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. vicinity of band edges or resonances, a fact that leads to the dramatic complication of a rigorous analysis (see review [9]).Another effect which needs specific analytical tools in its analysis is the influence of various correlations in a disorder. As is already understood, underlying correlations either of the short or long-range type can significantly suppress or enhance the localization length, therefore, strongly influence the global characteristics of the transmission [9]. In particular, by imposing specific long-range correlations one can practically create the devices with given transmission/reflection characteristics. This effect of enhancement or suppression of the Anderson localization was recently observed experimentally in waveguides with both the bulk [10] or surface [11] correlated disorder. It should be stressed that specific long-range correlations can emerge naturally in physical systems, not only in systems with intentionally included correlations; one such example will be discussed in this paper.New perspectives in creating devices with unusual transport characteristics are related to specific optic properties of metamaterials embedded in periodic structures [12][13][14][15][16][17][18][19][20][21][22]. A particular system of increasing interest is an array of two alternating a -and b -layers with equal optical path lengths, n a d a = |n b |d b . Here the a -layers are made of the right-handed (RH) material with positive refractive index n a > 0 and thickness d a . In contrast, the b -layers refer to the lefthanded (LH) material with negative index n b < 0 and thickness d b . In such a model the phase shift of the wave gained in the a -layer is fully compensated by the subsequent shift in the nex...

Faraday and Kerr Effects in Right and Left-Handed Films and Layered Materials

“…. (17) This is again a general expression valid for the arbitrary parameters L, n and Z. As expected, θ T 2 is even in n, and θ T 2 → 0 when L tends to zero.…”

“…This is unlike the well-known quadratic asymptotic behaviour ξ ∼ λ 2 for standard isotropic layers (see, e.g. [17]). It can be seen that the metamaterial configurations have an effect on the magneto-optical transport properties of the electromagnetic waves.…”

“…where θ T 2 is defined by Eq. (17). When L tends to zero (the thin film approximation), the above expression reduces to…”

Resonant enhancement of Anderson localization: Analytical approach

“…The features of optic wave localization in the non-dispersive array, where the refractive index is independent of the wave frequency, were analyzed in detail in Refs. [2,[14][15][16][17][18][19]. As one can recognize below, the effect of energy/frequency dispersion drastically changes the localization properties of both quantum and optic disordered systems.…”

“…This result is in a complete correspondence with those previously obtained for discrete optic systems with randomized refractive index (see, e.g., Refs. [14][15][16]18]). However, the flat distribution (3.3) may not be valid for the resonant values of ϕ, namely, for ϕ = 2πr/q with r, q integers.…”