A non-Monte Carlo approach to analyzing 1D Anderson localization in dispersive metamaterials

Abstract

Monte Carlo simulations have long been used to study Anderson localization in models of one-dimensional random stacks. Because such simulations use substantial computational resources and because the randomness of random number generators for such simulations has been called into question, a non-Monte Carlo approach is of interest. This paper uses a non-Monte Carlo methodology, limited to discrete random variables, to determine the Lyapunov exponent, or its reciprocal, known as the localization length, for a one-dimensional random stack model, proposed by Asatryan, et al., consisting of various combinations of negative, imaginary and positive index materials that include the effects of dispersion and absorption, as well as off-axis incidence and polarization effects. Dielectric permittivity and magnetic permeability are the two variables randomized in the models. In the paper, Furstenberg’s integral formula is used to calculate the Lyapunov exponent of an infinite product of random matrices modeling the one-dimensional stack. The integral formula requires integration with respect to the probability distribution of the randomized layer parameters, as well as integration with respect to the so-called invariant probability measure of the direction of the vector propagated by the long chain of random matrices. The non-Monte Carlo approach uses a numerical procedure of Froyland and Aihara which calculates the invariant measure as the left eigenvector of a certain sparse row-stochastic matrix, thus avoiding the use of any random number generator. The results show excellent agreement with the Monte Carlo generated simulations which make use of continuous random variables, while frequently providing reductions in computation time.