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]. 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]. The incoherent soliton consists of a phenomenon of self-trapping of spatially and temporally incoherent light in a noninstantaneous response nonlinear medium [3]. The remarkable simplicity of experiments performed in photorefractive media allowed for a fruitful investigation of the dynamics of incoherent nonlinear waves [4, 5], as witnessed by several important achievements, such as, e.g., the modulational instability of incoherent optical fields [6, 7, 8, 9], the existence of incoherent dark solitons [10, 11], incoherent pattern formation [12], or incoherent solitons in periodic lattices [13, 14]. 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, 16, 17, 18, 19, 20, 21, 22].

 

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.

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Several theories have been developed to provide a description of incoherent solitons in slowly responding nonlinear media [4, 5]. The most established methods are the mutual coherence function approach [23], the self-consistent multimode theory [24], the coherent density method [25], and the Wigner’s transform approach [26]. All these methods may be considered as nonlinear generalizations of previous classical methods for analyzing linear propagation of partially coherent light [27]. It was recently shown that these four methods are in fact equivalent [28, 29] and the choice of the most suitable representation rather depends on the nature of the physical problem to be investigated. Note that, in addition to these theories, a simplified ray-optics approach was suggested in Ref.[30], which is in some respects similar to that developed to describe random-phase solitons in plasma physics [31, 32].

Although these theoretical approaches proved efficient and powerful, they do not provide an answer to the important question regarding the long term evolution of a partially incoherent optical field propagating in usual nonlinear media. Indeed, in complete analogy with a system of classical particles, one may expect that the incoherent optical field could evolve, owing to nonlinearity, towards a thermodynamic equilibrium state (see Fig. 1). Thermal wave relaxation to equilibrium is actually a fundamental question related to the vast issue of fully developed turbulence, which still constitutes a challenging unsolved problem. However, a considerable simplification of the problem takes place when wave propagation is essentially dominated by linear dispersive effects, so that a (weakly nonlinear) kinetic description of the field becomes possible [33, 34, 35]. The kinetic wave theory has been the subject of a detailed investigation in the context of plasma physics [36, 37, 38], and then generalized in the framework of the weak-turbulence theory [39, 40, 41]. For most purposes the kinetic theory is based on the random phase approximation, which is justified when phase information becomes irrelevant to the wave interaction due to the strong tendency of the waves to decohere. The random phases can thus be averaged out to obtain a weak turbulence description of the incoherent wave interaction, which is formally based on irreversible kinetic equations [39]. It results that, in spite of the formal reversibility of the equation governing wave propagation, the kinetic equation describes an irreversible evolution of the field to thermodynamic equilibrium. The mathematical statement of this irreversible behavior relies on the H-theorem of entropy growth, whose origin is analogous to the Boltzmanns H-theorem relevant for gas kinetics [42].

A conspicuous prediction of weak-turbulence theory is the existence of a condensation process for the nonlinear optical field [40, 43, 19]. 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], 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], 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, 44, 45, 46].

The paper is organized as follows. We begin by exposing the problem of achieving a closure of the second-order moment equation for the optical field. It is shown that the closure depends crucially on whether the field is assumed to exhibit a homogenous or an inhomogenous statistics. In the case of inhomogenous statistics, the kinetic equation is shown to take a form analogous to the Vlasov’s equation (Sec. 2), while it takes the form of the Boltzmann’s equation when the field exhibits a homogenous statistics (Sec. 3). In Sec. 4 we review the essential properties of the condensation process of an optical field propagating in three dimensions. We next show that, provided one goes beyond the standard thermodynamic limit, wave condensation is shown to occur in the pure two-dimensional case, a remarkable feature confirmed by the numerical simulations. Finally, Sec. 5 gives a summary of our results, and discusses some interesting related issues. In particular, it is shown that the fundamental T dS equation of thermodynamics may be easily derived from the equilibrium distribution inherent to weak-turbulence theory.

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, 45, 46]

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 v_g. The diffraction parameter reads $\alpha =\frac{1}{2k_0}$ , and the dispersion parameter $\beta =\frac{1}{2}\frac{{\partial }^{2}k}{\partial {\omega }^2}{|}_{{\omega }_0}$ . 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 L_α = λ_c ²/α and the dispersive length L_β = τ_c ²/|β|, where τ_c and λ_c 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

and the total energy

which has a linear (dispersive) kinetic contribution

and a nonlinear contribution

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]. 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

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]

where ψ_j = 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, 34, 35].

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,β ≪ L_nl or ε = H_nl/H_l ≪ 1 [39]. Under these conditions, one may exploite the property of factorizability of stochastic Gaussian fields [48], 〈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)〉,

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, 5], 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 τ_R is much greater than the time correlation of the field (t_c ≪ τ_R), the medium responds to the time-averaged intensity and not to the instantaneous speckles that constitute the incoherent beam [4, 5]. Accordingly, the nonlinear index n_nl 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

where the nonlinear term involves the averaged power Ñ(z, r) = 〈|ψ|²〉 (z, r). Let us remark the important aspect that, because of the noninstantaneous nonlinearity, Gaussian statistics is preserved under nonlinear evolution of the field [49]. Now, we emphasize that, in contrast with the instantaneous response NLS Eq.(6), equation (9) does not lead to an infinite hierarchy of moment equations. Indeed, the derivation of the second-order moment equation from (9) does not require any additional assumption on the nature of the statistics of the field. In other terms, the presence of the averaged power in Eq.(9) leads to an exact closure of the second-order moment equation. As discussed in the next Section, because of this property, the equation governing the evolution of the spectrum does not describe an irreversible evolution of the field towards an equilibrium state.

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 = - 2α∇_r.∇_y B [27].

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

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, 26]

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 ∂_kn_k ≈ n_k/λ_c and ∂_rÑ ≈ ÑÑ/λ_inh, where λ_inh, denotes the characteristic length of inhomogeneity of the statistics of the field. Accordingly, the second term of the sum may be neglected whenever λ_inh ≫ λ_c [50]. Under this assumption of quasi-homogenous statistics, one obtains the following Vlasov’s like equation governing the evolution of the spectrum of the field

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]. 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

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

with the effective Hamiltonian

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 V_eff(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 + V_eff, which results to be the sum of the original dispersion relation ω ₀(k) = αk ² and of the mean-field potential V_eff.

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 V_eff(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, 5, 24]. More specifically, a soliton forms when the optical beam induces an effective attractive potential V_eff < 0 (waveguide) owing to a focusing nonlinearity (g < 0). In turn, the optical beam is guided in its own induced potential V_eff. Along this line, it was remarkably shown that incoherent solitons may even exhibit a dark structure [10,11,51, 52,53, 54]. 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, 56, 57, 58, 59, 60, 61, 62, 63, 64].

The study of modulational instability of incoherent light in focusing media revealed that a statistically homogenous incoherent field is unstable with respect to weakly statistical inhomogenities [6, 7, 8, 9, 65, 66]. Remarkably, this study revealed that wave incoherence may suppress modulational instability. The existence of a threshold for incoherent modulational instability was shown to be formally related to the existence of an effective Landau’s damping [26]. Incoherent modulational instability in instantaneous response nonlinear media was also demonstrated experimentally in optical fibers [18]. In this respect, the presence of the factor 2 in front of the nonlinear term in Eq.(8) was shown to lead to a modulation frequency that is substantially increased with respect to the corresponding value of coherent modulational instability. Note that this property was not reported in previous works of incoherent modulational instability in instantaneous response Kerr media [67, 68], because in those works continuous-wave partially incoherent beams were considered. Because those incoherent waves are free from intensity fluctuations, the modulational instability that they induce does not exhibit an enhancement of the modulation frequency [18].

We finally note that, as for the original NLS Eq.(6), the kinetic Eq.(11) as well as the Vlasov’s Eq.(12,13) are reversible with respect to the propagation direction z [they are invariant under the transformation (z, r, k) → (- z, r, - k)]. This important property significates that these kinetic equations could not describe an irreversible evolution of the field towards an equilibrium state. The process of thermal wave relaxation to equilibrium is the subject of the next Section.

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]. 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, 41]. The derivation is essentially based on a perturbation expansion theory in which linear dispersive effects dominate nonlinear effects, L _α,β ≪ L_nl, i.e., ε = H_nl/H_l ≪ 1. Accordingly, an effective large separation of the linear and the nonlinear lengths scales takes place [33, 34, 35]. 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].

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], 〈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 (k_x,k_y).

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

where J ^k,l _i,j = 〈ψ̃(k _i,ω_i, z)ψ̃^*(k _j, ω_j,z)ψ̃(k _k, ω_k,z)ψ̃^*(k _l,ω_l,z)〉 refers to the fourth-order moment of the field, J̄ = J ^* being the complex conjugate of J, and δ _i,j;k,l = δ _{ωi+ωj-ωk-ωl} δ _ki+kj-kk-kl, d k _1-3 = d k ₁ d k ₂ d k ₃, dω _1-3 = dω ₁ dω ₂ dω ₃. It is apparent from Eq.(18) that the second-order moment of the field depends on the fourth-order moment. If one assumes that the field obeys a Gaussian statistics, the right-hand side of Eq.(18) vanishes exactly, simply because the statistics of the field has been assumed to be homogenous, which corroborates the results of the previous Section. Accordingly, in contrast with the inhomogenous statistics problem, the closure of the moment equations for statistically homogenous fields requires a second order perturbation theory in ε = H_nl/H_l ≪ 1.

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

where the collision term reads

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

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

K = k_z 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]. 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

and the kinetic (linear) energy

where V refers to the system “volume” (V = AT_p, A referring to the characteristic beam area and T_p to the pulse duration). Let us remark that Eq.(19–21) do not conserve the total energy H, but only its linear contribution H_l. This results from the fact that the nonlinear energy has a negligible contribution in the perturbation expansion procedure of the kinetic theory (ε = H_nl/H_l ≪ 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, 70]. 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, 70].

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

As in standard statistical mechanics, the thermodynamic equilibrium state is determined from the postulate of maximum entropy [42]. The equilibrium spectra n ^eq(k,ω) realizing the maximum of S[n(k,ω)], subject to the constraints of conservation of H_l and N, may readily be calculated by introducing the corresponding Lagrange’s multipliers, 1/T and -μ/T. One obtains

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[n^eq] = 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 n^eq(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, 19, 40, 21], in which the constants T and μ can be determined from the conserved quantities H_l 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

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]).

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 $\omega \simeq \pm \sqrt{\frac{\alpha }{\mathrm{\beta k}}}$ . This confers a hyperbolic structure to the dispersion relation K(ω,k) [4]. Note that this property is at the origin of the existence of nonlinear X-waves [71, 72, 73]. 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]. 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, 75].

5. Condensation of classical optical waves

In the following we shall see that a classical optical wave exhibits a condensation process [40, 76], whose thermodynamic properties are analogous to those of the genuine Bose-Einstein condensation [19], 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), n^eq _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: z → zL_nl,t → tτ ₀,$\overrightarrow{r}\to \overrightarrow{r}{\lambda }_0$, with τ ₀ = (|β| L_nl)^1/2 and λ ₀ = (αL_nl)^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(k_x,ω,z) =|ψ̃|² (k_x,k_y = 0, ω,z), or S(k_y,ω,z) =|ψ̃|² (k_x = 0,k_y,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 500L_nl. (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).

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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 = H_l +H_nl [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 H_l, 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 H_l is necessarily accompanied by a reduction of the nonlinear contribution H_nl, as confirmed by the numerical simulations (see Fig. 3c-d). The lower the nonlinear contribution H_nl, the higher the kinetic contribution H_l. Actually, H_nl 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 H_nl and the conserved total energy H [77, 78, 79].

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] 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[n_k^eq] = 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 H_l and the power N in the limits k → ∞ and ω) → ∞. To regularize such unphysical divergence, we introduce an ultraviolet cut-off k_c and ω_c. 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]. 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 H_nl to the linear contribution H_l, while the total energy H = H_nl +H_l 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).

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where we assumed ${\omega }_c=k_c\sqrt{\frac{\alpha }{\mid \beta \mid }}$ so as to get an explicit analytical expression for N and H_l. 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 τ_c and length λ_c 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 H_l), may be calculated by setting μ = 0 in the equilibrium distribution (26). Note that the assumption μ = 0 for T ≤ T_c can be justified rigorously in the thermodynamic limit (i.e., V → ∞, N → ∞, keeping N/V constant). One readily obtains (N - N ₀)/V = 4πTk_c/α |β|^1/2 and H_l/V = 4πTk_c ³/3α |β|^1/2, which gives

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,

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

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

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 - k_c 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]. In other terms, the critical temperature T_c 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 dk_x = d_ky = 2π/√A. The expression for the power of the optical field at equilibrium (23) may thus be written in the following form

where ρ = N/A is the power density, and ∑_kx,ky denotes the discrete sum for - k_c ≤ k_x, k_y ≤ k_c. 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/T_c 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).

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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(k_x ² + k_y ²) - μ], where N ₀ = T/(-μ) and ∑′_k excludes the origin k = 0. The fraction of condensed power then reads

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

where the critical temperature T_c = 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 T_c = ρA/[∑′_k/(αk ²)] clearly shows that the discrete sum in frequency space provides a non-vanishing value of T_c, while T_c 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] 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).

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where ω_B(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 $\left|\psi \right|=\sqrt{{\rho }_0}$, 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 = H_c ≈ 1.7. Such a disagreement may be ascribed to the assumption μ = 0 (for H ≤ H_c) 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]. 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, 83]. In line with the quantum aspects of light-field condensation, let us finally mention the recently introduced concept of “photon fluid” [84, 85]. 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, 87, 88].

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 = H_nl/H_l ≪ 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.

We reviewed some essential properties of the process of classical wave condensation in three dimensions. We showed that, provided one goes beyond the standard thermodynamic limit, wave condensation is re-established in the pure two-dimensional case. Weak-turbulence theory proved efficient and powerful in describing the condensation process, as attested by the agreement with the numerical simulations of the NLS equation (see Ref.[19] and Fig. 5 above). Besides the process of wave condensation, the kinetic theory also allowed to elucidate an intriguing process of velocity-locking of incoherent waves, which was shown to be fundamentally related to the process of thermal wave relaxation to equilibrium [21,70]. This effect of velocity-locking has been observed experimentally in an optical fiber system [21].

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]. 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].

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, 90, 91, 92]. 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, 16].

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]. 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 dn_k, dH_l = ∑_k K(k)dn_k, dS = ∑_k dn_k/n_k. Making use of the expression of the equilibrium distribution given in (26), n^eq_k = T/[K(k) -μ], one obtains the following relation between the differentials

which corresponds to the familiar T dS equation of thermodynamics for a fixed volume of interaction [42]. Furthermore, by including the conservation of (linear) momentum of the optical field, P = ∑_k k n_k, the equilibrium distribution reads,

where v = P/N represents the spatial beam walk-off (“transverse velocity”) of the field at equilibrium [21]. Equation (36) then takes the form T dS = dH_l - 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 z ≥ z 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.