Electromagnetic scattering by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate

We consider a two-dimensional problem of scattering of a time harmonic electromagnetic plane wave by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate. The magnetic permeability is assumed to be a fixed positive constant in the media. The material properties of the media are characterized completely by an index of refraction, which is a bounded measurable function in the layer and a positive constant above the layer corresponding to a homogeneous dielectric medium. In this paper, we only examine the TM (transverse magnetic) polarization case. A radiation condition is introduced and equivalence with a second kind, Lippmann-Schwinger-type integral equation is shown. With additional assumptions on the index of refraction in the layer, uniqueness of solution is proved. Existence of solution is then established by employing a form of Fredholm alternative using a general result on the solvability of integral equations on unbounded domains. An approximate analytic solution for the case of a thin inhomogeneous layer is obtained from the integral equation formulation and is used to show that, if the index of refraction is appropriately chosen, the scattered held can grow with distance from the plate.