The inverse electromagnetic scattering problem in a piecewise homogeneous medium

This paper is concerned with the problem of scattering of time-harmonic electromagnetic waves from an impenetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is established, employing the integral equation method. Inspired by a novel idea developed by Hahner [11], we prove that the penetrable interface between layers can be uniquely determined from a knowledge of the electric far field pattern for incident plane waves. Then, using the idea developed by Liu and Zhang [21], a new mixed reciprocity relation is obtained and used to show that the impenetrable obstacle with its physical property can also be recovered. Note that the wave numbers in the corresponding medium may be different and therefore this work can be considered as a generalization of the uniqueness result of [20].Comment: 19 pages, 2 figures, submitted for publicatio

Scattering of time-harmonic electromagnetic waves involving perfectly conducting and conductive transmission conditions

The focus of the research described herein is the scattering of time-harmonic electromagnetic waves when encountering with impenetrable and penetrable obstacles. We study both the direct and inverse problems. In the case of an impenetrable obstacle, we assume perfectly conducting boundary condition and apply the integral equation method to show well-posedness of the direct problem. In the case of a penetrable obstacle, we assume conducting transmission conditions and apply both the integral equation and variational method to show well-posedness. The inverse problem we consider is determining the shape of an obstacle from the knowledge of the far field pattern. Specifically, we concentrated on uniqueness issues, that is, we examined under what conditions an obstacle can be identified from a knowledge of its far far field patterns for incident plane waves. We conclude this thesis with a discussion of an interior eigenvalue problem motivated by the penetrable case with conducting boundary conditions and show that the set of transmission eigenvalues form at most a discrete set

Invisibility and Inverse Problems

This survey of recent developments in cloaking and transformation optics is an expanded version of the lecture by Gunther Uhlmann at the 2008 Annual Meeting of the American Mathematical Society.Comment: 68 pages, 12 figures. To appear in the Bulletin of the AM