Mathematical Models for Laser-plasma Interaction

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. Received: June 25, 2004. Introduction This paper has two goals. First, it is a review for readers involved in applied mathematics in order to describe some models used in laser-plasma interaction and to show the links between different models; we also try to give precisely the assumptions which allow to switch from one model to another. Secondly, we give some features aiming at understanding the mathematical properties of these models, we emphasize specially the boundary conditions for different models and we give some enlightments on the methods for solving them numerically. Of course, we only deal with very few models used in laser-plasma interaction which is a very intense research area in physics. For instance we do not consider any kinetic effects in the plasma and any phenomena related to ultra-high laser intensity or ultra-short laser pulse (we neither consider any relativity effect); electron population is always assumed to be at local thermodynamical equilibrium (the distribution function is always a Maxwell distribution). In this kind of problems, some typical lengths and some typical speeds occur which are related to different physical phenomena and which range over several orders of magnitude. Considering the spatial variable, four scales are relevant: i) the typical length Lpl of variation of the mean electron density of the plasma;