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Library of Congress Preassigned Control Number Data 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 535–dc23 2014958708 Published by SPIE P.O. Box 10 Bellingham, Washington 98227-0010 USA Phone: 360.676.3290 Fax: 360.647.1445 Email: Books@spie.org 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 SeriesWelcome 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 fieldguides@SPIE.org. John E. Greivenkamp, Series Editor Optical Sciences Center The University of Arizona The Field Guide SeriesKeep information at your fingertips with the SPIE Field Guides:
Field Guide to Linear Systems in OpticsThe 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 ContentsGlossary 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 GlossaryFunctions 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. BPF Bandpass filter CTF Coherent transfer function D Pupil diameter do, di Object and image distances
Discrete Fourier transform of sequence fk E Complex vector electric field E[f(x)] Expected value of f(x) f Focal length f/# F-number f/#w Working F-number fs (x) Ideally sampled function f(x)
Fourier transform of f(x) h(x) Impulse response H(ξ) Transfer function
Optical transfer function HPF High-pass filter J0(x) Zeroth-order Bessel function of first kind k Wave vector L Spatial extent of a function
Linear shift invariant operator
Laplace transform of f(t) LPF Low-pass filter mn (f(x)) nth moment of f(x) MTF Modulation transfer function OTF Optical transfer function PSD Power spectra density PSF Point spread function r Polar coordinate radius r 2D vector
Mathematical operator SNR Signal to noise ratio T Temporal period t(x, y) Transmission function u Complex scalar optical field amplitude W Spatial frequency bandwidth W(x, y) Wavefront aberration function X Spatial period x, y Cartesian coordinates xs Spatial sampling interval
Z-transform of sequence fk δ(x − x0) Impulse function at x = x0 Δx Sampling resolution in the space domain Δξ Sampling resolution in the frequency domain γfg (x) Correlation between functions f(x) and g(x) γx, γy, γz Direction cosines η Spatial frequency in y λ Wavelength ν Temporal frequency θ Polar coordinate angle ρ Radial distance in frequency plane ρ
ν Normalized frequency ξ/ξs ξ Spatial frequency in x |
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