On Integral Equation Methods for Electromagnetic Scattering by Biperiodic Structures

Abstract

In this thesis, we use integral equation methods to study the scattering of timeharmonic electromagnetic plane waves by biperiodic multilayered structures. Here, such structures are modeled by vertically arranging finitely many non-self-intersecting polyhedral Lipschitz interfaces. We distinguish single profile scatterers and scatterers composed of at least two surfaces. The electromagnetic scattering problem, which arises from the presence of obstacles of the first type, serves as a basic model that we reuse in the multilayered case. By treating it with a combined potential ansatz, we obtain an equivalent new periodic singular integral equation with Fredholm index zero. The Fredholmness is verified by Garding inequalities. This property is crucial for proving one of the main results of our analysis in the single interface setting: an existence theorem for the derived integral equation under certain assumptions on the electromagnetic material parameters. The second central result, a uniqueness theorem, is proven by a variational argumentation and again depends on the values of the electromagnetic material parameters. In the main part of this thesis, we then extend the considered obstacles to “real” multilayered obstacles consisting of at least two grating interfaces. For these, we propose two recursive integral equation algorithms to determine the generated reflected and transmitted waves. On the one hand, the recursive scattering matrix method and, on the other hand, a recursive algorithm based on the transmission conditions across the interfaces of the multilayered structure. In addition to their accurate derivation, new existence and uniqueness results, depending on the electromagnetic properties of the considered obstacle, are presented for each algorithm. Their proofs partially exploit the insights originating from the treatment of the single interface model problem. Our analytical investigation is intended to provide a basis for new efficient numerical implementations of the biperiodic multilayered electromagnetic scattering problem.