Self-focusing in the presence of small time dispersion and nonparaxiality

We analyze the combined effect of small time dispersion and nonparaxiality on self-focusing and its ability to arrest the blowup of laser pulses by deriving reduced equations that depend on only the propagation distance and time. We calculate the pulse duration for which time dispersion dominates over nonparaxiality, or vice versa. We identify additional terms (shock term, group-velocity nonparaxiality, etc.) that should be retained when time dispersion and nonparaxiality are of comparable magnitude. These additional terms lead to temporal asymmetry, and in the visible spectrum they can dominate over both time dispersion and nonparaxiality. © 1997 Optical Society of America The simplest model for optical self-focusing is the nonlinear Schrödinger equation (NLS): icz 1D�c1jcj 2 c�0, c�0, r � � c0�r �. (1) Here c�z, r � is the electric field envelope of a laser beam propagating in a medium with Kerr nonlinearity, z is the distance in the direction of the propagation, r � �x2 1 y2�1/2 is the radial coordinate, and D � � ≠2�≠r2 1 �1�r��≠�≠r�is the Laplacian in the transvers