Photonic Band Gaps in Periodic Dielectric Structures: Relation to the Single-Scatterer Mie Resonances

Abstract

A lot of theoretical and experimental work is being done in the area of propagation of classical waves in periodic and disordered structures. The interest in this subject has grown, particularly in the last several years, due to a variety of fundamental and practical reasons. The possibility of the observation[1] of Anderson localization of EM waves in disordered dielectric structures and frequency gaps in periodic structures, in analogy to the electron waves, is of fundamental interest. The very large number of potential practical applications[2] of such photonic band gaps, such as the enhanced performance of semiconductor lasers, has also spurred interest in this topic. Studies have been done using scalar waves[3–7], EM waves[8–10] and elastic waves[11]. The existence of band gaps and localized states have been reported in a variety of cases, particularly in periodic and disordered arrays of spherical scatterers. However, the relative importance of the roles of two differrent mechanisms, single scatterer resonances and macroscopic Bragg-like resonances, in the formation of gaps and localized states is still being debated. The resolution of this question is of interest for the following reason. Most theoretical treatments of the problem involve a lot of complicated calculations. In the plane wave expansion method[7–8] that we have used, a large number of plane waves have to be used to ensure accuracy necessitating the diagonalization of large matrices and the expending of a lot of computational effort.