Ulam's method to estimate invariant measures and Lyapunov exponents for one-dimensional discretely randomized photonic structures
Glen J. Kissel · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 2009
In the one-dimensional optical analog to Anderson localization, a periodically layered medium has one or more parameters randomly disordered. Such a medium can be modeled by an infinite product of 2x2 random transfer matrices with the upper Lyapunov exponent of the matrix product identified as the localization factor (inverse localization length). Furstenberg's integral formula for the Lyapunov exponent requires integration with respect to both the probability measure of the random matrices and the invariant probability measure of the direction of the vector propagated by the random matrix product. This invariant measure is difficult to find analytically, so one of several numerical techniques must be used in its calculation. Here, we focus on one of those techniques, Ulam's method, which sets up a sparse matrix of the probabilities that an entire interval of possible directions will be transferred to some other interval of directions. The left eigenvector of this sparse matrix forms the estimated invariant measure. While Ulam's method is shown to produce results as accurate as others, it suffers from long computation times. The Ulam method, along with other approaches, is demonstrated on a random Fibonacci sequence having a known answer, and on a quarter-wave stack model with discrete disorder in layer thickness.