Inverse obstacle scattering for Maxwell’s equations in an unbounded structure

Abstract

This paper is concerned with analysis of electromagnetic wave scattering by an obstacle which is embedded in a two-layered lossy medium separated by an unbounded rough surface. Given a dipole point source, the direct problem is to determine the electromagnetic wave field for the given obstacle and unbounded rough surface; the inverse problem is to reconstruct simultaneously the obstacle and unbounded rough surface from the electromagnetic field measured on a plane surface above the obstacle. For the direct problem, a new boundary integral equation is proposed and its well-posedness is established. The analysis is based on the exponential decay of the dyadic Green function for Maxwell's equations in a lossy medium. For the inverse problem, the global uniqueness is proved and a local stability is discussed. A crucial step in the proof of the stability is to obtain the existence and characterization of the domain derivative of the electric field with respect to the shape of the obstacle and unbounded rough surface. , J j denote the electric field, the magnetic field, the electric current density, respectively, and ρ j = (iω) −1 ∇ · J j is the electric charge density. The external current source is assumed to be located in Ω 1 . The relation between the electric current density and the electric field is given bywhere J cs stands for the current source.

Simultaneous recovery of a locally rough interface and the embedded obstacle with its surrounding medium

“…It was then shown in [31] that an embedded electromagnetic obstacle was uniquely determined in a two-layered lossy medium separated by a planar surface. A similar result was obtained in [29] for Maxwell's equations on determining an interface with a perfectly conducting obstacle embedded in Ω 1 if the background medium is lossy. Furthermore, the uniqueness result was also obtained in [38] for inverse acoustic scattering by an embedded penetrable obstacle in the lower-half space, where the background medium is allowed to be lossless.…”

An Integral Equation Method for the Scattering by Core-Shell Structures in a Layered Medium

“…More recently, a boundary integral equation has been proposed for solving 2D homogeneous obstacle acoustic (electromagnetic) composite scattering problems. Based on the energy estimates, the uniqueness of the solution or the scattering problem is established [2,18]. To our best knowledge, no mathematical study is available for the scattering of the core-shell structures in a layered medium.…”

The time‐domain scattering by the elastic shell in a two‐layered unbounded structure

“…In Arens 22 and Elschner and Hu, 23 the integral equation method and variation method are used to prove the existence of solution in elastic wave scattering by unbounded rough surfaces. In Li et al 24 and Bao et al, 25 a homogeneous obstacle composite acoustic (electromagnetic) scattering and inverse scattering problems have been considered. The well‐posedness is proved by using the integral equation method for the scattering problem, and the obstacle and the infinite rough surface can be uniquely determined by the measured wave fields are also obtained.…”