Electromagnetic wave scattering from local perturbed periodic inhomogeneous layers

TL;DR

This paper develops a mathematical framework for analyzing electromagnetic wave scattering in locally perturbed periodic dielectric layers, employing advanced transforms and operator theory to establish solution existence and regularity.

Contribution

It introduces a novel variational formulation and solution approach for Maxwell's equations in complex layered media with local perturbations, using Bloch-Floquet transform and Fredholm theory.

Findings

Unique existence of solutions established

Regularity results for the transformed solutions

Method handles boundary singularities effectively

Abstract

We consider the scattering problem on locally perturbed periodic penetrable dielectric layers, which is formulated in terms of the full vector-valued time-harmonic Maxwell's equations. The right-hand side is not assumed to be periodic. At first, we derive a variational formulation for the electromagnetic scattering problem in a suitable Sobolev space on an unbounded domain and reformulate the problem into a family of bounded domain problems using the Bloch-Floquet transform. For this family we can show the unique existence of the solution by applying a carefully designed Helmholtz decomposition. Afterwards, we split the differential operator into a coercive part and a compact perturbation and apply the Fredholm theory. Having that, the Sherman-Morrison-Woodbury formula allows to construct the solution of the whole problem handling the singularities of the Calderon operator on the boundary. Moreover, we show some regularity results of the Bloch-Floquet transformed solution w.r.t. the quasi-periodicity.