Mathematical models for laser-plasma interaction
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) no. 2, pp. 275-318.

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.

DOI : https://doi.org/10.1051/m2an:2005014
Classification : 35Q55,  35Q60,  65M06,  34E20,  82D10
Mots clés : Euler-Maxwell system, numerical plasma simulation, geometrical optics, paraxial approximation, Schrödinger equation, three-wave coupling system, Brillouin instability
@article{M2AN_2005__39_2_275_0,
author = {Sentis, R\'emi},
title = {Mathematical models for laser-plasma interaction},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {275--318},
publisher = {EDP-Sciences},
volume = {39},
number = {2},
year = {2005},
doi = {10.1051/m2an:2005014},
zbl = {1080.35157},
mrnumber = {2143950},
language = {en},
url = {http://archive.numdam.org/item/M2AN_2005__39_2_275_0/}
}
Sentis, Rémi. Mathematical models for laser-plasma interaction. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) no. 2, pp. 275-318. doi : 10.1051/m2an:2005014. http://archive.numdam.org/item/M2AN_2005__39_2_275_0/

[1] M.R. Amin, C.E. Capjack, P. Fricz, W. Rozmus and V.T. Tikhonchuk, Two-dimensional studies of stimulated Brillouin scattering, filamentation. Phys. Fluids B 5 (1993) 3748-3764.

[2] A. Arnold and M. Ehrhardt, Discrete transparent boundary conditions for wide angle parabolic equations. J. Comput. Phys. 145 (1998) 611-638. | Zbl 0915.76081

[3] P. Ballereau, M. Casanova, F. Duboc, D. Dureau, H. Jourdren, P. Loiseau, J. Metral, O. Morice and R. Sentis, Coupling hydrodynamics with a paraxial solver for laser propagation. CEA internal report (2005).

[4] J.D. Benamou, An introduction to Eulerian geometrical optics. J. Sci. Comp. 19 (2003) 63-95. | Zbl 1042.78001

[5] J.D. Benamou, F. Castella, T. Katsaounis and B. Perthame, High Frequency limit of the Helmholtz equations. Rev. Mat. Iberoamericana 18 (2002) 187-209. | Zbl 1090.35165

[6] J.D. Benamou, O. Lafitte, R. Sentis and I. Solliec, A geometrical optics based numerical method for high frequency electromagnetic fields computations near fold caustics (part I). J. Comput. Appl. Math. 156 (2003) 93-125. | Zbl 1027.78011

[7] J.D. Benamou, O. Lafitte, R. Sentis and I. Solliec, A geometrical optics based numerical method for high frequency electromagnetic fields computations near fold caustics (part II, the Energy). J. Comput. Appl. Math. 167 (2004) 91-134. | Zbl 1054.78003

[8] J.P. Berenger. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114 (1994) 185-200. | Zbl 0814.65129

[9] R.L. Berger, C.H. Still, E.A. Williams and A.B. Langdon, On the dominant subdominant behavior of stimulated Raman and Brillouin scattering. Phys. Plasmas 5 (1998) 4337.

[10] R.L. Berger et al., Theory and three-dimensional simulation of light filamentation. Phys. Fluids B 5 (1993) 2243.

[11] C. Besse, N.J. Mauser and H.P. Stimming, Numerical study of the Davey-Stewartson System. ESAIM: M2AN 38 (2004) 1035-1054. | Numdam | Zbl 1080.65095

[12] H. Brezis, F. Golse and R. Sentis, Analyse asymptotique de l'équation de Poisson couplée à la relation de Boltzmann. Quasi-neutralité dans les plasmas. Note C. R. Acad. Sci. Paris Sér. I 321 (1995) 953-959. | Zbl 0839.76096

[13] F. Castella, B. Perthame and O. Runborg, High frequency limit of the Helmholtz equations, II. Source on a manifold. Comm. Partial Differential Equations 27 (2002) 607-651.

[14] F.F. Chen, Introduction to Plasmas Physics. Plenum, New York (1974).

[15] M. Colin and T. Colin, On a Quasilinear Zakharov system describing Laser-Plasma Interaction. Differential Integral Equations 17 (2004) 297-330.

[16] M. Colin and T. Colin, Cauchy problem and numerical simulation for a quasi-linear Zakharov system. Accepted for publication in Nonlinear Analysis.

[17] F. Collino, Perfectly matched absorbing layers for the paraxial equation. J. Comput. Phys. 131 (1997) 164-180. | Zbl 0866.73013

[18] A. Decoster, Fluid equations and transport coefficient of plasmas, in Modelling of collisions. P.-A. Raviart Ed., Masson, Paris (1997).

[19] S. Desroziers, Modelisation de la propagation laser par résolution de l'équation d'Helmholtz, CEA internal report (2005).

[20] M. Doumic, F. Golse and R. Sentis, Propagation laser paraxiale en coordonnées obliques: équation d'advection-Schrödinger. Note C. R. Acad. Sci. Paris Sér. I 336 (2003) 23-28. | Zbl 1038.35132

[21] M. Doumic, F. Duboc, F. Golse and R. Sentis, Numerical simulation for paraxial model of light propagation in a tilted frame: the advection-Schrödinger equation. CEA internal report (2005), preprint.

[22] M.R. Dorr, F.X. Garaizar and J.A. Hittinger, Simuation of laser-plasma filamentation. J. Comput. Phys. 17 (2002) 233-263. | Zbl 1045.76024

[23] V.V. Eliseev, W. Rozmus, V.T. Tikhonchuk and C.E. Capjack, Phys. Plasmas 2 (1996) 2215 and Phys. Plasmas 3 (1996) 3754.

[24] M.D. Feit and J.A. Fleck, Beam nonparaxiality, filament formation. J. Opt. Soc. Amer. B 5 (1988) 633-640.

[25] F.G. Friedlander and J.B. Keller, Asymptotic expansion of solutions of $\left(\Delta +{k}^{2}\right)u=0.$ Comm. Pure Appl. Math. 5 (1955) 387. | MR 70833 | Zbl 0064.34902

[26] S. Hüller, Ph. Mounaix, V.T. Tikhonchuk and D. Pesme, Interaction of two neighboring laser beams. Phys. Plasmas 4 (1997) 2670-2680.

[27] J.D. Jackson, Classical Electrodynamics. Wiley, New York (1962). | MR 436782 | Zbl 0997.78500

[28] H. Jourdren, HERA hydrodynamics AMR Plateform for multiphysics simulation, in Proc. of Chicago workshop on AMR methods (Sept. 2003). Springer Verlag, Berlin (2004).

[29] J.B. Keller and R.M. Lewis, Asymptotic Methods for P.D.E: The reduced Wave Equation. Research report Courant Inst. (1964); reprinted in Surveys Appl. Math. 1, J.B. Keller, W. McLaughlin, G.C. Papanicolaou, Eds. Plenum, New York (1995). | MR 1366207

[30] J.B. Keller and J.S. Papadakis, Eds., Wave Propagation and underwater Accoustics. Springer, Berlin. Lecture Notes in Phys. 70 (1977). | MR 464871 | Zbl 0399.76079

[31] Y.A. Krastsov and Y.I. Orlov, Geometric optics for Inhomogeneous Media. Springer, Berlin (1990). | MR 1113261

[32] W.L. Kruer, The Physics of Laser-Plasma Interaction. Addison-Wesley, New York (1988).

[33] D. Lee, A.D. Pierce, E.S. Shang, Parabolic equation development in the twentieth century. J. Comput. Acoust. 8 (2000) 527-637.

[34] P. Loiseau, O. Morice et al., Laser-beam smoothing induced by stimulated Brillouin scattering. CEA internal report (2005).

[35] P. Mounaix, D. Pesme and M. Casanova, Nonlinear reflectivity of an inhomogeneous plasma. Phys. Rev. E 55 (1997) 4653-4664.

[36] J.S. Papadakis, M.I. Taroudakis, P.J. Papadakis and B. Mayfield, A new method for a realistic treatement of the sea bottom in parabolic approximation. J. Acoust. Soc. Amer. 92 (1992) 2030-2038.

[37] G.C. Papanicolaou, C. Sulem, P.L. Sulem and X.P. Wang, Singular solutions of the Zaharov equations for Langmuir turbulence. Phys. Fluids B 3 (1991) 969-980.

[38] D. Pesme, Interaction collisionnelle et collective (Chap. 2) in La fusion par Confinement Inertiel I. Interaction laser-matière. R. Dautray-Watteau Ed., Eyrolles, Paris (1995).

[39] D. Pesme et al., Fluid-type Effects in the nonlinear Stimulated Brillouin Scatter, in Laser-Plasma Interaction Workshop at Wente, L. Divol Ed., Lawrence Livermore Nat. Lab. report UCRL-JC-148983 (2002).

[40] G. Riazuelo and G. Bonnaud, Coherence properties of a smoothed laser beam in a hot plasma. Phys. Plasmas 7 (2000) 3841.

[41] H.A. Rose, Laser beam deflection. Phys. Plasmas 3 (1996) 1709-1727.

[42] Shao et al., Spectral methods simulations of light scattering. IEEE J. Quantum Electronics 37 (2001) 617.

[43] G. Schurtz, Les codes numériques en FCI (Chap. 13), in La fusion par Confinement Inertiel, III. Techniques exp. et numériques, R. Dautray-Watteau Ed., Eyrolles, Paris (1995).

[44] W.W. Symes and J. Qian, A slowness matching eulerian method. J. Sci. Comput. 19 (2003) 501-526. | Zbl 1035.78017

[45] F.D. Tappert, The parabolic equation approximation method, in Wave Propagation and underwater Accoustics, J.B. Keller and J.S. Papadakis Eds., Springer, Berlin. Lecture Notes in Phys. 70 (1977). | MR 475274

[46] V.E. Zakharov, Collapse of Langmuir waves. Sov. Phys. JETP 35 (1972) 908.