On Anderson Localization of Quasi-Optical Wave Beams

We study analytically the statistical (Anderson) localization of the wave beam propagating in a structure of evenly placed waveguides with random parameters. We explore in detail the suppression of the diffraction broadening of the wave field in a randomly inhomogeneous medium in the continuum limit. A dispersion equation for the mean field is obtained from statistical averaging of the Helmholtz equation. The structure of the dispersion equation considerably differs from the parabolic dispersion law (typical of homogeneous media in the quasioptical approximation). The surface of the wave vectors defined by this equation (the refractive-index surface) has points where the curvature changes sign. In the theory of wave processes, this indicates that the diffraction is suppressed for the wave beams propagating in the so-called special directions in space, which corresponds to the Anderson localization. Evolution of such wave beams is considered on the basis of presenting the field as the Fourier integral of the spectrum of initial distribution over the transverse wave number. The asymptotic evaluation of the integral shows that the Anderson localization is described by the Airy beams. The transition region is studied numerically. Similar conclusions are also drawn in the case of a discrete problem.