ON THE WELL-POSEDNESS OF THE INVERSE ELECTROMAGNETIC SCATTERING PROBLEM FOR A DISPERSIVE MEDIUM

The study of the inverse scattering problem for electromagnetic waves, especially under consideration of such realistic effects as spatial inhomogeneity, 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 paper, we will 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) via the invariant imbedding techniques. 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.