Anderson localization of classical waves in weakly scattering metamaterials

Abstract

We study the propagation and localization of classical waves in one-dimensional disordered structures composed of alternating layers of left- and right-handed materials (mixed stacks) and compare them with structures composed of different layers of the same material (homogeneous stacks). For weakly scattering layers, we have developed an effective analytical approach and have calculated the transmission length within a wide range of the input parameters. This enables us to describe, in a unified way, the localized and ballistic regimes as well as the crossover between them. When both refractive index and layer thickness of a mixed stack are random, the transmission length in the long-wave range of the localized regime exhibits a quadratic power wavelength dependence with different coefficients of proportionality for mixed and homogeneous stacks. Moreover, the transmission length of a mixed stack differs from the reciprocal of the Lyapunov exponent of the corresponding infinite stack. In both the ballistic regime of a mixed stack and in the near long-wave region of a homogeneous stack, the transmission length of a realization is a strongly fluctuating quantity. In the far long-wave part of the ballistic region, the homogeneous stack becomes effectively uniform and the transmission length fluctuations are weaker. The crossover region from the localization to the ballistic regime is relatively narrow for both mixed and homogeneous stacks. In mixed stacks with only refractive-index disorder, Anderson localization at long wavelengths is substantially suppressed, with the localization length growing with wavelength much faster than for homogeneous stacks. The crossover region becomes essentially wider and transmission resonances appear only in much longer stacks. The effects of absorption on one-dimensional transport and localization have also been studied, both analytically and numerically. Specifically, it is shown that the crossover region is particularly sensitive to losses, so that even small absorption noticeably suppresses frequency dependent oscillations in the transmission length. All theoretical predictions are in an excellent agreement with the results of numerical simulations.