Localization of Mono- and Multichannel Waves in Disordered Periodic Systems

Analytical solution is obtained for the localization factor, namely, the exponential spatial decay rate, in the propagation of a disturbance through a chain of nearly periodic cell units. The localization is attributable to the departure from perfect spatial periodicity in the chain, known as disorder. In the case where multiple waves are permissible in the chain, each wave is characterized by a spatial Lyapunov exponent. These Lyapunov exponents constitute a spectrum of plus-minus pairs, and localization is governed by the largest negative Lyapunov exponent, which characterizes the least attenuated wave. This is then obtained by invoking Oseledec's multiplicative ergodic theorem, assuming that random disorders in different cell units are independent and identically distributed.