Electrical and Computer Engineering
Department of Electrical and Computer EngineeringUniversity of Illinois Urbana-Champaign



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This book is an outgrowth of a graduate-level course taught at the University of Illinois, Urbana-Champaign. The course was conceived to fill a need for educating students working with waves and fields in their research. As technology grows more sophisticated, the need for solutions to more complex problems becomes greater. Fortunately, in the last two decades, the development of computer technology has provided a new means for solving many of these problems. This book is intended to elucidate the methods and techniques pertinent to wave and field problems in inhomogeneous media, so that students may apply them in order to solve practical problems. Hence, the reader may use a computer to implement many of the techniques described in this book.

This is not a book on numerical techniques, however. For that, the reader is referred to many excellent works written on numerical methods. The emphasis here is on understanding the basic foundations of the many mathematical methods employed in solving problems of waves and fields in inhomogeneous media. Hence, the reader shall learn to formulate many complicated problems using such methods. A straightforward application of numerical methods to many wave and field problems involves using intensive computation and extensive computer time, and may saturate the resources of even present-day supercomputers. Good judgement that adapts a combination of numerical and analytical methods, however, can usually generate computer codes that are many times more efficient. The details of the numerical analyses of these techniques, namely, numerical integration, matrix analysis, or finite-element analysis, are beyond the scope of this book, but they can be gleaned from many other relevant sources. Furthermore, this book does not discuss random inhomogeneous media, a subject that could span another volume of similar size.

Analytic techniques were in vogue before computers became available. They were often the sole technique for analyses in the days of ``pencil-and-paper" physics or engineering. For instance, many classical topics, like waves in planarly layered media, the Cagniard-de Hoop method for transient analysis, and the Gel'fand-Levitan-Marchenko methods for inverse scattering, are elegantly developed for solving one-dimensional, inhomogeneous-medium problems. Such techniques usually have limited scope for applications; however, to ignore the discussion of such topics in this book would render it incomplete. They are included to pay homage to those predecessors who developed them and to give the reader a historical perspective.

Despite the preponderance of electromagnetic examples contained herein, many of the techniques are easily adapted to other wave phenomena, such as acoustic waves, Schr\"odinger waves, or elastic waves. In order to readily convey the underlying concept of a technique to the reader, some are illustrated in the form of scalar wave problems. The extension to the vector electromagnetic wave case is often almost perfunctory.

Throughout the book, it is assumed that the reader knows the rudiments of electromagnetic theory. Chapter 1 reviews this required background cursorily. Planarly layered media remain the most studied of the inhomogeneous media because of its simplicity. Many results can be obtained without computationally intensive calculations. Furthermore, closed-form solutions in terms of spectral integrals (Sommerfeld integrals) allow for their asymptotic approximations, providing more physical insight into the problems. Hence, this topic, with its longer history, abounds in analytical techniques. Consequently, Chapter 2 will concentrate on waves in planarly layered media. The relatively greater length of this chapter reflects the abundance of interesting material on planarly layered media that has been developed over the years. Although this topic has been covered by many other books, sometimes as the sole subject, only a few of the more important topics are included here. Moreover, numerical schemes like the propagator-matrix approach and the numerical integration of Sommerfeld integrals are also discussed.

Chapter 3 covers waves in cylindrically and spherically layered media. This kind of wave problem finds applications in optics, geophysical probing, and propagation of radio waves on the earth's surface and the ionosphere. It also finds application in multilayer coatings on aircraft to reduce the radar cross-sections of aircraft. It is surprising to observe that this subject has not been developed as extensively as waves in planarly layered media.

Transient measurements are routine in radar, bioacoustic, or sonar measurements. Chapter 4 considers the propagation of transient waves in inhomogeneous media. First, the Cagniard-de Hoop method is presented, which finds closed-form solutions for the response of a line source on top of a layered, dispersionless medium. Then, for a general inhomogeneous medium, the celebrated time-domain finite-difference method is discussed in conjunction with the use of absorbing boundary conditions to obtain transient solutions.

Chapter 5 presents the variational methods for the scalar wave equation and the electromagnetic wave equation. Such methods are often the foundation of many modern numerical methods in waves and fields, such as the finite-element method and the method of moments. Moreover, many of these numerical methods can be applied in a recipe-like manner to obtain the solution to a problem. It is still important, however, that a student understands the foundations upon which these numerical methods are based. Therefore, the beginning of the chapter explains the basic concepts of a linear vector space. Subsequently, the formulations of variational expressions for self-adjoint as well as non-self-adjoint problems are presented followed by a step-by-step derivation of the matrix equation from a variational expression (using the Rayleigh-Ritz method). This derivation is the foundation of the finite-element method, and the reader, by following it, will understand the basis of the finite-element method. But the detailed implementation of the finite-element method, and the many bookkeeping techniques developed, are beyond the scope of this book.

Chapter 6 describes the mode matching technique. This technique is particularly useful for studying scattering at junction discontinuities encountered in dielectric slab waveguides or many geophysics problems. In order to understand this technique, the readers are first introduced to the concepts of the radiation modes and the guided modes of a dielectric slab. Then, the canonical scattering solution from a single junction discontinuity is discussed. This canonical solution is, in turn, used to find the scattering solution from many junction discontinuities in a composite problem using a recursive algorithm. Here, vector notations are used to write the solutions in a compact fashion.

The dyadic Green's function has been an interesting topic in electromagnetic theory. In this subject, many scalar wave concepts cannot simply be extended to the vector wave case. Even though most problems can be solved without the use of dyadic Green's functions, the symbolic simplicity with which they could be used to express relationships makes the formulations of many problems simpler and more compact. Moreover, it is easier to conceptualize many problems with the dyadic Green's functions. Consequently, the dyadic Green's functions in layered inhomogeneous media are discussed in Chapter 7, beginning with the spatial and spectral representations of the dyadic Green's function in a homogeneous medium. Next, the singularity of the dyadic Green's function is discussed; this has been a topic of heated debate in recent years. Then, vector wave functions are introduced to derive the dyadic Green's functions in different coordinate systems. Finally, the dyadic Green's function is generalized to layered media of planar, cylindrical, and spherical configurations.

Chapter 8 presents integral equation methods. These methods are especially useful for inhomogeneous media that are piecewise constant or when the inhomogeneous region has a finite support in space. Surface integral equations and the way they are solved (using the method of moments and the extended-boundary-condition method) are described. Along with the extended-boundary-condition method, the $\dyad T$ matrix and the $\dyad S$ matrix for the scattering due to one scatterer are derived. Then, it is shown how such a solution can be used to find the scattering from many scatterers, or multilayered scatterers. Moreover, the next topic covers the case of the unimoment method, a hybrid method, which combines the use of the finite-element method and the surface integral equation method. Finally, the volume integral equation and techniques for its solution are included here. The regimes of validity of the approximate solution technique, like the Born or the Rytov approximations, are also considered.

Many people who study waves in inhomogeneous media are undoubtedly interested in the inverse scattering problem, which finds many applications in biological sensing, geophysics, remote sensing, nondestructive evaluation, target identifications, and so on. Chapter 9 presents the solution techniques of several inverse scattering problems. The first discussion covers the area of linear inverse scattering, which has had a tremendous impact on medical tomography. In particular, back-projection tomography and diffraction tomography are reviewed. The scattered field of a scattering experiment is, however, more often nonlinearly related to the object that causes the scattering. Hence, different techniques for obtaining solutions to such nonlinear inverse scattering problems are given. Most advances in the past have been made in the area of the one-dimensional inverse scattering problem. Consequently, the method of characteristics, the Gel'fand-Levitan, and the Marchenko methods in solving such one-dimensional problems are discussed here. For higher dimensions, however, one has to resort to quasi-Newton type methods in solving the nonlinear inverse scattering problem. In addition, the use of the distorted Born iterative method, and the Born iterative method in seeking the solutions to such problems, are presented. It is imperative to develop new ideas and make advances in the multidimensional problem, for many technologies can benefit greatly from them.

Many topics within this book are from the published literature. Some were developed while writing the book. In some cases, the formulations are altered slightly---but without sacrificing accuracy---from the published literature to conform to the uniformity of presentation and style.

A book of this size cannot cover all the topics that have been written on waves and fields in inhomogeneous media, but it will make many more topics accessible to the reader. Also, the reader is encouraged to make use of the reference sections for further study in any of the topics presented.


I am indebted to all colleagues who have contributed in this area. I am also indebted to many from whom I have had the opportunity to learn about the theory of waves and fields. I am particularly grateful to Professor Jin Au Kong, and also to Leung Tsang, who taught me much about electromagnetic theory when I was at MIT. At Schlumberger-Doll Research, I had the opportunity to learn the Cagniard-de Hoop method first-hand from Adrianus de Hoop. Much of the information on the finite-difference method was brought to my attention by Curt Randall. I also owe a great deal to Mike Ekstrom, Bob Kleinberg, and Pabitra Sen, who influenced my view on science. Feedbacks on the first draft of the book from Allen Howard, Adrianus de Hoop, John Lovell, and Jim Wait are much appreciated. At the University of Illinois, I am grateful for productive interactions with all my colleagues.

The support for my research from the National Science Foundation throu\-gh the Presidential Young Investigator Program, the Army Research Office\linebreak through the Advanced Construction Technology Center at the University of Illinois, and the Office of Naval Research are gratefully acknowledged. I am also thankful to support from a number of industrial organizations, especially Schlumberger, General Electric, Northrop, and TRW.

Finally, I wish to thank Li Tong for working relentlessly and skillfully to typeset the manuscript of this book. I am also thankful to many of my students, who help proofread the book (they are Jim Friedrich, Levent Gurel, Mahta Moghaddam, Qinghuo Liu, Muhammad Nasir, Greg Otto, Rob Wagner, and Yiming Wang), and in particular to Jim Friedrich and Qinghuo Liu, who proofread every chapter. Fred Daab is gratefully acknowledged for drafting all the figures. I am also deeply grateful to my wife and two children, who displayed such patience and love while I wrote this book.

For the second printing of the book, I like to thank Jiun-Hwa Lin, Qinghuo Liu, Caicheng Lu, Zaiping Nie, Greg Otto, Yiming Wang, and Bill Weedon for pointing out numerous typographic errors and suggestions for improvements in the first printing of the book.