Anderson localization in one-dimensional randomly disordered optical systems that are periodic on average

We compute the frequency dependence of the localization length in a one-dimensional randomly disordered optical system, which on average is periodic, by studying the dependence of the transmissivity on the length of a finite random sample. Specifically, we consider a layered system of dielectric slabs with electromagnetic waves propagating perpendicular to the interfaces and compute the localization length for frequencies of these waves in and around the neighborhood of the band gaps in the photonic band structure of the average periodic system. The localization length is found to be very small in the gaps and much larger in the bands. We also compute the dependence of the localization length in the presence of dissipation (complex dielectric constant) and obtain a simple relationship, for frequencies of the electromagnetic waves in the allowed bands, between the localization length in the nondissipative system, the decay length in the nonrandom periodic system with dissipative terms, and the localization length in the presence of dissipation. For frequencies in the gaps the localization length appears to be insensitive to the presence of dissipation.