Anderson localization of classical waves in weakly scattering one-dimensional Levy lattices

Abstract

Anderson localization of classical waves in weakly scattering one-dimensional Levy lattices is studied analytically and numerically. The disordered medium is composed of layers with alternating refractive indices and with thickness disorder distributed according to the Pareto distribution $\ensuremath{\sim}1/{x}^{(\ensuremath{\alpha}+1)}$. In Levy lattices the variance (or both variance and mean) of a random parameter does not exist, which leads to a different functional form for the localization length. In this study an equation for the localization length is obtained, and it is found to be in excellent agreement with the numerical calculations throughout the spectrum. The explicit asymptotic equations for the localization lengths for both short and long wavelengths have been deduced. It is shown that the localization length tends to a constant at short wavelengths and it is determined by the layer interface Fresnel coefficient. At the long wavelengths the localization length is proportional to the power of the wavelength $\ensuremath{\ell}\ensuremath{\sim}{\ensuremath{\lambda}}^{\ensuremath{\alpha}}$ for $1<\ensuremath{\alpha}<2$, and it has a transcendental behavior $\ensuremath{\ell}\ensuremath{\sim}{\ensuremath{\lambda}}^{2}/ln\ensuremath{\lambda}$ for $\ensuremath{\alpha}=2$. For $\ensuremath{\alpha}>2$, where the variance of the random distribution exists, the localization length attains its classical long-wavelength asymptotic form $\ensuremath{\ell}\ensuremath{\sim}{\ensuremath{\lambda}}^{2}$.