Non-Rayleigh distribution of reflected speckle intensities from localized states inside the gap of disordered photonic crystals

The field distributions of reflected speckles arising from localized states inside the gap of disordered photonic crystals in two dimensions were studied through numerical simulations using the multiple-scattering method. The statistics of the Lyapunov exponent of the transmitted waves were also studied. Similar to the case of disordered photonic crystals in one dimension, two types of localized states were found depending on the degree of disorder and the frequency inside the gap. Our simulation results indicate that the reflection statistics depend on whether or not the localized states are of the normal type. They also depend on whether the reflected angles are in the Bragg direction or not. For the non-Bragg angles, we found that the intensity distribution arising from the normal-type localized states follows Rayleigh statistics, in agreement with random matrix theory. However, deviations from Rayleigh statistics were found for second type of localized states. The crossover behavior from non-Rayleigh to Rayleigh statistics was studied as a function of the degree of disorder. By separating the field into coherent and diffuse parts, we have studied the statistics of field and phase distributions for both diffuse and total fields as well as their speckle contrasts. It is found that the crossover behavior is very similar to behavior in ballistic to diffusive wave propagation for the transmitted waves and can be described by the random-phasor-sum model (RPS). For the Bragg angle, non-Rayleigh statistics were found for both kinds of localized states. The statisics are sensitive to the degree of disorder. It is found that both the RPS and K distribution have limited ranges of validity in this case.