In this chapter, we review the basic equations that model electromagnetic wave propagation-the Maxwell equations-and 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.

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.

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 properties-the propagation of the incident fields, Ei(r, t) and Hi(r, t).

There is a special interest in finding solutions of the Maxwell equations in source-free, 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 symmetries-we have already Chapter 4 that the scattered field in the far zone is a spherical wave. For these reasons, study solutions to the source-free Maxwell equations.

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 282-285, and a comprehensive database of Null-field approach is given in References 189-194, 303. Alternative approaches to find the transition matrix based on the far field are presented in References 76-78. 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 T-matrix 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.

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.

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.

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 space-time coordinate systems. This section reviews some of the properties of this transformation that are relevant for electromagnetic waves.

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.

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.