The invariant measure for Anderson localized negative index metamaterials continuously disordered

摘要

We consider one-dimensional photonic bandgap structures with negative index of refraction materials modeled in everylayer, or in every other layer. When the index of refraction is randomized, and the number of layers becomes large, thelight waves undergo Anderson localization, resulting in confinement of the transmitted energy. Such a photonicbandgap structure can be modeled by a long product of random transfer matrices, from which the (upper) Lyapunovexponent can be calculated to characterize the localization effect. Furstenberg’s theorem gives a precise formula tocalculate the Lyapunov exponent when the random matrices, under general conditions, are independent and identicallydistributed. Specifically, Furstenberg’s integral formula can be used to calculate the Lyapunov exponent via integrationwith respect to the probability measure of the random matrices, and with respect to the so-called invariant probabilitymeasure of the direction of the vector propagated by the long chain of random matrices. It is this latter invariantprobability measure, so fundamental to Furstenberg’s theorem, which is generally impossible to determine analytically.Here we use a bin counting technique with Monte Carlo chosen random parameters from a continuous distribution tonumerically estimate the invariant measure and then calculate Lyapunov exponents from Furstenberg’s integral formula.This result, one of the first times an invariant measure has been calculated for a continuously disordered structure madeof alternating layers of positive and negative index materials, is compared to results for all negative index or equivalentlyall positive index structures.© (2012) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.