The effect of disorder on the wave propagation in one-dimensional periodic optical systems

The influence of disorder on the transmission through periodic waveguides is studied. Using a canonical form of the transfer matrix we investigate dependence of the Lyapunov exponent $\gamma$ on the frequency $\nu$ and magnitude of the disorder $\sigma$. It is shown that in the bulk of the bands $\gamma \sim \sigma^2$, while near the band edges it has the order $\gamma \sim \sigma^{2/3}$. This dependence is illustrated by numerical simulations.