Abstract

In an optical experiment, we report a wave turbulence regime that, starting with weakly nonlinear waves with randomized phases, shows an inverse cascade of photons toward the lowest wavenumbers. We show that the cascade is induced by a six-wave resonant interaction process and is characterized by increasing nonlinearity. At low wavenumbers the nonlinearity becomes strong and leads to modulational instability developing into solitons, whose number is decreasing farther along the beam.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57, 96-160 (1992).
    [CrossRef]
  2. S. L. Musher, A. M. Rubenchik, and V. E. Zakharov, “Hamiltonian approach to the description of nonlinear plasma phenomena,” Phys. Rep. 129, 285-366 (1985).
    [CrossRef]
  3. S. Nazarenko and V. Zakharov, “Dynamics of the Bose-Einstein condensation,” Physica D 201, 203-211 (2005).
    [CrossRef]
  4. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
    [CrossRef]
  5. S. Nazarenko and M. Onorato, “Wave turbulence and vortices in Bose-Einstein condensation,” Physica D 219, 1 (2006).
    [CrossRef]
  6. F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749 (1991).
    [CrossRef] [PubMed]
  7. G. A. Swartzlander, Jr., and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503 (1992).
    [CrossRef] [PubMed]
  8. C. Barsi, W. Wan, C. Sun, and J. W. Fleischer, “Dispersive shock waves with nonlocal nonlinearity,” Opt. Lett. 32, 2930 (2007).
    [CrossRef] [PubMed]
  9. V. E. Zakharov and N. N. Filonenko, “The energy spectrum for stochastic oscillations of a fluid surface,” Sov. Phys. Dokl. 11, 881-884 (1967).
  10. V. E. Zakharov, V. S. Lvov, and G. Falkovich, Kolmogorov Spectra of Turbulence (Springer-Verlag, 1992).
  11. F. Dias, A. Pushkarev, and V. Zakharov, “One-dimensional wave turbulence,” Phys. Rep. 398, 1-65 (2004).
    [CrossRef]
  12. A. J. Majda, D. W. McLaughlin, and E. G. Tabak, “A one-dimensional model for dispersive wave turbulence,” J. Nonlinear Sci. 6, 9-44 (1997).
    [CrossRef]
  13. D. Cai, A. J. Majda, D. W. McLaughlin, and E. G. Tabak, “Dispersive wave turbulence in one dimension,” Physica D 152-153, 551-572 (2001).
    [CrossRef]
  14. V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan'kov, “Soliton turbulence,” JETP Lett. 48, No. 2, 83-87 (1988).
  15. S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity locking of inocherent nonlinear wave packets,” Phys. Rev. Lett. 97, 033902 (2006).
    [CrossRef] [PubMed]
  16. A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
    [CrossRef] [PubMed]
  17. R. Jordan, B. Turkington, and C. L. Zirbel, “A mean-field statistical theory for the nonlinear Schrödinger equation,” Physica D 137, 353-378 (2000).
    [CrossRef]
  18. K. Ø. Rasmussen, T. Cretegny, P. G. Kevrekidis, and N. Grønbech-Jensen, “Statistical mechanics of a discrete nonlinear system,” Phys. Rev. Lett. 84, 3740-3743 (2000).
    [CrossRef] [PubMed]
  19. R. Jordan and C. Josserand, “Self-organization in nonlinear wave turbulence,” Phys. Rev. E 61, 1527-1539 (2000).
    [CrossRef]
  20. A. Eisner and B. Turkington, “Nonequilibrium statistical behavior of nonlinear Schrödinger equations,” Physica D 213, 85-97 (2006).
    [CrossRef]
  21. B. Barviau, B. Kibler, A. Kudlinski, A. Mussot, H. Millot, and A. Picozzi, “Experimental signature of optical wave thermalization through supercontinuum generation in photonic crystal fiber,” Opt. Express 17, 7392-7406 (2009).
    [CrossRef] [PubMed]
  22. B. Rumpf and A. C. Newell, “Localization and coherence in nonintegrable systems,” Physica D 184, 162-191 (2003).
    [CrossRef]
  23. B. Rumpf and A. C. Newell, “Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems,” Phys. Rev. Lett. 87, 054102 (2001).
    [CrossRef] [PubMed]
  24. V. I. Petviashvili and V. V. Yan'kov, “Solitons and turbulence,” Rev. Plasma Phys. 14, 1-62 (1989).
  25. M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198-201 (1995).
    [CrossRef] [PubMed]
  26. L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Clarendon, 2003).
  27. R. Y. Chiao and J. Boyce, “Bogoliubov dispersion relation and the possibility of superfluidity for weakly interacting photons in a two-dimensional photon fluid,” Phys. Rev. A 60, 4114-4121 (1999).
    [CrossRef]
  28. N. V. Tabiryan, A. V. Sukhov, and V. Y. Zeldovich, “The orientational optical nonlinearity of liquid crystals,” Mol. Cryst. Liq. Cryst. 136, 1-140 (1986).
    [CrossRef]
  29. I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, 1995).
  30. E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611-622 (1993).
    [CrossRef] [PubMed]
  31. M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature 432, 733-737 (2004).
    [CrossRef] [PubMed]
  32. M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003).
    [CrossRef]
  33. C. Conti, M. Peccianti, and G. Assanto, “Complex dynamics and configurational entropy of spatial optical solitons in nonlocal media,” Opt. Lett. 31, 2030 (2006).
    [CrossRef] [PubMed]
  34. P. G. De Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford Science Publications, Clarendon, 1993).
  35. U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori are preparing a larger manuscript to be called “One-dimensional optical wave turbulence.”

2009 (1)

2008 (1)

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (4)

S. Nazarenko and M. Onorato, “Wave turbulence and vortices in Bose-Einstein condensation,” Physica D 219, 1 (2006).
[CrossRef]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity locking of inocherent nonlinear wave packets,” Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef] [PubMed]

A. Eisner and B. Turkington, “Nonequilibrium statistical behavior of nonlinear Schrödinger equations,” Physica D 213, 85-97 (2006).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, “Complex dynamics and configurational entropy of spatial optical solitons in nonlocal media,” Opt. Lett. 31, 2030 (2006).
[CrossRef] [PubMed]

2005 (2)

S. Nazarenko and V. Zakharov, “Dynamics of the Bose-Einstein condensation,” Physica D 201, 203-211 (2005).
[CrossRef]

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

2004 (2)

F. Dias, A. Pushkarev, and V. Zakharov, “One-dimensional wave turbulence,” Phys. Rep. 398, 1-65 (2004).
[CrossRef]

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature 432, 733-737 (2004).
[CrossRef] [PubMed]

2003 (2)

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003).
[CrossRef]

B. Rumpf and A. C. Newell, “Localization and coherence in nonintegrable systems,” Physica D 184, 162-191 (2003).
[CrossRef]

2001 (2)

B. Rumpf and A. C. Newell, “Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems,” Phys. Rev. Lett. 87, 054102 (2001).
[CrossRef] [PubMed]

D. Cai, A. J. Majda, D. W. McLaughlin, and E. G. Tabak, “Dispersive wave turbulence in one dimension,” Physica D 152-153, 551-572 (2001).
[CrossRef]

2000 (3)

R. Jordan, B. Turkington, and C. L. Zirbel, “A mean-field statistical theory for the nonlinear Schrödinger equation,” Physica D 137, 353-378 (2000).
[CrossRef]

K. Ø. Rasmussen, T. Cretegny, P. G. Kevrekidis, and N. Grønbech-Jensen, “Statistical mechanics of a discrete nonlinear system,” Phys. Rev. Lett. 84, 3740-3743 (2000).
[CrossRef] [PubMed]

R. Jordan and C. Josserand, “Self-organization in nonlinear wave turbulence,” Phys. Rev. E 61, 1527-1539 (2000).
[CrossRef]

1999 (1)

R. Y. Chiao and J. Boyce, “Bogoliubov dispersion relation and the possibility of superfluidity for weakly interacting photons in a two-dimensional photon fluid,” Phys. Rev. A 60, 4114-4121 (1999).
[CrossRef]

1997 (1)

A. J. Majda, D. W. McLaughlin, and E. G. Tabak, “A one-dimensional model for dispersive wave turbulence,” J. Nonlinear Sci. 6, 9-44 (1997).
[CrossRef]

1995 (1)

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198-201 (1995).
[CrossRef] [PubMed]

1993 (1)

E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611-622 (1993).
[CrossRef] [PubMed]

1992 (2)

G. A. Swartzlander, Jr., and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503 (1992).
[CrossRef] [PubMed]

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57, 96-160 (1992).
[CrossRef]

1991 (1)

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749 (1991).
[CrossRef] [PubMed]

1989 (1)

V. I. Petviashvili and V. V. Yan'kov, “Solitons and turbulence,” Rev. Plasma Phys. 14, 1-62 (1989).

1988 (1)

V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan'kov, “Soliton turbulence,” JETP Lett. 48, No. 2, 83-87 (1988).

1986 (1)

N. V. Tabiryan, A. V. Sukhov, and V. Y. Zeldovich, “The orientational optical nonlinearity of liquid crystals,” Mol. Cryst. Liq. Cryst. 136, 1-140 (1986).
[CrossRef]

1985 (1)

S. L. Musher, A. M. Rubenchik, and V. E. Zakharov, “Hamiltonian approach to the description of nonlinear plasma phenomena,” Phys. Rep. 129, 285-366 (1985).
[CrossRef]

1967 (1)

V. E. Zakharov and N. N. Filonenko, “The energy spectrum for stochastic oscillations of a fluid surface,” Sov. Phys. Dokl. 11, 881-884 (1967).

Anderson, M. H.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198-201 (1995).
[CrossRef] [PubMed]

Arecchi, F. T.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749 (1991).
[CrossRef] [PubMed]

Assanto, G.

C. Conti, M. Peccianti, and G. Assanto, “Complex dynamics and configurational entropy of spatial optical solitons in nonlocal media,” Opt. Lett. 31, 2030 (2006).
[CrossRef] [PubMed]

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature 432, 733-737 (2004).
[CrossRef] [PubMed]

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003).
[CrossRef]

Barsi, C.

Barviau, B.

Bortolozzo, U.

U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori are preparing a larger manuscript to be called “One-dimensional optical wave turbulence.”

Boyce, J.

R. Y. Chiao and J. Boyce, “Bogoliubov dispersion relation and the possibility of superfluidity for weakly interacting photons in a two-dimensional photon fluid,” Phys. Rev. A 60, 4114-4121 (1999).
[CrossRef]

Braun, E.

E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611-622 (1993).
[CrossRef] [PubMed]

Cai, D.

D. Cai, A. J. Majda, D. W. McLaughlin, and E. G. Tabak, “Dispersive wave turbulence in one dimension,” Physica D 152-153, 551-572 (2001).
[CrossRef]

Chiao, R. Y.

R. Y. Chiao and J. Boyce, “Bogoliubov dispersion relation and the possibility of superfluidity for weakly interacting photons in a two-dimensional photon fluid,” Phys. Rev. A 60, 4114-4121 (1999).
[CrossRef]

Connaughton, C.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

Conti, C.

C. Conti, M. Peccianti, and G. Assanto, “Complex dynamics and configurational entropy of spatial optical solitons in nonlocal media,” Opt. Lett. 31, 2030 (2006).
[CrossRef] [PubMed]

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature 432, 733-737 (2004).
[CrossRef] [PubMed]

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003).
[CrossRef]

Cornell, E. A.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198-201 (1995).
[CrossRef] [PubMed]

Cretegny, T.

K. Ø. Rasmussen, T. Cretegny, P. G. Kevrekidis, and N. Grønbech-Jensen, “Statistical mechanics of a discrete nonlinear system,” Phys. Rev. Lett. 84, 3740-3743 (2000).
[CrossRef] [PubMed]

De Gennes, P. G.

P. G. De Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford Science Publications, Clarendon, 1993).

De Luca, A.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature 432, 733-737 (2004).
[CrossRef] [PubMed]

Dias, F.

F. Dias, A. Pushkarev, and V. Zakharov, “One-dimensional wave turbulence,” Phys. Rep. 398, 1-65 (2004).
[CrossRef]

Dyachenko, S.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57, 96-160 (1992).
[CrossRef]

Eisner, A.

A. Eisner and B. Turkington, “Nonequilibrium statistical behavior of nonlinear Schrödinger equations,” Physica D 213, 85-97 (2006).
[CrossRef]

Ensher, J. R.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198-201 (1995).
[CrossRef] [PubMed]

Falkovich, G.

V. E. Zakharov, V. S. Lvov, and G. Falkovich, Kolmogorov Spectra of Turbulence (Springer-Verlag, 1992).

Faucheux, L. P.

E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611-622 (1993).
[CrossRef] [PubMed]

Filonenko, N. N.

V. E. Zakharov and N. N. Filonenko, “The energy spectrum for stochastic oscillations of a fluid surface,” Sov. Phys. Dokl. 11, 881-884 (1967).

Fleischer, J. W.

Giacomelli, G.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749 (1991).
[CrossRef] [PubMed]

Grønbech-Jensen, N.

K. Ø. Rasmussen, T. Cretegny, P. G. Kevrekidis, and N. Grønbech-Jensen, “Statistical mechanics of a discrete nonlinear system,” Phys. Rev. Lett. 84, 3740-3743 (2000).
[CrossRef] [PubMed]

Jauslin, H. R.

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity locking of inocherent nonlinear wave packets,” Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef] [PubMed]

Jordan, R.

R. Jordan, B. Turkington, and C. L. Zirbel, “A mean-field statistical theory for the nonlinear Schrödinger equation,” Physica D 137, 353-378 (2000).
[CrossRef]

R. Jordan and C. Josserand, “Self-organization in nonlinear wave turbulence,” Phys. Rev. E 61, 1527-1539 (2000).
[CrossRef]

Josserand, C.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

R. Jordan and C. Josserand, “Self-organization in nonlinear wave turbulence,” Phys. Rev. E 61, 1527-1539 (2000).
[CrossRef]

Kevrekidis, P. G.

K. Ø. Rasmussen, T. Cretegny, P. G. Kevrekidis, and N. Grønbech-Jensen, “Statistical mechanics of a discrete nonlinear system,” Phys. Rev. Lett. 84, 3740-3743 (2000).
[CrossRef] [PubMed]

Khoo, I. C.

I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, 1995).

Kibler, B.

Kudlinski, A.

Lagrange, S.

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity locking of inocherent nonlinear wave packets,” Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef] [PubMed]

Laurie, J.

U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori are preparing a larger manuscript to be called “One-dimensional optical wave turbulence.”

Law, C. T.

G. A. Swartzlander, Jr., and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503 (1992).
[CrossRef] [PubMed]

Libchaber, A.

E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611-622 (1993).
[CrossRef] [PubMed]

Lvov, V. S.

V. E. Zakharov, V. S. Lvov, and G. Falkovich, Kolmogorov Spectra of Turbulence (Springer-Verlag, 1992).

Majda, A. J.

D. Cai, A. J. Majda, D. W. McLaughlin, and E. G. Tabak, “Dispersive wave turbulence in one dimension,” Physica D 152-153, 551-572 (2001).
[CrossRef]

A. J. Majda, D. W. McLaughlin, and E. G. Tabak, “A one-dimensional model for dispersive wave turbulence,” J. Nonlinear Sci. 6, 9-44 (1997).
[CrossRef]

Matthews, M. R.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198-201 (1995).
[CrossRef] [PubMed]

McLaughlin, D. W.

D. Cai, A. J. Majda, D. W. McLaughlin, and E. G. Tabak, “Dispersive wave turbulence in one dimension,” Physica D 152-153, 551-572 (2001).
[CrossRef]

A. J. Majda, D. W. McLaughlin, and E. G. Tabak, “A one-dimensional model for dispersive wave turbulence,” J. Nonlinear Sci. 6, 9-44 (1997).
[CrossRef]

Millot, G.

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef] [PubMed]

Millot, H.

Musher, S. L.

S. L. Musher, A. M. Rubenchik, and V. E. Zakharov, “Hamiltonian approach to the description of nonlinear plasma phenomena,” Phys. Rep. 129, 285-366 (1985).
[CrossRef]

Mussot, A.

Nazarenko, S.

S. Nazarenko and M. Onorato, “Wave turbulence and vortices in Bose-Einstein condensation,” Physica D 219, 1 (2006).
[CrossRef]

S. Nazarenko and V. Zakharov, “Dynamics of the Bose-Einstein condensation,” Physica D 201, 203-211 (2005).
[CrossRef]

U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori are preparing a larger manuscript to be called “One-dimensional optical wave turbulence.”

Newell, A. C.

B. Rumpf and A. C. Newell, “Localization and coherence in nonintegrable systems,” Physica D 184, 162-191 (2003).
[CrossRef]

B. Rumpf and A. C. Newell, “Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems,” Phys. Rev. Lett. 87, 054102 (2001).
[CrossRef] [PubMed]

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57, 96-160 (1992).
[CrossRef]

Onorato, M.

S. Nazarenko and M. Onorato, “Wave turbulence and vortices in Bose-Einstein condensation,” Physica D 219, 1 (2006).
[CrossRef]

Peccianti, M.

C. Conti, M. Peccianti, and G. Assanto, “Complex dynamics and configurational entropy of spatial optical solitons in nonlocal media,” Opt. Lett. 31, 2030 (2006).
[CrossRef] [PubMed]

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature 432, 733-737 (2004).
[CrossRef] [PubMed]

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003).
[CrossRef]

Petviashvili, V. I.

V. I. Petviashvili and V. V. Yan'kov, “Solitons and turbulence,” Rev. Plasma Phys. 14, 1-62 (1989).

Picozzi, A.

B. Barviau, B. Kibler, A. Kudlinski, A. Mussot, H. Millot, and A. Picozzi, “Experimental signature of optical wave thermalization through supercontinuum generation in photonic crystal fiber,” Opt. Express 17, 7392-7406 (2009).
[CrossRef] [PubMed]

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef] [PubMed]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity locking of inocherent nonlinear wave packets,” Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef] [PubMed]

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

Pitaevskii, L. P.

L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Clarendon, 2003).

Pitois, S.

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef] [PubMed]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity locking of inocherent nonlinear wave packets,” Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef] [PubMed]

Pomeau, Y.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

Prost, J.

P. G. De Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford Science Publications, Clarendon, 1993).

Pushkarev, A.

F. Dias, A. Pushkarev, and V. Zakharov, “One-dimensional wave turbulence,” Phys. Rep. 398, 1-65 (2004).
[CrossRef]

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57, 96-160 (1992).
[CrossRef]

Pushkarev, A. N.

V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan'kov, “Soliton turbulence,” JETP Lett. 48, No. 2, 83-87 (1988).

Ramazza, P. L.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749 (1991).
[CrossRef] [PubMed]

Rasmussen, K. Ø.

K. Ø. Rasmussen, T. Cretegny, P. G. Kevrekidis, and N. Grønbech-Jensen, “Statistical mechanics of a discrete nonlinear system,” Phys. Rev. Lett. 84, 3740-3743 (2000).
[CrossRef] [PubMed]

Residori, S.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749 (1991).
[CrossRef] [PubMed]

U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori are preparing a larger manuscript to be called “One-dimensional optical wave turbulence.”

Rica, S.

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

Rubenchik, A. M.

S. L. Musher, A. M. Rubenchik, and V. E. Zakharov, “Hamiltonian approach to the description of nonlinear plasma phenomena,” Phys. Rep. 129, 285-366 (1985).
[CrossRef]

Rumpf, B.

B. Rumpf and A. C. Newell, “Localization and coherence in nonintegrable systems,” Physica D 184, 162-191 (2003).
[CrossRef]

B. Rumpf and A. C. Newell, “Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems,” Phys. Rev. Lett. 87, 054102 (2001).
[CrossRef] [PubMed]

Shvets, V. F.

V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan'kov, “Soliton turbulence,” JETP Lett. 48, No. 2, 83-87 (1988).

Stringari, S.

L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Clarendon, 2003).

Sukhov, A. V.

N. V. Tabiryan, A. V. Sukhov, and V. Y. Zeldovich, “The orientational optical nonlinearity of liquid crystals,” Mol. Cryst. Liq. Cryst. 136, 1-140 (1986).
[CrossRef]

Sun, C.

Swartzlander, G. A.

G. A. Swartzlander, Jr., and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503 (1992).
[CrossRef] [PubMed]

Tabak, E. G.

D. Cai, A. J. Majda, D. W. McLaughlin, and E. G. Tabak, “Dispersive wave turbulence in one dimension,” Physica D 152-153, 551-572 (2001).
[CrossRef]

A. J. Majda, D. W. McLaughlin, and E. G. Tabak, “A one-dimensional model for dispersive wave turbulence,” J. Nonlinear Sci. 6, 9-44 (1997).
[CrossRef]

Tabiryan, N. V.

N. V. Tabiryan, A. V. Sukhov, and V. Y. Zeldovich, “The orientational optical nonlinearity of liquid crystals,” Mol. Cryst. Liq. Cryst. 136, 1-140 (1986).
[CrossRef]

Turkington, B.

A. Eisner and B. Turkington, “Nonequilibrium statistical behavior of nonlinear Schrödinger equations,” Physica D 213, 85-97 (2006).
[CrossRef]

R. Jordan, B. Turkington, and C. L. Zirbel, “A mean-field statistical theory for the nonlinear Schrödinger equation,” Physica D 137, 353-378 (2000).
[CrossRef]

Umeton, C.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature 432, 733-737 (2004).
[CrossRef] [PubMed]

Wan, W.

Wieman, C. E.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198-201 (1995).
[CrossRef] [PubMed]

Yan'kov, V. V.

V. I. Petviashvili and V. V. Yan'kov, “Solitons and turbulence,” Rev. Plasma Phys. 14, 1-62 (1989).

V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan'kov, “Soliton turbulence,” JETP Lett. 48, No. 2, 83-87 (1988).

Zakharov, V.

S. Nazarenko and V. Zakharov, “Dynamics of the Bose-Einstein condensation,” Physica D 201, 203-211 (2005).
[CrossRef]

F. Dias, A. Pushkarev, and V. Zakharov, “One-dimensional wave turbulence,” Phys. Rep. 398, 1-65 (2004).
[CrossRef]

Zakharov, V. E.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57, 96-160 (1992).
[CrossRef]

V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan'kov, “Soliton turbulence,” JETP Lett. 48, No. 2, 83-87 (1988).

S. L. Musher, A. M. Rubenchik, and V. E. Zakharov, “Hamiltonian approach to the description of nonlinear plasma phenomena,” Phys. Rep. 129, 285-366 (1985).
[CrossRef]

V. E. Zakharov and N. N. Filonenko, “The energy spectrum for stochastic oscillations of a fluid surface,” Sov. Phys. Dokl. 11, 881-884 (1967).

V. E. Zakharov, V. S. Lvov, and G. Falkovich, Kolmogorov Spectra of Turbulence (Springer-Verlag, 1992).

Zeldovich, V. Y.

N. V. Tabiryan, A. V. Sukhov, and V. Y. Zeldovich, “The orientational optical nonlinearity of liquid crystals,” Mol. Cryst. Liq. Cryst. 136, 1-140 (1986).
[CrossRef]

Zirbel, C. L.

R. Jordan, B. Turkington, and C. L. Zirbel, “A mean-field statistical theory for the nonlinear Schrödinger equation,” Physica D 137, 353-378 (2000).
[CrossRef]

J. Nonlinear Sci. (1)

A. J. Majda, D. W. McLaughlin, and E. G. Tabak, “A one-dimensional model for dispersive wave turbulence,” J. Nonlinear Sci. 6, 9-44 (1997).
[CrossRef]

JETP Lett. (1)

V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan'kov, “Soliton turbulence,” JETP Lett. 48, No. 2, 83-87 (1988).

Mol. Cryst. Liq. Cryst. (1)

N. V. Tabiryan, A. V. Sukhov, and V. Y. Zeldovich, “The orientational optical nonlinearity of liquid crystals,” Mol. Cryst. Liq. Cryst. 136, 1-140 (1986).
[CrossRef]

Nature (1)

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature 432, 733-737 (2004).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rep. (2)

S. L. Musher, A. M. Rubenchik, and V. E. Zakharov, “Hamiltonian approach to the description of nonlinear plasma phenomena,” Phys. Rep. 129, 285-366 (1985).
[CrossRef]

F. Dias, A. Pushkarev, and V. Zakharov, “One-dimensional wave turbulence,” Phys. Rep. 398, 1-65 (2004).
[CrossRef]

Phys. Rev. A (2)

R. Y. Chiao and J. Boyce, “Bogoliubov dispersion relation and the possibility of superfluidity for weakly interacting photons in a two-dimensional photon fluid,” Phys. Rev. A 60, 4114-4121 (1999).
[CrossRef]

E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611-622 (1993).
[CrossRef] [PubMed]

Phys. Rev. E (2)

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003).
[CrossRef]

R. Jordan and C. Josserand, “Self-organization in nonlinear wave turbulence,” Phys. Rev. E 61, 1527-1539 (2000).
[CrossRef]

Phys. Rev. Lett. (7)

B. Rumpf and A. C. Newell, “Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems,” Phys. Rev. Lett. 87, 054102 (2001).
[CrossRef] [PubMed]

K. Ø. Rasmussen, T. Cretegny, P. G. Kevrekidis, and N. Grønbech-Jensen, “Statistical mechanics of a discrete nonlinear system,” Phys. Rev. Lett. 84, 3740-3743 (2000).
[CrossRef] [PubMed]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity locking of inocherent nonlinear wave packets,” Phys. Rev. Lett. 97, 033902 (2006).
[CrossRef] [PubMed]

A. Picozzi, S. Pitois, and G. Millot, “Spectral incoherent solitons: a localized soliton behavior in the frequency domain,” Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef] [PubMed]

C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
[CrossRef]

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749 (1991).
[CrossRef] [PubMed]

G. A. Swartzlander, Jr., and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503 (1992).
[CrossRef] [PubMed]

Physica D (7)

S. Nazarenko and M. Onorato, “Wave turbulence and vortices in Bose-Einstein condensation,” Physica D 219, 1 (2006).
[CrossRef]

S. Nazarenko and V. Zakharov, “Dynamics of the Bose-Einstein condensation,” Physica D 201, 203-211 (2005).
[CrossRef]

R. Jordan, B. Turkington, and C. L. Zirbel, “A mean-field statistical theory for the nonlinear Schrödinger equation,” Physica D 137, 353-378 (2000).
[CrossRef]

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation,” Physica D 57, 96-160 (1992).
[CrossRef]

D. Cai, A. J. Majda, D. W. McLaughlin, and E. G. Tabak, “Dispersive wave turbulence in one dimension,” Physica D 152-153, 551-572 (2001).
[CrossRef]

A. Eisner and B. Turkington, “Nonequilibrium statistical behavior of nonlinear Schrödinger equations,” Physica D 213, 85-97 (2006).
[CrossRef]

B. Rumpf and A. C. Newell, “Localization and coherence in nonintegrable systems,” Physica D 184, 162-191 (2003).
[CrossRef]

Rev. Plasma Phys. (1)

V. I. Petviashvili and V. V. Yan'kov, “Solitons and turbulence,” Rev. Plasma Phys. 14, 1-62 (1989).

Science (1)

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198-201 (1995).
[CrossRef] [PubMed]

Sov. Phys. Dokl. (1)

V. E. Zakharov and N. N. Filonenko, “The energy spectrum for stochastic oscillations of a fluid surface,” Sov. Phys. Dokl. 11, 881-884 (1967).

Other (5)

V. E. Zakharov, V. S. Lvov, and G. Falkovich, Kolmogorov Spectra of Turbulence (Springer-Verlag, 1992).

L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Clarendon, 2003).

I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, 1995).

P. G. De Gennes and J. Prost, The Physics of Liquid Crystals, 2nd ed. (Oxford Science Publications, Clarendon, 1993).

U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori are preparing a larger manuscript to be called “One-dimensional optical wave turbulence.”

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 (7)

Fig. 1
Fig. 1

Experimental setup: a laminar-shaped input beam propagates inside the LC layer; random space modulations are imposed at the entrance of the cell by means of a spatial light modulator (SLM).

Fig. 2
Fig. 2

Experimental spectrum of the light intensity, N k = | I k | 2 at two different distances z.

Fig. 3
Fig. 3

Numerical spectrum of the light intensity, N k = | I k | 2 at two different distances z. Averaging is taken over a small finite time window and over ten realizations.

Fig. 4
Fig. 4

Experimental results for intensity distribution I ( x , z ) . The area enclosed by the dashed line is shown at a higher resolution (using a larger magnification objective).

Fig. 5
Fig. 5

Numerical results for intensity distribution I ( x , z ) . The frame on the left is a magnified section of the initial propagation of the beam.

Fig. 6
Fig. 6

Linear intensity profiles I ( x ) taken at different propagation distances, z = 0.3 , 4.5 , 7.5 mm .

Fig. 7
Fig. 7

The k ω spectrum of the wave field at z = 2.1 m . ω * = 1 256 q l ξ 2 and k * = 1 128 l ξ . The Bogoliubov dispersion relation is shown by the solid curve.

Equations (10)

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

2 i q ψ z + 2 ψ x 2 + k 0 2 n a 2 a ψ = 0 ,
2 a x 2 1 l ξ 2 a + ε 0 n a 2 4 K | ψ | 2 = 0 ,
2 i q ψ z = 2 ψ x 2 ε 0 n a 4 l ξ 2 k 0 2 4 K ( ψ | ψ | 2 + l ξ 2 ψ 2 | ψ | 2 x 2 ) = δ H δ ψ * .
H = | ψ x | 2 ε 0 n a 4 l ξ 2 k 0 2 8 K [ | ψ | 4 l ξ 2 ( | ψ | 2 x ) 2 ] d x ,
J = 4 K k 2 ϵ 0 n a 4 k 0 2 l ξ 2 I ,
n k z = A f k 12345 δ ( k + k 1 + k 2 k 3 k 4 k 5 ) δ ( ω k + ω 1 + ω 2 ω 3 ω 4 ω 5 ) d k 1 d k 2 d k 3 d k 4 d k 5 ,
f k 12345 = n k n 1 n 2 n 3 n 4 n 5 ( 1 n k + 1 n 1 + 1 n 2 1 n 3 1 n 4 1 n 5 )
n k = C | k | 1 ,
n k = C | k | 3 5 .
N k | k | 1 5 .

Metrics