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

This paper is concerned with the electromagnetic scattering of a point source by a perfectly electrically conducting 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 reflected and transmitted fields measured on a plane surface which is above and below the unknown objects, respectively. For the direct problem, its well-posedness is established and a new boundary integral equation is proposed. 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.