Advertisement

Letters in Mathematical Physics

, Volume 96, Issue 1–3, pp 415–447 | Cite as

Kinetic Description of Random Optical Waves and Anomalous Thermalization of a Nearly Integrable Wave System

  • Claire Michel
  • Josselin Garnier
  • Pierre Suret
  • Stéphane Randoux
  • Antonio PicozziEmail author
Article

Abstract

This article is composed of two parts. The first part is aimed at providing an overview on the kinetic description of random nonlinear waves considering the one-dimensional nonlinear Schrödinger (NLS) equation as a representative model of optical wave propagation. We expose, in particular, the key problem of achieving a closure of the infinite hierarchy of moment equations for the random field. The hierarchy is closed at the first order when the statistics of the random wave is non-stationary or when the response time of the nonlinearity is non-instantaneous, which, respectively, leads to the Vlasov kinetic equation and the weak-Langmuir turbulence equation. When the amount of non-stationary statistics is comparable to the amount of non-instantaneous nonlinearity, we derive a generalized Vlasov–Langmuir equation that provides a unified formulation of the Vlasov and Langmuir approaches. On the other hand, when the statistics of the random wave is stationary and the nonlinear response instantaneous, the closure of the hierarchy of moment equations requires a second-order perturbation expansion procedure, which leads to the Hasselmann (or wave turbulence) kinetic equation. Contrarily to the Vlasov and Langmuir equations, the Hasselmann equation is irreversible, a feature which is expressed by a H-theorem of entropy growth that describes wave thermalization toward the thermodynamic equilibrium distribution, i.e. the Rayleigh–Jeans (RJ) spectrum. In the second part of the paper, we discuss a process of anomalous thermalization by considering the example of the scalar NLS equation whose integrability is broken by the presence of third-order dispersion. The anomalous thermalization is characterized by an irreversible evolution of the wave toward an equilibrium state of a fundamental different nature than the conventional RJ equilibrium state. The wave turbulence kinetic equation reveals that the anomalous thermalization is due to the existence of a local invariant in frequency space J ω, which originates in degenerate resonances of the system. In contrast to integral invariants that lead to a generalized RJ distribution, here, it is the local nature of the invariant J ω that makes the new equilibrium states fundamentally different than the usual RJ equilibrium states. We study in detail the anomalous thermalization by means of numerical simulations of the NLS equation and of the wave turbulence equation by using an improved criterion of applicability of the kinetic theory. The spectrum of the field is shown to exhibit an intriguing asymmetric deformation, which is characterized by the unexpected emergence of a constant spectral pedestal in the long-term evolution of the field. It turns out that the local invariant J ω explains all the essential properties of the anomalous thermalization of the wave.

Mathematics Subject Classification (2000)

74A25 76Fxx 78A10 

Keywords

kinetic wave theory nonlinear random waves thermalization nonlinear optics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Mandel L., Wolf E.: Optical Coherence and Quantum Optics. Cambridge University Press, New York (1995)Google Scholar
  2. 2.
    Mitchell M., Chen Z., Shih M., Segev M.: Self-trapping of partially spatially incoherent light. Phys. Rev. Lett. 77, 490 (1996)CrossRefADSGoogle Scholar
  3. 3.
    Mitchell M., Segev M.: Self-trapping of incoherent white light. Nature (London) 387, 880 (1997)CrossRefADSGoogle Scholar
  4. 4.
    Christodoulides D.N., Coskun T.H., Mitchell M., Chen Z., Segev M.: Theory of incoherent dark solitons. Phys. Rev. Lett. 80, 5113 (1998)CrossRefADSGoogle Scholar
  5. 5.
    Chen Z., Mitchell M., Segev M., Coskun T.H., Christodoulides D.N.: Self-trapping of dark incoherent light beams. Science 280, 889 (1998)CrossRefADSGoogle Scholar
  6. 6.
    Akhmediev N.N., Ankiewicz A.: Coherent and incoherent contributions to multisoliton complexes. Phys. Rev. Lett. 83, 4736 (1999)CrossRefADSGoogle Scholar
  7. 7.
    Peccianti M., Assanto G.: Incoherent spatial solitary waves in nematic liquid crystals. Opt. Lett. 26, 1791 (2001)CrossRefADSGoogle Scholar
  8. 8.
    Pasmanik G.A.: Self-interaction of incoherent light beams. Sov. Phys. JETP 39, 234 (1974)ADSGoogle Scholar
  9. 9.
    Mitchell M., Segev M., Coskun T., Christodoulides D.N.: Theory of self-trapped spatially incoherent light beams. Phys. Rev. Lett. 79, 4990 (1997)CrossRefADSGoogle Scholar
  10. 10.
    Christodoulides D.N., Coskun T.H., Mitchell M., Segev M.: Theory of incoherent self-focusing in biased photorefractive media. Phys. Rev. Lett. 78, 646 (1997)CrossRefADSGoogle Scholar
  11. 11.
    Kivshar Y.S., Agrawal G.P.: Optical Solitons: From Fibers to Photonic Crystals. Academic Press, San Diego (2003)Google Scholar
  12. 12.
    Soljacic M., Segev M., Coskun T., Christodoulides D.N., Vishwanath A.: Modulation instability of incoherent beams in noninstantaneous nonlinear media. Phys. Rev. Lett. 84, 467 (2000)CrossRefADSGoogle Scholar
  13. 13.
    Kip D., Soljacic M., Segev M., Eugenieva E., Christodoulides D.N.: Modulation instability and pattern formation in spatially incoherent light beams. Science 290, 495–498 (2000)CrossRefADSGoogle Scholar
  14. 14.
    Hall B., Lisak M., Anderson D., Fedele R., Semenov V.E.: Statistical theory for incoherent light propagation in nonlinear media. Phys. Rev. E 65, 035602 (2000)CrossRefADSGoogle Scholar
  15. 15.
    Dylov D.V., Fleischer J.W.: Observation of all-optical bump-on-tail instability. Phys. Rev. Lett. 100, 103903 (2008)CrossRefADSGoogle Scholar
  16. 16.
    Picozzi P., Haelterman M.: Parametric three-wave soliton generated from incoherent light. Phys. Rev. Lett. 86, 2010 (2001)CrossRefADSGoogle Scholar
  17. 17.
    Picozzi A., Montes C., Haelterman M.: Coherence properties of the parametric three-wave interaction driven from an incoherent pump. Phys. Rev. E 66, 056605 (2002)CrossRefADSGoogle Scholar
  18. 18.
    Picozzi A., Haelterman M., Pitois S., Millot G.: Incoherent solitons in instantaneous response nonlinear media. Phys. Rev. Lett. 92, 143906 (2004)CrossRefADSGoogle Scholar
  19. 19.
    Wu M., Krivosik P., Kalinikos B.A., Patton C.E.: Random generation of coherent solitary waves from incoherent waves. Phys. Rev. Lett. 96, 227202 (2006)CrossRefADSGoogle Scholar
  20. 20.
    Cohen O., Buljan H., Schwartz T., Fleischer J., Segev M.: Incoherent solitons in instantaneous nonlocal nonlinear media. Phys. Rev. E 73, 015601 (2006)CrossRefADSGoogle Scholar
  21. 21.
    Rotschild C., Schwartz T., Cohen O., Segev M.: Incoherent spatial solitons in effectively-instantaneous nonlocal nonlinear media. Nat. Photon. 2, 371 (2008)CrossRefADSGoogle Scholar
  22. 22.
    Alfassi B., Rotschild C., Segev M.: Incoherent surface solitons in effectively instantaneous nonlocal nonlinear media. Phys. Rev. A 80, 041808 (2009)CrossRefADSGoogle Scholar
  23. 23.
    Sauter A., Pitois S., Millot G., Picozzi A.: Incoherent modulation instability in instantaneous nonlinear Kerr media. Opt. Lett. 30, 2143 (2005)CrossRefADSGoogle Scholar
  24. 24.
    Bernstein I.B., Green J.M., Kruskal M.D.: Exact nonlinear plasma oscillations. Phys. Rev. 108, 546 (1957)CrossRefzbMATHADSMathSciNetGoogle Scholar
  25. 25.
    Hasegawa A.: Dynamics of an ensemble of plane waves in nonlinear dispersive media. Phys. Fluids 18, 77 (1975)CrossRefADSMathSciNetGoogle Scholar
  26. 26.
    Hasegawa A.: Envelope soliton of random phase waves. Phys. Fluids 20, 2155 (1977)CrossRefzbMATHADSGoogle Scholar
  27. 27.
    Zakharov V.E., Musher S.L., Rubenchik A.M.: Hamiltonian approach to the description of non-linear plasma phenomena. Phys. Reports 129, 285–366 (1985)CrossRefADSMathSciNetGoogle Scholar
  28. 28.
    Christodoulides D.N., Eugenieva E.D., Coskun T.H., Segev M., Mitchell M.: Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media. Phys. Rev. E 63, 035601 (2001)CrossRefADSGoogle Scholar
  29. 29.
    Lisak M., Helczynski L., Anderson D.: Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media. Opt. Commun. 220, 321 (2003)CrossRefADSGoogle Scholar
  30. 30.
    Picozzi A., Pitois S., Millot G.: Spectral incoherent solitons: a localized soliton behavior in the frequency domain. Phys. Rev. Lett. 101, 093901 (2008)CrossRefADSGoogle Scholar
  31. 31.
    Barviau B., Kibler B., Kudlinski A., Mussot A., Millot G., Picozzi A.: Experimental signature of optical wave thermalization through supercontinuum generation in photonic crystal fiber. Opt. Express 17, 7392 (2009)CrossRefGoogle Scholar
  32. 32.
    Musher S.L., Rubenchik A.M., Zakharov V.E.: Weak Langmuir turbulence. Phys. Reports 252, 177 (1995)CrossRefADSGoogle Scholar
  33. 33.
    Zel’dovich Ya.B., Syunyaev R.A.: Shock wave structure in the radiation spectrum during Bose condensation of photons. Sov. Phys. JETP 35, 81 (1972)ADSGoogle Scholar
  34. 34.
    Zel’dovich Ya.B., Levich E.V., Syunyaev R.A.: Stimulated Compton interaction between Maxwellian electrons and spectrally narrow radiation. Sov. Phys. JETP 35, 733 (1972)ADSGoogle Scholar
  35. 35.
    Zakharov V.E., Musher S.L., Rubenchik A.M.: Weak Langmuir turbulence of an isothermal plasma. Sov. Phys. JETP 42, 80 (1976)ADSGoogle Scholar
  36. 36.
    Montes C., Peyraud J., Hénon M.: One-dimensional boson soliton collisions. Phys. Fluids 22, 176 (1979)CrossRefADSGoogle Scholar
  37. 37.
    Montes C.: Photon soliton and fine structure due to nonlinear Compton scattering. Phys. Rev. A 20, 1081 (1979)CrossRefADSGoogle Scholar
  38. 38.
    Degtyarev L.M., Nakhan’kov V.G., Rudakov L.I.: Dynamics of the formation and interaction of Langmuir solitons and strong turbulence. Zh. Eksp. Teor. Fiz. 67, 533 (1974) [Sov. Phys. JETP 40, 264 (1974)]ADSGoogle Scholar
  39. 39.
    Pereira N.R.: Collisions between Langmuir solitons. Phys. Fluids 20, 750 (1976)CrossRefADSGoogle Scholar
  40. 40.
    Abdulloev Kh.O., Bogoljubskii I.L., Makhankov V.G.: Dynamics of Langmuir turbulence. Formation and interaction of solitons. Phys. Lett. A 48A, 161 (1974)CrossRefADSGoogle Scholar
  41. 41.
    Barviau B., Randoux S., Suret P.: Spectral broadening of a multimode continuous-wave optical field propagating in the normal dispersion regime of a fiber. Opt. Lett. 31, 1696 (2006)CrossRefADSGoogle Scholar
  42. 42.
    Picozzi A.: Towards a nonequilibrium thermodynamic description of incoherent nonlinear optics. Opt. Express 15, 9063 (2007)CrossRefADSGoogle Scholar
  43. 43.
    Picozzi A.: Spontaneous polarization induced by natural thermalization of incoherent light. Opt. Express 16, 17171 (2008)CrossRefADSGoogle Scholar
  44. 44.
    Picozzi A.: Entropy and degree of polarization for nonlinear optical waves. Opt. Lett. 29, 1653 (2004)CrossRefADSGoogle Scholar
  45. 45.
    Babin S.A., Churkin D.V., Ismagulov A.E., Kablukov S.I., Podivilov E.V.: Four-wave-mixing-induced turbulent spectral broadening in a long Raman fiber laser. J. Opt. Soc. Am. B 24, 1729 (2007)CrossRefADSGoogle Scholar
  46. 46.
    Babin S.A., Karalekas V., Podivilov E., Mezentsev V., Harper P., Ania-Castañòn J., Turitsyn S.: Turbulent broadening of optical spectra in ultralong Raman fiber lasers. Phys. Rev. A 77, 033803 (2008)CrossRefADSGoogle Scholar
  47. 47.
    Turitsyna E.G., Falkovich G., Mezentsev V.K., Turitsyn S.K.: Optical turbulence and spectral condensate in long-fiber lasers. Phys. Rev. A 80, 031804 (2009)CrossRefADSGoogle Scholar
  48. 48.
    Picozzi A., Rica S.: Coherence absorption and condensation induced by the thermalization of incoherent fields. Europhys. Lett. 84, 34004 (2008)CrossRefADSGoogle Scholar
  49. 49.
    Picozzi A., Haelterman M.: Condensation in Hamiltonian parametric wave interaction. Phys. Rev. Lett. 92, 103901 (2004)CrossRefADSGoogle Scholar
  50. 50.
    Picozzi A., Aschieri P.: Influence of dispersion on the resonant interaction between three incoherent waves. Phys. Rev. E 72, 046606 (2005)CrossRefADSMathSciNetGoogle Scholar
  51. 51.
    Bortolozzo U., Laurie J., Nazarenko S., Residori S.: Optical wave turbulence and the condensation of light. J. Opt. Soc. Am. B 26, 2280 (2009)CrossRefADSGoogle Scholar
  52. 52.
    Hammani K., Kibler B., Finot C., Picozzi A.: Emergence of rogue waves from optical turbulence. Phys. Lett. A 374, 3585 (2010)CrossRefADSGoogle Scholar
  53. 53.
    Pitois S., Lagrange S., Jauslin H.R., Picozzi A.: Velocity locking of incoherent nonlinear wave packets. Phys. Rev. Lett. 97, 033902 (2006)CrossRefADSGoogle Scholar
  54. 54.
    Lagrange S., Jauslin H.R., Picozzi A.: Thermalization of the dispersive three-wave interaction. Europhys. Lett. 79, 64001 (2007)CrossRefADSGoogle Scholar
  55. 55.
    Barviau B., Kibler B., Coen S., Picozzi A.: Towards a thermodynamic description of supercontinuum generation. Opt. Lett. 33, 2833 (2008)CrossRefADSGoogle Scholar
  56. 56.
    Barviau B., Kibler B., Picozzi A.: Wave turbulence description of supercontinuum generation: influence of self-steepening and higher-order dispersion. Phys. Rev. A 79, 063840 (2009)CrossRefADSGoogle Scholar
  57. 57.
    Levi L., Schwartz T., Manela O., Segev M., Buljan H.: Spontaneous pattern formation upon incoherent waves: From modulation-instability to steady-state. Opt. Express 16, 7818 (2008)CrossRefADSGoogle Scholar
  58. 58.
    Zakharov V.E., L’vov V.S., Falkovich G.: Kolmogorov Spectra of Turbulence I. Springer, Berlin (1992)zbMATHGoogle Scholar
  59. 59.
    Hasselmann K.: On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory. J. Fluid Mech. (12), 481 (1962)Google Scholar
  60. 60.
    Hasselmann K.: On the non-linear energy transfer in a gravity-wave spectrum. Part 2. Conservation theorems; wave-particle analogy; irreversibility. J. Fluid Mech. 15, 273 (1963)CrossRefzbMATHADSMathSciNetGoogle Scholar
  61. 61.
    Tsytovich V.N.: Nonlinear Effects in Plasma. Plenum, New York (1970)Google Scholar
  62. 62.
    Hasegawa A.: Plasma Instabilities and Nonlinear Effects. Springer, Berlin (1975)CrossRefGoogle Scholar
  63. 63.
    Dyachenko S., Newell A.C., Pushkarev A., Zakharov V.E.: Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation. Physica D 57, 96 (1992)CrossRefzbMATHADSMathSciNetGoogle Scholar
  64. 64.
    Zakharov V.E., Dias F., Pushkarev A.: One-dimensional wave turbulence. Phys. Rep. 398, 1 (2004)CrossRefADSMathSciNetGoogle Scholar
  65. 65.
    Benney D.J., Saffman P.G.: Nonlinear interactions of random waves in a dispersive medium. Proc. R. Soc. Lond. Ser. A 289, 301 (1966)CrossRefADSGoogle Scholar
  66. 66.
    Newell A.C.: The closure problem in a system of random gravity waves. Rev. Geophys. 6, 1 (1968)CrossRefADSGoogle Scholar
  67. 67.
    Newell A.C., Nazarenko S., Biven L.: Wave turbulence and intermittency. Phys. D 152, 520 (2001)CrossRefMathSciNetGoogle Scholar
  68. 68.
    Lvov Y.V., Nazarenko S.: Noisy spectra, long correlations, and intermittency in wave turbulence. Phys. Rev. E 69, 066608 (2004)CrossRefADSMathSciNetGoogle Scholar
  69. 69.
    Choi Y., Lvov Y.V., Nazarenko S.: Probability densities and preservation of randomness in wave turbulence. Phys. Lett. A 332, 230 (2004)CrossRefzbMATHADSGoogle Scholar
  70. 70.
    Choi Y., Lvov Y.V., Nazarenko S.: Joint statistics of amplitudes and phases in wave turbulence. Phys. D 201, 121 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  71. 71.
    Suret P., Randoux S., Jauslin H.R., Picozzi A.: Anomalous thermalization of nonlinear wave systems. Phys. Rev. Lett. 104, 054101 (2010)CrossRefADSGoogle Scholar
  72. 72.
    Michel C., Suret P., Randoux S., Jauslin H.R., Picozzi A.: Influence of third-order dispersion on the propagation of incoherent light in optical fibers. Opt. Lett. 35, 2367 (2010)CrossRefADSGoogle Scholar
  73. 73.
    Zakharov V.E., Schulman E.I.: Degenerative dispersion laws, motion invariants and kinetic equations. Phys. D 1, 192 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  74. 74.
    Zakharov V.E., Schulman E.I.: To the integrability of the system of two coupled nonlinear Schrödinger equations. Phys. D 4, 270 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  75. 75.
    Zakharov V.E., Schulman E.I.: On additionaal motion invariants of classical Hamiltonian wave systems. Phys. D 29, 283 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  76. 76.
    Balk, A.M., Ferapontov, E.E.: Invariants of wave systems and Web geometry. In: Zakharov, V.E. (ed.) Nonlinear Waves and Weak Turbulence, vol. 182. American Mathematical Society Translation 2 (1998)Google Scholar
  77. 77.
    Boyd R.W.: Nonlinear Optics. Academic Press, San Diego (2008)Google Scholar
  78. 78.
    Le Bellac M., Mortessagne F., Batrouni G.: Equilibrium and Nonequilibrium. Cambridge University Press, Cambridge (2004)Google Scholar
  79. 79.
    Kompaneets A.S.: The establishment of thermal equilibrium between quanta and electrons. Zh. Eksp. Teor. Fiz. 31, 871 (1956) [Sov. Phys. JETP 4, 730 (1957)]Google Scholar
  80. 80.
    Galeev A.A., Karpman V.I., Sagdeev R.Z.: One problem now under solution in the theory of plasma turbulence. Dokl. Akad. Nauk SSSR 157, 1088 (1964) [Sov. Phys. Dokl. 9, 681 (1965)]Google Scholar
  81. 81.
    Zel’dovich Ya.B., Levich E.V.: Bose Condensation and Shock Waves in Photon Spectra. Zh. Eksp. Teor. Fiz. 55, 2423 (1968) [Sov. Phys. JETP 28, 1287 (1969)]Google Scholar
  82. 82.
    Zakharov, V.E.: Collapse of Langmuir waves. Zh. Eksp. Teor. Fiz. 62, 1745–1751 (1972) [Sov. Phys. JETP 35, 908–914 (1972)]Google Scholar
  83. 83.
    Dreicer H.: Kinetic theory of an electron-photon gas. Phys. Fluids 7, 735 (1964)CrossRefADSMathSciNetGoogle Scholar
  84. 84.
    Garnier J., Picozzi A.: Unified kinetic formulation of incoherent waves propagating in nonlinear media with noninstantaneous response. Phys. Rev. A 81, 033831 (2010)CrossRefADSGoogle Scholar
  85. 85.
    Picozzi A., Haelterman M.: Hidden coherence along space-time trajectories in parametric wave mixing. Phys. Rev. Lett. 88, 083901 (2002)CrossRefADSGoogle Scholar
  86. 86.
    Jedrkiewicz O., Picozzi A., Clerici M., Faccio D., Di Trapani P.: Emergence of X-shaped spatiotemporal coherence in optical waves. Phys. Rev. Lett. 97, 243903 (2006)CrossRefADSGoogle Scholar
  87. 87.
    Sagdeev R.Z., Usikov D.A., Zaslavsky G.M.: Nonlinear Physics. Harwood Academic Publications, Switzerland (1988)zbMATHGoogle Scholar
  88. 88.
    Jordan R., Turkington B., Zirbel C.L.: A mean-field statistical theory for the nonlinear Schrödinger equation. Phys. D 137, 353 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  89. 89.
    Jordan R., Josserand C.: Self-organization in nonlinear wave turbulence. Phys. Rev. E 61, 1527 (2000)CrossRefADSMathSciNetGoogle Scholar
  90. 90.
    Eisner A., Turkington B.: Nonequilibrium statistical behavior of nonlinear Schrödinger equations. Phys. D 213, 85 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  91. 91.
    Rumpf R., Newell A.C.: Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems. Phys. Rev. Lett. 87, 054102 (2001)CrossRefADSGoogle Scholar
  92. 92.
    Rumpf R., Newell A.C.: Localization and coherence in nonintegrable systems. Phys. D 184, 162 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  93. 93.
    Fratalocchi A., Conti C., Ruocco G., Trillo S.: Free-energy transition in a gas of noninteracting nonlinear wave particles. Phys. Rev. Lett. 101, 044101 (2008)CrossRefADSGoogle Scholar
  94. 94.
    Agrawal G.P.: Nonlinear Fiber Optics, 4th edn. Academic Press, San Diego (2006)Google Scholar
  95. 95.
    Dudley J.M., Genty G., Coen S.: Supercontinuum generation in photonic crystal fiber. Rev. Mod. Phys. 78, 1135 (2006)CrossRefADSGoogle Scholar
  96. 96.
    Davis M.J., Morgan S.A., Burnett K.: Simulations of Bose fields at finite temperature. Phys. Rev. Lett. 87, 160402 (2001)CrossRefADSGoogle Scholar
  97. 97.
    Connaughton C., Josserand C., Picozzi A., Pomeau Y., Rica S.: Condensation of classical nonlinear waves. Phys. Rev. Lett. 95, 263901 (2005)CrossRefADSGoogle Scholar
  98. 98.
    Biven L., Nazarenko S.V., Newell A.C.: Breakdown of wave turbulence and the onset of intermittency. Phys. Lett. A 280, 28 (2001)CrossRefzbMATHADSMathSciNetGoogle Scholar
  99. 99.
    Düring G., Picozzi A., Rica S.: Breakdown of weak-turbulence and nonlinear wave condensation. Phys. D 238, 1524–1549 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  100. 100.
    Taki M., Mussot A., Kudlinski A., Louvergneaux E., Kolobov M., Douay M.: Third-order dispersion for generating optical rogue solitons. Phys. Lett. A 374, 691 (2010)CrossRefADSGoogle Scholar
  101. 101.
    Mussot A., Kudlinski A., Kolobov M., Louvergneaux E., Douay M., Taki M.: Observation of extreme temporal events in CW-pumped supercontinuum. Opt. Express 17, 17010 (2009)CrossRefADSGoogle Scholar

Copyright information

© Springer 2010

Authors and Affiliations

  • Claire Michel
    • 1
  • Josselin Garnier
    • 2
  • Pierre Suret
    • 3
  • Stéphane Randoux
    • 3
  • Antonio Picozzi
    • 1
    Email author
  1. 1.Institut Carnot de Bourgogne, UMR-CNRS 5029Université de BourgogneDijonFrance
  2. 2.Laboratoire de Probabilités et Modèles Aléatoires, UMR-CNRS 7599Université de Paris VIIParisFrance
  3. 3.Laboratoire de Physique des Lasers, Atomes et Molecules, UMR-CNRS 8523Université de LilleVilleneuve d’Ascq CedexFrance

Personalised recommendations