Outgoing wave conditions in photonic crystals and transmission properties at interfaces

TL;DR

This paper develops an outgoing wave condition for unbounded photonic crystals using Bloch wave expansion, enabling analysis of wave transmission and negative refraction at interfaces.

Contribution

It introduces a novel radiation condition for x-dependent photonic crystals and applies it to study wave transmission and refraction phenomena.

Findings

Established a weak uniqueness result for the radiation condition.

Derived conservation of vertical wave number at interfaces.

Demonstrated conditions leading to negative refraction.

Abstract

We analyze the propagation of waves in unbounded photonic crystals, the waves are described by a Helmholtz equation with $x$-dependent coefficients. The scattering problem must be completed with a radiation condition at infinity, which was not available for $x$-dependent coefficients. We develop an outgoing wave condition with the help of a Bloch wave expansion. Our radiation condition admits a (weak) uniqueness result, formulated in terms of the Bloch measure of solutions. We use the new radiation condition to analyze the transmission problem where, at fixed frequency, a wave hits the interface between free space and a photonic crystal. We derive that the vertical wave number of the incident wave is a conserved quantity. Together with the frequency condition for the transmitted wave, this condition leads (for appropriate photonic crystals) to the effect of negative refraction at the interface.