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, the intensities of the diffracted beams are calculated from the Raman-Nath equation. In the limit when a semiclassical parameter is small, the long-distance average agrees very well with the ergodic ray momentum density. Semiclassical scalings of the fluctuations of beam intensities about the ergodic average are predicted for the caustic curves and their cusp points.