Mathematical models for laser-plasma interaction

Abstract

Abstract. We address here mathematical models related to the Laser-Plasma Interaction. After a simplified introduction to the physical background concerning the modelling of the laser propagation and its interaction with a plasma, we recall some classical results about the geometrical optics in plasmas. Then we deal with the well known paraxial approximation of the solution of the Maxwell equation; we state a coupling model between the plasma hydrodynamics and the laser propagation. Lastly, we consider the coupling with the ion acoustic waves which has to be taken into account to model the so called Brillouin instability. Here, besides the macroscopic density and the velocity of the plasma, one has to handle the space-time envelope of the main laser wave, the space-time envelope of the stimulated Brillouin backscattered laser wave and the space envelope of the Brillouin ion acoustic waves. Numerical methods are also described to deal with the paraxial model and the three-wave coupling system related to the Brillouin instability.Mathematics Subject Classification. 35Q55, 35Q60, 65M06, 34E20, 82D10.

A multi-D model for Raman amplification

“…The aim of this paper is to construct (as rigorously as possible) such a quasilinear system. There exists other works related to propagation of beams in transverse directions see [9,13] for example. In [1], the authors elaborate a formalism for continuous spectrum related to Raman amplification.…”

On the Boyd-Kadomstev System for a Three-Wave Coupling Problem and its Asymptotic Limit

“…We recall in appendix a glance of the derivation of the models which are handled for such simulations (see also [20], [4], [5], [6] for mathematical justifications in the framework of the Brillouin instability or of the Raman instability and [7], [8], [14] for systematic and rigorous derivation of geometric optics models). For example in the HERA code (cf.…”

“…The derivation of the coupling model has been performed for a long time but it is quite tricky; for a recent physical introduction to this three-wave coupling modeling, see for example [2], [13], [19] (see also [20] for an mathematical introduction to this derivation). We only give the outlines of this derivation.…”

On the Boyd–Kadomtsev system for the three-wave coupling problem

“…Notons que pour traiter de façon réaliste l'instabilité Brillouin, il convient de se placer dans un cadre tri-dimensionnel et de rajouter au système précédent des termes pour prendre en compte la diffraction, la réfraction et les effets hydrodynamiques du plasma (cf. [2,6,8,4,9]). Cependant, les difficultés majeures tant sur le plan mathématique que numérique peuvent être vues sur le système précédent.…”