INTERMEDIATE MODEL FOR SPATIAL EVOLUTION IN NONLINEAR OPTICS
Abstract
This paper follows the work of Colin–Gallice–Laurioux6 in which a new model generalizing the Schrödinger (NLS) model of the diffractive optics is derived for the laser propagation in nonlinear media. In particular, it provides good approximate solutions of the Maxwell–Lorentz system for highly oscillating initial data with broad spectrum. In real situations one is given boundary data. We propose to derive a similar evolution model but in the variable associated to the direction of propagation. However, since the space directions for the Maxwell equations are not hyperbolic, the boundary problem is ill-posed and one needs to apply a cutoff defined in the Fourier space, selecting those frequencies for which the operator is hyperbolic. The model we obtain is nearly L2 conservative on its domain of validity.
We then give a justification of the derivation. For this purpose we introduce a related well-posed initial boundary value problem. Finally, we perform numerical computations on the example of Maxwell with Kerr nonlinearity in some cases of short or spectrally chirped data where our model outperforms the Schrödinger one.
