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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 14 — Jul. 9, 2007
  • pp: 9063–9083
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Towards a nonequilibrium thermodynamic description of incoherent nonlinear optics

Antonio Picozzi  »View Author Affiliations


Optics Express, Vol. 15, Issue 14, pp. 9063-9083 (2007)
http://dx.doi.org/10.1364/OE.15.009063


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Abstract

This concise review is aimed at providing an introduction to the kinetic theory of partially coherent optical waves propagating in nonlinear media. The subject of incoherent nonlinear optics received a renewed interest since the first experimental demonstration of incoherent solitons in slowly responding photorefractive crystals. Several theories have been successfully developed to provide a detailed description of the novel dynamical features inherent to partially coherent nonlinear optical waves. However, such theories leave unanswered the following important question: Which is the long term (spatiotemporal) evolution of a partially incoherent optical field propagating in a nonlinear medium? In complete analogy with kinetic gas theory, one may expect that the incoherent field may evolve, owing to nonlinearity, towards a thermodynamic equilibrium state. Weak-turbulence theory is shown to describe the essential properties of this irreversible process of thermal wave relaxation to equilibrium. Precisely, the theory describes an irreversible evolution of the spectrum of the field towards a thermodynamic equilibrium state. The irreversible behavior is expressed through the H-theorem of entropy growth, whose origin is analogous to the celebrated Boltzmann’s H-theorem of kinetic gas theory. It is shown that thermal wave relaxation to equilibrium may be characterized by the existence of a genuine condensation process, whose thermodynamic properties are analogous to those of Bose-Einstein condensation, despite the fact that the considered optical wave is completely classical. In spite of the formal reversibility of optical wave propagation, the condensation process occurs by means of an irreversible evolution of the field towards a homogeneous plane-wave (condensate) with small-scale fluctuations superimposed (uncondensed particles), which store the information necessary for the reversible propagation. As a remarkable result, an increase of entropy (“disorder”) in the optical field requires the generation of a coherent structure (plane-wave). We show that, beyond the standard thermodynamic limit, wave condensation also occurs in two spatial dimensions. The numerical simulations are in quantitative agreement with the kinetic wave theory, without any adjustable parameter.

© 2007 Optical Society of America

1. Introduction

The coherence properties of partially incoherent optical waves propagating in nonlinear media have been studied since the advent of nonlinear optics in the 1960s, because of the natural poor degree of coherence of laser sources available at that time [1

1. See, e.g., J. Ducuing and N. Bloembergen, “Statistical Fluctuations in Nonlinear Optical Processes,” Phys. Rev. 133, A1493 – A1502 (1964). [CrossRef]

]. However, it is only recently that the dynamics of incoherent nonlinear optical waves received a renewed interest. The main motive for this renewal of interest is essentially due to the first experimental demonstration of incoherent solitons in photorefractive crystals [2

2. M. Mitchell, Z. Chen, Ming-feng Shih, and M. Segev, “Self-Trapping of Partially Spatially Incoherent Light,” Phys. Rev. Lett. 77, 490 (1996). [CrossRef] [PubMed]

]. The incoherent soliton consists of a phenomenon of self-trapping of spatially and temporally incoherent light in a noninstantaneous response nonlinear medium [3

3. M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature (London) 387, 880 (1997). [CrossRef]

]. The remarkable simplicity of experiments performed in photorefractive media allowed for a fruitful investigation of the dynamics of incoherent nonlinear waves [4

4. Y.S. Kivshar and G.P. Agrawal, “Optical Solitons : From Fibers to Photonic Crystals” (Ac. Press, 2003).

, 5

5. M. Segev and D. N. Christodoulides, “Incoherent Solitons,” Eds. S. Trillo and W. Torruellas, Spatial Solitons (Springer, Berlin, 2001).

], as witnessed by several important achievements, such as, e.g., the modulational instability of incoherent optical fields [6

6. M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath “Modulation Instability of Incoherent Beams in Noninstantaneous Nonlinear Media,” Phys. Rev. Lett. 84, 467 (2000). [CrossRef] [PubMed]

, 7

7. D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation instability and pattern formation in spatially incoherent light beams,” Science 290, 495–498 (2000). [CrossRef] [PubMed]

, 8

8. C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, “Eliminating the Transverse Instabilities of Kerr Solitons,” Phys. Rev. Lett. 85, 4888 (2000). [CrossRef] [PubMed]

, 9

9. S. M. Sears, M. Soljacic, D. N. Christodoulides, and M. Segev, “Pattern formation via symmetry breaking in nonlinear weakly correlated systems,” Phys. Rev. E 65, 036620 (2002). [CrossRef]

], the existence of incoherent dark solitons [10

10. D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of Incoherent Dark Solitons,” Phys. Rev. Lett. 80, 5113 (1998). [CrossRef]

, 11

11. Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889 (1998). [CrossRef] [PubMed]

], incoherent pattern formation [12

12. H. Buljan, M. Soljacic, T. Carmon, and M. Segev, “Cavity pattern formation with incoherent light,” Phys. Rev. E 68, 016616 (2003). [CrossRef]

], or incoherent solitons in periodic lattices [13

13. H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides “Random-Phase Solitons in Nonlinear Periodic Lattices,” Phys. Rev. Lett. 92, 223901 (2004). [CrossRef] [PubMed]

, 14

14. O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature (London) 433, 500 (2005). [CrossRef] [PubMed]

]. Let us note that incoherent solitons and remarkable dynamical features inherent to incoherent nonlinear waves have also been recently investigated in instantaneous response nonlinear media [15

15. A. Picozzi and M. Haelterman, “Parametric Three-Wave Soliton Generated from Incoherent Light,” Phys. Rev. Lett. 86, 2010–2013 (2001). [CrossRef] [PubMed]

, 16

16. A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent Solitons in Instantaneous Response Nonlinear Media,” Phys. Rev. Lett. 92, 143906 (2004). [CrossRef] [PubMed]

, 17

17. A. Picozzi and M. Haelterman, “Condensation in Hamiltonian Parametric Wave Interaction,” Phys. Rev. Lett. 92, 103901 (2004). [CrossRef] [PubMed]

, 18

18. A. Sauter, S. Pitois, G. Millot, and A. Picozzi, “Incoherent modulation instability in instantaneous nonlinear Kerr media,” Opt. Lett. 30, 2143–2145 (2005). [CrossRef] [PubMed]

, 19

19. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005). [CrossRef]

, 20

20. A. Picozzi, “Nonequilibrated Oscillations of Coherence in Coupled Nonlinear Wave Systems,” Phys. Rev. Lett. 96, 013905 (2006). [CrossRef] [PubMed]

, 21

21. S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity Locking of Incoherent Nonlinear Wave Packets,” Phys. Rev. Lett. 97, 033902 (2006). [CrossRef] [PubMed]

, 22

22. O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, “Incoherent solitons in instantaneous nonlocal nonlinear media,” Phys. Rev. E 73, 015601 (2006). [CrossRef]

].

Fig. 1. Analogy between a system of classical particles (two-particle collision) and the propagation of an optical wave in a cubic (e.g., Kerr) nonlinear medium. As described by the kinetic gas theory, collisions between particles are responsible for an irreversible evolution of the gas towards thermodynamic equilibrium. In complete analogy, an optical wave propagating in a nonlinear medium should exhibit an irreversible evolution to equilibrium, as a result of the four-wave mixing between frequency components of the incoherent field.

A conspicuous prediction of weak-turbulence theory is the existence of a condensation process for the nonlinear optical field [40

40. 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 (1992). [CrossRef]

, 43

43. Y. Pomeau, “Long time behavior of solutions of nonlinear classical field equations: the example of NLS defocusing,” Physica D 61, 227 (1992). [CrossRef]

, 19

19. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005). [CrossRef]

]. More precisely, it is shown that, in spite of the formal reversibility of the NonLinear Schrödinger (NLS) equation, the condensation process manifests itself through an irreversible evolution of the field towards a homogeneous plane wave (condensate) with small-scale fluctuations superimposed (uncondensed particles), which store the information necessary for the reversible propagation of the field. The kinetic theory reveals that the thermodynamic properties of this condensation process are analogous to those of the genuine Bose-Einstein condensation [19

19. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005). [CrossRef]

], despite the fact that the considered optical field is completely classical. We emphasize that the generation of the plane-wave solely results from the natural tendency of the isolated wave system (conservative and Hamiltonian) to approach the equilibrium state, i.e., the state that realizes the maximum of entropy. This signifies that an increase of entropy (“disorder”) in the field requires the generation of a coherent structure (plane-wave condensate). In other terms, the theory reveals that it is thermodynamically advantageous for the optical field to generate a plane wave. A simple explanation of this counterintuitive and remarkable result will be discussed.

Let us finally notice that, although the process of thermal relaxation to equilibrium is undeniably considered as a basic generic property of a molecular gas system [42

42. K. Huang, “Statistical Mechanics” (Wiley, 1963).

], its role in the nonlinear evolution of a pure wave system has not been precisely established experimentally. This situation is mainly due to the fact that the irreversible relaxation process is predicted in a lossless (conservative and reversible) wave system, while any practical system unavoidably exhibits dissipation. In this way, one encounters the practical difficulties to maintain the model equations valid over the long durations that require the process of relaxation to equilibrium. Nonetheless, nonlinear optics appears to be a promising field of investigation of thermal wave relaxation because of the availability of low-loss nonlinear media (e.g., silica optical fibers) in which light propagation is accurately ruled by conservative (e.g., NLS-like) equations over long distances [4

4. Y.S. Kivshar and G.P. Agrawal, “Optical Solitons : From Fibers to Photonic Crystals” (Ac. Press, 2003).

, 44

44. R.W. Boyd, “Nonlinear Optics” (Acad. Press, 2002).

, 45

45. N.N. Akhmediev and A. Ankiewicz, “Solitons - Non-linear pulses and beams” (Springer, 1997).

, 46

46. A. C. Newell and J. V. Moloney, “Nonlinear Optics” (Addison-Wesley Publ. Comp., 1992).

].

2. Model equation

To provide an introduction to the kinetic wave theory, we shall consider the concrete example of the propagation of a partially coherent optical field in a cubic nonlinear medium (e.g., Kerr medium). In the framework of the paraxial and slowly-varying envelope approximations, the evolution of the field envelope ψ(z,r,t), of central frequency ω 0 and wavenumber k 0, is known to obey the NLS equation [44

44. R.W. Boyd, “Nonlinear Optics” (Acad. Press, 2002).

, 45

45. N.N. Akhmediev and A. Ankiewicz, “Solitons - Non-linear pulses and beams” (Springer, 1997).

, 46

46. A. C. Newell and J. V. Moloney, “Nonlinear Optics” (Addison-Wesley Publ. Comp., 1992).

]

izψ=α2ψ+βttψ+gψ2ψ
(1)

where ∇2 = ∂2 x + ∂2 y represents the transverse spatial derivatives, and g denotes the nonlinear Kerr coefficient, g < 0 (g > 0) referring to a focusing (defocusing) nonlinear medium. The variable z denotes the propagation length in the nonlinear medium and t the retarded time in a reference frame moving at the group velocity of the field vg. The diffraction parameter reads α=12k0 , and the dispersion parameter β=122kω2|ω0 . According to the model Eq.(1), the propagation of the field is characterized by three lengths scales, the nonlinear length L nl = 1/|g| 〈|ψ|2〉 (the bracket 〈.〉 referring to an average over the statistical realizations), the diffraction length Lα = λc 2/α and the dispersive length Lβ = τc 2/|β|, where τc and λc respectively refer to the time and length correlation of the partially incoherent optical field.

The NLS Eq.(1) conserves two important quantities. The power of the optical field

N=ψ2drdt,
(2)

and the total energy

H=Hl+Hnl,
(3)

which has a linear (dispersive) kinetic contribution

Hl=(αψ2βtψ2)drdt,
(4)

and a nonlinear contribution

Hnl=g2ψ4drdt.
(5)

The kinetic equation consists of an equation describing the evolution of the spectrum of the field during its propagation in the nonlinear medium. Note that, in the particolar case in which dispersion effects may be neglected (α = β = 0), an expression for the evolution of the second order correlation function was calculated explicitly in Ref. [47

47. J. T. Manassah, “Self-phase modulation of incoherent light revisited,” Opt. Lett. 16, 1638 (1991). [CrossRef] [PubMed]

]. The structure of the kinetic equation crucially depends on the nature of the statistics of the optical field. The statistics is said to be homogenous (stationary), if the correlation function B(z,r1,r2,t 1,t 2) = 〈ψ(z, r1,t 1)ψ *(z, r2,t 2)〉 only depends on the distance |r1 - r2| (delay |t 1 - t 2|). In the following we consider separately the situation in which the statistics is assumed to be homogenous or inhomogenous.

3. Inhomogenous statistics

For the sake of clarity, let us consider the pure spatial evolution of the field, in which Eq.(1) takes the simplified form

izψ=α2ψ+gψ2ψ.
(6)

Note that the analysis we are going to expose may easily be extended to the full spatio-temporal evolution of the field [Eq.(1)]. In the same way, the analysis can easily be transposed to the pure temporal evolution of the field, which is relevant for the study of a guided optical wave configuration, e.g., in an optical fiber.

3.1. The closure of the moment equations

Making use of Eq.(6), we follow the usual procedure to derive a second-order moment equation for the correlation function B(z,r1,r2) [27

27. L. Mandel and E. Wolf, “Optical Coherence and Quantum Optics” (Cambridge University Press, New York, 1995).

]

izB=α(2212)B+g(ψ12ψ1*ψ2*ψ2*2ψ1ψ2),
(7)

where ψj = ψ(z,r j), j = 1,2. It becomes apparent in Eq.(7) that, because of the nonlinear character of Eq.(6), the evolution of the second-order moment of the field depends on its fourth-order moment. Naturally, the equation governing the evolution of the fourth-oder moment will depend on the six-order moment of the field, and so on. Accordingly, one obtains an infinite hierarchy of moment equations, in which the n-th order moment depends on the (n+2)-th order moment of the field. This makes the equations impossible to solve unless some way can be found to truncate the hierarchy. This refers to the fundamental problem of achieving a closure of the infinite hierarchy of the moment equations [33

33. K. Hasselmann, “On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory,” J. Fluid Mech. 12, 481–500 (1962). [CrossRef]

, 34

34. K. Hasselmann, “On the non-linear energy transfer in a gravity-wave spectrum. Part 2. Conservation theorems; wave-particle analogy; irreversibility,” J. Fluid Mech. 15, 273–281 (1963). [CrossRef]

, 35

35. A. C. Newell, “The closure problem in a system of random gravity waves,” Rev. of Geophys. 6, 1–31 (1968). [CrossRef]

].

A simple way to achieve such a closure is to assume that the field obeys a Gaussain statistics, which is justified when linear effects dominate nonlinear effects, i.e., L α,βLnl or ε = Hnl/Hl ≪ 1 [39

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

]. Under these conditions, one may exploite the property of factorizability of stochastic Gaussian fields [48

48. A. Papoulis and S.U. Pillai “Probability, Random Variables, and Stochastic Processes” (McGraw-Hill, Fourth Edition, 2002).

], 〈ψ 1 2 ψ 1 * ψ 2 *) =2〈|ψ 1|2〉 〈ψ 1 ψ 2 *|〉, and 〈ψ 2 *2 ψ 1 ψ 2〉 = 2〈|ψ 2|2〉 〈ψ 1 ψ 2 *〉, to obtain a closed equation for the evolution of the second-order moment B(z,r,y) = 〈ψ(z,r+y/2)ψ *(z,r-y/2)〉,

izB=2αr.yB+2g[B(z,r1,0)B(z,r2,0)]B,
(8)

where r = (r 1 + r 2)/2 and y = r 1 - r 2. Note that B(z, r, 0) ≡ Ñ(z, r) = 〈|ψ|2〉 (z, r) refers to the averaged “power” of the field, which depends on the spatial variable r because the statistics has been assumed to be inhomogenous.

3.2. Instantaneous vs noninstantaneous nonlinearity

Before discussing further Eq.(8), let us make an important observation. Notice that Eq. (8) is almost identical to that derived in the context of noninstantaneous response nonlinear media [4

4. Y.S. Kivshar and G.P. Agrawal, “Optical Solitons : From Fibers to Photonic Crystals” (Ac. Press, 2003).

, 5

5. M. Segev and D. N. Christodoulides, “Incoherent Solitons,” Eds. S. Trillo and W. Torruellas, Spatial Solitons (Springer, Berlin, 2001).

], the only difference being the presence of the factor 2 in front of the nonlinear term. Indeed, when the response time of the nonlinear medium τR is much greater than the time correlation of the field (tcτR), the medium responds to the time-averaged intensity and not to the instantaneous speckles that constitute the incoherent beam [4

4. Y.S. Kivshar and G.P. Agrawal, “Optical Solitons : From Fibers to Photonic Crystals” (Ac. Press, 2003).

, 5

5. M. Segev and D. N. Christodoulides, “Incoherent Solitons,” Eds. S. Trillo and W. Torruellas, Spatial Solitons (Springer, Berlin, 2001).

]. Accordingly, the nonlinear index nnl of refraction only depends on the time averaged intensity. Assuming furthermore that the field exhibits a temporal stationary statistics, the time averaged intensity is equal, by ergodicity, to the (local) statistical average of the intensity. The equation governing the evolution of the field then takes the following simplified form

izψ=α2ψ+gψ2ψ,
(9)

3.3. Vlasov’s like equation

Let us now gain some insight into the physics described by Eq.(8). First of all, it is important to remark that if the statistics were homogenous, the correlation function would no longer depend on the spatial variable r and the nonlinear term in Eq.(8) results to vanish. In this limit, the equation for B reduces to the well-known Wolf’s equation describing the evolution of the correlation function in a linear medium, i∂zB = - 2αr.∇y B [27

27. L. Mandel and E. Wolf, “Optical Coherence and Quantum Optics” (Cambridge University Press, New York, 1995).

].

This observation indicates that the role of the nonlinear term in Eq.(8) may be conveniently analyzed in term of the degree of homogeneity of the statistics of the field. For this purpose, we note that the power difference in Eq.(8) may be expanded as follows, Ñ(z, r + y/2) - Ñ(z, r - y/2) =2∑ p=0(y/2)2p+1. r 2p+1 Ñ(z,r)/(2p+1)! [we recall that Ñ(z, r) = B(z, r, 0)]. Let us also define the local spectrum of the field by means of the Wigner’s transform

nk(z,r)=B(z,r,y)exp(ik.y)dy.
(10)

Note that the spectrum of the field depends on the spatial variable r whenever the field exhibits an inhomogenous statistics. By means of the Wigner’s expansion of Eq.(8), one thus readily obtains [50

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

, 26

26. B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002). [CrossRef]

]

znk(z,r)=2αk.rnk(z,r)+2gp=0(1)p22p(2p+1)!2p+1N˜(z,r)r2p+1.2p+1nk(z,r)k2p+1.
(11)

When the statistics of the field may be assumed to be quasi-homogenous, one may retain only the first term in the sum, i.e., Ñ(z, r + y/2) - Ñ(z, r - y/2)≃y.∇r Ñ(z,r). More precisely, one may approximate knknk/λc and rÑÑÑ/λinh, where λinh, denotes the characteristic length of inhomogeneity of the statistics of the field. Accordingly, the second term of the sum may be neglected whenever λinhλc [50

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

]. Under this assumption of quasi-homogenous statistics, one obtains the following Vlasov’s like equation governing the evolution of the spectrum of the field

znk(z,r)=2αk.rnk+2grN˜.knk,
(12)
N˜(z,r)=nk(z,r)dk.
(13)

The analogy between the Vlasov’s equation originally derived for the description of dilute non-equilibrium plasma, and Eq.(12,13) describing partially incoherent optical waves was explicitly pointed out in Ref.[26

26. B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E 65, 035602 (2002). [CrossRef]

]. To interpret the physics underlying this equation, we remark that it can be recast in Hamiltonian form by means of the following Liouville’s equation

dznk(z,r)zn+k˙.kn+r˙.rn=0,
(14)

where the variables k and r appear as canonical conjugate variables,

k˙=zk=rω˜=r(2gN˜),
(15)
r˙=zr=kω˜=2αk,
(16)

with the effective Hamiltonian

ω˜(r,k)=αk2+2gN˜(r).
(17)

Note that, as for the NLS Eq.(6), Eq.(12–13) conserves the total power of the field N = ∫Ñ(z,r)d r. This property naturally results from the Liouville’s equation (14), which implies conservation of the area N=∬ d r d k n k(z, r) occupied by the quasi-particle distribution n k(z, r) in the phase-space (r,k).

Equations (14–17) remarkably reveal that the partially incoherent optical wave could be modeled as an ensemble of independent “quasi-particles”, in which their (nonlinear) interaction manifests itself by means of an effective potential Veff(z,r) = 2gÑ(z,r). The quasi-particles then evolve as if they were independent, their evolution being characterized by an effective dispersion relation ω̃ = ω 0 + Veff, which results to be the sum of the original dispersion relation ω 0(k) = αk 2 and of the mean-field potential Veff.

We remark that, as for the original Vlasov’s equation derived in plasmas, Eqs.(12–13) are self-consistent, in the sense that the effective potential Veff(z) depends itself on the quasi-particle distribution n k(z, r). Such a nontrivial self-consistent interaction is the key property responsible for the existence of incoherent solitons in inertial nonlinear media, as explicitly described by the self-consistent multimode theory [4

4. Y.S. Kivshar and G.P. Agrawal, “Optical Solitons : From Fibers to Photonic Crystals” (Ac. Press, 2003).

, 5

5. M. Segev and D. N. Christodoulides, “Incoherent Solitons,” Eds. S. Trillo and W. Torruellas, Spatial Solitons (Springer, Berlin, 2001).

, 24

24. M. Mitchell, M. Segev, T. Coskun, and D.N. Christodoulides, “Theory of Self-Trapped Spatially Incoherent Light Beams,” Phys. Rev. Lett. 79, 4990 (1997). [CrossRef]

]. More specifically, a soliton forms when the optical beam induces an effective attractive potential Veff < 0 (waveguide) owing to a focusing nonlinearity (g < 0). In turn, the optical beam is guided in its own induced potential Veff. Along this line, it was remarkably shown that incoherent solitons may even exhibit a dark structure [10

10. D. N. Christodoulides, T. H. Coskun, M. Mitchell, Z. Chen, and M. Segev, “Theory of Incoherent Dark Solitons,” Phys. Rev. Lett. 80, 5113 (1998). [CrossRef]

,11

11. Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889 (1998). [CrossRef] [PubMed]

,51

51. T. H. Coskun, D. N. Christodoulides, Z. Chen, and M. Segev “Dark incoherent soliton splitting and phase-memory effects: Theory and experiment,” Phys. Rev. E 59, R4777 (1999). [CrossRef]

, 52

52. C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar “Partially Incoherent Optical Vortices in Self-Focusing Nonlinear Media,” Phys. Rev. Lett. 92, 043904 (2004). [CrossRef] [PubMed]

,53

53. K. Motzek, F. Kaiser, J. R. Salgueiro, Y. Kivshar, and C. Denz, “Incoherent vector vortex-mode solitons in self-focusing nonlinear media,” Opt. Lett. 29, 2285 (2004). [CrossRef] [PubMed]

, 54

54. A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, “Observation of Attraction between Dark Solitons,” Phys. Rev. Lett. 96, 043901 (2006). [CrossRef] [PubMed]

]. More generally, incoherent solitons have been the subject of an intense investigation and were shown to exhibit unique properties not found for their coherent counterpart [55

55. N. Akhmediev, W. Krolikowski, and A. W. Snyder, “Partially Coherent Solitons of Variable Shape,” Phys. Rev. Lett. 81, 4632 (1998). [CrossRef]

, 56

56. A. A. Sukhorukov and N. N. Akhmediev “Coherent and Incoherent Contributions to Multisoliton Complexes,” Phys. Rev. Lett. 83, 4736 (1999). [CrossRef]

, 57

57. O. Bang, D. Edmundson, and W. Krolikowski, “Collapse of Incoherent Light Beams in Inertial Bulk Kerr Media,” Phys. Rev. Lett. 83, 5479 (1999). [CrossRef]

, 58

58. T. H. Coskun, A. G. Grandpierre, D. N. Christodoulides, and M. Segev “Coherence enhancement of spatially incoherent light beams through soliton interactions,” Opt. Lett. 25, 826 (2000). [CrossRef]

, 59

59. W. Krolikowski, D. Edmundson, and O. Bang, “Unified model for partially coherent solitons in logarithmically nonlinear media,” Phys. Rev. E 61, 3122 (2000). [CrossRef]

, 60

60. M. Peccianti and G. Assanto “Incoherent spatial solitary waves in nematic liquid crystals,” Opt. Lett. 26, 1791–1793 (2001). [CrossRef]

, 61

61. S. A. Ponomarenko, N. M. Litchinitser, and G. P. Agrawal “Theory of incoherent optical solitons: Beyond the mean-field approximation,” Phys. Rev. E 70, 015603 (2004). [CrossRef]

, 62

62. S. A. Ponomarenko and G. P. Agrawal “Asymmetric incoherent vector solitons,” Phys. Rev. E 69, 036604 (2004). [CrossRef]

, 63

63. Ting-Sen Ku, Ming-Feng Shih, A. A. Sukhorukov, and Y. S. Kivshar “Coherence Controlled Soliton Interactions,” Phys. Rev. Lett. 94, 063904 (2005). [CrossRef] [PubMed]

, 64

64. K. G. Makris, H. Sarkissian, D. N. Christodoulides, and G. Assanto “Nonlocal incoherent spatial solitons in liquid crystals,” J. Opt. Soc. Am. B 22, 1371–1377 (2005). [CrossRef]

].

4. Homogenous statistics

4.1. Boltzmann’s like equation

For infinite dimensional Hamiltonian systems like classical optical wave fields, the relationship between formal reversibility and actual dynamics can be rather complex. In integrable systems, such as the one-dimensional NLS equation, the dynamics is essentially periodic in time, reflecting the underlying regular phase-space structure of nested tori. This recurrent behavior is broken in nonintegrable systems, where the dynamics is in general governed by an irreversible process of diffusion in phase space [69

69. R. Z. Sagdeev, D. A. Usikov, and G. M. Zaslavsky, “Nonlinear Physics” (Harwood Academic Publ., Chur, Switzerland, 1988).

]. The essential properties of this irreversible evolution to equilibrium are described by the kinetic equation of weak-turbulence theory.

In the following we briefly outline the derivation of the kinetic equation for an incoherent field characterized by a homogenous statistics, whose evolution is assumed to be governed by the NLS equation. In particular, we enlight the important differences with respect to the derivation of the Vlasov’s equation. The details of the procedure for the derivation of the kinetic equation may be found in Refs. [39

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

, 41

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

]. The derivation is essentially based on a perturbation expansion theory in which linear dispersive effects dominate nonlinear effects, L α,βLnl, i.e., ε = Hnl/Hl ≪ 1. Accordingly, an effective large separation of the linear and the nonlinear lengths scales takes place [33

33. K. Hasselmann, “On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory,” J. Fluid Mech. 12, 481–500 (1962). [CrossRef]

, 34

34. K. Hasselmann, “On the non-linear energy transfer in a gravity-wave spectrum. Part 2. Conservation theorems; wave-particle analogy; irreversibility,” J. Fluid Mech. 15, 273–281 (1963). [CrossRef]

, 35

35. A. C. Newell, “The closure problem in a system of random gravity waves,” Rev. of Geophys. 6, 1–31 (1968). [CrossRef]

]. In this way the statistics of the field may be assumed to be Gaussian, which allows one to achieve the closure of the hierarchy of moment equations. Note that the statistics does not need to be Gaussian initially. Because linear effects dominate nonlinear effects, it is the linear behavior which brings the system close to a state of Gaussian statistics. For most purposes this approach is equivalent to the so called “random-phase approximation” [39

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

].

When the optical field exhibits a homogenous statistics, it proves convenient to derive the kinetic equation in Fourier’s space. In this respect we remark that, because the statistics is homogenous, the spectrum of the field no longer depend on the space-time variables (r,t). More precisely, the spectrum of a field characterized by a homogenous statistics is δ-correlated [48

48. A. Papoulis and S.U. Pillai “Probability, Random Variables, and Stochastic Processes” (McGraw-Hill, Fourth Edition, 2002).

], 〈ψ̃(k 1,ω 1,z)ψ̃*(k 2,ω 2,z)〉 = n k1,ω1(z) δ k1-k2 δ ω1-ω2, where ψ̃(k, ω,z) = ∫ψ(r,t,z) exp(-iωt - i k.r) d r dt is the spatio-temporal Fourier’s transform of the field amplitude, k being the transverse wave-vector of components (kx,ky).

By means of a Fourier’s expansion of the NLS Eq. (1), one may readily derive the following equation for the evolution of the spatio-temporal spectrum of the field

iznk0,ω0δk4k0δω4ω0=gdk13dω13(J1,23,4δ0,2;1,3J¯1,23,0δ4,2;1,3)
(18)

The details of the derivation of the kinetic equation can be found in Ref. [39

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

]. In substance, one derives an equation for the fourth-order moment J that depends on the six-order moment of the field. Owing to the factorizability property of Gaussian fields, the six-order moment is expanded in terms of products of second-order moments, which gives the following kinetic equation describing the evolution of the spatio-temporal spectrum n k,ω (z) of the optical field

znk1,ω1(z)=𝒞oll[nk1,ω1],
(19)

where the collision term reads

𝒞oll[nk1,ω1]=dk24dω24W0(nk3,ω3nk4,ω4nk1,ω1+nk3,ω3nk4,ω4nk2,ω2nk1,ω1nk2,ω2nk3,ω3nk1,ω1nk2,ω2nk4,ω4).
(20)

Basically, the kinetic approach models the four-wave interaction as a collisional gas of quasi-particles satisfying the resonant conditions of energy and momentum conservation at each elementary collision (see Fig. 1), as expressed through the presence of Dirac’s δ-functions in the cross-section term

W0=g2δ(1)(ω1+ω2ω3ω4)δ(2)(k1+k2k3k4)×δ(1)[K(k1,ω1)+K(k2,ω2)K(k3,ω3)K(k4,ω4)],
(21)

where the spatio-temporal dispersion relation is given by the linearized NLS Eq.(1)

K(k,ω)=αk2βω2,
(22)

K = kz being the projection of the wave-vector along the longitudinal axis z of propagation.

4.2. Properties of the kinetic equation

The kinetic Eq.(19–21) has a structure anagolous to that of the celebrated Boltzmann’s equation, which is known to describe the evolution of a dilute classical gas far from the equilibrium state [42

42. K. Huang, “Statistical Mechanics” (Wiley, 1963).

]. For this reason the kinetic Eq.(19–21) exhibits properties similar to those of the Boltzmann’s equation. It conserves the total power (or quasi-particle number) of the field

N=Vnk,ω(z)dk,
(23)

and the kinetic (linear) energy

Hl=VK(k,ω)nk,ω(z)dk,
(24)

where V refers to the system “volume” (V = ATp, A referring to the characteristic beam area and Tp to the pulse duration). Let us remark that Eq.(19–21) do not conserve the total energy H, but only its linear contribution Hl. This results from the fact that the nonlinear energy has a negligible contribution in the perturbation expansion procedure of the kinetic theory (ε = Hnl/Hl ≪ 1). Notice that, as for the NLS Eq.(1), the kinetic Eq.(19–21) also conserves linear momentum. This property becomes essential for the study of the interaction between several optical fields, as recently studied in Ref. [21

21. S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity Locking of Incoherent Nonlinear Wave Packets,” Phys. Rev. Lett. 97, 033902 (2006). [CrossRef] [PubMed]

, 70

70. S. Lagrange, H. R. Jauslin, and A. Picozzi, “Thermalization of the dispersive three-wave interaction” (submitted).

]. It has been shown in these works that distinct wave-packets thermalize towards an equilibrium state in which they propagate with an identical group-velocity [21

21. S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity Locking of Incoherent Nonlinear Wave Packets,” Phys. Rev. Lett. 97, 033902 (2006). [CrossRef] [PubMed]

, 70

70. S. Lagrange, H. R. Jauslin, and A. Picozzi, “Thermalization of the dispersive three-wave interaction” (submitted).

].

In complete analogy with the Boltzmann’s equation, the kinetic wave equation is not reversible with respect to the propagation distance z. The irreversible character of Eq.(19–21) is expressed through the H-theorem of entropy growth, dS/dz ≥ 0, where the nonequilibrium entropy reads

S(z)=VLog[nk,ω(z)]dk.
(25)

As in standard statistical mechanics, the thermodynamic equilibrium state is determined from the postulate of maximum entropy [42

42. K. Huang, “Statistical Mechanics” (Wiley, 1963).

]. The equilibrium spectra n eq(k,ω) realizing the maximum of S[n(k,ω)], subject to the constraints of conservation of Hl and N, may readily be calculated by introducing the corresponding Lagrange’s multipliers, 1/T and -μ/T. One obtains

nk,ωeq=TK(k,ω)μ,
(26)

where T and μ respectively denote the temperature and the chemical potential by analogy with thermodynamics. The equilibrium distribution (26) yields an exactly vanishing collision term (20), 𝒞oll[neq] = 0. This means that once the spectrum has reached the equilibrium distribution (26), it no longer evolve during the propagation, zn = 0.

It is important to note that, depending on the sign of the dispersion coefficient β, two qualitatively different equilibrium spectra are found in Eq.(26). In the case of anomalous dispersion (β < 0), the spectrum neq(k, ω) exhibits a symmetric spatio-temporal structure reflecting the symmetric roles of space and time. In this case, the equilibrium spectrum recovers the standard Rayleigh-Jeans lorentzian distribution [39

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

, 19

19. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005). [CrossRef]

, 40

40. 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 (1992). [CrossRef]

, 21

21. S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity Locking of Incoherent Nonlinear Wave Packets,” Phys. Rev. Lett. 97, 033902 (2006). [CrossRef] [PubMed]

], in which the constants T and μ can be determined from the conserved quantities Hl and N. The chemical potential characterizes the typical bandwidths of the spatio-temporal lorentzian spectrum, so that the correlation time and length of the optical field at equilibrium are proportional to

λceqαμ,τceqβμ.
(27)

The irreversible evolution of the spectrum of the field towards the equilibrium state (26) is clearly visible in the numerical simulations of the NLS equation (see, e.g., Ref. [21

21. S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity Locking of Incoherent Nonlinear Wave Packets,” Phys. Rev. Lett. 97, 033902 (2006). [CrossRef] [PubMed]

]).

The situation is completely different in the normal dispersion regime (β > 0). In this case, temporal dispersion and spatial diffraction act in opposite ways and compensate each other for those frequencies lying along the lines ω±αβk . This confers a hyperbolic structure to the dispersion relation K(ω,k) [4

4. Y.S. Kivshar and G.P. Agrawal, “Optical Solitons : From Fibers to Photonic Crystals” (Ac. Press, 2003).

]. Note that this property is at the origin of the existence of nonlinear X-waves [71

71. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear Electromagnetic X-Waves,” Phys. Rev. Lett. 90, 170406 (2003). [CrossRef] [PubMed]

, 72

72. P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously Generated X-Shaped Light Bullets,” Phys. Rev. Lett. 91, 093904 (2003); [CrossRef] [PubMed]

, 73

73. M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic Nonlinear X Waves for Femtosecond Pulse Propagation in Water,” Phys. Rev. Lett. 92, 253901 (2004). [CrossRef] [PubMed]

]. The corresponding equilibrium spectrum n eq(k, ω) then exhibits a spatio-temporal X-shape, a peculiar property that has been confirmed by the numerical simulations in Ref.[70

70. S. Lagrange, H. R. Jauslin, and A. Picozzi, “Thermalization of the dispersive three-wave interaction” (submitted).

]. Note that this intriguing spectrum was recently shown to witness the presence of a field characterized by an X-shaped spatio-temporal coherence, i.e., coherence is neither spatial nor temporal, but skewed along spatio-temporal trajectories [74

74. A. Picozzi and M. Haelterman “Hidden Coherence Along Space-Time Trajectories in Parametric Wave Mixing,” Phys. Rev. Lett. 88, 083901 (2002). [CrossRef] [PubMed]

, 75

75. O. Jedrkiewicz, A. Picozzi, M. Clerici, D. Faccio, and P. Di Trapani, “Emergence of X-Shaped Spatiotemporal Coherence in Optical Waves,” Phys. Rev. Lett. 97, 243903 (2006). [CrossRef]

].

5. Condensation of classical optical waves

In the following we shall see that a classical optical wave exhibits a condensation process [40

40. 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 (1992). [CrossRef]

, 76

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

], whose thermodynamic properties are analogous to those of the genuine Bose-Einstein condensation [19

19. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005). [CrossRef]

], in spite of the classical nature of the NLS equation. The condensation process consists of an irreversible evolution of the nonlinear wave towards an equilibrium state characterized by the presence of a large scale coherent structure, i.e., a plane-wave. We emphasize that the formation of the coherent structure solely results from the natural irreversible evolution of the field towards its “most disordered state,” i.e., the state that realizes the maximum of entropy.

We consider the propagation of the nonlinear wave in a defocusing (g > 0) Kerr medium characterized by an anomalous group-velocity dispersion (β < 0). This ensures that the monochromatic plane-wave solution to the three-dimensional NLS Eq.(1) is modulationally stable with respect to perturbations. In this way, it is apparent that the positivity of the spectrum (26), neq k,ω > 0, imposes the following constraints, T ≥ 0 and μ. ≤0.

Before discussing the thermodynamic properties of the condensation process, let us gain some physical insight into the mechanism underlying wave condensation. We realized numerical simulations of the 3D normalized NLS Eq.(1) in a cubic box of 643 points with periodic boundary conditions. The variables can be recovered in real units through the transformation: zzLnl,t 0,rrλ0, with τ 0 = (|β| Lnl)1/2 and λ 0 = (αLnl)1/2. The movies of Figs. 2 represent a typical evolution of the spatial spectrum of the optical field during its propagation in the nonlinear medium (note that for the sake of clarity, only the central part of the spectrum is visible in the movie, i.e., 322 points). The spatial spectrum S(k,z) has been obtained by selecting the frequency plane ω = 0 in the full 3D spatio-temporal spectrum, S(k,z) =|ψ̃|2 (k, ω = 0,z). Note that the same evolution is obtained by plotting the spatio-temporal spectrum S(kx,ω,z) =|ψ̃|2 (kx,ky = 0, ω,z), or S(ky,ω,z) =|ψ̃|2 (kx = 0,ky,ω,z). The initial condition ψ(r,t,z = 0) corresponds to an incoherent field of zero mean(〈ψ〉 = 0), characterized by a δ-correlated spatio-temporal spectrum.

The simulations clearly show that the spectrum of the field spontaneously concentrates in the fundamental mode k = ω = 0 during its propagation. Actually, the field exhibits an irreversible evolution towards equilibrium, as witnessed by the saturation of the growth of entropy (d zS ≃ 0), as well as the saturation of the fraction of condensed power, i.e., the normalized power in the fundamental mode k = ω = 0 (see Figs. 3). In other terms, the optical field tends to evolve towards a monochromatic plane-wave of frequency ω 0 and wave-number k 0 [see Eq.(1)]. However, we emphasize that the field cannot evolve towards a pure monochromatic plane-wave, because such evolution would imply a loss of information for the field, which would thus violate the reversibility of the system. Actually, the monochromatic plane-wave remains immersed in a sea of small scale fluctuations, which contain, in principle, all the information necessary for a reversible propagation of the field. Such small scale fluctuations are clearly visible in the movie of Fig. 2b, which represents the evolution of the (spatial) spectrum in logarithmic scale. We remark that, contrary to other disciplines (e.g., ultra-cold gases experiments) where the arrow of time could not be reversed in practice, in optics the formal reversibility of the condensation process could be demonstrated by means of a phase-conjugation experiment.

Fig. 2. Optical wave condensation: Numerical simulation of the normalized three-dimensional NLS Eq.(1) showing the evolution of the spatial spectrum [S(k,ω = 0,z) =|ψ̃|2 (k,ω= 0,z)] of the field [in normal (a) [Media 1], and logarithmic (b) [Media 2], scales] during its propagation. The concentration of the power of the field in the fundamental mode k =ω= 0 solely results from its natural irreversible evolution towards the equilibrium state. The evolution of some relevant quantities of the field corresponding to this numerical simulation are illustrated in Fig. 3. The complete movie corresponds to a propagation over 500Lnl. (The space-time discretization of the normalized NLS Eq.(1) is dx = dt = 1, the power density is N/V = 1/2, with a computational box of 643).

It follows from this discussion that the irreversible evolution of the field towards equilibrium is characterized by the generation of a coherent structure, i.e., the monochromatic plane wave. Let us try to provide a simple explanation of this counterintuitive result. We recall that the total energy of the field has a kinetic (linear) contribution and a nonlinear contribution, H = Hl +Hnl [Eq. (3)]. The natural tendency of the field is to evolve towards equilibrium, i.e., the “most disordered state”. One may reasonably expect that such state would be characterized by the presence of rapid fluctuations of the field amplitude ψ(r,t). A natural measure of the amount of fluctuations is naturally provided by the kinetic energy Hl, since it measures the gradients of the spatio-temporal variations of the field ψ(r,t) [see Eq.(4)]. Accordingly, the kinetic energy is expected to increase during the process of relaxation to equilibrium. However, the total energy H must remain constant during the propagation, so that an increase of the kinetic contribution Hl is necessarily accompanied by a reduction of the nonlinear contribution Hnl, as confirmed by the numerical simulations (see Fig. 3c-d). The lower the nonlinear contribution Hnl, the higher the kinetic contribution Hl. Actually, Hnl is minimized for a constant amplitude of the field, ψ(r,t) =const, i.e., for the monochromatic plane wave solution. This merely explains why it is advantageous for the field to generate a coherent structure so as to reach the most disordered state. In other terms, an increase of the amount of disorder in the optical field requires the generation of the monochromatic plane wave. Note that the same reasoning may be transposed to the focusing regime of interaction (g < 0). The field generates a soliton so as to minimize H nl, while the kinetic energy of small scales fluctuations compensates for the difference between Hnl and the conserved total energy H [77

77. V.E. Zakharov, A.N. Pushkarev, V.F. Shvetz, and V.V. Yan’kov, “Solitonic turbulence,” Pis’ma v Zh. Eksp. Teor. Fiz. 48, 79–81 (1988) [JETP Lett. 48, 83–85 (1988)].

, 78

78. B. Rumpf and A. C. Newell, “Coherent Structures and Entropy in Constrained, Modulationally Unstable Nonintegrable Systems,” Phys. Rev. Lett. 87, 054102 (2001). [CrossRef] [PubMed]

, 79

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

].

5.1. Condensation in 3D

Below we summarize the essential thermodynamic properties of the condensation process in three dimensions. We refer the interested reader to Ref. [19

19. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005). [CrossRef]

] for more details. Let us first re-mark that the distribution (26) realizes the maximum of the entropy S[n k] and vanishes exactly the collision term, Coll[nkeq] = 0. However, it is important to note that Eq.(26) is only a formal solution, because it does not lead to converging expressions for the energy Hl and the power N in the limits k → ∞ and ω) → ∞. To regularize such unphysical divergence, we introduce an ultraviolet cut-off kc and ωc. Note that this frequency cut-off appears naturally in any numerical simulation through the spatiotemporal discretization of the NLS Eq.(1). A frequency cut-off also arises naturally in a guided wave configuration of the optical field [80

80. P. Aschieri and A. Picozzi (to be published).

]. Combining Eqs.(23,24) and (26), the expression for the power of the field at equilibrium then takes the following form

Fig. 3. Thermal wave relaxation to equilibrium: Evolution of the fraction of condensed power (a), entropy (b), and energy (c,d) corresponding to the numerical simulation illustrated in Fig. 2. The fraction of condensed power irreversibly evolves towards the equilibrium value predicted by the theory (N 0/N ≃ 71% for a total energy of H = 1 and a power density N/V = 1/2). The process of entropy growth is saturated once the equilibrium state is reached, as described by the H-theorem of entropy growth. Figs. (c) and (d) show that a transfer of energy occurs from the nonlinear contribution Hnl to the linear contribution Hl, while the total energy H = Hnl +Hl remains constant. This process of energy transfer explains why an increase of entropy in the field requires the generation of a the coherent structure (i.e., the plane-wave condensate).
NV=4πTkcαβ[1μkcarctan(kcμ)],
(28)
HlV=4πTkc33αβ[1+3μkc2+3(μkc2)32arctan(kcμ)],
(29)

where we assumed ωc=kcαβ so as to get an explicit analytical expression for N and Hl. Note that this assumption does not affect the generality of the results we are going to expose. An inspection of Eq.(28) reveals that μ tends to 0- for a non-vanishing temperature T, keeping a constant power density N/V. According to Eq.(27), this means that the correlation time τc and length λc diverge to infinity. By analogy with the Bose-Einstein transition in quantum systems, such a divergence of the equilibrium distribution at k = ω = 0 reveals the existence of a condensation process.

As in standard Bose-Einstein condensation, the fraction of condensed power N 0/N vs the temperature T (or the energy Hl), may be calculated by setting μ = 0 in the equilibrium distribution (26). Note that the assumption μ = 0 for TTc can be justified rigorously in the thermodynamic limit (i.e., V → ∞, N → ∞, keeping N/V constant). One readily obtains (N - N 0)/V = 4πTkc/α |β|1/2 and Hl/V = 4πTkc 3/3α |β|1/2, which gives

N0N=1HlHl,c,
(30)

where the critical energy reads H l,c = Nk 2 c/3. Alternatively, the fraction of condensed power may be expressed as a function of the temperature,

N0N=1TTc,
(31)

where Tc = 3α |β|1/2 H l,c/(4πVkc 3). As in standard Bose-Einstein condensation, N 0 vanishes at the critical temperature Tc, and N 0 becomes the total number of particles as T tends to 0.

The linear behavior of N 0 vs Hl in Eq.(30) was found consistent with the results of numerical simulations (see Fig. 2 of Ref. [19

19. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005). [CrossRef]

]). However, the approach outlined above does not take into account the “interactions between the quasi-particles”. To include the nonlinear (interaction) contribution, the Bogoliubov’s expansion procedure of a weakly interacting Bose gas has been adapted to the classical wave problem considered here. The interested reader may find the details of the analysis in Ref.[19

19. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005). [CrossRef]

]. Let us emphasize that a quantitative agreement was obtained between the theory and the numerical simulations of the three-dimensional NLS Eq.(1), without any adjustable parameter [19

19. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005). [CrossRef]

]. In particular, the fraction of condensed power N 0/N ≃ 0.71 for H = 1 illustrated in Figs. 1–2, is in excellent agreement with the theory (see Fig. 2 of Ref [19

19. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005). [CrossRef]

]).

5.2. Beyond the thermodynamic limit: Condensation in 2D

Let us now consider the condensation process in two dimensions. For concreteness, we consider the purely spatial evolution of the optical field in the framework of the two-dimensional NLS Eq.(6). The analysis exposed above in 3D may readily be applied to 2D, which gives N/A = πTLog(1 - kc 2/μ)/α, where A refers to the area of the system (i.e., the transverse area of an optical waveguide). It becomes apparent from this expression that, for a fixed power density N/A, μ reaches zero for a vanishing temperature T. In complete analogy with the Bose-Einstein condensation, this indicates that condensation no longer take place in 2D [19

19. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005). [CrossRef]

]. In other terms, the critical temperature Tc tends to zero because of the infrared divergence of the equilibrium distribution n k eq. Actually, this result is rigorously correct in the thermodynamic limit (i.e., A → ∞, N → ∞, keeping N/A constant). Nevertheless, we shall see below that, for situations of physical interest in which N and A are finite, wave condensation is re-established in two dimensions, a remarkable property confirmed by the numerical simulations.

For a finite value of the system size A, one has to replace the continuous integral in frequency space by a discrete sum (∫d k → (4π 2/A)∑k), with the frequency discretization dkx = dky = 2π/√A. The expression for the power of the optical field at equilibrium (23) may thus be written in the following form

1T=1kx,ky1[α(kx2+ky2)μ],
(32)

where ρ = N/A is the power density, and ∑kx,ky denotes the discrete sum for - kckx, kykc. For a fixed value of ρ, Eq.(32) provides a closed relation between the temperature and the chemical potential, μ = μ(T).

Fig. 4. Wave condensation in two-dimensions beyond the thermodynamic limit: Fraction of condensed power N 0/N vs T/Tc calculated by solving the coupled Eqs.(32,33). Even for small values of the area A, the condensation curve exhibits properties analogous to those of the three-dimensional condensation in the thermodynamic limit, as described by Eq.(31).

Let us now analyze the power condensed in the fundamental mode k = 0 by decomposing the sum as follows, N = N 0 + T∑′k 1/[α(kx 2 + ky 2) - μ], where N 0 = T/(-μ) and ∑′k excludes the origin k = 0. The fraction of condensed power then reads

N0N=1Tkx,ky1[α(kx2ky2)μ(T)].
(33)

This equation provides a closed relation between N 0/N and T, where the function μ(T) is given through Eq.(32). The fraction of the condensed power N 0/N of Eq.(33) has been evaluated numerically and the results are represented in Fig. 4 for different values of the system size A. The figure remarkably reveals that the fundamental mode k = 0 becomes macroscopically occupied at low temperature, as for a genuine condensation process. More precisely, for increasing values of A, the condensation curve N 0/N tends to a straight line

N0N1TTc,
(34)

where the critical temperature Tc = ρA/[∑′k 1/(αk 2)] is obtained by setting μ = 0 in Eq.(32). This remarkably reveals that, even for small values of A, the condensation curve exhibits properties analogous to those of the three-dimensional condensation in the thermodynamic limit [see Eq.(31]. The expression of the critical temperature Tc = ρA/[∑′k/(αk 2)] clearly shows that the discrete sum in frequency space provides a non-vanishing value of Tc, while Tc tends to zero in the thermodynamic limit [∑k → (A/4π 2) ∫ d k] because of the (infrared) logarithmic divergence of the continuous integral ∫d k/k 2.

In summary, the critical behavior of the two-dimensional condensation curve looks similar to that of a genuine “phase transition”. Note however that, strictly speaking, “phase transitions” only occur in the thermodynamic limit, so that such terminology is not appropriate for the two-dimensional problem considered here. Nevertheless, if one takes the macroscopic occupation of the fundamental mode k = 0 as the essential characteristic of condensation, then one may conclude that wave condensation do occur in two dimensions.

The existence of the condensation process in 2D has been confirmed by the numerical simulations of Eq.(6). The numerical results have been reported in Fig. 5. Each point (◇) was obtained by averaging N 0/N over 1000 length units, once the equilibrium state was reached. The theoretical curve N 0/N vs H has been obtained by adapting the 3D-theory developed in Ref.[19

19. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of Classical Nonlinear Waves,” Phys. Rev. Lett. 95, 263901 (2005). [CrossRef]

] to the two-dimensional problem considered here. One obtains the following closed relation between the total energy and the fraction of condensed power

Fig. 5. Wave condensation in two-dimensions: the continuous line refers to the condensation curve calculated theoretically [Eq.(35)], and points (◇) refer to the numerical simulation of the 2D NLS Eq.(6). Each point (◇) corresponds to an averaging of N 0/N over 1000 length units, once the equilibrium state is reached. (The spatial discretization of the normalized NLS Eq.(6) is dx = 1/2, the power density is N/A = 1/4, the number of points 322).
HA=ρ2+(ρρ0)22+(ρρ0)k1k[(αk2+ρ0)ωB2(k)],
(35)

where ωB(k) = (a 2 k 4 + 2αgρ 0 k 2)1/2 refers to the (Bogoliubov’s) dispersion relation of small-scale fluctuations superposed to the plane-wave amplitude |ψ|=ρ0, ρ 0 = N 0/A being the density of condensed power.

As remarkably illustrated in Fig. 5, the numerical results are in quantitative agreement with the theory, without any adjustable parameter. Note however an appreciable discrepancy near by the transition point, at H = Hc ≈ 1.7. Such a disagreement may be ascribed to the assumption μ = 0 (for HHc) inherent to the derivation of Eq.(35). Indeed, as discussed above through Eqs.(32,33), a non vanishing chemical potential makes the transition to condensation “smoother”, in qualitative agreement with the numerical results reported in Fig. 5. Important to note, a detailed analysis of Eq.(35) reveals that, as for the 3D case, the nonlinear interaction changes the nature of the transition to condensation, which becomes of the first order. However, such a subcritical behavior has not been identified in the numerical simulations. Work is currently in progress in order to elucidate this important problem.

To conclude, let us make the general remark that classical wave condensation is in some sense reminiscent of the genuine Bose-Einstein condensation of quantum systems [81

81. R. Lacaze, P. Lallemand, Y. Pomeau, and S. Rica, “Dynamical formation of a BoseEinstein condensate,” Physica D 152–153, 779–786 (2001). [CrossRef]

]. Indeed, it is worth noting that the equilibrium distribution (26) may be deduced from the quantum Bose distribution in the limit where the modes are highly occupied, n k ≫ 1. More precisely, the kinetic wave Eq.(19,20) could be derived from the quantum kinetic equation governing a quantum Bose gas in the limit n k ≫ 1. Note that, in the opposite limit where the modes are scarcely occupied n k ≪ 1, the quantum kinetic equation recovers the Boltzmann’s equation describing a classical gas far from equilibrium. The idea that highly occupied modes of a quantum Bose field are well described by a classical field is well known, e.g., in laser physics, where highly occupied modes are described by classical equations. In this way, it has been proposed that a modified (“projected”) NLS equation can be used to model the nonequilibrium dynamics of finite-temperature Bose gases [82

82. M. J. Davis, S. A. Morgan, and K. Burnett, “Simulations of Bose Fields at Finite Temperature,” Phys. Rev. Lett. 87, 160402 (2001). [CrossRef] [PubMed]

, 83

83. M. J. Davis, S. A. Morgan, and K. Burnett, “Simulations of thermal Bose fields in the classical limit,” Phys. Rev. A 66, 053618 (2002). [CrossRef]

]. In line with the quantum aspects of light-field condensation, let us finally mention the recently introduced concept of “photon fluid” [84

84. 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 (1999). [CrossRef]

, 85

85. R. Y. Chiao, T. H. Hansson, J. M. Leinaas, and S. Viefers, “Effective photon-photon interaction in a two-dimensional photon fluid,” Phys. Rev. A 69, 063816 (2004). [CrossRef]

]. It is shown that the Bogoliubov’s dispersion relation for the elementary excitations of the weakly interacting Bose gas holds for the case of the weakly interacting photon gas in a nonlinear Fabry-Perot cavity. In this context, the chemical potential of a photon in the cavity does not vanish. One indication of the existence of a photon superfluid would be that there exists a critical transition from a dissipationless state of superflow, i.e., a laminar flow of the photon fluid below a critical velocity past an obstacle, into a turbulent state of superflow associated to the generation of quantized vortices [86

86. T. Frisch, Y. Pomeau, and S. Rica “Transition to dissipation in a model of superflow,” Phys. Rev. Lett. 69, 1644 (1992). [CrossRef] [PubMed]

, 87

87. N. G. Berloff and B. V. Svistunov “Scenario of strongly nonequilibrated Bose-Einstein condensation,” Phys. Rev. A 66, 013603 (2002). [CrossRef]

, 88

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

].

6. Perspectives and conclusion

Considering the NLS equation as a representative model of optical wave propagation, we showed that the assumption of (quasi-)Gaussian statistics allows one to derive a kinetic equation describing the evolution of the spectrum of the optical field. The structure of the kinetic equation depends crucially on the nature of the statistics of the field. When the field exhibits an inhomogenous statistics, the closure of the hierarchy of moment equations is achieved through a first order perturbation theory in ε = Hnl/Hl ≪ 1. The corresponding kinetic equation preserves the “time” reversal symmetry (i.e., they are invariant with respect to the direction of propagation) of the original NLS equation. Accordingly, the kinetic equations could not describe an irreversible evolution of the optical field towards an equilibrium state. Conversely, if the field exhibits a homogenous statistics, the closure of the moment equations requires a second-order perturbation theory in ε. The corresponding kinetic equation is characterized by the presence of a collision term describing an irreversible evolution towards equilibrium. Important to note, if the nonlinear medium is characterized by a noninstantaneous response time, the hierarchy of moment equations results to be closed at the first order in ε, so that the corresponding kinetic equation does not describe thermal wave relaxation to equilibrium.

6.1. Importance of coherent phase effects in incoherent optical interactions

The weak-turbulence theory may easily be applied to the three-wave interaction describing optical wave propagation in quadratic nonlinear crystals. The corresponding kinetic equations exhibit properties similar to those of the four-wave kinetic Eq.(19,20) [39

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

]. The kinetic theory then seems to provide a genuine description of incoherent wave interaction in usual quadratic or cubic nonlinear media. In this respect, we note that this theory could be exploited to shed new light on the mechanism underlying the process of supercontinuum generation [89

89. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]

].

However, caution should be exercised when drawing hastened conclusions as regard the domain of validity of weak-turbulence theory. Let us recall that the theory is essentially based on the random phase approximation, and in this respect it is inherently unable to account for the existence of phase correlations between distinct incoherent wave-packets. There are indeed particular situations in which phase correlations spontaneously emerge in a system of fully incoherent optical waves. This remarkable feature has been identified in the resonant three-wave interaction. In this framework, the incoherence of the pump field may absorb the incoherence of the idler wave, then allowing the signal field to evolve towards a fully coherent state [15

15. A. Picozzi and M. Haelterman, “Parametric Three-Wave Soliton Generated from Incoherent Light,” Phys. Rev. Lett. 86, 2010–2013 (2001). [CrossRef] [PubMed]

, 90

90. A. Picozzi, C. Montes, and M. Haelterman, “Coherence properties of the parametric three-wave interaction driven from an incoherent pump,” Phys. Rev. E 66, 056605 (2002). [CrossRef]

, 91

91. A. Picozzi and P. Aschieri, “Influence of dispersion on the resonant interaction between three incoherent waves,” Phys. Rev. E 72, 046606 (2005). [CrossRef]

, 92

92. C. Montes, A. Picozzi, and K. Gallo, “Ultra-coherent signal output from an incoherent cw-pumped singly resonant optical parametric oscillator,” Opt. Comm. 237, 437–449 (2004). [CrossRef]

]. The spontaneous emergence of a mutual coherence between incoherent wave-packets is the essential mechanism underlying the existence of incoherent solitons in instantaneous response nonlinear media [15

15. A. Picozzi and M. Haelterman, “Parametric Three-Wave Soliton Generated from Incoherent Light,” Phys. Rev. Lett. 86, 2010–2013 (2001). [CrossRef] [PubMed]

, 16

16. A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, “Incoherent Solitons in Instantaneous Response Nonlinear Media,” Phys. Rev. Lett. 92, 143906 (2004). [CrossRef] [PubMed]

].

Along the same way, a pair of coupled incoherent nonlinear waves propagating in a Kerr medium was shown to exhibit huge oscillations of coherence, characterized by a reversible transfer of coherence between the coupled waves. This effect relies on the emergence of a phase correlation between the two waves. Such a sustained oscillatory behavior is in contradiction with the expected irreversible evolution towards equilibrium. As a result, the coherence transfer process was shown to be characterized by a reduction of nonequilibrium entropy, which violates the H-theorem of entropy growth inherent to the kinetic theory [20

20. A. Picozzi, “Nonequilibrated Oscillations of Coherence in Coupled Nonlinear Wave Systems,” Phys. Rev. Lett. 96, 013905 (2006). [CrossRef] [PubMed]

]. These recent works indicate that weak-turbulence theory should be extended so as to account for the existence of coherent phase effects in the dynamics of incoherent nonlinear fields.

6.2. Thermodynamics of a pure wave system?

Let us finally remark that the process of thermal wave relaxation to equilibrium paves the way for the study of the thermodynamics of a pure optical wave system. Let us illustrate this aspect by showing that the equilibrium distribution (26) allows one to derive the T dS equation of thermodynamics. To simplify the notations, let us focus on the purely spatial evolution of the optical field. Accordingly, the differentials of power (23), energy (24), and entropy (25), may be written as follows dN = ∑k dnk, dHl = ∑k K(k)dnk, dS = ∑k dnk/nk. Making use of the expression of the equilibrium distribution given in (26), neqk = T/[K(k) -μ], one obtains the following relation between the differentials

TdS=dHlμdN,
(36)

which corresponds to the familiar T dS equation of thermodynamics for a fixed volume of interaction [42

42. K. Huang, “Statistical Mechanics” (Wiley, 1963).

]. Furthermore, by including the conservation of (linear) momentum of the optical field, P = ∑k k nk, the equilibrium distribution reads,

nkeq=T[K(k)2v.kμ],
(37)

where v = P/N represents the spatial beam walk-off (“transverse velocity”) of the field at equilibrium [21

21. S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity Locking of Incoherent Nonlinear Wave Packets,” Phys. Rev. Lett. 97, 033902 (2006). [CrossRef] [PubMed]

]. Equation (36) then takes the form T dS = dHl - μdN -2v.d P, where the last term may be used to define an analogous of the thermodynamic pressure for a pure wave system.

Along this line, one may imagine a way to analyze the second principle of thermodynamics in a pure wave system. Consider for instance an optical wave propagating in a waveguide characterized by a large number of transverse spatial modes (large transverse area A). Let us assume that the optical wave that propagates in such waveguide has reached an equilibrium state characterized by some entropy S 1. Then suppose that the waveguide is realized in such a way that its area A(z) increases suddenly at some propagation distance z 0. For zz 0, the optical wave then relaxes irreversibly towards a novel state of equilibrium, with a novel value of entropy S 2. In analogy with thermodynamics, this experiment may be considered as the analogous of the expansion of a gas enclosed in a box, in which a piston is removed at some time t 0, the propagation distance z playing the role of time t. According to the second principle of thermodynamics, the lift of the constraint on the system “volume” implies that the entropy S 2 must be grater than the entropy S 1. As illustrated by this simple example, the kinetic wave theory then seems to open a variety of novel fascinating opportunities for the study of nonlinear optics with partially incoherent waves.

Acknowledgments

I’m especially grateful to S. Rica for illuminating discussions and for his valuable comments on the weak-turbulence theory. I thank G. Millot, S. Pitois, C. Josserand, P. Aschieri, M. Haelterman, C. Montes, S. Lagrange, and H. R. Jauslin for fruitful discussions, as well as I. Bongrand and I. Luna for their support. I also thank S. Pitois for his help in realizing the movies of Fig. 2.

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M. J. Davis, S. A. Morgan, and K. Burnett, “Simulations of Bose Fields at Finite Temperature,” Phys. Rev. Lett. 87, 160402 (2001). [CrossRef] [PubMed]

83.

M. J. Davis, S. A. Morgan, and K. Burnett, “Simulations of thermal Bose fields in the classical limit,” Phys. Rev. A 66, 053618 (2002). [CrossRef]

84.

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 (1999). [CrossRef]

85.

R. Y. Chiao, T. H. Hansson, J. M. Leinaas, and S. Viefers, “Effective photon-photon interaction in a two-dimensional photon fluid,” Phys. Rev. A 69, 063816 (2004). [CrossRef]

86.

T. Frisch, Y. Pomeau, and S. Rica “Transition to dissipation in a model of superflow,” Phys. Rev. Lett. 69, 1644 (1992). [CrossRef] [PubMed]

87.

N. G. Berloff and B. V. Svistunov “Scenario of strongly nonequilibrated Bose-Einstein condensation,” Phys. Rev. A 66, 013603 (2002). [CrossRef]

88.

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

89.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]

90.

A. Picozzi, C. Montes, and M. Haelterman, “Coherence properties of the parametric three-wave interaction driven from an incoherent pump,” Phys. Rev. E 66, 056605 (2002). [CrossRef]

91.

A. Picozzi and P. Aschieri, “Influence of dispersion on the resonant interaction between three incoherent waves,” Phys. Rev. E 72, 046606 (2005). [CrossRef]

92.

C. Montes, A. Picozzi, and K. Gallo, “Ultra-coherent signal output from an incoherent cw-pumped singly resonant optical parametric oscillator,” Opt. Comm. 237, 437–449 (2004). [CrossRef]

OCIS Codes
(030.6600) Coherence and statistical optics : Statistical optics
(190.0190) Nonlinear optics : Nonlinear optics
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

ToC Category:
Theoretical Concepts and Methods

History
Original Manuscript: February 8, 2007
Revised Manuscript: May 9, 2007
Manuscript Accepted: May 11, 2007
Published: July 9, 2007

Virtual Issues
Focus Serial: Frontiers of Nonlinear Optics (2007) Optics Express

Citation
Antonio Picozzi, "Towards a nonequilibrium thermodynamic description of incoherent nonlinear optics," Opt. Express 15, 9063-9083 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-14-9063


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  88. S. Nazarenko and M. Onorato "Wave turbulence and vortices in Bose-Einstein condensation," Physica D 219, 1 (2006). [CrossRef]
  89. J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006). [CrossRef]
  90. A. Picozzi, C. Montes, and M. Haelterman, "Coherence properties of the parametric three-wave interaction driven from an incoherent pump," Phys. Rev. E 66, 056605 (2002). [CrossRef]
  91. A. Picozzi and P. Aschieri, "Influence of dispersion on the resonant interaction between three incoherent waves," Phys. Rev. E 72, 046606 (2005). [CrossRef]
  92. C. Montes, A. Picozzi and K. Gallo, "Ultra-coherent signal output from an incoherent cw-pumped singly resonant optical parametric oscillator," Opt. Comm. 237, 437-449 (2004). [CrossRef]

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