Inverse scattering for anisotropic mirror image symmetric media

Abstract

An inverse medium scattering problem for a homogcnwus slab of anisotropic dispersive medium is presented. The inverse problem is to recover the dyadic susceptibility kernel from the dyadic reflection kernels for obliquely incident transient electromagnetic plane waves. A time domain technique, based on wave splitting and invariant imbedding, is used, Mirror images enter into the fomalism in a natural way and are therefore discussed.This leads to a simplified problem for a class of mirror image symmetric media. This is solved numerically and the stability of the algorithm is tested using noisy data.

Existence, Uniqueness, and Causality Theorems for Wave Propagation in Stratified, Temporally Dispersive, Complex Media

“…Theorem 2.2 shows that the regularity of the solution to the canonical problem is increased with the regularity of the memory function a. This admits the definition of the Green functions employed in, e.g., [8,9,17,19,18].…”

“…As mentioned in section 1, the results of this paper have already been used by the scientific community in a number of papers on direct and inverse scattering in dispersive, complex media [8,9,17,19,18]. In these articles, the Green functions equations are employed rather than the canonical functions equations (2.2).…”

“…Observe that knowledge of the unique existence of a weak solution to the Green functions equations in the sense of Theorem 4.1 is sufficient for numerical purposes [8,9,17,19]. Notice also that the finite jumps in g ± across L ∓ , respectively, cannot be obtained from the integro-differential equation (4.1).…”

“…The investigations in, e.g., [8,9,17,19,18] suggest the study of the following mixed initial-boundary value problem, defined in the product set (x, s) ∈ (0, 1) × R: The independent variables s and x are the travel-time coordinates for time and slab depth, respectively. For each ordered pair (x, s), e i (s), e ± (x, s) ∈ M 2×1 (R) and a(x, s), b(x) ∈ M 4×4 (R), where M m×n (R) is the linear space over R consisting of the m × n matrices with real entries.…”

“…The vector fields e ± have been obtained by (optical) wave splitting. A wave splitting is a change of the dependent variables, such that in the simple media outside the complex slab, the split vector fields e + and e − represent the right-going and the left-going waves, respectively; see, e.g., [8,9,17,19,18] or section 5. Throughout space, the sum e + + e − and the difference e + − e − represent the total electric and magnetic fields, respectively; see, e.g., section 5.…”