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Handbook of optics third edition volume II design fabrication and testing sources and detectors radiometry and photometry

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Michael Bass is professor emeritus at CREOL, The College of Optics and Photonics, University of Central Florida, Orlando, Florida.. Guifang Li is a professor at CREOL, The College of Op

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www.elsolucionario.net

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OF OPTICS

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ABOUT THE EDITORS

Editor-in-Chief: Dr Michael Bass is professor emeritus at CREOL, The College of Optics and

Photonics, University of Central Florida, Orlando, Florida

Dr Vasudevan Lakshminarayanan is professor of Optometry, Physics, and Electrical Engineering

at the University of Waterloo, Ontario, Canada

Dr Guifang Li is a professor at CREOL, The College of Optics and Photonics, University of Central Florida, Orlando, Florida

Dr Carolyn MacDonald is a professor at the University at Albany, and director of the Center for X-Ray Optics

Dr Virendra N Mahajan is a distinguished scientist at The Aerospace Corporation

Dr Eric Van Stryland is a professor at CREOL, The College of Optics and Photonics, University

of Central Florida, Orlando, Florida

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OF OPTICS

Volume II Design, Fabrication, and Testing;

Sources and Detectors;

Radiometry and Photometry

THIRD EDITION

Sponsored by the OPTICAL SOCIETY OF AMERICA

Michael Bass Editor-in-Chief

CREOL, The College of Optics and Photonics University of Central Florida Orlando, Florida

Virendra N Mahajan Associate Editor

The Aerospace Corporation

El Segundo, California

Eric Van Stryland Associate Editor

CREOL, The College of Optics and Photonics University of Central Florida Orlando, Florida

New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul

Singapore Sydney Toronto

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the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in

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COVER ILLUSTRATIONS

Left: Telescope such as used by Galileo to discover lunar craters and Jupiter’s moons The basic

design is still used in opera and sports glasses See Chap 1

Middle: Simplifi ed schematic of a laser showing the gain medium which amplifi es the light,

and the resonator which defi nes the light’s direction and spatial distribution The third critical part, the source to excite the gain medium, is not shown See Chap 16

Right: Zernike circle polynomial representing balanced astigmatism with a standard deviation

of one wave illustrated as an isometric plot on the top, interferogram on the left, and spread function on the right See Chap 11

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This page intentionally left blank.

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CONTENTS

Contributors xvii Brief Contents of All Volumes xix Editors’ Preface xxv

Preface to Volume II xxvii Glossary and Fundamental Constants xxix

Chapter 2 Aberration Curves in Lens Design Donald C O’Shea

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viii CONTENTS

Chapter 4 Optical Specifi cations Robert R Shannon 4.1

4.1 Glossary / 4.1

4.2 Introduction / 4.1

4.3 Preparation of Optical Specifi cations / 4.5

4.4 Image Specifi cations / 4.6

4.5 Element Description / 4.8

4.6 Environmental Specifi cations / 4.10

4.7 Presentation of Specifi cations / 4.10

4.8 Problems with Specifi cation Writing / 4.11

6.2 Introduction and Summary / 6.1

6.3 Mounting Individual Rotationally Symmetric Optics / 6.2

6.4 Multicomponent Lens Assemblies / 6.5

6.5 Mounting Windows and Domes / 6.11

6.6 Mounting Small Mirrors and Prisms / 6.11

6.7 Mounting Moderate-Sized Mirrors / 6.17

6.8 Contact Stresses in Optics / 6.21

6.9 Temperature Effects on Mounted Optics / 6.21

8.3 Homogeneous Thermal Effects / 8.2

8.4 Tolerable Homogeneous Temperature Change (No Compensation) / 8.5

8.5 Effect of Thermal Gradients / 8.6

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CONTENTS ix

Part 2 Fabrication

Chapter 9 Optical Fabrication Michael P Mandina 9.3

9.1 Introduction / 9.3

9.2 Material Forms of Supply / 9.3

9.3 Basic Steps in Spherical Optics Fabrication / 9.4

9.4 Plano Optics Fabrication / 9.7

9.5 Asphere Optics Fabrication / 9.7

9.6 Crystalline Optics / 9.8

9.7 Purchasing Optics / 9.9

9.8 Conclusion / 9.9

9.9 References / 9.9

Chapter 10 Fabrication of Optics by Diamond Turning

10.1 Glossary / 10.1

10.2 Introduction / 10.1

10.3 The Diamond-Turning Process / 10.2

10.4 The Advantages of Diamond Turning / 10.2

10.5 Diamond-Turnable Materials / 10.4

10.6 Comparison of Diamond Turning and Traditional Optical Fabrication / 10.6

10.7 Machine Tools for Diamond Turning / 10.6

10.8 Basic Steps in Diamond Turning / 10.8

10.9 Surface Finish of Diamond-Turned Optics / 10.9

10.10 Metrology of Diamond-Turned Optics / 10.12

11.4 Zernike Circle Polynomials / 11.6

11.5 Zernike Annular Polynomials / 11.13

11.14 Discussion and Conclusions / 11.39 11.15 References / 11.40

Chapter 12 Optical Metrology Zacarías Malacara and

12.1 Glossary / 12.1

12.2 Introduction and Defi nitions / 12.2

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16.3 Laser Properties Associated with the Laser Gain Medium / 16.4

16.4 Laser Properties Associated with Optical Cavities or Resonators / 16.19

16.5 Special Laser Cavities / 16.25 16.6 Specifi c Types of Lasers / 16.29

16.7 References / 16.37

Chapter 17 Light-Emitting Diodes Roland H Haitz,

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Chapter 18 High-Brightness Visible LEDs

18.1 The Materials Systems / 18.1

18.2 Substrates and Epitaxial Growth / 18.2

18.3 Processing / 18.3

18.4 Solid-State Lighting / 18.4

18.5 Packaging / 18.5

Chapter 19 Semiconductor Lasers Pamela L Derry,

19.1 Glossary / 19.1

19.2 Introduction / 19.3

19.3 Applications for Semiconductor Lasers / 19.3

19.4 Basic Operation / 19.4

19.5 Fabrication and Confi gurations / 19.6

19.6 Quantum Well Lasers / 19.9

19.7 High-Power Semiconductor Lasers / 19.18

19.8 High-Speed Modulation / 19.30

19.9 Spectral Properties / 19.36

19.10 Surface-Emitting Lasers / 19.39 19.11 Conclusion / 19.41

19.12 References / 19.43

Chapter 20 Ultrashort Optical Sources and Applications

20.1 Introduction / 20.1

20.2 Description of Optical Pulses and Pulse Trains / 20.2

20.3 Pulse Evolution toward Steady State / 20.9

20.4 Coupling Circulating Pulses Inside a Cavity / 20.12

20.5 Designs of Cavities with Two Circulating Pulses / 20.15

20.6 Analogy of a Two-Level System / 20.22

21.3 The Driving Laser / 21.4

21.4 Attosecond Pulse Generation / 21.6

21.5 Attosecond Pulse Characterization / 21.8

21.6 Acknowledgments / 21.10

21.7 References / 21.10

Chapter 22 Laser Stabilization John L Hall,

22.1 Introduction and Overview / 22.1

22.2 Servo Principles and Issues / 22.5

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xii CONTENTS

22.3 Practical Issues / 22.12

22.4 Summary and Outlook / 22.23

22.5 Conclusions and Recommendations / 22.24

22.6 Acknowledgments / 22.24

22.7 References / 22.24

Chapter 23 Quantum Theory of the Laser János A Bergou,

23.1 Glossary / 23.1

23.2 Introduction / 23.5

23.3 Some History of the Photon Concept / 23.6

23.4 Quantum Theory of the Laser / 23.14

23.5 The Laser Phase-Transition Analogy / 23.35

23.6 Exotic Masers and Lasers / 23.40

24.5 Detector Performance and Sensitivity / 24.13

24.6 Other Performance Parameters / 24.18

24.7 Detector Performance / 24.21

24.8 References / 24.101

24.9 Suggested Readings / 24.102

Chapter 25 Photodetection Abhay M Joshi

Chapter 26 High-Speed Photodetectors

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Part 6 Imaging Detectors

Chapter 29 Photographic Films Joseph H Altman 29.3

29.14 Image Structure / 29.17 29.15 Acutance / 29.17 29.16 Graininess / 29.19 29.17 Sharpness and Graininess Considered Together / 29.22 29.18 Signal-to-Noise Ratio and Detective Quantum Effi ciency / 29.22

29.19 Resolving Power / 29.24

29.20 Information Capacity / 29.24 29.21 List of Photographic Manufacturers / 29.25 29.22 References / 29.25

Chapter 30 Photographic Materials John D Baloga 30.1

30.1 Introduction / 30.1

30.2 The Optics of Photographic Films and Papers / 30.2

30.3 The Photophysics of Silver Halide Light Detectors / 30.7

30.4 The Stability of Photographic Image Dyes toward Light Fade / 30.10

30.5 Photographic Spectral Sensitizers / 30.13

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xiv CONTENTS

30.6 General Characteristics of Photographic Films / 30.18

30.7 References / 30.28

Chapter 31 Image Tube Intensifi ed Electronic Imaging

C Bruce Johnson and Larry D Owen 31.1

31.1 Glossary / 31.1

31.2 Introduction / 31.2

31.3 The Optical Interface / 31.3

31.4 Image Intensifi ers / 31.7

31.5 Image Intensifi ed Self-Scanned Arrays / 31.19

Chapter 33 Infrared Detector Arrays Lester J Kozlowski

33.1 Glossary / 33.1

33.2 Introduction / 33.3

33.3 Monolithic FPAs / 33.10

33.4 Hybrid FPAs / 33.14

33.5 Performance: Figures of Merit / 33.23

33.6 Current Status and Future Trends / 33.28

33.7 References / 33.31

Part 7 Radiometry and Photometry

Chapter 34 Radiometry and Photometry Edward F Zalewski 34.3

34.1 Glossary / 34.3

34.2 Introduction / 34.5

34.3 Radiometric Defi nitions and Basic Concepts / 34.7

34.4 Radiant Transfer Approximations / 34.13

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CONTENTS xv

35.10 Measurement of Absorptance / 35.10 35.11 Measurement of Refl ectance / 35.10 35.12 Measurement of Emittance / 35.14 35.13 References / 35.16

35.14 Further Reading / 35.23

Chapter 36 Radiometry and Photometry: Units and Conversions James M Palmer 36.1

36.1 Glossary / 36.1

36.2 Introduction and Background / 36.2

36.3 Symbols, Units, and Nomenclature in Radiometry / 36.4

36.4 Symbols, Units, and Nomenclature in Photometry / 36.5

36.5 Conversion of Radiometric Quantities to Photometric Quantities / 36.11

36.6 Conversion of Photometric Quantities to Radiometric Quantities / 36.12

37.2 Basis of Physical Photometry / 37.1

37.3 Photometric Base Unit—the Candela / 37.3

37.4 Quantities and Units in Photometry and Radiometry / 37.3

37.5 Principles in Photometry and Radiometry / 37.8

37.6 Practice in Photometry and Radiometry / 37.11

37.7 References / 37.12

Chapter 38 Spectroradiometry Carolyn J Sher DeCusatis 38.1

38.1 Introduction / 38.1

38.2 Defi nitions, Calculations, and Figures of Merit / 38.1

38.3 General Features of Spectroradiometry Systems / 38.7

38.4 Typical Spectroradiometry System Designs / 38.13

39.3 Software Modeling of Nonimaging Systems / 39.6

39.4 Basic Building Blocks / 39.8

40.3 Vision Biology and Perception / 40.3

40.4 The Science of Lighting Design / 40.6

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CONTRIBUTORS

Joseph H Altman Institute of Optics, University of Rochester, Rochester, New York (CHAP 29)

Ladan Arissian Texas A&M University, College Station, Texas, and National Research Council of Canada,

John D Baloga Imaging Materials and Media, Eastman Kodak Company, Rochester, New York (CHAP 30)

János A Bergou Institute for Quantum Studies and Department of Physics, Texas A&M University, College

Station, Texas, and Department of Physics and Astronomy, Hunter College of the City University of New York, New

John E Bowers Department of Electrical and Computer Engineering, University of California, Santa Barbara,

Robert P Breault Breault Research Organization, Tucson, Arizona (CHAP.7)

William Cassarly Optical Research Associates, Pasadena, California (CHAP.39)

Zenghu Chang Department of Physics, Kansas State University, Cardwell Hall, Manhattan, Kansas (CHAP.21)

M George Craford Hewlett-Packard Co., San Jose, California (CHAP 17)

Katherine Creath Optineering, Tucson, Arizona, and College of Optical Sciences, University of Arizona, Tucson,

Pamela L Derry Boeing Defense & Space Group, Seattle, Washington (CHAP 19)

Jean-Claude Diels Departments of Physics and Electrical Engineering, University of New Mexico, Albuquerque,

Berthold-Georg Englert Institute for Quantum Studies and Department of Physics, Texas A&M University,

College Station, Texas, and Max-Planck-Institut für Quantenoptik, Garching bei München, Germany, and Abteilung

Chris J Evans Zygo Corporation, Middlefield, Connecticut (CHAP 10)

Luis Figueroa Boeing Defense & Space Group, Seattle, Washington (CHAP 19)

Anurag Gupta Optical Research Associates, Tucson, Arizona (CHAP 40)

Roland H Haitz Hewlett-Packard Co., San Jose, California (CHAP 17)

John L Hall JILA, University of Colorado and National Institute of Standards and Technology, Boulder, Colorado

(CHAP 22)

Michael E Harrigan Harrigan Optical Design, Victor, New York (CHAP 2)

Chi-Shain Hong Boeing Defense & Space Group, Seattle, Washington (CHAP 19)

C Bruce Johnson Johnson Scientific Group, Inc., Phoenix, Arizona (CHAP 31)

Abhay M Joshi Discovery Semiconductors, Inc., Cranbury, New Jersey (CHAP 25)

R John Koshel Photon Engineering LLC, and College of Optical Sciences, University of Arizona, Tucson,

Walter F Kosonocky * New Jersey Institute of Technology, University Heights, Newark, New Jersey (CHAP 33)

Lester J Kozlowski Altasens, Inc., Westlake Village, California (CHAP 33)

Paul W Kruse Consultant, Edina, Minnesota (CHAP 28)

Anthony LaRocca † General Dynamics, Advanced Information Systems, Ypsilanti, Michigan (CHAP 15)

∗ Deceased.

† Retired.

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Melvin Lax * Department of Physics, City College of the City University of New York, New York, New York (CHAP 23)

Virendra N Mahajan The Aerospace Corporation, El Segundo, California (CHAP 11)

Zacarías Malacara Centro de Investigaciones en Óptica, A C., León, Gto., México (CHAP 12)

Daniel Malacara-Hernández Centro de Investigaciones en Óptica, A C., León, Gto., México (CHAPS 12, 13)

Michael P Mandina Brandon Light, Optimax Systems, Inc., Ontario, New York (CHAP 9)

Paul R Norton U.S Army Night Vision and Electronics Directorate, Fort Belvoir, Virginia (CHAP 24)

Donald C O’Shea Georgia Institute of Technology, School of Physics, Atlanta, Georgia (CHAP 2)

Yoshi Ohno Optical Technology Division, National Institute of Standards and Technology, Gaithersburg,

Gregory H Olsen Sensors Unlimited, Inc., Princeton, New Jersey (CHAP 25)

Larry D Owen NuOptics International, Phoenix, Arizona (CHAP 31)

James M Palmer * College of Optical Sciences, University of Arizona, Tucson, Arizona (CHAPS 35, 36)

Richard L Rhorer National Institute of Standards and Technology, Gaithersburg, Maryland (CHAP 10)

Michael Roberts Pilkington Optronics, Wales, United Kingdom (CHAP 8)

Philip J Rogers Pilkington Optronics, Wales, United Kingdom (CHAP 8)

Winston V Schoenfeld CREOL, The College of Optics and Photonics, University of Central Florida, Orlando,

Marian O Scully Institute for Quantum Studies and Department of Physics, Texas A&M University, College

Robert R Shannon † College of Optical Sciences, University of Arizona, Tucson, Arizona (CHAPS 4, 5)

Carolyn J Sher DeCusatis Pace University, White Plains, New York (CHAP 38)

William T Silfvast CREOL, The College of Optics and Photonics, University of Central Florida, Orlando,

Douglas C Sinclair Sinclair Optics, Inc., Fairport, New York (CHAP 3)

Warren J Smith * Kaiser Electro-Optics, Inc., Carlsbad, California (CHAP 1)

Matthew S Taubman JILA, University of Colorado and National Institute of Standards and Technology,

Timothy J Tredwell Sensor Systems Division, Imager Systems Development Laboratory, Eastman Kodak

Herbert Walther * Max-Planck-Institut für Quantenoptik, Garching bei München, Germany, and Sektion Physik

Robert H Weissman Hewlett-Packard Co., San Jose, California (CHAP 17)

Yih G Wey Department of Electrical and Computer Engineering, University of California, Santa Barbara,

John R Willison Stanford Research Systems, Inc., Sunnyvale, California (CHAP.27)

William L Wolfe College of Optical Sciences, University of Arizona, Tucson, Arizona (CHAP 28)

James C Wyant College of Optical Sciences, University of Arizona, Tucson, Arizona (CHAP 14)

Jun Ye JILA, University of Colorado and National Institute of Standards and Technology, Boulder, Colorado (CHAP 22)

Paul R Yoder, Jr Consultant in Optical Engineering, Norwalk, Connecticut (CHAP 6)

Edward F Zalewski College of Optical Sciences, University of Arizona, Tucson, Arizona (CHAP 34)

M Suhail Zubairy Institute for Quantum Studies and Department of Physics, Texas A&M University, College

xviii CONTRIBUTORS

∗ Deceased.

† Retired.

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BRIEF CONTENTS OF

ALL VOLUMES

VOLUME I GEOMETRICAL AND PHYSICAL OPTICS, POLARIZED LIGHT,

COMPONENT AND INSTRUMENTSPART 1 GEOMETRICAL OPTICS

Chapter 1 General Principles of Geometrical Optics Douglas S Goodman

PART 2 PHYSICAL OPTICSChapter 2 Interference John E Greivenkamp

Chapter 3 Diffraction Arvind S Marathay and John F McCalmont

Chapter 4 Transfer Function Techniques Glenn D Boreman

Chapter 5 Coherence Theory William H Carter

Chapter 6 Coherence Theory: Tools and Applications Gisele Bennett, William T Rhodes, and

J Christopher James

Chapter 7 Scattering by Particles Craig F Bohren

Chapter 8 Surface Scattering Eugene L Church and Peter Z Takacs

Chapter 9 Volume Scattering in Random Media Aristide Dogariu and Jeremy Ellis Chapter 10 Optical Spectroscopy and Spectroscopic Lineshapes Brian Henderson Chapter 11 Analog Optical Signal and Image Processing Joseph W Goodman

PART 3 POLARIZED LIGHT

Chapter 12 Polarization Jean M Bennett Chapter 13 Polarizers Jean M Bennett Chapter 14 Mueller Matrices Russell A Chipman Chapter 15 Polarimetry Russell A Chipman Chapter 16 Ellipsometry Rasheed M A Azzam

PART 4 COMPONENTS

Chapter 17 Lenses R Barry Johnson Chapter 18 Afocal Systems William B Wetherell Chapter 19 Nondispersive Prisms William L Wolfe Chapter 20 Dispersive Prisms and Gratings George J Zissis Chapter 21 Integrated Optics Thomas L Koch, Frederick J Leonberger, and Paul G Suchoski Chapter 22 Miniature and Micro-Optics Tom D Milster and Tomasz S Tkaczyk

Chapter 23 Binary Optics Michael W Farn and Wilfrid B Veldkamp Chapter 24 Gradient Index Optics Duncan T Moore

PART 5 INSTRUMENTS

Chapter 25 Cameras Norman Goldberg Chapter 26 Solid-State Cameras Gerald C Holst Chapter 27 Camera Lenses Ellis Betensky, Melvin H Kreitzer, and Jacob Moskovich Chapter 28 Microscopes Rudolf Oldenbourg and Michael Shribak

Chapter 29 Reflective and Catadioptric Objectives Lloyd Jones

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Chapter 30 Scanners Leo Beiser and R Barry Johnson Chapter 31 Optical Spectrometers Brian Henderson Chapter 32 Interferometers Parameswaran Hariharan Chapter 33 Holography and Holographic Instruments Lloyd Huff Chapter 34 Xerographic Systems Howard Stark

Chapter 35 Principles of Optical Disk Data Storage Masud Mansuripur

VOLUME II DESIGN, FABRICATION, AND TESTING; SOURCES AND

DETECTORS; RADIOMETRY AND PHOTOMETRYPART 1 DESIGN

Chapter 1 Techniques of First-Order Layout Warren J Smith

Chapter 2 Aberration Curves in Lens Design Donald C O’Shea and Michael E Harrigan

Chapter 3 Optical Design Software Douglas C Sinclair

Chapter 4 Optical Specifications Robert R Shannon

Chapter 5 Tolerancing Techniques Robert R Shannon

Chapter 6 Mounting Optical Components Paul R Yoder, Jr.

Chapter 7 Control of Stray Light Robert P Breault

Chapter 8 Thermal Compensation Techniques Philip J Rogers and Michael Roberts

PART 2 FABRICATIONChapter 9 Optical Fabrication Michael P Mandina Chapter 10 Fabrication of Optics by Diamond Turning Richard L Rhorer and Chris J Evans

PART 3 TESTING

Chapter 11 Orthonormal Polynomials in Wavefront Analysis Virendra N Mahajan Chapter 12 Optical Metrology Zacarías Malacara and Daniel Malacara-Hernández Chapter 13 Optical Testing Daniel Malacara-Hernández

Chapter 14 Use of Computer-Generated Holograms in Optical Testing Katherine Creath and

James C Wyant

PART 4 SOURCES

Chapter 15 Artificial Sources Anthony LaRocca Chapter 16 Lasers William T Silfvast Chapter 17 Light-Emitting Diodes Roland H Haitz, M George Craford, and Robert H Weissman Chapter 18 High-Brightness Visible LEDs Winston V Schoenfeld

Chapter 19 Semiconductor Lasers Pamela L Derry, Luis Figueroa, and Chi-shain Hong Chapter 20 Ultrashort Optical Sources and Applications Jean-Claude Diels and Ladan Arissian Chapter 21 Attosecond Optics Zenghu Chang

Chapter 22 Laser Stabilization John L Hall, Matthew S Taubman, and Jun Ye Chapter 23 Quantum Theory of the Laser János A Bergou, Berthold-Georg Englert, Melvin Lax,

Marian O Scully, Herbert Walther, and M Suhail Zubairy

PART 5 DETECTORS

Chapter 24 Photodetectors Paul R Norton Chapter 25 Photodetection Abhay M Joshi and Gregory H Olsen Chapter 26 High-Speed Photodetectors John E Bowers and Yih G Wey Chapter 27 Signal Detection and Analysis John R Willison

Chapter 28 Thermal Detectors William L Wolfe and Paul W Kruse

PART 6 IMAGING DETECTORS

Chapter 29 Photographic Films Joseph H Altman Chapter 30 Photographic Materials John D Baloga

xx BRIEF CONTENTS OF ALL VOLUMES

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Chapter 31 Image Tube Intensified Electronic Imaging C Bruce Johnson and Larry D Owen Chapter 32 Visible Array Detectors Timothy J Tredwell

Chapter 33 Infrared Detector Arrays Lester J Kozlowski and Walter F Kosonocky

PART 7 RADIOMETRY AND PHOTOMETRY

Chapter 34 Radiometry and Photometry Edward F Zalewski Chapter 35 Measurement of Transmission, Absorption, Emission, and Reflection James M Palmer Chapter 36 Radiometry and Photometry: Units and Conversions James M Palmer

Chapter 37 Radiometry and Photometry for Vision Optics Yoshi Ohno Chapter 38 Spectroradiometry Carolyn J Sher DeCusatis

Chapter 39 Nonimaging Optics: Concentration and Illumination William Cassarly Chapter 40 Lighting and Applications Anurag Gupta and R John Koshel

VOLUME III VISION AND VISION OPTICS

Chapter 1 Optics of the Eye Neil Charman

Chapter 2 Visual Performance Wilson S Geisler and Martin S Banks

Chapter 3 Psychophysical Methods Denis G Pelli and Bart Farell

Chapter 4 Visual Acuity and Hyperacuity Gerald Westheimer

Chapter 5 Optical Generation of the Visual Stimulus Stephen A Burns and Robert H Webb

Chapter 6 The Maxwellian View with an Addendum on Apodization Gerald Westheimer

Chapter 7 Ocular Radiation Hazards David H Sliney

Chapter 8 Biological Waveguides Vasudevan Lakshminarayanan and Jay M Enoch

Chapter 9 The Problem of Correction for the Stiles-Crawford Effect of the First Kind in Radiometry and

Photometry, a Solution Jay M Enoch and Vasudevan Lakshminarayanan Chapter 10 Colorimetry David H Brainard and Andrew Stockman

Chapter 11 Color Vision Mechanisms Andrew Stockman and David H Brainard

Chapter 12 Assessment of Refraction and Refractive Errors and Their Influence on

Optical Design B Ralph Chou Chapter 13 Binocular Vision Factors That Influence Optical Design Clifton Schor Chapter 14 Optics and Vision of the Aging Eye John S Werner, Brooke E Schefrin, and Arthur Bradley Chapter 15 Adaptive Optics in Retinal Microscopy and Vision Donald T Miller and Austin Roorda Chapter 16 Refractive Surgery, Correction of Vision, PRK and LASIK L Diaz-Santana and Harilaos Ginis Chapter 17 Three-Dimensional Confocal Microscopy of the Living Human Cornea Barry R Masters Chapter 18 Diagnostic Use of Optical Coherence Tomography in the Eye Johannes F de Boer Chapter 19 Gradient Index Optics in the Eye Barbara K Pierscionek

Chapter 20 Optics of Contact Lenses Edward S Bennett Chapter 21 Intraocular Lenses Jim Schwiegerling Chapter 22 Displays for Vision Research William Cowan Chapter 23 Vision Problems at Computers Jeffrey Anshel and James E Sheedy Chapter 24 Human Vision and Electronic Imaging Bernice E Rogowitz, Thrasyvoulos N Pappas, and

Jan P Allebach

Chapter 25 Visual Factors Associated with Head-Mounted Displays Brian H Tsou and Martin Shenker

VOLUME IV OPTICAL PROPERTIES OF MATERIALS, NONLINEAR

OPTICS, QUANTUM OPTICSPART 1 PROPERTIES

Chapter 1 Optical Properties of Water Curtis D Mobley

Chapter 2 Properties of Crystals and Glasses William J Tropf, Michael E Thomas, and

Eric W Rogala

Chapter 3 Polymeric Optics John D Lytle

Chapter 4 Properties of Metals Roger A Paquin

BRIEF CONTENTS OF ALL VOLUMES xxi

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Chapter 5 Optical Properties of Semiconductors David G Seiler, Stefan Zollner, Alain C Diebold, and Paul

M Amirtharaj

Chapter 6 Characterization and Use of Black Surfaces for Optical Systems Stephen M Pompea

and Robert P Breault

Chapter 7 Optical Properties of Films and Coatings Jerzy A Dobrowolski

Chapter 8 Fundamental Optical Properties of Solids Alan Miller

Chapter 9 Photonic Bandgap Materials Pierre R Villeneuve

PART 2 NONLINEAR OPTICS

Chapter 10 Nonlinear Optics Chung L Tang Chapter 11 Coherent Optical Transients Paul R Berman and D G Steel Chapter 12 Photorefractive Materials and Devices Mark Cronin-Golomb and Marvin Klein Chapter 13 Optical Limiting David J Hagan

Chapter 14 Electromagnetically Induced Transparency Jonathan P Marangos and Thomas Halfmann Chapter 15 Stimulated Raman and Brillouin Scattering John Reintjes and M Bashkansky

Chapter 16 Third-Order Optical Nonlinearities Mansoor Sheik-Bahae and Michael P Hasselbeck Chapter 17 Continuous-Wave Optical Parametric Oscillators M Ebrahim-Zadeh

Chapter 18 Nonlinear Optical Processes for Ultrashort Pulse Generation Uwe Siegner and Ursula Keller Chapter 19 Laser-Induced Damage to Optical Materials Marion J Soileau

PART 3 QUANTUM AND MOLECULAR OPTICS

Chapter 20 Laser Cooling and Trapping of Atoms Harold J Metcalf and Peter van der Straten Chapter 21 Strong Field Physics Todd Ditmire

Chapter 22 Slow Light Propagation in Atomic and Photonic Media Jacob B Khurgin Chapter 23 Quantum Entanglement in Optical Interferometry Hwang Lee, Christoph F Wildfeuer,

Sean D Huver, and Jonathan P Dowling

VOLUME V ATMOSPHERIC OPTICS, MODULATORS, FIBER OPTICS,

X-RAY AND NEUTRON OPTICSPART 1 MEASUREMENTS

Chapter 1 Scatterometers John C Stover

Chapter 2 Spectroscopic Measurements Brian Henderson

PART 2 ATMOSPHERIC OPTICSChapter 3 Atmospheric Optics Dennis K Killinger, James H Churnside, and Laurence S Rothman

Chapter 4 Imaging through Atmospheric Turbulence Virendra N Mahajan and Guang-ming Dai

Chapter 5 Adaptive Optics Robert Q Fugate

PART 3 MODULATORSChapter 6 Acousto-Optic Devices I-Cheng Chang

Chapter 7 Electro-Optic Modulators Georgeanne M Purvinis and Theresa A Maldonado

Chapter 8 Liquid Crystals Sebastian Gauza and Shin-Tson Wu

PART 4 FIBER OPTICSChapter 9 Optical Fiber Communication Technology and System Overview Ira Jacobs Chapter 10 Nonlinear Effects in Optical Fibers John A Buck

Chapter 11 Photonic Crystal Fibers Philip St J Russell and G J Pearce Chapter 12 Infrared Fibers James A Harrington

Chapter 13 Sources, Modulators, and Detectors for Fiber Optic Communication Systems Elsa Garmire Chapter 14 Optical Fiber Amplifiers John A Buck

Chapter 15 Fiber Optic Communication Links (Telecom, Datacom, and Analog) Casimer DeCusatis

and Guifang Li

xxii BRIEF CONTENTS OF ALL VOLUMES

Trang 25

Chapter 16 Fiber-Based Couplers Daniel Nolan Chapter 17 Fiber Bragg Gratings Kenneth O Hill Chapter 18 Micro-Optics-Based Components for Networking Joseph C Palais Chapter 19 Semiconductor Optical Amplifiers Jay M Wiesenfeld and Leo H Spiekman Chapter 20 Optical Time-Division Multiplexed Communication Networks Peter J Delfyett Chapter 21 WDM Fiber-Optic Communication Networks Alan E Willner, Changyuan Yu, Zhongqi Pan, and

PART 5 X-RAY AND NEUTRON OPTICS

Subpart 5.1 Introduction and Applications

Chapter 26 An Introduction to X-Ray and Neutron Optics Carolyn A MacDonald Chapter 27 Coherent X-Ray Optics and Microscopy Qun Shen

Chapter 28 Requirements for X-Ray diffraction Scott T Misture Chapter 29 Requirements for X-Ray Fluorescence George J Havrilla Chapter 30 Requirements for X-Ray Spectroscopy Dirk Lützenkirchen-Hecht and Ronald Frahm Chapter 31 Requirements for Medical Imaging and X-Ray Inspection Douglas Pfeiffer Chapter 32 Requirements for Nuclear Medicine Lars R Furenlid

Chapter 33 Requirements for X-Ray Astronomy Scott O Rohrbach Chapter 34 Extreme Ultraviolet Lithography Franco Cerrina and Fan Jiang Chapter 35 Ray Tracing of X-Ray Optical Systems Franco Cerrina and M Sanchez del Rio Chapter 36 X-Ray Properties of Materials Eric M Gullikson

Subpart 5.2 Refractive and Interference Optics

Chapter 37 Refractive X-Ray Lenses Bruno Lengeler and Christian G Schroer Chapter 38 Gratings and Monochromators in the VUV and Soft X-Ray Spectral Region Malcolm R Howells Chapter 39 Crystal Monochromators and Bent Crystals Peter Siddons

Chapter 40 Zone Plates Alan Michette Chapter 41 Multilayers Eberhard Spiller Chapter 42 Nanofocusing of Hard X-Rays with Multilayer Laue Lenses Albert T Macrander, Hanfei Yan,

Hyon Chol Kang, Jörg Maser, Chian Liu, Ray Conley, and G Brian Stephenson

Chapter 43 Polarizing Crystal Optics Qun Shen

Subpart 5.3 Reflective Optics

Chapter 44 Reflective Optics James Harvey Chapter 45 Aberrations for Grazing Incidence Optics Timo T Saha Chapter 46 X-Ray Mirror Metrology Peter Z Takacs

Chapter 47 Astronomical X-Ray Optics Marshall K Joy and Brian D Ramsey Chapter 48 Multifoil X-Ray Optics Ladislav Pina

Chapter 49 Pore Optics Marco Beijersbergen Chapter 50 Adaptive X-Ray Optics Ali Khounsary Chapter 51 The Schwarzschild Objective Franco Cerrina Chapter 52 Single Capillaries Donald H Bilderback and Sterling W Cornaby Chapter 53 Polycapillary X-Ray Optics Carolyn MacDonald and Walter M Gibson

Subpart 5.4 X-Ray Sources

Chapter 54 X-Ray Tube Sources Susanne M Lee and Carolyn MacDonald Chapter 55 Synchrotron Sources Steven L Hulbert and Gwyn P Williams Chapter 56 Laser Generated Plasmas Alan Michette

BRIEF CONTENTS OF ALL VOLUMES xxiii

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Chapter 57 Pinch Plasma Sources Victor Kantsyrev Chapter 58 X-Ray Lasers Greg Tallents

Chapter 59 Inverse Compton X-Ray Sources Frank Carroll

Subpart 5.5 X-Ray Detectors

Chapter 60 Introduction to X-Ray Detectors Walter M Gibson and Peter Siddons Chapter 61 Advances in Imaging Detectors Aaron Couture

Chapter 62 X-Ray Spectral Detection and Imaging Eric Lifshin

Subpart 5.6 Neutron Optics and Applications

Chapter 63 Neutron Optics David Mildner Chapter 64 Grazing-Incidence Neutron Optics Mikhail Gubarev and Brian Ramsey

xxiv BRIEF CONTENTS OF ALL VOLUMES

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EDITORS’ PREFACE

The third edition of the Handbook of Optics is designed to pull together the dramatic developments

in both the basic and applied aspects of the field while retaining the archival, reference book value

of a handbook This means that it is much more extensive than either the first edition, published

in 1978, or the second edition, with Volumes I and II appearing in 1995 and Volumes III and IV in

2001 To cover the greatly expanded field of optics, the Handbook now appears in five volumes Over

100 authors or author teams have contributed to this work

Volume I is devoted to the fundamentals, components, and instruments that make optics possible

Volume II contains chapters on design, fabrication, testing, sources of light, detection, and a new section devoted to radiometry and photometry Volume III concerns vision optics only and is printed entirely

in color In Volume IV there are chapters on the optical properties of materials, nonlinear, quantum and molecular optics Volume V has extensive sections on fiber optics and x ray and neutron optics, along with shorter sections on measurements, modulators, and atmospheric optical properties and turbulence

Several pages of color inserts are provided where appropriate to aid the reader A purchaser of the print

version of any volume of the Handbook will be able to download a digital version containing all of the

material in that volume in PDF format to one computer (see download instructions on bound-in card)

The combined index for all five volumes can be downloaded from www.HandbookofOpticsOnline.com

It is possible by careful selection of what and how to present that the third edition of the

Handbook could serve as a text for a comprehensive course in optics In addition, students who take

such a course would have the Handbook as a career-long reference.

Topics were selected by the editors so that the Handbook could be a desktop (bookshelf) general

reference for the parts of optics that had matured enough to warrant archival presentation New chapters were included on topics that had reached this stage since the second edition, and existing chapters from the second edition were updated where necessary to provide this compendium In selecting subjects to include, we also had to select which subjects to leave out The criteria we applied were: (1) was it a specific application of optics rather than a core science or technology and (2) was it

a subject in which the role of optics was peripheral to the central issue addressed Thus, such topics as medical optics, laser surgery, and laser materials processing were not included While applications of

optics are mentioned in the chapters there is no space in the Handbook to include separate chapters

devoted to all of the myriad uses of optics in today’s world If we had, the third edition would be much longer than it is and much of it would soon be outdated We designed the third edition of the

Handbook of Optics so that it concentrates on the principles of optics that make applications possible.

Authors were asked to try to achieve the dual purpose of preparing a chapter that was a while reference for someone working in the field and that could be used as a starting point to become acquainted with that aspect of optics They did that and we thank them for the outstanding

worth-results seen throughout the Handbook We also thank Mr Taisuke Soda of McGraw-Hill for his help

in putting this complex project together and Mr Alan Tourtlotte and Ms Susannah Lehman of the Optical Society of America for logistical help that made this effort possible

We dedicate the third edition of the Handbook of Optics to all of the OSA volunteers who, since

OSA’s founding in 1916, give their time and energy to promoting the generation, application, archiving, and worldwide dissemination of knowledge in optics and photonics

Michael Bass, Editor-in-Chief

Associate Editors:

Casimer M DeCusatis Jay M Enoch Vasudevan Lakshminarayanan

Guifang Li Carolyn MacDonald Virendra N Mahajan Eric Van Stryland

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PREFACE TO VOLUME II

Volume II of the Handbook of Optics is a continuation of Volume I It starts with optical system

design and covers first-order layout, aberration curves, design software, specifications and ances, component mounting, stray light control, and thermal compensation techniques Optical fabrication and testing are discussed next A new chapter on the use of orthonormal polynomials in optical design and testing has been added Such a polynomial representing balanced astigmatism is illustrated on the cover The section on sources includes different types of lasers, laser stabilization, laser theory, and a discussion of ultrashort laser sources Light-emitting diodes including the new

toler-“high-brightness” LEDs are presented Artificial sources of light for both the laboratory and field are described along with a discussion of light standards calibration The section on detectors includes high-speed and thermal detectors along with an analysis of signal detection Imaging using film, detector arrays, and image tubes is discussed This volume ends with a section on radiometry and photometry Two new chapters have been added in this area One is on spectroradiometry and the other is on lighting and applications

Every effort was made to contact all the authors of chapters in the second edition that would appear in this edition so that they could update their chapters However, the authors of several chapters could not be located or were not available Their chapters are reproduced without update

Every effort has been made to ensure that such chapters have been correctly reproduced

There are many other chapters in this edition of the Handbook that could have been included

in Volumes I and II However, page limitations prevented that For example, in Volume V there is

a section on Atmospheric Optics It consists of three chapters, one on transmission through the atmosphere, another on imaging through atmospheric turbulence, and a third on adaptive optics to overcome some of the deleterious effects of turbulence

The chapters are generally aimed at the graduate students, though practicing scientists and engineers will find them equally suitable as references on the topics discussed Each chapter has suf-ficient references for additional and/or further study

Virendra N Mahajan The Aerospace Corporation Eric Van Stryland CREOL, The College of Optics and Photonics

Associate Editors

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GLOSSARY AND FUNDAMENTAL

CONSTANTS

Introduction

This glossary of the terms used in the Handbook represents to a large extent the language of optics

The symbols are representations of numbers, variables, and concepts Although the basic list was compiled by the author of this section, all the editors have contributed and agreed to this set of sym-bols and definitions Every attempt has been made to use the same symbols for the same concepts

throughout the entire Handbook, although there are exceptions Some symbols seem to be used for

many concepts The symbol a is a prime example, as it is used for absorptivity, absorption

coeffi-cient, coefficient of linear thermal expansion, and more Although we have tried to limit this kind of redundancy, we have also bowed deeply to custom

Units

The abbreviations for the most common units are given first They are consistent with most of the established lists of symbols, such as given by the International Standards Organization ISO1 and the International Union of Pure and Applied Physics, IUPAP.2

The most commonly used symbols are then given Most chapters of the Handbook also have a

glos-sary of the terms and symbols specific to them for the convenience of the reader In the following list, the symbol is given, its meaning is next, and the most customary unit of measure for the quan-tity is presented in brackets A bracket with a dash in it indicates that the quantity is unitless Note that there is a difference between units and dimensions An angle has units of degrees or radians and

a solid angle square degrees or steradians, but both are pure ratios and are dimensionless The unit symbols as recommended in the SI system are used, but decimal multiples of some of the dimen-sions are sometimes given The symbols chosen, with some cited exceptions, are also those of the first two references

RATIONALE FOR SOME DISPUTED SYMBOLS

The choice of symbols is a personal decision, but commonality improves communication This

sec-tion explains why the editors have chosen the preferred symbols for the Handbook We hope that this

will encourage more agreement

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Fundamental Constants

It is encouraging that there is almost universal agreement for the symbols for the fundamental

con-stants We have taken one small exception by adding a subscript B to the k for Boltzmann’s constant.

Mathematics

We have chosen i as the imaginary almost arbitrarily IUPAP lists both i and j, while ISO does not

report on these

Spectral Variables

These include expressions for the wavelength l, frequency v, wave number s, ω for circular or

radian frequency, k for circular or radian wave number and dimensionless frequency x Although some use f for frequency, it can be easily confused with electronic or spatial frequency Some use

n~ for wave number, but, because of typography problems and agreement with ISO and IUPAP, we

have chosen s ; it should not be confused with the Stefan-Boltzmann constant For spatial

frequen-cies we have chosen x and h, although f x and f y are sometimes used ISO and IUPAP do not report

on these

Radiometry

Radiometric terms are contentious The most recent set of recommendations by ISO and IUPAP are L for

radiance [Wcm–2sr–1], M for radiant emittance or exitance [Wcm–2], E for irradiance or incidance [Wcm–2],

and I for intensity [Wsr–2] The previous terms, W, H, N, and J, respectively, are still in many texts, notably

Smith4 and Lloyd5 but we have used the revised set, although there are still shortcomings We have tried to

deal with the vexatious term intensity by using specific intensity when the units are Wcm–2sr–1, field intensity

when they are Wcm–2, and radiometric intensity when they are Wsr–1.There are two sets to terms for these radiometric quantities, which arise in part from the terms for different types of reflection, transmission, absorption, and emission It has been proposed that

the ion ending indicate a process, that the ance ending indicate a value associated with a lar sample, and that the ivity ending indicate a generic value for a “pure” substance Then one also

particu-has reflectance, transmittance, absorptance, and emittance as well as reflectivity, transmissivity, absorptivity, and emissivity There are now two different uses of the word emissivity Thus the words

exitance, incidence, and sterance were coined to be used in place of emittance, irradiance, and

radi-ance It is interesting that ISO uses radiance, exitance, and irradiance whereas IUPAP uses radiance

excitance [sic], and irradiance We have chosen to use them both, i.e., emittance, irradiance, and

radiance will be followed in square brackets by exitance, incidence, and sterance (or vice versa)

Individual authors will use the different endings for transmission, reflection, absorption, and sion as they see fit

emis-We are still troubled by the use of the symbol E for irradiance, as it is so close in meaning

to electric field, but we have maintained that accepted use The spectral concentrations of these

quantities, indicated by a wavelength, wave number, or frequency subscript (e.g., L l) represent

partial differentiations; a subscript q represents a photon quantity; and a subscript v indicates

a quantity normalized to the response of the eye Thereby, L v is luminance, E v illuminance, and

M v and I v luminous emittance and luminous intensity The symbols we have chosen are tent with ISO and IUPAP

consis-The refractive index may be considered a radiometric quantity It is generally complex and is

indicated by ñ = n – ik The real part is the relative refractive index and k is the extinction coefficient

These are consistent with ISO and IUPAP, but they do not address the complex index or extinction coefficient

xxx GLOSSARY AND FUNDAMENTAL CONSTANTS

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Optical Design

For the most part ISO and IUPAP do not address the symbols that are important in this area

There were at least 20 different ways to indicate focal ratio; we have chosen FN as

symmetri-cal with NA; we chose f and efl to indicate the effective fosymmetri-cal length Object and image distance, although given many different symbols, were finally called s o and s i since s is an almost universal

symbol for distance Field angles are q and f ; angles that measure the slope of a ray to the optical

axis are u; u can also be sin u Wave aberrations are indicated by W ijk, while third-order ray tions are indicated by s i and more mnemonic symbols

aberra-Electromagnetic Fields

There is no argument about E and H for the electric and magnetic field strengths, Q for quantity

of charge, r for volume charge density, s for surface charge density, etc There is no guidance from

Refs 1 and 2 on polarization indication We chose ⬜ and || rather than p and s, partly because s is

sometimes also used to indicate scattered light

There are several sets of symbols used for reflection transmission, and (sometimes) absorption, each with good logic The versions of these quantities dealing with field amplitudes are usually

specified with lower case symbols: r, t, and a The versions dealing with power are alternately given

by the uppercase symbols or the corresponding Greek symbols: R and T versus r and t We have

chosen to use the Greek, mainly because these quantities are also closely associated with Kirchhoff ’s law that is usually stated symbolically as a = ⑀ The law of conservation of energy for light on a sur-

face is also usually written as a + r + t = 1.

Derived SI Quantities

electric charge C coulombelectric potential V voltelectric capacitance F faradelectric resistance Ω ohmelectric conductance S siemensmagnetic flux Wb weber

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c speed of light vacuo [299792458 ms–1]

c1 first radiation constant = 2pc2h = 3.7417749 × 10–16 [Wm2]

c2 second radiation constant = hc/k = 0.014838769 [mK]

e elementary charge [1.60217733 × 10–19 C]

g n free fall constant [9.80665 ms–2]

h Planck’s constant [6.6260755 × 10–34 Ws]

kB Boltzmann constant [1.380658 × 10–23 JK–1]

m e mass of the electron [9.1093897 × 10–31 kg]

NA Avogadro constant [6.0221367 × 1023 mol–1]

c speed of light in vacuo [ms–1]

c1 first radiation constant [Wm2]

c2 second radiation constant [mK]

f c Fermi occupation function, conduction band

f v Fermi occupation function, valence band

xxxii GLOSSARY AND FUNDAMENTAL CONSTANTS

Trang 35

FN focal ratio (f/number) [—]

g gain per unit length [m–1]

gth gain threshold per unit length [m1]

H magnetic field strength [Am–1, Cs–1 m–1]

Im() imaginary part of

J current density [Am–2]

j total angular momentum [kg m2 s–1]

J1() Bessel function of the first kind [—]

k radian wave number =2p/l [rad cm–1]

k wave vector [rad cm–1]

n real part of the relative refractive index [—]

ñ complex index of refraction [—]

NA numerical aperture [—]

OPD optical path difference [m]

P macroscopic polarization [C m–2]Re() real part of [—]

u slope of ray with the optical axis [rad]

V Abbe reciprocal dispersion [—]

a (power) absorptance (absorptivity)

GLOSSARY AND FUNDAMENTAL CONSTANTS xxxiii

Trang 36

diclectric coefficient (constant) [—]

q, f angular coordinates [rad, °]

x, h rectangular spatial frequencies [m−1, r−1]

d(x) Dirac delta function of x

dij Kronecker delta

REFERENCES

1 Anonymous, ISO Standards Handbook 2: Units of Measurement, 2nd ed., International Organization for

Standardization, 1982

2 Anonymous, Symbols, Units and Nomenclature in Physics, Document U.I.P 20, International Union of Pure

and Applied Physics, 1978

3 E Cohen and B Taylor, “The Fundamental Physical Constants,” Physics Today, 9 August 1990.

4 W J Smith, Modern Optical Engineering, 2nd ed., McGraw-Hill, 1990.

5 J M Lloyd, Thermal Imaging Systems, Plenum Press, 1972.

William L Wolfe

College of Optical Sciences University of Arizona Tucson, Arizona

xxxiv GLOSSARY AND FUNDAMENTAL CONSTANTS

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PA RT

1

DESIGN

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TECHNIQUES OF FIRST-ORDER LAYOUT

l axial intercept distance

M angular magnifi cation

m linear, lateral magnifi cation

y height above optical axis

a radiometer fi eld of view, projector fi eld of view

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1.4 DESIGN

First-order layout is the determination of the arrangement of the components of an optical system

in order to satisfy the first-order requirements imposed on the system The term “first-order” means the paraxial image properties: the size of the image, its orientations, its location, and the illumination

or brightness of the image This also implies apertures, f-numbers, fields of view, physical size

limi-tations, and the like It does not ordinarily include considerations of aberration correction; these are usually third- and higher-order matters, not first-order However, ordinary chromatic aberration and secondary spectrum are first-order aberrations Additionally, the first-order layout can have an effect on the Petzval curvature of field, the cost of the optics, the sensitivity to misalignment, and the defocusing effects of temperature changes

The primary task of first-order layout is to determine the powers and spacings of the system nents so that the image is located in the right place and has the right size and orientation It is not neces-sary to deal with surface-by-surface ray-tracing here; the concern is with components “Components”

compo-may mean single elements, cemented doublets, or even complex assemblies of many elements The order properties of a component can be described by its Gauss points: the focal points and principal points For layout purposes, however, the initial work can be done assuming that each component is of zero thickness; then only the component location and its power (or focal length) need be defined

trac-the axial ray at trac-the object and at trac-the image; this yields trac-the magnification m u u= 0/ ; object height k′times magnification yields the image height

The ray-tracing equations are

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