Rare-event statistics of subthreshold self-focusing

Canonical nonlinear optics treats self-focusing as a deterministic beam evolution scenario, unfolding above the self-focusing threshold due to the intensity-dependent change in the refractive index. While this deterministic view is adequate for a vast class of nonlinear processes, its insight into stochastic nonlinear phenomena, including laser-induced damage and self-focusing-enhanced spectral transformations, is limited. Here, we present a stochastic treatment of self-focusing. We derive a closed-form analytical solution for the count rate of extreme self-focusing events in nonlinear beam dynamics below the self-focusing threshold. We show that the rare-event statistics of subthreshold self-focusing is highly sensitive to the signal-to-noise ratio and the bandwidth of the laser field waveform. For low-signal-to-noise beams, the rare-event distribution of deeply subthreshold self-focusing is shown to exhibit a manifestly nonexponential tail, thus indicating the enhancement of extreme self-focusing events. We show that subthreshold self-focusing is further enhanced by a broader bandwidth of the noise component of the laser field. It is such broadband, low-signal-to-noise laser fields that are especially prone to deeply subthreshold, rogue-wave self-focusing, lowering, via a laser-induced breakdown, the lifetime of downstream optics.