Linear Systems, Fourier Transforms, and Optics

This book is a textbook on linear systems, Fourier analysis, diffraction theory, and image formation. It is not a textbook on Fourier optics, but was intended to helps students with the basics before attempting that subject. This book might also be helpful to students that are studying linear systems theory or image processing alone and need an additional reference. There are problems at the end of each chapter, and the problems include both numerical calculations and derivations. No solutions to the problems are included. Numerous examples are shown with complete steps. Some examples are numerical, and many are not. Minus the optical material, I had already seen the rest of the material in the book before I used it, so perhaps I am not the best judge of how complete a textbook it was, but to me it seemed very complete and clear. Unlike many similar textbooks, the author did not assume much about the reader's background other than the Calculus, differential equations, and linear algebra that you would expect any graduate engineering student to have already mastered. I definitely recommend going through it or having access to it before you enroll in a class on Fourier optics.Chapter 2 presents an elementary review of various properties and classes of mathematical functons, as well as a description of the manner in which these functions represent physical quantities. Chapter 3 introduces a number of special functions that are of great use in later chapters. In particular the rectangle function, the sinc function, the delta function, and the comb function are very useful. Also, several special functions of two variables are described. In Chapter 4 the fundamentals of harmonic analysis are explored as well as how various arbitrary functions may be represented by linear combinations of other more elementary functions. Chapter 5 discusses the physical systems in term of linear operators, and the notions of linearity and shift invariance are introduced. Next, the impulse response function, the transfer function, and the eigenfunctions associated with linear shift-invariant systems are discussed. Chapter 6 is devoted to studies of the convolution, cross-correlation, and autocorrelation operations. The properties of these operations are explored in considerable depth. The fact that the output of a linear shift-invariant system is given by the convolution of the input with the impulse response of the system is derived and explored.In Chapter 7 the properties of the Fourier transformation is investigated, as well as the importance of this transform in the analysis of linear shift-invariant systems. In chapter 8 the characteristics of various types of linear filters are described. Their applications in various types of signal processing and recovery is discussed. Also discussed is the matched-filter problem and the various interpretations of the sampling theory.Chapter 9 extends the previous material on one-dimensional systems to two dimensions. In particular, an investigation of convolution and Fourier transformation in two dimensions is conducted, and the Hankel transform and its properties are studied. Also, the line response and edge response functions are introduced. In chapter 10 the propagation and diffraction of optical wave fields in both the Fresnel and Fraunhofer regions is explored. Also studied are the effects of lenses on the diffraction process. Special attention is paid to the curious properties of Gaussian beams in the last section of the chapter.Finally, in Chapter 11, the concepts of linear systems and Fourier analysis are combined with the theory of diffraction to describe the process of image formation in terms of a linear filtering operation. This is done for both coherent and incoherent imaging, and the corresponding impulse response functions and and transfer functions are discussed in detail.Several special functions are tabulated in the first appendix for those with little or no previous training in optics, and the fundamentals of geometrical image formation and abberations are presented in the second appendix.

I had optical FT class in my grad program and the professor was teaching it from Goodman's book... I wish I had this book then... I use it and O love it... very simple and it takes you back to the basics and fundamentals. It is not an advance FT book but it is a must have FT book for me... Love it and thanks for the first reviewer for the very helpful detailed review that made me buy this book :)

I consider Gaskill's book to be the best I've seen for advanced undergraduate and first-year graduate classes on linear systems. Gaskill approaches the subject in a clear and understandable style while dealing with the subject in a complete and quantitative manner. Though he does not eschew mathematical rigor by any means, the text is well written and logically formatted, making it refreshingly easy to follow what is, in other texts, a more difficult subject. Though I've filed Gaskill's book in my library alongside other dealing with optics, this is primarily a book on mathematics, but written more for engineers and scientists than for mathematicians.After a brief introduction, the author begins (in chapter 2) with a quick summary of mathematical concepts, including classes of functions, one and two-dimensional functions, complex numbers, phasors, and the scalar wave equation.The third chapter introduces useful functions (many of a discontinuous nature) that find application in modeling linear systems. These include step functions and the impulse function in both one and two dimensions. Development of these functions follows an intuitive path that reflects the way in which they are often used. The many figures are particularly useful in conveying concepts more effectively.Chapter four develops the theme of harmonic analysis by introducing the notion of orthogonal expansions and extending this development to the Fourier series, leading to development of the Fourier integral. The chapter finishes with some worked examples showing the spectra of simple functions. Chapter 7 seems a little out of place, since it deals with the Fourier transform, yet appears in the book several chapters later, after the author introduces the concepts of linear systems and the convolution.Though one of the shorter chapters, chapter five is pivotal, and develops the idea of mathematical operators and physical systems - with the crucial development of the impulse response. The application of the impulse response is extended by chapter 6, which develops the mathematics of convolution. For a linear, shift-invariant system the impulse response convolved with the input to the system gives the system's output.Chapter 8 pulls together the material in the previous chapters to mathematically describe the characteristics and applications of linear filters. Examples include amplitude filters, phase filters, combination amplitude and phase filters, and some interesting applications showing (for example) how to filter the noise from a signal of interest. All this development is strictly mathematical, with no real-world worked examples (except in the abstract). Nevertheless, this chapter is very useful and (in the author's style) easy to understand and follow.Chapter 9 deals with two-dimensional convolutions and the two-dimensional Fourier transform. This chapter is essentially an extension of the earlier one-dimensional developments in earlier chapters, but introduces some useful mathematical tools, including the convolution and Fourier transform in polar coordinates. The Hankel transform, developed in this chapter, is particularly useful for work in optics where many examples (laser beams, for instance) exhibit circular symmetry. In these examples the two-dimensional integrals may be greatly simplified by the Hankel transform to a one-dimensional form where (even in the absence of a closed-form equation) they are far more tractable. The chapter concludes with useful tables of common transforms.Chapter 10 leaves the almost purely mathematical forum of the previous chapters by introducing the subject of propagation and diffraction of optical waves. Gaskill first develops the mathematics of the optical waves and then derives the equations that show how these waves are diffracted. Not surprisingly, the diffraction fields are expressible in terms of the transforms developed earlier in the book. The chapter also describes the influence of optical lenses on the diffraction patterns and the very important subject of propagation of Gaussian beams (since many laser beams, and the fundamental mode in weakly guiding optical fibers have Gaussian profiles).Chapter 11 continues the optical theme by explaining image-forming systems. The student will be particularly enabled in this chapter if he or she has had prior exposure to the subject of diffraction and perhaps some exposure to the idea of image aberrations. The book ends with appendix 1, on special functions, and appendix 2, on elementary geometric optics. Each chapter has a list of references, and problems for the student, and the book has a complete index making it useful as a desk references book as well as a textbook for advanced undergraduate and first-year graduate coursework.Gaskill's book is mathematically intense, but the author's style and frequent use of figures makes the book surprisingly easy to read. Prerequisites for this book should include a couple of years of calculus, differential equations, and a smattering of linear algebra. Some exposure to concepts in optics, including diffraction and aberrations would also be helpful.Gaskill's book will be helpful far beyond optics, with applications in electrical engineering, mechanical engineering, digital image processing, or anywhere else that linear systems might be encountered.

Excellent book on Fourier Optics. Provides a unique view when compared to Goodman in the sense that is more linear systems based. Well written and provides excellent insight in the subject. Only downside is Gaskill stopped at one edition. Please update this excellent text.

If you want to survive a first year graduate class on Fourier Optics, get this book. Gaskill is precise and comprehensive, presenting concepts incrementally with ample diagrams to illustrate all along the way. I've got Goodman and Bracewell on my shelf, but it's Gaskill's that's saving my life this semester.

Super helpful for my Diffraction class! The book came from the seller in pretty much pristine condition minus some writing on the inner page. So happy, what a deal!

Excellent!