Nonlinear diffractive optics with curved phases: beam dispersion and transition between light and shadow

Abstract

We give asymptotic descriptions of smooth oscillating solutions of hyperbolic systems with variable coefficients, in the weakly nonlinear diffractive optics regime. The dependence of the coefficients of the system in the space–time variable (corresponding to propagation in a non‐homogeneous medium) implies that the rays are not parallel lines – the same occurs with non‐planar initial phases. Approximations are given by WKB asymptotics with 3‐scales profiles and curved phases. The fastest scale concerns oscillations, while the slowest one describes the modulation of the envelope, which is along rays for the oscillatory components. We consider two kinds of behaviors at the intermediate scale: ‘weakly decaying’ (Sobolev), giving the transverse evolution of a ‘ray packet’, and ‘shock‐type’ profiles describing a region of rapid transition for the amplitude.