Scattering of Electromagnetic Waves by Obstacles
This book is an introduction to some of the most important properties of electromagnetic waves and their interaction with passive materials and scatterers. The main purpose of the book is to give a theoretical treatment of these scattering phenomena, and to illustrate numerical computations of some canonical scattering problems for different geometries and materials. The scattering theory is also important in the theory of passive antennas, and this book gives several examples on this topic. Topics covered include an introduction to the basic equations used in scattering; the Green functions and dyadics; integral representation of fields; introductory scattering theory; scattering in the time domain; approximations and applications; spherical vector waves; scattering by spherical objects; the nullfield approach; and propagation in stratified media. The book is organised along two tracks, which can be studied separately or together. Track 1 material is appropriate for a first reading of the textbook, while Track 2 contains more advanced material suited for the second reading and for reference. Exercises are included for each chapter.
Inspec keywords: Green's function methods; approximation theory; electromagnetic wave scattering; timedomain analysis
Other keywords: spherical vector waves; electromagnetic waves scattering; time domain; integral representation; dyadics; introductory scattering theory; spherical objects; approximations; green functions; stratified media propagation
Subjects: General electrical engineering topics; Interpolation and function approximation (numerical analysis); Mathematical analysis; Electromagnetic wave propagation
 Book DOI: 10.1049/SBEW524E
 Chapter DOI: 10.1049/SBEW524E
 ISBN: 9781613532218
 eISBN: 9781613532225
 Page count: 757
 Format: PDF

Front Matter
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1 Basic Equations
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In this chapter, we review the basic equations that model electromagnetic wave propagationthe Maxwell equationsand we set the notation used in this book. The chapter examines not only the basic equations, but also a detailed modeling of the interaction of electromagnetic fields with materials. The material modeling is general with several explicit examples, and it provides the appropriate background setting for e.g., scattering by general linear stratified materials.

2 The Green Functions and Dyadics
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To analyze the electromagnetic fields, we need some results from the Green function techniques and the scalar and vector potentials. The analysis in this chapter covers both the timedomain and the frequencydomain results, but we start the investigation in the frequency domain, since this analysis is simpler. The frequencydomain results can then be used in the time domain.

3 Integral Representation of Fields
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The integral representations show that the electromagnetic field in a volume of an isotropic material is represented in terms of volume and surface integrals. The boundary surface is a fictitious surface, which can coincide as a limiting case with the real boundary surface between two different materials. These representations are very important and powerful tools for e.g., the development of integral equations that can be used to numerically solve electromagnetic field problems.

4 Introductory Scattering Theory
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When a timeharmonic, electromagnetic wave propagates in a homogeneous, lossless, isotropic material, characterized by the realvalued parameters ε and μ, the wave propagates unhindered. If there is a region where the electric or magnetic material parameters have a different value, then the propagation of the wave altersthe wave is scattered. The ultimate goal of the scattering theory is to analyze this alteration of the electromagnetic fields in a quantitative way.

5 Scattering in the Time Domain
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A scattering problem is a comparison between two different, but related, wave propagation situations.The underlying wave propagation problem deals with propagation of field quantities in space and time in a volume with undisturbed electromagnetic propertiesthe propagation of the incident fields, Ei(r, t) and Hi(r, t).

6 Approximations and Applications
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problem. In this chapter, we introduce some of the most popular approximations used to solve scattering problems. An overview of the different approximations and their domain of usage is given in Figure 6.1, where a is the typical length scale of the scatterer.

7 Spherical Vector Waves
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There is a special interest in finding solutions of the Maxwell equations in sourcefree, homogeneous, isotropic materials in the spherical coordinate system (r, 8, φ). The reason this is at least twofold: (1) we aim at developing efficient tools to solve scattering problems with spherical symmetries, and (2) outside the circumscribed sphere of the scatterer, the are naturally expanded in vector waves with spherical symmetrieswe have already Chapter 4 that the scattered field in the far zone is a spherical wave. For these reasons, study solutions to the sourcefree Maxwell equations.

8 Scattering by Spherical Objects
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The usefulness of spherical vector waves becomes obvious when we solve scattering problems by obstacles that have spherical symmetries. In Sections 8.18.5, we solve the scattering by perfectly conducting, dielectric, layered, anisotropic, and biisotropic sphere, respectively.These solutions are called the Mie solution1 or partial wave solution, see e.g., References 54, 58, 94. A method to compute scattering by nonspherical shapes is presented in Chapter 9. Scattering by more complex materials has also been investigated in the literature, e.g., scattering by a gyrotropic sphere.

9 The NullField Approach
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The transition matrix, Tnnι, contains all information that is necessary to solve the scattering problem, such as various cross sections etc., see Chapter 7. To find the transition matrix, in the general case, is a very difficult problem. In this section, we derive a general expression of the transition matrix in terms of surface integrals over the spherical vector waves. The derivation was originally presented by Peter C. Waterman.1 Some of his excellent scientific production found in References 282285, and a comprehensive database of Nullfield approach is given in References 189194, 303. Alternative approaches to find the transition matrix based on the far field are presented in References 7678. The derivation proceeds in a formal way, and does not address the delicate mathematical question concerning convergence, etc. These questions are addressed in e.g., References 45, 46, 56, 145, 222, 232. Several recent review articles on the method are published in the literature [188, 194, 231, 233]. We start by analyzing the single, homogeneous scatterer case Section 9.1, and then proceed, in Section 9.2, to find the Tmatrix for a collection of scatterers. The chapter ends with Section 9.3 that treats scattering by an obstacle above a perfectly conducting ground. This combination of finite structures with planar ones has many applications in the geophysical sciences.

10 Propagation in Stratified Media
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In this chapter, we develop the main tools for the analysis of wave propagation in stratified media, and we have the ambition to make a unified approach to scattering by a general bianisotropic media. The appealing part of this analysis is that all materials, from simple isotropic materials to general bianisotropic ones, are treated with the same uniform theory. This theory is built on a rather simple 2 × 2 matrix algebra, which is easily implemented on a computer. Of course, considerable simplifications occur if the medium is less complex, such as isotropic materials. These special cases are analyzed in detail.

Appendix A: Vectors and Linear Transformations
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This appendix contains a short overview of the concept of vectors and linear transformations (dyadics) of vectors, and how these are represented in terms of their components. Transformations between different rotated coordinate systems are also reviewed as well as the corresponding transformation of the components of a dyadic. The appendix ends with a short overview of quaternions and their use to represent rotations.

Appendix B: Bessel Functions
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In wave propagation problems, the Bessel differential equation often appears, especially in problems showing axial or spherical symmetries. This appendix collects some useful and important results for the solution of the Bessel differential equation. Moreover, the modified Bessel functions and the spherical Bessel and Hankel functions are presented.

Appendix C: Spherical Harmonics
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In this appendix, we define the Legendre polynomials that are used in the book, and moreover, we list some of their most important properties. In scattering theory, the spherical harmonics and the vector spherical harmonics play an important role, and after the definition of these functions, we also present some of their important properties. Specifically, the transformation properties between spherical and plane waves are derived.

Appendix D: The Fourier and Other Transforms
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A series of useful results related to Fourier transforms in one or several dimensions is collected in this appendix. This overview also contains the Poisson summation formula, the Hilbert transform, and Meiıman's theorem, that often are used in conjunction with Fourier transforms. The class of positivedefinite functions and Herglotz functions is also reviewed. The appendix ends with two sections on the Watson transformation and the location of zeros and poles.

Appendix E: Relativity
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If two observers are in motion relative to each other, they describe the same physical events with different space and time coordinates. The famous Lorentz transformation describes the relation between the two spacetime coordinate systems. This section reviews some of the properties of this transformation that are relevant for electromagnetic waves.

Appendix F: Some Useful Mathematical Results
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This appendix contains a series of useful mathematical results that are used in the book. Theorems are important when computing the action of an entire function of a square dyadic. The first theorem by Cayley and Hamilton is fundamental.

Appendix G: Asymptotic Evaluation of Integrals
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In this section, we compute the dominant contribution to the integral I(ξ) = ^{b}∫_{a} g(x)e^{iξf (x)} dx for large values of the real parameter . The limits a and b are assumed to be real and can be ±∞. The function (x) is assumed to be real valued in the interval [a, b]. The function g(x) can be complex valued.

Appendix H: The Nabla Operator in Curvilinear Coordinate Systems
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In this appendix, some important expressions with the nabla operator in the Cartesian, cylindrical, and spherical are collected.

Appendix I: Notation
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Appropriate notation leads to a more easily understood, systematic, and structured text, and in the same token, good notation implies a tendency of making less errors and slips in the analysis. Most of the notation is explained at the place in the text where it is introduced, but some more general notation that is often used is collected in this appendix.

Appendix J: Units and Constants
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The explicit form of the equations in electromagnetism varies depending on the system of units that we use. The SI system is the one that is used in most literature nowadays, and this textbook is no exception. The relevant constants in the SI system that are used in the text are collected in this appendix.

Back Matter
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