near the zero crossing of the laser electric field, its energy spectrum may get up- or downshifted by many electronvolts. This is in strong contrast with the orthogonal geometry used before, where the primary effect of the laser field is a broadening rather than a shift of the photoelectron energy spectrum. The predicted spectral shift relies not only on a precise timing of the XUV pulse to the zero transition of the laser field. This shift without substantial broadening happens only if the generated XUV burst (and therefore the generated photoelectron wave packet) is very short compared to the laser field half-oscillation period T0 /2. If the XUV pulse and therefore the electron wave packet approaches or exceeds T0 /2, different portions of the wave packet experience different momentum transfer, resulting in a large spread of energies. A recent experiment using XUV pulses of 8 eV bandwidth verified this prediction and allowed to set an upper limit of 500 as and 200 as on the duration of the XUV pulse and its timing jitter with respect to its generating laser wave, respectively [4.286].

4.12.4 Applications of Sub-femtosecond XUV Pulses: Time-Resolved Spectroscopy of Atomic Processes

Electron energy

hνx 0

Wbind

Fig. 4.54 Energy shift of XUV photoelectrons ejected nearly parallel to the polarization direction of a strong laser field near the instant of zero transition of the laser field by a sub-T0 /2-duration XUV pulse

Finally, we show, how pump–probe spectroscopy of atomic excitation and relaxation processes – such as e.g. optical-field ionization and inner-shell relaxation processes – can be traced directly in the time domain by drawing on low-energy isolated X-ray pulses in combination with synchronized strong few-cycle laser pulses, i. e. on tools that are available now. Pump–probe techniques offer the most direct experimental approach to tracing microscopic dynamics. The extension of time-resolved spectroscopy to ultrafast electronic processes taking place deep inside atoms has so far been frustrated by the simultaneous requirements of short wavelengths (i. e. high photon energy) and sub-femtosecond pulse duration. There is the additional difficulty that a straightforward interpretation of spectroscopic data requires isolated (single) pulses. With these pulses now available from few-cycle-driven highorder harmonic generation, extension of time-resolved spectroscopy into the attosecond domain can now be

Nonlinear Optics

4.12 Attosecond Pulses: Measurement and Application

Electron energy

E L (t) t=0

t>0

Sub-fs X-ray pump

235

Photoelectron spectrum versus td

Sub-fs X-ray probe O E x (t)

Fig. 4.55 Tracing inner-shell relaxation processes by

XUV–pump/XUV–probe spectroscopy. The kinetic energy of photoelectrons detached by the probe pulse from the atom following excitation by an XUV pump pulse is analyzed as a function of the delay between the pump and the probe pulse

Electron energy

E L (t)

O E x (t)

Fig. 4.56 Schematic representation of the Auger process and its temporal evolution displaying the decay of an innershell vacancy

time of this Auger electron corresponds exactly to the lifetime of the inner-shell vacancy. Hence, sampling the Auger electron emission in the same way as the primary photoelectron emission with the oscillating laser field enables researchers to gain direct time-domain access to inner-shell atomic processes with as resolution. The first proof-of-principle time-resolved inner-shell spectroscopic experiment was recently demonstrated [4.343]. Although the decay time measured in this experiment can also be inferred from energy-domain measurements, it served as a benchmark process for testing the feasibility of the methods described above for attosecond time-resolved spectroscopy. The frontiers of both time-resolved spectroscopy and the control of microscopic dynamics are about to be radically extended due to the new technical capability that allows scientists to synthesize intense pulses containing a few cycles evolving in a precisely determined way

Part A 4.12

tackled. Figure 4.55 shows the principle of an XUVpump/XUV-probe experiment. In this – conceptually most straightforward – implementation of as spectroscopy, the sub-femtosecond XUV pulses are used for both triggering and probing bound–bound or bound–free transitions in atoms or molecules. However, the sub-femtosecond XUV pulses currently available do not have sufficient flux for XUV–pump/XUV–probe spectroscopy yet. This is because in this kind of experiments the physical quantity measured as a function of delay between pump and probe pulse relies on a two-XUV-photon transition, the probability of which scales with σ Ipump Iprobe , where σ is the atomic cross section of the absorption process and Ipump and I probe are the intensities of the pump and probe XUV bursts, respectively. Both Ipump and Iprobe are many orders of magnitude less than what can be achieved in the optical regime. Moreover σ is also reduced by many orders of magnitude because it scales with λ6 . Extension of the concept of laser-field-assisted XUV photoelectron emission to sampling the emission of secondary (Auger) electrons offers an alternative solution to time-resolving inner-shell atomic processes with as accuracy. The underlying principle is illustrated in Fig. 4.56. The sub-femtosecond XUV pulse excites a core electron and produces thereby a shortlived inner-shell vacancy. This is rapidly filled by an electron from a higher energy level (outer shell). The energy lost by the electron undergoing this transition is carried away either by an energetic (XUV/X-ray) photon or by a secondary (Auger) electron. The emission

Auger spectrum versus td

236

Part A

Basic Principles and Materials

[4.330]. With this tool it should be possible to precisely control the motion of energetic electron wave packets around atoms on attosecond timescales just as the motion of nuclear wave packets in molecules can be controlled within a few femtoseconds. The single sub-femtosecond electron bunches and (XUV/X-ray) photon bursts that arise from the recently gained ability to control electron wave packets will enable the scientific community to excite and probe atomic dynamics on atomics time scales.

4.12.5 Some Recent Developments

Part A 4

Attosecond science is rapidly evolving [4.344] and the last three years have seen important progress both in the performances of femtosecond and attosecond light pulses based on high-order harmonic generation in gases and in their applications in different scientific areas. Harmonic sources now reach pulse energies in the microjoule range [4.345] and their spectra extend to energies of several keV (though with a lower

throughput) [4.346]. Applications are flourishing, going from the determination of vibration frequencies in molecules [4.347] to microscopy [4.348, 349] and even recently to seeding of X-ray laser plasmas [4.350] and possibly, in the future, free-electron lasers. Attosecond pulses have been studied in more details with different techniques [4.351, 352] and the method to isolate a single pulse refined [4.353, 354]. Their time-frequency characteristics have been mapped out [4.355–357] and ways to control both the individual pulses [4.358] and the train structure [4.359] have been developed. The shortest isolated pulse produced to date is 130 as, using the polarization gating technique for the temporal confinement [4.360, 361]. Applications now include characterization of electromagnetic fields [4.362], tomography of molecular orbitals [4.363], molecular dynamics studies of simple molecules [4.364, 365], dynamical studies and interferometric measurements of electron wavepackets [4.366, 367], and time-resolved inner-shell spectroscopy in atoms and solids [4.368].

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Nonlinear Optics

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4.330

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References

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Part A 4

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249

Optical Mater 5. Optical Materials and Their Properties

This chapter provides an extended overview on today’s optical materials, which are commonly used for optical components and systems. In Sect. 5.1 the underlying physical background on light–matter interaction is presented, where the phenomena of refraction (linear and nonlinear), reflection, absorption, emission and scattering are introduced. Sections 5.2 through 5.8 focus on the detailed properties of the most common types of optical materials, such as glass, glass ceramics, crystals, and plastics. In addition, special materials displaying “unusual nonlinear” or “quasi-nonreversible” optical behavior such as photorefractive or photorecording solids are described in Sect. 5.9. The reader could use this chapter as either a comprehensive introduction to the field of optical materials or as a reference text for the most relevant material information.

5.4 Laser Glass ........................................... 5.4.1 Common Laser Glasses and Properties ............................. 5.4.2 Laser Damage .............................. 5.4.3 Storage and Handling of Laser Glass

293 297 300

5.5 Glass–Ceramics for Optical Applications .. 5.5.1 Overview ..................................... 5.5.2 Properties of Glass–Ceramics ......... 5.5.3 Applications ................................

300 300 301 306

5.6 Nonlinear Materials .............................. 5.6.1 Overview on Nonlinear Optical Materials..................................... 5.6.2 Application: All Optical Switching ..................... 5.6.3 Second Harmonic Generation in Glass....................................... 5.6.4 Glass Systems Investigated for Nonlinear Effects ..................... 5.6.5 NL-Effects in Doped Glasses...........

307

5.2

5.3

Interaction of Light with Optical Materials ........................... 5.1.1 Dielectric Function ....................... 5.1.2 Linear Refraction.......................... 5.1.3 Absorption .................................. 5.1.4 Optical Anisotropy ........................ 5.1.5 Nonlinear Optical Behavior and Optical Poling ........................ 5.1.6 Emission ..................................... 5.1.7 Volume Scattering ........................ 5.1.8 Surface Scattering ........................ 5.1.9 Other Effects ................................ Optical Glass ........................................ 5.2.1 Chronological Development ........... 5.2.2 Compositions of Modern Optical Glass ................ 5.2.3 Environmentally Friendly Glasses ... 5.2.4 How to Choose Appropriate Optical Glasses .......................................

250 250 255 258 261 265 269 271 275 278 282 282 283 287 288

Colored Glasses .................................... 290 5.3.1 Basics ......................................... 290 5.3.2 Color in Glass ............................... 292

307 312 313 313 314

Plastic Optics ........................................ 5.7.1 Moulding Materials ...................... 5.7.2 Manufacturing Methods ................ 5.7.3 Manufacturing Process .................. 5.7.4 Coating and Component Assembly ............. 5.7.5 New Developments.......................

317 317 319 320

5.8 Crystalline Optical Materials................... 5.8.1 Halides, CaF2 ............................... 5.8.2 Semiconductors ........................... 5.8.3 Sapphire ..................................... 5.8.4 Optic Anisotropy in Cubic Crystals ...........................

323 323 325 325

5.9 Special Optical Materials........................ 5.9.1 Tunable Liquid Crystal Electronic Lens ........................................... 5.9.2 OLEDs.......................................... 5.9.3 Photorefractive Crystals ................. 5.9.4 Metal Mirrors ...............................

327

322 322

326

327 333 339 346

5.10 Selected Data ....................................... 354 References .................................................. 360

Part A 5

5.7 5.1

293

250

Part A

Basic Principles and Materials

5.1 Interaction of Light with Optical Materials In this section the general physics of the interaction of light with matter is briefly presented. A detailed insight into theoretical electrodynamics cannot be given here. The interested reader might refer to standard textbooks on electrodynamics e.g. [5.1, 2].

where r = (x, y, z) are the three spatial coordinates. With these simplest possible boundary conditions the material equations (5.3) read

5.1.1 Dielectric Function

After applying a few vector operations, one gets the wave equation for the electromagnetic field E in vacuum:

The starting point for an analysis of any interaction between electromagnetic waves with matter is Maxwell’s equations. The static interaction for the dielectric displacement and the magnetic induction is described by ∆ ∆

·D=ρ, ·B=0,

(5.1)

whereas the dynamic interaction of the electric and magnetic fields is given by ˙ , × E = −B ˙ . ×H= j+D

∆

(5.2)

∆

Part A 5.1

E and B are the electric and magnetic fields; D and H are the electric displacement and the auxiliary magnetic fields; ρ and j are the charge and the current density. Material equations are needed to close Maxwell’s equations: D = ε0 E + P , B = µ0 H + M ,

(5.3)

where P and M are the polarization and magnetization densities. The vacuum permittivity (in SI units) is ε0 = 8.854 × 10−12 A s/V m and the vacuum permeability is µ0 = 4π × 10−7 V s/A m. The complete optical properties for any spatial combination of matter are included in the solution of (5.1), (5.2), which are closed by using the material equations (5.3) and by using appropriate boundary conditions. For only a few special cases such a solution can be written down directly. In the following we give a few examples. Wave Equation in Vacuum If we want to solve (5.1), (5.2) in infinite vacuum we have the following boundary conditions and material equations:

P(r) = 0 ,

M(r) = 0 ,

ρ(r) = 0 ,

j(r) = 0 , (5.4)

D = ε0 E , B = µ0 H .

∆E − µ0 ε0 E¨ = 0 .

(5.5)

(5.6)

An identical wave equation can be derived for the magnetic field B. (5.6) immediately defines the speed of light c (in vacuum). 1 c= . (5.7) µ0 ε0 Equation (5.6) is generally solved by all fields which fulfill E(r, t) = E0 · f (kr ± ωt) involving any arbitrary scalar function f . The most common systems of function f are plane waves: Es (r, t) = E0 Re e−i(kr−ωt) . (5.8) These plane waves, with a time and spatial dependent phase θ = kr, are described by a wave vector k, an angular frequency ω, and a corresponding wavelength λ = 2π/k = 2πc/ω, where k = |k| is the absolute value of the wave vector. Describing an arbitrary field E in terms of plane waves is identical to decomposing this electrical field into its Fourier components. Wave Propagation in an Ideal Transparent Medium We can describe an ideal material by simply replacing the speed of light in vacuum by that of the medium. c c→ , (5.9) n where n is the (in this case only real) refractive index of the material. Wave propagation in a dispersing or weakly absorbing medium is considered at the end of the present section. In fact, most parts of an optical design can be done by treating optical glasses as such ideal transparent materials (see Sect. 5.1.2). Even though such an ideal material cannot exist in reality, optical glasses come very close to it (for electromagnetic radiation in the visible range). For such an ideal material the wave equation (5.6) reads:

∆E −

n2 E¨ = 0 . c2

(5.10)

Optical Materials and Their Properties

It is solved again by plane transverse waves. Where the speed of light is now reduced to the speed of light in the transparent medium cmed = c/n and the wavelength of the lightwave is reduced to λmed = λ/n. Refraction and Reflection We now derive the laws of refraction and reflection for the ideal transparent medium just described. They are obtained by solving Maxwell’s equations at the (infinite) boundary between two materials of different refractive indices n 1 and n 2 (see Fig. 5.1). As boundary conditions one obtains that the normal component of the electric displacement (and magnetic induction) and the tangential component of the electric (and magnetic) field have to be continuous at the interface:

D1n = D2n ,

E 1t = E 2t .

(5.11)

Further, a phase shift of an incoming wave occurs upon reflection θr = π − θi ,

(5.12)

where θr,i are the phases of the reflected and incident wave, respectively. If we solve Maxwell’s equations for an incoming plane wave (applying the boundary conditions stated above), Snell’s law of refraction is obtained (5.13)

together with that of reflection αr = α1 .

(5.14)

E ||0i E⊥0r

E⊥0i

α1 α1 ki

n1 n2

α2 E ||0t E⊥0t

Now, the electric field E is decomposed into its components which are defined relative to the plane outlined by the three beams of incoming, transmitted and reflected light. This decomposition is shown in Fig. 5.1. The coefficients for reflection and transmission are defined as:

r = r⊥ =

E 0r

E 0i

⊥ E 0r ⊥ E 0i

;

t =

;

t⊥ =

Fig. 5.1 The polarization directions of the E and B fields

for reflection and refraction at an interface between two optical materials of different refractive indices are shown. A circle indicates that the vector is perpendicular to the plane shown

E 0t

E 0i

⊥ E 0t

⊥ E 0i

, .

(5.15)

The Fresnel formula for these coefficients can be derived as sin(α1 − α2 ) n 1 cos(α1 ) − n 2 cos α2 =− , r⊥ = n 1 cos(α1 ) + n 2 cos(α2 ) sin(α1 + α2 ) tan(α1 − α2 ) n 2 cos(α1 ) − n 1 cos α2 =− , r = n 1 cos(α2 ) + n 2 cos(α1 ) tan(α1 + α2 ) 2n 1 cos(α1 ) t⊥ = n 1 cos(α1 ) + n 2 cos(α2 ) 2 sin(α2 ) cos(α1 ) , =− sin(α1 + α2 ) 2n 1 cos(α1 ) t = (5.16) n 1 cos(α2 ) + n 2 cos(α1 ) 2 sin(α2 ) cos(α1 ) . =− sin(α1 + α2 ) cos(α1 − α2 ) Here the usual convention has been used that the coefficients of reflectivity obtain an additional minus sign in order to indicate backtraveling of light. The quantities that are measured in an experiment are intensities. The relationship between the intensities defines the reflectivity and transmissivity of a material R⊥ := |r⊥ |2 ;

R := |r |2 ,

T⊥ := |t⊥ | ;

T := |t |2 .

E ||

251

(5.17)

The angular-dependent coefficients of reflection from (5.16) are displayed in Fig. 5.2. In Fig. 5.2a the case of light propagating from an optically thin medium with refractive index n 1 to an optically thicker medium with refractive index n 2 > n 1 is plotted. At the so called Brewster angle αB the reflected light is completely polarized; αB is given by the condition α1 + α2 = π/2. Therefore, the Brewster angle αB results as a solution of π n2 α1 = − arccos cos α1 (5.18) 2 n1 which gives αB = arctan nn 21 . In Fig. 5.2b the case of light propagating from an optically thick to an opti-

Part A 5.1

n 1 sin α1 = n 2 sin α2

5.1 Interaction of Light with Optical Materials

252

Part A

Basic Principles and Materials

absorbing media discussed at the end of the present section. Here, only the refractive indices have to be replaced by the complex quantities of (5.36). It is further helpful to define a transmittivity and reflectivity for unpolarized light

a) R 1

Runpol =

r⊥

⊥2 E 0r + E 0r 2

⊥2 E 0i + E 0i

;

T unpol =

⊥2 E 0t + E 0t 2

⊥2 E 0i + E 0i

. (5.21)

r ||

0 α1

b) R

αB

π/2

unpol Rall

r ||

αB

2 tan2 (α1 −α2 ) ⊥2 sin2 (α1 −α2 ) + E 0i tan2 (α1 +α2 ) sin2 (α1 +α2 ) 2 ⊥2 E 0i + E 0i

E 0i

With the definitions from (5.21) the following sum rule must be fulfilled: n 2 cos α2 unpol Runpol + T =1. (5.22) n 1 cos α1 The rule provides an easy check for transmitted and reflected total intensities, especially for normal incidence.

r⊥

Inserting the expressions for the reflection coefficients we obtain e.g. the total reflectivity as a function of the incident and refracted angular

Part A 5.1

αT α1

π/2

Fig. 5.2a,b The reflection coefficients are plotted as a func-

tion of incident scattering angle for light propagating from a medium of (a) smaller refractive index into a medium of larger index and (b) larger refractive index into a medium of smaller index. Here, total reflection occurs at an angle αT and αB is the Brewster angle

cally thin medium is plotted. Here an additional special angle occurs – the angle of total reflection, αT . All light approaching the surface at an angle larger than αT is totally reflected. At α1 = αT the angle for refraction in the medium with refractive index n 2 is α2 = π/2. For αT it follows that: n2 . αT = arcsin (5.19) n1 Evaluating (5.16) for the special case of incident light as limα→0 allows one to calculate the reflectivity for normal incidence n 1 − n 2 2 . Rnorm = (5.20) n1 + n2 For practical applications it is important to note that the Fresnel equations remain valid in the case of weakly

Wave Propagation in an Isotropic, Homogeneous Medium We now consider wave propagation in an ideal optical material. This is a nonmagnetic, homogeneous, isotropic, perfectly insulating medium, which is further a perfectly linear optical material. Considering time dependence including retardation in the materials (5.2) leads to:

D(r, t) = ε0 E(r, t) + P(r, t) .

(5.23)

The polarizability is related to the electric field via the susceptibility χ. In the case of a homogeneous isotropic material χ is a scalar function. In Sect. 5.1.4 we will consider the case of optically anisotropic media, where χ becomes a second-rank tensor. P(r, t) =

dt χ r − r , t − t E r, t .

−∞

(5.24)

Fourier transformation in time and space deconvolutes the integral and leads to: P(k, ω) = χ(k, ω)E(k, ω) ,

(5.25)

where χ(k, ω) is, in general, a complex analytic function of the angular frequency ω. The complex function χ(k, ω) unifies the two concepts of a low-frequency

Optical Materials and Their Properties

polarizability χ and a low-frequency conductivity σ of mobile charges to a single complex quantity lim χ(ω) = χ (ω) + 4πi

ω→0

σ(ω) . ω

(5.26)

At larger frequencies the separation of the two concepts breaks down, since above the frequencies of optical phonon modes in the IR the bound charges are unable to follow the electric field, whereas below the phonon modes the charges can follow this motion (Sect. 5.1.3). The usual form in which the susceptibility enters the equations for optical purposes is via the dielectric function ε(k, ω) = 1 + χ(k, ω) .

(5.27)

Inserting the dielectric function into the material equation (5.23) gives: D(k, ω) = ε0 ε(k, ω)E(k, ω) .

(5.28)

Dielectric function ε IR

VIS

Re(ε) Im(ε) ωIR

ωUV Frequency ω

Fig. 5.3 Dielectric function ε(ω) for the “model optical solid” with one generic absorption in the infrared ωIR and a second one in the ultraviolet ωUV . The dielectric function is plotted on a logarithmic energy scale. The solid line is the real part and the dashed line is the imaginary part of ε(ω)

253

where the relations ∂ 2 /∂t 2 Es (r, t) → −ω2 Es (r, t) and ∆Es (r, t) → −k2 Es (r, t) have been used. The expression in brackets in (5.29) defines the dispersion relation for an optically linear, homogeneous, isotropic material. Poynting Vector and Energy Transport The energy flux density of the electric field is obtained via the Poynting vector, given by

S= E× H .

(5.30)

It gives the rate at which electromagnetic energy crosses a unit area and has the unit W/m2 . It points in the direction of energy propagation. The time average of the absolute value of the Poynting vector |S| is called the intensity I of the electromagnetic wave 1 I = |S| = |E × H| (5.31) 2 and is the energy flux density of the electromagnetic radiation. In thespecial case of propagation of transverse plane waves [as given in (5.8)], it simplifies to 1 n I= |E0 |2 , (5.32) 2 cµ0 where in vacuum n = 1 is valid. General Form of the Dielectric Function For most optical materials the dielectric function has a form in which a transparent frequency (or wavelength) window is bounded at the high energy site by electron– hole excitations (dominating the UV edge) and at the low energy site by IR absorptions given by optical phonon modes (lattice vibrations). The general form of the dielectric function is given by the Kramers–Heisenberg equation [5.3] αk, j ε(k, ω) = 1 + . (5.33) 2 − ω2 − iωη ω k, j k, j j

Here αk, j is the amplitude, ωk, j the frequency and ηk, j the damping of the particular excitation j. A schematic view of the dielectric function is plotted in Fig. 5.3. Here we use a model for a transparent homogeneous, isotropic solid (such as glass) with one generic absorption at low energies (ωIR in the infrared, IR) and another one at large phonon energies (ωUV in the ultraviolet spectral range, UV). In the following this model solid is used to discuss optical material properties. Dispersion Relation Solving (5.29) gives two frequency-dependent solutions for the wave vector k as a function of ω since the lefthand side of (5.29) is quadratic in k. Far away from

Part A 5.1

Here, we restrict ourselves to ideal optically isotropic materials by neglecting the nature of the dielectric function as a second-rank tensor. In Sect. 5.1.4 we will extend our considerations to optically anisotropic materials. With the same steps as in Eqs. (5.5–5.6) a wave equation can be derived which has the following form in Fourier space:

ω2 k2 − ε(k, ω) 2 E0 = 0 , (5.29) c

5.1 Interaction of Light with Optical Materials

254

Part A

Basic Principles and Materials

Velocity v IR

Vacuum

VIS

Im(k) ωUV

vph Re(k)

vgr

ωIR ωIR

Fig. 5.4 Dispersion relation ω(k) for the “model optical solid” on a double logarithmic scale. The solid line is the real part of k and the dashed line the imaginary part of k. For comparison the simply linear dispersion relation for light propagating in vacuum ω = ck is shown with a long-dashed line

ωUV

Fig. 5.5 The group velocity vgr (k) = ∂ω/∂k (solid line) and the phase velocity vph (k) = ω/k (dashed line) are plotted on a logarithmic frequency scale. Far away from absorptions both velocities approach each other while the group velocity is always smaller than the phase velocity

velocity

Part A 5.1

absorptions the dielectric function is real and positive. Here only one solution exists, which describes the wave propagating with the speed of light in the medium. Close to an absorption two solutions exist which are even more complicated. This means that near a resonance the dispersion of the light cannot be considered independently from the dispersion of the excitations in the material. They both form a composite “new” entity propagating in the medium. This is called the polariton [5.4]. For our model solid with two generic absorptions (ωIR and ωUV ), the dispersion is plotted in Fig. 5.4. Wave Propagation, Phase and Group Velocity When an electromagnetic wave propagates through a medium one can define two velocities. The phase velocity is the speed with which a certain phase propagates. It is, for example, the velocity of the wavefront maxima moving through the medium. The phase velocity is given as

vph =

ω . k

(5.34)

In Fig. 5.5 the phase velocity of the model solid is plotted on a logarithmic frequency scale. Close to the absorption edges of the material it loses its meaning because attenuation due to the absorption processes will dominate most processes. Far away from absorptions it reaches a nearly constant value. The second velocity is the group

∂ω . (5.35) ∂k This is the velocity at which a complete wave packet travels through the medium and is, hence, the speed with which information can travel through the system. It is plotted in Fig. 5.5 for the model solid. Reasonably far away from material absorptions the group and phase velocities approach each other. However, the group velocity is always smaller than the phase velocity. vgr (k) =

Refractive Index The refractive index n is the most widely used physical quantity in optical design. It is the square root of the dielectric function. The dynamic refractive index is generally a complex quantity

n(ω) = n(ω) + iκ(ω) ˜

(5.36)

and must fulfill the Kramers–Kronig relations [5.3]. The refractive index for our generic model solid is plotted as a function of logarithmic frequency in Fig. 5.6. In practical use, the wavelength dependence is often exploited

n(k, (5.37) ˜ λ) = ε(k, 2πc/λ) . With a few basic steps, (5.33) can be rewritten as a function of wavelength alone. If one further restricts to wavelengths which are far away from absorptions ω2 − ω2k, j ωηk, j , the Sellmeier formula (see

Optical Materials and Their Properties

Refraction Index n

5.1 Interaction of Light with Optical Materials

255

Reflectivity R

VIS

~ Re(n)

~ Im(n)

ωIR

ωUV Frequency ω

Fig. 5.6 The complex refractive index n(ω) = n ∗ (ω) + iκ(ω) is plotted on a logarithmic frequency scale. The real part is plotted with a solid line and the imaginary part with a dashed line

ωIR

ωUV Frequency ω

Fig. 5.7 Reflectivity R for normal incidence plotted on a logarithmic frequency scale

magnetic fields at point P2 are given by ω

Sect. 5.1.2), which is widely used for characterizing optical materials, is obtained n(λ)2 ≈ 1 +

B j λ2 j

λ2 − λ2j

(5.38)

Wave Propagation in Weakly Absorbing Medium In this subsection the link between the attenuation of a wave and the imaginary part of the refractive index is given. A weakly absorbing medium is defined by the imaginary part of the refractive index (5.37) being much smaller than the real part

κn

(5.39)

(the coefficient κ/n is also called the attenuation index). In this case light propagates as transverse waves through the medium. We consider two points in our medium: P1 and P2 . Between these points the light travels the distance l. In the absence of absorption, the electric and

˜ H2 = H1 ei c nl .

(5.40)

Using (5.31) we obtain for weak absorption the radiation intensity at point P1 . The radiation intensity at P2 is: 1 I2 = |E2 × H2 | , 2 ω 1 = |E1 × H1 | e−2 c κl , 2 ω = I1 e−2 c κl = I1 e−αl .

(5.41)

The absorption coefficient α is connected to the complex refractive index by ω α=2 κ; (5.42) c α can easily be measured and its importance for optical properties is discussed in Sect. 5.1.3. It is also important to note that the Fresnel equations (5.16) remain valid in the case of a weakly absorbing medium if the complex refractive indices are used. As an example, we plot the reflectivity (at the interface air– model solid) near an absorption resulting from (5.20) for a complex refractive index n 2 → n˜ 2 . In Fig. 5.7 the reflectivity is plotted on a logarithmic frequency scale. Note that the absorption seems to be shifted compared to the plots of the complex dielectric function or the complex refractive index. Measurement of the reflectivity is of importance for reflection spectroscopy.

5.1.2 Linear Refraction As already introduced in Sect. 5.1.1, two phenomena occur when light impinges upon the surface of any optical material: reflection and refraction [5.5]. The reflected

Part A 5.1

where λi = 2πc/ωi with i ∈ {(k, j)} is used and B j = ak, j λk, j /(2πc)2 . Normally B j and λ j are just fitting constants to describe the dispersion of the refractive index over a certain wavelength range. They are, however, connected to the microscopic fundamental absorption behavior of the material. Sometimes also n(λ) and not n(λ)2 is approximated with a Sellmeier formula. Since as well n(λ) as well as n(λ)2 are complex differential (analytic) functions both formula give refractive indices and dispersions with the same accuracy. However care has to be taken, which quantity is expressed when using a Sellmeier formula.

˜ , E2 = E1 ei c nl

256

Part A

Basic Principles and Materials

light bounces off the glass surface, while the refracted light travels through the material. The amount of light that is reflected depends on the refractive index of the sample, which also affects the refractive behavior of the sample [5.6]. The refractive index of optical materials turns out to be one of the most important factors that must be considered when designing systems to transmit and modulate light [5.7]. The refractive index is a complex material property that depends on temperature and wavelength [5.8]. The wavelength dependence of the refractive index is the dispersion [5.5]. Law of Refraction When a light ray impinges upon a glass surface, a portion is reflected and the rest is either transmitted or absorbed. The material modulates the light upon transmission. The light travels at a different velocity as it is transmitted through the glass as compared to through vacuum. As introduced in Sect. 5.1.1, the index of refraction (n), is defined as the ratio of the speed of light in vacuum (c) to that in the material (cm ) [5.11]:

c . cm

(5.43)

Part A 5.1

Most commonly, the reported refractive indices are relative to the speed of light in air, rather than in vacuum, no matter which technique is used to measure the refractive index [5.12]. The index of refraction for vacuum, by definition, must be exactly 1. The index of refraction of air is 1.00029 at standard temperature (25 ◦ C) and pressure (1 atm) (STP). Therefore, the index of refraction of optical matter (n rel ) relative to air (n air ), rather than vacuum is [5.11]: n rel =

nm . n air

(5.44)

Table 5.1 Indices of common materials at standard tem-

perature and pressure at 587.56 nm (helium d line) [5.9] Material

Material

Vacuum Air Water Acetone Ethanol Sugar solution (30 wt %) Fused silica Sugar solution (80 wt %)

1 1.00029 1.33 1.36 1.36 1.38

Crown glass Sodium chloride Polystyrene Carbon disulfide Flint glass Sapphire

1.52 1.54 1.55 1.63 1.65 1.77

1.46 1.49

Heavy flint glass Diamond

1.89 2.42

θi

θr

θt

Fig. 5.8 Ray tracings of incident, reflected, and transmitted

light from one medium to another representing the angles and indices necessary to apply Snell’s law (after [5.10])

The STP indices of some common compounds, and classes of compounds, are shown in Table 5.1 [5.9]. As discussed in Sect. 5.1.1, when light hits a glass surface at an angle αi , it is Fresnel-reflected back at an angle αt . The angle of incidence is equal to the angle of reflection (αi = αr ), as shown in Fig. 5.8 [5.10]. The percentage of light reflected for αi = 0 at each interface (R) relative to the incident intensity (Sect. 5.1.1) is dependent on the index of refraction of the two media the light is passing through, typically air (n 2 ) and glass (n 1 ) Fig. 5.8 and (5.20) [5.10] n1 − n2 2 . (5.45) Rnorm = 100 n1 + n2 Fresnel’s formula (5.45) assumes smooth surfaces that produce only specular reflection. Diffuse reflection occurs when the surface is rough, so the incident light is reflected through a range of angles, thereby reducing the intensity of the reflected light at any given angle [5.10] (see Sect. 5.1.8). The specular reflection that is taken into consideration by Fresnel’s relationship can be monitored and used to estimate the refractive index of samples in situ [5.13]. The angle of the light transmitted within the material (αt ), relative to the incident light transmitted through air, is dependent on the refractive indices of the air (n air ) and solid (n m ) and the incident light angle (αi ) [5.10]: n air sin αi = n m sin αt .

(5.46)

Optical Materials and Their Properties

1.53 BK-7 n F = 0.486 ␮m 1.52

n D = 0.589 ␮m n C = 0.656 ␮m

1.51

1.50

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.9 2.0 λ (␮m)

Fig. 5.9 Dispersion present in BK-7 optical glass. Common index of refraction measurement wavelengths are indicated

This is the general form of Snell’s law of refraction to predict transmitted angles in media [5.14].

Anomalous dispersion is an increase in refractive index with an increase in wavelength. Anomalous dispersion typically occurs at frequencies that represent a crossover between the polarization mechanisms (dipolar, ionic, electronic etc.) that are responsible for absorption of electromagnetic radiation. This is further discussed in [5.11]. Due to dispersion, the index of refraction must be reported with the wavelength of measurement. The most common wavelengths at which n is measured are reported in Table 5.2. These wavelengths most often correspond to common sharp emission lines. The index can be determined most accurately ± 1 × 10−6 by measuring the angle of minimum deviation of light in a prism [5.15]. However, a Pulfrich refractometer ± 1 × 10−5 is most commonly used in industry. Details of measurement techniques are given in [5.16]. The index at various wavelengths is commonly referred to by the designations in Table 5.2: i. e. n d is the refractive index measured at the yellow helium d line of 582.5618 nm. The dispersion is often given as a difference in n at two wavelengths. For instance, the primary dispersion is given by n F − n C (hydrogen lines) and n F − n C (cadmium lines). The most commonly reported measure of dispersion is the Abbe number (ν), which is Table 5.2 Wavelength of spectral lines used for measur-

ing index of refraction, with the common designation and spectral line source [5.11] Wavelength (nm) 2325.4 1970.1 1529.6 1013.98 852.1101 706.5188 656.2725 643.8469 632.8 589.2938 587.5618 546.074 486.1327 479.9914 435.8343 404.6561 365.0146

Designation

t s r C C D d e F F g h i

Spectral line Hg IR line Hg IR line Hg IR line H IR line Cs IR line He red line H red line Cd red line He–Ne laser line Na yellow line (center of doublet) He yellow line Hg green line H blue line Cd blue line Hg blue line Hg violet line Hg UV line

257

Part A 5.1

Dispersion Relationships in Glass The refractive index of a medium is dependent upon the wavelength of the light being transmitted (see Sect. 5.1.1). This wavelength dependence is dispersion, which means that different wavelengths of light will be modulated differently by the same piece of matter [5.10]. Each wavelength of light will be subject to a different index of refraction. One ramification of dispersion is that white light can be separated into its principal visible components through a glass prism, or a simple raindrop. It is the dispersion of white light through raindrops that causes rainbows. The dispersion of light through optical materials results in the light being refracted at various angles because of Snell’s law (5.46). The various components of white light experience different indices of refraction, which leads to different angles of exiting light. The difference in refractive index with wavelength is illustrated in Fig. 5.9 for BK-7 optical glass, which is a high dispersion material. In normal dispersion, the index increases for shorter wavelengths of light [5.15]. Normal dispersion is valid only far away from absorption bands (see Fig. 5.6). Water has a normal dispersion response in the visible, so the red light is refracted by a lower index, and thus a greater angle, which is why red is on top in a rainbow. Discussion of rainbow formation is presented elsewhere in depth [5.10].

5.1 Interaction of Light with Optical Materials

258

Part A

Basic Principles and Materials

Part A 5.1

commonly given for two sets of conditions nd − 1 ne − 1 , νe = . (5.47) νd = nF − nC n F − n C The Abbe number is a measure of the ratio of the refractive power to the dispersion. In most optical materials catalogs, a six-digit number is assigned to the solid that is dependent upon the index and the Abbe number: 1000(n d − 1) + 10νd . Using this property, e.g. optical glasses are divided into two general categories: crowns and flints (see Sect. 5.2). Crown glasses typically have a low index of refraction and a high Abbe number (n d < 1.60 and νd > 55), whereas flint glasses have high indices and low Abbe numbers (n d > 1.60 and νd > 50) [5.15]. The terms crown and flint have historical significance in that flint glasses typically had lead oxide added to them to increase the refractive index (see Sect. 5.2) and crown glasses were typically blown and had curvature – a crown. Typically, to make an achromatic optical system crown and flint lenses are combined in series. The n d and νd are plotted for various types of optical glasses in Fig. 5.87. The n d and νd are listed with glass type and manufacture in Table 5.37. Often it is desired to have a mathematical representation of the index as a function of wavelength. A considerable number of models exist for just this purpose. Perhaps the best known, and most widely used, is the Sellmeier form [5.15] n 2 (λ) − 1 =

B1 λ2 2 λ −C

B2 λ2 2 λ −C

B3 λ2 2 λ −C

average effective resonance wavelength for the temperature coefficients in microns. The constants E 0 , E 1 , D0 , D1 , and D2 are provided on the manufacture product data sheets for each composition. The index of refraction must be measured.

5.1.3 Absorption Based on the discussion in Sect. 5.1.1 the most common processes causing absorption in optical materials will be discussed here. Introduction to Absorption Absorption in glass is characterized by a decrease in transmitted light intensity through the sample that is not accounted for by reflection losses at the surface or scattering by inclusions [5.14]. As already introduced in Sect. 5.1.1 absorption is not uniform across all wavelengths of interest (UV–VIS–IR; ≈ 200–2000 nm) and can be characterized by absorption bands [5.17]. The absorption bands are due to both intrinsic and extrinsic effects [5.14, 18, 19]. The quantity used to discuss absorption as a function of wavelength in glass is the transmi