Generalized self-similar propagation and amplification of optical pulses in nonlinear media with high-order normal dispersion

Abstract

We investigate theoretically and numerically the self-similar propagation of optical pulses in the presence of gain, positive Kerr nonlinearity, and positive (i.e., normal) dispersion of even order $m$. Starting from a modified nonlinear Schr\"odinger equation, separating the evolution of amplitude and phase, we find that the resulting equations simplify considerably in the asymptotic limit. Exact solutions to the resulting equations indicate that the temporal intensity profile follows a $1\ensuremath{-}{T}^{m/(m\ensuremath{-}1)}$ function with an $m$-dependent scaling relation, with a ${T}^{1/(m\ensuremath{-}1)}$ chirp, where $T$ is the pulse's local time. These correspond to a triangle and a step function, respectively, as $m\ensuremath{\rightarrow}\ensuremath{\infty}$. These results are borne out by numerical simulations, although we do observe indications of nonasymptotic behavior.