A general framework for diffractive optics and its applications to lasers with large spectrums and short pulses

The aim of this article is to generalize the usual tools of diffractive optics in order to allow the study of phenomena which are out of their range. This generalization relies on the algebra of oscillations with a continuous oscillatory spectrum, which is wider than the usual spaces of periodic or almost-periodic functions. We perform the analysis for general nonlinear hyperbolic systems, both in the dispersive and in the nondispersive cases, and particularly focus on the behavior of the nonlinearities. Our tools yield considerable simplifications in these nonlinearities, which allows us to point out qualitative differences between the dispersive and the nondispersive cases. Finally, we study in detail two physical examples which can be modeled with the present tools: lasers with large spectrums, and those with ultrashort pulses.