超短脉冲强激光在大气中的传输特性
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摘要
超短超强脉冲激光能够在大气中传输形成一条稳定的等离子体通道,并由此引发超连续白光、三次谐波、锥角辐射等很多物理现象。这些现象在激光遥感、激光引雷、激光诱导闪电、激光诱导核反应、强激光远距离能量输送等领域具有广阔的应用前景。因而超高功率和超短脉冲强激光在大气中的传输成为近年来激光物理学研究的新热点。
本文基于一个由空间动态补偿模型发展而来的模型,主要讨论了超短超强激光脉冲在大气中传输形成的等离子体通道的演变和控制。目前人们普遍认为,超短超强脉冲激光在空气中自聚焦传输形成等离子体通道主要是衍射、色散、克尔效应和多光子吸收共同作用的结果。本文在考虑以上重要机制的基础上,还引入了拉曼散射、等离子体尾波场和相对论自聚焦等多种效应,并通过采用合理的近似,修正了表征超短脉冲强激光大气中传输的非线性薛定谔方程。采用分步傅里叶方法和差分法得到了二维和三维的数值仿真结果。主要讨论光强通量和电子密度在空间中的分布,光脉冲在时间域和空间域的演变情况,通过数值仿真,揭示不同波长、输入脉冲能量、脉宽、束腰半径等初始参数对成丝的影响,并对其他较为复杂的改变激光能量分布的方法提出了展望,如采用时域啁啾的初始脉冲、构造后继脉冲形成双脉冲结构等。这些工作为探索超短脉冲强激光在大气中成丝位置和形态控制提供了可能的新途径。
A stable plasma channel is formed when ultra-short high power laser propagates in the atmosphere, which meanwhile leads to a variety of physical phenomena, such as supercontinuum spectrum generation, third-order harmonic generation, conical emission. These phenomena have broad application prospects in laser triggered lightning, laser-induced lightning, laser-induced nuclear reactions, long-distance energy transmission of high power laser, etc. Therefore, the research of propagation of ultra-high power and ultrashort pulse laser in the atmosphere has become a new hot spot in laser physics.
Based on an extended spatial dynamic compensation model, this paper mainly discusses the evolution and control of laser-plasma channel which formed by the propagation of ultrashort intense laser pulse in the atmosphere. It is now widely believed that the formation of plasma channel in air is mainly due to the combined action of diffraction, dispersion, Kerr effect and multi-photon absorption. In this paper, besides the basis important mechanism above, Raman scattering, plasma wake fields, relativistic focusing and other effects are also taken into consideration. Using a reasonable approximation, we amend the nonlinear Schr(o|¨)dinger equation that governs the propagation of ultrashort laser pulse in the atmosphere. Split-step Fourier and differential method have been applied to gain both two-dimensional and three-dimensional numerical simulation results. Flux of light intensity and electron density distribution in space as well as pulse evolution in time and space are discussed. Through numerical simulation, it reveals that different initial parameters of pulse such as wavelength, input pulse energy, pulse width and waist, may have influence on the filamentation. A prospect of other complicated methods that may vary the filament distribution was also made, such as adopting temporally chirped initial pulse, inducing subsequent pulse to form twin laser pulses. These work might add a new path to the exploration of the position and shape control of ultra-short laser pulses filamentation in the atmosphere.
本文基于一个由空间动态补偿模型发展而来的模型,主要讨论了超短超强激光脉冲在大气中传输形成的等离子体通道的演变和控制。目前人们普遍认为,超短超强脉冲激光在空气中自聚焦传输形成等离子体通道主要是衍射、色散、克尔效应和多光子吸收共同作用的结果。本文在考虑以上重要机制的基础上,还引入了拉曼散射、等离子体尾波场和相对论自聚焦等多种效应,并通过采用合理的近似,修正了表征超短脉冲强激光大气中传输的非线性薛定谔方程。采用分步傅里叶方法和差分法得到了二维和三维的数值仿真结果。主要讨论光强通量和电子密度在空间中的分布,光脉冲在时间域和空间域的演变情况,通过数值仿真,揭示不同波长、输入脉冲能量、脉宽、束腰半径等初始参数对成丝的影响,并对其他较为复杂的改变激光能量分布的方法提出了展望,如采用时域啁啾的初始脉冲、构造后继脉冲形成双脉冲结构等。这些工作为探索超短脉冲强激光在大气中成丝位置和形态控制提供了可能的新途径。
A stable plasma channel is formed when ultra-short high power laser propagates in the atmosphere, which meanwhile leads to a variety of physical phenomena, such as supercontinuum spectrum generation, third-order harmonic generation, conical emission. These phenomena have broad application prospects in laser triggered lightning, laser-induced lightning, laser-induced nuclear reactions, long-distance energy transmission of high power laser, etc. Therefore, the research of propagation of ultra-high power and ultrashort pulse laser in the atmosphere has become a new hot spot in laser physics.
Based on an extended spatial dynamic compensation model, this paper mainly discusses the evolution and control of laser-plasma channel which formed by the propagation of ultrashort intense laser pulse in the atmosphere. It is now widely believed that the formation of plasma channel in air is mainly due to the combined action of diffraction, dispersion, Kerr effect and multi-photon absorption. In this paper, besides the basis important mechanism above, Raman scattering, plasma wake fields, relativistic focusing and other effects are also taken into consideration. Using a reasonable approximation, we amend the nonlinear Schr(o|¨)dinger equation that governs the propagation of ultrashort laser pulse in the atmosphere. Split-step Fourier and differential method have been applied to gain both two-dimensional and three-dimensional numerical simulation results. Flux of light intensity and electron density distribution in space as well as pulse evolution in time and space are discussed. Through numerical simulation, it reveals that different initial parameters of pulse such as wavelength, input pulse energy, pulse width and waist, may have influence on the filamentation. A prospect of other complicated methods that may vary the filament distribution was also made, such as adopting temporally chirped initial pulse, inducing subsequent pulse to form twin laser pulses. These work might add a new path to the exploration of the position and shape control of ultra-short laser pulses filamentation in the atmosphere.
引文
[1]Braun A.,et al.,Self-channeling of high-peak-power femtosecond laser pulses in air,Opt.Lett.,20(1),73-75,1995.
[2]Woste L,Wedekind C,Wille H,et al.Femtosecond atmospheric lamp[J]Las.optoelektron,1997,29(5):51-53.
[3]Yang H,Zhang J,Zhang Q J,et al,Polarization.dependent supercontinuum generation from light filam ents in air[J]opt.Lett.,2005,30(5):534-536.
[4]Kasparian J,Sauerbrey R,Mondelain D,et al.Infrared extension of the supercontinuum generated by femtosecond terawatt laser pulses propagating in the atmosphere[J].Opt.Lett.,2000,25(18):1397-139
[5]Yang H,Zhang J,Zhang J,et al.Third-order harmonic generation by self-guided femtosecond pulses in air[J].Phys.Rev.E2003,67(1):R15401.
[6]AkSzbek N,lwasaki A,Becker A,et al.Third-harmonic generation and self-channeling in air using high.powerfemtosecond laser pulses[J].Phys.Rev.Lett.,2002,89(14):143901.
[7]Nibbering E T J,Curley P F,Grillon G,et al.Conical emission from self-guided femtosecond pulses inair[J].Opt.Lett.,1996,21(1):62-64.
[8]Kasparian J,Rodriguez M,et al.White.light filaments for atmospheric analysis[J].Science,2003,301(5629):61-64.
[9]Zhao M X,Diels J C,et al.Femtosecond ultraviolet laser pulse induced lightning discharges in gases[J].IEEEJ.Quantum Electron.1995,31(3):599-612
[10]郝作强等,空气中激光等离子体通道的荧光测量和声学诊断两种方法的比较实验研究,物理学报,55(1),0299-0303,2006。
[11]张杰等,超强飞秒激光脉冲在空气中的传输研究,量子电子学报,23(3),282-294,2006。
[12]董全力等,大气中激光等离子体通道寿命的延长及测量分析,物理学报,54(7),3247-3-250,2005。
[13]郝作强等,空气中激光等离子体通道的三次谐波辐射研究,物理学报,54(7),3173-3176,2005。
[14]朱佳斌等,外加高压电场下空气中激光等离子体通道寿命研究,光学学报,27(6),1059-1062.2007
[15]Yu W,et al.,Plasma channeling by multiple short-pulse lasers,Laser and Particle Beams,27,109-114,2009
[16]Meyand R.G.and Haught A.F.,Gas breakdown at optical frequencies,Phys.Rev.Lett.11(9),401-403,1963.
[17]Jin Z,Zhang J,Xu M H,et al.Control of filamentation induced by femtosecond laser pulses propagating in air[J].Optics Express,2005,13(25):10424-10430.
[18]J.Kasparian,R.Bourayou,V.Boutoul,et al.12th SQE,(2002).
[19]A Couairon,G Mechain,S Tzortzakis,et al.Propagation of twin laser pulses in air and concatenation of plasma strings produced by femtosecond infrared filaments[J].Optics Communications,2003,225:177-192.
[20]Pavel Polynkin,Miroslav Kolesik,Jerome Moloney.Extended filamentation with temporally chirped femtosecond Bessel-Gauss beams in air[J]. Optics Express, 2009,17(2): 575-584
[21] Pavel Polynkin, Miroslav Kolesik, Adam Roberts. Generation of extended plasma channels in air using femtosecond Bessel beams[J]. Optics Express, 2008, 16(20): 15733-15740
[22] P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, "Filamentation in Kerr media from pulsed Bessel beams", PHYSICAL REVIEW A 77, 043814 (2008).
[23] D. Abdollahpourl, P. Panagiotopoulosl, M. Turconi,etc. "Long spatio-temporally stationary filaments in air using short pulse UV laser Bessel beams", OPTICS EXPRESS, Vol. 17, No. 7 (2009).
[24] Selcuk Akturk, Bing Zhou, Benjamin Pasquiou, et al. Generation of long plasma channels in air by using axicongenerated Bessel beams. 2008 OSA / CLEO/QELS 2008
[25] Selcuk Akturk , Bing Zhou, Michel Franco, et al. Arnaud Couairon Generation of long plasma channels in air by focusing ultrashort laser pulses with an axicon[J] Optics Communications 2009, 282: 129-134.
[26] Yang H, Zhang J, Li Y J, et al. Characteristics of self-guided laser plasma channels generated by femtosecond laser pulses in air[J]. Phys. Rev. E, 2002, 66: 016406.
[27] Schwarz J, Rambo P, Diels J C, et al. Ultraviolet filamentation in air[J]. Optics Communications, 2000,180:383-390.
[28] Miguel R, Riad B, Guillaume M, et al. Kilometer-range nonlinear propagation of femtosecond laser pulses[J]. Phys. Rev. E, 2004, 69: 036607.
[29] Mayo V M, Hugo S, Enrique C. Shadowgraphy and Interferometry Using a CW Laser and a CCD of a Laser-Induced Plasma in Atmospheric Air[J]. IEEE. Trans. Plasma. Sci, 2001, 29(4): 613-616.
[30] Magesh Thiyagarajan, John Scharer, Experimental investigation of ultraviolet laser induced plasma density and temperature evolution in air[J] Appl. Phys, 2008,104: 013303.
[31] Magesh T. and John S., Experimental investigation of ultraviolet laser induced plasma density and temperature evolution in air, J. of Applied Physics, 104, 013303, 2008.
[32] Yang H, Zhang J, Li Y J, et al. Characteristics of self-guided laser plasma channels generated by femtosecond laser pulses in air[J]. Phys. Rev. E, 2002, 66: 016406.
[33] G. Fibich, Y. Sivan, Y. Erlich, E. Louzon, M. Fraenkel, S. Eisenmann, Y. Katzir, and A. Zigler, "Control of the collapse distance in atmospheric propagation," Opt. Express 14,4946 (2006).
[34] W. Liu, F. Th'eberge, J.-F. Daigle, P. T. Simrad, Y. Kamali, H. L. Xu, and S. L. Chin, "An efficient control of ultrashort laser filament location in air for the purpose of remote sensing " Appl. Phys. B 85, 55 (2006).
[35] Shmuel Eisenmann, Einat Louzon, Yiftach Katzir, et al. Control of the filamentation distance and pattern in long-range atmospheric propagation[J]. Optics Express, 2007, 15(6): 2779-2784.
[36] M. Chateauneuf, S. Payeur, J. Dubois, and J.-C. Kieffer, "Microwave guiding in air by a cylindrical filament array waveguide", APPLIED PHYSICS LETTERS 92, 091104 (2008).
[37] Donatas Majus, Vytautas Jukna, Gintaras Valiulis, and Audrius Dubietis, "Generation of periodic filament arrays by self-focusing of highly elliptical ultrashort pulsed laser beams", PHYSICAL REVIEW A 79, 033843 (2009).
[38] Marburger, J.H., 1975. Self-focusing: Theory. Prog. Quant. Electr. 4, 35-110.
[39] Dawes, EX., Marburger, J.H., 1969. Computer studies in self-focusing. Phys. Rev. 179 (3), 862-868.
[40] Fibich, G., Eisenmann, S., Ilan, B., Erlich,Y, Fraenkel,M., Henis, Z., Gaeta,A.L., Zigler,A., 2005a. Self-focusing distance of very high power laser pulses. Opt. Express 13(15), 5897.
[41] Fibich, G., 1996. Adiabatic law for self-focusing of optical beams. Opt. Lett. 21 (21), 1735-1737.
[42]Akozbek,N.,Bowden,C.M.,Talebpour,A.,Chin,S.L.,2000.Femtosecond pulse propagation in air:variational analysis.Phys.Rev.E 61(4),4540-4549.
[43]Couairon,A.,2003a.Dynamics of femtosecond filamentation from saturation of self-focusing laser pulses.Phys.Rev.A 68,015,801.
[44]Kolesik,M.,Moloney,J.V.,2004b.Self-healing femtosecond light filaments.Opt.Lett.29(6),590-592.
[45]Braun,A.,Korn,G.,Liu,X.,Du,D.,Squier,J.,Mourou,G.,1995.Self-channeling of high-peak-power femtosecond laser pulses in air.Opt.Lett.20(1),73-75.
[46]Kandidov,V.P.,Kosareva,O.G.,Mozhaev,E.I.,Tamarov,M.P.,2000.Femtosecond nonlinear optics in the atmosphere.Atmos.Oceanic Opt.13(5),394-401.
[47]Becker,A.,Akozbek,N.,Vijayalakshmi,K.,Oral,E.,Bowden,C.M.,Chin,S.L.,2001 a.Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas.Appl.Phys.B 73,287-290.
[48]Vidal,F.,Johnston,T.W.,1996.Electromagnetic beam breakup:multiple filaments,single beam equilibria,and radiation.Phys.Rev.Lett.77(7),1282-1285.
[49]Xi T T,Lu X,Zhang J.Interaction of Light Filaments Generated by Femtosecond Laser Pulses in Air[J].PHYSICAL REVIEW LETTERS.2006,96(2)
[50]Sprangle P,Penano J R,Hafizi B.Propagation of intense short laser pulses in the atmosphere[J].PHYSICAL REVIEW E.2002,66(4).
[51]Govind P.Agrawal,非线性光纤光学原理及应用,北京:电子工业出版社,2002,33-35
[52]Couairon A,Berge L.Light filaments in air for ultraviolet and infrared wavelengths[J].PHYSICAL REVIEW LETTERS.2002,88(13).
[53]苗润才,沈常宇,等.环形斑光束在非线性克尔介质中的自聚焦效应[J].光子学报.2003,32(2):204-208.
[54]Couairon A,Mysyrowicz A.Femtosecond filamentation in transparent media[J].
[2]Woste L,Wedekind C,Wille H,et al.Femtosecond atmospheric lamp[J]Las.optoelektron,1997,29(5):51-53.
[3]Yang H,Zhang J,Zhang Q J,et al,Polarization.dependent supercontinuum generation from light filam ents in air[J]opt.Lett.,2005,30(5):534-536.
[4]Kasparian J,Sauerbrey R,Mondelain D,et al.Infrared extension of the supercontinuum generated by femtosecond terawatt laser pulses propagating in the atmosphere[J].Opt.Lett.,2000,25(18):1397-139
[5]Yang H,Zhang J,Zhang J,et al.Third-order harmonic generation by self-guided femtosecond pulses in air[J].Phys.Rev.E2003,67(1):R15401.
[6]AkSzbek N,lwasaki A,Becker A,et al.Third-harmonic generation and self-channeling in air using high.powerfemtosecond laser pulses[J].Phys.Rev.Lett.,2002,89(14):143901.
[7]Nibbering E T J,Curley P F,Grillon G,et al.Conical emission from self-guided femtosecond pulses inair[J].Opt.Lett.,1996,21(1):62-64.
[8]Kasparian J,Rodriguez M,et al.White.light filaments for atmospheric analysis[J].Science,2003,301(5629):61-64.
[9]Zhao M X,Diels J C,et al.Femtosecond ultraviolet laser pulse induced lightning discharges in gases[J].IEEEJ.Quantum Electron.1995,31(3):599-612
[10]郝作强等,空气中激光等离子体通道的荧光测量和声学诊断两种方法的比较实验研究,物理学报,55(1),0299-0303,2006。
[11]张杰等,超强飞秒激光脉冲在空气中的传输研究,量子电子学报,23(3),282-294,2006。
[12]董全力等,大气中激光等离子体通道寿命的延长及测量分析,物理学报,54(7),3247-3-250,2005。
[13]郝作强等,空气中激光等离子体通道的三次谐波辐射研究,物理学报,54(7),3173-3176,2005。
[14]朱佳斌等,外加高压电场下空气中激光等离子体通道寿命研究,光学学报,27(6),1059-1062.2007
[15]Yu W,et al.,Plasma channeling by multiple short-pulse lasers,Laser and Particle Beams,27,109-114,2009
[16]Meyand R.G.and Haught A.F.,Gas breakdown at optical frequencies,Phys.Rev.Lett.11(9),401-403,1963.
[17]Jin Z,Zhang J,Xu M H,et al.Control of filamentation induced by femtosecond laser pulses propagating in air[J].Optics Express,2005,13(25):10424-10430.
[18]J.Kasparian,R.Bourayou,V.Boutoul,et al.12th SQE,(2002).
[19]A Couairon,G Mechain,S Tzortzakis,et al.Propagation of twin laser pulses in air and concatenation of plasma strings produced by femtosecond infrared filaments[J].Optics Communications,2003,225:177-192.
[20]Pavel Polynkin,Miroslav Kolesik,Jerome Moloney.Extended filamentation with temporally chirped femtosecond Bessel-Gauss beams in air[J]. Optics Express, 2009,17(2): 575-584
[21] Pavel Polynkin, Miroslav Kolesik, Adam Roberts. Generation of extended plasma channels in air using femtosecond Bessel beams[J]. Optics Express, 2008, 16(20): 15733-15740
[22] P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, "Filamentation in Kerr media from pulsed Bessel beams", PHYSICAL REVIEW A 77, 043814 (2008).
[23] D. Abdollahpourl, P. Panagiotopoulosl, M. Turconi,etc. "Long spatio-temporally stationary filaments in air using short pulse UV laser Bessel beams", OPTICS EXPRESS, Vol. 17, No. 7 (2009).
[24] Selcuk Akturk, Bing Zhou, Benjamin Pasquiou, et al. Generation of long plasma channels in air by using axicongenerated Bessel beams. 2008 OSA / CLEO/QELS 2008
[25] Selcuk Akturk , Bing Zhou, Michel Franco, et al. Arnaud Couairon Generation of long plasma channels in air by focusing ultrashort laser pulses with an axicon[J] Optics Communications 2009, 282: 129-134.
[26] Yang H, Zhang J, Li Y J, et al. Characteristics of self-guided laser plasma channels generated by femtosecond laser pulses in air[J]. Phys. Rev. E, 2002, 66: 016406.
[27] Schwarz J, Rambo P, Diels J C, et al. Ultraviolet filamentation in air[J]. Optics Communications, 2000,180:383-390.
[28] Miguel R, Riad B, Guillaume M, et al. Kilometer-range nonlinear propagation of femtosecond laser pulses[J]. Phys. Rev. E, 2004, 69: 036607.
[29] Mayo V M, Hugo S, Enrique C. Shadowgraphy and Interferometry Using a CW Laser and a CCD of a Laser-Induced Plasma in Atmospheric Air[J]. IEEE. Trans. Plasma. Sci, 2001, 29(4): 613-616.
[30] Magesh Thiyagarajan, John Scharer, Experimental investigation of ultraviolet laser induced plasma density and temperature evolution in air[J] Appl. Phys, 2008,104: 013303.
[31] Magesh T. and John S., Experimental investigation of ultraviolet laser induced plasma density and temperature evolution in air, J. of Applied Physics, 104, 013303, 2008.
[32] Yang H, Zhang J, Li Y J, et al. Characteristics of self-guided laser plasma channels generated by femtosecond laser pulses in air[J]. Phys. Rev. E, 2002, 66: 016406.
[33] G. Fibich, Y. Sivan, Y. Erlich, E. Louzon, M. Fraenkel, S. Eisenmann, Y. Katzir, and A. Zigler, "Control of the collapse distance in atmospheric propagation," Opt. Express 14,4946 (2006).
[34] W. Liu, F. Th'eberge, J.-F. Daigle, P. T. Simrad, Y. Kamali, H. L. Xu, and S. L. Chin, "An efficient control of ultrashort laser filament location in air for the purpose of remote sensing " Appl. Phys. B 85, 55 (2006).
[35] Shmuel Eisenmann, Einat Louzon, Yiftach Katzir, et al. Control of the filamentation distance and pattern in long-range atmospheric propagation[J]. Optics Express, 2007, 15(6): 2779-2784.
[36] M. Chateauneuf, S. Payeur, J. Dubois, and J.-C. Kieffer, "Microwave guiding in air by a cylindrical filament array waveguide", APPLIED PHYSICS LETTERS 92, 091104 (2008).
[37] Donatas Majus, Vytautas Jukna, Gintaras Valiulis, and Audrius Dubietis, "Generation of periodic filament arrays by self-focusing of highly elliptical ultrashort pulsed laser beams", PHYSICAL REVIEW A 79, 033843 (2009).
[38] Marburger, J.H., 1975. Self-focusing: Theory. Prog. Quant. Electr. 4, 35-110.
[39] Dawes, EX., Marburger, J.H., 1969. Computer studies in self-focusing. Phys. Rev. 179 (3), 862-868.
[40] Fibich, G., Eisenmann, S., Ilan, B., Erlich,Y, Fraenkel,M., Henis, Z., Gaeta,A.L., Zigler,A., 2005a. Self-focusing distance of very high power laser pulses. Opt. Express 13(15), 5897.
[41] Fibich, G., 1996. Adiabatic law for self-focusing of optical beams. Opt. Lett. 21 (21), 1735-1737.
[42]Akozbek,N.,Bowden,C.M.,Talebpour,A.,Chin,S.L.,2000.Femtosecond pulse propagation in air:variational analysis.Phys.Rev.E 61(4),4540-4549.
[43]Couairon,A.,2003a.Dynamics of femtosecond filamentation from saturation of self-focusing laser pulses.Phys.Rev.A 68,015,801.
[44]Kolesik,M.,Moloney,J.V.,2004b.Self-healing femtosecond light filaments.Opt.Lett.29(6),590-592.
[45]Braun,A.,Korn,G.,Liu,X.,Du,D.,Squier,J.,Mourou,G.,1995.Self-channeling of high-peak-power femtosecond laser pulses in air.Opt.Lett.20(1),73-75.
[46]Kandidov,V.P.,Kosareva,O.G.,Mozhaev,E.I.,Tamarov,M.P.,2000.Femtosecond nonlinear optics in the atmosphere.Atmos.Oceanic Opt.13(5),394-401.
[47]Becker,A.,Akozbek,N.,Vijayalakshmi,K.,Oral,E.,Bowden,C.M.,Chin,S.L.,2001 a.Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas.Appl.Phys.B 73,287-290.
[48]Vidal,F.,Johnston,T.W.,1996.Electromagnetic beam breakup:multiple filaments,single beam equilibria,and radiation.Phys.Rev.Lett.77(7),1282-1285.
[49]Xi T T,Lu X,Zhang J.Interaction of Light Filaments Generated by Femtosecond Laser Pulses in Air[J].PHYSICAL REVIEW LETTERS.2006,96(2)
[50]Sprangle P,Penano J R,Hafizi B.Propagation of intense short laser pulses in the atmosphere[J].PHYSICAL REVIEW E.2002,66(4).
[51]Govind P.Agrawal,非线性光纤光学原理及应用,北京:电子工业出版社,2002,33-35
[52]Couairon A,Berge L.Light filaments in air for ultraviolet and infrared wavelengths[J].PHYSICAL REVIEW LETTERS.2002,88(13).
[53]苗润才,沈常宇,等.环形斑光束在非线性克尔介质中的自聚焦效应[J].光子学报.2003,32(2):204-208.
[54]Couairon A,Mysyrowicz A.Femtosecond filamentation in transparent media[J].