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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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. Berg´e, 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 Schr¨odinger 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, N. Gavish, and X.-P. Wang, "New singular solutions of the nonlinear Schrödinger equation,"
    [CrossRef]
  14. PhysicaD 211, 193-220 (2005).
    [CrossRef]
  15. 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]
  16. 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).
  17. 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).
    [CrossRef] [PubMed]
  18. 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]
  19. M. D. Feit and J. A. Fleck, Jr., "Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams," J. Opt. Soc. Am. B 5, 633-640 (1988).
    [CrossRef] [PubMed]
  20. 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).
  21. V. I. Bespalov and V. I. Talanov, "Filamentary structure of light beams in non-linear liquids," JETP Lett. 3,307-310 (1966).
    [CrossRef]
  22. A. J. Campillo, S. L. Shapiro, B. R. Suydam, "Relationship of self-focusing to spatial instability modes," Appl. Phys. Lett. 24, 178-180 (1974).
    [CrossRef] [PubMed]
  23. 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]
  24. 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]
  25. R. Y. Chiao, E. Garmire, and C. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964). 25. L. Berg´e, C. Gou´edard, 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. Tamǒsauskas, 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. M´echain, A. Couairon, Y.-B. Andé, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, and A. Mysyrowicz, 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]

2005

2004

G. M´echain, A. Couairon, Y.-B. Andé, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, and A. Mysyrowicz, 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]

2003

R. Y. Chiao, E. Garmire, and C. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964). 25. L. Berg´e, C. Gou´edard, 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]

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]

2000

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]

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]

1999

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

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

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

1992

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).

1991

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]

1988

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

B. LeMesurier, P. Papanicolaou, C. Sulem, and P. Sulem, "Local structure of the self-focusing singularity of the nonlinear Schr¨odinger equation," Physica D 32, 210-226 (1988).
[CrossRef]

1985

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

1974

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

1973

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).
[CrossRef] [PubMed]

1966

V. I. Bespalov and V. I. Talanov, "Filamentary structure of light beams in non-linear liquids," JETP Lett. 3,307-310 (1966).
[CrossRef]

1965

P. L. Kelley, "Self-focusing of optical beams," Phys. Rev. Lett. 15, 1005-1008 (1965).
[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).

Andé, Y.-B.

G. M´echain, A. Couairon, Y.-B. Andé, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, and A. Mysyrowicz, 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]

André, Y.-B.

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]

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).
[CrossRef]

Bourayou, R.

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, B. R. Suydam, "Relationship of self-focusing to spatial instability modes," Appl. Phys. Lett. 24, 178-180 (1974).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964). 25. L. Berg´e, C. Gou´edard, 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]

Couairon, A.

G. M´echain, A. Couairon, Y.-B. Andé, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, and A. Mysyrowicz, 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. M´echain, A. Couairon, Y.-B. Andé, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, and A. Mysyrowicz, 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]

Eisenmann, S.

Erlich, Y.

Feit, M. D.

Fibich, G.

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. M´echain, A. Couairon, Y.-B. Andé, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, and A. Mysyrowicz, 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.

Garmire, E.

R. Y. Chiao, E. Garmire, and C. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964). 25. L. Berg´e, C. Gou´edard, 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]

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]

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]

Henis, Z.

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.

Kasparian, J.

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]

LeMesurier, B.

B. LeMesurier, P. Papanicolaou, C. Sulem, and P. Sulem, "Local structure of the self-focusing singularity of the nonlinear Schr¨odinger equation," Physica D 32, 210-226 (1988).
[CrossRef]

M´echain, G.

G. M´echain, A. Couairon, Y.-B. Andé, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, and A. Mysyrowicz, 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.

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]

Mysyrowicz, A.

G. M´echain, A. Couairon, Y.-B. Andé, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, and A. Mysyrowicz, 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 Schr¨odinger equation," Physica D 32, 210-226 (1988).
[CrossRef]

Prade, B.

G. M´echain, A. Couairon, Y.-B. Andé, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, and A. Mysyrowicz, 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.

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.

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. M´echain, A. Couairon, Y.-B. Andé, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, and A. Mysyrowicz, 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]

Shapiro, S. L.

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

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).
[CrossRef] [PubMed]

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]

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).

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]

Sulem, C.

B. LeMesurier, P. Papanicolaou, C. Sulem, and P. Sulem, "Local structure of the self-focusing singularity of the nonlinear Schr¨odinger 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]

Sulem, P.

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 Schr¨odinger equation," Physica D 32, 210-226 (1988).
[CrossRef]

Suydam, B. R.

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

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).
[CrossRef] [PubMed]

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).
[CrossRef]

Townes, C.

R. Y. Chiao, E. Garmire, and C. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964). 25. L. Berg´e, C. Gou´edard, 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]

Tzortzakis, S.

G. M´echain, A. Couairon, Y.-B. Andé, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, and A. Mysyrowicz, 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]

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.

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.

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, 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).

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]

Yu, J.

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

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).
[CrossRef] [PubMed]

G. M´echain, A. Couairon, Y.-B. Andé, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, and A. Mysyrowicz, 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.

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

Geom. Funct. Anal.

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

JETP Lett.

V. I. Bespalov and V. I. Talanov, "Filamentary structure of light beams in non-linear liquids," JETP Lett. 3,307-310 (1966).
[CrossRef]

Nature (London)

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

Opt. Lett.

Phys. Rev. A

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, 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).

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]

Phys. Rev. E

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]

Phys. Rev. Lett.

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]

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

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

Physica D

R. Y. Chiao, E. Garmire, and C. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964). 25. L. Berg´e, C. Gou´edard, 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]

B. LeMesurier, P. Papanicolaou, C. Sulem, and P. Sulem, "Local structure of the self-focusing singularity of the nonlinear Schr¨odinger equation," Physica D 32, 210-226 (1988).
[CrossRef]

Rev. Mod. Phys.

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

Science

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

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

Other

L. Berg´e, 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]

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).

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, N. Gavish, and X.-P. Wang, "New singular solutions of the nonlinear Schrödinger equation,"
[CrossRef]

PhysicaD 211, 193-220 (2005).
[CrossRef]

A. Dubietis, E. Gaižauskas, G. Tamǒsauskas, and P. D. Trapani, "Light filaments without self-channeling," Phys. Rev. Lett. 92, 253903 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]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


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)

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

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 ) ,

Metrics