Ergodicity in wave-wave diffraction

Ray propagation over long distances through a sinusoidal volume grating, where the refractive index varies perpendicular to the direction of incidence, is characterized by a proliferation of caustics. These dominate the distribution of directions (momentum density) of emerging rays. For this integrable system with one-dimensional transverse motion, the ergodic average of the momentum density is calculated exactly. For the corresponding waves, …