| Issue |
ESAIM: M2AN
Volume 52, Number 5, September–October 2018
|
|
|---|---|---|
| Page(s) | 1913 - 1945 | |
| DOI | https://doi.org/10.1051/m2an/2018026 | |
| Published online | 11 December 2018 | |
Outgoing wave conditions in photonic crystals and transmission properties at interfaces
Technische Universität Dortmund, Fakultät für Mathematik,
Vogelpothsweg 87,
44227
Dortmund, Germany.
* Corresponding author: ben.schweizer@tu-dortmund.de
Received:
19
December
2016
Accepted:
16
April
2018
We analyze the propagation of waves in unbounded photonic crystals. Waves are described by a Helmholtz equation with x-dependent coefficients, the scattering problem must be completed with a radiation condition at infinity. We develop an outgoing wave condition with the help of a Bloch wave expansion. Our radiation condition admits a 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 show 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.
Mathematics Subject Classification: 35Q60 / 35P25 / 35B27
Key words: Helmholtz equation / radiation / waveguide / Bloch analysis / outgoing wave condition / photonic crystal / transmission problem / negative refraction
© EDP Sciences, SMAI 2018
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