Anderson localization for discretely disordered metamaterials: polarization and off-axis incidence effects

Abstract

We consider polarization and off-axis incidence effects for one-dimensional random stacks consisting of alternating and non-alternating layers of positive and negative index materials, with index of refraction and thickness discretely disordered. Such long randomly disordered systems exhibit Anderson localization, whose effects can be studied via the Lyapunov exponent of the product of independent identically distributed random transfer matrices modeling the stack. We use Furstenberg’s integral formula to calculate Lyapunov exponents for s and p polarizations, and for a range of angles of incidence for these random matrix models. 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 algorithm of Froyland-Aihara, especially suited for discretely disordered parameters, to calculate the invariant measure. This algorithm produces 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. The Froyland-Aihara algorithm thus provides a non-Monte Carlo method to calculate localization effects, with potential reduction in computation time compared to traditional layer or vector iteration methods.