A Bloch wave numerical scheme for scattering problems in periodic wave-guides

TL;DR

This paper introduces a novel numerical scheme using Bloch wave functions to efficiently solve Helmholtz scattering problems in periodic wave-guides with complex media interfaces.

Contribution

It proposes a new domain truncation method combined with Bloch wave ansatz functions and provides theoretical stability analysis for the scheme.

Findings

Effective handling of media interfaces in wave-guides

Stable numerical scheme with proven existence results

Application to negative refraction in photonic crystals

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 $\varepsilon$-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.