Abstract

We investigate the self-focusing dynamics of super-Gaussian optical beams in a Kerr medium. We find that up to several times the critical power for self-focusing, super-Gaussian beams evolve towards a Townes profile. At higher powers the super-Gaussian beams form rings which break into filaments as a result of noise. Our results are consistent with the observed self-focusing dynamics of femtosecond laser pulses in air [1] in which filaments are formed along a ring about the axis of the initial beam where the initial beam did not form a ring.

© 2006 Optical Society of America

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  1. J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
    [CrossRef] [PubMed]
  2. P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
    [CrossRef]
  3. B.W. Zeff, B. Kleber, J. Fineberg, and D. P. Lathrop, “Singularity dynamics in curvature collapse and jet eruption on a fluid surface,” Nature (London) 403, 401–404 (2000).
    [CrossRef]
  4. P. A. Robinson, “Nonlinear wave collapse and strong turbulence,” Rev. Mod. Phys. 69, 507–573 (1997).
    [CrossRef]
  5. C. A. Sackett, J.M. Gerton, M. Welling, and R. G. Hulet, “Measurements of collective collapse in a Bose-Einstein condensate with attractive interactions,” Phys. Rev. Lett. 82, 876–879 (1999).
    [CrossRef]
  6. L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92, 225002 1–4 (2004).
    [CrossRef]
  7. G. Fibich and A. L. Gaeta, “On the critical power for self-focusing in bulk media and hollow waveguides,” Opt. Lett. 25, 335–337 (2000).
    [CrossRef]
  8. G. Fraiman, “Asymptotic stability of manifold of self-similar solutions in self-focusing,” Sov. Phys. JETP 61, 228–233 (1985).
  9. M. Landman, G. Papanicolaou, C. Sulem, and P. Sulem, “Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension,” Phys. Rev. A 38, 3837–3843 (1988).
    [CrossRef] [PubMed]
  10. B. LeMesurier, P. Papanicolaou, C. Sulem, and P. Sulem, “Local structure of the self-focusing singularity of the nonlinear Schrödinger equation,” Physica D 32, 210–226 (1988).
    [CrossRef]
  11. F. Merle and P. Raphael, “Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation,” Geom. Funct. Anal. 13, 591–642 (2003).
    [CrossRef]
  12. K. D. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902 1–4 (2003).
    [CrossRef]
  13. G. Fibich, Nir Gavish, and Xiao-Ping Wang, “New singular solutions of the nonlinear Schrödinger equation,” Physica D 211, 193–220 (2005).
    [CrossRef]
  14. D. V. Skryabin and W. J. Firth, “Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,” Phys. Rev. E 58, 3916–3930 (1998).
    [CrossRef]
  15. L. T. Vuong, T. D. Grow, A. Ishaaya, A.L. Gaeta, G. W. Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 1–4 (2006).
    [CrossRef]
  16. A. J. Campillo, S. L. Shapiro, and B. R. Suydam, “Periodic breakup of optical beams due to self-focusing,” Appl. Phys. B 23, 628–630 (1973).
  17. J. M. Soto-Crespo, D. R. Heatley, and E. M. Wright, “Stability of the higher-bound states in a saturable selffocusing medium,” Phys. Rev. A 44, 636–644 (1991).
    [CrossRef] [PubMed]
  18. M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
    [CrossRef]
  19. J.M. Soto-Crespo, E.M. Wright, and N.N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3174 (1992).
    [CrossRef] [PubMed]
  20. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in non-linear liquids,” JETP Lett. 3, 307–310 (1966).
  21. A. J. Campillo, S. L. Shapiro, and B. R. Suydam, “Relationship of self-focusing to spatial instability modes,” Appl. Phys. Lett. 24, 178–180 (1974).
    [CrossRef]
  22. G. Fibich, S. Eisenmann, B. Ilan, Y. Erlich, M. Fraenkel, Z. Henis, A. L. Gaeta, and A. Zigler, “Self-focusing distances of very high power laser pulses,” Opt. Express 13, 5897–5903 (2005).
    [CrossRef] [PubMed]
  23. 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 1–15 (2004).
    [CrossRef]
  24. R. Y. Chiao, E. Garmire, and C. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
    [CrossRef]
  25. L. Bergé, C. Gouédard, J. Schjodt-Eriksen, and H. Ward, “Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states,” Physica D 176, 181–211 (2003).
    [CrossRef]
  26. A. Dubietis, E. Gaižauskas, G. Tamošauskas, and P. D. Trapani, “Light filaments without self-channeling,” Phys. Rev. Lett. 92, 253903 1–4 (2004).
    [CrossRef]
  27. A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (2000).
    [CrossRef] [PubMed]
  28. G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new propagation regime without ionization,” Appl. Phys. B 79, 379–382 (2004).
    [CrossRef]

2006 (1)

L. T. Vuong, T. D. Grow, A. Ishaaya, A.L. Gaeta, G. W. Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 1–4 (2006).
[CrossRef]

2005 (2)

2004 (4)

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 1–15 (2004).
[CrossRef]

A. Dubietis, E. Gaižauskas, G. Tamošauskas, and P. D. Trapani, “Light filaments without self-channeling,” Phys. Rev. Lett. 92, 253903 1–4 (2004).
[CrossRef]

G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new propagation regime without ionization,” Appl. Phys. B 79, 379–382 (2004).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92, 225002 1–4 (2004).
[CrossRef]

2003 (4)

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

F. Merle and P. Raphael, “Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation,” Geom. Funct. Anal. 13, 591–642 (2003).
[CrossRef]

K. D. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902 1–4 (2003).
[CrossRef]

L. Bergé, C. Gouédard, J. Schjodt-Eriksen, and H. Ward, “Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states,” Physica D 176, 181–211 (2003).
[CrossRef]

2000 (3)

G. Fibich and A. L. Gaeta, “On the critical power for self-focusing in bulk media and hollow waveguides,” Opt. Lett. 25, 335–337 (2000).
[CrossRef]

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

B.W. Zeff, B. Kleber, J. Fineberg, and D. P. Lathrop, “Singularity dynamics in curvature collapse and jet eruption on a fluid surface,” Nature (London) 403, 401–404 (2000).
[CrossRef]

1999 (1)

C. A. Sackett, J.M. Gerton, M. Welling, and R. G. Hulet, “Measurements of collective collapse in a Bose-Einstein condensate with attractive interactions,” Phys. Rev. Lett. 82, 876–879 (1999).
[CrossRef]

1998 (1)

D. V. Skryabin and W. J. Firth, “Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,” Phys. Rev. E 58, 3916–3930 (1998).
[CrossRef]

1997 (1)

P. A. Robinson, “Nonlinear wave collapse and strong turbulence,” Rev. Mod. Phys. 69, 507–573 (1997).
[CrossRef]

1992 (1)

J.M. Soto-Crespo, E.M. Wright, and N.N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3174 (1992).
[CrossRef] [PubMed]

1991 (1)

J. M. Soto-Crespo, D. R. Heatley, and E. M. Wright, “Stability of the higher-bound states in a saturable selffocusing medium,” Phys. Rev. A 44, 636–644 (1991).
[CrossRef] [PubMed]

1988 (3)

M. Landman, G. Papanicolaou, C. Sulem, and P. Sulem, “Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension,” Phys. Rev. A 38, 3837–3843 (1988).
[CrossRef] [PubMed]

B. LeMesurier, P. Papanicolaou, C. Sulem, and P. Sulem, “Local structure of the self-focusing singularity of the nonlinear Schrödinger equation,” Physica D 32, 210–226 (1988).
[CrossRef]

M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
[CrossRef]

1985 (1)

G. Fraiman, “Asymptotic stability of manifold of self-similar solutions in self-focusing,” Sov. Phys. JETP 61, 228–233 (1985).

1974 (1)

A. J. Campillo, S. L. Shapiro, and B. R. Suydam, “Relationship of self-focusing to spatial instability modes,” Appl. Phys. Lett. 24, 178–180 (1974).
[CrossRef]

1973 (1)

A. J. Campillo, S. L. Shapiro, and B. R. Suydam, “Periodic breakup of optical beams due to self-focusing,” Appl. Phys. B 23, 628–630 (1973).

1966 (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in non-linear liquids,” JETP Lett. 3, 307–310 (1966).

1965 (1)

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

1964 (1)

R. Y. Chiao, E. Garmire, and C. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Akhmediev, N.N.

J.M. Soto-Crespo, E.M. Wright, and N.N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3174 (1992).
[CrossRef] [PubMed]

André, Y.-B.

G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new propagation regime without ionization,” Appl. Phys. B 79, 379–382 (2004).
[CrossRef]

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

Bergé, L.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92, 225002 1–4 (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 1–15 (2004).
[CrossRef]

L. Bergé, C. Gouédard, J. Schjodt-Eriksen, and H. Ward, “Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states,” Physica D 176, 181–211 (2003).
[CrossRef]

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in non-linear liquids,” JETP Lett. 3, 307–310 (1966).

Bourayou, R.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92, 225002 1–4 (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 1–15 (2004).
[CrossRef]

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

Campillo, A. J.

A. J. Campillo, S. L. Shapiro, and B. R. Suydam, “Relationship of self-focusing to spatial instability modes,” Appl. Phys. Lett. 24, 178–180 (1974).
[CrossRef]

A. J. Campillo, S. L. Shapiro, and B. R. Suydam, “Periodic breakup of optical beams due to self-focusing,” Appl. Phys. B 23, 628–630 (1973).

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Couairon, A.

G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new propagation regime without ionization,” Appl. Phys. B 79, 379–382 (2004).
[CrossRef]

D’Amico, C.

G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new propagation regime without ionization,” Appl. Phys. B 79, 379–382 (2004).
[CrossRef]

Dubietis, A.

A. Dubietis, E. Gaižauskas, G. Tamošauskas, and P. D. Trapani, “Light filaments without self-channeling,” Phys. Rev. Lett. 92, 253903 1–4 (2004).
[CrossRef]

Eisenmann, S.

Eliel, E. R.

L. T. Vuong, T. D. Grow, A. Ishaaya, A.L. Gaeta, G. W. Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 1–4 (2006).
[CrossRef]

Erlich, Y.

Feit, M. D.

Fibich, G.

L. T. Vuong, T. D. Grow, A. Ishaaya, A.L. Gaeta, G. W. Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 1–4 (2006).
[CrossRef]

G. Fibich, S. Eisenmann, B. Ilan, Y. Erlich, M. Fraenkel, Z. Henis, A. L. Gaeta, and A. Zigler, “Self-focusing distances of very high power laser pulses,” Opt. Express 13, 5897–5903 (2005).
[CrossRef] [PubMed]

G. Fibich, Nir Gavish, and Xiao-Ping Wang, “New singular solutions of the nonlinear Schrödinger equation,” Physica D 211, 193–220 (2005).
[CrossRef]

K. D. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902 1–4 (2003).
[CrossRef]

G. Fibich and A. L. Gaeta, “On the critical power for self-focusing in bulk media and hollow waveguides,” Opt. Lett. 25, 335–337 (2000).
[CrossRef]

Fineberg, J.

B.W. Zeff, B. Kleber, J. Fineberg, and D. P. Lathrop, “Singularity dynamics in curvature collapse and jet eruption on a fluid surface,” Nature (London) 403, 401–404 (2000).
[CrossRef]

Firth, W. J.

D. V. Skryabin and W. J. Firth, “Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,” Phys. Rev. E 58, 3916–3930 (1998).
[CrossRef]

Fleck, J. A.

Fraenkel, M.

Fraiman, G.

G. Fraiman, “Asymptotic stability of manifold of self-similar solutions in self-focusing,” Sov. Phys. JETP 61, 228–233 (1985).

Franco, M.

G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new propagation regime without ionization,” Appl. Phys. B 79, 379–382 (2004).
[CrossRef]

Frey, S.

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

Gaeta, A. L.

G. Fibich, S. Eisenmann, B. Ilan, Y. Erlich, M. Fraenkel, Z. Henis, A. L. Gaeta, and A. Zigler, “Self-focusing distances of very high power laser pulses,” Opt. Express 13, 5897–5903 (2005).
[CrossRef] [PubMed]

K. D. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902 1–4 (2003).
[CrossRef]

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

G. Fibich and A. L. Gaeta, “On the critical power for self-focusing in bulk media and hollow waveguides,” Opt. Lett. 25, 335–337 (2000).
[CrossRef]

Gaeta, A.L.

L. T. Vuong, T. D. Grow, A. Ishaaya, A.L. Gaeta, G. W. Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 1–4 (2006).
[CrossRef]

Gaižauskas, E.

A. Dubietis, E. Gaižauskas, G. Tamošauskas, and P. D. Trapani, “Light filaments without self-channeling,” Phys. Rev. Lett. 92, 253903 1–4 (2004).
[CrossRef]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Gavish, Nir

G. Fibich, Nir Gavish, and Xiao-Ping Wang, “New singular solutions of the nonlinear Schrödinger equation,” Physica D 211, 193–220 (2005).
[CrossRef]

Gerton, J.M.

C. A. Sackett, J.M. Gerton, M. Welling, and R. G. Hulet, “Measurements of collective collapse in a Bose-Einstein condensate with attractive interactions,” Phys. Rev. Lett. 82, 876–879 (1999).
[CrossRef]

Gouédard, C.

L. Bergé, C. Gouédard, J. Schjodt-Eriksen, and H. Ward, “Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states,” Physica D 176, 181–211 (2003).
[CrossRef]

Grow, T. D.

L. T. Vuong, T. D. Grow, A. Ishaaya, A.L. Gaeta, G. W. Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 1–4 (2006).
[CrossRef]

Heatley, D. R.

J. M. Soto-Crespo, D. R. Heatley, and E. M. Wright, “Stability of the higher-bound states in a saturable selffocusing medium,” Phys. Rev. A 44, 636–644 (1991).
[CrossRef] [PubMed]

Henis, Z.

Hooft, G. W.

L. T. Vuong, T. D. Grow, A. Ishaaya, A.L. Gaeta, G. W. Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 1–4 (2006).
[CrossRef]

Hulet, R. G.

C. A. Sackett, J.M. Gerton, M. Welling, and R. G. Hulet, “Measurements of collective collapse in a Bose-Einstein condensate with attractive interactions,” Phys. Rev. Lett. 82, 876–879 (1999).
[CrossRef]

Ilan, B.

Ishaaya, A.

L. T. Vuong, T. D. Grow, A. Ishaaya, A.L. Gaeta, G. W. Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 1–4 (2006).
[CrossRef]

Kasparian, J.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92, 225002 1–4 (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 1–15 (2004).
[CrossRef]

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

Kelley, P. L.

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

Kleber, B.

B.W. Zeff, B. Kleber, J. Fineberg, and D. P. Lathrop, “Singularity dynamics in curvature collapse and jet eruption on a fluid surface,” Nature (London) 403, 401–404 (2000).
[CrossRef]

Landman, M.

M. Landman, G. Papanicolaou, C. Sulem, and P. Sulem, “Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension,” Phys. Rev. A 38, 3837–3843 (1988).
[CrossRef] [PubMed]

Lathrop, D. P.

B.W. Zeff, B. Kleber, J. Fineberg, and D. P. Lathrop, “Singularity dynamics in curvature collapse and jet eruption on a fluid surface,” Nature (London) 403, 401–404 (2000).
[CrossRef]

Lederer, F.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92, 225002 1–4 (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 1–15 (2004).
[CrossRef]

LeMesurier, B.

B. LeMesurier, P. Papanicolaou, C. Sulem, and P. Sulem, “Local structure of the self-focusing singularity of the nonlinear Schrödinger equation,” Physica D 32, 210–226 (1988).
[CrossRef]

Méchain, G.

G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new propagation regime without ionization,” Appl. Phys. B 79, 379–382 (2004).
[CrossRef]

Méjean, G.

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 1–15 (2004).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92, 225002 1–4 (2004).
[CrossRef]

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

Merle, F.

F. Merle and P. Raphael, “Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation,” Geom. Funct. Anal. 13, 591–642 (2003).
[CrossRef]

Moll, K. D.

K. D. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902 1–4 (2003).
[CrossRef]

Mysyrowicz, A.

G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new propagation regime without ionization,” Appl. Phys. B 79, 379–382 (2004).
[CrossRef]

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

Papanicolaou, G.

M. Landman, G. Papanicolaou, C. Sulem, and P. Sulem, “Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension,” Phys. Rev. A 38, 3837–3843 (1988).
[CrossRef] [PubMed]

Papanicolaou, P.

B. LeMesurier, P. Papanicolaou, C. Sulem, and P. Sulem, “Local structure of the self-focusing singularity of the nonlinear Schrödinger equation,” Physica D 32, 210–226 (1988).
[CrossRef]

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 1–15 (2004).
[CrossRef]

Prade, B.

G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new propagation regime without ionization,” Appl. Phys. B 79, 379–382 (2004).
[CrossRef]

Raphael, P.

F. Merle and P. Raphael, “Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation,” Geom. Funct. Anal. 13, 591–642 (2003).
[CrossRef]

Robinson, P. A.

P. A. Robinson, “Nonlinear wave collapse and strong turbulence,” Rev. Mod. Phys. 69, 507–573 (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 1–15 (2004).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92, 225002 1–4 (2004).
[CrossRef]

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

Sackett, C. A.

C. A. Sackett, J.M. Gerton, M. Welling, and R. G. Hulet, “Measurements of collective collapse in a Bose-Einstein condensate with attractive interactions,” Phys. Rev. Lett. 82, 876–879 (1999).
[CrossRef]

Salmon, E.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92, 225002 1–4 (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 1–15 (2004).
[CrossRef]

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

Sauerbrey, R.

G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new propagation regime without ionization,” Appl. Phys. B 79, 379–382 (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 1–15 (2004).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92, 225002 1–4 (2004).
[CrossRef]

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

Schjodt-Eriksen, J.

L. Bergé, C. Gouédard, J. Schjodt-Eriksen, and H. Ward, “Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states,” Physica D 176, 181–211 (2003).
[CrossRef]

Shapiro, S. L.

A. J. Campillo, S. L. Shapiro, and B. R. Suydam, “Relationship of self-focusing to spatial instability modes,” Appl. Phys. Lett. 24, 178–180 (1974).
[CrossRef]

A. J. Campillo, S. L. Shapiro, and B. R. Suydam, “Periodic breakup of optical beams due to self-focusing,” Appl. Phys. B 23, 628–630 (1973).

Skryabin, D. V.

D. V. Skryabin and W. J. Firth, “Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,” Phys. Rev. E 58, 3916–3930 (1998).
[CrossRef]

Skupin, S.

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 1–15 (2004).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92, 225002 1–4 (2004).
[CrossRef]

Soto-Crespo, J. M.

J. M. Soto-Crespo, D. R. Heatley, and E. M. Wright, “Stability of the higher-bound states in a saturable selffocusing medium,” Phys. Rev. A 44, 636–644 (1991).
[CrossRef] [PubMed]

Soto-Crespo, J.M.

J.M. Soto-Crespo, E.M. Wright, and N.N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3174 (1992).
[CrossRef] [PubMed]

Sulem, C.

M. Landman, G. Papanicolaou, C. Sulem, and P. Sulem, “Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension,” Phys. Rev. A 38, 3837–3843 (1988).
[CrossRef] [PubMed]

B. LeMesurier, P. Papanicolaou, C. Sulem, and P. Sulem, “Local structure of the self-focusing singularity of the nonlinear Schrödinger equation,” Physica D 32, 210–226 (1988).
[CrossRef]

Sulem, P.

B. LeMesurier, P. Papanicolaou, C. Sulem, and P. Sulem, “Local structure of the self-focusing singularity of the nonlinear Schrödinger equation,” Physica D 32, 210–226 (1988).
[CrossRef]

M. Landman, G. Papanicolaou, C. Sulem, and P. Sulem, “Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension,” Phys. Rev. A 38, 3837–3843 (1988).
[CrossRef] [PubMed]

Suydam, B. R.

A. J. Campillo, S. L. Shapiro, and B. R. Suydam, “Relationship of self-focusing to spatial instability modes,” Appl. Phys. Lett. 24, 178–180 (1974).
[CrossRef]

A. J. Campillo, S. L. Shapiro, and B. R. Suydam, “Periodic breakup of optical beams due to self-focusing,” Appl. Phys. B 23, 628–630 (1973).

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in non-linear liquids,” JETP Lett. 3, 307–310 (1966).

Tamošauskas, G.

A. Dubietis, E. Gaižauskas, G. Tamošauskas, and P. D. Trapani, “Light filaments without self-channeling,” Phys. Rev. Lett. 92, 253903 1–4 (2004).
[CrossRef]

Townes, C.

R. Y. Chiao, E. Garmire, and C. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Trapani, P. D.

A. Dubietis, E. Gaižauskas, G. Tamošauskas, and P. D. Trapani, “Light filaments without self-channeling,” Phys. Rev. Lett. 92, 253903 1–4 (2004).
[CrossRef]

Tzortzakis, S.

G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new propagation regime without ionization,” Appl. Phys. B 79, 379–382 (2004).
[CrossRef]

Vuong, L. T.

L. T. Vuong, T. D. Grow, A. Ishaaya, A.L. Gaeta, G. W. Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 1–4 (2006).
[CrossRef]

Wang, Xiao-Ping

G. Fibich, Nir Gavish, and Xiao-Ping Wang, “New singular solutions of the nonlinear Schrödinger equation,” Physica D 211, 193–220 (2005).
[CrossRef]

Ward, H.

L. Bergé, C. Gouédard, J. Schjodt-Eriksen, and H. Ward, “Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states,” Physica D 176, 181–211 (2003).
[CrossRef]

Welling, M.

C. A. Sackett, J.M. Gerton, M. Welling, and R. G. Hulet, “Measurements of collective collapse in a Bose-Einstein condensate with attractive interactions,” Phys. Rev. Lett. 82, 876–879 (1999).
[CrossRef]

Wille, H.

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

Wolf, J. P.

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 1–15 (2004).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92, 225002 1–4 (2004).
[CrossRef]

Wolf, J.-P.

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

Wöste, L.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92, 225002 1–4 (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 1–15 (2004).
[CrossRef]

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

Wright, E. M.

J. M. Soto-Crespo, D. R. Heatley, and E. M. Wright, “Stability of the higher-bound states in a saturable selffocusing medium,” Phys. Rev. A 44, 636–644 (1991).
[CrossRef] [PubMed]

Wright, E.M.

J.M. Soto-Crespo, E.M. Wright, and N.N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3174 (1992).
[CrossRef] [PubMed]

Yu, J.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92, 225002 1–4 (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 1–15 (2004).
[CrossRef]

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

Zeff, B.W.

B.W. Zeff, B. Kleber, J. Fineberg, and D. P. Lathrop, “Singularity dynamics in curvature collapse and jet eruption on a fluid surface,” Nature (London) 403, 401–404 (2000).
[CrossRef]

Zigler, A.

Appl. Phys. B (2)

A. J. Campillo, S. L. Shapiro, and B. R. Suydam, “Periodic breakup of optical beams due to self-focusing,” Appl. Phys. B 23, 628–630 (1973).

G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, “Long-range self-channeling of infrared laser pulses in air: a new propagation regime without ionization,” Appl. Phys. B 79, 379–382 (2004).
[CrossRef]

Appl. Phys. Lett. (1)

A. J. Campillo, S. L. Shapiro, and B. R. Suydam, “Relationship of self-focusing to spatial instability modes,” Appl. Phys. Lett. 24, 178–180 (1974).
[CrossRef]

Geom. Funct. Anal. (1)

F. Merle and P. Raphael, “Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation,” Geom. Funct. Anal. 13, 591–642 (2003).
[CrossRef]

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

JETP Lett. (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in non-linear liquids,” JETP Lett. 3, 307–310 (1966).

Nature (London) (1)

B.W. Zeff, B. Kleber, J. Fineberg, and D. P. Lathrop, “Singularity dynamics in curvature collapse and jet eruption on a fluid surface,” Nature (London) 403, 401–404 (2000).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (3)

J.M. Soto-Crespo, E.M. Wright, and N.N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A 45, 3168–3174 (1992).
[CrossRef] [PubMed]

M. Landman, G. Papanicolaou, C. Sulem, and P. Sulem, “Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension,” Phys. Rev. A 38, 3837–3843 (1988).
[CrossRef] [PubMed]

J. M. Soto-Crespo, D. R. Heatley, and E. M. Wright, “Stability of the higher-bound states in a saturable selffocusing medium,” Phys. Rev. A 44, 636–644 (1991).
[CrossRef] [PubMed]

Phys. Rev. E (2)

D. V. Skryabin and W. J. Firth, “Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,” Phys. Rev. E 58, 3916–3930 (1998).
[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 1–15 (2004).
[CrossRef]

Phys. Rev. Lett. (8)

R. Y. Chiao, E. Garmire, and C. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

A. Dubietis, E. Gaižauskas, G. Tamošauskas, and P. D. Trapani, “Light filaments without self-channeling,” Phys. Rev. Lett. 92, 253903 1–4 (2004).
[CrossRef]

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

L. T. Vuong, T. D. Grow, A. Ishaaya, A.L. Gaeta, G. W. Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 1–4 (2006).
[CrossRef]

K. D. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90, 203902 1–4 (2003).
[CrossRef]

C. A. Sackett, J.M. Gerton, M. Welling, and R. G. Hulet, “Measurements of collective collapse in a Bose-Einstein condensate with attractive interactions,” Phys. Rev. Lett. 82, 876–879 (1999).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, “Multiple filamentation of terawatt laser pulses in air,” Phys. Rev. Lett. 92, 225002 1–4 (2004).
[CrossRef]

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

Physica D (3)

G. Fibich, Nir Gavish, and Xiao-Ping Wang, “New singular solutions of the nonlinear Schrödinger equation,” Physica D 211, 193–220 (2005).
[CrossRef]

B. LeMesurier, P. Papanicolaou, C. Sulem, and P. Sulem, “Local structure of the self-focusing singularity of the nonlinear Schrödinger equation,” Physica D 32, 210–226 (1988).
[CrossRef]

L. Bergé, C. Gouédard, J. Schjodt-Eriksen, and H. Ward, “Filamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states,” Physica D 176, 181–211 (2003).
[CrossRef]

Rev. Mod. Phys. (1)

P. A. Robinson, “Nonlinear wave collapse and strong turbulence,” Rev. Mod. Phys. 69, 507–573 (1997).
[CrossRef]

Science (1)

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

Sov. Phys. JETP (1)

G. Fraiman, “Asymptotic stability of manifold of self-similar solutions in self-focusing,” Sov. Phys. JETP 61, 228–233 (1985).

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Figures (8)

Fig. 1.
Fig. 1.

Profiles of the solutions to the NLSE for a super-Gaussian input beam at two powers for various propagation distances z. (a) P=5Pcr , the beam collapses towards the Townes profile, (inset) three-dimensional plot for z=0.1. (b) P=10Pcr , the beam forms a ring shape, (inset) three-dimensional plot for z=0.065.

Fig. 2.
Fig. 2.

Nonlinear propagation of rays (horizontal lines) and phase fronts (vertical lines) for (a) Gaussian ( S H = ( 1 + 1 9 e ρ 2 ) ζ ) and (b) super-Gaussian ( S H = ( 1 + 1 9 e ρ 4 ) ζ ) input beams.

Fig. 3.
Fig. 3.

Solutions to the NLSE with a super-Gaussian input beam with 5% amplitude noise (phase noise produces similar results) and an input power of 20 Pcr at distances (a) ξ=0, (b) ξ=0.025, (c) ξ=0.0375, and (d) ξ=0.05.

Fig. 4.
Fig. 4.

Left: Comparison of our modulational instability analysis with the approximate number of filaments observed in the numerical simulations. Right: Numerically simulated filamentation of an initially super-Gaussian beam with 10% amplitude noise for input powers of (a) 10 Pcr , (b) 15 Pcr , (c) 20 Pcr , (d) 30 Pcr , (e) 40 Pcr , (f) 50 Pcr .

Fig. 5.
Fig. 5.

Schematic of the experimental set up used to observe beam collapse. In the experiment the beam is reflected from the spatial light modulator at a near-normal incidence.

Fig. 6.
Fig. 6.

Images of the input and output intensity beam profiles for a 7-cm propagation distance (0.9 mm X 0.9 mm). (a) Gaussian input profile, (b) output beam with the Gaussian input and an input energy of E=5.6 µJ, (c) super-Gaussian input profile, (d) output beam with a super-Gaussian input and an input energy of E=5.0 µJ. The pulse energies were just below the threshold for supercontinuum generation.

Fig. 7.
Fig. 7.

Experimental intensity distributions of an initially super-Gaussian beam (E=13.3 µJ) as it propagates in the water cell. The image area is 0.3 mm X 0.3 mm. Left: Input profile. Right: Output profiles with the length of the water cell set at, (a) 1.3 cm, (b) 2.0 cm, (c) 3.0 cm, and (d) 4.3 cm.

Fig. 8.
Fig. 8.

Input super-Gaussian beams (a-c) for increasing energy and corresponding output profiles (d-f) just before the super-continuum threshold in a 10-cm cell. (d) E=4.2 µJ, I=3.3×1010 W/cm2, (e) E=9.1 µJ, I=4.6×1010 W/cm2, (f) E=17.5 µJ, I=6.2×1010 W/cm2. The image area is 1 mm X 1 mm.

Equations (6)

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i ψ ζ + 2 ψ + ψ 2 ψ = 0 ,
i ψ ζ + ψ 2 ψ = 0 .
ψ = ψ 0 ( μ , ν ) e i k 0 S ,
S ( ρ , ζ ) = 2 n 2 n 0 ψ 0 2 ζ .
E ( ρ , ζ ) = A H ( ρ , ζ ) e i k 0 S H ,
η = 2 e 1 ( 2 P w 2 α e ) ,

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