Condensation and thermalization of classsical optical waves in a waveguide

We consider the long-term evolution of a random nonlinear wave that propagates in a multimode optical waveguide. The optical wave exhibits a thermalization process characterized by an irreversible evolution toward an equilibrium state. The tails of the equilibrium distribution satisfy the property of energy equipartition among the modes of the waveguide. As a consequence of this thermalization, the optical field undergoes a process of classical wave condensation, which is characterized by a macroscopic occupation of the fundamental mode of the waveguide. Considering the nonlinear Schr\"odinger equation with a confining potential, we formulate a wave turbulence description of the random wave into the basis of the eigenmodes of the waveguide. The condensate amplitude is calculated analytically as a function of the wave energy, and it is found in quantitative agreement with the numerical simulations. The analysis reveals that the waveguide configuration introduces an effective physical frequency cutoff, which regularizes the ultraviolet catastrophe inherent to the ensemble of classical nonlinear waves. The numerical simulations have been performed in the framework of a readily accessible nonlinear fiber optics experiment.