Abstract

Experiments in which several-meter-long infrared ultrashort laser pulses are guided in air are compared with numerical simulations. During a first self-focusing stage that is affected by modulational instability, the beam is shown to break up into two channels of light that finally coalesce into a narrow filament that is able to propagate over several Rayleigh lengths. The filament propagation is associated with the generation of an electron plasma, whose density is greater than 1016 cm-3. Electron generation persists well beyond the focal region. The simulations restore the global dynamics of the pulse, including the main stages of Kerr focusing, light guiding driven by ionization, and the ultimate diffraction of the beam.

© 2002 Optical Society of America

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  1. F. Vidal, D. Comtois, C.-Y. Chien, A. Desparois, B. La Fontaine, T. W. Johnston, J.-C. Kieffer, H. P. Mercure, and F. A. Rizk, “Modeling the triggering of streamers in air by ultrashort laser pulses,” IEEE Trans. Plasma Sci. 28, 418–433 (2000).
    [CrossRef]
  2. L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Neirdeimer, F. Ronneberger, H. Schilinger, and R. Sauerbrey, “Femtosecond atmospheric lamp,” Laser Optoelektron. 29, 51–53 (1997).
  3. 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, 73–75 (1995).
    [CrossRef] [PubMed]
  4. 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] [PubMed]
  5. S. Henz and J. Herrmann, “Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: variational approach” Phys. Rev. E 53, 4092–4097 (1996).
    [CrossRef]
  6. 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, 304–306 (1997).
    [CrossRef] [PubMed]
  7. O. G. Kosareva, V. P. Kandidov, A. Brodeur, C. Y. Chien, and S. L. Chin, “Conical emission from laser–plasma interactions in the filamentation of powerful ultrashort laser pulses in air,” Opt. Lett. 22, 1332–1334 (1997).
    [CrossRef]
  8. H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lamouroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron, “Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding?” Opt. Lett. 23, 120–122 (1998).
    [CrossRef]
  9. B. La Fontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J.-C. Kieffer, and H. Pépin, “Filamentation of ultrashort pulse laser beams resulting from their propagation over long distances in air,” Phys. Plasmas 6, 1615–1621 (1999).
    [CrossRef]
  10. V. P. Kandidov, O. G. Kosareva, and S. A. Shlenov, “Influence of transient self-defocusing on the propagation of high-power femtosecond laser pulses in gases under ionization conditions,” Quantum Electron. 24, 905 (1994).
    [CrossRef]
  11. M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23, 382–384 (1998).
    [CrossRef]
  12. A. Chiron, B. Lamouroux, H. R. Lange, J.-F. Ripoche, M. A. Franco, B. S. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
    [CrossRef]
  13. A. Couairon and L. Bergé, “Modeling the filamentation of ultra-short pulses in ionizing media,” Phys. Plasmas 7, 193–209 (2000).
    [CrossRef]
  14. L. Bergé and A. Couairon, “Nonlinear propagation of self-guided ultra-short pulses in ionized gases,” Phys. Plasmas 7, 210–230 (2000).
    [CrossRef]
  15. L. Bergé and A. Couairon, “Gas-induced solitons,” Phys. Rev. Lett. 86, 1003–1006 (2001).
    [CrossRef] [PubMed]
  16. S. Tzortzakis, L. Bergé, A. Couairon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Break-up and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett. 86, 5470–5473 (2001).
    [CrossRef] [PubMed]
  17. M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, “Optically turbulent femtosecond light guide in air,” Phys. Rev. Lett. 83, 2938–2941 (1999).
    [CrossRef]
  18. S. Tzortzakis, B. Lamouroux, A. Chiron, M. A. Franco, B. S. Prade, A. Mysyrowicz, and S. D. Moustaizis, “Nonlinear propagation of subpicosecond ultraviolet laser pulses in air,” Opt. Lett. 25, 1270–1272 (2000).
    [CrossRef]
  19. J. Schwarz, P. Rambo, J.-C. Diels, M. Kolesik, E. M. Wright, and J. V. Moloney, “Ultraviolet filamentation in air,” Opt. Commun. 180, 383–390 (2000).
    [CrossRef]
  20. S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propogation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902–1–4 (2001).
    [CrossRef]
  21. J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
    [CrossRef]
  22. L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Zh. Eksp. Teor. Fiz. 47, 1945–1957 (1964) [Sov. Phys. JETP 20, 1307–1314 (1965)].
  23. A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 23, 924–934 (1966) [Zh. Eksp. Teor. Fiz. 50, 1393–1409 (1966)].
  24. M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 [Zh. Eksp. Teor. Fiz. 91, 2008–2013 (1986)].
  25. F. A. Ilkov, J. E. Decker, and S. L. Chin, “Ionization of atoms in the tunneling regime with experimental evidence using Hg atoms,” J. Phys. B 25, 4005–4020 (1992).
    [CrossRef]
  26. C. S. Milsted and C. D. Cantrell, “Vector effects in self-focusing,” Phys. Rev. A 53, 3536–3542 (1996).
    [CrossRef] [PubMed]
  27. J. Schjødt-Eriksen, J. V. Moloney, E. M. Wright, and P. L. Christiansen, “Pulse splitting of modulationally unstable femtosecond pulse,” submitted to Opt. Express.
  28. S. Petit, A. Talebpour, A. Proulx, and S. L. Chin, “Polarization dependence of the propagation of intense laser pulses in air,” Opt. Commun. 175, 323–327 (2000).
    [CrossRef]
  29. J. F. Ripoche, G. Grillon, B. S. Prade, M. A. Franco, E. Nibbering, H. R. Lange, and A. Mysyrowicz, “Determination of the time dependence of n2 in air,” Opt. Commun. 135, 310–314 (1997).
    [CrossRef]
  30. A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (2000).
    [CrossRef] [PubMed]
  31. A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
    [CrossRef]
  32. P. Mora and T. M. Antonsen, Jr., “Kinetic modeling of intense, short laser pulses propagating in tenuous plasmas,” Phys. Plasmas 4, 217–229 (1997).
    [CrossRef]
  33. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
    [CrossRef]
  34. 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 60, R3505–R3507 (1999).
    [CrossRef]
  35. H. D. Ladouceur, A. P. Baronavski, D. Lohrman, P. W. Grounds, and P. G. Girardi, “Electrical conductivity of a femtosecond laser generated plasma channel in air,” Opt. Commun. 189, 107–111 (2001).
    [CrossRef]
  36. S. Tzortzakis, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Time evolution of the plasma channel at the trail of a self-guided IR femtosecond laser pulse in air,” Opt. Commun. 181, 123–127 (2000).
    [CrossRef]
  37. V. I. Bespalov and V. I. Talanov, Zh. Eksp. Teor. Fiz. Pis'ma Red. 3, 471–476 (1966) [JETP Lett. 3, 307–310 (1966)].
  38. L. Bergé, M. R. Schmidt, J. J. Rasmussen, P. L. Christiansen, and K. Ø. Rasmussen, “Amalgamation, of interacting light beamlets in Kerr-type media,” J. Opt. Soc. Am. B 14, 2550–2562 (1997).
    [CrossRef]
  39. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

2001 (4)

L. Bergé and A. Couairon, “Gas-induced solitons,” Phys. Rev. Lett. 86, 1003–1006 (2001).
[CrossRef] [PubMed]

S. Tzortzakis, L. Bergé, A. Couairon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Break-up and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett. 86, 5470–5473 (2001).
[CrossRef] [PubMed]

S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propogation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902–1–4 (2001).
[CrossRef]

H. D. Ladouceur, A. P. Baronavski, D. Lohrman, P. W. Grounds, and P. G. Girardi, “Electrical conductivity of a femtosecond laser generated plasma channel in air,” Opt. Commun. 189, 107–111 (2001).
[CrossRef]

2000 (8)

S. Tzortzakis, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Time evolution of the plasma channel at the trail of a self-guided IR femtosecond laser pulse in air,” Opt. Commun. 181, 123–127 (2000).
[CrossRef]

A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (2000).
[CrossRef] [PubMed]

S. Petit, A. Talebpour, A. Proulx, and S. L. Chin, “Polarization dependence of the propagation of intense laser pulses in air,” Opt. Commun. 175, 323–327 (2000).
[CrossRef]

S. Tzortzakis, B. Lamouroux, A. Chiron, M. A. Franco, B. S. Prade, A. Mysyrowicz, and S. D. Moustaizis, “Nonlinear propagation of subpicosecond ultraviolet laser pulses in air,” Opt. Lett. 25, 1270–1272 (2000).
[CrossRef]

J. Schwarz, P. Rambo, J.-C. Diels, M. Kolesik, E. M. Wright, and J. V. Moloney, “Ultraviolet filamentation in air,” Opt. Commun. 180, 383–390 (2000).
[CrossRef]

A. Couairon and L. Bergé, “Modeling the filamentation of ultra-short pulses in ionizing media,” Phys. Plasmas 7, 193–209 (2000).
[CrossRef]

L. Bergé and A. Couairon, “Nonlinear propagation of self-guided ultra-short pulses in ionized gases,” Phys. Plasmas 7, 210–230 (2000).
[CrossRef]

F. Vidal, D. Comtois, C.-Y. Chien, A. Desparois, B. La Fontaine, T. W. Johnston, J.-C. Kieffer, H. P. Mercure, and F. A. Rizk, “Modeling the triggering of streamers in air by ultrashort laser pulses,” IEEE Trans. Plasma Sci. 28, 418–433 (2000).
[CrossRef]

1999 (5)

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

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, “Optically turbulent femtosecond light guide in air,” Phys. Rev. Lett. 83, 2938–2941 (1999).
[CrossRef]

A. Chiron, B. Lamouroux, H. R. Lange, J.-F. Ripoche, M. A. Franco, B. S. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
[CrossRef]

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 60, R3505–R3507 (1999).
[CrossRef]

1998 (2)

1997 (7)

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, 304–306 (1997).
[CrossRef] [PubMed]

O. G. Kosareva, V. P. Kandidov, A. Brodeur, C. Y. Chien, and S. L. Chin, “Conical emission from laser–plasma interactions in the filamentation of powerful ultrashort laser pulses in air,” Opt. Lett. 22, 1332–1334 (1997).
[CrossRef]

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Neirdeimer, F. Ronneberger, H. Schilinger, and R. Sauerbrey, “Femtosecond atmospheric lamp,” Laser Optoelektron. 29, 51–53 (1997).

P. Mora and T. M. Antonsen, Jr., “Kinetic modeling of intense, short laser pulses propagating in tenuous plasmas,” Phys. Plasmas 4, 217–229 (1997).
[CrossRef]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

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

L. Bergé, M. R. Schmidt, J. J. Rasmussen, P. L. Christiansen, and K. Ø. Rasmussen, “Amalgamation, of interacting light beamlets in Kerr-type media,” J. Opt. Soc. Am. B 14, 2550–2562 (1997).
[CrossRef]

1996 (3)

C. S. Milsted and C. D. Cantrell, “Vector effects in self-focusing,” Phys. Rev. A 53, 3536–3542 (1996).
[CrossRef] [PubMed]

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] [PubMed]

S. Henz and J. Herrmann, “Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: variational approach” Phys. Rev. E 53, 4092–4097 (1996).
[CrossRef]

1995 (1)

1994 (1)

V. P. Kandidov, O. G. Kosareva, and S. A. Shlenov, “Influence of transient self-defocusing on the propagation of high-power femtosecond laser pulses in gases under ionization conditions,” Quantum Electron. 24, 905 (1994).
[CrossRef]

1992 (1)

F. A. Ilkov, J. E. Decker, and S. L. Chin, “Ionization of atoms in the tunneling regime with experimental evidence using Hg atoms,” J. Phys. B 25, 4005–4020 (1992).
[CrossRef]

1986 (1)

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 [Zh. Eksp. Teor. Fiz. 91, 2008–2013 (1986)].

1975 (1)

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

1966 (2)

A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 23, 924–934 (1966) [Zh. Eksp. Teor. Fiz. 50, 1393–1409 (1966)].

V. I. Bespalov and V. I. Talanov, Zh. Eksp. Teor. Fiz. Pis'ma Red. 3, 471–476 (1966) [JETP Lett. 3, 307–310 (1966)].

1965 (1)

L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Zh. Eksp. Teor. Fiz. 47, 1945–1957 (1964) [Sov. Phys. JETP 20, 1307–1314 (1965)].

Ammosov, M. V.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 [Zh. Eksp. Teor. Fiz. 91, 2008–2013 (1986)].

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 60, R3505–R3507 (1999).
[CrossRef]

Antonsen Jr., T. M.

P. Mora and T. M. Antonsen, Jr., “Kinetic modeling of intense, short laser pulses propagating in tenuous plasmas,” Phys. Plasmas 4, 217–229 (1997).
[CrossRef]

Baronavski, A. P.

H. D. Ladouceur, A. P. Baronavski, D. Lohrman, P. W. Grounds, and P. G. Girardi, “Electrical conductivity of a femtosecond laser generated plasma channel in air,” Opt. Commun. 189, 107–111 (2001).
[CrossRef]

Bergé, L.

S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propogation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902–1–4 (2001).
[CrossRef]

L. Bergé and A. Couairon, “Gas-induced solitons,” Phys. Rev. Lett. 86, 1003–1006 (2001).
[CrossRef] [PubMed]

S. Tzortzakis, L. Bergé, A. Couairon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Break-up and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett. 86, 5470–5473 (2001).
[CrossRef] [PubMed]

A. Couairon and L. Bergé, “Modeling the filamentation of ultra-short pulses in ionizing media,” Phys. Plasmas 7, 193–209 (2000).
[CrossRef]

L. Bergé and A. Couairon, “Nonlinear propagation of self-guided ultra-short pulses in ionized gases,” Phys. Plasmas 7, 210–230 (2000).
[CrossRef]

L. Bergé, M. R. Schmidt, J. J. Rasmussen, P. L. Christiansen, and K. Ø. Rasmussen, “Amalgamation, of interacting light beamlets in Kerr-type media,” J. Opt. Soc. Am. B 14, 2550–2562 (1997).
[CrossRef]

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, Zh. Eksp. Teor. Fiz. Pis'ma Red. 3, 471–476 (1966) [JETP Lett. 3, 307–310 (1966)].

Bonnaud, G.

A. Chiron, B. Lamouroux, H. R. Lange, J.-F. Ripoche, M. A. Franco, B. S. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

Brabec, T.

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

Braun, A.

Brodeur, A.

Cantrell, C. D.

C. S. Milsted and C. D. Cantrell, “Vector effects in self-focusing,” Phys. Rev. A 53, 3536–3542 (1996).
[CrossRef] [PubMed]

Chien, C. Y.

Chien, C.-Y.

F. Vidal, D. Comtois, C.-Y. Chien, A. Desparois, B. La Fontaine, T. W. Johnston, J.-C. Kieffer, H. P. Mercure, and F. A. Rizk, “Modeling the triggering of streamers in air by ultrashort laser pulses,” IEEE Trans. Plasma Sci. 28, 418–433 (2000).
[CrossRef]

Chin, S. L.

S. Petit, A. Talebpour, A. Proulx, and S. L. Chin, “Polarization dependence of the propagation of intense laser pulses in air,” Opt. Commun. 175, 323–327 (2000).
[CrossRef]

O. G. Kosareva, V. P. Kandidov, A. Brodeur, C. Y. Chien, and S. L. Chin, “Conical emission from laser–plasma interactions in the filamentation of powerful ultrashort laser pulses in air,” Opt. Lett. 22, 1332–1334 (1997).
[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, 304–306 (1997).
[CrossRef] [PubMed]

F. A. Ilkov, J. E. Decker, and S. L. Chin, “Ionization of atoms in the tunneling regime with experimental evidence using Hg atoms,” J. Phys. B 25, 4005–4020 (1992).
[CrossRef]

Chiron, A.

S. Tzortzakis, B. Lamouroux, A. Chiron, M. A. Franco, B. S. Prade, A. Mysyrowicz, and S. D. Moustaizis, “Nonlinear propagation of subpicosecond ultraviolet laser pulses in air,” Opt. Lett. 25, 1270–1272 (2000).
[CrossRef]

A. Chiron, B. Lamouroux, H. R. Lange, J.-F. Ripoche, M. A. Franco, B. S. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

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 60, R3505–R3507 (1999).
[CrossRef]

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lamouroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron, “Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding?” Opt. Lett. 23, 120–122 (1998).
[CrossRef]

Christiansen, P. L.

Clement, T. S.

A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
[CrossRef]

Comtois, D.

F. Vidal, D. Comtois, C.-Y. Chien, A. Desparois, B. La Fontaine, T. W. Johnston, J.-C. Kieffer, H. P. Mercure, and F. A. Rizk, “Modeling the triggering of streamers in air by ultrashort laser pulses,” IEEE Trans. Plasma Sci. 28, 418–433 (2000).
[CrossRef]

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

Couairon, A.

L. Bergé and A. Couairon, “Gas-induced solitons,” Phys. Rev. Lett. 86, 1003–1006 (2001).
[CrossRef] [PubMed]

S. Tzortzakis, L. Bergé, A. Couairon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Break-up and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett. 86, 5470–5473 (2001).
[CrossRef] [PubMed]

S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propogation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902–1–4 (2001).
[CrossRef]

A. Couairon and L. Bergé, “Modeling the filamentation of ultra-short pulses in ionizing media,” Phys. Plasmas 7, 193–209 (2000).
[CrossRef]

L. Bergé and A. Couairon, “Nonlinear propagation of self-guided ultra-short pulses in ionized gases,” Phys. Plasmas 7, 210–230 (2000).
[CrossRef]

Curley, P. F.

Decker, J. E.

F. A. Ilkov, J. E. Decker, and S. L. Chin, “Ionization of atoms in the tunneling regime with experimental evidence using Hg atoms,” J. Phys. B 25, 4005–4020 (1992).
[CrossRef]

Delone, N. B.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 [Zh. Eksp. Teor. Fiz. 91, 2008–2013 (1986)].

Desparois, A.

F. Vidal, D. Comtois, C.-Y. Chien, A. Desparois, B. La Fontaine, T. W. Johnston, J.-C. Kieffer, H. P. Mercure, and F. A. Rizk, “Modeling the triggering of streamers in air by ultrashort laser pulses,” IEEE Trans. Plasma Sci. 28, 418–433 (2000).
[CrossRef]

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

Diddams, S. A.

A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
[CrossRef]

Diels, J.-C.

J. Schwarz, P. Rambo, J.-C. Diels, M. Kolesik, E. M. Wright, and J. V. Moloney, “Ultraviolet filamentation in air,” Opt. Commun. 180, 383–390 (2000).
[CrossRef]

Du, D.

Franco, M.

S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propogation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902–1–4 (2001).
[CrossRef]

Franco, M. A.

S. Tzortzakis, L. Bergé, A. Couairon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Break-up and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett. 86, 5470–5473 (2001).
[CrossRef] [PubMed]

S. Tzortzakis, B. Lamouroux, A. Chiron, M. A. Franco, B. S. Prade, A. Mysyrowicz, and S. D. Moustaizis, “Nonlinear propagation of subpicosecond ultraviolet laser pulses in air,” Opt. Lett. 25, 1270–1272 (2000).
[CrossRef]

S. Tzortzakis, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Time evolution of the plasma channel at the trail of a self-guided IR femtosecond laser pulse in air,” Opt. Commun. 181, 123–127 (2000).
[CrossRef]

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 60, R3505–R3507 (1999).
[CrossRef]

A. Chiron, B. Lamouroux, H. R. Lange, J.-F. Ripoche, M. A. Franco, B. S. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lamouroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron, “Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding?” Opt. Lett. 23, 120–122 (1998).
[CrossRef]

J. F. Ripoche, G. Grillon, B. S. Prade, M. A. Franco, E. Nibbering, H. 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] [PubMed]

Gaeta, A. L.

A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (2000).
[CrossRef] [PubMed]

Girardi, P. G.

H. D. Ladouceur, A. P. Baronavski, D. Lohrman, P. W. Grounds, and P. G. Girardi, “Electrical conductivity of a femtosecond laser generated plasma channel in air,” Opt. Commun. 189, 107–111 (2001).
[CrossRef]

Grillon, G.

Grounds, P. W.

H. D. Ladouceur, A. P. Baronavski, D. Lohrman, P. W. Grounds, and P. G. Girardi, “Electrical conductivity of a femtosecond laser generated plasma channel in air,” Opt. Commun. 189, 107–111 (2001).
[CrossRef]

Henz, S.

S. Henz and J. Herrmann, “Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: variational approach” Phys. Rev. E 53, 4092–4097 (1996).
[CrossRef]

Herrmann, J.

S. Henz and J. Herrmann, “Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: variational approach” Phys. Rev. E 53, 4092–4097 (1996).
[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, 304–306 (1997).
[CrossRef] [PubMed]

F. A. Ilkov, J. E. Decker, and S. L. Chin, “Ionization of atoms in the tunneling regime with experimental evidence using Hg atoms,” J. Phys. B 25, 4005–4020 (1992).
[CrossRef]

Jiang, Z.

B. La Fontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J.-C. Kieffer, and H. Pépin, “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.

F. Vidal, D. Comtois, C.-Y. Chien, A. Desparois, B. La Fontaine, T. W. Johnston, J.-C. Kieffer, H. P. Mercure, and F. A. Rizk, “Modeling the triggering of streamers in air by ultrashort laser pulses,” IEEE Trans. Plasma Sci. 28, 418–433 (2000).
[CrossRef]

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

Kandidov, V. P.

Keldysh, L. V.

L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Zh. Eksp. Teor. Fiz. 47, 1945–1957 (1964) [Sov. Phys. JETP 20, 1307–1314 (1965)].

Kieffer, J.-C.

F. Vidal, D. Comtois, C.-Y. Chien, A. Desparois, B. La Fontaine, T. W. Johnston, J.-C. Kieffer, H. P. Mercure, and F. A. Rizk, “Modeling the triggering of streamers in air by ultrashort laser pulses,” IEEE Trans. Plasma Sci. 28, 418–433 (2000).
[CrossRef]

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

Kolesik, M.

J. Schwarz, P. Rambo, J.-C. Diels, M. Kolesik, E. M. Wright, and J. V. Moloney, “Ultraviolet filamentation in air,” Opt. Commun. 180, 383–390 (2000).
[CrossRef]

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, “Optically turbulent femtosecond light guide in air,” Phys. Rev. Lett. 83, 2938–2941 (1999).
[CrossRef]

Korn, G.

Kosareva, O. G.

Krainov, V. P.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 [Zh. Eksp. Teor. Fiz. 91, 2008–2013 (1986)].

Krausz, F.

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

La Fontaine, B.

F. Vidal, D. Comtois, C.-Y. Chien, A. Desparois, B. La Fontaine, T. W. Johnston, J.-C. Kieffer, H. P. Mercure, and F. A. Rizk, “Modeling the triggering of streamers in air by ultrashort laser pulses,” IEEE Trans. Plasma Sci. 28, 418–433 (2000).
[CrossRef]

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

Ladouceur, H. D.

H. D. Ladouceur, A. P. Baronavski, D. Lohrman, P. W. Grounds, and P. G. Girardi, “Electrical conductivity of a femtosecond laser generated plasma channel in air,” Opt. Commun. 189, 107–111 (2001).
[CrossRef]

Lamouroux, B.

S. Tzortzakis, B. Lamouroux, A. Chiron, M. A. Franco, B. S. Prade, A. Mysyrowicz, and S. D. Moustaizis, “Nonlinear propagation of subpicosecond ultraviolet laser pulses in air,” Opt. Lett. 25, 1270–1272 (2000).
[CrossRef]

A. Chiron, B. Lamouroux, H. R. Lange, J.-F. Ripoche, M. A. Franco, B. S. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

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 60, R3505–R3507 (1999).
[CrossRef]

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lamouroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron, “Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding?” Opt. Lett. 23, 120–122 (1998).
[CrossRef]

Lange, H. R.

A. Chiron, B. Lamouroux, H. R. Lange, J.-F. Ripoche, M. A. Franco, B. S. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lamouroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron, “Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding?” Opt. Lett. 23, 120–122 (1998).
[CrossRef]

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

Liu, X.

Lohrman, D.

H. D. Ladouceur, A. P. Baronavski, D. Lohrman, P. W. Grounds, and P. G. Girardi, “Electrical conductivity of a femtosecond laser generated plasma channel in air,” Opt. Commun. 189, 107–111 (2001).
[CrossRef]

Marburger, J. H.

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

Mercure, H. P.

F. Vidal, D. Comtois, C.-Y. Chien, A. Desparois, B. La Fontaine, T. W. Johnston, J.-C. Kieffer, H. P. Mercure, and F. A. Rizk, “Modeling the triggering of streamers in air by ultrashort laser pulses,” IEEE Trans. Plasma Sci. 28, 418–433 (2000).
[CrossRef]

Milsted, C. S.

C. S. Milsted and C. D. Cantrell, “Vector effects in self-focusing,” Phys. Rev. A 53, 3536–3542 (1996).
[CrossRef] [PubMed]

Mlejnek, M.

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, “Optically turbulent femtosecond light guide in air,” Phys. Rev. Lett. 83, 2938–2941 (1999).
[CrossRef]

M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23, 382–384 (1998).
[CrossRef]

Moloney, J. V.

J. Schwarz, P. Rambo, J.-C. Diels, M. Kolesik, E. M. Wright, and J. V. Moloney, “Ultraviolet filamentation in air,” Opt. Commun. 180, 383–390 (2000).
[CrossRef]

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, “Optically turbulent femtosecond light guide in air,” Phys. Rev. Lett. 83, 2938–2941 (1999).
[CrossRef]

M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23, 382–384 (1998).
[CrossRef]

Mora, P.

P. Mora and T. M. Antonsen, Jr., “Kinetic modeling of intense, short laser pulses propagating in tenuous plasmas,” Phys. Plasmas 4, 217–229 (1997).
[CrossRef]

Mourou, G.

Moustaizis, S. D.

Mysyrowicz, A.

S. Tzortzakis, L. Bergé, A. Couairon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Break-up and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett. 86, 5470–5473 (2001).
[CrossRef] [PubMed]

S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propogation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902–1–4 (2001).
[CrossRef]

S. Tzortzakis, B. Lamouroux, A. Chiron, M. A. Franco, B. S. Prade, A. Mysyrowicz, and S. D. Moustaizis, “Nonlinear propagation of subpicosecond ultraviolet laser pulses in air,” Opt. Lett. 25, 1270–1272 (2000).
[CrossRef]

S. Tzortzakis, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Time evolution of the plasma channel at the trail of a self-guided IR femtosecond laser pulse in air,” Opt. Commun. 181, 123–127 (2000).
[CrossRef]

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 60, R3505–R3507 (1999).
[CrossRef]

A. Chiron, B. Lamouroux, H. R. Lange, J.-F. Ripoche, M. A. Franco, B. S. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lamouroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron, “Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding?” Opt. Lett. 23, 120–122 (1998).
[CrossRef]

J. F. Ripoche, G. Grillon, B. S. Prade, M. A. Franco, E. Nibbering, H. 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] [PubMed]

Neirdeimer, S.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Neirdeimer, F. Ronneberger, H. Schilinger, and R. Sauerbrey, “Femtosecond atmospheric lamp,” Laser Optoelektron. 29, 51–53 (1997).

Nibbering, E.

J. F. Ripoche, G. Grillon, B. S. Prade, M. A. Franco, E. Nibbering, H. 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.

Nikolov, S.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Neirdeimer, F. Ronneberger, H. Schilinger, and R. Sauerbrey, “Femtosecond atmospheric lamp,” Laser Optoelektron. 29, 51–53 (1997).

Pépin, H.

B. La Fontaine, F. Vidal, Z. Jiang, C. Y. Chien, D. Comtois, A. Desparois, T. W. Johnston, J.-C. Kieffer, and H. Pépin, “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, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 23, 924–934 (1966) [Zh. Eksp. Teor. Fiz. 50, 1393–1409 (1966)].

Petit, S.

S. Petit, A. Talebpour, A. Proulx, and S. L. Chin, “Polarization dependence of the propagation of intense laser pulses in air,” Opt. Commun. 175, 323–327 (2000).
[CrossRef]

Popov, V. S.

A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 23, 924–934 (1966) [Zh. Eksp. Teor. Fiz. 50, 1393–1409 (1966)].

Prade, B.

S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propogation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902–1–4 (2001).
[CrossRef]

Prade, B. S.

S. Tzortzakis, L. Bergé, A. Couairon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Break-up and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett. 86, 5470–5473 (2001).
[CrossRef] [PubMed]

S. Tzortzakis, B. Lamouroux, A. Chiron, M. A. Franco, B. S. Prade, A. Mysyrowicz, and S. D. Moustaizis, “Nonlinear propagation of subpicosecond ultraviolet laser pulses in air,” Opt. Lett. 25, 1270–1272 (2000).
[CrossRef]

S. Tzortzakis, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Time evolution of the plasma channel at the trail of a self-guided IR femtosecond laser pulse in air,” Opt. Commun. 181, 123–127 (2000).
[CrossRef]

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 60, R3505–R3507 (1999).
[CrossRef]

A. Chiron, B. Lamouroux, H. R. Lange, J.-F. Ripoche, M. A. Franco, B. S. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lamouroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron, “Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding?” Opt. Lett. 23, 120–122 (1998).
[CrossRef]

J. F. Ripoche, G. Grillon, B. S. Prade, M. A. Franco, E. Nibbering, H. 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] [PubMed]

Proulx, A.

S. Petit, A. Talebpour, A. Proulx, and S. L. Chin, “Polarization dependence of the propagation of intense laser pulses in air,” Opt. Commun. 175, 323–327 (2000).
[CrossRef]

Rairoux, P.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Neirdeimer, F. Ronneberger, H. Schilinger, and R. Sauerbrey, “Femtosecond atmospheric lamp,” Laser Optoelektron. 29, 51–53 (1997).

Rambo, P.

J. Schwarz, P. Rambo, J.-C. Diels, M. Kolesik, E. M. Wright, and J. V. Moloney, “Ultraviolet filamentation in air,” Opt. Commun. 180, 383–390 (2000).
[CrossRef]

Rasmussen, J. J.

Rasmussen, K. Ø.

Riazuelo, G.

A. Chiron, B. Lamouroux, H. R. Lange, J.-F. Ripoche, M. A. Franco, B. S. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

Ripoche, J. F.

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

Ripoche, J.-F.

A. Chiron, B. Lamouroux, H. R. Lange, J.-F. Ripoche, M. A. Franco, B. S. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lamouroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron, “Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding?” Opt. Lett. 23, 120–122 (1998).
[CrossRef]

Rizk, F. A.

F. Vidal, D. Comtois, C.-Y. Chien, A. Desparois, B. La Fontaine, T. W. Johnston, J.-C. Kieffer, H. P. Mercure, and F. A. Rizk, “Modeling the triggering of streamers in air by ultrashort laser pulses,” IEEE Trans. Plasma Sci. 28, 418–433 (2000).
[CrossRef]

Ronneberger, F.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Neirdeimer, F. Ronneberger, H. Schilinger, and R. Sauerbrey, “Femtosecond atmospheric lamp,” Laser Optoelektron. 29, 51–53 (1997).

Salin, F.

Sauerbrey, R.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Neirdeimer, F. Ronneberger, H. Schilinger, and R. Sauerbrey, “Femtosecond atmospheric lamp,” Laser Optoelektron. 29, 51–53 (1997).

Schilinger, H.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Neirdeimer, F. Ronneberger, H. Schilinger, and R. Sauerbrey, “Femtosecond atmospheric lamp,” Laser Optoelektron. 29, 51–53 (1997).

Schmidt, M. R.

Schwarz, J.

J. Schwarz, P. Rambo, J.-C. Diels, M. Kolesik, E. M. Wright, and J. V. Moloney, “Ultraviolet filamentation in air,” Opt. Commun. 180, 383–390 (2000).
[CrossRef]

Shlenov, S. A.

V. P. Kandidov, O. G. Kosareva, and S. A. Shlenov, “Influence of transient self-defocusing on the propagation of high-power femtosecond laser pulses in gases under ionization conditions,” Quantum Electron. 24, 905 (1994).
[CrossRef]

Squier, J.

Stein, B.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Neirdeimer, F. Ronneberger, H. Schilinger, and R. Sauerbrey, “Femtosecond atmospheric lamp,” Laser Optoelektron. 29, 51–53 (1997).

Sudrie, L.

S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propogation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902–1–4 (2001).
[CrossRef]

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, Zh. Eksp. Teor. Fiz. Pis'ma Red. 3, 471–476 (1966) [JETP Lett. 3, 307–310 (1966)].

Talebpour, A.

S. Petit, A. Talebpour, A. Proulx, and S. L. Chin, “Polarization dependence of the propagation of intense laser pulses in air,” Opt. Commun. 175, 323–327 (2000).
[CrossRef]

Terent’ev, M. V.

A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 23, 924–934 (1966) [Zh. Eksp. Teor. Fiz. 50, 1393–1409 (1966)].

Tzortzakis, S.

S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propogation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902–1–4 (2001).
[CrossRef]

S. Tzortzakis, L. Bergé, A. Couairon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Break-up and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett. 86, 5470–5473 (2001).
[CrossRef] [PubMed]

S. Tzortzakis, B. Lamouroux, A. Chiron, M. A. Franco, B. S. Prade, A. Mysyrowicz, and S. D. Moustaizis, “Nonlinear propagation of subpicosecond ultraviolet laser pulses in air,” Opt. Lett. 25, 1270–1272 (2000).
[CrossRef]

S. Tzortzakis, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Time evolution of the plasma channel at the trail of a self-guided IR femtosecond laser pulse in air,” Opt. Commun. 181, 123–127 (2000).
[CrossRef]

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 60, R3505–R3507 (1999).
[CrossRef]

Van Engen, A. G.

A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
[CrossRef]

Vidal, F.

F. Vidal, D. Comtois, C.-Y. Chien, A. Desparois, B. La Fontaine, T. W. Johnston, J.-C. Kieffer, H. P. Mercure, and F. A. Rizk, “Modeling the triggering of streamers in air by ultrashort laser pulses,” IEEE Trans. Plasma Sci. 28, 418–433 (2000).
[CrossRef]

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

Wedekind, C.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Neirdeimer, F. Ronneberger, H. Schilinger, and R. Sauerbrey, “Femtosecond atmospheric lamp,” Laser Optoelektron. 29, 51–53 (1997).

Werner, C.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Neirdeimer, F. Ronneberger, H. Schilinger, and R. Sauerbrey, “Femtosecond atmospheric lamp,” Laser Optoelektron. 29, 51–53 (1997).

Wille, H.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Neirdeimer, F. Ronneberger, H. Schilinger, and R. Sauerbrey, “Femtosecond atmospheric lamp,” Laser Optoelektron. 29, 51–53 (1997).

Wöste, L.

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Neirdeimer, F. Ronneberger, H. Schilinger, and R. Sauerbrey, “Femtosecond atmospheric lamp,” Laser Optoelektron. 29, 51–53 (1997).

Wright, E. M.

J. Schwarz, P. Rambo, J.-C. Diels, M. Kolesik, E. M. Wright, and J. V. Moloney, “Ultraviolet filamentation in air,” Opt. Commun. 180, 383–390 (2000).
[CrossRef]

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, “Optically turbulent femtosecond light guide in air,” Phys. Rev. Lett. 83, 2938–2941 (1999).
[CrossRef]

M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23, 382–384 (1998).
[CrossRef]

Zozulya, A. A.

A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
[CrossRef]

Eur. Phys. J. D (1)

A. Chiron, B. Lamouroux, H. R. Lange, J.-F. Ripoche, M. A. Franco, B. S. Prade, G. Bonnaud, G. Riazuelo, and A. Mysyrowicz, “Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases,” Eur. Phys. J. D 6, 383–396 (1999).
[CrossRef]

IEEE Trans. Plasma Sci. (1)

F. Vidal, D. Comtois, C.-Y. Chien, A. Desparois, B. La Fontaine, T. W. Johnston, J.-C. Kieffer, H. P. Mercure, and F. A. Rizk, “Modeling the triggering of streamers in air by ultrashort laser pulses,” IEEE Trans. Plasma Sci. 28, 418–433 (2000).
[CrossRef]

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

J. Phys. B (1)

F. A. Ilkov, J. E. Decker, and S. L. Chin, “Ionization of atoms in the tunneling regime with experimental evidence using Hg atoms,” J. Phys. B 25, 4005–4020 (1992).
[CrossRef]

Laser Optoelektron. (1)

L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Neirdeimer, F. Ronneberger, H. Schilinger, and R. Sauerbrey, “Femtosecond atmospheric lamp,” Laser Optoelektron. 29, 51–53 (1997).

Opt. Commun. (5)

J. Schwarz, P. Rambo, J.-C. Diels, M. Kolesik, E. M. Wright, and J. V. Moloney, “Ultraviolet filamentation in air,” Opt. Commun. 180, 383–390 (2000).
[CrossRef]

S. Petit, A. Talebpour, A. Proulx, and S. L. Chin, “Polarization dependence of the propagation of intense laser pulses in air,” Opt. Commun. 175, 323–327 (2000).
[CrossRef]

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

H. D. Ladouceur, A. P. Baronavski, D. Lohrman, P. W. Grounds, and P. G. Girardi, “Electrical conductivity of a femtosecond laser generated plasma channel in air,” Opt. Commun. 189, 107–111 (2001).
[CrossRef]

S. Tzortzakis, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Time evolution of the plasma channel at the trail of a self-guided IR femtosecond laser pulse in air,” Opt. Commun. 181, 123–127 (2000).
[CrossRef]

Opt. Lett. (7)

M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23, 382–384 (1998).
[CrossRef]

S. Tzortzakis, B. Lamouroux, A. Chiron, M. A. Franco, B. S. Prade, A. Mysyrowicz, and S. D. Moustaizis, “Nonlinear propagation of subpicosecond ultraviolet laser pulses in air,” Opt. Lett. 25, 1270–1272 (2000).
[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, 73–75 (1995).
[CrossRef] [PubMed]

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] [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, 304–306 (1997).
[CrossRef] [PubMed]

O. G. Kosareva, V. P. Kandidov, A. Brodeur, C. Y. Chien, and S. L. Chin, “Conical emission from laser–plasma interactions in the filamentation of powerful ultrashort laser pulses in air,” Opt. Lett. 22, 1332–1334 (1997).
[CrossRef]

H. R. Lange, G. Grillon, J.-F. Ripoche, M. A. Franco, B. Lamouroux, B. S. Prade, A. Mysyrowicz, E. T. J. Nibbering, and A. Chiron, “Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding?” Opt. Lett. 23, 120–122 (1998).
[CrossRef]

Phys. Plasmas (4)

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

A. Couairon and L. Bergé, “Modeling the filamentation of ultra-short pulses in ionizing media,” Phys. Plasmas 7, 193–209 (2000).
[CrossRef]

L. Bergé and A. Couairon, “Nonlinear propagation of self-guided ultra-short pulses in ionized gases,” Phys. Plasmas 7, 210–230 (2000).
[CrossRef]

P. Mora and T. M. Antonsen, Jr., “Kinetic modeling of intense, short laser pulses propagating in tenuous plasmas,” Phys. Plasmas 4, 217–229 (1997).
[CrossRef]

Phys. Rev. A (1)

C. S. Milsted and C. D. Cantrell, “Vector effects in self-focusing,” Phys. Rev. A 53, 3536–3542 (1996).
[CrossRef] [PubMed]

Phys. Rev. E (2)

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 60, R3505–R3507 (1999).
[CrossRef]

S. Henz and J. Herrmann, “Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: variational approach” Phys. Rev. E 53, 4092–4097 (1996).
[CrossRef]

Phys. Rev. Lett. (7)

L. Bergé and A. Couairon, “Gas-induced solitons,” Phys. Rev. Lett. 86, 1003–1006 (2001).
[CrossRef] [PubMed]

S. Tzortzakis, L. Bergé, A. Couairon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Break-up and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett. 86, 5470–5473 (2001).
[CrossRef] [PubMed]

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, “Optically turbulent femtosecond light guide in air,” Phys. Rev. Lett. 83, 2938–2941 (1999).
[CrossRef]

S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propogation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902–1–4 (2001).
[CrossRef]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (2000).
[CrossRef] [PubMed]

A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
[CrossRef]

Prog. Quantum Electron. (1)

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

Quantum Electron. (1)

V. P. Kandidov, O. G. Kosareva, and S. A. Shlenov, “Influence of transient self-defocusing on the propagation of high-power femtosecond laser pulses in gases under ionization conditions,” Quantum Electron. 24, 905 (1994).
[CrossRef]

Sov. Phys. JETP (3)

L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Zh. Eksp. Teor. Fiz. 47, 1945–1957 (1964) [Sov. Phys. JETP 20, 1307–1314 (1965)].

A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 23, 924–934 (1966) [Zh. Eksp. Teor. Fiz. 50, 1393–1409 (1966)].

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 [Zh. Eksp. Teor. Fiz. 91, 2008–2013 (1986)].

Zh. Eksp. Teor. Fiz. Pis'ma Red. (1)

V. I. Bespalov and V. I. Talanov, Zh. Eksp. Teor. Fiz. Pis'ma Red. 3, 471–476 (1966) [JETP Lett. 3, 307–310 (1966)].

Other (2)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

J. Schjødt-Eriksen, J. V. Moloney, E. M. Wright, and P. L. Christiansen, “Pulse splitting of modulationally unstable femtosecond pulse,” submitted to Opt. Express.

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Figures (14)

Fig. 1
Fig. 1

Autocorrelation of the initial laser pulse (solid curve) and fit by the autocorrelation function of a sech2 profile with 50-fs duration (dotted curve).

Fig. 2
Fig. 2

Initial laser beam profile.

Fig. 3
Fig. 3

Numerical results for a focused beam with energies 1 mJ (dotted curves), 2.5 mJ (dashed curves) and 5 mJ (solid curves) for the two-species model and for the single-species model at 5 mJ (dashed–dotted curves). (a) Peak intensity on axis r=0 as a function of propagation distance z. (b) Electron density as a function of propagation distance. The bold curve at level 1014 cm-3 corresponds to the electron density that is due to nitrogen ionization by a 5-mJ pulse. Electrons produced by oxygen ionization dominate in the total electron density. (c) Mean radius R(z). (d) Energy fraction in the filament with the radius plotted in (c).

Fig. 4
Fig. 4

Transverse beam profiles. Top: Ein=1 mJ and, left to right, z=125 cm, z=150 cm, z=197 cm, z=232 cm; middle: Ein=2.5 mJ and z=150 cm, z=190 cm, z=199 cm, z=237 cm; bottom: Ein=5 mJ and z=150 cm, z=190 cm, z=223 cm, z=313 cm. Arrow, 1.2 mm.

Fig. 5
Fig. 5

Fluence distributions (in joules per square centimeter) obtained numerically along the direction of propagation from incident beams shaped on the experimental beam. The scale lengths on the x and y axes are millimeters. Top: Ein=1 mJ and, left to right, z=150 cm, z=180 cm, z=200 cm; middle: Ein=2.5 mJ and z=130 cm, z=150 cm, z=200 cm; bottom: Ein=5 mJ and z=120 cm, z=140 cm, z=200 cm.

Fig. 6
Fig. 6

Experimental (filled squares) and numerical (curves) results for a focused beam with 5-mJ energy. (a) Mean radius R(z), (b) peak (dashed curve) and average (solid curve) electron density, (c) peak (dashed curve) and average (solid curve) intensity as a function of propagation distance.

Fig. 7
Fig. 7

(a) Pulse power spectra (integrated over 500 shots) measured at z=225 cm for a focused beam with energies of 1, 2.5, and 5 mJ. (b) Pulse power spectra for Ein=5 mJ recorded at propagation distances z=0 (dotted curve), z=190 cm (dashed–dotted curve), z=225 cm (solid curve), and z=250 cm (dashed curve). Their numerical counterparts are shown in (c) at z=170 cm for the same energies as in (a) and in (d) at 5 mJ for the same distances as in (b).

Fig. 8
Fig. 8

Pulse temporal profiles from numerical simulations with a 5-mJ input beam: (a) z=120 cm, (b) z=140 cm, (c) z=160 cm, (d) z=200 cm.

Fig. 9
Fig. 9

Propagation patterns of 50-fs unfocused beams in air at the propagation distances from left to right: z=2.5, 4.5, 6.5, 8.5 m. Top, Ein=1 mJ; middle, Ein=2.5 mJ; bottom, Ein=5 mJ. Arrow, 2 mm.

Fig. 10
Fig. 10

3D plot of the intensity distribution at z=4.5 m for an initially collimated beam with 5-mJ energy.

Fig. 11
Fig. 11

(a) Mean radius, (b) peak electron density, (c) intensity, and (d) fluence as functions of propagation distance for a collimated input beam with 5-mJ energy.

Fig. 12
Fig. 12

Fluence distribution F(r, z) for a collimated input beam with 5-mJ energy.

Fig. 13
Fig. 13

Intensity profiles of the filaments (a) along the y axis, (b) along the x axis, and for a collimated input beam with Ein=5 mJ.

Fig. 14
Fig. 14

Comparison of general ionization rates [Eq. (A2), solid curves] with tunnel rates [Eq. (A11), dashed curves] and their counterpart in the multiphoton limit [Eq. (A13), dashed–dotted curves] for oxygen (bold curves) and nitrogen (fine curves) as functions of laser intensity.

Equations (31)

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-×(×E)-1c2 2t2 -tn2(r, t-t)E(t)dt
=μ0Jt+2PNLt2,
Jt=e2mρE-0c t nρnWnKnω0|E|2 E.
PNL=2n2|E|2E,
ρ=ρat-ρO-ρN,
ρNt=-WNρN,
ρOt=-WOρO.
Ez=i2k0T-12r2+1r rE-i k2 2Eτ2-i k02ρcT-1(ρE)+ik0n2T(1-xdK)|E|2+xdKτdK ×-τ exp-τ-ττK|E(τ)|2dτE-12 n ρnWnKnω0|E|2 E,
ρt=σair IKair(ρat-ρ),
E(r, t, 0)=(2Pin/πw02)1/2×exp(-r2/w02-t2/tp2-ikr2/2 f).
2ik0 Ez+ΔE+k02(n2|E|2-n2K|E|2K)E=0,
Λ=2πPcrI0(1-αI0K-1)1/2,Λz=Λ2π2λ0,
E(r, 0)=j=1,22Pjπwj21/2 exp-(r-ri)2wj2,
P=|E|2dr=2Pj[1+exp(-δ2/2wj2)],
dz2r2=2(H+K-1K+1k02n2K|E|2(K+1)dr)/k02P,
H=|E|2-k02n22|E|4+k02n2KK+1|E|2(K+1)dr,
H4Pjwj2 1+1-δ22wj2exp-δ22wj2-Pj2Pcr×1+3 exp(-δ2/wj2)+4 exp-3δ24wj2+1+2(K+1)exp-2K+1K+1 δ22wj2,
γ=ω0eF 2mU0
W(ω0, F)=ωa.u.3/2π|Cn*,l*|2f(l, m)×U0UH2F0F1+γ22n-|m|-3/2×Am(ω0, γ)exp-2F03Fg(γ),
|Cn*,l*|2=22n*n*Γ(n*+l*+1)Γ(n*-l*),
f(l, m)=(2l+1)(l+|m|)!2|m|(|m|)!(l-|m|)!,f(0, 0)=1.
Am(ω0, γ)=43π 1|m|! γ21+γ2×κν+ exp[-α(κ-ν)]Φm[β(κ-ν)],
Φm(x)=exp(-x2)0x(x2-y2)|m| exp(y2)dy,
β(γ)=2γ1+γ2,
α(γ)=2sinh-1 γ-γ1+γ2=2γ3/3γ12(log 2γ-1)γ1,
g(γ)=32γ 1+12γ2sinh-1 γ-1+γ22γ=1-γ210+9γ4280γ132γ log 2γ-12γ1,
ν=U0ω0 1+12γ2,κ0=mod(ν+1).
W(ω0, F)=ωa.u.3/2π|Cn*,l*|2f(l, m)×U0UH 2F0F2n-|m|-3/2 exp-2F03F.
W(ω0, F)=ωa.u. 22π|Cn*,l*|2×U0UH f(l, m)|m|!(4ν0)2n-|m|-3/2 1(4γ2)κ0×exp[2κ0-ν0(1+1/γ2)]×Φm[2(κ0-ν)],
w(ω0, F)=σIK,
σ=ωa.u. 42n-|m|2π2|Cn*,l*|2 f(l, m)|m|! U0UH(ν0)2(n+K)-|m|-3/2×exp(2K-ν0)Φ0[2(K-ω0)] 1F02κ0.

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