Abstract

We examine two analytical models describing the photoionization of O2 molecules in both tunnel and multiphoton regimes. Applying these models to the nonlinear propagation of chirped laser beams in the atmosphere, we demonstrate how chirp phases can be used as a tool to monitor the self-guiding range of femtosecond filaments or even to shorten the pulse duration. The good agreement between simulations and experimental measurements validates our numerical code for ultrashort-pulse propagation in air.

© 2006 Optical Society of America

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    [CrossRef] [PubMed]
  4. B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J. C. Kieffer, H. Pepin, and H. P. Mercure, "Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air," Phys. Plasmas 6, 1615-1621 (1999).
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    [CrossRef]
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  8. A. M. Perelomov and V. S. Popov, "Ionization of atoms in alternating electric field. III," Sov. Phys. JETP 25, 336-343 (1967).
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    [CrossRef]
  12. M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
    [CrossRef]
  13. I. S. Golubtsov, V. P. Kandidov, and O. G. Kosareva, "Initial phase modulation of a high-power femtosecond laser pulse as a tool for controlling its filamentation and generation of a supercontinuum in air," Quantum Electron. 33, 525-530 (2003).
    [CrossRef]
  14. V. P. Kandidov, I. S. Golubtsov, and O. G. Kosareva, "Supercontinuum sources in a high-power femtosecond laser pulse propagating in liquids and gases," Quantum Electron. 34, 348-354 (2004).
    [CrossRef]
  15. I. Alexeev, A. Ting, D. F. Gordon, E. Briscoe, J. R. Peñano, R. F. Hubbard, and Ph. Sprangle, "Longitudinal compression of short laser pulses in air," Appl. Phys. Lett. 84, 4080-4082 (2004).
    [CrossRef]
  16. J. F. Ripoche, G. Grillon, B. Prade, M. Franco, E. Nibbering, R. Lange, and A. Mysyrowicz, "Determination of the time dependence of n2 in air," Opt. Commun. 135, 310-314 (1997).
    [CrossRef]
  17. L. Bergé, "Boosted propagation of femtosecond filaments in air by double-pulse combination," Phys. Rev. E 69, 065601 (2004).
    [CrossRef]
  18. N. Aközbek, M. Scalora, C. M. Bowden, and S. L. Chin, "White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air," Opt. Commun. 191, 353-362 (2001).
    [CrossRef]
  19. S. Champeaux and L. Bergé, "Femtosecond pulse compression in pressure-gas cells filled with argon," Phys. Rev. E 68, 066603 (2003).
    [CrossRef]
  20. L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov. Phys. JETP 20, 1307-1314 (1965).
  21. M. J. DeWitt, E. Wells, and R. R. Jones, "Radiometric comparison of intense field ionization of atoms and diatomic molecules," Phys. Rev. Lett. 87, 153001 (2001).
    [CrossRef] [PubMed]
  22. L. Bergé, S. Skupin, G. Méjean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P. Wolf, "Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air," Phys. Rev. E 71, 016602 (2005).
    [CrossRef]
  23. J. H. Marburger, "Self-focusing: theory," Prog. Quantum Electron. 4, 35-110 (1975).
    [CrossRef]
  24. A. Vinçotte and L. Bergé, "Chi(5) susceptibility stabilizes the propagation of ultra-short laser pulses in air," Phys. Rev. A 70, 061802 (2004).
    [CrossRef]
  25. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).
  26. S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, "Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters," Phys. Rev. E 70, 046602 (2004).
    [CrossRef]

2005 (1)

L. Bergé, S. Skupin, G. Méjean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P. Wolf, "Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air," Phys. Rev. E 71, 016602 (2005).
[CrossRef]

2004 (6)

A. Vinçotte and L. Bergé, "Chi(5) susceptibility stabilizes the propagation of ultra-short laser pulses in air," Phys. Rev. A 70, 061802 (2004).
[CrossRef]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, "Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters," Phys. Rev. E 70, 046602 (2004).
[CrossRef]

M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
[CrossRef]

V. P. Kandidov, I. S. Golubtsov, and O. G. Kosareva, "Supercontinuum sources in a high-power femtosecond laser pulse propagating in liquids and gases," Quantum Electron. 34, 348-354 (2004).
[CrossRef]

I. Alexeev, A. Ting, D. F. Gordon, E. Briscoe, J. R. Peñano, R. F. Hubbard, and Ph. Sprangle, "Longitudinal compression of short laser pulses in air," Appl. Phys. Lett. 84, 4080-4082 (2004).
[CrossRef]

L. Bergé, "Boosted propagation of femtosecond filaments in air by double-pulse combination," Phys. Rev. E 69, 065601 (2004).
[CrossRef]

2003 (2)

I. S. Golubtsov, V. P. Kandidov, and O. G. Kosareva, "Initial phase modulation of a high-power femtosecond laser pulse as a tool for controlling its filamentation and generation of a supercontinuum in air," Quantum Electron. 33, 525-530 (2003).
[CrossRef]

S. Champeaux and L. Bergé, "Femtosecond pulse compression in pressure-gas cells filled with argon," Phys. Rev. E 68, 066603 (2003).
[CrossRef]

2002 (2)

2001 (2)

N. Aközbek, M. Scalora, C. M. Bowden, and S. L. Chin, "White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air," Opt. Commun. 191, 353-362 (2001).
[CrossRef]

M. J. DeWitt, E. Wells, and R. R. Jones, "Radiometric comparison of intense field ionization of atoms and diatomic molecules," Phys. Rev. Lett. 87, 153001 (2001).
[CrossRef] [PubMed]

1999 (2)

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, 29-32 (1999).
[CrossRef]

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J. C. Kieffer, H. Pepin, and H. P. Mercure, "Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air," Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

1998 (1)

1997 (2)

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

J. F. Ripoche, G. Grillon, B. Prade, M. Franco, E. Nibbering, R. Lange, and A. Mysyrowicz, "Determination of the time dependence of n2 in air," Opt. Commun. 135, 310-314 (1997).
[CrossRef]

1996 (1)

1995 (1)

1986 (1)

M. V. Ammosov, N. B. Delone, and V. P. Krainov, "Tunnel ionization of complex atoms and atomic ions in an alternating electromagnetic field," Sov. Phys. JETP 64, 1191-1194 (1986).

1975 (1)

J. H. Marburger, "Self-focusing: theory," Prog. Quantum Electron. 4, 35-110 (1975).
[CrossRef]

1967 (1)

A. M. Perelomov and V. S. Popov, "Ionization of atoms in alternating electric field. III," Sov. Phys. JETP 25, 336-343 (1967).

1966 (1)

A. M. Perelomov, V. S. Popov, and M. V. Terent'ev, "Ionization of atoms in alternating electric field," Sov. Phys. JETP 23, 924-934 (1966).

1965 (1)

L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov. Phys. JETP 20, 1307-1314 (1965).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

Aközbek, N.

N. Aközbek, M. Scalora, C. M. Bowden, and S. L. Chin, "White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air," Opt. Commun. 191, 353-362 (2001).
[CrossRef]

Alexeev, I.

I. Alexeev, A. Ting, D. F. Gordon, E. Briscoe, J. R. Peñano, R. F. Hubbard, and Ph. Sprangle, "Longitudinal compression of short laser pulses in air," Appl. Phys. Lett. 84, 4080-4082 (2004).
[CrossRef]

Ammosov, M. V.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, "Tunnel ionization of complex atoms and atomic ions in an alternating electromagnetic field," Sov. Phys. JETP 64, 1191-1194 (1986).

Bergé, L.

L. Bergé, S. Skupin, G. Méjean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P. Wolf, "Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air," Phys. Rev. E 71, 016602 (2005).
[CrossRef]

A. Vinçotte and L. Bergé, "Chi(5) susceptibility stabilizes the propagation of ultra-short laser pulses in air," Phys. Rev. A 70, 061802 (2004).
[CrossRef]

L. Bergé, "Boosted propagation of femtosecond filaments in air by double-pulse combination," Phys. Rev. E 69, 065601 (2004).
[CrossRef]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, "Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters," Phys. Rev. E 70, 046602 (2004).
[CrossRef]

S. Champeaux and L. Bergé, "Femtosecond pulse compression in pressure-gas cells filled with argon," Phys. Rev. E 68, 066603 (2003).
[CrossRef]

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

Bourayou, R.

M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
[CrossRef]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, "Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters," Phys. Rev. E 70, 046602 (2004).
[CrossRef]

Bowden, C. M.

N. Aközbek, M. Scalora, C. M. Bowden, and S. L. Chin, "White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air," Opt. Commun. 191, 353-362 (2001).
[CrossRef]

Braun, A.

Briscoe, E.

I. Alexeev, A. Ting, D. F. Gordon, E. Briscoe, J. R. Peñano, R. F. Hubbard, and Ph. Sprangle, "Longitudinal compression of short laser pulses in air," Appl. Phys. Lett. 84, 4080-4082 (2004).
[CrossRef]

Brodeur, A.

Champeaux, S.

S. Champeaux and L. Bergé, "Femtosecond pulse compression in pressure-gas cells filled with argon," Phys. Rev. E 68, 066603 (2003).
[CrossRef]

Chien, C. Y.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J. C. Kieffer, H. Pepin, and H. P. Mercure, "Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air," Phys. Plasmas 6, 1615-1621 (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 ultra-short laser pulses in air," Opt. Lett. 22, 304-306 (1997).
[CrossRef] [PubMed]

Chin, S. L.

N. Aközbek, M. Scalora, C. M. Bowden, and S. L. Chin, "White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air," Opt. Commun. 191, 353-362 (2001).
[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, 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 ultra-short laser pulses in air," Opt. Lett. 22, 304-306 (1997).
[CrossRef] [PubMed]

Comtois, D.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J. C. Kieffer, H. Pepin, and H. P. Mercure, "Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air," Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Couairon, A.

Curley, P. F.

Delone, N. B.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, "Tunnel ionization of complex atoms and atomic ions in an alternating electromagnetic field," Sov. Phys. JETP 64, 1191-1194 (1986).

Desparois, A.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J. C. Kieffer, H. Pepin, and H. P. Mercure, "Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air," Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

DeWitt, M. J.

M. J. DeWitt, E. Wells, and R. R. Jones, "Radiometric comparison of intense field ionization of atoms and diatomic molecules," Phys. Rev. Lett. 87, 153001 (2001).
[CrossRef] [PubMed]

Du, D.

Eislöffel, J.

M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
[CrossRef]

Franco, M.

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

J. F. Ripoche, G. Grillon, B. Prade, M. Franco, E. Nibbering, R. Lange, and A. Mysyrowicz, "Determination of the time dependence of n2 in air," Opt. Commun. 135, 310-314 (1997).
[CrossRef]

Franco, M. A.

Frey, S.

L. Bergé, S. Skupin, G. Méjean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P. Wolf, "Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air," Phys. Rev. E 71, 016602 (2005).
[CrossRef]

Golubtsov, I. S.

V. P. Kandidov, I. S. Golubtsov, and O. G. Kosareva, "Supercontinuum sources in a high-power femtosecond laser pulse propagating in liquids and gases," Quantum Electron. 34, 348-354 (2004).
[CrossRef]

I. S. Golubtsov, V. P. Kandidov, and O. G. Kosareva, "Initial phase modulation of a high-power femtosecond laser pulse as a tool for controlling its filamentation and generation of a supercontinuum in air," Quantum Electron. 33, 525-530 (2003).
[CrossRef]

Gordon, D. F.

I. Alexeev, A. Ting, D. F. Gordon, E. Briscoe, J. R. Peñano, R. F. Hubbard, and Ph. Sprangle, "Longitudinal compression of short laser pulses in air," Appl. Phys. Lett. 84, 4080-4082 (2004).
[CrossRef]

Grillon, G.

J. F. Ripoche, G. Grillon, B. Prade, M. Franco, E. Nibbering, R. Lange, and A. Mysyrowicz, "Determination of the time dependence of n2 in air," Opt. Commun. 135, 310-314 (1997).
[CrossRef]

E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysyrowicz, "Conical emission from self-guided femtosecond pulses in air," Opt. Lett. 21, 62-64 (1996).
[CrossRef]

Hatzes, A. P.

M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
[CrossRef]

Hubbard, R. F.

I. Alexeev, A. Ting, D. F. Gordon, E. Briscoe, J. R. Peñano, R. F. Hubbard, and Ph. Sprangle, "Longitudinal compression of short laser pulses in air," Appl. Phys. Lett. 84, 4080-4082 (2004).
[CrossRef]

Ilkov, F. A.

Jiang, Z.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J. C. Kieffer, H. Pepin, and H. P. Mercure, "Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air," Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Johnston, T. W.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J. C. Kieffer, H. Pepin, and H. P. Mercure, "Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air," Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Jones, R. R.

M. J. DeWitt, E. Wells, and R. R. Jones, "Radiometric comparison of intense field ionization of atoms and diatomic molecules," Phys. Rev. Lett. 87, 153001 (2001).
[CrossRef] [PubMed]

Kandidov, V. P.

V. P. Kandidov, I. S. Golubtsov, and O. G. Kosareva, "Supercontinuum sources in a high-power femtosecond laser pulse propagating in liquids and gases," Quantum Electron. 34, 348-354 (2004).
[CrossRef]

I. S. Golubtsov, V. P. Kandidov, and O. G. Kosareva, "Initial phase modulation of a high-power femtosecond laser pulse as a tool for controlling its filamentation and generation of a supercontinuum in air," Quantum Electron. 33, 525-530 (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 ultra-short laser pulses in air," Opt. Lett. 22, 304-306 (1997).
[CrossRef] [PubMed]

Kasparian, J.

L. Bergé, S. Skupin, G. Méjean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P. Wolf, "Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air," Phys. Rev. E 71, 016602 (2005).
[CrossRef]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, "Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters," Phys. Rev. E 70, 046602 (2004).
[CrossRef]

M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
[CrossRef]

Keldysh, L. V.

L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov. Phys. JETP 20, 1307-1314 (1965).

Kieffer, J. C.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J. C. Kieffer, H. Pepin, and H. P. Mercure, "Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air," Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Korn, G.

Kosareva, O. G.

V. P. Kandidov, I. S. Golubtsov, and O. G. Kosareva, "Supercontinuum sources in a high-power femtosecond laser pulse propagating in liquids and gases," Quantum Electron. 34, 348-354 (2004).
[CrossRef]

I. S. Golubtsov, V. P. Kandidov, and O. G. Kosareva, "Initial phase modulation of a high-power femtosecond laser pulse as a tool for controlling its filamentation and generation of a supercontinuum in air," Quantum Electron. 33, 525-530 (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 ultra-short laser pulses in air," Opt. Lett. 22, 304-306 (1997).
[CrossRef] [PubMed]

Krainov, V. P.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, "Tunnel ionization of complex atoms and atomic ions in an alternating electromagnetic field," Sov. Phys. JETP 64, 1191-1194 (1986).

LaFontaine, B.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J. C. Kieffer, H. Pepin, and H. P. Mercure, "Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air," Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Lange, R.

J. F. Ripoche, G. Grillon, B. Prade, M. Franco, E. Nibbering, R. Lange, and A. Mysyrowicz, "Determination of the time dependence of n2 in air," Opt. Commun. 135, 310-314 (1997).
[CrossRef]

Laux, U.

M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
[CrossRef]

Lederer, F.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, "Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters," Phys. Rev. E 70, 046602 (2004).
[CrossRef]

Lin, C. D.

X. M. Tong, Z. X. Zhao, and C. D. Lin, "Theory of molecular tunneling ionization," Phys. Rev. A 66, 033402 (2002).
[CrossRef]

Liu, X.

Marburger, J. H.

J. H. Marburger, "Self-focusing: theory," Prog. Quantum Electron. 4, 35-110 (1975).
[CrossRef]

Méjean, G.

L. Bergé, S. Skupin, G. Méjean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P. Wolf, "Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air," Phys. Rev. E 71, 016602 (2005).
[CrossRef]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, "Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters," Phys. Rev. E 70, 046602 (2004).
[CrossRef]

M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
[CrossRef]

Mercure, H. P.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J. C. Kieffer, H. Pepin, and H. P. Mercure, "Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air," Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Mlejnek, M.

Moloney, J. V.

Mourou, G.

Mysyrowicz, A.

Nibbering, E.

J. F. Ripoche, G. Grillon, B. Prade, M. Franco, E. Nibbering, R. Lange, and A. Mysyrowicz, "Determination of the time dependence of n2 in air," Opt. Commun. 135, 310-314 (1997).
[CrossRef]

Nibbering, E. T. J.

Peñano, J. R.

I. Alexeev, A. Ting, D. F. Gordon, E. Briscoe, J. R. Peñano, R. F. Hubbard, and Ph. Sprangle, "Longitudinal compression of short laser pulses in air," Appl. Phys. Lett. 84, 4080-4082 (2004).
[CrossRef]

Pepin, H.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J. C. Kieffer, H. Pepin, and H. P. Mercure, "Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air," Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Perelomov, A. M.

A. M. Perelomov and V. S. Popov, "Ionization of atoms in alternating electric field. III," Sov. Phys. JETP 25, 336-343 (1967).

A. M. Perelomov, V. S. Popov, and M. V. Terent'ev, "Ionization of atoms in alternating electric field," Sov. Phys. JETP 23, 924-934 (1966).

Peschel, U.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, "Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters," Phys. Rev. E 70, 046602 (2004).
[CrossRef]

Popov, V. S.

A. M. Perelomov and V. S. Popov, "Ionization of atoms in alternating electric field. III," Sov. Phys. JETP 25, 336-343 (1967).

A. M. Perelomov, V. S. Popov, and M. V. Terent'ev, "Ionization of atoms in alternating electric field," Sov. Phys. JETP 23, 924-934 (1966).

Prade, B.

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

J. F. Ripoche, G. Grillon, B. Prade, M. Franco, E. Nibbering, R. Lange, and A. Mysyrowicz, "Determination of the time dependence of n2 in air," Opt. Commun. 135, 310-314 (1997).
[CrossRef]

Prade, B. S.

Ripoche, J. F.

J. F. Ripoche, G. Grillon, B. Prade, M. Franco, E. Nibbering, R. Lange, and A. Mysyrowicz, "Determination of the time dependence of n2 in air," Opt. Commun. 135, 310-314 (1997).
[CrossRef]

Rodriguez, M.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, "Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters," Phys. Rev. E 70, 046602 (2004).
[CrossRef]

M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
[CrossRef]

Salin, F.

Salmon, E.

L. Bergé, S. Skupin, G. Méjean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P. Wolf, "Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air," Phys. Rev. E 71, 016602 (2005).
[CrossRef]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, "Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters," Phys. Rev. E 70, 046602 (2004).
[CrossRef]

M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
[CrossRef]

Sauerbrey, R.

M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
[CrossRef]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, "Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters," Phys. Rev. E 70, 046602 (2004).
[CrossRef]

Scalora, M.

N. Aközbek, M. Scalora, C. M. Bowden, and S. L. Chin, "White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air," Opt. Commun. 191, 353-362 (2001).
[CrossRef]

Scholz, A.

M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
[CrossRef]

Skupin, S.

L. Bergé, S. Skupin, G. Méjean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P. Wolf, "Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air," Phys. Rev. E 71, 016602 (2005).
[CrossRef]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, "Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters," Phys. Rev. E 70, 046602 (2004).
[CrossRef]

Sprangle, Ph.

I. Alexeev, A. Ting, D. F. Gordon, E. Briscoe, J. R. Peñano, R. F. Hubbard, and Ph. Sprangle, "Longitudinal compression of short laser pulses in air," Appl. Phys. Lett. 84, 4080-4082 (2004).
[CrossRef]

Squier, J.

Stecklum, B.

M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
[CrossRef]

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, 29-32 (1999).
[CrossRef]

Terent'ev, M. V.

A. M. Perelomov, V. S. Popov, and M. V. Terent'ev, "Ionization of atoms in alternating electric field," Sov. Phys. JETP 23, 924-934 (1966).

Ting, A.

I. Alexeev, A. Ting, D. F. Gordon, E. Briscoe, J. R. Peñano, R. F. Hubbard, and Ph. Sprangle, "Longitudinal compression of short laser pulses in air," Appl. Phys. Lett. 84, 4080-4082 (2004).
[CrossRef]

Tong, X. M.

X. M. Tong, Z. X. Zhao, and C. D. Lin, "Theory of molecular tunneling ionization," Phys. Rev. A 66, 033402 (2002).
[CrossRef]

Tzortzakis, S.

Vidal, F.

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J. C. Kieffer, H. Pepin, and H. P. Mercure, "Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air," Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Vinçotte, A.

A. Vinçotte and L. Bergé, "Chi(5) susceptibility stabilizes the propagation of ultra-short laser pulses in air," Phys. Rev. A 70, 061802 (2004).
[CrossRef]

Wells, E.

M. J. DeWitt, E. Wells, and R. R. Jones, "Radiometric comparison of intense field ionization of atoms and diatomic molecules," Phys. Rev. Lett. 87, 153001 (2001).
[CrossRef] [PubMed]

Wolf, J. P.

L. Bergé, S. Skupin, G. Méjean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P. Wolf, "Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air," Phys. Rev. E 71, 016602 (2005).
[CrossRef]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, "Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters," Phys. Rev. E 70, 046602 (2004).
[CrossRef]

Wolf, J.-P.

M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
[CrossRef]

Wöste, L.

M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
[CrossRef]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, "Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters," Phys. Rev. E 70, 046602 (2004).
[CrossRef]

Wright, E. M.

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, 29-32 (1999).
[CrossRef]

Yu, J.

L. Bergé, S. Skupin, G. Méjean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P. Wolf, "Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air," Phys. Rev. E 71, 016602 (2005).
[CrossRef]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, "Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters," Phys. Rev. E 70, 046602 (2004).
[CrossRef]

M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
[CrossRef]

Zhao, Z. X.

X. M. Tong, Z. X. Zhao, and C. D. Lin, "Theory of molecular tunneling ionization," Phys. Rev. A 66, 033402 (2002).
[CrossRef]

Appl. Phys. Lett. (1)

I. Alexeev, A. Ting, D. F. Gordon, E. Briscoe, J. R. Peñano, R. F. Hubbard, and Ph. Sprangle, "Longitudinal compression of short laser pulses in air," Appl. Phys. Lett. 84, 4080-4082 (2004).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

J. F. Ripoche, G. Grillon, B. Prade, M. Franco, E. Nibbering, R. Lange, and A. Mysyrowicz, "Determination of the time dependence of n2 in air," Opt. Commun. 135, 310-314 (1997).
[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, 29-32 (1999).
[CrossRef]

N. Aközbek, M. Scalora, C. M. Bowden, and S. L. Chin, "White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air," Opt. Commun. 191, 353-362 (2001).
[CrossRef]

Opt. Lett. (4)

Phys. Plasmas (1)

B. LaFontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J. C. Kieffer, H. Pepin, and H. P. Mercure, "Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air," Phys. Plasmas 6, 1615-1621 (1999).
[CrossRef]

Phys. Rev. A (2)

X. M. Tong, Z. X. Zhao, and C. D. Lin, "Theory of molecular tunneling ionization," Phys. Rev. A 66, 033402 (2002).
[CrossRef]

A. Vinçotte and L. Bergé, "Chi(5) susceptibility stabilizes the propagation of ultra-short laser pulses in air," Phys. Rev. A 70, 061802 (2004).
[CrossRef]

Phys. Rev. E (5)

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, "Filamentation of femtosecond light pulses in the air: turbulent cells versus long-range clusters," Phys. Rev. E 70, 046602 (2004).
[CrossRef]

S. Champeaux and L. Bergé, "Femtosecond pulse compression in pressure-gas cells filled with argon," Phys. Rev. E 68, 066603 (2003).
[CrossRef]

L. Bergé, S. Skupin, G. Méjean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P. Wolf, "Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air," Phys. Rev. E 71, 016602 (2005).
[CrossRef]

L. Bergé, "Boosted propagation of femtosecond filaments in air by double-pulse combination," Phys. Rev. E 69, 065601 (2004).
[CrossRef]

M. Rodriguez, R. Bourayou, G. Méjean, J. Kasparian, J. Yu, E. Salmon, A. Scholz, B. Stecklum, J. Eislöffel, U. Laux, A. P. Hatzes, R. Sauerbrey, L. Wöste, and J.-P. Wolf, "Kilometer-range nonlinear propagation of femtosecond laser pulses," Phys. Rev. E 69, 036607 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

M. J. DeWitt, E. Wells, and R. R. Jones, "Radiometric comparison of intense field ionization of atoms and diatomic molecules," Phys. Rev. Lett. 87, 153001 (2001).
[CrossRef] [PubMed]

Prog. Quantum Electron. (1)

J. H. Marburger, "Self-focusing: theory," Prog. Quantum Electron. 4, 35-110 (1975).
[CrossRef]

Quantum Electron. (2)

I. S. Golubtsov, V. P. Kandidov, and O. G. Kosareva, "Initial phase modulation of a high-power femtosecond laser pulse as a tool for controlling its filamentation and generation of a supercontinuum in air," Quantum Electron. 33, 525-530 (2003).
[CrossRef]

V. P. Kandidov, I. S. Golubtsov, and O. G. Kosareva, "Supercontinuum sources in a high-power femtosecond laser pulse propagating in liquids and gases," Quantum Electron. 34, 348-354 (2004).
[CrossRef]

Sov. Phys. JETP (4)

A. M. Perelomov, V. S. Popov, and M. V. Terent'ev, "Ionization of atoms in alternating electric field," Sov. Phys. JETP 23, 924-934 (1966).

A. M. Perelomov and V. S. Popov, "Ionization of atoms in alternating electric field. III," Sov. Phys. JETP 25, 336-343 (1967).

M. V. Ammosov, N. B. Delone, and V. P. Krainov, "Tunnel ionization of complex atoms and atomic ions in an alternating electromagnetic field," Sov. Phys. JETP 64, 1191-1194 (1986).

L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov. Phys. JETP 20, 1307-1314 (1965).

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

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Figures (10)

Fig. 1
Fig. 1

Top row, temporal dynamics in the ( t , z ) plane exhibits the same thickness for pulse durations (a) with as well as (b) without full chromatic dispersion including space–time focusing and self-steepening effects. Intensity levels increase from blue to red. (c) Peak intensities, (d) electron densities along the propagation axis, (e) temporal profiles at z = 0.8 m , (f) associated spectra with (solid curve) and without (dashed–dotted curve) full chromatic dispersion.

Fig. 2
Fig. 2

Ionization rate versus laser intensity obtained from the PPT theory (solid curve), ADK molecular model (dashed curve), fitting curve from Talebpour et al.[11] (dotted curve), and the MPI-like formulation used in Ref. [6] (dashed–dotted curve).

Fig. 3
Fig. 3

(a) Peak intensities, (b) pulse radii, (c) electron densities, (d) beam energies for 70 fs pulses and different ionization models: PPT (solid curves), ADK molecular (dashed curves), ADK molecular + χ ( 5 ) (dotted curves, see text), and MPI-like approximation (dashed–dotted curves).

Fig. 4
Fig. 4

(a) and (d) peak intensities, (b) and (e) pulse radii, (c) and (f) peak electron densities with P in = 1.05 P cr 0.81 P cf eff (left column) and P in = 2 P cr 1.54 P cf eff (right column) for different chirp values C > 0 (dotted curves), C = 0 (dashed–dotted curves), and C < 0 (solid curves).

Fig. 5
Fig. 5

Beam radius versus z 5 m for C = 0 (dash–dotted curve), C = 1.02 (dotted curve), and C = 1.02 (solid curve).

Fig. 6
Fig. 6

(a) Peak intensities, (b) pulse radii, (c) electron densities, (d) on-axis beam fluences considering different input energies: E in = 1.3 mJ (dotted curves), E in = 1.9 mJ (dashed curves), and E in = 2.5 mJ (solid curves) for C = 3 .

Fig. 7
Fig. 7

FWHM temporal width (left column) and associated normalized temporal profile (right column) of the on-axis intensity at z = 0 m (dotted curves), z = z c (solid curves), and z = 105 m (dashed curves) for (a) and (e) E in = 1.3 mJ , (b) and (f) E in = 1.5 mJ , (c) and (g) E in = 1.9 mJ , (d) and (h) E in = 2.5 mJ .

Fig. 8
Fig. 8

(a) Peak intensity and (b) peak electron density for 3 mm pulses [Eq. (15)] with C = 0 (dashed–dotted curves) and C = 0.35 (solid curves).

Fig. 9
Fig. 9

Intensity profiles at y = 0 near z = 4.2 m for (a) C = 0 and (b) C = 0.35 .

Fig. 10
Fig. 10

Intensity profiles at y = 0 evaluated at z = 6.2 m for (a) C = 0 and (b) C = 0.35 .

Tables (4)

Tables Icon

Table 1 Displacements of z c for Chirped Beams Relative to Unchirped Beams [ ( z c C 0 z c C = 0 ) z c C = 0 ]

Tables Icon

Table 2 First ( z c ) and Last ( z c , 2 ) Compression Points for Negatively Chirped Beams ( C = 20 ) According to the Input Power Ratio ( P cr eff = P cr 0.77 )

Tables Icon

Table 3 Last Compression Point, z c , 2 (in Meters), versus Chirp Parameter

Tables Icon

Table 4 FWHM Temporal Widths in Femtoseconds Computed at z = 105 m for Four Energy Values

Equations (27)

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

z E = i 2 k 0 2 E i k 2 2 t 2 E i k 0 2 ρ c ρ E + i k 0 n 2 2 [ E 2 + 1 τ k t e ( t t ) τ k E ( t ) 2 d t ] E σ 2 ρ E W ( E ) 2 U i ρ nt E E 2 ,
t ρ = W ( E ) ρ nt + σ U i ρ E 2 ,
W ( E ) W MPI ( E ) = σ ( K ) E 2 K ,
W ( E ) = 4 2 π Ξ [ 2 E 0 E 1 + γ 2 ] 2 n * ( 3 2 ) m U i γ 2 1 + γ 2 × e 2 ν [ sinh 1 ( γ ) [ γ 1 + γ 2 ( 1 + 2 γ 2 ) ] ] κ ν 0 + e α [ κ ν ] Φ m [ β ( κ ν ) ] ,
E 0 = [ 2 U i ] 3 2 ,
γ = ω 0 2 U i E ,
ν = U i ω 0 ( 1 + 1 2 γ 2 ) ,
β = 2 γ 1 + γ 2 ,
α = 2 × [ sinh 1 ( γ ) γ 1 + γ 2 ] ,
ν 0 = ν + 1 ,
Φ m ( x ) = e x 2 0 x ( x 2 y 2 ) m e y 2 d y ,
n * = Z 2 U i ,
Ξ = C n * , l * 2 f ( l , m ) m ! ,
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 ) ! ,
f ( 0 , 0 ) = 1 ,
l * = n * 1 .
Ξ = 1 2 m ( m ! ) 2 [ l C l ( 2 U i ) ( n * 2 ) + ( 1 4 ) ( 2 l + 1 ) ( l + m ) ! ( l m ) ! ] 2 ,
E ( r , t , z = 0 ) = 2 P in π w 0 2 e ( r 2 w 0 2 ) ( t 2 t p 2 ) i C ( t 2 t p 2 ) ,
z c = 0.367 × z r ( P in P cr 0.852 ) 2 0.0219
n nl = n 2 2 [ E 2 + 1 τ k t e ( t t ) τ k E ( t ) 2 d t ] ρ 2 ρ c n 4 E 4 ,
z E = i 2 k 0 2 E i k 2 2 t 2 E .
U ( r , t , z = 0 ) = U 0 e t 2 t p 2 i C ( t 2 t p 2 ) .
U ( r , t , z ) = U 0 t p e ( 1 + i C ) t 2 [ t p 2 2 i k z ( 1 + i C ) ] t p 2 2 i k z ( 1 + i C ) ,
T ( z ) = T 0 ( 1 + 2 k C z t p 2 ) 2 + ( 2 k z t p 2 ) 2 .
z min = C 1 + C 2 T 0 2 4 k ln 2 ,
E ( r , t , z = 0 ) = I 0 ( 1 + 0.1 × noise ) e [ ( x 2 + 2 y 2 ) 3 2 w 0 3 ] ( t 2 t p 2 ) i C ( t 2 t p 2 ) ,

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