Anderson localization effects via the invariant measure for discretely disordered negative and positive index structures

Abstract

For one-dimensional photonic bandgap structures consisting of alternating layers of positive and negative index materials, Anderson localization effects will appear when one or more parameters is disordered. Such long randomly disordered systems can be modeled via a long chain of independent identically distributed random matrices. The Lyapunov exponent of such a random matrix product characterizes the energy confinement due to Anderson localization. Furstenberg’s integral formula gives, at least theoretically, the Lyapunov exponent precisely. Furstenberg’s integral formula requires integration with respect to the probability distribution of the randomized layer parameters, and integration with respect to the so-called invariant probability measure of the direction of the vector propagated by the long chain of random matrices. This invariant measure can rarely be calculated analytically, so some numerical technique must be used to produce the invariant measure for a given random matrix product model. Here we use the Froyland-Aihara method to find the invariant measure. This method estimates the invariant measure from the left eigenvector of a certain sparse row-stochastic matrix. This sparse matrix represents the probabilities that a vector in one of a number of discrete directions will be transferred to another discrete direction via the random transfer matrix. This paper, possibly for the first time, presents the numerically calculated invariant measure for a discretely disordered one-dimensional photonic bandgap structure which includes negative index material in alternating layers. Results are compared with the structure containing all positive index layers, as well as with the counterpart structure in which random variables are drawn from a uniform probability density function.