Introduction to Fourier Optics
`
`FNC 1021
`
`

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`McGraw-Hill Series in Electrical and Computer Engineering
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`SENIOR CONSULTING EDITOR
`Stephen W. Director, Carnegie Mellon University
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`Elec tromagnetics
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`SENIOR CONSULTING EDITOR
`Stephen W. Director, Carnegie Mellon University
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`Dearhold and McSpadden: Electromagnetic Wave Propagation
`Goodman: Introduction to Fourier Optics
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`

`

`Introduction
`to Fourier Optics
`
`S E C O N D E D I T I O N
`
`Joseph W. Goodman
`Stanford University
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`T H E McGRAW-HILL C O M P A N I E S , INC.
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`INTRODUCTION TO FOURIER OPTICS
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`Copyright ©1996, 1968 by The McGraw-Hill Companies, Inc. Reissued 1988 by The McGraw—Hill
`Companies. All rights reserved. Printed in the United States of America. Except as permitted under
`the United States Copyright Act of I976, no part of tlfis publication may be reproduced or distributed
`in any form or by any means, or stored in a data base or retrieval system, without the prior written
`permission of the publisher.
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`'Hrisbookisprintedon—acrd-freepaper.
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`34567890FGRFGR90987
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`ISBN 0-07—024254—2
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`This book was set in Times Roman by Publication Services, Inc.
`The editors were Lynn Cox and John M. Morriss;
`the production supervisor was Paula Keller.
`The cover was designed by Anthony Paccione.
`Quebecor Printing/Fairfield was printer and binder:
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`Librm of Congress Catalog Card Number:
`95-82033
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`

`

`A B O U T THE AUTHOR
`
`JOSEPH W. GOODMAN received the A.B. degree in Engineering and Applied
`Physics from Harvard University and the M.S and Ph.D. degrees in Electrical Engi-
`neering from Stanford University. He has been a member of the Stanford faculty since
`1967, and served as the Chairman of the Department of Electrical Engineering from
`1988 through 1996.
`Dr. Goodman's contributions to optics have been recognized in many ways. He has
`served as President of the International Commission for Optics and of the Optical So-
`ciety of America (OSA). He received the F.E. Terman award of the American Society
`for Engineering Education (1971), the Max Born Award of the OSA for contributions
`to physical optics (1983), the Dennis Gabor Award of the International Society for Op-
`tical Engineering (SPIE, 1987), the Education Medal of the Institute of Electrical and
`Electronics Engineers (IEEE, 1987), the Frederic Ives Medal of the OSA for overall
`distinction in optics (1990), and the Esther Hoffman Beller Medal of the OSA for con-
`tributions to optics education (1995). He is a Fellow of the OSA, the SPIE, and the
`IEEE. In 1987 he was elected to the National Academy of Engineering.
`In addition to Introduction to Fourier Optics, Dr. Goodman is the author of Statis-
`tical Optics ( J . Wiley & Sons, 1985) and the editor of International Trends in Optics
`(Academic Press, 1991). He has authored more than 200 scientific and technical articles
`in professional journals and books.
`
`

`

`To the memory of my Mothel; Doris Ryan Goodman,
`and my Fathel; Joseph Goodman, Jr:
`
`

`

`CONTENTS
`
`Preface
`
`xvii
`
`1 Introduction
`1 . Optics, Information, and Communication
`1.2 The Book
`
`2 Analysis of Two-Dimensional Signals and Systems
`2.1 Fourier Analysis in Two Dimensions
`2.1.1 Dejinition and Existence Conditions / 2.1.2 The Fourier
`Transform as a Decomposition / 2.1.3 Fourier Transform
`Theorems / 2.1.4 Separable Functions / 2.1.5 Functions with
`Circular Symmetry: Fourier- Bessel Transforms / 2.1.6 Some
`Frequently Used Functions and Some Useful Fourier Transform
`Pairs
`2.2 Local Spatial Frequency and Space-Frequency Localization
`2.3 Linear Systems
`2.3.1 Lineurity and the Superposition Integral / 2.3.2 Invuriunt
`Linear Systems: Transfer Functions
`2.4 Two-Dimensional Sampling Theory
`2.4.1 The Whittaker-Shannon Sampling Theorem / 2.4.2 Spacse-
`Bandwidth Producf
`Problems-Chapter 2
`
`3 Foundations of Scalar Diffraction Theory
`3.1 Historical Introduction
`3.2 From a Vector to a Scalar Theory
`3.3 Some Mathematical Preliminaries
`3.3.1 The Helmholtz Equation / 3.3.2 Green :s Theorem /
`3.3.3 The Intrgrul Theorem oj Helmholtz and Kirchhofl
`3.4 The Kirchhoff Formulation of Diffraction by a Planar
`Screen
`3.4.1 Applicution cfrhr Integral Theorem / 3.4.2 The Kirchhoff
`Boundary Conditiorzs / 3.4.3 The L'resnel-Kirchhoff D;ffrclction
`Formula
`3.5 The Rayleigh-Somrnerfeld Formulation of Diffraction
`3.5.1 Choice of Alternative Green :s Furzction.~ / 3.5.2 The
`Kuylc.igh-Sornmerfeld 1)iffruc.tion Fornzulu
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`

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`xii Contents
`
`3.6 Comparison of the Kirchhoff and Rayleigh-Sommerfeld
`Theories
`3.7 Further Discussion of the Huygens-Fresnel Principle
`3.8 Generalization to Nonmonochromatic Waves
`3.9 Diffraction at Boundaries
`3.10 The Angular Spectrum of Plane Waves
`3.10.1 The Angular Spectrum and Its Physical Interpretation /
`3.10.2 Propagation of the Angular Spectrum / 3.10.3 Effects
`of a Diffracting Aperture on the Angular Spectrum / 3.10.4
`The Propagation Phenomenon as a Linear Spatial Filter
`Problems-Chapter 3
`
`4 Fresnel and Fraunhofer Diffraction
`4.1 Background
`4.1. I The Intensity of a Wave Field / 4.1.2 The Huygens-Fresnel
`Principle in Rectangular Coordinates
`4.2 The Fresnel Approximation
`4.2.1 Positive vs. Negative Phases / 4.2.2 Accuracy of the
`Fresnel Approximation / 4.2.3 The Fresnel Approximation and
`the Angular Spectrum / 4.2.4 Fresnel Diffraction Between
`Confocal Spherical Sur$aces
`4.3 The Fraunhofer Approximation
`4.4 Examples of Fraunhofer Diffraction Patterns
`4.4.1 Rectangular Aperture / 4.4.2 Circular Aperture /
`4.4.3 Thin Sinusoidal Amplitude Grating / 4.4.4 Thin
`Sinusoidal Phase Grating
`4.5 Examples of Fresnel Diffraction Calculations
`4.5.1 Fresnel Diffraction by a Square Aperture /
`4.5.2 Fresnel Diffraction by a Sinusoidal Amplitude
`Grating-Talbot Images
`Problems--Chapter 4
`
`5 Wave-Optics Analysis of Coherent Optical Systems
`5.1 A Thin Lens as a Phase Transformation
`5.1. I The Thickness Function / 5.1.2 The Paraxial
`Approximation / 5.1.3 The Phase Transformation and
`Its Physical Meaning
`5.2 Fourier Transforming Properties of Lenses
`5.2.1 Input Placed Against the Lens / 5.2.2 Input Placed in Front
`of the Lens / 5.2.3 Input Placed Behind the Lens / 5.2.4 Example
`of an Optical Fourier Transform
`
`

`

`Contents
`
`5.3
`
`Image Formation: Monochromatic Illumination
`5.3.1 The Impulse Response of a Positive Lens / 5.3.2 Eliminating
`Quadratic Phase Factors: The Lens Law / 5.3.3 The Relation
`Between Object and Image
`5.4 Analysis of Complex Coherent Optical Systems
`5.4. I An Operator Notation / 5.4.2 Application of the Operator
`Approach to Some Optical Systems
`Problems--Chapter 5
`
`6 Frequency Analysis of Optical Imaging Systems
`6.1 Generalized Treatment of Imaging Systems
`6.1.1 A Generalized Model / 6.1.2 Effects of Diffraction on the
`Image / 6.1.3 Polychromatic Illumination: The Coherent and
`Incoherent Cases
`6.2 Frequency Response for Diffraction-Limited Coherent
`Imaging
`6.2.1 The Amplitude Transfer Function / 6.2.2 Examples of
`Amplitude Transfer Functions
`6.3 Frequency Response for Diffraction-Limited Incoherent
`Imaging
`6.3.1 The Optical Transfer Function / 6.3.2 General Properties
`of the OTF / 6.3.3 The OTF of an Aberration-Free System /
`6.3.4 Examples of Diffraction-Limited OTFs
`6.4 Aberrations and Their Effects on Frequency Response
`6.4.1 The Generalized Pupil Function / 6.4.2 EfSects of
`Aberrations on the Amplitude Transfer Function / 6.4.3 Effects
`of Aberrations on the OTF / 6.4.4 Example of a Simple
`Aberration: A Focusing Error / 6.4.5 Apodization and Its
`Effects on Frequency Response
`6.5 Comparison of Coherent and Incoherent Imaging
`6.5.1 Frequency Spectrum of the Image Intensity / 6.5.2
`Two-Point Resolution / 6.5.3 Other Effects
`6.6 Resolution Beyond the Classical Diffraction Limit
`6.6.1 Underlying Mathematical Fundamentals / 6.6.2 Intuitive
`Explanation of Bandwidth Extrapolation / 6.6.3 An Extrapolation
`Method Based on the Sampling Theorem / 6.6.4 An Iterative
`Extrapolation Method / 6.6.5 Practical Limitations
`Problems--Chapter 6
`
`7 Wavefront Modulation
`7.1 Wavefront Modulation with Photographic Film
`7.1.1 The Physical Processes of Exposure, Development, and
`Fixing / 7.1.2 Dejinition of Terms / 7.1.3 Film in an Incoherent
`
`

`

`xiv Contents
`
`Optical System / 7.1.4 Film in a Coherent Optical System /
`7.1.5 The Modulation Transfer Function / 7.1.6 Bleaching of
`Photographic Emulsions
`7.2 Spatial Light Modulators
`7.2.1 Properties of Liquid Crystals / 7.2.2 Spatial Light
`Modulators Based on Liquid Crystals / 7.2.3 Magneto-Optic
`Spatial Light Modulators / 7.2.4 Deformable Mirror Spatial
`Light Modulators / 7.2.5 Multiple Quantum Well Spatial Light
`Modulators / 7.2.6 Acousto-Optic Spatial Light Modulators
`7.3 Diffractive Optical Elements
`7.3.1 Binary Optics / 7.3.2 Other Types of DifSractive Optics /
`7.3.3 A Word of Caution
`Problems--Chapter 7
`
`8.2
`
`8 Analog Optical Information Processing
`8.1 Historical Background
`8.1.1 The Abbe-Porter Experiments / 8.1.2 The Zemike
`Phase-Contrast Microscope / 8.1.3 Improvement of Photographs:
`Mardchal / 8.1.4 The Emergence of a Communications
`Viewpoint / 8.1.5 Application of Coherent Optics to More
`General Data Processing
`Incoherent Image Processing Systems
`8.2.1 Systems Based on Geometrical Optics / 8.2.2 Systems That
`Incorporate the Effects of Diffraction
`8.3 Coherent Optical Information Processing Systems
`8.3.1 Coherent System Architectures / 8.3.2 Constraints on Filter
`Realization
`8.4 The VanderLugt Filter
`8.4.1 Synthesis of the Frequency-Plane Mask / 8.4.2 Processing
`the Input Data / 8.4.3 Advantages of the VanderLugt Filter
`8.5 The Joint Transform Correlator
`8.6 Application to Character Recognition
`8.6.1 The Matched Filter / 8.6.2 A Character-Recognition
`Problem / 8.6.3 Optical Synthesis of a Character-Recognition
`Machine / 8.6.4 Sensitivity to Scale Size and Rotation
`8.7 Optical Approaches to Invariant Pattern Recognition
`8.7.1 Mellin Correlators / 8.7.2 Circular Harmonic Correlation /
`8.7.3 Synthetic Discriminant Functions
`Image Restoration
`8.8.1 The Inverse Filter / 8.8.2 The Wiener Filtec or the Least-
`Mean-Square-Error Filter / 8.8.3 Filter Realization
`8.9 Processing Synthetic-Aperture Radar (SAR) Data
`8.9.1 Formation of the Synthetic Aperture / 8.9.2 The Collected
`Data and the Recording Format / 8.9.3 Focal Properties of the
`
`8.8
`
`

`

`Contents xv
`
`Film Transparency / 8.9.4 Forming a Two-Dimensional Image /
`8.9.5 The Tilted Plane Processor
`8.10 Acousto-Optic Signal Processing Systems
`8.10.1 Bragg Cell Spectrum Analyzer / 8.10.2 Space-Integrating
`Correlator / 8.10.3 Time-Integrating Correlator / 8.10.4 Other
`Acousto-Optic Signal Processing Architectures
`8.11 Discrete Analog Optical Processors
`8.11.1 Discrete Representation of Signals and Systems /
`8.11.2 A Serial Matrix-Vector Multiplier / 8.11.3 A Parallel
`Incoherent Matrix-Vector Multiplier / 8.11.4 An Outer
`Product Processor / 8.11.5 Other Discrete Processing
`Architectures / 8.11.6 Methods for Handling Bipolar and
`Complex Data
`Problems-Chapter 8
`
`9 Holography
`9.1 Historical Introduction
`9.2 The Wavefront Reconstruction Problem
`9.2.1 Recording Amplitude and Phase / 9.2.2 The Recording
`Medium / 9.2.3 Reconstruction of the Original Wavefront /
`9.2.4 Linearity of the Holographic Process / 9.2.5 Image
`Formation by Holography
`9.3 The Gabor Hologram
`9.3.1 Origin of the Reference Wave / 9.3.2 The Twin Images /
`9.3.3 Limitations of the Gabor Hologram
`9.4 The Leith-Upatnieks Hologram
`9.4.1 Recording the Hologram / 9.4.2 Obtaining the
`Reconstructed Images / 9.4.3 The Minimum Reference
`Angle / 9.4.4 Holography of Three-Dimensional Scenes /
`9.4.5 Practical Problems in Holography
`Image Locations and Magnification
`9.5.1 Image Locations / 9.5.2 Axial and Transverse
`Magn$cations / 9.5.3 An Example
`Some Different Types of Holograms
`9.6.1 Fresnel, Fraunhofer, Image, and Fourier Holograms /
`9.6.2 Transmission and Reflection Holograms / 9.6.3 Holographic
`Stereograms / 9.6.4 Rainbow Holograms / 9.6.5 Multiplex
`Holograms / 9.6.6 Embossed Holograms
`9.7 Thick Holograms
`9.7.1 Recording a Volume Holographic Grating /
`9.7.2 Reconstructing Wavefronts from a Volume Grating /
`9.7.3 Fringe Orientations for More Complex Recording
`Geometries / 9.7.4 Gratings of Finite Size / 9.7.5 Diffraction
`ESficiency-Coupled Mode Theory
`
`9.6
`
`9.5
`
`276
`
`282
`
`290
`
`295
`295
`296
`
`302
`
`304
`
`3 14
`
`3 19
`
`329
`
`

`

`xvi Contents
`
`9.8 Recording Materials
`9.8.1 Silver Halide Emulsions / 9.8.2 Photopolymer Films /
`9.8.3 Dichromated Gelatin / 9.8.4 Photorefractive Materials
`9.9 Computer-Generated Holograms
`9.9.1 The Sampling Problem / 9.9.2 The Computational
`Problem / 9.9.3 The Representational Problem
`9.10 Degradations of Holographic Images
`9.10.1 Effects of Film MTF / 9.10.2 Effects of Film
`Nonlinearities / 9.10.3 Effects of Film-Grain Noise /
`9.10.4 Speckle Noise
`9.11 Holography with Spatially Incoherent Light
`9.12 Applications of Holography
`9.12.1 Microscopy and High-Resolution Volume Imagery /
`9.12.2 Inte$erometry / 9.12.3 Imaging Through Distorting
`Media / 9.12.4 Holographic Data Storage / 9.12.5 Holographic
`Weights for Artijicial Neural Networks / 9.12.6 Other
`Applications
`Problems--Chapter 9
`
`A Delta Functions and Fourier Transform Theorems
`A.l Delta Functions
`A.2 Derivation of Fourier Transform Theorems
`
`B Introduction to Paraxial Geometrical Optics
`B.l The Domain of Geometrical Optics
`B.2 Refraction, Snell's Law, and the Paraxial Approximation
`B.3 The Ray-Transfer Matrix
`B.4 Conjugate Planes, Focal Planes, and Principal Planes
`B.5 Entrance and Exit Pupils
`
`C Polarization and Jones Matrices
`C.1 Definition of the Jones Matrix
`C.2 Examples of Simple Polarization Transformations
`C.3 Reflective Polarization Devices
`
`Bibliography
`
`Index
`
`346
`
`35 1
`
`363
`
`369
`372
`
`388
`
`393
`393
`395
`
`401
`40 1
`403
`404
`407
`41 1
`
`415
`415
`417
`4 18
`
`42 1
`
`433
`
`

`

`PREFACE
`
`Fourier analysis is a ubiquitous tool that has found application to diverse areas of
`physics and engineering. This book deals with its applications in optics, and in partic-
`ular with applications to diffraction, imaging, optical data processing, and holography.
`Since the subject covered is Fourier Optics, it is natural that the methods of Fourier
`analysis play a key role as the underlying analytical structure of our treatment. Fourier
`analysis is a standard part of the background of most physicists and engineers. The
`theory of linear systems is also familiar, especially to electrical engineers. Chapter 2
`reviews the necessary mathematical background. For those not already familiar with
`Fourier analysis and linear systems theory, it can serve as the outline for a more detailed
`study that can be made with the help of other textbooks explicitly aimed at this subject.
`Ample references are given for more detailed treatments of this material. For those
`who have already been introduced to Fourier analysis and linear systems theory, that
`experience has usually been with functions of a single independent variable, namely
`time. The material presented in Chapter 2 deals with the mathematics in two spatial
`dimensions (as is necessary for most problems in optics), yielding an extra richness not
`found in the standard treatments of the one-dimensional theory.
`The original edition of this book has been considerably expanded in this second
`edition, an expansion that was needed due to the tremendous amount of progress in
`the field since 1968 when the first edition was published. The book can be used as a
`textbook to satisfy the needs of several different types of courses. It is directed towards
`both physicists and engineers, and the portions of the book used in the course will in
`general vary depending on the audience. However, by properly selecting the material to
`be covered, the needs of any of a number of different audiences can be met. This Preface
`will make several explicit suggestions for the shaping of different kinds of courses.
`First a one-quarter or one-semester course on diffraction and image formation can
`be constructed from the materials covered in Chapters 2 through 6, together with all
`three appendices. If time is short, the following sections of these chapters can be omitted
`or left as reading for the advanced student: 3.8, 3.9,5.4, and 6.6.
`A second type of one-quarter or one-semester course would cover the basics of
`Fourier Optics, but then focus on the application area of analog optical signal process-
`ing. For such a course, I would recommend that Chapter 2 be left to the reading of
`the student, that the material of Chapter 3 be begun with Section 3.7, and followed
`by Section 3.10, leaving the rest of this chapter to a reading by those students who
`are curious as to the origins of the Huygens-Fresnel principle. In Chapter 4, Sections
`4.2.2 and 4.5.1 can be skipped. Chapter 5 can begin with Eq. (5-10) for the amplitude
`transmittance function of a thin lens, and can include all the remaining material, with
`the exception that Section 5.4 can be left as reading for the advanced students. If time
`is short, Chapter 6 can be skipped entirely. For this course, virtually all of the material
`presented in Chapter 7 is important, as is much of the material in Chapter 8. If it is nec-
`essary to reduce the amount of material, I would recommend that the following sections
`be omitted: 8.2,8.8, and 8.9. It is often desirable to include some subset of the material
`
`

`

`xviii Preface
`
`on holography from Chapter 9 in this course. I would include sections 9.4,9.6.1,9.6.2,
`9.7.1, 9.7.2, 9.8, 9.9, and 9.12.5. The three appendices should be read by the students
`but need not be covered in lectures.
`A third variation would be a one-quarter or one-semester course that covers the
`basics of Fourier Optics but focuses on holography as an application. The course can
`again begin with Section 3.7 and be followed by Section 3.10. The coverage through
`Chapter 5 can be identical with that outlined above for the course that emphasizes op-
`tical signal processing. In this case, the material of Sections 6.1, 6.2, 6.3, and 6.5 can
`be included. In Chapter 7, only Section 7.1 is needed, although Section 7.3 is a useful
`addition if there is time. Chapter 8 can now be skipped and Chapter 9 on holography
`can be the focus of attention. If time is short, Sections 9.10 and 9.11 can be omitted.
`The first two appendices should be read by the students, and the third can be skipped.
`In some universities, more than one quarter or one semester can be devoted to this
`material. In two quarters or two semesters, most of the material in this book can be
`covered.
`The above suggestions can of course be modified to meet the needs of a particular
`set of students or to emphasize the material that a particular instructor feels is most ap-
`propriate. I hope that these suggestions will at least give some ideas about possibilities.
`There are many people to whom I owe a special word of thanks for their help with
`this new edition of the book. Early versions of the manuscript were used in courses at
`several different universities. I would in particular like to thank Profs. A.A. Sawchuk,
`J.F. Walkup, J. Leger, P. Pichon, D. Mehrl, and their many students for catching so many
`typographical errors and in some cases outright mistakes. Helpful comments were also
`made by I. Erteza and M. Bashaw, for which I am grateful. Several useful suggestions
`were also made by anonymous manuscript reviewers engaged by the publisher. A spe-
`cial debt is owed to Prof. Emmett Leith, who provided many helpful suggestions. I
`would also like to thank the students in my 1995 Fourier Optics class, who competed
`fiercely to see who could find the most mistakes. Undoubtedly there are others to whom
`I owe thanks, and I apologize for not mentioning them explicitly here.
`Finally, I thank Hon Mai, without whose patience, encouragement and support this
`book would not have have been possible.
`
`Joseph W. Goodman
`
`

`

`Introduction to Fourier Optics
`
`

`

`C H A P T E R 1
`
`Introduction
`
`1.1
`OPTICS, INFORMATION, AND COMMUNICATION
`
`Since the late 1930s, the venerable branch of physics known as optics has gradually
`developed ever-closer ties with the communication and information sciences of elec-
`trical engineering. The trend is understandable, for both communication systems and
`imaging systems are designed to collect or convey information. In the former case, the
`information is generally of a temporal nature (e.g. a modulated voltage or current wave-
`form), while in the latter case it is of a spatial nature (e.g. a light amplitude or intensity
`distribution over space), but from an abstract point of view, this difference is a rather
`superficial one.
`Perhaps the strongest tie between the two disciplines lies in the similar mathemat-
`ics which can be used to describe the respective systems of interest - the mathematics
`of Fourier analysis and systems theory. The fundamental reason for the similarity is not
`merely the common subject of "information", but rather certain basic properties which
`communication systems and imaging systems share. For example, many electronic net-
`works and imaging devices share the properties called linearity and invariance (for def-
`initions see Chapter 2). Any network or device (electronic, optical, or otherwise) which
`possesses these two properties can be described mathematically with considerable ease
`using the techniques of frequency analysis. Thus, just as it is convenient to describe an
`audio amplifier in terms of its (temporal) frequency response, so too it is often conve-
`nient to describe an imaging system in terms of its (spatial) frequency response.
`The similarities do not end when the linearity and invariance properties are absent.
`Certain nonlinear optical elements (e-g. photographic film) have input-output relation-
`ships which are directly analogous to the corresponding characteristics of nonlinear
`electronic components (diodes, transistors, etc.), and similar mathematical analysis can
`be applied in both cases.
`
`

`

`2
`
`Introduction to Fourier Optics
`
`It is particularly important to recognize that the similarity of the mathematical
`structures can be exploited not only for analysis purposes but also for synthesis pur-
`poses. Thus, just as the spectrum of a temporal function can be intentionally manipu-
`lated in a prescribed fashion by filtering, so too can the spectrum of a spatial function
`be modified in various desired ways. The history of optics is rich with examples of im-
`portant advances achieved by application of Fourier synthesis techniques - the Zernike
`phase-contrast microscope is an example that was worthy of a Nobel prize. Many other
`examples can be found in the fields of signal and image processing.
`
`1.2
`THE BOOK
`
`The readers of this book are assumed at the start to have a solid foundation in Fourier
`analysis and linear systems theory. Chapter 2 reviews the required background; to
`avoid boring those who are well grounded in the analysis of temporal signals and sys-
`tems, the review is conducted for functions of two independent variables. Such func-
`tions are, of course, of primary concern in optics, and the extension from one to two
`independent variables provides a new richness to the mathematical theory, introducing
`many new properties which have no direct counterpart in the theory of temporal signals
`and systems.
`The phenomenon called difSraction is of the utmost importance in the theory of
`optical systems. Chapter 3 treats the foundations of scalar diffraction theory, including
`the Kirchhoff, Rayleigh-Sommerfeld, and angular spectrum approaches. In Chapter 4,
`certain approximations to the general results are introduced, namely the Fresnel and
`Fraunhofer approximations, and examples of diffraction-pattern calculations are pre-
`sented.
`Chapter 5 considers the analysis of coherent optical systems which consist of lenses
`and free-space propagation. The approach is that of wave optics, rather than the more
`common geometrical optics method of analysis. A thin lens is modeled as a quadratic
`phase transformation; the usual lens law is derived from this model, as are certain
`Fourier transforming properties of lenses.
`Chapter 6 considers the application of frequency analysis techniques to both co-
`herent and incoherent imaging systems. Appropriate transfer functions are defined and
`their properties discussed for systems with and without aberrations. Coherent and in-
`coherent systems are compared from various points of view. The limits to achievable
`resolution are derived.
`In Chapter 7 the subject of wavefront modulation is considered. The properties
`of photographic film as an input medium for incoherent and coherent optical systems
`are discussed. Attention is then turned to spatial light modulators, which are devices
`for entering information into optical systems in real time or near real time. Finally,
`diffractive optical elements are described in some detail.
`Attention is turned to analog optical information processing in Chapter 8. Both
`continuous and discrete processing systems are considered. Applications to image
`
`

`

`1 Introduction 3
`C H A ~ R
`
`enhancement, pattern recognition, and processing of synthetic-aperture radar data are
`considered.
`The final chapter is devoted to the subject of holography. The techniques devel-
`oped by Gabor and by Leith and Upatnieks are considered in detail and compared.
`Both thin and thick holograms are treated. Extensions to three-dimensional imaging
`are presented. Various applications of holography are described, but emphasis is on the
`fundamentals.
`
`

`

`C H A P T E R 2
`
`Analysis of Two-Dimensional Signals
`and Systems
`
`Many physical phenomena are found experimentally to share the basic property that
`their response to several stimuli acting simultaneously is identically equal to the sum of
`the responses that each component stimulus would produce individually. Such phenom-
`ena are called lineal; and the property they share is called linearity. Electrical networks
`composed of resistors, capacitors, and inductors are usually linear over a wide range of
`inputs. In addition, as we shall soon see, the wave equation describing the propagation
`of light through most media leads us naturally to regard optical imaging operations as
`linear mappings of "object" light distributions into "image" light distributions.
`The single property of linearity leads to a vast simplification in the mathematical
`description of such phenomena and represents the foundation of a mathematical struc-
`ture which we shall refer to here as linear systems theory. The great advantage afforded
`by linearity is the ability to express the response (be it voltage, current, light amplitude,
`or light intensity) to a complicated stimulus in terms of the responses to certain "elemen-
`tary" stimuli. Thus if a stimulus is decomposed into a linear combination of elementary
`stimuli, each of which produces a known response of convenient form, then by virtue
`of linearity, the total response can be found as a corresponding linear combination of
`the responses to the elementary stimuli.
`In this chapter we review some of the mathematical tools that are useful in describ-
`ing linear phenomena, and discuss some of the mathematical decompositions that are
`often employed in their analysis. Throughout the later chapters we shall be concerned
`with stimuli (system inputs) and responses (system outputs) that may be either of two
`different physical quantities. If the illumination used in an optical system exhibits a
`property called spatial coherence, then we shall find that it is appropriate to describe
`the light as a spatial distribution of complex-valued field amplitude. When the illumi-
`nation is totally lacking in spatial coherence, it is appropriate to describe the light as a
`spatial distribution of real-valued intensity. Attention will be focused here on the anal-
`ysis of linear systems with complex-valued inputs; the results for real-valued inputs are
`thus included as special cases of the theory.
`
`

`

`CHAPTER 2 Analysis of Two-Dimensional Signals and Systems 5
`
`2.1
`FOURIER ANALYSIS IN TWO DIMENSIONS
`
`A mathematical tool of great utility in the analysis of both linear and nonlinear phenom-
`ena is Fourier analysis. This tool is widely used in the study of electrical networks and
`communication systems; it is assumed that the reader has encountered Fourier theory
`previously, and therefore that he or she is familiar with the analysis of functions of one
`independent variable (e.g. time). For a review of the fundamental mathematical con-
`cepts, see the books by Papoulis [226], Bracewell [32], and Gray and Goodman [131].
`A particularly relevant treatment is by Bracewell [33]. Our purpose here is limited to
`extending the reader's familiarity to the analysis of functions of two independent vari-
`ables. No attempt at great mathematical rigor will be made, but rather, an operational
`approach, characteristic of most engineering treatments of the subject, will be adopted.
`
`2.1.1 Definition and Existence Conditions
`
`The Fourier transform (alternatively the Fourier spectrum or frequency spectrum) of
`a (in general, complex-valued) function g of two independent variables x and y will be
`represented here by F{g} and is defined by1
`
`The transform so defined is itself a complex-valued function of two independent vari-
`ables fx and fr, which we generally refer to as frequencies. Similarly, the inverse
`Fourier transform of a function G(fx, fy) will be represented by F 1 { G } and is de-
`fined as
`
`Note that as mathematical operations the transform and inverse transform are very sim-
`ilar, differing only in the sign of the exponent appearing in the integrand. The inverse
`Fourier transform is sometimes referred to as the Fourier integral representation of a
`function g(x, y).
`Before discussing the properties of the Fourier transform and its inverse, we must
`first decide when (2-1) and (2-2) are in fact meaningful. For certain functions, these
`integrals may not exist in the usual mathematical sense, and therefore this discussion
`would be incomplete without at least a brief mention of "existence conditions". While
`a variety of sets of suficient conditions for the existence of (2- 1) are possible, perhaps
`the most common set is the following:
`
`'When a single limit of integration appears above or below a double integral, then that limit applies to both
`integrations.
`
`

`

`6
`
`Introduction to Fourier Optics
`
`1. g must be absolutely integrable over the infinite ( x , y ) plane.
`2. g must have only a finite number of discontinuities and a finite number of maxima
`and minima in any finite rectangle.
`3. g must have no infinite discontinuities.
`In general, any one of these conditions can be weakened at the price of strengthen-
`ing one or both of the companion conditions, but such considerations lead us rather far
`afield from our purposes here.
`As Bracewell [32] has pointed out, "physical possibility is a valid sufficient condi-
`tion for the existence of a transform." However, it is often convenient in the analysis of
`systems to represent true physical waveforms by idealized mathematical functions, and
`for such functions one or more of the above existence conditions may be violated. For
`example, it is common to represent a strong, narrow time pulse by the so-called Dirac
`delta function2 often represented by
`2
`2
`6(t) = lim Nexp(-N ~t ),
`N + m
`where the limit operation provides a convenient mental construct but is not meant to be
`taken literally. See Appendix A for more details. Similarly, an idealized point source of
`light is oft