# Looking at optics with Fourier analysis

Fourier analysis permeates every branch of engineering and science. But, the application of Fourier analysis to optics stands out because of its intuitive appeal and simplicity of experimental realization.

Looking at optics with Fourier analysis

Jeff Korn

Fourier Series and Optical Transform Techniques in Contemporary Optics, An Introduction

Raymond G. Wilson, John Wiley and Sons, New York, NY, 1995, 325 pages, $59.95

Fourier analysis permeates every branch of engineering and science. But, the application of Fourier analysis to optics stands out because of its intuitive appeal and simplicity of experimental realization.

Fourier Series and Optical Transform Techniques in Contemporary Optics is an elementary study of Fourier methods with an emphasis on visual representation of the inner workings of the transforms. Rigor and theorems are not an important aspect of this book. As a consequence, the author suggests that the book be used as a supplement to a more rigorous text in a first course on Fourier series and transforms.

The first chapter introduces the reader to the concept of periodic functions in both the time and spatial domains. This discussion is made concrete by many physical examples and hard numbers on all the salient parameters. There is very little, if any, abstraction.

The next three chapters deal with the Fourier series and transform. These constructions are described well, and there are a lot of plots to illustrate how a change in one space affects the other space. One point of possible confusion is how phase is represented. Instead of plotting power spectra and phase separately, three-dimensional plots of the real and imaginary parts are shown. This results in spiral-like graphs that are often cluttered or unclear.

Chapters five and six provide a brief introduction to scalar diffraction and linear system theory as applied to optical systems. Even though there isn`t much formal development of diffraction theory, the author does an excellent job explaining the various regimes of diffraction (that is, Fresnel and Fraunhofer.) The only theorem treated in any depth, however, is the convolution theorem. The author includes many computer- generated plots illustrating the process of convolution.

The remaining chapters deal with advanced concepts: computer techniques to calculate Fourier transforms, apodization, complex apertures, and spatial filtering. These topics are treated in the same clear manner as the rest of the book.

A weakness of the book is the lack of real-world applications. Given the plethora of practical examples (for example, optical computing and spectroscopy) and the level of the text, the author could have used any one of them to give the reader the real-world connection that students sometime desire.

On the whole, I found this book to be well organized and very readable. The many graphs and clear text are a definite strength, making this book suitable even for students of Fourier analysis in disciplines other than optics. I highly recommend Fourier Series and Optical Transform Techniques in Contemporary Optics as a supplement to a first, or even advanced, course in Fourier analysis.

JEFF KORN is a senior engineer at Lasertron Inc., 23 Fourth Ave., Burlington, MA 01803; e-mail: [email protected]