2011, Schmidt, Gunther

In this paper we consider an integral equation algorithm to study the scattering of plane waves by multilayer diffraction gratings under oblique incidence. The scattering problem is described by a system of Helmholtz equations with piecewise constant coefficients in $R^2$ coupled by special transmission conditions at the interfaces between different layers. Boundary integral methods lead to a system of singular integral equations, containing at least two equations for each interface. To deal with an arbitrary number of material layers we present the extension of a recursive procedure developed by Maystre for normal incidence, which transforms the problem to a sequence of equations with $2 times 2$ operator matrices on each interface. Necessary and sufficient conditions for the applicability of the algorithm are derived.

2014, Bugert, Beatrice, Schmidt, Gunther

In this paper, we propose a new recursive integral equation algorithm to solve the direct problem of electromagnetic scattering by biperiodic multilayered structures with polyhedral Lipschitz regular interfaces. We work with a combined potential approach that involves one unknown density on each of the grating profiles of the multilayered scatterer. Justified by the transmission conditions of the underlying electromagnetic scattering problem, we assume that densities in adjacent layers are linearly linked by a boundary integral operator and derive a recursion for these densities. It comprehends the inversion of one boundary integral equation on each scattering interface. Our algorithm is shown to be equivalent to the biperiodic multilayered electromagnetic scattering problem. Moreover, we obtain new existence and uniqueness results for our recursive integral equation algorithm, which promises to lead to an efficient numerical implementation of the considered scattering problem. These solvability results depend on the regularity of the grating interfaces and the values of the electromagnetic material parameters of the biperiodic multilayered structure at hand.