Localization in a one-dimensional alloy with an arbitrary distribution of spacing between impurities: Application to L\'{e}vy glass

We have studied the localization of waves in a one-dimensional lattice consisting of impurities where the spacing between consecutive impurities can take certain values with given probabilities. In general, such a distribution of impurities induces correlations in the disorder. In particular with a power-law distribution of spacing, this system is used as a model for light propagation in L\'{e}vy glasses. We introduce a method of calculating the Lyapunov exponent which overcomes limitations in the previous studies and can be easily extended to higher orders of perturbation theory. We obtain the Lyapunov exponent up to fourth order of perturbation and discuss the range of validity of perturbation theory, transparent states, and anomalous energies which are characterized by divergences in different orders of the expansion. We also carry out numerical simulations which are in agreement with our analytical results.