Wave Propagation in Periodic and Random Time-Dependent Media

We study wave propagation in materials in which the refractive index varies in time. We compare the situations in which the variations are periodic and random. If the modulation period (in the periodic case) or the coherence time (in the random case) of the refractive index is much smaller than the central wave period, then the two situations are analogous, and effective dispersion is the main effect. If the modulation period or coherence time is of the same order as the central wave period, then wavenumber-dependent amplification and dispersion are the main effects, but they have different characteristics: the amplification is very narrowband in the periodic case and broadband in the random case. In the random case, we get an analog of the O'Doherty--Anstey theory in randomly spatially varying media: the wave front experiences a random shift and a deterministic deformation, but contrary to the spatially varying case, the deterministic deformation induced by random time variations involves amplification and dispersion; moreover, the incoherent wave fluctuations grow faster than the coherent wave front, so that the former eventually dominate the latter.