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

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 $n_b$. Main 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, $n_ad_a=|n_b| d_b$, with $d_a$, $d_b$ 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 with 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 $\omega$. In the limit $\omega \rightarrow 0$ one gets a quite specific dependence, $L^{-1}_{loc}\propto\sigma^4\omega^8$ 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 $n_a$, $n_b$ the thicknesses $d_a$, $d_b$ slightly fluctuate as well (positional disorder). We show how the conventional localization recovers with an addition of positional disorder.