Loss of phase and universality of stochastic interactions between laser beams

We show that all laser beams gradually lose their initial phase information in nonlinear propagation. 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 interaction outcome become impossible. Because the relative phase between the two beams becomes uniformly distributed in $[0,2\pi]$, however, the statistics of the interaction outcome are universal, and can be efficiently computed using a polynomial-chaos approach, even when the distributions of the noise sources are unknown.