Criteria for stochastic self-focusing

Analysis of extreme-value statistics of stochastic laser pulses suggests a closed-form, quantitative criterion of self-focusing avoidance. We present an analytical solution for the excess kurtosis of the statistics of nonlinear-optical processes, which is shown to be a rapidly growing function of the nonlinearity order, thus indicating a physically significant redistribution of statistical weight within the probability distribution of the respective nonlinear readouts from its central part to its tails. Unlike deterministic self-focusing, whose criterion is expressed in terms of a well-defined self-focusing threshold ${P}_{\mathrm{cr}}$, its stochastic counterpart is a probabilistic process whose combined probability for a sample of $N$ laser pulses builds up as a function of $N$, leading to $N$-dependent self-focusing avoidance criteria. Specifically, for $N$ \ensuremath{\gg} 1 laser pulses with a signal-to-noise ratio $a$, the criterion of self-focusing avoidance is shown to shift as ${a}^{2}{P}_{\mathrm{cr}}/(2lnN)$. Instead of dealing with a question as to how to completely avoid self-focusing, stochastic analysis has to deal with a question of how to effectively manage the self-focusing probability over a finite sample of laser shots. The occurrence of self-focusing in stochastic nonlinear optics is thus not a question of $if$, but a question of when.