Self-focusing of cosh-Gaussian laser beam and its effect on the excitation of ion-acoustic wave and stimulated Brillouin backscattering in collisionless plasma

  • Gunjan PurohitEmail author
  • Bineet Gaur


An analytical and numerical study has been carried out for self-focusing of an intense cosh-Gaussian laser beam in collisionless plasma and its impact on the excitation of ion-acoustic wave and stimulated Brillouin backscattering process. The analytical model has been developed under Wentzel–Kramers–Brillouin and paraxial ray approximations. The nonlinearities of ponderomotive force on electron and the relativistic oscillation of the electron mass have been used in this study. The nonlinear differential equations have been set up for the beam width parameters of the main beam, ion-acoustic wave, backscattered wave and back reflectivity of stimulated Brillouin scattering (SBS). These equations have been solved numerically for different values of decentred parameter (b), relative plasma density (ωp0/ω0) and incident laser intensity (a). The results have been compared with only relativistic nonlinearity and Gaussian profile of laser beam. The focusing of laser beam, ion-acoustic wave and scattered wave are found to be strong under relativistic-ponderomotive regime compared to only relativistic regime. Further, it is observed that focusing/intensity of main laser beam, ion acoustic wave and SBS back reflectivity increases with increasing the values of b and ωp0/ω0. Itis also found that back reflectivity of SBS process gets suppressed with the increase in the value of a. This study may be useful in laser induced fusion scheme where back scattering of SBS plays very important role.


Self-focusing Relativistic-ponderomotive nonlinearity Collisionless plasma Cosh-Gaussian laser beam Ion-acoustic wave Stimulated Brillouin scattering 



The authors are thankful to the Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India for providing financial assistance for carrying out this research work vide project file No. EMR/2016/000112.


  1. Akhmanov, S.A., Sukhorukov, A.P., Khokhlov, R.V.: Self-focusing and diffraction of light in a nonlinear medium. Sov. Phys. Usp. 10(5), 609–636 (1968)ADSGoogle Scholar
  2. Albright, B.J., Yin, L., Bowers, K.J., Bergen, B.: Multi-dimensional dynamics of stimulated Brillouin scattering in a laser speckle: ion acoustic wave bowing, breakup, and laser-seeded two-ion-wave decay. Phys. Plasmas 23(3), 032703 (2016)ADSGoogle Scholar
  3. Aleksandrov, V.V., Brenner, M.V., Koval’skii, N.G., Loburev, S.V., Rubenchik, A.M.: Brillouin scattering in a laser plasma at moderate intensities 1012-1014 W/cm2. Sov. Phys. JETP 61(3), 459–463 (1985)Google Scholar
  4. Al-Rashed, A.A.R., Saleh, B.E.A.: Decentered Gaussian beams. Appl. Opt. 34(30), 6819–6825 (1995)ADSGoogle Scholar
  5. Amin, M.R., Capjack, C.E., Frycz, P., Rozmus, W., Tikhonchuk, V.T.: Two-dimensional studies of stimulated Brillouin scattering, filamentation, and self-focusing instabilities of laser light in plasmas. Phys. Fluids B 5(10), 3748–3764 (1993)ADSGoogle Scholar
  6. Baldis, H.A., Villeneuve, D.M., La Fontaine, B., Enright, G.D., Labaune, C., Baton, S., Mounaix, P., Pesme, D., Casanova, M., Rozmus, W.: Stimulated Brillouin scattering in picosecond time scales: experiments and modelling. Phys. Fluids B 5(9), 3319–3327 (1993)ADSGoogle Scholar
  7. Baton, S.D., Amiranoff, F., Malka, V., Modena, A., Salvati, M., Coulaud, C., Rousseaux, C., Renard, N., Mounaix, P.H., Stenz, C.: Measurement of the stimulated Brillouin scattering reflectivity from a spatially smoothed laser beam in a homogeneous large-scale plasma. Phys. Rev. E 57(5), R4895 (1998)ADSGoogle Scholar
  8. Borisov, A.M., Borovskiy, A.V., Shiryaev, O.B., Korobkin, V.V., Prokhorov, A.M.: Relativistic and charge-displacement self-channeling of intense ultrashort laser pulses in plasmas. Phys. Rev. A 45(8), 5830 (1992)ADSGoogle Scholar
  9. Brandi, H.S., Manus, C., Mainfray, G.: Relativistic self-focusing of ultraintense laser pulses in inhomogeneous underdense plasmas. Phys. Rev. E 47(5), 3780 (1993a)ADSGoogle Scholar
  10. Brandi, H.S., Manus, C., Mainfray, G.: Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. I. Paraxial approximation. Phys. Fluids B 5(10), 3539–3550 (1993b)ADSGoogle Scholar
  11. Casperson, L.W., Hall, D.G., Tovar, A.A.: Sinusoidal-Gaussian beams in complex optical systems. J. Opt. Soc. Am. A 14(12), 3341–3348 (1997)ADSMathSciNetGoogle Scholar
  12. Chirokikh, A., Seka, W., Simon, A., Craxton, R.S., Tikhonchuk, V.T.: Stimulated Brillouin scattering in long-scale-length laser plasmas. Phys. Plasmas 5(4), 1104–1109 (1998)ADSGoogle Scholar
  13. Eliseev, V.V., Rozmus, W., Tikhonchuk, V.T., Capjack, C.E.: Stimulated Brillouin scattering and ponderomotive self-focusing from a single laser hot spot. Phys. Plasmas 2(5), 1712–1724 (1995)ADSGoogle Scholar
  14. Fuchs, J., Labuane, C., Depierreux, D., Baldis, H.A., Michard, A., James, G.: Experimental evidence of plasma-induced incoherence of an intense laser beam propagating in an underdense plasma. Phys. Rev. Lett. 86(3), 432–435 (2001)ADSGoogle Scholar
  15. Gao, W., Lu, Z.W., Wang, S.Y., He, W.M., Hasi, W.L.J.: Measurement of stimulated Brillouin scattering threshold by the optical limiting of pump output energy. Laser Part. Beams 28(1), 179–184 (2010)ADSGoogle Scholar
  16. Gauniyal, R., Ahmad, N., Rawat, P., Gaur, B., Mahmoud, S.T., Purohit, G.: Stimulated Brillouin backscattering of hollow Gaussian laser beam in collisionless plasma under relativistic–ponderomotive regime. Laser Part. Beams 35(1), 81–91 (2017)ADSGoogle Scholar
  17. Gill, T.S., Mahajan, R., Kaur, R.: Self-focusing of cosh-Gaussian laser beam in a plasma with weakly relativistic and ponderomotive regime. Phys. Plasmas 18(3), 033110 (2011)ADSGoogle Scholar
  18. Giulietti, A., Macchi, A., Schifano, E., Biancalana, V., Danson, C., Giulietti, D., Gizzi, L.A., Willi, O.: Stimulated Brillouin backscattering from underdense expanding plasmas in a regime of strong filamentation. Phys. Rev. E 59(1), 1038–1046 (1999)ADSGoogle Scholar
  19. Gupta, N., Singh, A.: Dynamics of quadruple laser beams in collisionless plasmas. Waves Random Complex Media 28, 1–18 (2017)Google Scholar
  20. Habibi, M., Ghamari, F.: Significant enhancement in self-focusing of high-power laser beam through dense plasmas by ramp density profile. J. Opt. Soc. Am. B 32(7), 1429–1434 (2015)ADSGoogle Scholar
  21. Huller, S., Masson-Laborde, P.E., Pesme, D., Labaune, C., Bandulet, H.: Modeling of stimulated Brillouin scattering in expanding plasmas. J. Phys. Conf. Ser. 112(2), 022031 (2008)Google Scholar
  22. Kaw, P.K., Schmidt, G., Wilcox, T.: Filamentation and trapping of electromagnetic radiation in plasmas. Phys. Fluids 16(9), 1522–1525 (1973)ADSGoogle Scholar
  23. Konar, S., Mishra, M., Jana, S.: Nonlinear evolution of cosh-Gaussian laser beams and generation of flat top spatial solitons in cubic quintic nonlinear media. Phys. Lett. A 362(5–6), 505–510 (2007)ADSGoogle Scholar
  24. Krall, N.A., Trivelpicec, A.W.: Principle of Plasma Physics. McGraw Hill-Kogakusha, Tokyo (1973)Google Scholar
  25. Kruer, W.L.: The Physics of Laser Plasma Interactions. Addison-Wesley, Redwood City (1988)Google Scholar
  26. Labaune, C., Fabre, E., Michard, A., Briand, F.: Evidence of stimulated Brillouin backscattering from a plasma at short laser wavelengths. Phys. Rev. A 32(1), 577 (1985)ADSGoogle Scholar
  27. Labaune, C., Rozmus, W., Baldis, H.A., Mounaix, P., Pesme, D., Baton, S.D., Fontaine, B.L., Villeneuve, D.M., Enright, G.D.: Proceedings of SPIE 1413, Short-Pulse High-Intensity Lasers and Applications (1991)Google Scholar
  28. Labaune, C., Baldis, H.A., Schifano, E., Bauer, B.S., Michard, A., Renard, N., Seka, W., Moody, J.D., Estabrook, K.G.: Location of ion-acoustic waves from back and side stimulated Brillouin scattering. Phys. Rev. Lett. 76(20), 3727 (1996)ADSGoogle Scholar
  29. Labaune, C., Baldis, H.A., Renard, N., Schifano, E., Michard, A.: Interplay between ion acoustic waves and electron plasma waves associated with stimulated Brillouin and Raman scattering. Phys. Plasmas 4(2), 423–427 (1997)ADSGoogle Scholar
  30. Lu, B., Luo, S.: Beam propagation factor of hard-edge diffracted cosh-Gaussian beams. Opt. Commun. 178(4–6), 275–281 (2000)ADSGoogle Scholar
  31. Lu, B., Ma, H., Zhang, B.: Propagation properties of cosh-Gaussian beams. Opt. Commun. 164(4–6), 165–170 (1999)ADSGoogle Scholar
  32. Mahmoud, S.T., Sharma, R.P., Kumar, A., Yadav, S.: Effect of pump depletion and self-focusing on stimulated Brillouin scattering process in laser-plasma interactions. Phys. Plasmas 6(3), 927–931 (1999)ADSGoogle Scholar
  33. Masson-Laborde, P.E., Hüller, S., Pesme, D., Labaune, Ch., Depierreux, S., Loiseau, P., Bandulet, H.: Stimulated Brillouin scattering reduction induced by self-focusing for a single laser speckle interacting with an expanding plasma. Phys. Plasmas 21(3), 032703 (2014)ADSGoogle Scholar
  34. Mounaix, P., Divol, L., Huller, S., Tikhonchuk, V.T.: Effects of spatial and temporal smoothing on stimulated Brillouin scattering in the independent-hot-spot model limit. Phys. Rev. Lett. 85(21), 4526–4529 (2000)ADSGoogle Scholar
  35. Myatt, J., Pesme, D., Huller, S., Maximov, A.V., Rozmus, W., Capjack, C.E.: Nonlinear propagation of a randomized laser beam through an expanding plasma. Phys. Rev. Lett. 87(25), 255003 (2001)Google Scholar
  36. Nanda, V., Kant, N.: Strong self-focusing of a cosh-Gaussian laser beam in collisionless magneto-plasma under plasma density ramp. Phys. Plasmas 21(7), 072111 (2014)ADSGoogle Scholar
  37. Neumayer, P., Berger, R.L., Divol, L., Froula, D.H., London, R.A., MacGowan, B.J., Meezan, N.B., Ross, J.S., Sorce, C., Suter, L.J., Glenzer, S.H.: Suppression of stimulated Brillouin scattering by increased Landau damping in multiple-ion-species hohlraum plasmas. Phys. Rev. Lett. 100(10), 105001 (2008)Google Scholar
  38. Niknam, A.R., Barzegar, S., Hashemzadeh, M.: Self-focusing and stimulated Brillouin back-scattering of a long intense laser pulse in a finite temperature relativistic plasma. Phys. Plasmas 20, 122117 (2013)ADSGoogle Scholar
  39. Purohit, G., Rawat, P.: Stimulated Brillouin backscattering of a ring-rippled laser beam in collisionless plasma. Laser Part. Beams 33(3), 499–509 (2015)ADSGoogle Scholar
  40. Rozmus, W., Sharma, R.P., Samson, J.C., Tighe, W.: Nonlinear evolution of stimulated Raman scattering in homogeneous plasmas. Phys. Fluids 30(7), 2181–2193 (1987)ADSGoogle Scholar
  41. Sharma, R.P., Sharma, P., Rajput, S., Bhardwaj, A.K.: Suppression of stimulated Brillouin scattering in laser beam hot spots. Laser Part. Beams 27(4), 619–627 (2009)ADSGoogle Scholar
  42. Singh, A., Walia, K.: Self-focusing of Gaussian laser beam in collisionless plasma and its effect on stimulated Brillouin scattering process. Opt. Commun. 290, 175–182 (2013)ADSGoogle Scholar
  43. Sodha, M.S., Ghatak, A.K., Tripathi, V.K.: Self-focusing of Laser Beams. Tata-McGraw-Hill, New Delhi (1974)Google Scholar
  44. Thakur, V., Kant, N.: Stronger self-focusing of cosh-Gaussian laser beam under exponential density ramp in plasma with linear absorption. Optik 183, 912–917 (2019)ADSGoogle Scholar
  45. Varaki, M.A., Jafari, S.: Relativistic self-focusing of an intense laser pulse with hot magnetized plasma in the presence of a helical magnetostatic wiggler. Phys. Plasmas 24, 082309 (2017)Google Scholar
  46. Wang, Y.L., Lu, Z.W., He, W.M., Zheng, Z.X., Zhao, Y.H.: A new measurement of stimulated Brillouin scattering phase conjugation fidelity for high pump energies. Laser Part. Beams 27(2), 297–302 (2009)ADSGoogle Scholar
  47. Wei, M.S., Beg, F.N., Clark, E.L., Dangor, A.E., Evans, R.G., Gopal, A., Ledingham, K.W.D., McKenna, P., Norreys, P.A., Tatarakis, M., Zepf, M., Krushelnick, K.: Observations of the filamentation of high-intensity laser-produced electron beams. Phys. Rev. E 70(5), 056412 (2004)Google Scholar
  48. Yahia, V., Masson-Laborde, P.E., Depierreux, S., Goyon, C., Loisel, G., Baccou, C., Borisenko, N.G., Orekhov, A., Rienecker, T., Rosmej, O., Teychenné, D., Labaune, C.: Reduction of stimulated Brillouin backscattering with plasma beam smoothing. Phys. Plasmas 22(4), 042707 (2015)ADSGoogle Scholar
  49. Zhou, G.: Propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere. Opt. Express 19(5), 3945–3951 (2011)ADSGoogle Scholar

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Authors and Affiliations

  1. 1.Laser Plasma Computational Laboratory, Department of PhysicsDAV (PG) CollegeDehradunIndia

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