Beyond leading order logarithmic scaling in the catastrophic self-focusing (collapse) of a laser beam in Kerr media

TL;DR

This paper investigates the self-focusing collapse of laser beams in Kerr media, revealing that the traditional loglog scaling law is only valid at unrealistically high amplitudes and proposing a new perturbative approach for more moderate amplitudes.

Contribution

The authors derive a new equation for the slow parameter governing self-focusing and develop a perturbation theory that extends the scaling law validity to realistic amplitudes.

Findings

Loglog scaling law requires unattainably large amplitudes.

New perturbation theory matches numerical results for moderate initial powers.

Proposed model extends understanding of self-focusing dynamics at realistic conditions.

Abstract

We study the catastrophic stationary self-focusing (collapse) of laser beam in nonlinear Kerr media. The width of a self-similar solutions near collapse distance $z=z_c$ obeys $(z_c-z)^{1/2}$ scaling law with the well-known leading order modification of loglog type $\propto (\ln|\ln(z_c-z)|)^{-1/2}$. We show that the validity of the loglog modification requires double-exponentially large amplitudes of the solution $\sim {10^{10}}^{100}$, which is unrealistic to achieve in either physical experiments or numerical simulations. We derive a new equation for the adiabatically slow parameter which determines the system self-focusing across a large range of solution amplitudes. Based on this equation we develop a perturbation theory for scaling modifications beyond the leading loglog. We show that for the initial pulse with the optical power moderately above ($\lesssim 1.2$) the critical power of self-focusing, the new scaling agrees with numerical simulations beginning with amplitudes around only three times above of the initial pulse.