In this paper, conceptual points regarding electron elastic (Kapitza–Dirac effect) and inelastic diffraction effects on the slowed electromagnetic wave-structures/light-gratings are considered. From the unified point of view it is analyzed the main works on this subject for last four decades in chronological order, pointing out the essential peculiarity inherent in induced Cherenkov, Compton, and undulator/wiggler processes too. This review paper has also purpose to resolve confusion in scientific literature connected with the recently appeared paper Hayrapetyan et al. in 2015 regarding electron diffraction effect on a traveling wave in a dielectric medium.

A second-order nonlinear frequency-domain model extending the linear complementary mild-slope equation (CMSE) is presented. The nonlinear model uses the same streamfunction formulation as the CMSE. This allows the vertical profile assumption to accurately satisfy the kinematic bottom boundary condition in the case of nonlinear triad interactions as well as for the linear refraction–diffraction part. The result is a model with higher accuracy of wave–bottom interactions including wave–wave interaction. The model's validity is confirmed by comparison with accurate numerical models, laboratory experiments over submerged obstacles and analytical perturbation solutions for class III Bragg resonance.