The effect of longitudinal electric field components on the propagation of intense ultrashort optical pulses

In this paper we investigate the effect of longitudinal electric field components on the propagation of intense ultrashort optical pulses. We find that the longitudinal electric field components have contributions both from modes that in the paraxial limit are polarized transversely to the beam axis and traveling along the beam axis and from a mode that is polarized along the beam axis even in the paraxial limit and that travel transversely to the beam axis. We show that the amplitude of this last mode in general satisfies a dispersive wave equation in the plane transverse to the beam axis and that the source term in this equation depends on the amplitude of both types of modes. The source is large if the fields are intense, the pulse is short or the fields are nonparaxial. Thus the effect of the mode that is polarized along the beam axis is to transport energy away from the collapse region and in this way make the system less prone to self-focusing collapse. In the weakly nonlinear slowly varying limit we show explicitly that the effect of the longitudinally polarized mode is to restrict the range of transverse modulationally unstable wavenumbers and to act as a defocusing lens in the collapse region.