Inverse scattering by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium

This paper is concerned with the inverse problem of scattering of time-harmonic acoustic waves by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is first established by using the integral equation method. We then proceed to establish two tools that play important roles for the inverse problem: one is a mixed reciprocity relation and the other is a priori estimates of the solution on some part of the interfaces between the layered media. For the inverse problem, we prove in this paper that both the penetrable interfaces and the possible inside inhomogeneity can be uniquely determined from a knowledge of the far field pattern for incident plane waves.     

A uniqueness result for the inverse electromagnetic scattering problem in a piecewise homogeneous medium

The scattering of time-harmonic electromagnetic plane waves by an impenetrable obstacle in a piecewise homogeneous medium is considered. The well-posedness of the direct problem is proved by the variational method. Under the condition that the wave numbers in the innermost and outermost homogeneous layers coincide, we then establish a uniqueness result for the inverse problem, that is, the unique determination of the obstacle and its boundary condition from a knowledge of the electric far field pattern for incident plane waves. The proof is based on a generalization of the mixed reciprocity relation.     

INVERSE MEDIUM SCATTERING PROBLEMS IN NEAR-FIELD OPTICS

A regularized recursive linearization method is developed for a two-dimensional in-verse medium scattering problem that arises in near-field optics, which reconstructs the scatterer of an inhomogeneous medium deposited on a homogeneous substrate from data accessible through photon scanning tunneling microscopy experiments. In addition to the ill-posedness of the inverse scattering problems, two difficulties arise from the layered back-ground medium and limited aperture data. Based on multiple frequency scattering data, the method starts from the Born approximation corresponding to the weak scattering at a low frequency, each update is obtained via recursive linearization with respect to the wavenumber by solving one forward problem and one adjoint problem of the Helmholtz equation. Numerical experiments are included to illustrate the feasibility of the proposed method.