Wave condensation with weak disorder versus beam self-cleaning in multimode fibers

Classical nonlinear waves exhibit a phenomenon of condensation that results from the natural irreversible process of thermalization toward the Rayleigh-Jeans equilibrium spectrum. Wave condensation originates in the divergence of the thermodynamic equilibrium Rayleigh-Jeans distribution, which is responsible for the macroscopic population of the fundamental mode of the system. Several recent experiments revealed a remarkable phenomenon of spatial organization of an optical beam that propagates through a graded-index multimode optical fiber (MMF), a phenomenon termed beam self-cleaning. Our aim in this article is to provide some physical insight into the mechanism underlying optical beam self-cleaning through the analysis of wave condensation in the presence of structural disorder inherent to MMFs. We consider experiments of beam self-cleaning where long pulses are injected and populate many modes of a 10-20 m MMF, for which the dominant contribution of disorder originates from polarization random fluctuations (weak disorder). On the basis of the wave turbulence theory, we derive nonequilibrium kinetic equations describing the random waves in a regime where disorder dominates nonlinear effects. The theory reveals that the presence of a conservative weak disorder introduces an effective dissipation in the system, which is shown to inhibit wave condensation in the usual continuous wave turbulence approach. On the other hand, the experiments of beam cleaning are described by a discrete wave turbulence approach, where the effective dissipation induced by disorder modifies the regularization of wave resonances, which leads to an acceleration of condensation that can explain the effect of beam self-cleaning. By considering different models of weak disorder in MMFs, we show that a model where the modes experience a partially correlated noise is sufficient to accelerate the thermalization, whereas a fully mode-correlated noise does not lead to a dissipation-induced acceleration of condensation. The simulations are in quantitative agreement with the theory, and evidence an effect of beam cleaning even in a regime of moderate weak disorder. At the leading order linear regime, random mode coupling among degenerate modes (strong disorder) can enforce thermalization and condensation. The analysis also reveals that the effect of beam cleaning is characterized by a partial repolarization as a natural consequence of the condensation process. In addition, the discrete wave turbulence approach explains why optical beam self-cleaning has not been observed in step-index multimode fibers.