**Preface**

Diffraction and imaging are central topics in many modern and scientific

fields. Fourier analysis and sythesis techniques are a unifying theme

throughout this subject matter. For example, many modern imaging

techniques have evolved through research and development with the

Fourier methods.

This textbook has its origins in courses, research, and development

projects spanning a period of more than 30 years. It was a pleasant experience to observe

over the years how the topics relevant to this book evolved and became more

significant as the technology progressed. The topics involved are many and are highly

multidisciplinary.

Even though Fourier theory is central to understanding, it needs to be supplemented

with many other topics such as linear system theory, optimization, numerical methods,

imaging theory, and signal and image processing. The implementation issues and

materials of fabrication also need to be coupled with the theory. Consequently, it is

difficult to characterize this field in simple terms. Increasingly, progress in technology

makes it of central significance, resulting in a need to introduce courses, which cover

the major topics together of both science and technology. There is also a need to help

students understand the significance of such courses to prepare for modern technology.

This book can be used as a textbook in courses emphasizing a number of the

topics involved at both senior and graduate levels. There is room for designing

several one-quarter or one-semester courses based on the topics covered.

The book consists of 20 chapters and three appendices. The first three chapters

can be considered introductory discussions of the fundamentals. Chapter 1 gives a

brief introduction to the topics of diffraction, Fourier optics and imaging, with

examples on the emerging techniques in modern technology.

Chapter 2 is a summary of the theory of linear systems and transforms needed in

the rest of the book. The continous-space Fourier transform, the real Fourier

transform and their properties are described, including a number of examples. Other

topics involved are covered in the appendices: the impulse function in Appendix A,

linear vector spaces in Appendix B, the discrete-time Fourier transform, the discrete

Fourier transform, and the fast Fourier transform (FFT) in Appendix C.

Chapter 3 is on fundamentals of wave propagation. Initially waves are described

generally, covering all types of waves. Then, the chapter specializes into

electromagnetic waves and their properties, with special emphasis on plane waves.

The next four chapters are fundamental to scalar diffraction theory. Chapter 4

introduces the Helmholtz equation, the angular spectrum of plane waves, the

Fresnel-Kirchoff and Rayleigh-Sommerfeld theories of diffraction. They represent

wave propagation as a linear integral transformation closely related to the Fourier

transform.

Chapter 5 discusses the Fresnel and Fraunhofer approximations that allow

diffraction to be expressed in terms of the Fourier transform. As a special application

area for these approximations, diffraction gratings with many uses are described.

Diffraction is usually discussed in terms of forward wave propagation. Inverse

diffraction covered in Chapter 6 is the opposite, involving inverse wave propagation.

It is important in certain types of imaging as well as in iterative methods of

optimization used in the design of optical elements. In this chapter, the emphasis is

on the inversion of the Fresnel, Fraunhofer, and angular spectrum representations.

The methods discussed so far are typically valid for wave propagation near the *z*-

axis, the direction of propagation. In other words, they are accurate for wave

propagation directions at small angles with the *z*-axis. The Fresnel and Fraunhofer

approximations are also not valid at very close distances to the diffraction plane.

These problems are reduced to a large extent with a new method discussed in

Chapter 7. It is called the near and far field approximation (NFFA) method. It

involves two major topics: the first one is the inclusion of terms higher than second

order in the Taylor series expansion; the second one is the derivation of equations to

determine the semi-irregular sampling point positions at the output plane so that the

FFT can still be used for the computation of wave propagation. Thus, the NFFA

method is fast and valid for wide-angle, near and far field wave propagation

applications.

When the diffracting apertures are much larger than the wavelength, geometrical

optics discussed in Chapter 8 can be used. Lens design is often done by using

geometric optics. In this chapter, the rays and how they propagate are described with

equations for both thin and thick lenses. The relationship to waves is also addressed.

Imaging with lenses is the most classical type of imaging. Chapters 9 and 10 are

reserved to this topic in homogeneous media, characterizing such imaging as a linear

system. Chapter 9 discusses imaging with coherent light in terms of the 2-D Fourier

transform. Two important applications, phase contrast microscopy and scanning

confocal microscopy, are described to illustrate how the theory is used in practice.

Chapter 10 is the continuation of Chapter 9 to the case of quasimonochromatic

waves. Coherent imaging and incoherent imaging are explained. The theoretical

basis involving the Hilbert transform and the analytic signal is covered in detail.

Optical aberrations and their evaluation with Zernike polynomials are also

described.

The emphasis to this point is on the theory. The implementation issues are

introduced in Chapter 11. There are many methods of implementation. Two major

ones are illustrated in this chapter, namely, photographic films and plates and

electron-beam lithography for diffractive optics.

In Chapters 9 and 10, the medium of propagation is assumed to be homogeneous

(constant index of refraction). Chapter 12 discusses wave propagation in

inhomogeneous media. Then, wave propagation becomes more difficult to compute

numerically. The Helmholtz equation and the paraxial wave equation are

generalized to inhomogeneous media. The beam propagation method (BPM) is

introduced as a powerful numerical method for computing wave propagation in

inhomogenous media. The theory is illustrated with the application of a directional

coupler that allows light energy to be transferred from one waveguide to another.

Holography as the most significant 3-D imaging technique is the topic of Chapter

13. The most basic types of holographic methods including analysis of holographic

imaging, magnification, and aberrations are described in this chapter.

In succeeding chapters, diffractive optical elements (DOEs), new modes of

imaging, and diffraction in the subwavelength scale are considered, with extensive

emphasis on numerical methods of computation. These topics are also related to

signal/image processing and iterative optimization techniques discussed in Chapter

14. These techniques are also significant for the topics of previous chapters,

especially when optical images are further processed digitally.

The next two chapters are devoted to diffractive optics, which is creation of

holograms, more commonly called DOEs, in a digital computer, followed by a

recording system to create the DOE physically. Generation of a DOE under

implementational constraints involves coding of amplitude and phase of an

incoming wave, a topic borrowed from communication engineering. There are many

such methods. Chapter 15 starts with Lohmann’s method, which is the first such

method historically. This is followed by two methods, which are useful in a variety

of waves such as 3-D image generation, and a method called one-image-only

holography, which is capable of generating only the desired image while suppressing

the harmonic images due to sampling and nonlinear coding of amplitude and phase.

The final section of the chapter is on the binary Fresnel zone plate, which is a DOE

acting as a flat lens.

Chapter 16 is a continuation of Chapter 15, and covers new methods of coding

DOEs and their further refinements. The method of projections onto convex sets

(POCS) discussed in Chapter 14 is used in several ways for this purpose. The

methods discussed are virtual holography, which makes implementation easier,

iterative interlacing technique (IIT), which makes use of POCS for optimizing a

number of subholograms, the ODIFIIT, which is a further refinement of IIT by

making use of the decimation-in-frequency property of the FFT, and the hybrid

Lohmann–ODIFIIT method, resulting in considerably higher accuracy.

Chapters 17 and 18 are on computerized imaging techniques. The first such

technique is synthetic aperture radar (SAR) covered in Chapter 17. In a number of

ways, a raw SAR image is similar to the image of a DOE. Only further processing,

perhaps more appropriately called decoding, results in a reconstructed image of a

terrain of the earth. The images generated are very useful in remote sensing of the

earth. The principles involved are optical and diffractive, such as the use of the

Fresnel approximation.

In the second part of computerized imaging, computed tomography (CT) is

covered in Chapter 18. The theoretical basis for CT is the Radon transform, a cousin

of the Fourier transform. The projection slice theorem shows how the 1-D Fourier

transforms of projections are used to generate slices of the image spectrum in the 2-

D Fourier transform plane. CT is highly numerical, as evidenced by a number of

algorithms for image reconstruction in the rest of the chapter.

Optical Fourier techniques have become very important in optical communications

and networking. One such area covered in Chapter 19 is arrayed waveguide

gratings (AWGs) used in dense wavelength division multiplexing (DWDM). AWG

is also called phased array (PHASAR). It is an imaging device in which an array of

waveguides are used. The waveguides are different in length by an integer *m* times

the central wavelength so that a large phase difference is achieved from one

waveguide to the next. The integer *m* is quite large, such as 30, and is responsible for

the large resolution capability of the phasar device, meaning that the small changes

in wavelength can be resolved in the output plane. This is the reason why

waveguides are used rather than free space. However, it is diffraction that is used

past the waveguides to generate images of points at different wavelengths at the

output plane. This is similar to a DOE, which is a sampled device. Hence, the images

repeat at certain intervals. This limits the number of wavelengths that can be imaged

without interference from other wavelengths. A method called irregularly sampled

zero-crossings (MISZCs) is discussed to avoid this problem. The MISZC has its

origin in one-image-only holography discussed in Chapter 15.

Scalar diffraction theory becomes less accurate when the sizes of the diffracting

apertures are smaller than the wavelength of the incident wave. Then, the Maxwell

equations need to be solved by numerical methods. Some emerging approaches for

this purpose are based on the method of finite differences, the Fourier modal

analysis, and the method of finite elements. The first two approaches are discussed in

Chapter 20. First, the paraxial BPM method discussed in Section 12.4 is

reformulated in terms of finite differences using the Crank-Nicholson method.

Next, the wide-angle BPM using the Pade approximation is discussed. The final

sections highlight the finite difference time domain and the Fourier modal method.

Many colleagues, secretaries, friends, and students across the globe have been

helpful toward the preparation of this manuscript. I am especially grateful to them

for keeping me motivated under all circumstances for a lifetime. I am also very

fortunate to have worked with John Wiley & Sons, on this project. They have been

amazingly patient with me. Without such patience, I would not have been able to

finish the project. Special thanks to George Telecki, the editor, for his patience and

support throughout the project.