Large-Scale Self-Trapping of Optical Beams in the Paraxial Ray Approximation

We present a systematic discussion of time-independent large-scale self-focusing in a paraxial ray "constant-shape" approximation. An equation is obtained for the variance of a self-focusing beam with a Gaussian transverse intensity profile as a function of axial distance. This equation, valid for an arbitrary nonlinear constitutive relation, is solved for several important situations, including the case of a saturable nonlinear refractive index. We also derive closed expressions for such a saturable index arising from reorientation of axially symmetric molecules in an external field, ignoring local-field effects. Self-focusing of an elliptically polarized beam is considered in the same approximation, and numerical results are presented which show that the self-focusing length is quite sensitive to small departures from circular polarization.