The mathematics of scattering by unbounded, rough, inhomogeneous layers

In this paper we study, via variational methods, a boundary value problem for the Helmholtz equation modelling scattering of time harmonic waves by a layer of spatially varying refractive index above an unbounded rough surface on which the field vanishes. In particular, in the 2D case with TE polarization, the boundary value problem models the scattering of time harmonic electromagnetic waves by an inhomogeneous conducting or dielectric layer above a perfectly conducting unbounded rough surface, with the magnetic permeability a fixed positive constant in the medium. Via analysis of an equivalent variational formulation, we show that this problem is well-posed in two important cases: when the frequency is small enough; and when the medium in the layer has some energy absorption. In this latter case we also establish exponential decay of the solution with depth in the layer. An attractive feature is that all constants in our estimates are bounded by explicit functions of the index of refraction and the geometry of the scatterer.