Existence, Uniqueness and Regularity for Solutionsv of the Conical Diffraction Problem

Abstract

This paper is devoted to the analysis of two Helmholtz equations in ℝ2 coupled via quasiperiodic transmission conditions on a set of piecewise smooth interfaces. The solution of this system is quasiperiodic in one direction and satisfies outgoing wave conditions with respect to the other direction. It is shown that Maxwell's equations for the diffraction of a time-harmonic oblique incident plane wave by periodic interfaces can be reduced to problems of this kind. The analysis is based on a strongly elliptic variational formulation of the differential problem in a bounded periodic cell involving nonlocal boundary operators. We obtain existence and uniqueness results for solutions corresponding to electromagnetic fields with locally finite energy. Special attention is paid to the regularity and leading asymptotics of solutions near the edges of the interface.

Conical diffraction by multilayer gratings: a recursive integral equation approach

“…[2,11]) that the finite energy condition (2.2) is satisfied only if the z-components of E and B are H 1 -regular. Moreover, E z , B z determine the other components of the electric and magnetic fields and are solutions of the Helmholtz equations…”

“…To prove that the Rayleigh coefficientsû ± n ,v ± n vanish for arbitrary non-trivial solutions (u, v) we proceed as in [2,11]. Choose a periodic cell Ω H , which has in the x-direction the width d, is bounded by the straight lines {y = ±H} and contains Γ j , j = k, .…”

“…Their quasiperiodic solutions have to satisfy radiation conditions and are coupled by transmission conditions at the interfaces between different grating materials. A variational formulation of this problem has been studied in [2] based on strong ellipticity estimates, which are valid under some restrictions on the permittivities ε, 0 arg ε < π, of the non-magnetic grating materials, which are specified in Section 2. Using layer potentials with the quasiperiodic fundamental solution of the Helmholtz equation the diffraction problem for multilayer gratings can be transformed to a system of integral equations over the interfaces.…”

Elastic Scattering by Unbounded Rough Surfaces

“…Ammari, Bao & Wood [3], Bao & Dobson [9], Bonnet-Bendhia & Starling [10], Elschner & Schmidt [26], Elschner, Hinder, Penzel & Schmidt [27], Elschner & Yamamoto [28] and Kirsch [33]. In the case of elastic scattering by periodic surfaces, the variational approach is established by Elschner & Hu in [24,25] for the boundary value problems of the first, second, third and fourth kind as well as for transmission problems with non-smooth interfaces in R n (n = 2, 3).…”

Domain decomposition method for Maxwell’s equations: Scattering off periodic structures

“…In Section 4 we introduce a variational formulation to couple the PML to the interior problem. In contrast to Elschner et al [11,10] the electromagnetic field E = (E 1 , E 2 , E 3 ) is discretized with higher order Whitney elements for the (E 1 , E 2 ) component and Lagrange elements of the same order in the E 3 component. This allows for the accurate evaluation of Fourier coefficients needed for the coupling to a multi-layer stack as well for the computation of the far field coefficients.…”