## Wave turbulence in integrable systems: nonlinear propagation of incoherent optical waves in single-mode fibers |

Optics Express, Vol. 19, Issue 18, pp. 17852-17863 (2011)

http://dx.doi.org/10.1364/OE.19.017852

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### 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

## 1. Introduction

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]

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]

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]

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]

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

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]

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]

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

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

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]

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]

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]

**31**, 1696–1698 (2006). [CrossRef] [PubMed]

*et al.*, who proposed to modify the traditional approach of the WT theory [23

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]

*et al.*[23

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]

## 2. Theoretical Treatment

*L*is defined in the usual way as

_{NL}*L*= 1/(

_{NL}*γP*

_{0}), where

*P*

_{0}represents the power carried by the incoherent wave. The NLS equation [Eq. (1)] conserves the power

*N*= ∫|

*ψ*(

*z,t*)|

^{2}

*dt*and the total energy

*H*=

*H*+

_{L}*H*that has a linear (kinetic) contribution

_{NL}*H*= ∫

_{L}*k*(

*ω*)

*(*ψ ˜

*z*,

*ω*)

*dω*and a nonlinear contribution

*(*ψ ˜

*z*,

*ω*) is the Fourier transform of

*ψ*(

*z,t*) defined as

*𝒩*(

*z*) =

*n*

_{ω1}(

*z*)

*n*

_{ω3}(

*z*)

*n*

_{ω4}(

*z*) +

*n*

_{ω2}(

*z*)

*n*

_{ω3}(

*z*)

*n*

_{ω4}(

*z*) –

*n*

_{ω1}(

*z*)

*n*

_{ω2}(

*z*)

*n*

_{ω3}(

*z*) –

*n*

_{ω1}(

*z*)

*n*

_{ω2}(

*z*)

*n*

_{ω4}(

*z*) and Δ

*k*=

*k*(

*ω*

_{1}) +

*k*(

*ω*

_{2}) –

*k*(

*ω*

_{3}) –

*k*(

*ω*

_{4}).

*k*= 0). For Δ

*k*≠ 0, the contribution of the fast oscillating function

*e*

^{iΔkz}is considered as being non essential for propagation distances larger than 1/Δ

*k*. With these assumptions, one obtains the following standard WT kinetic equation [10

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]

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

*z*= 0 has been supposed gaussian so that the fourth-order moment

*(0,*ψ ˜

*ω*

_{1})

*(0,*ψ ˜

*ω*

_{2})

ψ ˜

^{*}(0,

*ω*

_{3})

ψ ˜

^{*}(0,

*ω*

_{4})〉 =

*n*

_{ω1}(0)

*n*

_{ω2}(0)[

*δ*(

*ω*

_{1}–

*ω*

_{3}).

*δ*(

*ω*

_{2}–

*ω*

_{4}) +

*δ*(

*ω*

_{1}–

*ω*

_{4}).

*δ*(

*ω*

_{2}–

*ω*

_{3})]). The fourth-order moment is therefore a real quantity having a vanishing contribution when Eq. (6) is substituted into Eq. (4).

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

*𝒩*(

*z*) vanishes as a result of the integration over the Dirac delta functions. In opposition with two- or three-dimensional geometry, the phase-matching conditions found in 1D scalar NLS equation only permit trivial interactions among frequency components (i.e.

*ω*

_{3}=

*ω*

_{1,2}and

*ω*

_{4}=

*ω*

_{2,1}). This results in a collision term that is identically equal to zero. Therefore the only possible way to describe an evolution of

*n*

_{ω1}in a one-dimensional geometry from WT theory consists in taking into account non-phase matched interactions among frequency components (i.e. Δ

*k*≠ 0).

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]

*et al.*have proposed a theoretical treatment of nonlinear propagation of an incoherent light wave inside an optical fiber. Although full details of their calculation are not explicitly given in Ref. [23

**18**, 22393–22405 (2010). [CrossRef] [PubMed]

*et al.*establish their kinetic equation (Eq. (5) of Ref. [23

**18**, 22393–22405 (2010). [CrossRef] [PubMed]

**18**, 22393–22405 (2010). [CrossRef] [PubMed]

*et al.*in Ref. [23

**18**, 22393–22405 (2010). [CrossRef] [PubMed]

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]

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

*ϕ*(

*ω*) of the Fourier components of the field taken as initial condition (

*ψ*(

*z*= 0,

*ω*) = |

*ψ*(

*z*= 0,

*ω*)|

*e*

^{iϕ(ω)}) are uniformly distributed between 0 and 2

*π*[21

**31**, 1696–1698 (2006). [CrossRef] [PubMed]

*z*= 0 transiently changes from nonlinear propagation. It reaches a statistical steady state after a transient stage whose duration and amplitude critically depend on the shape of the power spectrum taken as initial condition. The most significant evolutions are observed when the tails of the power spectrum decay more sharply than an exponential function (i.e. for power spectra such as

*n*(

_{ω}*z*= 0) =

*n*

_{0}

*exp*(−|

*ω*/Δ

*ω*|

*) with*

^{α}*α*> 1). This is illustrated in Fig. 1 that shows power spectra (black lines) computed from numerical simulations of Eq. (1) with the following gaussian power spectrum (dashed black lines) taken as initial condition: With the definition Eq. (9), the half-width Δ

*ω*at 1/

*e*of the power spectrum is unity and with our normalization, the parameter

*n*

_{0}becomes a measure of the ratio between the linear dispersive length

*L*and the nonlinear length

_{D}*L*:

_{NL}*z*∼ 0.1

*L*∼ 0.2/(

_{D}*β*

_{2}Δ

*ω*

^{2}) in physical units). The central part of the spectrum does not change much from nonlinear propagation whereas the the wings of the spectrum are significantly modified. Their shape does not remain gaussian but it becomes exponential. Let us emphasize that the power carried by frequency components lying in the wings of the spectrum increases by several orders of magnitude. Note that this phenomenon has already been illustrated in Ref. [23

**18**, 22393–22405 (2010). [CrossRef] [PubMed]

*n*

_{0}(see spectra plotted in red lines in Fig. 1). As previously mentioned,

*n*

_{0}is proportional to the ratio between linear and nonlinear lengths. Changes in

*n*

_{0}are therefore associated with changes in the ratio

*ɛ*between the

*H*and

_{NL}*H*(

_{L}*ɛ*=

*H*/

_{NL}*H*). Figure 1(a) shows that numerical simulations of Eqs. (4) and (5) are in quantitative agreement with simulations of Eq. (1) over more than 20 decades in the linear (kinetic) regime in which

_{L}*ɛ*=

*H*/

_{NL}*H*≃ 0.05. As shown in Fig. 1(b), this quantitative agreement is preserved over ∼ 15 decades even in the nonlinear regime in which

_{L}*ɛ*=

*H*/

_{NL}*H*≃ 0.5. Whatever the interaction regime explored in our numerical simulations (i.e. from

_{L}*ɛ*∼ 0.01 to

*ɛ*∼ 1), the central part of the spectrum that carries the essential of the power of the incoherent wave is not significantly modified from nonlinear propagation.

*n*(

_{ω}*z*= 0) of the incoherent field only slightly changes with propagation distance

*z*. With this assumption, the term

*𝒩*(

*z*′) in the integral of Eq. (6) is approximated by its value

*𝒩*(

*z*= 0) at

*z*= 0 and it is extracted from the integral. Substituting Eq. (6) into Eq. (4), we obtain the following kinetic equation If

*z*≫ 1/Δ

*k*, the function

*δ*(Δ

*k*) =

*δ*(

*k*(

*ω*

_{1}) +

*k*(

*ω*

_{2}) –

*k*(

*ω*

_{3}) –

*k*(

*ω*

_{4})) and the collision term found in the kinetic equation [Eq. (7)] vanishes. However as long as

*z*∼ 1/Δ

*k*, non-phase-matched interactions among spectral components cannot be neglected. The collision integral found in Eq. (10) is not identically equal to zero and changes in the power spectrum

*n*

_{ω1}can be observed despite the integrability of the wave system.

*ω*

_{2}. This gives: with Δ

*k*=

*k*(

*ω*

_{1}) +

*k*(

*ω*

_{3}+

*ω*

_{4}–

*ω*

_{1}) –

*k*(

*ω*

_{3}) –

*k*(

*ω*

_{4}) = 2

*σ*(

*ω*

_{1}–

*ω*

_{3})(

*ω*

_{1}–

*ω*

_{4}). The term

*ℳ*(

*z*= 0) is given by

*i*= 1, 3, 4). Further simplification of the kinetic equation can be achieved by considering that two parts of the collision term found in Eq. (11) balance each other. We thus obtain:

*n*

_{0}= 0.1,

*ɛ*=

*H*/

_{NL}*H*= 0.05). In the slightly nonlinear regime (Fig. 1(b) computed for

_{L}*n*

_{0}= 1,

*ɛ*=

*H*/

_{NL}*H*= 0.5, the approximation is less effective but a quantitative agreement with the simulation of Eq. (1) is nevertheless preserved over ∼ 8 decades. The first integral found in the right-hand side of Eq. (12) mainly contributes to change occurring in the wings of the spectrum whereas the second integral essentially contributes to changes occurring around the center of the spectrum (i. e. around

_{L}*ω*

_{1}∼ 0). The aim of this paper is to study the deep quantitative changes found in the tails of the spectrum (see Fig. 1). We now restrict our theoretical analysis to the only study of this specific point (

*ω*

_{1}>> Δ

*ω*). The problem thus reduces to If the power spectrum is initially gaussian

*x*=

*ω*

_{3}–

*ω*

_{1}/3 and

*y*=

*ω*

_{4}–

*ω*

_{1}/3. Using the variables

*x*and

*y*, the function

*x*=

*y*= 0. With a gaussian power spectrum taken as initial condition, the growth rate of the power carried by a spectral component at a frequency

*ω*

_{1}falling in the tails of the spectrum is determined by the interaction among spectral components at frequencies ±

*ω*

_{1}/3 falling in the center of the gaussian spectrum. This result is not intuitive : the evolution of the component at a frequency

*ω*

_{1}is not driven by degenerate four-wave mixing among the pairs of frequencies (0, 0) and (−

*ω*

_{1}, +

*ω*

_{1}). Indeed, the dominant contribution corresponds to degenerate four-wave mixing among the pairs of frequencies (+

*ω*

_{1}/3, +

*ω*

_{1}/3) and (−

*ω*

_{1}/3, +

*ω*

_{1}). Note that this result is valid for any initial spectrum with an hypergaussian shape (i. e.

*n*(

_{ω}*z*= 0) =

*n*

_{0}

*exp*(−

*ω*

^{2}

*) where*

^{p}*p*> 0 is an integer)

*k*found in the denominator of Eq. (13) can be taken constant

*k*in the sine function of Eq. (14) by

*ω*>> Δ

*ω*) with propagation distance

*z*. For the sake of clarity, we rephrase its expression in physical units: with

*β*=

*β*

_{2}/2.

*n*

_{ω1}of a spectral component taken in the wings of the gaussian spectrum grows with the propagation distance

*z*. Moreover Eq. (15) shows that

*n*

_{ω1}will reach a steady value after damped oscillations. This phenomenon is illustrated in Fig. 2 that shows the decaying oscillations of the power of two different spectral components. As illustrated in Fig. 2, there is a good quantitative agreement between the results obtained from the numerical integrations of Eq. (1) and of Eq. (15). Note that simulations of Eq. (1) have been realized over an ensemble of 100 realizations and that the curves plotted with black lines in Fig. 2 represent an averaged result. Note also that the decaying oscillations plotted in Fig. 2 are those leading to the steady-state spectrum plotted in Fig. 1(a).

*L*. Let us recall that these oscillations have their physical origin in transient and non-phase-matched interactions among frequency components of the incoherent light field (see Eq. (10)). The period of the oscillations is determined by the fact that the spectral component at

_{D}*ω*

_{1}dominantly interact with the frequency components

*k*is around

*βω*

_{1}Δ

*ω*)

^{−1}.

*et al*in Ref. [23

**18**, 22393–22405 (2010). [CrossRef] [PubMed]

**18**, 22393–22405 (2010). [CrossRef] [PubMed]

*et al.*in Ref. [23

**18**, 22393–22405 (2010). [CrossRef] [PubMed]

*B*in a parameter depending on

*γ*and

*P*

_{0}in a way that does not need to be explicited for our discussion. According to Soh

*et al.*,

^{2}is a constant quantity defined as the “averaged root-mean squared full optical bandwidth”. With the averaging procedure proposed by Soh

*et al*, the upper-bound |

*Bsinc*(

^{2}

*z*)| is an oscillating and decaying function of

*z*that is independent of

*ω*

_{1}. In our detailed analysis, we have shown that spectral components at frequencies

*ω*

_{1}falling in the wings of the spectrum relax to the statistical steady state from damped oscillations whose properties (period and damping rate) fundamentally depend on the frequency

*ω*

_{1}.

*σ*on the mean spectra found from numerical integration of Eq. (1) (see Fig. 1). The results determined above from WT theory are inherently insensitive to the sign of

*σ*in linear (kinetic) regime (

*ɛ*<< 1). In the linear regime, coherent structures such as solitons do not emerge from the propagation of incoherent waves. Moreover the frequencies at which modulational instability can appear in anomalous dispersion regime are localized in the vinicity of the center of the spectrum. In other words, modulational instability does not influence the growth of the tails of the spectrum in the linear regime. We have checked this behavior from numerical integration of Eq. (1) and mean spectra of Fig. 1(a) do not depend of the sign of

*σ*in this linear regime (

*ɛ*= 0.05).

*ɛ*= 0.5 (see Fig. 1(b)), numerical simulations of Eq. (1) contrarily show that the mean spectrum

*n*(

_{ω}*z*= 1) depends on the sign of

*σ*. Numerical simulations made in the anomalous dispersion regime (

*σ*= −1) show that the mean spectrum

*n*(

_{ω}*z*= 1) reached at

*z*= 1 is broader than the one plotted in black line in Fig. 1(b) for the normal dispersion regime (

*σ*= +1). This feature can be interpreted from the fact that soliton-like structures and modulational instability begin to play a role in the anomalous dispersion regime at

*ɛ*= 0.5. We have shown that our analytical results obtained from kinetic theory are still in good agreement with numerical integration of NLS equation [Eq. (1)] at

*ɛ*= 0.5 (see Fig. 1(b)). In other words, the range of values of

*ɛ*=

*H*/

_{NL}*H*over which WT theory provides results that are quantitatively correct is wider in the normal dispersion regime than in the anomalous dispersion regime.

_{L}## 3. Experiments

**31**, 1696–1698 (2006). [CrossRef] [PubMed]

*γ*= 6 W

^{−1}km

^{−1}and its second-order dispersion coefficient is

*β*

_{2}= 20 ps

^{2}km

^{−1}at 1064 nm.

*ɛ*=

*H*/

_{NL}*H*<< 1). Up to now, only the nonlinear interaction regime has been explored in experiments (

_{L}*ɛ*=

*H*/

_{NL}*H*>> 1) [21

_{L}**31**, 1696–1698 (2006). [CrossRef] [PubMed]

**31**, 1696–1698 (2006). [CrossRef] [PubMed]

*μ*m and with spectra spreading over ∼ 1 nm, the nonlinear interaction regime in a standard singlemode fiber typically refers to optical powers around ∼ 1 W [21

**31**, 1696–1698 (2006). [CrossRef] [PubMed]

^{10}times (equivalently 100 dB) lower than the power of spectral components lying in the center of the spectrum. The possibility to explore the linear interaction regime therefore critically depends on the sensitivity and the dynamic range of the optical spectrum analyzer (OSA) used in the measurement.

*ɛ*=

*H*/

_{NL}*H*≃ 1. To explore the linear interaction regime, the optical power should be decreased by at least one order of magnitude but the detection of the slight changes occurring in the wings of the spectrum then becomes impossible with our OSA.

_{L}*A*is adjusted to get the best fit between the hyper-gaussian profile and the experimental profile. As shown in Fig. 4(a), the power spectrum found in numerical simulations of Eq. (1) is in good quantitative agreement with the power spectrum recorded in experiments. Despite the fact that the experiment is made in a slightly nonlinear regime (

*ɛ*=

*H*/

_{NL}*H*≃ 1), a quantitative agreement over approximately four decades is obtained between numerical results computed from the integration of simplified kinetic Eq. (12) (blue line in Fig. 4(b)) and numerical results computed from integration of Eq. (1) (red lines in Fig. 4(b)). We have also made additional numerical simulations in order to study the influence of the exact shape of the laser power spectrum on the shape of the spectrum found at the output end of the PMF. As shown in Fig. 4(a), the power spectrum of the NdYVO4 laser is well fitted by an hyper-gaussian profile over approximately two decades but its tails deviates from this profile. Numerical simulations show that these deviations do not significantly influence the spectrum found at the output of the PMF. This can be understood by recalling that the evolution of the tails of the spectrum is only determined by the interaction among spectral components lying in the center of the spectrum (see Sec. 2)

_{L}## 4. Conclusion

## Acknowledgment

## References and links

1. | G. P. Agrawal, |

2. | M. J. Ablowitz and H. Segur, |

3. | W. Zhao and E. Bourkoff, “Interactions between dark solitons,” Opt. Lett. |

4. | C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. |

5. | A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo, “Free-energy transition in a gas of noninteracting nonlinear wave particles,” Phys. Rev. Lett. |

6. | G. A. El and A. M. Kamchatnov, “Kinetic equation for a dense soliton gas,” Phys. Rev. Lett. |

7. | J. T. Manassah, “Self-phase modulation of incoherent light,” Opt. Lett. |

8. | V. E. Zakharov, “Turbulence in integrable systems,” Stud. Appl. Math. |

9. | A. C. Newell, S. Nazarenko, and L. Biven, “Wave turbulence and intermittency,” Physica D |

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 |

11. | C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. |

12. | A. Picozzi, “Towards a nonequilibrium thermodynamic description of incoherent nonlinear optics,” Opt. Express |

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 |

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. |

15. | P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, “Anomalous thermalization of nonlinear wave systems,” Phys. Rev. Lett. |

16. | V. E. Zakharov, F. Dias, and A. Pushkarev, “One-dimensional wave turbulence,” Phys. Rep. |

17. | U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori, “Optical wave turbulence and the condensation of light,” J. Opt. Soc. Am. B |

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 |

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 |

20. | M. Tabor, |

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. |

22. | B. M. Herbst and M. J. Ablowitz, “Numerically induced chaos in the nonlinear Schrodinger equation,” Phys. Rev. Lett. |

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 |

24. | V. E. Zakharov, V. S. L’vov, and G. Falkovich, |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(190.4370) Nonlinear optics : Nonlinear optics, fibers

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: April 26, 2011

Revised Manuscript: July 14, 2011

Manuscript Accepted: July 24, 2011

Published: August 26, 2011

**Citation**

Pierre Suret, Antonio Picozzi, and Stéphane Randoux, "Wave turbulence in integrable systems: nonlinear propagation of incoherent optical waves in single-mode fibers," Opt. Express **19**, 17852-17863 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-17852

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### References

- 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]
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