Vector effects in self-focusing

Using a Hertz vector formulation, it is possible to simplify the equations for beam propagation in nonlinear media without resorting to the scalar approximation. A numerical method to take advantage of the formulation is developed and applied to the problem of self-focusing. It is found that filament-like solutions can result for some values of the nonlinear index of refraction when the scalar approximation is not made even though the paraxial approximation is still used. {} 1996 The American Physical Society.