Abstract : The nonlinear propagation of coherent optical fields has been extensively explored in the framework of nonlinear optics, while the linear propagation of incoherent fields has been widely studied in the framework of statistical optics. However, these two fundamental fields of optics have been mostly developed independently of each other, so that a satisfactory understanding of statistical nonlinear optics is still lacking. This article is aimed at reviewing a unified theoretical formulation of statistical nonlinear optics on the basis of the wave turbulence theory, which provides a nonequilibrium thermodynamic description of the system of incoherent nonlinear waves. We consider the nonlinear Schrödinger equation as a representative model accounting either for a nonlocal or a noninstantaneous nonlinearity, as well as higher-order dispersion effects. Depending on the amount of nonlocal (noninstantaneous) nonlinear interaction and the amount of inhomogeneous (nonstationary) statistics of the incoherent wave, different types of kinetic equations are derived and discussed.
In the spatial domain, when the incoherent wave exhibits inhomogeneous statistical fluctuations, different forms of the (Hamiltonian) Vlasov equation are obtained depending on the amount of nonlocality. This Vlasov approach describes the processes of incoherent modulational instability and localized incoherent soliton structures.
In the temporal domain, the causality property inherent to the response function leads to a kinetic formulation analogous to the weak Langmuir turbulence equation, which describes nonlocalized spectral incoherent solitons. In the presence of a highly noninstantaneous response, this formulation reduces to a family of singular integro-differential kinetic equations (e.g., Benjamin–Ono equation), which describe incoherent dispersive shock waves. Conversely, a non-stationary statistics leads to a (non-Hamiltonian) long-range Vlasov formulation, whose self-consistent potential is constrained by the causality condition of the response function.
The spatio-temporal domain will be considered in the limit of an inertial nonlinearity. We review different theories developed to describe bright and dark spatial incoherent solitons experimentally observed in slowly responding nonlinear media: The coherent density method, the mutual coherence function approach, the modal theory and the Wigner–Moyal formulation.
When the incoherent wave exhibits homogeneous fluctuations, the relevant kinetic equation is the wave turbulence (Hasselmann) equation. It describes wave condensation and the underlying irreversible process of thermalization to thermodynamic equilibrium, as well as genuine nonequilibrium turbulent regimes. In this way different remarkable phenomena associated to wave thermalization are reviewed, e.g., wave condensation or supercontinuum generation in photonic crystal fibers, as well as different mechanisms of breakdown of thermalization. Finally, recent developments aimed at providing a wave turbulence formulation of Raman fiber lasers and passive optical cavities are reviewed in relation with condensation-like phenomena.