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Tyo, J. Scott, author.

Field Guide to Linear Systems in Optics / J. Scott Tyo and Andrey S. Alenin

pages cm. - (SPIE Field Guide series; FG35)

Includes bibliographical references and index.

ISBN 978-1-62841-547-6

1. Linear systems. 2. Optics. 3. Fourier transformations. I. Title.

QC355.2 2015



Published by SPIE

P.O. Box 10

Bellingham, Washington 98227-0010 USA

Phone: 360.676.3290

Fax: 360.647.1445


The content of this book reflects the thought of the author(s). Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon.

Printed in the United States of America.

Last updated 06/15/2015

Introduction to the Series

Welcome to the SPIE Field Guides — a series of publications written directly for the practicing engineer or scientist. Many textbooks and professional reference books cover optical principles and techniques in depth. The aim of the SPIE Field Guides is to distill this information, providing readers with a handy desk or briefcase reference that provides basic, essential information about optical principles, techniques, or phenomena, including definitions and descriptions, key equations, illustrations, application examples, design considerations, and additional resources. A significant effort will be made to provide a consistent notation and style between volumes in the series.

Each SPIE Field Guide addresses a major field of optical science and technology. The concept of these Field Guides is a format-intensive presentation based on figures and equations supplemented by concise explanations. In most cases, this modular approach places a single topic on a page, and provides full coverage of that topic on that page. Highlights, insights, and rules of thumb are displayed in sidebars to the main text. The appendices at the end of each Field Guide provide additional information such as related material outside the main scope of the volume, key mathematical relationships, and alternative methods. While complete in their coverage, the concise presentation may not be appropriate for those new to the field.

The SPIE Field Guides are intended to be living documents. The modular page-based presentation format allows them to be easily updated and expanded. We are interested in your suggestions for new Field Guide topics as well as what material should be added to an individual volume to make these Field Guides more useful to you. Please contact us at

John E. Greivenkamp, Series Editor

Optical Sciences Center

The University of Arizona

The Field Guide Series

Keep information at your fingertips with the SPIE Field Guides:
  • Adaptive Optics, Second Edition, Robert Tyson & Benjamin Frazier

  • Atmospheric Optics, Larry Andrews

  • Binoculars and Scopes, Paul Yoder, Jr. & Daniel Vukobratovich

  • Diffractive Optics, Yakov Soskind

  • Digital Micro-Optics, Bernard Kress

  • Displacement Measuring Interferometry, Jonathan Ellis

  • Fiber Optic Sensors, William Spillman, Jr. & Eric Udd

  • Geometrical Optics, John Greivenkamp

  • Holography, Pierre-Alexandre Blanche

  • Illumination, Angelo Arecchi, Tahar Messadi, & John Koshel

  • Image Processing, Khan M. Iftekharuddin & Abdul Awwal

  • Infrared Systems, Detectors, and FPAs, Second Edition, Arnold Daniels

  • Interferometric Optical Testing, Eric Goodwin & Jim Wyant

  • Laser Pulse Generation, Rüdiger Paschotta

  • Lasers, Rüdiger Paschotta

  • Lens Design, Julie Bentley & Craig Olson

  • Linear Systems in Optics, J. Scott Tyo & Andrey Alenin

  • Microscopy, Tomasz Tkaczyk

  • Nonlinear Optics, Peter Powers

  • Optical Fabrication, Ray Williamson

  • Optical Fiber Technology, Rüdiger Paschotta

  • Optical Lithography, Chris Mack

  • Optical Thin Films, Ronald Willey

  • Optomechanical Design and Analysis, Katie Schwertz & James Burge

  • Physical Optics, Daniel Smith

  • Polarization, Edward Collett

  • Probability, Random Processes, and Random Data Analysis, Larry Andrews

  • Radiometry, Barbara Grant

  • Special Functions for Engineers, Larry Andrews

  • Spectroscopy, David Ball

  • Terahertz Sources, Detectors, and Optics, Créidhe O'Sullivan & J. Anthony Murphy

  • Visual and Ophthalmic Optics, Jim Schwiegerling

Field Guide to Linear Systems in Optics

The College of Optical Sciences (OSC) at the University of Arizona has long offered a course called “OPTI512R: Fourier Transforms, Linear Systems, and Optics” in its graduate program. The course was initiated and designed by Prof. Jack Gaskill, and was taught largely out of a textbook by the same name that was published in 1978. When Prof. Tyo joined OSC in 2006, he was asked to take over the course, as Prof. Gaskill had retired some years earlier.

Tyo came to the class with an electrical engineer’s classic understanding of linear systems in time and frequency. Tyo quickly came to realize that, at that time, Prof. Gaskill’s textbook was the only one written from the perspective of an optical engineer who needs to take 2D spatial Fourier transforms instead of 1D temporal ones. This difference gives rise to several subtle but important stylistic requirements that Prof. Gaskill captured well in his text. As with most instructors, Tyo began to add his own take on the material over the years.

Andrey Alenin joined his group in 2010, and he showed a strong interest in both the pedagogy and the presentation of the course material; the two authors have since worked together to refine the presentation. As of the writing of this Field Guide, Prof. Gaskill’s text is still the primary reference in the class. However, when John Greivenkamp discussed with us the possibility of writing a Field Guide on this topic, he gave the authors the opportunity to go through the notes and reorganize them into a sequence more suited for this handbook format.

The process is, of course, circular. During the current semester of teaching OPTI512R, while completing this Field Guide, the authors have realized that the entire structure of the course will need to be revised going forward. The efforts undertaken to write this book have provided a new perspective on the classic course content.

We would like to extend our gratitude to the following individuals who aided in the preparation of parts of this book. Series editor John Greivenkamp was invaluable for his guidance on style and his tips about what to include and what to omit. Brian Anderson from the University of Arizona read and commented on several of the pages that discuss topics from quantum mechanics. Scott McNeill from SPIE was of help in setting up the formatting of the book.

We are grateful to the owners and staff of the Cartel Coffee Lab and the Dragoon Brewery, who allowed us to occupy power outlets, seats, and their Wi-Fi connections for hours on end as we tried to escape the campus and hide in order finish the book.

Prof. Tyo would like to express his gratitude to his wife, Elizabeth Ritchie, for her patience while he worked on the book during their sabbatical.

Andrey Alenin would like to thank Geraldine Longo for her continuous encouragement, as well as comments and advice on aesthetics of presentation.

J. Scott Tyo

College of Optical Sciences

The University of Arizona

Andrey S. Alenin

College of Optical Sciences

The University of Arizona

Table of Contents

Glossary of Symbols and Acronyms x

Mathematical Background and Notation 1

Complex Numbers and Complex Plane 1

Complex Arithmetic 2

Specialized Complex Operations 3

Complex Sinusoidal Functions and Phasors 4

Idealized Models and the Unit Step Function 5

Pulse-Like Functions 6

Impulse Function 7

Impulse Function Properties 8

Integrals and Derivatives of the Delta Function 9

Comb Function 10

Orthonormal Basis Functions 11

Fourier Analysis 12

Harmonic Analysis and Fourier Series 12

Square Wave and Truncated Fourier Series 13

Fourier Transform 14

Fourier Transform Properties 15 Symmetry of Functions and Fourier Transforms 16

Parseval's Theorem and Moment Theorem 17

Laplace Transform 18

2D Functions 19

Impulse Functions in Two Dimensions 20

Fourier Transforms of 2D Functions 21

Hankel Transform 22

Hankel Transform Pairs and Properties 23

Skew Functions 24

Linear Shift-Invariant Systems 25

Operators and LSI Systems 25

Convolution and Impulse Response 26

Causality 27

Graphical Convolution 28

Convolution Theorem 29

Correlation 30

Convolution and Correlation in Two Dimensions 31

Sampling, Discrete Systems, and the DFT 32

Band-Limited and Space-Limited Functions 32

Ideal Sampling 33

Sampling in Two Dimensions 34

Non-Ideal Sampling 35

Aliasing 36

Band-Limited Reconstruction 37

Discrete-Space Fourier Transform (DSFT) 38

z-Transform 39

Discrete Fourier Transform (DFT) 40

DFT Properties 41

DFT Evaluation 42

Continuous and Discrete Fourier Domains 43

Gibbs Phenomenon and Frequency Leakage 44

Windowing of Sequences 45

Fast Fourier Transform (FFT) 46

Discrete Convolution 47

Interpolation and Decimation 48

Signal and Image Processing 50

Filters 50

Amplitude-Only Filters 51

Phase-Only Filters 52

Special Classes of Phase Filters 53

Equalization 54

Matched Filtering 55

Projection-Slice Theorem 56

Random Functions and Sequences 57

Power Spectral Density (PSD) Function 58

Filtering Random Signals 59

Wiener–Helstrom Filter 60

Propagation of Optical Fields 61

Modes 61

Plane Wave Spectrum 62 Transfer Function/Impulse Response of Free Space 63

Propagation of Optical Beams 64

Spatial and Temporal Coherence 65

Diffraction 66

Paraxial Approximation and Scalar Diffraction 67

Fresnel Diffraction 68

Fraunhofer Diffraction 69

Fraunhofer/Fresnel Basis Functions 70

Fourier Transforming Properties of Lenses 71

Fourier Description of Optical Cavity Modes 72

Higher-Order Cavity Modes 73

Slab Waveguides 74

Optical Fiber Waveguides 75

Image-Forming Systems 76

Diffraction-Limited Focal Imaging Systems 76

Airy Disk 77

Coherent Transfer Function (CTF) 78

Optical Transfer Function (OTF) 79

Aberrated Systems 80

Comparisons of Coherent and Incoherent Output 81

Two-Point Resolution with Coherent Light 82

Roughness and Scattered Light 83

Applications of Linear Systems and Fourier Analysis 84

Fourier Transform Spectroscopy (FTS) 84

Multiplexing 85

Sampled Color Imaging Systems 86

RGB Detector and Display Arrays 87

Channeled Spectropolarimetry 88

Optical Signal Processing 89

Green's Functions 90

Moment Method 91

Array Apertures 92

Crystal Lattices and Reciprocal Lattices 94

Fourier Transform Tables 95

Equation Summary 98

Bibliography 102

Index 103


Functions in this Field Guide are functions of spatial variables x and y unless noted otherwise. Lowercase letters are used to denote functions of the spatial variables (f(x), g(x)), whereas capital letters represent their Fourier transforms (F(ξ), G(ξ)).

Sequences of discrete samples of a function are denoted with the subscript k (fk, gk) and samples of the corresponding DFTs are denoted with subscript n (Fn, Gn).

Primed variables (x′, y′, ξ′, η′, etc.) denote variables of integration.


Bandpass filter


Coherent transfer function


Pupil diameter

do, di

Object and image distances


Discrete Fourier transform of sequence fk


Complex vector electric field


Expected value of f(x)


Focal length




Working F-number

fs (x)

Ideally sampled function f(x)


Fourier transform of f(x)


Impulse response


Transfer function


Optical transfer function


High-pass filter


Zeroth-order Bessel function of first kind


Wave vector


Spatial extent of a function


Linear shift invariant operator


Laplace transform of f(t)


Low-pass filter

mn (f(x))

nth moment of f(x)


Modulation transfer function


Optical transfer function


Power spectra density


Point spread function


Polar coordinate radius


2D vector xx^+yy^


Mathematical operator


Signal to noise ratio


Temporal period

t(x, y)

Transmission function


Complex scalar optical field amplitude


Spatial frequency bandwidth

W(x, y)

Wavefront aberration function


Spatial period

x, y

Cartesian coordinates


Spatial sampling interval


Z-transform of sequence fk


Impulse function at x = x0


Sampling resolution in the space domain


Sampling resolution in the frequency domain

γfg (x)

Correlation between functions f(x) and g(x)

γx, γy, γz

Direction cosines


Spatial frequency in y




Temporal frequency


Polar coordinate angle


Radial distance in frequency plane




Normalized frequency ξ/ξs


Spatial frequency in x


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