A variational method for wave scattering from penetrable rough layers

We consider time-harmonic wave scattering from unbounded penetrable rough layers and provide existence theory for this problem via variational formulations. A rough layer is a model for a stratified and unbounded inhomogeneous medium where the refractive index varies. For our variational approach, the refractive index can be real or (partially) complex valued and is allowed to jump across interfaces. However, the index needs to satisfy a non-trapping condition, which requires, roughly speaking, monotonicity in the direction normal to the layer. In the half-space above and below the rough layer, we set up a radiation condition using the angular spectrum representation. Due to the unbounded setting, we establish integral formulas similar to Rellich's identity to obtain an a priori bound for a variational solution of the rough-layer scattering problem. This a priori bound is the basis for our existence result. We further provide regularity theory for the rough-layer scattering problem and give bounds on its frequency dependence. Our existence results hold for all wave numbers and apply to acoustic scattering in dimension two and three as well as to electromagnetic scattering for the electric and the magnetic mode.