Loss of phase and universality of stochastic interactions between laser beams

Traditionally, interactions between laser beams or filaments were considered to be deterministic. We show, however, that in most physical settings, these interactions ultimately become stochastic. Specifically, we show that in the nonlinear propagation of laser beams, the shot-to-shot variation of the nonlinear phase shift increases with distance, and ultimately becomes uniformly distributed in [0, 2π]. Therefore, if two beams travel a sufficiently long distance before interacting, it is not possible to predict whether they would intersect in- or out-of-phase. Hence, if the underlying propagation model is non-integrable, deterministic predictions and control of the outcome of the interaction become impossible. Because the relative phase between the two beams becomes uniformly distributed in [0, 2π], however, the statistics of these stochastic interactions are universal and fully predictable. These statistics can be efficiently computed using a novel universal model for stochastic interactions, even when the noise distribution is unknown.