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Crystals 2015, 5(1), 14-44; doi:10.3390/cryst5010014

Bloch Modes and Evanescent Modes of Photonic Crystals: Weak Form Solutions Based on Accurate Interface Triangulation

1
Theoretische Physik, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstr. 7B, 91058 Erlangen, Germany
2
Applied Maths, Research School of Physical Sciences and Engineering, The Australian National University, Canberra 0200 ACT, Australia
*
Author to whom correspondence should be addressed.
Academic Editor: Yuri Kivshar
Received: 20 August 2014 / Accepted: 10 December 2014 / Published: 5 January 2015
(This article belongs to the Special Issue Photonic Crystals)
View Full-Text   |   Download PDF [6548 KB, uploaded 5 January 2015]   |  

Abstract

We propose a new approach to calculate the complex photonic band structure, both purely dispersive and evanescent Bloch modes of a finite range, of arbitrary three-dimensional photonic crystals. Our method, based on a well-established plane wave expansion and the weak form solution of Maxwell’s equations, computes the Fourier components of periodic structures composed of distinct homogeneous material domains from a triangulated mesh representation of the inter-material interfaces; this allows substantially more accurate representations of the geometry of complex photonic crystals than the conventional representation by a cubic voxel grid. Our method works for general two-phase composite materials, consisting of bi-anisotropic materials with tensor-valued dielectric and magnetic permittivities ε and μ and coupling matrices ς. We demonstrate for the Bragg mirror and a simple cubic crystal closely related to the Kelvin foam that relatively small numbers of Fourier components are sufficient to yield good convergence of the eigenvalues, making this method viable, despite its computational complexity. As an application, we use the single gyroid crystal to demonstrate that the consideration of both conventional and evanescent Bloch modes is necessary to predict the key features of the reflectance spectrum by analysis of the band structure, in particular for light incident along the cubic [111] direction. View Full-Text
Keywords: photonic crystals; band structure; evanescent modes; weak form solutions; single gyroid; reflectance; mesh representations; band gap material photonic crystals; band structure; evanescent modes; weak form solutions; single gyroid; reflectance; mesh representations; band gap material
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Saba, M.; Schröder-Turk, G.E. Bloch Modes and Evanescent Modes of Photonic Crystals: Weak Form Solutions Based on Accurate Interface Triangulation. Crystals 2015, 5, 14-44.

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