The inverse electromagnetic scattering problem for a spatially homogeneous, dispersive and dissipative medium

Abstract

The study of the inverse scattering problem for electromagnetic waves, especially under consideration of such realistic effects as spatial inhomogeniety, dispersion, dissipativity and multiple spatial dimensions, is based on the use of such scattering waves for probing the interior of systems not readily accessible to direct examination (e.g., biological systems). The significant biological interest is concerned with the absorption and dissipation of nonionizing radiation in living organisms. As a result, one is interested in knowing the electrical properties (e.g., the dielectric constants and conductivities) of the living tissue. However, it is well known that both the dielectric constant and the conductance (i.e., the displacement and conduction current susceptibilities) are frequency dependent because living tissue is composed largely of water. In this dissertation, we will consider a method for solving the direct and inverse scattering of electromagnetic waves in the time domain for spatially homogeneous media in which both the displacement susceptibility kernel and the conduction current susceptibility kernel of the medium are time dependent. The method is based on the invariant imbedding techniques. The idea is to derive a set of nonlinear integrodifferential equations relating the displacement susceptibility and conduction current susceptibility kernels to the scattering operators (i.e., reflection and transmission operators). From these nonlinear integrodifferential equations, we will prove theorems for the existence, uniqueness and continuous dependence on data for the direct and inverse scattering problems for both the semi-infinite medium and the finite slab. For the inverse scattering problem, we obtain an equivalent nonlinear Volterra equation of the second kind. Numerical results are given for both the direct and inverse scattering problems. Finally, the extension of the inverse scattering problem to a higher dimensional case for a vertical magnetic dipole excitation with axial symmetry is discussed.