Non-conventional Anderson localization in a matched quarter stack with metamaterials

Abstract

We study the problem of non-conventional Anderson localization emerging in bilayer periodic-on-average structures with alternating layers of materials, with positive and negative refraction indices n _a and n _b . Attention is paid to the model of the so-called quarter stack with perfectly matched layers (the same unperturbed by disorder impedances, Z _a = Z _b , and optical path lengths, n _a d _a = | n _b | d _b , with d _a and d _b being the thicknesses of basic layers). As was recently numerically discovered, in such structures with weak fluctuations of refractive indices (compositional disorder), the localization length L _loc is enormously large in comparison to the conventional localization occurring in the structures with positive refraction indices only. In this paper we develop a new approach, which allows us to derive the expression for L _loc for weak disorder and any wave frequency ω . In the limit ω → 0 one gets a quite specific dependence, L ^−1 _loc ∝ σ ^4 ω ^8 , which is obtained within the fourth order of perturbation theory. We also analyze the interplay between two types of disorder, when in addition to the fluctuations of n _a and n _b , the thicknesses d _a and d _b slightly fluctuate as well (positional disorder). We show how conventional localization recovers with the addition of positional disorder.