Optical localization: using fixed point placement to simplify the invariant measure

Abstract

Optical localization in one-dimensional very long disordered photonic bandgap structures is characterized by the upper Lyapunov exponent of the infinite random matrix product model. This upper (and positive) Lyapunov exponent, or localization factor (inverse localization length), is, at least theoretically, calculable from Furstenberg's formula. This formula not only requires the probability density function for the random variables of the random transfer matrix, but also requires the invariant probability measure of the direction of the vector propagated by the long chain of random matrices. This invariant measure is generally only calculable numerically, and when the transfer matrices are in the usual real basis, or even a plane wave basis, the invariant measure takes no consistent form. However, by using a similarity transformation which moves the fixed points of the bilinear transformation corresponding to the long random matrix product, we can put the random matrix product in the so-called real hyperbolic-canonical basis. The aim of this change of basis is to place the attracting fixed point at the origin and the repelling fixed point at infinity. The invariant measure, as a result, is dramatically simplified, approaching a probability mass function with mass concentrated around the angle zero.