Paraxial theory of slow self-focusing

We present a theory of slow self-focusing that is paraxial in nature, gives the field including the phase and eikonal explicitly, while it also agrees with the results of variational and moments theories. After presenting the features of the theory, particularly its similarity to the central force problem, we go on to reformulate the theory for an absorbing medium. We find that the laser beam focuses to a constant beamwidth with a small phase-front curvature depending on the extent of absorption. The theory is applicable to a whole range of saturating nonlinearities although it specializes to two plasma cases, the ponderomotive force based and the relativistic electron quiver based nonlinearities, for definitive results.