The invariant measure for Anderson localized negative index metamaterials continuously disordered

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We consider one-dimensional photonic bandgap structures with negative index of refraction materials modeled in every layer, or in every other layer. When the index of refraction is randomized, and the number of layers becomes large, the light waves undergo Anderson localization, resulting in confinement of the transmitted energy. Such a photonic bandgap structure can be modeled by a long product of random transfer matrices, from which the (upper) Lyapunov exponent can be calculated to characterize the localization effect. Furstenberg’s theorem gives a precise formula to calculate the Lyapunov exponent when the random matrices, under general conditions, are independent and identically distributed. Specifically, Furstenberg’s integral formula can be used to calculate the Lyapunov exponent via integration with respect to the probability measure of the random matrices, and with respect to the so-called invariant probability measure of the direction of the vector propagated by the long chain of random matrices. It is this latter invariant probability 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 to numerically 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 made of alternating layers of positive and negative index materials, is compared to results for all negative index or equivalently all positive index structures.