Bloch dispersion and high frequency homogenization for separable doubly-periodic structures

Bloch waves are considered for a class of explicitly solvable two-dimensional periodic structures, as models of photonic structures; the class of structures chosen reduce to coupled one-dimensional problems. These provide benchmarks upon which asymptotic techniques can be tested and are also of interest in their own right. Two specific cases are considered: a generalized two-dimensional Mathieu-like equation and that of piecewise constant checkerboard media; the latter provides an ideal paradigm as the resulting dispersion relations are explicit and give virtually the only two-dimensional, non-trivial, dispersion relations for Bloch waves. The dispersion relations demonstrate many features of topical interest such as stop-bands and flat dispersion curves corresponding to slow light. Illustrative calculations show all-angle-negative refraction at higher frequencies than normal, so lensing and cloaking effects are obtained. The separable structures are used to illustrate the efficacy of homogenization theory, near the edges of the Brillouin zone, when the wavelength and microstructural lengthscales are of similar order. Conventional homogenization is widely assumed to be ineffective for modeling photonic crystals as it is limited to low frequencies when the wavelength is long relative to the microstructural lengthscales. Here high frequency homogenization theory, which is free of the conventional limitations, is used to generate effective partial differential equations on a macroscale, that have the microscale embedded within them through averaged quantities. (c) 2011 Elsevier B.V. All rights reserved.