Threshold of stochastic self-focusing from the Poisson property of extreme-event statistics

Statistics of self-focusing induced by a stochastic laser driver is shown to converge, in the large-sample-size limit, to a generalized Poisson distribution whose mean is given by the exponent of the respective extreme-value statistics. For a given ratio of the laser peak power to the self-focusing threshold P_cr, the mean number of self-focusing counts in a large sample of laser pulses is shown to depend on the number of pulses in the sample, N, and the signal-to-noise ratio of laser pulses, a. We derive a closed-form solution for the threshold of stochastic self-focusing, which, unlike its deterministic counterpart, P_cr, is a function of the sample size N and the signal-to-noise ratio a. The parameter N_a = exp (a²/2) is shown to set a borderline between the deterministic and stochastic regimes of self-focusing. When the number of laser pulses in a sample becomes comparable to N_a, self-focusing can no longer be viewed as deterministic even for high signal-to-noise laser beams.