Analyzing light localization using Iwasawa-canonical transfer matrices

Abstract

The transfer matrix formalism has proven to be a powerful tool for analyzing one-dimensional photonic bandgap structures, whether their multilayers are perfectly periodic or randomized in some fashion. In the randomized structure, as the number of layers tends to infinity, Furstenberg's formula can be used, at least theoretically, to find the deterministic Lyapunov exponent (localization factor, sometimes called the inverse localization length) governing the confinement of energy transmission in the model. The challenge in using Furstenberg's formula is that it requires the calculation of the invariant probability measure of the direction of the vector propagated by the chain of random matrices. This invariant measure is usually impossible to find analytically, and so one must resort to numerical simulation or some other approximating assumption. To aid in the numerical determination of this invariant probability measure, we consider matrix similarity transformations based on the average plane wave transfer matrix at a given frequency. These transformations, like the original transfer matrix, are elements of SU(1,1), the special pseudo-unitary group, and are obtained by moving the fixed points of the bilinear (or Möbius) transformation of the original transfer matrix to the corresponding fixed points of the canonical forms known from the Iwasawa decomposition. Amazingly, in some situations, including a quarter-wave stack, such a transformation can cause the invariant probability measure to become a nearly uniform probability density function, making the Furstenberg formula more readily useable.