Skip to main content Accessibility help

Stability and dynamics of a cosh-Gaussian laser beam in relativistic thermal quantum plasma

  • Ranju Mahajan (a1), Tarsem Singh Gill (a2), Ravinder Kaur (a3) and Munish Aggarwal (a4)


This paper presents an investigation on the self-focusing of a cosh-Gaussian laser beam in the thermal quantum plasma (TQP) by taking into account the effects of relativistic nonlinearity. An appropriate nonlinear Schrödinger equation has been solved analytically by applying the variational approach. The self-focusing and the self-phase modulation are examined under a variety of parameters. The self-trapping of a cosh-Gaussian laser beam is further studied at various values of the decentered parameter, b with different absorption levels, ${k}^{\prime}_i$ . Numerical analysis shows that these parameters play a vital role in propagation characteristics. The significant contribution of the quantum effects to enhance the self-focusing and minimize the longitudinal phase has been observed. Further, a comparison has been made with the classical relativistic (CR), the relativistic cold quantum (RCQ), and the thermal quantum (TQ) regimes. The self-focusing is found to occur earlier and is strongest for the case of TQP in the present analysis.


Corresponding author

Author for correspondence: Ranju Mahajan and Tarsem Singh Gill, Department of Physics, Lyallpur Khalsa College, Jalandhar 144001, India; Department of Physics, Guru Nanak Dev University, Amritsar 143005, India E-mail:,


Hide All
Akhmanov, SA, Sukhorukov, AP and Khokhlov, RV (1968) Self-focusing and diffraction of light in a nonlinear medium. Soviet Physics Uspekhi 10, 609636.
Ali, S and Shukla, PK (2006) Potential distributions around a moving test charge in quantum plasmas. Physics of Plasmas 13, 102112(1–7).
Anderson, D (1978) Stationary self-trapped laser beams in plasma. Physica Scripta 18, 3536.
Anderson, D and Bonnedal, M (1979) Variational approach to nonlinear self-focusing of Gaussian laser beams. Physics of Fluids 22, 105119.
Andreev, AV (2000) Self-consistent equations for the interaction of an atom with an electromagnetic field of arbitrary intensity. JETP Letters 72, 238240.
Askar'yan, GA (1962) Effect of the gradient of a strong electromagnetic beam on electron and atoms. Journal of Experimental and Theoretical Physics 42, 15671570.
Azechi, H and FIREX Project (2006) Present status of the FIREX programme for the demonstration of ignition and burn. Plasma Physics and Controlled Fusion 48, 267.
Azechi, H, Jitsuno, T, Kanabe, T, Katayama, M, Mima, K, Miyanaga, N, Nakai, M, Nakai, S, Nakaishi, H, Nakatsuka, M, Nishiguchi, A, Norrays, PA, Setsuhara, Y, Takagi, M, Yamanaka, M and Yamanaka, C (1991) High-density compression experiments at ILE, Osaka. Laser and Particle. Beams 9, 193207.
Baykal, Y (2004) Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere. Journal of the Optical Society of America A: Optics and Image Science 21, 12901299.
Belafhal, A and Ibnchaikh, M (2000) Comment on propagation properties of Hermite-cosh-Gaussian laser beams. Optics Communications 186, 269.
Bokaei, B and Niknam, AR (2014) Increasing the upper-limit intensity and temperature range for thermal self-focusing of a laser beam by using plasma density ramp-up. Physics of Plasmas 21, 032309.
Chabrier, G, Douchin, F and Potekhin, AY (2002) Dense astrophysical plasmas. Condensed Matter: An Institute of Physics Journal 14, 9133.
Chiao, RY, Garmire, E and Townes, CH (1964) Self-trapping of optical beams. Physical Review Letters 13, 479482.
Eliasson, B and Shukla, PK (2012) Relativistic x-ray quantum free-electron lasers: a collective KleinGordon model. Plasma Physics and Controlled Fusion 54, 124011.
Esarey, E, Sprangle, P, Krall, J and Ting, A (1997) Self-focusing and guiding of short laser pulses in ionizing gases and plasmas. IEEE Journal of Quantum Electronics 33, 18791914.
Faenov, A Ya, Colgan, J, Hansen, SB, Zhidkov, A, Pikuz, TA, Nishiuchi, M, Pikuz, SA, Skobelev, I Yu, Abdallah, J, Sakaki, H, Sagisaka, A, Pirozhkov, AS, Ogura, K, Fukuda, Y, Kanasaki, M, Hasegawa, N, Nishikino, M, Kando, M, Watanabe, Y, Kawachi, T, Masuda, S, Hosokai, T, Kodama, R and Kondo, K (2015) Nonlinear increase of X-ray intensities from thin foils irradiated with a 200 TW femtosecond laser. Scientific Reports 5, 13436.
Firth, WJ (1977) Propagation of laser beams through inhomogeneous media. Optics Communications 22, 226230.
Gill, TS, Kaur, R and Mahajan, R (2010a) Propagation of high power electromagnetic beam in relativistic magnetoplasma: Higher order paraxial ray theory. Physics of Plasmas 17, 093101.
Gill, TS, Mahajan, R and Kaur, R (2010b) Relativistic and ponderomotive effects on evolution of laser beam in a non-uniform plasma channel. Laser and Particle Beams 28, 1120.
Gill, TS, Mahajan, R and Kaur, R (2010c) Relativistic and ponderomotive effects on evolution of dark hollow Gaussian electromagnetic beams in a plasma. Laser and Particle Beams 28, 521529.
Gill, TS, Mahajan, R and Kaur, R (2011) Self-focusing of cosh-Gaussian laser beam in a plasma with weakly relativistic and ponderomotive regime. Physics of Plasmas 18, 033110.
Glenzer, SH and Redmer, R (2009) X-ray Thomson scattering in high energy density plasmas. Reviews of Modern Physics 81, 1625.
Glenzer, SH, Landen, OL, Neumayer, P, Lee, RW, Widmann, K, Pollaine, SW, Wallace, RJ, Gregori, G, Höll, A, Bornath, T, Thiele, R, Schwarz, V, Kraeft, W-D and Redmer, R (2007) Observations of plasmons in warm dense matter. Physical Review Letters 98, 065002.
Gondarenko, NA, Ossakow, SL and Milikh, GM (2005) Generation and evolution of density irregularities due to self-focusing in ionospheric modifications. Journal of Geophysical Research 110, A09304113.
Gupta, DN and Suk, H (2007) Electron acceleration to high energy by using two chirped lasers. Laser and Particle Beams 25, 31.
Guzdar, PN, Chaturvedi, PK, Papadopoulos, K and Ossakow, SL (1998) The thermal self-focusing instability near the critical surface in the high-latitude ionosphere. Journal of Geophysical Research 103, 2231.
Haas, FB (2011) An introduction to quantum plasmas. Brazilian Journal of Physics 41, 349363.
Habibi, M and Ghamari, F (2012a) Stationary self-focusing of intense laser beam in cold quantum plasma using ramp density profile. Physics of Plasmas 19, 103110.
Habibi, M and Ghamari, F (2012b) Investigation of non-stationary self-focusing of intense laser pulse in cold quantum plasma using ramp density profile. Physics of Plasmas 19, 113109.
Habibi, M and Ghamari, F (2014) Relativistic self-focusing of ultra-high intensity x-ray laser beams in warm quantum plasma with upward density profile. Physics of Plasmas 21, 052705.
Habibi, M and Ghamari, F (2015a) Improved focusing of a cosh-Gaussian laser beam in quantum plasma: higher order paraxial theory. IEEE Transactions on Plasma Science 43, 21602165.
Habibi, M and Ghamari, F (2015b) Significant enhancement in self-focusing of high-power laser beam through dense plasmas by ramp density profile. Journal of Optical Society of America B 32, 1429.
Hefferon, G, Sharma, A and Kourakis, I (2010) Electromagnetic pulse compression and energy localization in quantum plasmas. Physics Letters A 208, 012087.
Hora, H (2007) New aspects for fusion energy using inertial confinement. Laser and Particle Beams 25, 3745.
Honda, M, Meyer-Ter-Vehn, J and Pukhov, A (2000) Two-dimensional particle-in-cell simulation for magnetized transport of ultra-high relativistic currents in plasma. Physics of Plasmas 7, 1302.
Jung, YD and Murakami, I (2009) Quantum effects on magnetization due to ponderomotive force in cold quantum plasmas. Physics Letters A 373, 969971.
Karlsson, M, Anderson, D and Desaix, M (1992) Dynamics of self-focusing and self-phase modulation in a parabolic index optical fiber. Optics Letters 17, 22.
Karlsson, M, Anderson, D, Desaix, M and Lisak, M (1991) Dynamic effects of Kerr nonlinearity and spatial diffraction on self-phase modulation of optical pulses. Optics Letters 16, 1373.
Kaur, R, Gill, TS and Mahajan, R (2010) Self-focusing, self modulation and stability properties of laser beam propagating in plasma: A variational approach. Journal of Physics: Conference Series, 969971.
Kaur, R, Gill, TS and Mahajan, R (2011) Steady state self-focusing, self-phase modulation of laser beam in an inhomogeneous plasma. Optik 122, 375380.
Kelley, PL (1965) Self-focusing of laser beams and stimulated Raman gain in liquids. Physical Review Letters 15, 10101012.
Kodama, R, Norreys, PA, Mima, K, Dangor, AE, Evans, RG, Fujita, H, Kitagawa, Y, Krushelnick, K, Miyakoshi, T, Miyanaga, N, Norimatsu, T, Rose, SJ, Shozaki, T, Shigemori, K, Sunahara, A, Tampo, M, Tanaka, KA, Toyama, Y, Yamanaka, T and Zepf, M (2001) Fast heating of ultrahigh-density plasma as a step towards laser fusion ignition. Nature 412, 798802.
Kremp, D, Bornath, T, Bonitz, M and Schlanges, M (1999) Quantum kinetic theory of plasmas in strong laser fields. Physical Review E 60, 4725.
Lam, JF, Lippmann, B and Tappert, F (1977) Self-trapped laser beams in plasma. Physics of Fluids 20, 1176.
Lakshman, M and Rajasekar, S (2003) Nonlinear Dynamics. Springer Verlag.
Landen, OL, Farley, DR, Glendinning, SG, Logory, LM, Bell, PM, Koch, JA, Lee, FD, Bradley, DK, Kalantar, DH, Back, CA and Tarner, RE (2001) X-ray backlighting for the National Ignition Facility. Review of Scientific Instruments 72, 627.
Lee, RW, Moon, SJ, Chung, HK, Rozmus, W, Baldis, HA, Gregori, G, Cauble, RC, Landen, OL, Wark, JS, Ng, A, Rose, SJ, Lewis, CL, Riley, D, Gauthier, Jean-Claude and Audebert, P (2003) Finite temperature dense matter studies on next-generation light sources. Journal of the Optical Society of America B: Optical Physics 20, 770778.
Lindl, JD, Amendt, P, Berger, LR, Glendinning, SG, Glenzer, HS, Haan, WS, Kauffman, LR, Landen, LO and Suter, JL (2004) The physics basis for ignition using indirect-drive targets on the National Ignition Facility. Physics of Plasmas 11, 339491.
Liu, CS and Tripathi, VK (2001) Self-focusing and frequency broadening of an intense short-pulse laser in plasmas. Journal of the Optical Society of America A 18, 1714.
Lugiato, LA and Narducci, LM (1985) Single-mode and multimode instabilities in lasers and related optical systems. Physical Review A 32, 15761587.
Malkin, VM and Fisch, NJ (2007) Relic crystal-lattice effects on Raman compression of powerful X-ray pulses in plasmas. Physical Review Letters 99, 205001.
Malkin, VM, Fisch, NJ and Wurtele, JS (2007) Compression of powerful x-ray pulses to attosecond durations by stimulated Raman backscattering in plasmas. Physical Review E 75, 026404.
Mahajan, R, Gill, TS and Kaur, R (2010) Nonlinear dynamics of intense EM pulses in plasma. JPCS:IOP 208, 012087.
Manassah, JT, Baldeck, PL and Alfano, RR (1988) Self-focusing and self-phase modulation in a parabolic graded-index optical fiber. Optics Letters 13, 589591.
Manfredi, G (2005) How to model quantum plasmas. Fields Institute Communications Series 46, 263287.
Marklund, M and Shukla, PK (2006) Nonlinear collective effects in photon-photon and photon-plasma interactions. Reviews of Modern Physics 78, 591.
Milchberg, HM, Durfee III, CG and Mcllrath, TJ (1995) High-order frequency conversion in the plasma waveguide. Physical Review Letters 75, 2494.
Mourou, GA, Tajima, T and Bulanov, SV (2006) Optics in the relativistic regime. Reviews of Modern Physics 78, 309.
Mulser, P and Bauer, D (2004) Fast ignition of fusion pellets with superintense lasers: Concepts, problems and prospectives. Laser and Particle Beams 22, 512.
Na, SC and Jung, YD (2009) Temperature effects on the nonstationary Karpman–Washimi ponderomotive magnetization in quantum plasmas. Physics of Plasmas 16, 074504(1–4).
Neumayer, P, Fortmann, C, Döppner, T, Davis, P, Falcone, RW, Kritcher, AL, Landen, OL, Lee, HJ, Lee, RW, Niemann, C, Le Pape, S and Glenzer, SH (2010) Plasmons in strongly coupled shock-compressed matter. Physical Review Letters 105, 075003.
Neumayer, P, Gregori, G, Ravasio, A, Koenig, M, Price, D, Widmann, K, Bastea, M, Landen, OL and Glenzer, SH (2006) Solid-density plasma characterization with x-ray scattering on the 200J Janus laser. Review of Scientific Instruments 77, 10F317.
Niknam, AR, Barzegar, S and Hashemzadeh, M (2013) Self-focusing and stimulated Brillouin back-scattering of a long intense laser pulse in a finite temperature relativistic plasma. Physics of Plasmas. 20, 122117.
Opher, M, Silva, LO, Dauger, DE, Decyk, VK and Dawson, JM (2001) Nuclear reaction rates and energy in stellar plasmas: The effect of highly damped modes. Physics of Plasmas 8, 24542460.
Parashar, J, Pandey, HD and Tripathi, VK (1997) Two-dimensional effects in a tunnel ionized plasma. Physics of Plasmas 4, 3040.
Patil, SD and Takale, MV (2013) Self-focusing of Gaussian laser beam in relativistic cold quantum plasma. Physics of Plasmas 20, 072703.
Patil, SD and Takale, MV (2014) Response to “Comment on ‘Stationary self-focusing of Gaussian laser beam in relativistic thermal quantum plasma’. Physics of Plasmas 21, 064702.
Patil, SD, Takale, MV, Navare, ST and Dongare, MB (2010) Focusing of Hermite-cosh-Gaussian laser beams in collisionless magnetoplasma. Laser and Particle Beams 28, 343349.
Patil, SD, Takale, MV, Navare, ST, Fulari, VJ and Dongare, MB (2012) Relativistic self-focusing of cosh-Gaussian laser beams in a plasma. Optics and Laser Technology 44, 314.
Patil, SD, Takale, MV, Fulari, VJ, Gupta, DN and Suk, H (2013a) Combined effect of ponderomotive and relativistic self-focusing on laser beam propagation in a plasma. Applied Physics B: Photophysics and Laser Chemistry 111, 1.
Patil, SD, Takale, MV, Navare, ST, Dongare, MB and Fulari, VJ (2013b) Self-focusing of Gaussian laser beam in relativistic cold quantum plasma. Optik 124, 180.
Peyrusse, O, Busquet, M, Kieffer, JC, Jiang, Z and Cote, CY (1995) Generation of hot solid-density plasmas by laser radiation pressure confinement. Physical Review Letters 75, 3862.
Regan, SP, Bradley, DK, Chirokikh, AV, Craxton, RS, Meyerhofer, DD, Seka, W, Short, RW, Simon, A, Town, RPJ and Yaakobi, B (1999) Laser-plasma interactions in long-scale-length plasmas under direct-drive National Ignition Facility conditions. Physics of Plasmas 6, 20722080.
Remington, BA, Drake, RP and Ryutov, DD (2006) Experimental astrophysics with high power lasers and Z pinches. Reviews of Modern Physics 78, 755.
Ren, H, Wu, Z and Chu, PK (2007) Dispersion of linear waves in quantum plasmas. Physics of Plasmas 14, 062102.
Saini, NS and Gill, TS (2006) Self-focusing and self-phase modulation of an elliptic Gaussian laser beam in collisionless magnetoplasma. Laser and Particle Beams 24, 447.
Sarkisov, GS, Bychenkov, V.Yu, Novikov, VN, Tikhonchuk, VT, Maksimchuk, A, Chen, S-Y, Wagner, R, Mourou, G and Umstadter, D (1999) Self-focusing, channel formation, and high-energy ion generation in interaction of an intense short laser pulse with a He jet. Physical Review E 59, 70427054.
Sharma, A, Verma, MP and Sodha, MS (2004) Self-focusing of electromagnetic beams in a collisional plasmas with nonlinear absorption. Physics of Plasmas 11, 42754279.
Shpatakovskaya, G (2006) Semiclassical model of a one-dimensional quantum dot. Journal of Experimental and Theoretical Physics 102, 466.
Shukla, PK and Eliasson, B (2007) Nonlinear interactions between electromagnetic waves and electron plasma oscillations in quantum plasmas. Physical Review Letters 99, 096401.
Shukla, PK and Eliasson, B (2010) Nonlinear aspects of quantum plasma physics. Physics Uspekhi 53, 5176.
Shukla, PK and Eliasson, B (2011) Nonlinear collective interactions in quantum plasmas with degenerate electron fluids. Reviews of Modern Physics 83, 885.
Shukla, PK, Ali, S, Stenflo, L and Marklund, M (2006) Physics of Plasmas 13, 112111.
Skarka, V and Aleksic, NB (2006) Stability criterion for dissipative soliton solutions of one-, two-, and three dimensional complex cubic quintic Ginzburg-Landau equations. Physical Review Letters 96, 013903-1013903-4.
Skarka, V, Berezhiani, VI and Miklaszewski, R (1997) Spatiotemporal soliton propagation in saturating nonlinear optical media. Physical Review E 56, 10801087.
Skarka, V, Berezhiani, VI and Miklaszewski, R (1999) Generation of light spatiotemporal solitons from asymmetric pulses in saturating nonlinear media. Physical Review E 59, 12701273.
Sodha, MS, Ghatak, AK and Tripathi, VK (1974) Self-Focusing of Laser Beams in Dielectric Plasma and Semi Conductors. New York: Tata McGraw-Hill.
Sodha, MS, Ghatak, AK and Tripathi, VK (1976) Self-focusing of laser beams in plasmas and semiconductors. Progress in Optics 13, 171365.
Sprangle, P, Hafizi, B and Penano, JR (2000) Laser pulse modulation instabilities in plasma channels. Physical Review E 61, 43814393.
Subbarao, D, Uma, R and Singh, H (1998) Paraxial theory of self-focusing of cylindrical laser beams. I. ABCD laws. Physics of Plasmas 5, 3440.
Tabak, M, Hammer, J, Glinsky, ME, Kruer, WL, Wilks, SC, Woodworth, J, Campbell, EM, Perry, MD and Mason, RJ (1994) Ignition and high gain with ultrapowerful lasers. Physics of Plasmas 1, 16261634.
Takale, MV, Navare, ST, Patil, SD, Fulari, VJ and Dongare, MB (2009) Self-focusing and defocusing of TEM0p Hermite-Gaussian laser beams in collisionless plasma. Optics Communications 282, 31573162.
Uhm, HS, Nam, IH and Suk, HI (2012) Scaling laws of design parameters for plasma wakefield accelerators. Physics Letters A 376, 165168.
Vinko, SM, Ciricosta, O, Cho, BI, Engelhorn, K, Chung, HK, Brown, CRD, Burian, T, Chalpusky, J, Falcone, RW, Graves, C, Hajkova, V, Higginbotham, A, Juha, L, Krzywinski, J, Lee, HJ, Messerschmidt, M, Murphy, CD, Ping, Y, Scherz, A, Schlotter, W, Toleikis, S, Turner, JJ, Vysin, L, Wang, T, Wu, B, Zastrau, U, Zhu, D, Lee, RW, Heimann, PA, Nagler, B and Wark, JS (2012) Creation and diagnosis of a solid-density plasma with an X-ray free-electron laser. Nature 482, 5962.
Wang, NQ (1990) Chaotic behaviour in an electron-beam plasma. Physics Letters A 145, 2932.
Winterberg, F (2008) Lasers for inertial confinement fusion driven by high explosives. Laser and Particle Beams 26, 127135.
Yu, W, Yu, MY, Xu, H, Tian, YW, Chen, J and Wong, AY (2007) Intense local plasma heating by stopping of ultrashort ultraintense laser pulse in dense plasma. Laser and Particle Beams 25, 631638.
Zare, S, Rezaee, S, Yazdani, S, Anvari, A and Sadighi-Bonabi, R (2015) Relativistic Gaussian laser beam self-focusing in collisional quantum plasmas. Laser and Particle Beams 33, 397.
Zhou, Z, Wang, Y, Yuan, C and Du, Y (2011) Self-focusing and defocusing of Gaussian laser beams in plasmas with linear temperature ramp. Physics of Plasmas 18, 073107.



Altmetric attention score

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