Resolution of inverse scattering problems for the full three-dimensional Maxwell-equations in inhomogeneous media using the approximate inverse

Abstract

A new method is developed in this work to solve the inverse electromagnetic scattering problem in inhomogeneous media using near-field measurements. The modeling is based on the formulation as contrast source integral equations of the full three-dimensional time-harmonic Maxwell-model. This inverse problem is ill-posed and nonlinear. The known idea of using equivalent sources splits inverse scattering into two subproblems: the inverse source problem, which is linear and ill-posed, and the inverse medium problem, which is more stable but nonlinear. We introduce the concept of generalized induced source to recast the system of intertwined vector equations, describing the electromagnetic inverse source problem, into decoupled scalar scattering problems. We utilize the method of the approximate inverse to recover the induced source for each experiment. We consider in three-dimensional setting the spherical scattering operator introduced by Abbdullah and Louis [Abd98] for 2-D acoustic waves. We derive its singular-value decomposition and determine a basis for its null space. We further apply some results about error estimate from [Lou99] to the scalar problem in three-dimensions with spherical set-up. The nonlinear version of the algorithm of Kaczmarz is then adapted, using the generalized induced source, to derive an iterative scheme for the resolution of the inverse medium problem. Numerical simulations illustrate the efficiency and practical usefulness of the developed method.