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### Spie Press Book

Field Guide to Linear Systems in Optics
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Book Description

"Linear systems" is a broad and important area in many scientific and engineering disciplines, and it is especially important in optics because it forms the basis for Fourier optics, diffraction theory, image-quality assessment, and many other areas. This Field Guide provides the practicing optical engineer with a reference for the basic concepts and techniques of linear systems, including Fourier series, continuous and discrete Fourier transforms, convolution, sampling and aliasing, and MTF/PSF using the language, notation, and applications from optics, imaging, and diffraction.

Book Details

Date Published: 21 January 2015
Pages: 120
ISBN: 9781628415476
Volume: FG35
Errata

Mathematical Background and Notation
Complex Numbers and Complex Plane
Complex Arithmetic
Specialized Complex Operations
Complex Sinusoidal Functions and Phasors
Idealized Models and the Unit Step Function
Pulse-Like Functions
Impulse Function
Impulse Function Properties
Integrals and Derivatives of the Delta Function
Comb Function
Orthonormal Basis Functions

Fourier Analysis
Harmonic analysis and Fourier Series
Square Wave and Truncated Fourier Series
Fourier Transform
Fourier Transform Properties
Symmetry of Functions and Fourier Transforms
Parseval's Theorem and Moment Theorem
Laplace Transform
2D Functions
Impulse Functions in Two Dimensions
Fourier Transforms of 2D Functions
Hankel Transform
Hankel Transform Pairs and Properties
Skew Functions

Linear Shift-Invariant Systems
Operators and LSI Systems
Convolution and Impulse Response
Causality
Graphical Convolution
Convolution Theorem
Correlation
Convolution and Correlation in Two Dimensions

Sampling, Discrete Systems, and the DFT
Band-Limited and Space-Limited Functions
Ideal Sampling
Sampling in Two Dimensions
Non-Ideal Sampling
Aliasing
Band-Limited Reconstruction
Discrete-Space Fourier Transform (DSFT)
z-Transform
Discrete Fourier Transform (DFT)
DFT Properties
DFT Evaluation
Continuous and Discrete Fourier Domains
Gibbs Phenomenon and Frequency Leakage
Windowing of Sequences
Fast Fourier Transform (FFT)
Discrete Convolution
Interpolation and Decimation

Signal and Image Processing
Filters
Amplitude-Only Filters
Phase-Only Filters
Special Classes of Phase Filters
Equalization
Matched Filtering
Projection-Slice Theorem
Random Functions and Sequences
Power Spectral Density (PSD) Function
Filtering Random Signals
Wiener-Helstrom Filter

Propagation of Optical Fields
Modes
Plane Wave Spectrum
Transfer Function/Impulse Response of Free Space
Propagation of Optical Beams
Spatial and Temporal Coherence
Diffraction
Paraxial Approximation and Scalar Diffraction
Fresnel Diffraction
Fraunhofer Diffraction
Fraunhofer/Fresnel Basis Functions
Fourier Transforming Properties of Lenses
Fourier Description of Optical Cavity Modes
Higher-Order Cavity Modes
Slab Waveguides
Optical Fiber Waveguides

Image-Forming Systems
Diffraction-Limited Focal Imaging Systems
Airy Disk
Coherent Transfer Function (CTF)
Optical Transfer Function (OTF)
Aberrated Systems
Comparisons of Coherent and Incoherent Output
Two-Point Resolution With Coherent Light
Roughness and Scattered Light

Applications of Linear Systems and Fourier Analysis
Fourier Transform Spectroscopy (FTS)
Multiplexing
Sampled Color Imaging Systems
RGB Detector and Display Arrays
Channeled Spectropolarimetry
Optical Signal Processing
Green's Functions
Moment Method
Array Apertures
Crystal Lattices and Reciprocal Lattices

Fourier Transform Tables

Equation Summary

Bibliography

Index 103

## Preface

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 them 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
Andrey S. Alenin
College of Optical Sciences
The University of Arizona