Introduction to Fourier Optics
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Introduction
to Fourier Optics
SECOND EDITION
Joseph W. Goodman
Stanford University
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INTRODUCTION TO FOURIER OPTICS
Copyright ©1996, 1968 by The McGraw-Hill Companies, Inc. Reissued 1988 by The McGraw-Hill
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ABOUT 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 (OS A). He received the F.E. Terman award of the American Society
for Engineering Education (1971), the Max Bom Award of the OS A 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 Mother, Doris Ryan Goodman,
and my Fathel; Joseph Goodman, Jr,
CONTENTS
i urn " inr "in it" i mr i wMi ii i' iii ri i rr-nr'Tr - - — —
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 Definition 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 16
2.3 Lineal - Systems 1 9
2.3.1 Line urity and the Superposition Integral/ 2.3.2 Invariant
Linear Systems: Transfer Functions
2.4 Two-Dimensional Sampling Theory 22
2.4.1 The Whittaker-Shannon Sampling Theorem / 2.4.2 Space-
Bandwidth Product
Problems — Chapter 2 27
3 Foundations of Scalar Diffraction Theory 32
3.1 Historical Introduction 32
3.2 From a Vector to a Scalar Theory 36
3.3 Some Mathematical Preliminaries 38
3.3.1 The Helmholtz Equation / 3.3.2 Green's Theorem /
3.3.3 The Integral Theorem of Helmholtz and Kirchhoff
3.4 The Kirchhoff Formulation of Diffraction by a Planar
Screen 42
3.4.1 Application of the Integral Theorem / 3.4.2 The Kirchhoff
Boundary Conditions / 3.4.3 The Fresnel-Kirchhoff Diffraction
Formula
33 The Rayleigh-Sommerfeld Formulation of Diffraction
3.5.1 Choice of Alternative Green's Functions / 3.5.2 The
Rayleigh-Sommerfeld Diffraction Formula
xii Contents
3.6 Comparison of the Kirchhoff and Rayleigh-Sommerfeld
Theories 50
3.7 Further Discussion of the Huygens-Fresnel Principle 52
3.8 Generalization to Nonmonochromatic Waves 53
3.9 Diffraction at Boundaries 54
3.10 The Angular Spectrum of Plane Waves 55
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 6 1
4 Fresnel and Fraunhofer Diffraction 63
4.1 Background 63
4. 1. 1 The Intensity of a Wave Field / 4. 1.2 The Huygens-Fresnel
Principle in Rectangular Coordinates
4.2 The Fresnel Approximation 66
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 Surfaces
4.3 The Fraunhofer Approximation 73
4.4 Examples of Fraunhofer Diffraction Patterns 75
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 83
4.5.1 Fresnel Diffraction by a Square Aperture /
4.5.2 Fresnel Diffraction by a Sinusoidal Amplitude
Grating — Talbot Images
Problems — Chapter 4 90
5 Wave-Optics Analysis of Coherent Optical Systems 96
5.1 A Thin Lens as a Phase Transformation 96
5.1.1 The Thickness Function / 5.1.2 The Paraxial
Approximation / 5.1 .3 The Phase Transformation and
Its Physical Meaning
52 Fourier Transforming Properties of Lenses 101
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 xiii
5.3 Image Formation: Monochromatic Illumination 108
53.1 The Impulse Response of a Positive Lens/ 53.2 Eliminating
Quadratic Phase Factors: The Lens Law / 533 The Relation
Between Object and Image
5.4 Analysis of Complex Coherent Optical Systems 114
5.4.1 An Operator Notation / 5.42 Application of the Operator
Approach to Some Optical Systems
Problems— Chapter 5 120
6 Frequency Analysis of Optical Imaging Systems 126
6.1 Generalized Treatment of Imaging Systems 127
6.1.1 A Generalized Model / 6.1.2 Effects of Diffraction on the
Image / 6.13 Polychromatic Illumination: The Coherent and
Incoherent Cases
6.2 Frequency Response for Diffraction-Limited Coherent
Imaging 134
6.2.1 The Amplitude Transfer Function / 6.2.2 Examples of
Amplitude Transfer Functions
6.3 Frequency Response for Diffraction-Limited Incoherent
Imaging 1 37
63.1 The Optical Transfer Function / 63.2 General Properties
of the OTF / 633 The OTF of an Aberration-Free System /
63.4 Examples of Diffi-action-Limited OTFs
6.4 Aberrations and Their Effects on Frequency Response 145
6.4.1 The Generalized Pupil Function / 6.4.2 Effects of
Aberrations on the Amplitude Transfer Function / 6.43 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 154
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 160
6.6.1 Underlying Mathematical Fundamentals / 6.6.2 Intuitive
Explanation of Bandwidth Extrapolation / 6.63 An Extrapolation
Method Based on the Sampling Theorem / 6.6.4 An Iterative
Extrapolation Method / 6.6.5 Practical Limitations
Problems— Chapter 6 165
7 Wavefront Modulation 172
7.1 Wavefront Modulation with Photographic Film 173
7.1.1 The Physical Processes of Exposure , Development , and
Fixing / 7.1.2 Definition of Terms / 7.13 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 184
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 209
7.3. 1 Binary Optics / 7.3.2 Other Types of Diffractive Optics /
7.3.3 A Word of Caution
Problems- -Chapter 7 215
8 Analog Optical Information Processing 217
8.1 Historical Background 218
8.1.1 The Abbe-Porter Experiments / 8.1 .2 The Zemike
Phase-Contrast Microscope / 8. 1.3 Improvement of Photographs:
Marechal / 8.1.4 The Emergence of a Communications
Viewpoint / 8.1.5 Application of Coherent Optics to More
General Data Processing
8.2 Incoherent Image Processing Systems 224
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 232
8.3.1 Coherent System Architectures / 8.3.2 Constraints on Filter
Realization
8.4 The VanderLugt Filter 237
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 243
8.6 Application to Character Recognition 246
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 252
8.7.1 Mellin Correlators / 8.7.2 Circular Harmonic Correlation /
8.7.3 Synthetic Discriminant Functions
88 Image Restoration 257
8.8. 1 The Inverse Filter / 8.8.2 The Wiener Filter, or the Least-
Mean-Square-Error Filter / 8.8.3 Filter Realization
8.9 Processing Synthetic-Aperture Radar (SAR) Data 264
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
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
93 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
95 Image Locations and Magnification
9.5.1 Image Locations / 9.5.2 Axial and Transverse
Magnifications / 9.5.3 An Example
9.6 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
Efficiency-Coupled Mode Theory
276
282
290
295
295
296
302
304
314
319
329
xvi Contents
9.8 Recording Materials 346
9.8.1 Silver Halide Emulsions / 9.82 Photopolymer Films /
9.83 Dichromated Gelatin / 9.8.4 Photorefractive Materials
9.9 Computer-Generated Holograms 35 1
9.9.1 The Sampling Problem / 9 . 9.2 The Computational
Problem / 9.9.3 The Representational Problem
9.10 Degradations of Holographic Images 363
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 369
9.12 Applications of Holography 372
9.12.1 Microscopy and High-Resolution Volume Imagery /
9.12.2 Interferometry / 9.12.3 Imaging Through Distorting
Media / 9.12.4 Holographic Data Storage/ 9.12.5 Holographic
Weights for Artificial Neural Networks / 9.12.6 Other
Applications
Problems — Chapter 9 388
A Delta Functions and Fourier Transform Theorems 393
A. l Delta Functions 393
A 2 Derivation of Fourier Transform Theorems 395
B Introduction to Paraxial Geometrical Optics 401
B. l The Domain of Geometrical Optics 401
B 2 Refraction, Snell’s Law, and the Paraxial Approximation 403
B3 The Ray-Transfer Matrix 404
B.4 Conjugate Planes, Focal Planes, and Principal Planes 407
B. 5 Entrance and Exit Pupils 411
C Polarization and Jones Matrices 415
C. l Definition of the Jones Matrix 415
C 2 Examples of Simple Polarization Transformations 417
C3 Reflective Polarization Devices 418
Bibliography 421
Index
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
CHAPTER 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 diffraction 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
chapter i Introduction 3
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.
CHAPTER 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 orfrequency spectrum) of
a (in general, complex-valued) function g of two independent variables x and y will be
represented here by JT{g} and is defined by 1
Hg] =
00
— oo
g(x, y)exp[-j2TT(f x x + f Y y )] dxdy.
( 2 - 1 )
The transform so defined is itself a complex-valued function of two independent vari-
ables fx and fy, which we generally refer to as frequencies. Similarly, the inverse
Fourier transform of a function G(fx, fy) will be represented by j F -I {G} and is de-
fined as
oo
—oo
G(fx, fy) txp[j27T(f x x + f Y y)]df x dfy.
( 2 - 2 )
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 sufficient 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 function 2 often represented by
8(t ) = lim N exp(-N 2 7Tt 2 ), (2-3)
N->CO
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 often represented by the two-dimensional equivalent,
8(x, y ) = $im°N 2 exp[-N 2 'jr(x 1 + y 2 )]. (2-4)
Such "functions", being infinite at the origin and zero elsewhere, have an infinite dis-
continuity and therefore fail to satisfy existence condition 3. Qther important examples
are readily found; for example, the functions
fix, y ) = 1 and f(x, y) = cos {2irf x x) (2-5)
both fail to satisfy existence condition 1.
If the majority of functions of interest are to be included within the framework of
Fourier analysis, some generalization of the definition (2-1) is required. Fortunately, it
is often possible to find a meaningful transform of functions that do not strictly satisfy
the existence conditions, provided those functions can be defined as the limit of a se-
quence of functions that are transformable. By transforming each member function of
the defining sequence, a corresponding sequence of transforms is generated, and we
call the limit of this new sequence the generalized Fourier transform of the original
function. Generalized transforms can be manipulated in the same manner as conven-
tional transforms, and the distinction between the two cases can generally be ignored,
it being understood that when a function fails to satisfy the existence conditions and
yet is said to have a transform, then the generalized transform is actually meant. For a
more detailed discussion of this generalization of Fourier analysis the reader is referred
to the book by Lighthill [194].
To illustrate the calculation of a generalized transform, consider the Dirac delta
function, which has been seen to violate existence condition 3. Note that each member
function of the defining sequence (2-4) does satisfy the existence requirements and that
each, in fact, has a Fourier transform given by (see Table 2.1)
2 For a more detailed discussion of the delta function, including definitions, see Appendix A.
chapter 2 Analysis of Two-Dimensional Signals and Systems 7
F{N 2 exp[-N 2 Tr(x 2 + y 2 )]} = exp[- —
N z
Accordingly the generalized transform of 8(;t, y) is found to be
( 2 - 6 )
y)} = lim < exp
N — >00
vjfl t Ixl
N 2
= 1 .
(2-7)
Note that the spectrum of a delta function extends uniformly over the entire frequency
domain.
For other examples of generalized transforms, see Table 2.1.
2.1.2 The Fourier Transform as a Decomposition
As mentioned previously, when dealing with linear systems it is often useful to decom-
pose a complicated input into a number of more simple inputs, to calculate the response
of the system to each of these "elementary" functions, and to superimpose the individ-
ual responses to find the total response. Fourier analysis provides the basic means of
performing such a decomposition. Consider the familiar inverse transform relationship
8(0 = ( G(f) expijlTT fOdf (2-8)
J —00
expressing the time function g in terms of its frequency spectrum. We may regard this
expression as a decomposition of the function g(t) into a linear combination (in this
case an integral) of elementary functions, each with a specific form exp(y'27rf t). From
this it is clear that the complex number G(f) is simply a weighting factor that must
be applied to the elementary function of frequencyf in order to synthesize the desired
g(0-
In a similar fashion, we may regard the two-dimensional Fourier transform as a de-
composition of a function g(x, y) into a linear combination of elementary functions of
the form exp[/27r(/xx+ fyy)]. Such functions have a number of interesting properties.
Note that for any particular frequency pair ( fx , fy ) the corresponding elementary func-
tion has a phase that is zero or an integer multiple of 2tt radians along lines described
by the equation
y
fx
fr
(2-9)
where n is an integer. Thus, as indicated in Fig. 2.1, this elementary function may be
regarded as being "directed" in the (x,y) plane at an angle 8 (with respect to the x axis)
given by
6 = arctan^^j. (2-10)
In addition, the spatial period (i.e. the distance between zero-phase lines) is given by
8 Introduction to Fourier Optics
FIGURE 2.1
Lines of zero phase for the function
exp[j2ir(f x x + f Y y )].
L =
1
V fx + fy
( 2 - 11 )
In conclusion, then, we may again regard the inverse Fourier transform as providing a
means for decomposing mathematical functions . The Fourier spectrum G of a function g
is simply a description of the weighting factors that must be applied to each elementary
function in order to synthesize the desired g. The real advantage obtained from using
this decomposition will not be fully evident until our later discussion of invariant linear
systems.
2.13 Fourier Transform Theorems
The basic definition (2-1) of the Fourier transform leads to a rich mathematical
structure associated with the transform operation. We now consider a few of the
basic mathematical properties of the transform, properties that will find wide use in
later material. These properties are presented as mathematical theorems, followed
by brief statements of their physical significance. Since these theorems are direct
extensions of the analogous one-dimensional statements, the proofs are deferred to
Appendix A.
1. Linearity theorem. 3-{ag + (3h} — ct!F{g} + that is, the transform of a
weighted sum of two (or more) functions is simply the identically weighted sum of
their individual transforms.
2. Similarity theorem. If T{g(x, y)) = G(fx, fy ), then
)) = f Lc(A4) ; (2 -‘ 2)
that is, a "stretch" of the coordinates in the space domain (x, y) results in a contrac-
tion of the coordinates in the frequency domain (fx, fy ), plus a change in the overall
amplitude of the spectrum.
3. Shift theorem. If T{g(x, y)} = G(f x , fy), then
Hg(x — a, y — b)} = G(f x , f Y )exp[-j2Tr(f x a + fyb )]; (2-13)
that is, translation in the space domain introduces a linear phase shift in the fre-
quency domain.
chapter 2 Analysis of Two-Dimensional Signals and Systems 9
4. Rayleigh's theorem (Parseval's theorem). If F{g(x, y)} = G(fx> fy), then
J | g{x, y)| 2 dxdy = Jj | G(f x , fyf df x df Y . (2-14)
The integral on the left-hand side of this theorem can be interpreted as the energy
contained in the waveform g(x, y). This in turn leads us to the idea that the quantity
Wx.fy)? can be interpreted as an energy density in the frequency domain.
5. Convolution theorem. If !F{g(x, y)) = G(fx, fy) and T{h(x, y)} = H(fx, fy), then
T
JJ git V) Kx ~ t T - t?) di;dr) 1
G(f x , fy)H(f x , fy).
(2-15)
The convolution of two functions in the space domain (an operation that will be
found to arise frequently in the theory of linear systems) is entirely equivalent to
the more simple operation of multiplying their individual transforms and inverse
transforming.
6. Autocorrelation theorem. If F{g(x, y)} = G(fx, fy), then
T
git V) g*(£ - x, V - y) dg dig
y = \G{fx,fy)\ 2
Similarly,
t)| 2 } = G(t rj) G*(£ - fx, V - fy) d^ dr].
(2-16)
(2-17)
This theorem may be regarded as a special case of the convolution theorem in which
we convolve g(x, y) with g*(— x, -y).
7. Fourier integral theorem. At each point of continuity of g,
TT- { {gix, y)} = T-'HgU y)} = g(x, y). (2-18)
At each point of discontinuity of g, the two successive transforms yield the angular
average of the values of g in a small neighborhood of that point. That is, the suc-
cessive transformation and inverse transformation of a function yields that function
again, except at points of discontinuity.
The above transform theorems are of far more than just theoretical interest. They
will be used frequently, since they provide the basic tools for the manipulation of Fourier
transforms and can save enormous amounts of work in the solution of Fourier analysis
problems.
10 Introduction to Fourier Optics
2.1 .4 Separable F unctions
A function of two independent variables is called separable with respect to a specific
coordinate system if it can be written as a product of two functions, each of which
depends on only one of the independent variables. Thus the function g is separable in
rectangular coordinates (x, y) if
g(x,y) = gx(x)g Y (y), (2-19)
while it is separable in polar coordinates (r, 8) if
g(r, 0) = gR(r) g©(0). (2-20)
Separable functions are often more convenient to deal with than more general
functions, for separability often allows complicated two-dimensional manipulations to
be reduced to more simple one-dimensional manipulations. For example, a function
separable in rectangular coordinates has the particularly simple property that its two-
dimensional Fourier transform can be found as a product of one-dimensional Fourier
transforms, as evidenced by the following relation:
Hg(*> T)} = || g(*> y)exp[-j2TT(f x x + f Y y)]dxdy
— 00
gx(x) exp[-j2TTf x x]dx g Y (y) exp[- j2Trf Y y]dy
= ^x{gx}d r Y {gy}-
( 2 - 21 )
Thus the transform of g is itself separable into a product of two factors, one a function
of fx only and the second a function of f Y only, and the process of two-dimensional
transformation simplifies to a succession of more familiar one-dimensional manipula-
tions.
Functions separable in polar coordinates are not so easily handled as those sep-
arable in rectangular coordinates, but it is still generally possible to demonstrate that
two-dimensional manipulations can be performed by a series of one-dimensional ma-
nipulations. For example, the reader is asked to verify in the problems that the Fourier
transform of a general function separable in polar coordinates can be expressed as an
infinite sum of weighted Hankel transforms
00
Hg(r,6)} = ^ c k (-j) k exp(jk(f>)H k {g R (r)} (2-22)
k = -oo
where
1 f 2ir
Ck = 2 tt\q £®<W ex P (-jkd)dd
and H k { } is the Hankel transform operator of order k, defined by
chapter 2 Analysis of Two-Dimensional Signals and Systems 1 1
'HkigRir)} = 2 tt
i;
rg R (r)J k (2'irrp)fc.
Here the function J k is the kth-order Bessel function of the first kind.
(2-23)
2.15 Functions with Circular Symmetry: Fourier-Bessel Transforms
Perhaps the simplest class of functions separable in polar coordinates is composed of
those possessing circular symmetry. The function g is said to be circularly symmetric
if it can be written as a function of r alone, that is,
g(r,0) = g R (r). (2-24)
Such functions play an important role in the problems of interest here, since most optical
systems have precisely this type of symmetry. We accordingly devote special attention
to the problem of Fourier transforming a circularly symmetric function.
The Fourier transform of g in a system of rectangular coordinates is, of course,
given by
G(fx, fr)
g(x, y)exp[—j2ir(fxx + fry)] dxdy.
(2-25)
To fully exploit the circular symmetry of g, we make a transformation to polar coordi-
nates in both the (x, y) and the (fx, fr) planes as follows:
r = Jx 2 + y 2
6 - arctan^-j
P = J fx + fy
( fy\
0 = arc tan —
\jx )
x = r cos 9
y = r sin#
fx = PCOS0
fr = P sin <f>.
(2-26)
For the present we write the transform as a function of both radius and angle, 3
Hg} = G 0 (p,cf>). (2-27)
Applying the coordinate transformations (2-26) to Eq. (2-25), the Fourier transform
of g can be written
1 2? r
G 0 (p,<f>) = 1 dd drrgR(r)exp[-j2irrp(cosBcos<p + sin0sin<5)] (2-28)
o Jo
or equivalently,
[■co i-2t r
Go(p, 4>) = \ drrg R (r)\ dd exp[-y'27rrpcos(0 - <f>)]. (2-29)
Jo Jo
3 Note the subscript in G 0 is added simply because the functional form of the expression for the transform
in polar coordinates is in general different than the functional form for the same transform in rectangular
coordinates.
12 Introduction to Fourier Optics
Finally, we use the Bessel function identity
J o(a) = JL J ^ exp [-ja cos (0 - 0)] dd, (2-30)
where Jq is a Bessel function of the first kind, zero order, to simplify the expression for
the transform. Substituting (2-30) in (2-29), the dependence of the transform on angle
(}> is seen to disappear, leaving Go as the following function of radius p,
G 0 (p, <f>) = G 0 (p) = 277 [ rg R (r)Jo(2irrp)dr. (2-31)
Jo
Thus the Fourier transform of a circularly symmetric function is itself circularly
symmetric and can be found by performing the one-dimensional manipulation of (2-
31). This particular form of the Fourier transform occurs frequently enough to warrant
a special designation; it is accordingly referred to as the Fourier-Bessel transform, or
alternatively as the Hankel transform tf zero order (cf. Eq. (2-23)). For brevity, we
adopt the former terminology.
By means of arguments identical with those used above, the inverse Fourier trans-
form of a circularly symmetric spectrum G 0 (p ) can be expressed as
g R (r) = 2-Tr[ pG o (p)J 0 (2Trrp)dp. (2-32)
Jo
Thus for circularly symmetric functions there is no difference between the transform
and the inverse-transform operations.
Using the notation B{} to represent the Fourier-Bessel transform operation, it fol-
lows directly from the Fourier integral theorem that
BB~ y {g R {r)} = B- ] B{g R (r)} = BB{g R (r)} = g R (r) (2-33)
at each value of r where g R (r) is continuous. In addition, the similarity theorem can be
straightforwardly applied (see Prob. 2-6c) to show that
B{g R (ar)}= (2-34)
When using the expression (2-31) for the Fourier-Bessel transform, the reader should re-
member that it is no more than a special case of the two-dimensional Fourier transform,
and therefore any familiar property of the Fourier transform has an entirely equivalent
counterpart in the terminology of Fourier-Bessel transforms.
2.1.6 Some Frequently Used Functions and Some Useful Fourier Transform
Pairs
A number of mathematical functions will find such extensive use in later material that
considerable time and effort can be saved by assigning them special notations of their
own. Accordingly, we adopt the following definitions of some frequently used func-
tions:
chapter 2- Analysis of Two-Dimensional Signals and Systems 13
rect(x)
J
!
0 5
-1 -0 5 0 0 5 1
X
sgn(x)
1
-2
1
*
1 2
X
comb(x)
X
FIGURE 22
Special functions.
1
W < 5
Rectangle function
rect(jc) = <
1
2
\*\ = 3
.0
otherwise
Sine function
sinc(x)
sin(7rx)
TTX
f 1
x > 0
Signum function
sgnO) = <
0
x = 0
l-
1 x < 0
Triangle function
AW = {;-
- \x\
W* i
otherwise
Comb function
comb(x) =
00
^ S(x - n)
14 Introduction to Fourier Optics
TABLE 2.1
Transform pairs for some functions separable in
rectangular coordinates.
Function
ex- ( a 2 + b 2 y 2 )]
rect(ax) rect(£>_y)
A (ax) Mby )
Transform
r-vi sinc(/ x /a) sine (f Y /b)
r-TT sine 2 (f x /a) sine 2 (f Y lb)
\ab\
8(ax, by)
exp[jir(ax + by)]
I ab\
8(f x - all, f Y - b/2)
sgn(aj) sgn (by)
comb(cut) comb(/)}')
exp[jn(a 2 x 2 + b 2 y 2 )]
exp[-(a|x| + 6|.v|)]
ab 1 1
\ab\ j*nfx jirfy
|^j comb(f x /a) comb(fytb)
\ab\
exp|
J7ri ~ + V
\ab\ 1 +(2nf x /a) 2 1 + (l-ufylb) 1
Circle function circ( J x 1 + y 2 ) = J x 2 + y 2 = 1
[ 0 otherwise.
The first five of these functions, depicted in Fig. 2.2, are all functions of only one in-
dependent variable; however, a variety of separable functions can be formed in two
dimensions by means of products of these functions. The circle function is, of course,
unique to the case of two-dimensional variables; see Fig. 2.3 for an illustration of its
structure.
We conclude our discussion of Fourier analysis by presenting some specific two-
dimensional transform pairs. Table 2.1 lists a number of transforms of functions sep-
arable in rectangular coordinates. For the convenience of the reader, the functions are
presented with arbitrary scaling constants. Since the transforms of such functions can
be found directly from products of familiar one-dimensional transforms, the proofs of
these relations are left to the reader (cf. Prob. 2-2).
On the other hand, with a few exceptions (e.g. exp[— ir(x 2 + y 2 )], which is both
separable in rectangular coordinates and circularly symmetric), transforms of most
circularly symmetric functions cannot be found simply from a knowledge of one-
dimensional transforms. The most frequently encountered function with circular sym-
metry is:
f 1 r < 1
circ(r) = j r = 1
[ 0 otherwise
16 Introduction to Fourier Optics
Accordingly, some effort is now devoted to finding the transform of this function. Using
the Fourier-Bessel transform expression (2-31), the transform of the circle function can
be written
S{circ(r)} = 2tt [ rJo(277Tp)dr.
Jo
Using a change of variables r' = 2nrp and the identity
[Vote)# = xJ\(x),
Jo
we rewrite the transform as
B{circ(r)} = A-j r'Jo(r') dr' = (2-35)
2 TTp 1 Jo P
where J\ is a Bessel function of the first kind, order 1, Figure 2.3 illustrates the circle
function and its transform. Note that the transform is circularly symmetric, as expected,
and consists of a central lobe and a series of concentric rings of diminishing amplitude.
Its value at the origin is tt. As a matter of curiosity we note that the zeros of this trans-
form are not equally spaced in radius. A convenient normalized version of this function,
with value unity at the origin, is 2 J ' ( 2 2 J p p) . This particular function is called the "besinc"
function, or the “jinc” function.
For a number of additional Fourier-Bessel transform pairs, the reader is referred to
the problems (see Prob. 2-6).
22
LOCAL SPATIAL FREQUENCY
AND SPACE-FREQUENCY LOCALIZATION
Each Fourier component of a function is a complex exponential of a unique spatial fre-
quency. As such, every frequency component extends over the entire (x, y) domain.
Therefore it is not possible to associate a spatial location with a particular spatial fre-
quency. Nonetheless, we know that in practice certain portions of an image could con-
tain parallel grid lines at a certain fixed spacing, and we are tempted to say that the
particular frequency or frequencies represented by these grid lines are localized to cer-
tain spatial regions of the image. In this section we introduce the idea of local spatial
frequencies and their relation to Fourier components.
For the purpose of this discussion, we consider the general case of complex-valued
functions, which we will later see represent the amplitude and phase distributions of
monochromatic optical waves. For now, they are just complex functions. Any such func-
tion can be represented in the form
g(x,y) = a(x, y)exp[j<l>(x,y)] (2-36)
where a{x, y) is a real and nonnegative amplitude distribution, while (j>(x, y) is a real
phase distribution. For this discussion we assume that the amplitude distribution a(x, y)
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