Transient self-focusing of light

The aberration-free approximation is used in a theoretical analysis of self -focusing of wave beams in a medium of permittivity $\varepsilon(t)=\varepsilon_0(I+\delta\int_{-\infty}^t|E|^2\,dt)$. It is shown that two effects appear under transient self-focusing conditions: 1) filaments undergoing gradual constriction form behind a nonlinear focus in a medium with $\varepsilon=\varepsilon(t)$; 2) large frequency shifts result from transverse compression of a self-focusing channel when a beam “collapses” in a medium with a noninstantaneous nonlinearity of the aforementioned type or with an instantaneous nonlinearity of the $\varepsilon=\varepsilon_0(1 + \varepsilon'|E|^2)$ type.