Condensation of classical optical waves beyond the cubic nonlinear Schrödinger equation
Introduction
The propagation of incoherent nonlinear optical waves is a subject that is attracting a growing interest in various different fields of investigations, such as, e.g., wave propagation in homogeneous [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18] or periodic media [19], nonlinear imaging [20], cavity systems [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], or nonlinear interferometry [31]. In particular, the study of the long-term evolution of partially coherent nonlinear optical waves has been considered in various circumstances [3], [4], [13], [19], [32], [33], [34], as well as in various optical media characterized by different nonlinearities [2], [14], [15], [17]. When the propagation of the optical wave is almost conservative (the dissipation can be neglected) and it can be accurately described by a Hamiltonian model equation (e.g., NonLinear Schrödinger (NLS) equation), then the evolution of the incoherent wave is characterized by a nonequilibrium process of thermalization [8], [12], [13], [14], [16], [17]. This phenomenon of optical wave thermalization can be interpreted in analogy with the kinetic gas theory: the incoherent wave exhibits an irreversible evolution toward the thermodynamic equilibrium state, i.e., the Rayleigh–Jeans (RJ) equilibrium distribution that realizes the maximum of entropy (‘disorder’).
Nonlinear waves manifest themselves in a myriad of different forms, such as, e.g., sound, elastic, vibrational, surface or electromagnetic waves. However, in most of these wave systems dissipation plays a non-negligible role in the wave evolution. But electromagnetic light waves in nonlinear optics constitutes a promising field of investigation of thermal wave relaxation because of the availability of low-loss nonlinear media in which light propagation is accurately ruled by NLS-like equations over long distances (see, e.g., Ref. [35]).
A remarkable property of the phenomenon of thermalization is that it can be characterized by a ‘self-organization process,’ in the sense that it is thermodynamically advantageous for the optical wave to generate a large-scale coherent structure in order to reach the most disordered equilibrium state [36]. An important example of this type of self-organization is provided by the condensation of classical nonlinear waves in a defocusing Kerr-like nonlinear medium [1], [37], [38], [39], [40], [41], [42], [43]. Optical wave condensation is characterized by the spontaneous formation of an almost plane-wave starting from an initial incoherent field: the plane-wave solution (‘condensate’) remains immersed in a sea of small scale fluctuations (‘uncondensed quasiparticles’), which store the information necessary for the reversible evolution of the wave. The wave turbulence (WT) theory [38], [44], [45], [46], [47], [48], [49], [50], [51] is known to provide a detailed description of wave condensation, in both the weakly and the highly nonlinear regimes of condensation [41]. WT theory reveals that the thermodynamic properties of this condensation process are analogous to those of the genuine Bose–Einstein condensation [40], despite the fact that the considered optical wave is completely classical. Contrary to the recent observation of Bose–Einstein condensation of photons [52], wave condensation refers here to a pure classical regime. This is because the occupation number of photons is large compared to unity (see Section 2.3), and because the wave number range associated to the quantum behavior is much larger than the wave number range involved in an experiment of classical wave condensation (see Ref. [43]). We also note that, because this condensation process results from the natural thermalization of the optical wave to thermal equilibrium, it is of a different nature than the condensation processes recently discussed in dissipative optical cavities [21], [22], [27].
This phenomenon of classical wave condensation has been essentially studied in the framework of the NLS equation in the presence of a pure cubic Kerr nonlinearity [40], [41], [50], [53], [54], [55], [56], [57], [58]. In many cases, however, realistic optical experiments are not modelled by a cubic Kerr nonlinearity. An example is provided by the recent work [43], which reported experimental evidence of the phenomenon of wave condensation. Our aim in this Letter is to show that optical wave condensation can take place with more complex nonlinearities. We consider the examples of the nonlocal nonlinearity and of the saturable nonlinearity, which refer to natural extensions of the cubic nonlinearity that have been the subject of many studies in the nonlinear optics literature. A spatial nonlocal wave interaction means that the response of the nonlinearity at a particular point is not determined solely by the wave intensity at that point, but also depends on the wave intensity in the neighborhood of this point. Nonlocality, thus, constitutes a generic property of a large number of nonlinear wave systems [59], [60], [61], [62], [63], and the dynamics of nonlocal nonlinear waves has been widely investigated in this last decade [64], [65], [66], [67], [68]. On the other hand, a saturable nonlinearity is encountered whenever the optical field in a material is strong enough so that the higher-order nonlinearities become non-negligible [35].
We show that the generalized NLS equation accounting for a nonlocal or a saturable nonlinearity describes a process of wave condensation completely analogous to that described in the framework of the cubic Kerr nonlinearity. We extend the previous works [40], [41] in which the cubic nonlinearity was considered and derive analytical expressions of the condensate fraction, in both the weakly and the strongly nonlinear regimes of propagation. For both the saturable and the nonlocal nonlinearity, we obtain a quantitative agreement with the numerical simulations, without any adjustable parameter. We also show that the condensate amplitude exhibits strong fluctuations near by the transition to condensation, while the fluctuations are suppressed in the highly condensed regime. We believe that this work will stimulate new optical experiments aimed at studying the condensation of light in the framework of different accessible optical settings.
Section snippets
NLS model equation
In this section we study the transverse spatial evolution of a partially coherent wave that propagates in a nonlinear medium characterized by a nonlocal nonlinear response. A nonlocal nonlinear response is found in several systems such as,e.g., atomic vapors [59], nematic liquid crystals [60], thermal susceptibilities [61], [62] and plasmas physics [63]. For this reason the dynamics of nonlocal nonlinear waves has been widely investigated [64], [66], [67], [68]. We consider here the standard
NLS model equation
In this Section we study the case of a saturable nonlinearity by following the same presentation as that outlined above for the nonlocal nonlinearity. We consider the standard NLS model equation describing the propagation of on optical wave in a saturable nonlinear medium [35]where is the nonlinear law of the intensity of the optical wave, . Although may have different forms, we consider here the concrete example of a rational nonlinear saturation which
Discussion and conclusion
In summary, we have shown theoretically and numerically that the NLS equation accounting for a nonlocal or a saturable nonlinearity describes a process of wave condensation. The condensation curves derived analytically in both the small and strong condensation regimes have been found in quantitative agreement with the numerical simulations, without adjustable parameters. It is interesting to briefly comment the influence of the range of a nonlocal interaction on the process of thermalization.
Acknowledgments
The authors thank J.W. Fleischer for stimulating the study of wave condensation in the framework of generalized NLS model equations. SR is on leave from the Institute Non Linéaire de Nice (CNRS, France). He also acknowledges FONDECYT grant No1100289.
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