Thermalization of incoherent nonlinear waves: From incoherent solitons to a thermodynamic description of statistical nonlinear optics

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 dynamics of incoherent optical solitons. However, such theories leave unanswered the question regarding the long term evolution of a partially incoherent optical field propagating in a nonlinear medium. In analogy with kinetic gas theory, an incoherent optical field evolves, owing to nonlinearity, towards a thermodynamic equilibrium state. Wave turbulence theory describes the essential properties of this irreversible process of thermal wave relaxation to equilibrium. Such irreversible behavior is expressed by a H -theorem of entropy growth, whose origin is analogous to the celebrated Boltzmann’s H -theorem. In this review we analyze the thermalization of incoherent nonlinear optical fields in various circumstances. We shall begin with the simplest case where the optical field is ruled by the scalar NonLinear Schrödinger (NLS) equation. In this case wave thermalization is characterized by a condensation process of the optical field, whose thermodynamic properties are analogous to those of Bose-Einstein condensation, despite the fact that the considered optical wave is completely classical. We then study the thermalization of an ensemble of incoherent fields governed by the vector NLS equation. The influence of the relative intensity of the coupled fields reveals the existence of a process of coherence absorption: The condensation of the optical field is induced by the presence of another small-amplitude field, which absorbs almost all the noise of the system. Such a coherence absorption effect is shown to also occur in thermal quantum Bose gases. The influence of convection (group-velocity difference) on the thermalization process is also analyzed. A set of incoherent wave-packets is shown to irreversibly evolve towards an equilibrium state in which they propagate with an identical group-velocity. This velocity-locking effect has a thermodynamic origin and thus appears as a generic property of a system of incoherent nonlinear waves. The influence of a coherent coupling on wave thermalization leads to an unexpected process of spontaneous polarization of unpolarized incoherent light, without loss of energy. Finally, the influence of higher-order dispersion effects on optical thermalization