Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals

Abstract

In this paper, we extend the conventional plane wave expansion method in 3D anisotropic photonic crystal to be able to calculate the complex k even if permittivity and permeability are complex numbers or the functions of ω. There are some tricks in the derivation process, so we show the process in detail. Besides, we also provide an example for testing and explaining, and we also compare the results with the band structure derived from conventional plane wave expansion method, then we finally find that there is a good consistency between them.

Effect of loss on the dispersion relation of photonic and phononic crystals

“…Complex band structures can be found by searching for the wave number as a function of frequency, as demonstrated for photonic 17,18 and phononic [19][20][21] crystals. Whether the approach relies on the extended plane wave expansion (EPWE) or on FEM, a generalized eigenvalue problem of the form…”

Angular Band Gaps in Sonic Crystals: Evanescent Waves and Spatial Complex Dispersion Relation

“…We solve the inverted problem k(ω) and we compare both theoretically and experimentally the values of the scattering problem in finite structures. The eigenvalue problem is solved using the EPWE [42,43,46,51] and the scattering problem is solved using the multiple scattering theory [52,53]. We obtain complex isofrequency contours at angles in which angular BGs are predicted.…”

On the control of propagating acoustic waves in sonic crystals: analytical, numerical and optimization techniques

“…The complex band structures for phononic crystal were recently presented by Laude et al [Laude09] based on the work of Hsue et al [Hsue05]. In a similar way, the problem for the case of Sonic Crystal is extended in this Section, also showing the supercell approximation.…”

“…The description and the theoretical model have been richly complemented by very recent works devoted to the complex dispersion relation of SC [Sainidou05,Hsue05,Sainidou06,Laude09] Optimizing the scattering process in sonic crystals…”

“…Some recent techniques, like the extended plane wave expansion (EPWE) [Hsue05,Laude09,Romero10b], presented in Chapter 3, allow the analysis of the imaginary part of the band structures. Using EPWE method the relation k(ω) is obtained with k being probably complex and without any restriction of periodicity.…”