Abstract

We study theoretically, numerically and experimentally the nonlinear propagation of partially incoherent optical waves in single mode optical fibers. We revisit the traditional treatment of the wave turbulence theory to provide a statistical kinetic description of the integrable scalar NLS equation. In spite of the formal reversibility and of the integrability of the NLS equation, the weakly nonlinear dynamics reveals the existence of an irreversible evolution toward a statistically stationary state. The evolution of the power spectrum of the field is characterized by the rapid growth of spectral tails that exhibit damped oscillations, until the whole spectrum ultimately reaches a steady state. The kinetic approach allows us to derive an analytical expression of the damped oscillations, which is found in agreement with the numerical simulations of both the NLS and kinetic equations. We report the experimental observation of this peculiar relaxation process of the integrable NLS equation.

© 2011 OSA

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).
  2. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Society for Industrial and Applied Mathematics, 1981).
    [CrossRef]
  3. W. Zhao and E. Bourkoff, “Interactions between dark solitons,” Opt. Lett. 14, 1371–1373 (1989).
    [CrossRef] [PubMed]
  4. C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
    [CrossRef] [PubMed]
  5. A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo, “Free-energy transition in a gas of noninteracting nonlinear wave particles,” Phys. Rev. Lett. 101, 044101 (2008).
    [CrossRef] [PubMed]
  6. G. A. El and A. M. Kamchatnov, “Kinetic equation for a dense soliton gas,” Phys. Rev. Lett. 95, 204101 (2005).
    [CrossRef] [PubMed]
  7. J. T. Manassah, “Self-phase modulation of incoherent light,” Opt. Lett. 15, 329–331 (1990).
    [CrossRef] [PubMed]
  8. V. E. Zakharov, “Turbulence in integrable systems,” Stud. Appl. Math. 122, 219–234 (2009).
    [CrossRef]
  9. A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152–153, 520–550 (2001).
    [CrossRef]
  10. S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrodinger equation,” Physica D 57, 96–160 (1992).
    [CrossRef]
  11. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005).
    [CrossRef]
  12. A. Picozzi, “Towards a nonequilibrium thermodynamic description of incoherent nonlinear optics,” Opt. Express 15, 9063–9083 (2007).
    [CrossRef] [PubMed]
  13. P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classical optical waves in a waveguide,” Phys. Rev. A 83, 033838 (2011).
    [CrossRef]
  14. C. Michel, P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Influence of third-order dispersion on the propagation of incoherent light in optical fibers,” Opt. Lett. 35, 2367–2369 (2010).
    [CrossRef] [PubMed]
  15. P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Anomalous thermalization of nonlinear wave systems,” Phys. Rev. Lett. 104, 054101 (2010).
    [CrossRef] [PubMed]
  16. V. E. Zakharov, F. Dias, and A. Pushkarev, “One-dimensional wave turbulence,” Phys. Rep. 398, 1 (2004).
    [CrossRef]
  17. U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori, “Optical wave turbulence and the condensation of light,” J. Opt. Soc. Am. B 26, 2280 (2009).
    [CrossRef]
  18. B. Barviau, B. Kibler, A. Kudlinski, A. Mussot, G. Millot, and A. Picozzi, “Experimental signature of optical wave thermalization through supercontinuum generation in photonic crystal fiber,” Opt. Express 17, 7392 (2009).
    [CrossRef] [PubMed]
  19. B. Barviau, B. Kibler, and A. Picozzi, “Wave turbulence description of supercontinuum generation: influence of self-steepening and higher-order dispersion,” Phys. Rev. A 79, 063840 (2009).
    [CrossRef]
  20. M. Tabor, Chaos and Integrability in Nonlinear Dynamics (Wiley, 1989).
  21. B. Barviau, S. Randoux, and P. Suret, “Spectral broadening of a multimode continuous-wave optical field propagating in the normal dispersion regime of a fiber,” Opt. Lett. 31, 1696–1698 (2006).
    [CrossRef] [PubMed]
  22. B. M. Herbst and M. J. Ablowitz, “Numerically induced chaos in the nonlinear Schrodinger equation,” Phys. Rev. Lett. 62, 2065–2068 (1989).
    [CrossRef] [PubMed]
  23. D. B. S. Soh, J. P. Koplow, S. W. Moore, K. L. Schroder, and W. L. Hsu “The effect of dispersion on spectral broadening of incoherent continuous-wave light in optical fibers,” Opt. Express 18, 22393–22405 (2010).
    [CrossRef] [PubMed]
  24. V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, 1992).

2011 (1)

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classical optical waves in a waveguide,” Phys. Rev. A 83, 033838 (2011).
[CrossRef]

2010 (3)

2009 (5)

U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori, “Optical wave turbulence and the condensation of light,” J. Opt. Soc. Am. B 26, 2280 (2009).
[CrossRef]

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

B. Barviau, B. Kibler, and A. Picozzi, “Wave turbulence description of supercontinuum generation: influence of self-steepening and higher-order dispersion,” Phys. Rev. A 79, 063840 (2009).
[CrossRef]

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef] [PubMed]

V. E. Zakharov, “Turbulence in integrable systems,” Stud. Appl. Math. 122, 219–234 (2009).
[CrossRef]

2008 (1)

A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo, “Free-energy transition in a gas of noninteracting nonlinear wave particles,” Phys. Rev. Lett. 101, 044101 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (1)

2005 (2)

G. A. El and A. M. Kamchatnov, “Kinetic equation for a dense soliton gas,” Phys. Rev. Lett. 95, 204101 (2005).
[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]

2004 (1)

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

2001 (1)

A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152–153, 520–550 (2001).
[CrossRef]

1992 (1)

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

1990 (1)

1989 (2)

W. Zhao and E. Bourkoff, “Interactions between dark solitons,” Opt. Lett. 14, 1371–1373 (1989).
[CrossRef] [PubMed]

B. M. Herbst and M. J. Ablowitz, “Numerically induced chaos in the nonlinear Schrodinger equation,” Phys. Rev. Lett. 62, 2065–2068 (1989).
[CrossRef] [PubMed]

Ablowitz, M. J.

B. M. Herbst and M. J. Ablowitz, “Numerically induced chaos in the nonlinear Schrodinger equation,” Phys. Rev. Lett. 62, 2065–2068 (1989).
[CrossRef] [PubMed]

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Society for Industrial and Applied Mathematics, 1981).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Aschieri, P.

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classical optical waves in a waveguide,” Phys. Rev. A 83, 033838 (2011).
[CrossRef]

Barviau, B.

Biven, L.

A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152–153, 520–550 (2001).
[CrossRef]

Bortolozzo, U.

Bourkoff, E.

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, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef] [PubMed]

A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo, “Free-energy transition in a gas of noninteracting nonlinear wave particles,” Phys. Rev. Lett. 101, 044101 (2008).
[CrossRef] [PubMed]

Dias, F.

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

Doya, V.

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classical optical waves in a waveguide,” Phys. Rev. A 83, 033838 (2011).
[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 Schrodinger equation,” Physica D 57, 96–160 (1992).
[CrossRef]

El, G. A.

G. A. El and A. M. Kamchatnov, “Kinetic equation for a dense soliton gas,” Phys. Rev. Lett. 95, 204101 (2005).
[CrossRef] [PubMed]

Falkovich, G.

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, 1992).

Fratalocchi, A.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef] [PubMed]

A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo, “Free-energy transition in a gas of noninteracting nonlinear wave particles,” Phys. Rev. Lett. 101, 044101 (2008).
[CrossRef] [PubMed]

Garnier, J.

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classical optical waves in a waveguide,” Phys. Rev. A 83, 033838 (2011).
[CrossRef]

Herbst, B. M.

B. M. Herbst and M. J. Ablowitz, “Numerically induced chaos in the nonlinear Schrodinger equation,” Phys. Rev. Lett. 62, 2065–2068 (1989).
[CrossRef] [PubMed]

Hsu, W. L.

Jauslin, H. R.

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]

Kamchatnov, A. M.

G. A. El and A. M. Kamchatnov, “Kinetic equation for a dense soliton gas,” Phys. Rev. Lett. 95, 204101 (2005).
[CrossRef] [PubMed]

Kibler, B.

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

B. Barviau, B. Kibler, and A. Picozzi, “Wave turbulence description of supercontinuum generation: influence of self-steepening and higher-order dispersion,” Phys. Rev. A 79, 063840 (2009).
[CrossRef]

Koplow, J. P.

Kudlinski, A.

L’vov, V. S.

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, 1992).

Laurie, J.

Manassah, J. T.

Michel, C.

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classical optical waves in a waveguide,” Phys. Rev. A 83, 033838 (2011).
[CrossRef]

C. Michel, P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Influence of third-order dispersion on the propagation of incoherent light in optical fibers,” Opt. Lett. 35, 2367–2369 (2010).
[CrossRef] [PubMed]

Millot, G.

Moore, S. W.

Mussot, A.

Nazarenko, S.

U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori, “Optical wave turbulence and the condensation of light,” J. Opt. Soc. Am. B 26, 2280 (2009).
[CrossRef]

A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152–153, 520–550 (2001).
[CrossRef]

Newell, A. C.

A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152–153, 520–550 (2001).
[CrossRef]

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

Peccianti, M.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef] [PubMed]

Picozzi, A.

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classical optical waves in a waveguide,” Phys. Rev. A 83, 033838 (2011).
[CrossRef]

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Anomalous thermalization of nonlinear wave systems,” Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef] [PubMed]

C. Michel, P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Influence of third-order dispersion on the propagation of incoherent light in optical fibers,” Opt. Lett. 35, 2367–2369 (2010).
[CrossRef] [PubMed]

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

B. Barviau, B. Kibler, and A. Picozzi, “Wave turbulence description of supercontinuum generation: influence of self-steepening and higher-order dispersion,” Phys. Rev. A 79, 063840 (2009).
[CrossRef]

A. Picozzi, “Towards a nonequilibrium thermodynamic description of incoherent nonlinear optics,” Opt. Express 15, 9063–9083 (2007).
[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]

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]

Pushkarev, A.

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

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

Randoux, S.

Residori, S.

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]

Ruocco, G.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef] [PubMed]

A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo, “Free-energy transition in a gas of noninteracting nonlinear wave particles,” Phys. Rev. Lett. 101, 044101 (2008).
[CrossRef] [PubMed]

Schroder, K. L.

Segur, H.

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Society for Industrial and Applied Mathematics, 1981).
[CrossRef]

Soh, D. B. S.

Suret, P.

Tabor, M.

M. Tabor, Chaos and Integrability in Nonlinear Dynamics (Wiley, 1989).

Trillo, S.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef] [PubMed]

A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo, “Free-energy transition in a gas of noninteracting nonlinear wave particles,” Phys. Rev. Lett. 101, 044101 (2008).
[CrossRef] [PubMed]

Zakharov, V. E.

V. E. Zakharov, “Turbulence in integrable systems,” Stud. Appl. Math. 122, 219–234 (2009).
[CrossRef]

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

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

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, 1992).

Zhao, W.

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

Opt. Express (3)

Opt. Lett. (4)

Phys. Rep. (1)

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

Phys. Rev. A (2)

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, “Condensation and thermalization of classical optical waves in a waveguide,” Phys. Rev. A 83, 033838 (2011).
[CrossRef]

B. Barviau, B. Kibler, and A. Picozzi, “Wave turbulence description of supercontinuum generation: influence of self-steepening and higher-order dispersion,” Phys. Rev. A 79, 063840 (2009).
[CrossRef]

Phys. Rev. Lett. (6)

B. M. Herbst and M. J. Ablowitz, “Numerically induced chaos in the nonlinear Schrodinger equation,” Phys. Rev. Lett. 62, 2065–2068 (1989).
[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]

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Anomalous thermalization of nonlinear wave systems,” Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef] [PubMed]

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef] [PubMed]

A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo, “Free-energy transition in a gas of noninteracting nonlinear wave particles,” Phys. Rev. Lett. 101, 044101 (2008).
[CrossRef] [PubMed]

G. A. El and A. M. Kamchatnov, “Kinetic equation for a dense soliton gas,” Phys. Rev. Lett. 95, 204101 (2005).
[CrossRef] [PubMed]

Physica D (2)

A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D 152–153, 520–550 (2001).
[CrossRef]

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

Stud. Appl. Math. (1)

V. E. Zakharov, “Turbulence in integrable systems,” Stud. Appl. Math. 122, 219–234 (2009).
[CrossRef]

Other (4)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Society for Industrial and Applied Mathematics, 1981).
[CrossRef]

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, 1992).

M. Tabor, Chaos and Integrability in Nonlinear Dynamics (Wiley, 1989).

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Figures (4)

Fig. 1
Fig. 1

Numerical simulations. The gaussian power spectrum nω (z = 0) of the incoherent light wave chosen as initial condition is plotted in dashed black line. The averaged spectrum nω (z = 1) plotted in black line is obtained from the integration of Eq. (1) over an ensemble of 1000 realizations. The spectrum nω (z = 1) plotted in red line is obtained from the numerical integration of Eqs. (4) and (5). The spectrum nω (z = 1) plotted in blue line is obtained from the numerical integration of Eq. (12). Simulations plotted in (a) correspond to a linear (kinetic) regime in which ɛ = HNL /HL is equal to 0.05 (n 0 = 0.1, σ = +1). Simulations plotted in (b) correspond to a regime that is slightly more nonlinear: ɛ = HNL /HL = 0.5 (n 0 = 1, σ = +1).

Fig. 2
Fig. 2

Numerical simulations of Eq. (1) (black line) and of Eq. (15) (blue line) showing the decaying oscillations of the power of two spectral components taken in the wings of the spectrum plotted in Fig. 1(a). The initial condition is the gaussian power spectrum plotted in Fig. 1(a) (n 0 = 0.1, σ = +1). In (a), the spatial period Λ 9 π 4 ω 2 of the decaying oscillations is close to 1.96.10−2 for a spectral component at the frequency ω = 6. In (b), the period decreases to Λ ≃ 1.44.10−2 for a spectral component at the frequency ω = 7.

Fig. 3
Fig. 3

Shematic representation of the experimental setup. HWP: half-wave plate. OSA: optical spectrum analyzer.

Fig. 4
Fig. 4

(a): Power spectra recorded in experiments (black lines) and obtained from numerical simulations of Eq. (1). The narrow spectrum plotted in black line is the spectrum of the Nd:YVO4 laser launched inside the PMF. The wide spectrum plotted in black line is the spectrum recorded at the output of the PMF. In numerical simulations, the incoherent wave taken as initial condition has a power spectrum approximated by n ω 0 = n ω ( z = 0 ) = n 0 e x p ( ( ω A ) 4 ) (A = 0.73, n 0 = 4.72, σ = +1) (b): Numerical simulations of Eq. (1) (red lines) and Eq. (12) (blue line). Spectra plotted in red lines are identical to those plotted in (a).

Equations (17)

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i z ψ ( z , t ) = σ t 2 ψ ( z , t ) + | ψ ( z , t ) | 2 ψ ( z , t )
ψ ˜ ( z , ω ) ψ ˜ * ( z , ω ) = n ω ( z ) δ ( ω ω )
ψ ˜ ( z , ω 1 ) ψ ˜ ( z , ω 2 ) ψ ˜ * ( z , ω 3 ) ψ ˜ * ( z , ω 4 ) = J 1 , 2 3 , 4 ( z ) δ ( ω 1 + ω 2 ω 3 ω 4 )
n ω 1 ( z ) z = 1 π d ω 2 4 I m [ J 1 , 2 3 , 4 ( z ) ] δ ( ω 1 + ω 2 ω 3 ω 4 ) ,
J 1 , 2 3 , 4 ( z ) z i Δ k J 1 , 2 3 , 4 ( z ) = i π 𝒩 ( z )
J 1 , 2 3 , 4 ( z ) = J 1 , 2 3 , 4 ( z = 0 ) e i Δ k z + i π 0 z d z 𝒩 ( z ) e i Δ k ( z z ) .
n ω 1 ( z ) z = 1 π d ω 2 4 𝒩 ( z ) δ ( ω 1 + ω 2 ω 3 ω 4 ) δ ( Δ k )
n ω 1 ( z ) z = 1 π 2 0 z d z d ω 2 4 𝒩 ( z ) cos ( Δ k ( z z ) ) δ ( ω 1 + ω 2 ω 3 ω 4 )
n ω 0 = n ω ( z = 0 ) = n 0 exp [ ( ω Δ ω ) 2 ] = n 0 exp ( ω 2 ) .
n ω 1 ( z ) z = 1 π 2 d ω 2 4 𝒩 ( z = 0 ) sin ( Δ k z ) Δ k δ ( ω 1 + ω 2 ω 3 ω 4 ) .
n ω 1 ( z ) z = 1 π 2 d ω 3 4 ( z = 0 ) sin ( Δ k z ) Δ k
n ω 1 ( z ) z = 1 π 2 d ω 3 4 n ω 3 0 n ω 4 0 n ω 3 + ω 4 ω 1 0 sin ( Δ k z ) Δ k n ω 1 0 π 2 d ω 3 4 n ω 3 0 n ω 4 0 sin ( Δ k z ) Δ k
n ω 1 ( z ) z 1 π 2 d ω 3 4 n ω 3 0 n ω 4 0 n ω 3 + ω 4 ω 1 0 sin ( Δ k z ) Δ k .
n ω 1 ( z ) z 9 8 π 2 ω 1 2 d ω 3 4 n ω 3 0 n ω 4 0 n ω 3 + ω 4 ω 1 0 sin ( 2 ( ω 1 ω 3 ) ( ω 1 ω 4 ) z ) .
n ω 1 ( z ) z n 0 3 3 π 9 8 ω 1 2 exp ( ω 1 2 3 ( 1 + 8 z 2 9 ) ) sin ( 8 ω 1 2 z 9 )
n ω 1 ( z ) z γ 2 n 0 3 Δ ω 2 3 π 9 8 β ω 1 2 exp ( ω 2 3 Δ ω 2 ( 1 + 8 β 2 Δ ω 4 z 2 9 ) ) sin ( 8 β ω 1 2 z 9 )
| n ω 1 ( z ) z | | B n ω 1 ( z = 0 ) s i n c ( Ω ¯ 2 z ) |

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