Abstract

The propagation of tightly focused femtosecond laser pulse with numerical aperture of 0.12 in air is investigated experimentally. The formation and evolution of the filament bunch are recorded by time-resolved shadowgraph with laser energy from 2.4 mJ to 47 mJ. The distribution of electron density in breakdown area is retrieved using Nomarski interferometer. It is found that intensity clamping during filamentation effect still play a role even under strong external focusing. The electron density in some interaction zones is higher than 3×1019 cm-3, which indicates that each air molecule there is ionized.

© 2010 OSA

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  1. C. V. Gabel, “Femtosecond lasers in biology: nanoscale surgery with ultrafast optics,” Contemp. Phys. 49(6), 391–411 (2008).
    [CrossRef]
  2. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
    [CrossRef]
  3. K. Y. Kim, A. J. Taylor, J. H. Glownia, and G. Rodriguez, “Coherent control of terahertz supercontinuum generation in ultrafast laser-gas interactions,” Nat. Photonics 2(10), 605–609 (2008).
    [CrossRef]
  4. K. Stelmaszczyk, P. Rohwetter, G. Mejean, J. Yu, E. Salmon, J. Kasparian, R. Ackermann, J. P. Wolf, and L. Woste, “Long-distance remote laser-induced breakdown spectroscopy using filamentation in air,” Appl. Phys. Lett. 85(18), 3977–3979 (2004).
    [CrossRef]
  5. M. Lenzner, J. Kruger, S. Sartania, Z. Cheng, C. Spielmann, G. Mourou, W. Kautek, and F. Krausz, “Femtosecond optical breakdown in dielectrics,” Phys. Rev. Lett. 80(18), 4076–4079 (1998).
    [CrossRef]
  6. L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
    [CrossRef] [PubMed]
  7. W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, “Femtosecond laser pulse filamentation versus optical breakdown in H-2O,” Appl. Phys. B-Lasers Opt. 76(3), 215–229 (2003).
    [CrossRef]
  8. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
    [CrossRef] [PubMed]
  9. A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov, “Moving focus in the propagation of ultrashort laser pulses in air,” Opt. Lett. 22(5), 304–306 (1997).
    [CrossRef] [PubMed]
  10. J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
    [CrossRef] [PubMed]
  11. L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92(22), 225002 (2004).
    [CrossRef] [PubMed]
  12. S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Theberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
    [CrossRef]
  13. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep.-, Rev. Sec. Phys. Lett. 441, 47–189 (2007).
  14. F. Théberge, W. W. Liu, P. T. Simard, A. Becker, and S. L. Chin, “Plasma density inside a femtosecond laser filament in air: strong dependence on external focusing,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036406 (2006).
    [CrossRef] [PubMed]
  15. O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
    [CrossRef]
  16. J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B-Lasers Opt. 71, 877–879 (2000).
  17. A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B-Lasers Opt. 73, 287–290 (2001).
  18. J. Bernhardt, W. Liu, S. L. Chin, and R. Sauerbrey, “Pressure independence of intensity clamping during filamentation: theory and experiment,” Appl. Phys. B-Lasers Opt. 91(1), 45–48 (2008).
    [CrossRef]
  19. W. Liu, S. Petit, A. Becker, N. Akozbek, C. M. Bowden, and S. L. Chin, “Intensity clamping of a femtosecond laser pulse in condensed matter,” Opt. Commun. 202(1-3), 189–197 (2002).
    [CrossRef]
  20. Y. T. Li, T. T. Xi, Z. Q. Hao, Z. Zhang, X. Y. Peng, K. Li, Z. Jin, Z. Y. Zheng, Q. Z. Yu, X. Lu, and J. Zhang, “Oval-like hollow intensity distribution of tightly focused femtosecond laser pulses in air,” Opt. Express 15(26), 17973–17979 (2007).
    [CrossRef] [PubMed]
  21. P. P. Kiran, S. Bagchi, S. R. Krishnan, C. L. Arnold, G. R. Kumar, and A. Couairon, “Focal dynamics of multiple filaments: Microscopic imaging and reconstruction,” Phys. Rev. A 82, (2010).
    [CrossRef]
  22. H. Yang, J. Zhang, W. Yu, Y. J. Li, and Z. Y. Wei, “Long plasma channels generated by femtosecond laser pulses,” Phys. Rev. E 65, (2002).
  23. X. F. Li, L. Huang, and Y. Huang, “A new Abel inversion by means of the integrals of an input function with noise,” J. Phys. A-Math. Theor. 40, 347–360 (2007).
    [CrossRef]
  24. Z. Q. Hao, J. Zhang, Z. Zhang, X. H. Yuan, Z. Y. Zheng, X. Lu, Z. Jin, Z. H. Wang, J. Y. Zhong, and Y. Q. Liu, “Characteristics of multiple filaments generated by femtosecond laser pulses in air: Prefocused versus free propagation,” Phys. Rev. E 74, (2006).
    [CrossRef]
  25. S. Tzortzakis, M. A. Franco, Y. B. André, A. Chiron, B. Lamouroux, B. S. Prade, and A. Mysyrowicz, “Formation of a conducting channel in air by self-guided femtosecond laser pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(44 Pt A), R3505–R3507 (1999).
    [CrossRef]
  26. C. Y. Chien, B. La Fontaine, A. Desparois, Z. Jiang, T. W. Johnston, J. C. Kieffer, H. Pépin, F. Vidal, and H. P. Mercure, “Single-shot chirped-pulse spectral interferometry used to measure the femtosecond ionization dynamics of air,” Opt. Lett. 25(8), 578–580 (2000).
    [CrossRef]
  27. J. S. Liu, Z. L. Duan, Z. N. Zeng, X. H. Xie, Y. P. Deng, R. X. Li, Z. Z. Xu, and S. L. Chin, “Time-resolved investigation of low-density plasma channels produced by a kilohertz femtosecond laser in air,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 026412 (2005).
    [CrossRef] [PubMed]
  28. A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163(1-3), 29–32 (1999).
    [CrossRef]
  29. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
    [CrossRef]
  30. A. Couairon, S. Tzortzakis, L. Berge, M. Franco, B. Prade, and A. Mysyrowicz, “Infrared femtosecond light filaments in air: simulations and experiments,” J. Opt. Soc. Am. B 19(5), 1117–1131 (2002).
    [CrossRef]

2010

P. P. Kiran, S. Bagchi, S. R. Krishnan, C. L. Arnold, G. R. Kumar, and A. Couairon, “Focal dynamics of multiple filaments: Microscopic imaging and reconstruction,” Phys. Rev. A 82, (2010).
[CrossRef]

2009

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

2008

C. V. Gabel, “Femtosecond lasers in biology: nanoscale surgery with ultrafast optics,” Contemp. Phys. 49(6), 391–411 (2008).
[CrossRef]

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[CrossRef]

K. Y. Kim, A. J. Taylor, J. H. Glownia, and G. Rodriguez, “Coherent control of terahertz supercontinuum generation in ultrafast laser-gas interactions,” Nat. Photonics 2(10), 605–609 (2008).
[CrossRef]

J. Bernhardt, W. Liu, S. L. Chin, and R. Sauerbrey, “Pressure independence of intensity clamping during filamentation: theory and experiment,” Appl. Phys. B-Lasers Opt. 91(1), 45–48 (2008).
[CrossRef]

2007

X. F. Li, L. Huang, and Y. Huang, “A new Abel inversion by means of the integrals of an input function with noise,” J. Phys. A-Math. Theor. 40, 347–360 (2007).
[CrossRef]

Y. T. Li, T. T. Xi, Z. Q. Hao, Z. Zhang, X. Y. Peng, K. Li, Z. Jin, Z. Y. Zheng, Q. Z. Yu, X. Lu, and J. Zhang, “Oval-like hollow intensity distribution of tightly focused femtosecond laser pulses in air,” Opt. Express 15(26), 17973–17979 (2007).
[CrossRef] [PubMed]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep.-, Rev. Sec. Phys. Lett. 441, 47–189 (2007).

2006

F. Théberge, W. W. Liu, P. T. Simard, A. Becker, and S. L. Chin, “Plasma density inside a femtosecond laser filament in air: strong dependence on external focusing,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036406 (2006).
[CrossRef] [PubMed]

Z. Q. Hao, J. Zhang, Z. Zhang, X. H. Yuan, Z. Y. Zheng, X. Lu, Z. Jin, Z. H. Wang, J. Y. Zhong, and Y. Q. Liu, “Characteristics of multiple filaments generated by femtosecond laser pulses in air: Prefocused versus free propagation,” Phys. Rev. E 74, (2006).
[CrossRef]

2005

J. S. Liu, Z. L. Duan, Z. N. Zeng, X. H. Xie, Y. P. Deng, R. X. Li, Z. Z. Xu, and S. L. Chin, “Time-resolved investigation of low-density plasma channels produced by a kilohertz femtosecond laser in air,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 026412 (2005).
[CrossRef] [PubMed]

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Theberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
[CrossRef]

2004

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92(22), 225002 (2004).
[CrossRef] [PubMed]

K. Stelmaszczyk, P. Rohwetter, G. Mejean, J. Yu, E. Salmon, J. Kasparian, R. Ackermann, J. P. Wolf, and L. Woste, “Long-distance remote laser-induced breakdown spectroscopy using filamentation in air,” Appl. Phys. Lett. 85(18), 3977–3979 (2004).
[CrossRef]

2003

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, “Femtosecond laser pulse filamentation versus optical breakdown in H-2O,” Appl. Phys. B-Lasers Opt. 76(3), 215–229 (2003).
[CrossRef]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

2002

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
[CrossRef] [PubMed]

A. Couairon, S. Tzortzakis, L. Berge, M. Franco, B. Prade, and A. Mysyrowicz, “Infrared femtosecond light filaments in air: simulations and experiments,” J. Opt. Soc. Am. B 19(5), 1117–1131 (2002).
[CrossRef]

W. Liu, S. Petit, A. Becker, N. Akozbek, C. M. Bowden, and S. L. Chin, “Intensity clamping of a femtosecond laser pulse in condensed matter,” Opt. Commun. 202(1-3), 189–197 (2002).
[CrossRef]

H. Yang, J. Zhang, W. Yu, Y. J. Li, and Z. Y. Wei, “Long plasma channels generated by femtosecond laser pulses,” Phys. Rev. E 65, (2002).

2001

A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B-Lasers Opt. 73, 287–290 (2001).

2000

J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B-Lasers Opt. 71, 877–879 (2000).

C. Y. Chien, B. La Fontaine, A. Desparois, Z. Jiang, T. W. Johnston, J. C. Kieffer, H. Pépin, F. Vidal, and H. P. Mercure, “Single-shot chirped-pulse spectral interferometry used to measure the femtosecond ionization dynamics of air,” Opt. Lett. 25(8), 578–580 (2000).
[CrossRef]

1999

S. Tzortzakis, M. A. Franco, Y. B. André, A. Chiron, B. Lamouroux, B. S. Prade, and A. Mysyrowicz, “Formation of a conducting channel in air by self-guided femtosecond laser pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(44 Pt A), R3505–R3507 (1999).
[CrossRef]

A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163(1-3), 29–32 (1999).
[CrossRef]

1998

M. Lenzner, J. Kruger, S. Sartania, Z. Cheng, C. Spielmann, G. Mourou, W. Kautek, and F. Krausz, “Femtosecond optical breakdown in dielectrics,” Phys. Rev. Lett. 80(18), 4076–4079 (1998).
[CrossRef]

1997

A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov, “Moving focus in the propagation of ultrashort laser pulses in air,” Opt. Lett. 22(5), 304–306 (1997).
[CrossRef] [PubMed]

1995

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[CrossRef] [PubMed]

Ackermann, R.

K. Stelmaszczyk, P. Rohwetter, G. Mejean, J. Yu, E. Salmon, J. Kasparian, R. Ackermann, J. P. Wolf, and L. Woste, “Long-distance remote laser-induced breakdown spectroscopy using filamentation in air,” Appl. Phys. Lett. 85(18), 3977–3979 (2004).
[CrossRef]

Akozbek, N.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Theberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
[CrossRef]

W. Liu, S. Petit, A. Becker, N. Akozbek, C. M. Bowden, and S. L. Chin, “Intensity clamping of a femtosecond laser pulse in condensed matter,” Opt. Commun. 202(1-3), 189–197 (2002).
[CrossRef]

A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B-Lasers Opt. 73, 287–290 (2001).

Andre, Y. B.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

André, Y. B.

S. Tzortzakis, M. A. Franco, Y. B. André, A. Chiron, B. Lamouroux, B. S. Prade, and A. Mysyrowicz, “Formation of a conducting channel in air by self-guided femtosecond laser pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(44 Pt A), R3505–R3507 (1999).
[CrossRef]

Arnold, C. L.

P. P. Kiran, S. Bagchi, S. R. Krishnan, C. L. Arnold, G. R. Kumar, and A. Couairon, “Focal dynamics of multiple filaments: Microscopic imaging and reconstruction,” Phys. Rev. A 82, (2010).
[CrossRef]

Bagchi, S.

P. P. Kiran, S. Bagchi, S. R. Krishnan, C. L. Arnold, G. R. Kumar, and A. Couairon, “Focal dynamics of multiple filaments: Microscopic imaging and reconstruction,” Phys. Rev. A 82, (2010).
[CrossRef]

Becker, A.

F. Théberge, W. W. Liu, P. T. Simard, A. Becker, and S. L. Chin, “Plasma density inside a femtosecond laser filament in air: strong dependence on external focusing,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036406 (2006).
[CrossRef] [PubMed]

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Theberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
[CrossRef]

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, “Femtosecond laser pulse filamentation versus optical breakdown in H-2O,” Appl. Phys. B-Lasers Opt. 76(3), 215–229 (2003).
[CrossRef]

W. Liu, S. Petit, A. Becker, N. Akozbek, C. M. Bowden, and S. L. Chin, “Intensity clamping of a femtosecond laser pulse in condensed matter,” Opt. Commun. 202(1-3), 189–197 (2002).
[CrossRef]

A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B-Lasers Opt. 73, 287–290 (2001).

Berge, L.

A. Couairon, S. Tzortzakis, L. Berge, M. Franco, B. Prade, and A. Mysyrowicz, “Infrared femtosecond light filaments in air: simulations and experiments,” J. Opt. Soc. Am. B 19(5), 1117–1131 (2002).
[CrossRef]

Bergé, L.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92(22), 225002 (2004).
[CrossRef] [PubMed]

Bernhardt, J.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

J. Bernhardt, W. Liu, S. L. Chin, and R. Sauerbrey, “Pressure independence of intensity clamping during filamentation: theory and experiment,” Appl. Phys. B-Lasers Opt. 91(1), 45–48 (2008).
[CrossRef]

Bourayou, R.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92(22), 225002 (2004).
[CrossRef] [PubMed]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

Bowden, C. M.

W. Liu, S. Petit, A. Becker, N. Akozbek, C. M. Bowden, and S. L. Chin, “Intensity clamping of a femtosecond laser pulse in condensed matter,” Opt. Commun. 202(1-3), 189–197 (2002).
[CrossRef]

A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B-Lasers Opt. 73, 287–290 (2001).

Braun, A.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[CrossRef] [PubMed]

Brodeur, A.

A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov, “Moving focus in the propagation of ultrashort laser pulses in air,” Opt. Lett. 22(5), 304–306 (1997).
[CrossRef] [PubMed]

Cheng, Z.

M. Lenzner, J. Kruger, S. Sartania, Z. Cheng, C. Spielmann, G. Mourou, W. Kautek, and F. Krausz, “Femtosecond optical breakdown in dielectrics,” Phys. Rev. Lett. 80(18), 4076–4079 (1998).
[CrossRef]

Chien, C. Y.

C. Y. Chien, B. La Fontaine, A. Desparois, Z. Jiang, T. W. Johnston, J. C. Kieffer, H. Pépin, F. Vidal, and H. P. Mercure, “Single-shot chirped-pulse spectral interferometry used to measure the femtosecond ionization dynamics of air,” Opt. Lett. 25(8), 578–580 (2000).
[CrossRef]

A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov, “Moving focus in the propagation of ultrashort laser pulses in air,” Opt. Lett. 22(5), 304–306 (1997).
[CrossRef] [PubMed]

Chin, S. L.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

J. Bernhardt, W. Liu, S. L. Chin, and R. Sauerbrey, “Pressure independence of intensity clamping during filamentation: theory and experiment,” Appl. Phys. B-Lasers Opt. 91(1), 45–48 (2008).
[CrossRef]

F. Théberge, W. W. Liu, P. T. Simard, A. Becker, and S. L. Chin, “Plasma density inside a femtosecond laser filament in air: strong dependence on external focusing,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036406 (2006).
[CrossRef] [PubMed]

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Theberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
[CrossRef]

J. S. Liu, Z. L. Duan, Z. N. Zeng, X. H. Xie, Y. P. Deng, R. X. Li, Z. Z. Xu, and S. L. Chin, “Time-resolved investigation of low-density plasma channels produced by a kilohertz femtosecond laser in air,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 026412 (2005).
[CrossRef] [PubMed]

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, “Femtosecond laser pulse filamentation versus optical breakdown in H-2O,” Appl. Phys. B-Lasers Opt. 76(3), 215–229 (2003).
[CrossRef]

W. Liu, S. Petit, A. Becker, N. Akozbek, C. M. Bowden, and S. L. Chin, “Intensity clamping of a femtosecond laser pulse in condensed matter,” Opt. Commun. 202(1-3), 189–197 (2002).
[CrossRef]

A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B-Lasers Opt. 73, 287–290 (2001).

J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B-Lasers Opt. 71, 877–879 (2000).

A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163(1-3), 29–32 (1999).
[CrossRef]

A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov, “Moving focus in the propagation of ultrashort laser pulses in air,” Opt. Lett. 22(5), 304–306 (1997).
[CrossRef] [PubMed]

Chiron, A.

S. Tzortzakis, M. A. Franco, Y. B. André, A. Chiron, B. Lamouroux, B. S. Prade, and A. Mysyrowicz, “Formation of a conducting channel in air by self-guided femtosecond laser pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(44 Pt A), R3505–R3507 (1999).
[CrossRef]

Couairon, A.

P. P. Kiran, S. Bagchi, S. R. Krishnan, C. L. Arnold, G. R. Kumar, and A. Couairon, “Focal dynamics of multiple filaments: Microscopic imaging and reconstruction,” Phys. Rev. A 82, (2010).
[CrossRef]

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep.-, Rev. Sec. Phys. Lett. 441, 47–189 (2007).

A. Couairon, S. Tzortzakis, L. Berge, M. Franco, B. Prade, and A. Mysyrowicz, “Infrared femtosecond light filaments in air: simulations and experiments,” J. Opt. Soc. Am. B 19(5), 1117–1131 (2002).
[CrossRef]

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
[CrossRef] [PubMed]

Deng, Y. P.

J. S. Liu, Z. L. Duan, Z. N. Zeng, X. H. Xie, Y. P. Deng, R. X. Li, Z. Z. Xu, and S. L. Chin, “Time-resolved investigation of low-density plasma channels produced by a kilohertz femtosecond laser in air,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 026412 (2005).
[CrossRef] [PubMed]

Desparois, A.

C. Y. Chien, B. La Fontaine, A. Desparois, Z. Jiang, T. W. Johnston, J. C. Kieffer, H. Pépin, F. Vidal, and H. P. Mercure, “Single-shot chirped-pulse spectral interferometry used to measure the femtosecond ionization dynamics of air,” Opt. Lett. 25(8), 578–580 (2000).
[CrossRef]

Du, D.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[CrossRef] [PubMed]

Duan, Z. L.

J. S. Liu, Z. L. Duan, Z. N. Zeng, X. H. Xie, Y. P. Deng, R. X. Li, Z. Z. Xu, and S. L. Chin, “Time-resolved investigation of low-density plasma channels produced by a kilohertz femtosecond laser in air,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 026412 (2005).
[CrossRef] [PubMed]

Franco, M.

A. Couairon, S. Tzortzakis, L. Berge, M. Franco, B. Prade, and A. Mysyrowicz, “Infrared femtosecond light filaments in air: simulations and experiments,” J. Opt. Soc. Am. B 19(5), 1117–1131 (2002).
[CrossRef]

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
[CrossRef] [PubMed]

Franco, M. A.

S. Tzortzakis, M. A. Franco, Y. B. André, A. Chiron, B. Lamouroux, B. S. Prade, and A. Mysyrowicz, “Formation of a conducting channel in air by self-guided femtosecond laser pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(44 Pt A), R3505–R3507 (1999).
[CrossRef]

Frey, S.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

Gabel, C. V.

C. V. Gabel, “Femtosecond lasers in biology: nanoscale surgery with ultrafast optics,” Contemp. Phys. 49(6), 391–411 (2008).
[CrossRef]

Gattass, R. R.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[CrossRef]

Glownia, J. H.

K. Y. Kim, A. J. Taylor, J. H. Glownia, and G. Rodriguez, “Coherent control of terahertz supercontinuum generation in ultrafast laser-gas interactions,” Nat. Photonics 2(10), 605–609 (2008).
[CrossRef]

Golubtsov, I. S.

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, “Femtosecond laser pulse filamentation versus optical breakdown in H-2O,” Appl. Phys. B-Lasers Opt. 76(3), 215–229 (2003).
[CrossRef]

Hao, Z. Q.

Y. T. Li, T. T. Xi, Z. Q. Hao, Z. Zhang, X. Y. Peng, K. Li, Z. Jin, Z. Y. Zheng, Q. Z. Yu, X. Lu, and J. Zhang, “Oval-like hollow intensity distribution of tightly focused femtosecond laser pulses in air,” Opt. Express 15(26), 17973–17979 (2007).
[CrossRef] [PubMed]

Z. Q. Hao, J. Zhang, Z. Zhang, X. H. Yuan, Z. Y. Zheng, X. Lu, Z. Jin, Z. H. Wang, J. Y. Zhong, and Y. Q. Liu, “Characteristics of multiple filaments generated by femtosecond laser pulses in air: Prefocused versus free propagation,” Phys. Rev. E 74, (2006).
[CrossRef]

Hosseini, S. A.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Theberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
[CrossRef]

Huang, L.

X. F. Li, L. Huang, and Y. Huang, “A new Abel inversion by means of the integrals of an input function with noise,” J. Phys. A-Math. Theor. 40, 347–360 (2007).
[CrossRef]

Huang, Y.

X. F. Li, L. Huang, and Y. Huang, “A new Abel inversion by means of the integrals of an input function with noise,” J. Phys. A-Math. Theor. 40, 347–360 (2007).
[CrossRef]

Ilkov, F. A.

A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov, “Moving focus in the propagation of ultrashort laser pulses in air,” Opt. Lett. 22(5), 304–306 (1997).
[CrossRef] [PubMed]

Iwasaki, A.

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, “Femtosecond laser pulse filamentation versus optical breakdown in H-2O,” Appl. Phys. B-Lasers Opt. 76(3), 215–229 (2003).
[CrossRef]

Ji, Z.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

Jiang, Y.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

Jiang, Z.

C. Y. Chien, B. La Fontaine, A. Desparois, Z. Jiang, T. W. Johnston, J. C. Kieffer, H. Pépin, F. Vidal, and H. P. Mercure, “Single-shot chirped-pulse spectral interferometry used to measure the femtosecond ionization dynamics of air,” Opt. Lett. 25(8), 578–580 (2000).
[CrossRef]

Jin, Z.

Y. T. Li, T. T. Xi, Z. Q. Hao, Z. Zhang, X. Y. Peng, K. Li, Z. Jin, Z. Y. Zheng, Q. Z. Yu, X. Lu, and J. Zhang, “Oval-like hollow intensity distribution of tightly focused femtosecond laser pulses in air,” Opt. Express 15(26), 17973–17979 (2007).
[CrossRef] [PubMed]

Z. Q. Hao, J. Zhang, Z. Zhang, X. H. Yuan, Z. Y. Zheng, X. Lu, Z. Jin, Z. H. Wang, J. Y. Zhong, and Y. Q. Liu, “Characteristics of multiple filaments generated by femtosecond laser pulses in air: Prefocused versus free propagation,” Phys. Rev. E 74, (2006).
[CrossRef]

Johnston, T. W.

C. Y. Chien, B. La Fontaine, A. Desparois, Z. Jiang, T. W. Johnston, J. C. Kieffer, H. Pépin, F. Vidal, and H. P. Mercure, “Single-shot chirped-pulse spectral interferometry used to measure the femtosecond ionization dynamics of air,” Opt. Lett. 25(8), 578–580 (2000).
[CrossRef]

Ju, J.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

Kandidov, V. P.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Theberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
[CrossRef]

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, “Femtosecond laser pulse filamentation versus optical breakdown in H-2O,” Appl. Phys. B-Lasers Opt. 76(3), 215–229 (2003).
[CrossRef]

A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov, “Moving focus in the propagation of ultrashort laser pulses in air,” Opt. Lett. 22(5), 304–306 (1997).
[CrossRef] [PubMed]

Kasparian, J.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92(22), 225002 (2004).
[CrossRef] [PubMed]

K. Stelmaszczyk, P. Rohwetter, G. Mejean, J. Yu, E. Salmon, J. Kasparian, R. Ackermann, J. P. Wolf, and L. Woste, “Long-distance remote laser-induced breakdown spectroscopy using filamentation in air,” Appl. Phys. Lett. 85(18), 3977–3979 (2004).
[CrossRef]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B-Lasers Opt. 71, 877–879 (2000).

Kautek, W.

M. Lenzner, J. Kruger, S. Sartania, Z. Cheng, C. Spielmann, G. Mourou, W. Kautek, and F. Krausz, “Femtosecond optical breakdown in dielectrics,” Phys. Rev. Lett. 80(18), 4076–4079 (1998).
[CrossRef]

Kieffer, J. C.

C. Y. Chien, B. La Fontaine, A. Desparois, Z. Jiang, T. W. Johnston, J. C. Kieffer, H. Pépin, F. Vidal, and H. P. Mercure, “Single-shot chirped-pulse spectral interferometry used to measure the femtosecond ionization dynamics of air,” Opt. Lett. 25(8), 578–580 (2000).
[CrossRef]

Kim, K. Y.

K. Y. Kim, A. J. Taylor, J. H. Glownia, and G. Rodriguez, “Coherent control of terahertz supercontinuum generation in ultrafast laser-gas interactions,” Nat. Photonics 2(10), 605–609 (2008).
[CrossRef]

Kiran, P. P.

P. P. Kiran, S. Bagchi, S. R. Krishnan, C. L. Arnold, G. R. Kumar, and A. Couairon, “Focal dynamics of multiple filaments: Microscopic imaging and reconstruction,” Phys. Rev. A 82, (2010).
[CrossRef]

Korn, G.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[CrossRef] [PubMed]

Kosareva, O.

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, “Femtosecond laser pulse filamentation versus optical breakdown in H-2O,” Appl. Phys. B-Lasers Opt. 76(3), 215–229 (2003).
[CrossRef]

Kosareva, O. G.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Theberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
[CrossRef]

A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov, “Moving focus in the propagation of ultrashort laser pulses in air,” Opt. Lett. 22(5), 304–306 (1997).
[CrossRef] [PubMed]

Krausz, F.

M. Lenzner, J. Kruger, S. Sartania, Z. Cheng, C. Spielmann, G. Mourou, W. Kautek, and F. Krausz, “Femtosecond optical breakdown in dielectrics,” Phys. Rev. Lett. 80(18), 4076–4079 (1998).
[CrossRef]

Krishnan, S. R.

P. P. Kiran, S. Bagchi, S. R. Krishnan, C. L. Arnold, G. R. Kumar, and A. Couairon, “Focal dynamics of multiple filaments: Microscopic imaging and reconstruction,” Phys. Rev. A 82, (2010).
[CrossRef]

Kruger, J.

M. Lenzner, J. Kruger, S. Sartania, Z. Cheng, C. Spielmann, G. Mourou, W. Kautek, and F. Krausz, “Femtosecond optical breakdown in dielectrics,” Phys. Rev. Lett. 80(18), 4076–4079 (1998).
[CrossRef]

Kumar, G. R.

P. P. Kiran, S. Bagchi, S. R. Krishnan, C. L. Arnold, G. R. Kumar, and A. Couairon, “Focal dynamics of multiple filaments: Microscopic imaging and reconstruction,” Phys. Rev. A 82, (2010).
[CrossRef]

La Fontaine, B.

C. Y. Chien, B. La Fontaine, A. Desparois, Z. Jiang, T. W. Johnston, J. C. Kieffer, H. Pépin, F. Vidal, and H. P. Mercure, “Single-shot chirped-pulse spectral interferometry used to measure the femtosecond ionization dynamics of air,” Opt. Lett. 25(8), 578–580 (2000).
[CrossRef]

Lamouroux, B.

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
[CrossRef] [PubMed]

S. Tzortzakis, M. A. Franco, Y. B. André, A. Chiron, B. Lamouroux, B. S. Prade, and A. Mysyrowicz, “Formation of a conducting channel in air by self-guided femtosecond laser pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(44 Pt A), R3505–R3507 (1999).
[CrossRef]

Lederer, F.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92(22), 225002 (2004).
[CrossRef] [PubMed]

Leng, Y.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

Lenzner, M.

M. Lenzner, J. Kruger, S. Sartania, Z. Cheng, C. Spielmann, G. Mourou, W. Kautek, and F. Krausz, “Femtosecond optical breakdown in dielectrics,” Phys. Rev. Lett. 80(18), 4076–4079 (1998).
[CrossRef]

Li, K.

Y. T. Li, T. T. Xi, Z. Q. Hao, Z. Zhang, X. Y. Peng, K. Li, Z. Jin, Z. Y. Zheng, Q. Z. Yu, X. Lu, and J. Zhang, “Oval-like hollow intensity distribution of tightly focused femtosecond laser pulses in air,” Opt. Express 15(26), 17973–17979 (2007).
[CrossRef] [PubMed]

Li, R.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

Li, R. X.

J. S. Liu, Z. L. Duan, Z. N. Zeng, X. H. Xie, Y. P. Deng, R. X. Li, Z. Z. Xu, and S. L. Chin, “Time-resolved investigation of low-density plasma channels produced by a kilohertz femtosecond laser in air,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 026412 (2005).
[CrossRef] [PubMed]

Li, X. F.

X. F. Li, L. Huang, and Y. Huang, “A new Abel inversion by means of the integrals of an input function with noise,” J. Phys. A-Math. Theor. 40, 347–360 (2007).
[CrossRef]

Li, Y. J.

H. Yang, J. Zhang, W. Yu, Y. J. Li, and Z. Y. Wei, “Long plasma channels generated by femtosecond laser pulses,” Phys. Rev. E 65, (2002).

Li, Y. T.

Y. T. Li, T. T. Xi, Z. Q. Hao, Z. Zhang, X. Y. Peng, K. Li, Z. Jin, Z. Y. Zheng, Q. Z. Yu, X. Lu, and J. Zhang, “Oval-like hollow intensity distribution of tightly focused femtosecond laser pulses in air,” Opt. Express 15(26), 17973–17979 (2007).
[CrossRef] [PubMed]

Liang, X.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

Liu, J.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

Liu, J. S.

J. S. Liu, Z. L. Duan, Z. N. Zeng, X. H. Xie, Y. P. Deng, R. X. Li, Z. Z. Xu, and S. L. Chin, “Time-resolved investigation of low-density plasma channels produced by a kilohertz femtosecond laser in air,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 026412 (2005).
[CrossRef] [PubMed]

Liu, W.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

J. Bernhardt, W. Liu, S. L. Chin, and R. Sauerbrey, “Pressure independence of intensity clamping during filamentation: theory and experiment,” Appl. Phys. B-Lasers Opt. 91(1), 45–48 (2008).
[CrossRef]

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Theberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
[CrossRef]

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, “Femtosecond laser pulse filamentation versus optical breakdown in H-2O,” Appl. Phys. B-Lasers Opt. 76(3), 215–229 (2003).
[CrossRef]

W. Liu, S. Petit, A. Becker, N. Akozbek, C. M. Bowden, and S. L. Chin, “Intensity clamping of a femtosecond laser pulse in condensed matter,” Opt. Commun. 202(1-3), 189–197 (2002).
[CrossRef]

Liu, W. W.

F. Théberge, W. W. Liu, P. T. Simard, A. Becker, and S. L. Chin, “Plasma density inside a femtosecond laser filament in air: strong dependence on external focusing,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036406 (2006).
[CrossRef] [PubMed]

Liu, X.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[CrossRef] [PubMed]

Liu, Y. Q.

Z. Q. Hao, J. Zhang, Z. Zhang, X. H. Yuan, Z. Y. Zheng, X. Lu, Z. Jin, Z. H. Wang, J. Y. Zhong, and Y. Q. Liu, “Characteristics of multiple filaments generated by femtosecond laser pulses in air: Prefocused versus free propagation,” Phys. Rev. E 74, (2006).
[CrossRef]

Lu, X.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

Y. T. Li, T. T. Xi, Z. Q. Hao, Z. Zhang, X. Y. Peng, K. Li, Z. Jin, Z. Y. Zheng, Q. Z. Yu, X. Lu, and J. Zhang, “Oval-like hollow intensity distribution of tightly focused femtosecond laser pulses in air,” Opt. Express 15(26), 17973–17979 (2007).
[CrossRef] [PubMed]

Z. Q. Hao, J. Zhang, Z. Zhang, X. H. Yuan, Z. Y. Zheng, X. Lu, Z. Jin, Z. H. Wang, J. Y. Zhong, and Y. Q. Liu, “Characteristics of multiple filaments generated by femtosecond laser pulses in air: Prefocused versus free propagation,” Phys. Rev. E 74, (2006).
[CrossRef]

Luo, Q.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Theberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
[CrossRef]

Mazur, E.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[CrossRef]

Mejean, G.

K. Stelmaszczyk, P. Rohwetter, G. Mejean, J. Yu, E. Salmon, J. Kasparian, R. Ackermann, J. P. Wolf, and L. Woste, “Long-distance remote laser-induced breakdown spectroscopy using filamentation in air,” Appl. Phys. Lett. 85(18), 3977–3979 (2004).
[CrossRef]

Méjean, G.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92(22), 225002 (2004).
[CrossRef] [PubMed]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

Mercure, H. P.

C. Y. Chien, B. La Fontaine, A. Desparois, Z. Jiang, T. W. Johnston, J. C. Kieffer, H. Pépin, F. Vidal, and H. P. Mercure, “Single-shot chirped-pulse spectral interferometry used to measure the femtosecond ionization dynamics of air,” Opt. Lett. 25(8), 578–580 (2000).
[CrossRef]

Mourou, G.

M. Lenzner, J. Kruger, S. Sartania, Z. Cheng, C. Spielmann, G. Mourou, W. Kautek, and F. Krausz, “Femtosecond optical breakdown in dielectrics,” Phys. Rev. Lett. 80(18), 4076–4079 (1998).
[CrossRef]

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[CrossRef] [PubMed]

Mysyrowicz, A.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep.-, Rev. Sec. Phys. Lett. 441, 47–189 (2007).

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

A. Couairon, S. Tzortzakis, L. Berge, M. Franco, B. Prade, and A. Mysyrowicz, “Infrared femtosecond light filaments in air: simulations and experiments,” J. Opt. Soc. Am. B 19(5), 1117–1131 (2002).
[CrossRef]

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
[CrossRef] [PubMed]

S. Tzortzakis, M. A. Franco, Y. B. André, A. Chiron, B. Lamouroux, B. S. Prade, and A. Mysyrowicz, “Formation of a conducting channel in air by self-guided femtosecond laser pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(44 Pt A), R3505–R3507 (1999).
[CrossRef]

Nuter, R.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

Oral, E.

A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B-Lasers Opt. 73, 287–290 (2001).

Panov, N. A.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

Peng, X. Y.

Y. T. Li, T. T. Xi, Z. Q. Hao, Z. Zhang, X. Y. Peng, K. Li, Z. Jin, Z. Y. Zheng, Q. Z. Yu, X. Lu, and J. Zhang, “Oval-like hollow intensity distribution of tightly focused femtosecond laser pulses in air,” Opt. Express 15(26), 17973–17979 (2007).
[CrossRef] [PubMed]

Pépin, H.

C. Y. Chien, B. La Fontaine, A. Desparois, Z. Jiang, T. W. Johnston, J. C. Kieffer, H. Pépin, F. Vidal, and H. P. Mercure, “Single-shot chirped-pulse spectral interferometry used to measure the femtosecond ionization dynamics of air,” Opt. Lett. 25(8), 578–580 (2000).
[CrossRef]

Petit, S.

W. Liu, S. Petit, A. Becker, N. Akozbek, C. M. Bowden, and S. L. Chin, “Intensity clamping of a femtosecond laser pulse in condensed matter,” Opt. Commun. 202(1-3), 189–197 (2002).
[CrossRef]

Prade, B.

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
[CrossRef] [PubMed]

A. Couairon, S. Tzortzakis, L. Berge, M. Franco, B. Prade, and A. Mysyrowicz, “Infrared femtosecond light filaments in air: simulations and experiments,” J. Opt. Soc. Am. B 19(5), 1117–1131 (2002).
[CrossRef]

Prade, B. S.

S. Tzortzakis, M. A. Franco, Y. B. André, A. Chiron, B. Lamouroux, B. S. Prade, and A. Mysyrowicz, “Formation of a conducting channel in air by self-guided femtosecond laser pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(44 Pt A), R3505–R3507 (1999).
[CrossRef]

Rodriguez, G.

K. Y. Kim, A. J. Taylor, J. H. Glownia, and G. Rodriguez, “Coherent control of terahertz supercontinuum generation in ultrafast laser-gas interactions,” Nat. Photonics 2(10), 605–609 (2008).
[CrossRef]

Rodriguez, M.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92(22), 225002 (2004).
[CrossRef] [PubMed]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

Rohwetter, P.

K. Stelmaszczyk, P. Rohwetter, G. Mejean, J. Yu, E. Salmon, J. Kasparian, R. Ackermann, J. P. Wolf, and L. Woste, “Long-distance remote laser-induced breakdown spectroscopy using filamentation in air,” Appl. Phys. Lett. 85(18), 3977–3979 (2004).
[CrossRef]

Salmon, E.

K. Stelmaszczyk, P. Rohwetter, G. Mejean, J. Yu, E. Salmon, J. Kasparian, R. Ackermann, J. P. Wolf, and L. Woste, “Long-distance remote laser-induced breakdown spectroscopy using filamentation in air,” Appl. Phys. Lett. 85(18), 3977–3979 (2004).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92(22), 225002 (2004).
[CrossRef] [PubMed]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

Sartania, S.

M. Lenzner, J. Kruger, S. Sartania, Z. Cheng, C. Spielmann, G. Mourou, W. Kautek, and F. Krausz, “Femtosecond optical breakdown in dielectrics,” Phys. Rev. Lett. 80(18), 4076–4079 (1998).
[CrossRef]

Sauerbrey, R.

J. Bernhardt, W. Liu, S. L. Chin, and R. Sauerbrey, “Pressure independence of intensity clamping during filamentation: theory and experiment,” Appl. Phys. B-Lasers Opt. 91(1), 45–48 (2008).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92(22), 225002 (2004).
[CrossRef] [PubMed]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B-Lasers Opt. 71, 877–879 (2000).

Schroeder, H.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Theberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
[CrossRef]

Sharifi, M.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

Simard, P. T.

F. Théberge, W. W. Liu, P. T. Simard, A. Becker, and S. L. Chin, “Plasma density inside a femtosecond laser filament in air: strong dependence on external focusing,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036406 (2006).
[CrossRef] [PubMed]

Skupin, S.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92(22), 225002 (2004).
[CrossRef] [PubMed]

Spielmann, C.

M. Lenzner, J. Kruger, S. Sartania, Z. Cheng, C. Spielmann, G. Mourou, W. Kautek, and F. Krausz, “Femtosecond optical breakdown in dielectrics,” Phys. Rev. Lett. 80(18), 4076–4079 (1998).
[CrossRef]

Squier, J.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[CrossRef] [PubMed]

Stelmaszczyk, K.

K. Stelmaszczyk, P. Rohwetter, G. Mejean, J. Yu, E. Salmon, J. Kasparian, R. Ackermann, J. P. Wolf, and L. Woste, “Long-distance remote laser-induced breakdown spectroscopy using filamentation in air,” Appl. Phys. Lett. 85(18), 3977–3979 (2004).
[CrossRef]

Sudrie, L.

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
[CrossRef] [PubMed]

Talebpour, A.

A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163(1-3), 29–32 (1999).
[CrossRef]

Taylor, A. J.

K. Y. Kim, A. J. Taylor, J. H. Glownia, and G. Rodriguez, “Coherent control of terahertz supercontinuum generation in ultrafast laser-gas interactions,” Nat. Photonics 2(10), 605–609 (2008).
[CrossRef]

Theberge, F.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Theberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
[CrossRef]

Théberge, F.

F. Théberge, W. W. Liu, P. T. Simard, A. Becker, and S. L. Chin, “Plasma density inside a femtosecond laser filament in air: strong dependence on external focusing,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036406 (2006).
[CrossRef] [PubMed]

Tzortzakis, S.

A. Couairon, S. Tzortzakis, L. Berge, M. Franco, B. Prade, and A. Mysyrowicz, “Infrared femtosecond light filaments in air: simulations and experiments,” J. Opt. Soc. Am. B 19(5), 1117–1131 (2002).
[CrossRef]

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
[CrossRef] [PubMed]

S. Tzortzakis, M. A. Franco, Y. B. André, A. Chiron, B. Lamouroux, B. S. Prade, and A. Mysyrowicz, “Formation of a conducting channel in air by self-guided femtosecond laser pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(44 Pt A), R3505–R3507 (1999).
[CrossRef]

Vidal, F.

C. Y. Chien, B. La Fontaine, A. Desparois, Z. Jiang, T. W. Johnston, J. C. Kieffer, H. Pépin, F. Vidal, and H. P. Mercure, “Single-shot chirped-pulse spectral interferometry used to measure the femtosecond ionization dynamics of air,” Opt. Lett. 25(8), 578–580 (2000).
[CrossRef]

Vijayalakshmi, K.

A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B-Lasers Opt. 73, 287–290 (2001).

Wang, Z.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

Wang, Z. H.

Z. Q. Hao, J. Zhang, Z. Zhang, X. H. Yuan, Z. Y. Zheng, X. Lu, Z. Jin, Z. H. Wang, J. Y. Zhong, and Y. Q. Liu, “Characteristics of multiple filaments generated by femtosecond laser pulses in air: Prefocused versus free propagation,” Phys. Rev. E 74, (2006).
[CrossRef]

Wei, Z. Y.

H. Yang, J. Zhang, W. Yu, Y. J. Li, and Z. Y. Wei, “Long plasma channels generated by femtosecond laser pulses,” Phys. Rev. E 65, (2002).

Wille, H.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

Wolf, J. P.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92(22), 225002 (2004).
[CrossRef] [PubMed]

K. Stelmaszczyk, P. Rohwetter, G. Mejean, J. Yu, E. Salmon, J. Kasparian, R. Ackermann, J. P. Wolf, and L. Woste, “Long-distance remote laser-induced breakdown spectroscopy using filamentation in air,” Appl. Phys. Lett. 85(18), 3977–3979 (2004).
[CrossRef]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

Woste, L.

K. Stelmaszczyk, P. Rohwetter, G. Mejean, J. Yu, E. Salmon, J. Kasparian, R. Ackermann, J. P. Wolf, and L. Woste, “Long-distance remote laser-induced breakdown spectroscopy using filamentation in air,” Appl. Phys. Lett. 85(18), 3977–3979 (2004).
[CrossRef]

Wöste, L.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92(22), 225002 (2004).
[CrossRef] [PubMed]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

Xi, T. T.

Y. T. Li, T. T. Xi, Z. Q. Hao, Z. Zhang, X. Y. Peng, K. Li, Z. Jin, Z. Y. Zheng, Q. Z. Yu, X. Lu, and J. Zhang, “Oval-like hollow intensity distribution of tightly focused femtosecond laser pulses in air,” Opt. Express 15(26), 17973–17979 (2007).
[CrossRef] [PubMed]

Xie, X. H.

J. S. Liu, Z. L. Duan, Z. N. Zeng, X. H. Xie, Y. P. Deng, R. X. Li, Z. Z. Xu, and S. L. Chin, “Time-resolved investigation of low-density plasma channels produced by a kilohertz femtosecond laser in air,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 026412 (2005).
[CrossRef] [PubMed]

Xu, Z.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

Xu, Z. Z.

J. S. Liu, Z. L. Duan, Z. N. Zeng, X. H. Xie, Y. P. Deng, R. X. Li, Z. Z. Xu, and S. L. Chin, “Time-resolved investigation of low-density plasma channels produced by a kilohertz femtosecond laser in air,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 026412 (2005).
[CrossRef] [PubMed]

Yang, H.

H. Yang, J. Zhang, W. Yu, Y. J. Li, and Z. Y. Wei, “Long plasma channels generated by femtosecond laser pulses,” Phys. Rev. E 65, (2002).

Yang, J.

A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163(1-3), 29–32 (1999).
[CrossRef]

Yu, J.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92(22), 225002 (2004).
[CrossRef] [PubMed]

K. Stelmaszczyk, P. Rohwetter, G. Mejean, J. Yu, E. Salmon, J. Kasparian, R. Ackermann, J. P. Wolf, and L. Woste, “Long-distance remote laser-induced breakdown spectroscopy using filamentation in air,” Appl. Phys. Lett. 85(18), 3977–3979 (2004).
[CrossRef]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

Yu, Q. Z.

Y. T. Li, T. T. Xi, Z. Q. Hao, Z. Zhang, X. Y. Peng, K. Li, Z. Jin, Z. Y. Zheng, Q. Z. Yu, X. Lu, and J. Zhang, “Oval-like hollow intensity distribution of tightly focused femtosecond laser pulses in air,” Opt. Express 15(26), 17973–17979 (2007).
[CrossRef] [PubMed]

Yu, W.

H. Yang, J. Zhang, W. Yu, Y. J. Li, and Z. Y. Wei, “Long plasma channels generated by femtosecond laser pulses,” Phys. Rev. E 65, (2002).

Yuan, X. H.

Z. Q. Hao, J. Zhang, Z. Zhang, X. H. Yuan, Z. Y. Zheng, X. Lu, Z. Jin, Z. H. Wang, J. Y. Zhong, and Y. Q. Liu, “Characteristics of multiple filaments generated by femtosecond laser pulses in air: Prefocused versus free propagation,” Phys. Rev. E 74, (2006).
[CrossRef]

Zeng, Z. N.

J. S. Liu, Z. L. Duan, Z. N. Zeng, X. H. Xie, Y. P. Deng, R. X. Li, Z. Z. Xu, and S. L. Chin, “Time-resolved investigation of low-density plasma channels produced by a kilohertz femtosecond laser in air,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 026412 (2005).
[CrossRef] [PubMed]

Zhang, J.

Y. T. Li, T. T. Xi, Z. Q. Hao, Z. Zhang, X. Y. Peng, K. Li, Z. Jin, Z. Y. Zheng, Q. Z. Yu, X. Lu, and J. Zhang, “Oval-like hollow intensity distribution of tightly focused femtosecond laser pulses in air,” Opt. Express 15(26), 17973–17979 (2007).
[CrossRef] [PubMed]

Z. Q. Hao, J. Zhang, Z. Zhang, X. H. Yuan, Z. Y. Zheng, X. Lu, Z. Jin, Z. H. Wang, J. Y. Zhong, and Y. Q. Liu, “Characteristics of multiple filaments generated by femtosecond laser pulses in air: Prefocused versus free propagation,” Phys. Rev. E 74, (2006).
[CrossRef]

H. Yang, J. Zhang, W. Yu, Y. J. Li, and Z. Y. Wei, “Long plasma channels generated by femtosecond laser pulses,” Phys. Rev. E 65, (2002).

Zhang, Z.

Y. T. Li, T. T. Xi, Z. Q. Hao, Z. Zhang, X. Y. Peng, K. Li, Z. Jin, Z. Y. Zheng, Q. Z. Yu, X. Lu, and J. Zhang, “Oval-like hollow intensity distribution of tightly focused femtosecond laser pulses in air,” Opt. Express 15(26), 17973–17979 (2007).
[CrossRef] [PubMed]

Z. Q. Hao, J. Zhang, Z. Zhang, X. H. Yuan, Z. Y. Zheng, X. Lu, Z. Jin, Z. H. Wang, J. Y. Zhong, and Y. Q. Liu, “Characteristics of multiple filaments generated by femtosecond laser pulses in air: Prefocused versus free propagation,” Phys. Rev. E 74, (2006).
[CrossRef]

Zheng, Z. Y.

Y. T. Li, T. T. Xi, Z. Q. Hao, Z. Zhang, X. Y. Peng, K. Li, Z. Jin, Z. Y. Zheng, Q. Z. Yu, X. Lu, and J. Zhang, “Oval-like hollow intensity distribution of tightly focused femtosecond laser pulses in air,” Opt. Express 15(26), 17973–17979 (2007).
[CrossRef] [PubMed]

Z. Q. Hao, J. Zhang, Z. Zhang, X. H. Yuan, Z. Y. Zheng, X. Lu, Z. Jin, Z. H. Wang, J. Y. Zhong, and Y. Q. Liu, “Characteristics of multiple filaments generated by femtosecond laser pulses in air: Prefocused versus free propagation,” Phys. Rev. E 74, (2006).
[CrossRef]

Zhong, J. Y.

Z. Q. Hao, J. Zhang, Z. Zhang, X. H. Yuan, Z. Y. Zheng, X. Lu, Z. Jin, Z. H. Wang, J. Y. Zhong, and Y. Q. Liu, “Characteristics of multiple filaments generated by femtosecond laser pulses in air: Prefocused versus free propagation,” Phys. Rev. E 74, (2006).
[CrossRef]

Appl. Phys. B-Lasers Opt.

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, “Femtosecond laser pulse filamentation versus optical breakdown in H-2O,” Appl. Phys. B-Lasers Opt. 76(3), 215–229 (2003).
[CrossRef]

J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B-Lasers Opt. 71, 877–879 (2000).

A. Becker, N. Akozbek, K. Vijayalakshmi, E. Oral, C. M. Bowden, and S. L. Chin, “Intensity clamping and re-focusing of intense femtosecond laser pulses in nitrogen molecular gas,” Appl. Phys. B-Lasers Opt. 73, 287–290 (2001).

J. Bernhardt, W. Liu, S. L. Chin, and R. Sauerbrey, “Pressure independence of intensity clamping during filamentation: theory and experiment,” Appl. Phys. B-Lasers Opt. 91(1), 45–48 (2008).
[CrossRef]

Appl. Phys. Lett.

K. Stelmaszczyk, P. Rohwetter, G. Mejean, J. Yu, E. Salmon, J. Kasparian, R. Ackermann, J. P. Wolf, and L. Woste, “Long-distance remote laser-induced breakdown spectroscopy using filamentation in air,” Appl. Phys. Lett. 85(18), 3977–3979 (2004).
[CrossRef]

Can. J. Phys.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Theberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
[CrossRef]

Contemp. Phys.

C. V. Gabel, “Femtosecond lasers in biology: nanoscale surgery with ultrafast optics,” Contemp. Phys. 49(6), 391–411 (2008).
[CrossRef]

J. Opt. Soc. Am. B

A. Couairon, S. Tzortzakis, L. Berge, M. Franco, B. Prade, and A. Mysyrowicz, “Infrared femtosecond light filaments in air: simulations and experiments,” J. Opt. Soc. Am. B 19(5), 1117–1131 (2002).
[CrossRef]

J. Phys. A-Math. Theor.

X. F. Li, L. Huang, and Y. Huang, “A new Abel inversion by means of the integrals of an input function with noise,” J. Phys. A-Math. Theor. 40, 347–360 (2007).
[CrossRef]

Laser Phys.

O. G. Kosareva, W. Liu, N. A. Panov, J. Bernhardt, Z. Ji, M. Sharifi, R. Li, Z. Xu, J. Liu, Z. Wang, J. Ju, X. Lu, Y. Jiang, Y. Leng, X. Liang, V. P. Kandidov, and S. L. Chin, “Can we reach very high intensity in air with femtosecond PW laser pulses?” Laser Phys. 19(8), 1776–1792 (2009).
[CrossRef]

Nat. Photonics

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008).
[CrossRef]

K. Y. Kim, A. J. Taylor, J. H. Glownia, and G. Rodriguez, “Coherent control of terahertz supercontinuum generation in ultrafast laser-gas interactions,” Nat. Photonics 2(10), 605–609 (2008).
[CrossRef]

Opt. Commun.

W. Liu, S. Petit, A. Becker, N. Akozbek, C. M. Bowden, and S. L. Chin, “Intensity clamping of a femtosecond laser pulse in condensed matter,” Opt. Commun. 202(1-3), 189–197 (2002).
[CrossRef]

A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163(1-3), 29–32 (1999).
[CrossRef]

Opt. Express

Y. T. Li, T. T. Xi, Z. Q. Hao, Z. Zhang, X. Y. Peng, K. Li, Z. Jin, Z. Y. Zheng, Q. Z. Yu, X. Lu, and J. Zhang, “Oval-like hollow intensity distribution of tightly focused femtosecond laser pulses in air,” Opt. Express 15(26), 17973–17979 (2007).
[CrossRef] [PubMed]

Opt. Lett.

A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20(1), 73–75 (1995).
[CrossRef] [PubMed]

A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov, “Moving focus in the propagation of ultrashort laser pulses in air,” Opt. Lett. 22(5), 304–306 (1997).
[CrossRef] [PubMed]

C. Y. Chien, B. La Fontaine, A. Desparois, Z. Jiang, T. W. Johnston, J. C. Kieffer, H. Pépin, F. Vidal, and H. P. Mercure, “Single-shot chirped-pulse spectral interferometry used to measure the femtosecond ionization dynamics of air,” Opt. Lett. 25(8), 578–580 (2000).
[CrossRef]

Phys. Rep.-, Rev. Sec. Phys. Lett.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep.-, Rev. Sec. Phys. Lett. 441, 47–189 (2007).

Phys. Rev. A

P. P. Kiran, S. Bagchi, S. R. Krishnan, C. L. Arnold, G. R. Kumar, and A. Couairon, “Focal dynamics of multiple filaments: Microscopic imaging and reconstruction,” Phys. Rev. A 82, (2010).
[CrossRef]

Phys. Rev. E

H. Yang, J. Zhang, W. Yu, Y. J. Li, and Z. Y. Wei, “Long plasma channels generated by femtosecond laser pulses,” Phys. Rev. E 65, (2002).

Z. Q. Hao, J. Zhang, Z. Zhang, X. H. Yuan, Z. Y. Zheng, X. Lu, Z. Jin, Z. H. Wang, J. Y. Zhong, and Y. Q. Liu, “Characteristics of multiple filaments generated by femtosecond laser pulses in air: Prefocused versus free propagation,” Phys. Rev. E 74, (2006).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

J. S. Liu, Z. L. Duan, Z. N. Zeng, X. H. Xie, Y. P. Deng, R. X. Li, Z. Z. Xu, and S. L. Chin, “Time-resolved investigation of low-density plasma channels produced by a kilohertz femtosecond laser in air,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(2), 026412 (2005).
[CrossRef] [PubMed]

F. Théberge, W. W. Liu, P. T. Simard, A. Becker, and S. L. Chin, “Plasma density inside a femtosecond laser filament in air: strong dependence on external focusing,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036406 (2006).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics

S. Tzortzakis, M. A. Franco, Y. B. André, A. Chiron, B. Lamouroux, B. S. Prade, and A. Mysyrowicz, “Formation of a conducting channel in air by self-guided femtosecond laser pulses,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(44 Pt A), R3505–R3507 (1999).
[CrossRef]

Phys. Rev. Lett.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92(22), 225002 (2004).
[CrossRef] [PubMed]

M. Lenzner, J. Kruger, S. Sartania, Z. Cheng, C. Spielmann, G. Mourou, W. Kautek, and F. Krausz, “Femtosecond optical breakdown in dielectrics,” Phys. Rev. Lett. 80(18), 4076–4079 (1998).
[CrossRef]

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
[CrossRef] [PubMed]

Rep. Prog. Phys.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

Science

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

The experimental setup for shadowgraphic and interferometric diagnostics with the initial laser beam profile

Fig. 2
Fig. 2

Time-resolved shadowgraphs of filament bunch in air produced by 16.5 mJ laser pulse (The dash line indicates the position of the focal plane)

Fig. 3
Fig. 3

Shadowgraphs of filament bunch produced by laser pulse with different energy of 2.4 mJ, 4.5 mJ, 7.5 mJ, 9 mJ, 16.5 mJ, 22 mJ, 38 mJ and 47 mJ (The dash lines indicate the position of the focal plane).

Fig. 4
Fig. 4

The initial width of filament bunch (circle symbols) and the estimated laser intensity from the transverse size of the filament bunch (square symbols) and from the electron density (triangle symbols) as a function of laser energy.

Fig. 5
Fig. 5

Interferograms and the corresponding electron density (Ne) of air plasma induced by laser pulse with different energy: (a) 16.5 mJ. (b) 22 mJ. (c) 38 mJ. (d) 47 mJ. (The dash lines indicate the position of the focal plane)

Fig. 6
Fig. 6

The electron density as a function of the laser intensity.

Fig. 7
Fig. 7

The first (solid line) and second level (dash line) ionization rate of air.

Fig. 8
Fig. 8

The electron density generated by the first level ionization of air as a function of laser peak intensity with (dash line) and without (solid line) collisional ionization.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

Δ Φ = 2 π D = ( k p k 0 ) d l = ω 2 c n c n e d l
ρ 1 t = W a i r ( I ) ρ 1 + W a i r ( I ) ( ρ a i r ρ 1 ρ 2 ) ,
ρ 2 t = W a i r ( I ) ρ 1 ,
ρ ( t ) = ρ 1 ( t ) + 2 × ρ 2 ( t ) .
W = ω a . u . 3 / 2 π | C n * , l * | 2 f ( l , m ) U i U H ( 2 F 0 F 1 + γ 2 ) 2 n * | m | 3 / 2 A m ( ω 0 , γ ) exp [ 2 F 0 3 F g ( γ ) ]
| C n * , l * | 2 = 2 2 n * n * Γ ( n * + l * + 1 ) Γ ( n * l * )
f ( l , m ) = ( 2 l + 1 ) ( l + | m | ) ! 2 | m | | m | ! ( l | m | ) !
A m ( ω 0 , γ ) = 4 3 π 1 | m | ! γ 2 1 + γ 2 κ v + exp [ α ( κ v ) ] ϕ m [ β ( κ v ) ] ,
v = U i ω 0 ( 1 + 1 2 γ 2 ) ,
ϕ m [ β ( κ v ) ] = exp ( x 2 ) 0 x ( x 2 y 2 ) | m | exp ( y 2 ) d y ,
α ( γ ) = 2 [ sin h 1 γ γ 1 + γ 2 ] ,
β ( γ ) = 2 γ 1 + γ 2 ,
g ( γ ) = 3 2 γ [ ( 1 + 1 2 γ 2 ) sin h 1 γ 1 + γ 2 2 γ ] .
ρ t = W a i r ( I ) ( ρ a i r ρ ) + σ U i ρ I .

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