Skip to main content Accessibility help

Order statistics and extreme properties of spatially smoothed laser beams in laser-plasma interaction

  • S. Hüller (a1) and A. Porzio (a1) (a2)


The order statistics of intense speckles or “laser hot spots” are studied in the context of the so-called “optically smoothed” light beams of laser-matter interaction. We investigate theoretically and by means of numerical simulations the distribution function for the k-th most intense speckle maxima in the upper tail speckle distribution. From these distributions for each order k, a distribution function for the intense speckles as a function of their peak intensity can be established, which allows to compute their impact on nonlinear processes, like parametric instabilities. This is done for the example of stimulated Brillouin scattering, using the so-called independent hot spot model, for which the backscatter reactivity level is computed, which proves to be in very good agreements with numerical simulations. This result is of great interest for nonlinear processes, like instabilities, where extreme speckles play an important role.


Corresponding author

Address correspondence and reprint requests to: S. Hüller, Centre de Physique Théorique, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France. E-mail:


Hide All
Biancalana, V. & Chessa, P. (1994). Handling of quasi-Gaussian beams by phase plates: Far-field simulation. Appl. Opt. 33, 34653477.
Embrechts, P., Kluppelberg, C. & Mikosch, T. (1997). Modelling extremal events. New York: Springer.
Garnier, J. (1999). Statistical properties of the hot spots of smoothed beams produced by random phase plates revisited. Phys. Plasmas 6, 1601.
Goodman, J. (1985). Statistical Optics. New York: John Wiley & Sons.
Goodman, J. (1984). Statistical properties of laser speckle patterns. Laser Speckle and Related Phenomena. Vol. 9, 974, New York: Springer.
Hüller, S., Mounaix, Ph. & Tikhonchuk, V.T. (1998). SBS reflectivity from spatially smoothed laser beam: Random phase plates versus polarization smoothing. Phys. Plasmas 5, 2706.
Hüller, S. (2003). Interplay between parametric instabilities in fusion-relevant laser plasmas. Le Vide 307, 4460.
Hüller, S., Masson-Laborde, P.E., Pesme, D., Casanova, M., Detering, F. & Maximov, A. (2006). Harmonic decomposition to describe the nonlinear evolution of Stimulated Brillouin Scattering. Phys. Plasmas 13, 22703.
Kato, Y., Mima, K., Miyanaga, N., Arinaga, S., Kitagawa, Y., Nakatsuka, M. & Yamanaka, C. (1984). Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression. Phys. Rev. Lett. 53, 1057.
Kline, J.L. & Montgomery, D.S. (2005). Kinetic and fluid Langmuir wave nonlinearities driven by stimulated Raman scattering in a diffraction limited single-hot-spot. Laser and Particle Beams 23, 2731.
Leadbetter, M.R., Lindgren, G. & Rootzen, H. (1982). Extreme and Related Properties of Random Sequences and Processes. New York: Springer.
Leadbetter, M.R. (1983). Extreme and local dependence in stationary sequences. Z. Wahrscheinlichkeitstheorie verw. Gebiete 65, 291306.
Malka, V., Renard, N., Hüller, S., Pesme, D., Amiranoff, F., Baton, S.D., Modena, A., Mounaix, P., Rousseaux, C. & Salvati, M. (2000). Strong self-focusing and localization of SBS in quasi-stationary plasma. Phys. Plasmas 7, 42594265.
Masson-Laborde, P.E., Hüller, S., Pesme, D., Casanova, M. & Loiseau, P. (2006). Modeling parametric scattering instabilities in large-scale expanding plasmas. J. Phys. IV France 133, 247.
Middleton, D. (1960). An introduction to Statistical communications theory, New York: John Wiley & Sons.
Montgomery, D.S., Johnson, R.P., Cobble, J.A., Fernandez, J.C., Lindman, E.L., Rose, H.A. & Estabrook, K.G. (1999). Characterization of plasma and laser conditions for single hot spot experiments. Laser and Part. Beams 7, 349359.
Mounaix, Ph., Divol, L., Hüller, S. & Tikhonchuk, V.T. (2000). Effects of Spatial and Temporal Smoothing on Stimulated Brillouin Scattering in the Independent-Hot-Spot Model Limit. Phys. Rev. Lett. 85, 4526.
Myatt, J., Pesme, D., Hüller, S., Maximov, A.M., Rozmus, W. & Capjack, C.E. (2001). Nonlinear propagation of a randomized laser beam through an expanding plasma. Phys. Rev. Lett. 87, 5003.
Obenschain, S.P., Grun, J., Herbst, M.J., Kearney, K.J., Manka, C.K., McLean, E.A., Mostovych, A.N., Stamper, J.A., Whitlock, R.R., Bodner, S.E., Gardner, J.H. & Lehmberg, R.H. (1986). Laser-target interaction with induced spatial incoherence. Phys. Rev. Lett. 56, 28072810.
Ohtsubo, J. & Asakura, A. (1977). Statistical properties of laser speckle produced in the diffraction field. Applied Opt. 16, 1742.
Ohtsubo, J. & Asakura, A. (1977). Statistical properties of the sum of partially developed speckle patterns. Opt. Lett. 1, 98.
Pesme, D., Hüller, S., Myatt, J., Riconda, C., Maximov, A., Tikhonchuk, V.T., Labaune, C., Fuchs, J., Depierreux, S. & Baldis, H.A. (2002). Laser-plasma interaction studies in the context of megajoule lasers for inertial fusion. Plasma Phys. Contr. Fusion 44, B53.
Porzio, A. & Hüller, S. (2010). Extremal Properties of weakly correlated random variable arising in speckle patterns, J. Statist. Phys. 138, 10101044.
Rose, H.A. & Dubois, D.F. (1993). Statistical properties of hot spots produced by a random phase plate. Phys. Fluids B 5, 590596.
Rose, H.A. & DuBois, D.F. (1993). Initial development of ponderomotive filaments in plasma from intense hot spots produced by a random phase plate. Phys. Fluids B 5, 33373356.
Rose, H.A. & DuBois, D.F. (1994). Laser hot spots and the breakdown of linear instability theory with application to stimulated Brillouin scattering. Phys. Rev. Lett. 72, 2883.
Rose, H.A. (1997). Saturation of stimulated Brillouin scatter by self-consistent flow profile modification in laser hot spots Phys. Plasmas 4, 437442.
Rosenbluth, M.N. (1973). Parametric Instabilities in inhomogeneous media. Phys. Rev. Lett. 29, 565567.
Schmitt, A.J. & Afeyan, B.B. (1998). Time-dependent filamentation and stimulated Brillouin forward scattering in inertial confinement fusion plasmas. Phys. Plasmas 5, 503.
Tikhonchuk, V.T., Labaune, C. & Baldis, H.A. (1996). Modeling of a stimulated Brillouin scattering experiment with statistical distribution of speckles. Phys. Plasmas 3, 3777.
Tikhonchuk, V.T., Hüller, S., Mounaix, P. (1997). Effect of the speckle self-focusing on the stationary stimulated Brillouin scattering reflectivity from a randomized laser beam in an inhomogeneous plasma. Phys. Plasmas 4, 43694381.
Tikhonchuk, V.T., Fuchs, J., Labaune, C., Depierreux, S., Hüller, S., Myatt, J. & Baldis, H.A. (2001). Stimulated Brillouin and Raman scattering from a randomized laser beam in large inhomogeneous plasmas. II: Model description and comparison with experiments. Phys. Plasmas 8, 1636.
Weber, S., Riazuelo, G., Michel, P., Loubère, R., Walraet, F., Tikhonchuk, V.T., Malka, V., Ovadia, J. & Bonnaud, G. (2004). Modeling of laser-plasma interaction on hydrodynamic scales: Physics development and comparison with experiments. Laser Part. Beams 22, 189195.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed