A Bloch Wave Numerical Scheme for Scattering Problems in Periodic Wave-Guides

Abstract

We present a new numerical scheme to solve the Helmholtz equation in a wave-guide. We consider a medium that is bounded in the x 2 -direction, unbounded in the x 1 -direction and ε-periodic for large |x 1 |, allowing different media on the left and on the right. We suggest a new numerical method that is based on a truncation of the domain and the use of Bloch wave ansatz functions in radiation boxes. We prove the existence and a stability estimate for the infinite dimensional version of the proposed problem. The scheme is tested on several interfaces of homogeneous and periodic media and it is used to investigate the effect of negative refraction at the interface of a photonic crystal with a positive effective refractive index.

On a Limiting Absorption Principle for Sesquilinear Forms with an Application to the Helmholtz Equation in a Waveguide

“…Methods like non-reflecting boundary conditions or perfectly matched layers are Ben Schweizer e-mail: ben.schweizer@tu-dortmund.de Technische Universität Dortmund, Fakultät für Mathematik, Vogelpothsweg 87, D-44227 Maik Urban e-mail: maik.urban@tu-dortmund.de Technische Universität Dortmund, Fakultät für Mathematik, Vogelpothsweg 87, D-44227 successfully applied to problems with constant coefficients, but these methods cannot be used in a periodic medium. The approach of [4] is to introduce radiation boxes and to demand that the solution of the truncated problem is an outgoing wave in the radiation boxes. The problem can be formulated with the help of a sesquilinear form.…”

“…This program has been performed successfully in several settings, as recent examples we mention [7] and [11]. A radiation condition for truncated domains was introduced in [4], the article contains a well-posedness result for positive absorption δ > 0. Here, we study the limit δ → 0 of that system.…”

“…In the first part of this article, we derive the limiting absorption principle in an abstract setting, and consider sesquilinear forms on Hilbert spaces. We recall that the system of [4] is formulated with sesquilinear forms, hence our abstract results can be applied to this "radiation condition on truncated domains". Given a (real or complex) Hilbert space H , a sesquilinear form b, and ∈ H * , we look for an element u ∈ H that satisfies the equation…”

“…Application to the Helmholtz equation in periodic waveguide. Let us sketch the setting of the truncated problem of [4]. The geometry is described by the numbers ε, R > 0 and H ∈ N, the truncated domain is Ω R := (−εR, εR) × (0, εH).…”

“…The problem is to find a function u δ , which solves (1) in Ω R and which is radiating in radiation boxes. This problem is recast in [4] with a sesquilinear form β δ and is written as…”