Hostname: page-component-7479d7b7d-fwgfc Total loading time: 0 Render date: 2024-07-12T19:28:38.200Z Has data issue: false hasContentIssue false
  • English
  • Français

Mathematical models for laser-plasma interaction

Published online by Cambridge University Press:  15 April 2005

Rémi Sentis
Rémi Sentis*
Affiliation:
CEA/Bruyeres, B.P. 12, 91680 Bruyères-le-Chatel, France. remi.sentis@cea.fr
Get access

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.

Keywords

Euler-Maxwell systemnumerical plasma simulationgeometrical opticsparaxial approximationSchrödinger equationthree-wave coupling systemBrillouin instability.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 2 , March 2005 , pp. 275 - 318
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amin, M.R., Capjack, C.E., Fricz, P., Rozmus, W. and Tikhonchuk, V.T., Two-dimensional studies of stimulated Brillouin scattering, filamentation. Phys. Fluids B 5 (1993) 3748–3764. CrossRef
Arnold, A. and Ehrhardt, M., Discrete transparent boundary conditions for wide angle parabolic equations. J. Comput. Phys. 145 (1998) 611–638. CrossRef
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).
Benamou, J.D., An introduction to Eulerian geometrical optics. J. Sci. Comp. 19 (2003) 63–95. CrossRef
Benamou, J.D., Castella, F., Katsaounis, T. and Perthame, B., High Frequency limit of the Helmholtz equations. Rev. Mat. Iberoamericana 18 (2002) 187–209. CrossRef
Benamou, J.D., Lafitte, O., Sentis, R. and Solliec, I., 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. CrossRef
Benamou, J.D., Lafitte, O., Sentis, R. and Solliec, I., 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. CrossRef
J.P. Berenger. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114 (1994) 185–200.
Berger, R.L., Still, C.H., Williams, E.A. and Langdon, A.B., On the dominant subdominant behavior of stimulated Raman and Brillouin scattering. Phys. Plasmas 5 (1998) 4337. CrossRef
Berger, R.L. et al., Theory and three-dimensional simulation of light filamentation. Phys. Fluids B 5 (1993) 2243. CrossRef
Besse, C., Mauser, N.J. and Stimming, H.P., Numerical study of the Davey-Stewartson System. ESAIM: M2AN 38 (2004) 1035–1054. CrossRef
Brezis, H., Golse, F. and Sentis, R., 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.
Castella, F., Perthame, B. and Runborg, O., High frequency limit of the Helmholtz equations, II. Source on a manifold. Comm. Partial Differential Equations 27 (2002) 607–651. CrossRef
F.F. Chen, Introduction to Plasmas Physics. Plenum, New York (1974).
Colin, M. and Colin, T., On a Quasilinear Zakharov system describing Laser-Plasma Interaction. Differential Integral Equations 17 (2004) 297–330.
M. Colin and T. Colin, Cauchy problem and numerical simulation for a quasi-linear Zakharov system. Accepted for publication in Nonlinear Analysis.
F. Collino, Perfectly matched absorbing layers for the paraxial equation. J. Comput. Phys. 131 (1997) 164–180.
A. Decoster, Fluid equations and transport coefficient of plasmas, in Modelling of collisions. P.-A. Raviart Ed., Masson, Paris (1997).
S. Desroziers, Modelisation de la propagation laser par résolution de l'équation d'Helmholtz, CEA internal report (2005).
Doumic, M., Golse, F. and Sentis, R., Propagation laser paraxiale en coordonnées obliques: équation d'advection-Schrödinger. Note C. R. Acad. Sci. Paris Sér. I 336 (2003) 23–28. CrossRef
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.
Dorr, M.R., Garaizar, F.X. and Hittinger, J.A., Simuation of laser-plasma filamentation. J. Comput. Phys. 17 (2002) 233–263. CrossRef
V.V. Eliseev, W. Rozmus, V.T. Tikhonchuk and C.E. Capjack, Phys. Plasmas 2 (1996) 2215 and Phys. Plasmas 3 (1996) 3754.
Feit, M.D. and Fleck, J.A., Beam nonparaxiality, filament formation. J. Opt. Soc. Amer. B 5 (1988) 633–640. CrossRef
F.G. Friedlander and J.B. Keller, Asymptotic expansion of solutions of (Δ + k²)u = 0 Comm. Pure Appl. Math. 5 (1955) 387.
Hüller, S., Mounaix, Ph., Tikhonchuk, V.T. and Pesme, D., Interaction of two neighboring laser beams. Phys. Plasmas 4 (1997) 2670–2680. CrossRef
J.D. Jackson, Classical Electrodynamics. Wiley, New York (1962).
H. Jourdren, HERA hydrodynamics AMR Plateform for multiphysics simulation, in Proc. of Chicago workshop on AMR methods (Sept. 2003). Springer Verlag, Berlin (2004).
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).
J.B. Keller and J.S. Papadakis, Eds., Wave Propagation and underwater Accoustics. Springer, Berlin. Lecture Notes in Phys. 70 (1977).
Y.A. Krastsov and Y.I. Orlov, Geometric optics for Inhomogeneous Media. Springer, Berlin (1990).
W.L. Kruer, The Physics of Laser-Plasma Interaction. Addison-Wesley, New York (1988).
Lee, D., Pierce, A.D., Shang, E.S., Parabolic equation development in the twentieth century. J. Comput. Acoust. 8 (2000) 527–637. CrossRef
P. Loiseau, O. Morice et al., Laser-beam smoothing induced by stimulated Brillouin scattering. CEA internal report (2005).
P. Mounaix, D. Pesme and M. Casanova, Nonlinear reflectivity of an inhomogeneous plasma. Phys. Rev. E 55 (1997) 4653–4664.
Papadakis, J.S., Taroudakis, M.I., Papadakis, P.J. and Mayfield, B., A new method for a realistic treatement of the sea bottom in parabolic approximation. J. Acoust. Soc. Amer. 92 (1992) 2030–2038. CrossRef
Papanicolaou, G.C., Sulem, C., Sulem, P.L. and Wang, X.P., Singular solutions of the Zaharov equations for Langmuir turbulence. Phys. Fluids B 3 (1991) 969–980. CrossRef
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).
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).
Riazuelo, G. and Bonnaud, G., Coherence properties of a smoothed laser beam in a hot plasma. Phys. Plasmas 7 (2000) 3841. CrossRef
Rose, H.A., Laser beam deflection. Phys. Plasmas 3 (1996) 1709–1727. CrossRef
Shao, et al., Spectral methods simulations of light scattering. IEEE J. Quantum Electronics 37 (2001) 617. CrossRef
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).
Symes, W.W. and Qian, J., A slowness matching eulerian method. J. Sci. Comput. 19 (2003) 501–526. CrossRef
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).
Zakharov, V.E., Collapse of Langmuir waves. Sov. Phys. JETP 35 (1972) 908.