There has recently been great interest in the problem of localization-delocalization that governs the anomalous transport of electrons or classical waves through one-dimensional (1D) systems with correlated disorder of different nature (see, e.g., review [1] and references therein). The fundamental significance of this problem is due to exciting results that revise a commonly accepted belief that any random long enough 1D structure exhibits the Anderson localization. From the viewpoint of applications, many of obtained results may have a strong impact for the creation of a new class of electron nanodevices, optic fibers, acoustic and electromagnetic waveguides and stratified media with selective transport properties. As is known [2], the Anderson localization is controlled by a single scaling parameter called localization length. For a weak correlated disorder, the inverse value of the localization length is proportional to randomness power spectrum, which is the Fourier transform of the binary correlator. Based on this fact, in early papers [3,4] devoted to the problem of anomalous transport, it was shown that any desired combination of transparent and reflecting frequency windows can be observed by an appropriate choice of random delta-like scatterers with specific long-range correlations. The location and width of the windows are found to be controlled by the form of binary correlator of a disorder. The experimental realization [5] of such delta-like scatterers in a single-mode waveguide has confirmed the theoretical predictions. Then, the main ideas of long-range correlations were extended to both single-mode [6] and multimode [7] waveguides (or quasi-1D electron wires) with surface scattering. It was demonstrated analytically and by direct numerical simulations that waveguides with a prescribed transparency can be fabricated by a proper design of randomly corrugated surface profiles. Also, one should mention the article [8] in which the guiding systems with a continuously distributed stratification were analyzed in connection with the correlated disorder. The study of quasi-1D structures with surface or stratified disorder has revealed a quite unexpected phenomenon of coexistence of localized and ballistic transport regimes, as well as an effect of perfect transparency for a subset of waveguide modes. In spite of a remarkable progress, one should emphasize that the problem remains open for correlated random structures that are periodic on average, in the case of finite width and height of barriers. Note that the majority of studies are due to numerical simulations, and by assuming rapidly decaying correlations [9]. Main analytical results are obtained either for systems with random uncorrelated elements of finite size (with a disorder of white-noise type) [10], or for the patterns with correlated disorder, however, with delta-like potential wells [3] or barriers [4]. The aim of this contribution is to obtain analytically the localization length for an array of periodic on average dielectric bi-layers of finite sizes, distributed according to correlated disorder potentials. To this end we employ the widely used transfer-matrix approach generalizing it to random, arbitrarily correlated, systems. Although we treat the simplest case when only the width of one constitutive layer is randomly deviated from its mean value, the method and the results can be directly extended to more complicated disordered structures.
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