Localization and the invariant probability measure for photonic band gap structures

Abstract

Optical localization in a randomly disordered infinite length one-dimensional photonic band gap structure is studied using the transfer matrix formalism. Asymptotically, the infinite product of random matrices acting on a nonrandom input vector induces an invariant probability measure on the direction of the propagated vector. This invariant measure is numerically calculated for use in Furstenberg's master formula giving the upper Lyapunov exponent (localization factor) of the infinite random matrix product. A quarter-wave stack model with one of the bilayer thicknesses disordered is used for simulation purposes. In this plane wave model the invariant measure is rarely a uniform probability density function, as is sometimes assumed in the literature. Yet, the assumption of a uniform probability density function for the invariant measure gives surprisingly good results for a highly disordered system in the UV region.