A paraxial formulation of steady-state self-focusing is presented, taking due care of the correct paraxial approximation of the plasma frequency and the resultant nonlinear refractive index of the plasma in the presence of the electromagnetic field of a high-powered laser beam. For the laser beam and plasma refractive index description, the correct momentum space of the photons in circular cylindrical geometry of the laser beam is the transformation in terms of the Laguerre-Gauss modes that lead to the correct non-Taylor series paraxial approximation of the beam and its propagation equation. For self-trapping, the same conditions as the moments and variational approaches result. The natural way to express the results for self-focusing in this approximation is in terms of the so-called ABCD laws for the beam parameters, presented here for the self-similar beam propagation of a centrally humped (Gaussian) beam. These laws define a one-complex-parameter group of transformations [the SU(2) group for the absorptionless case and the restricted Lorentz group in general] that describe the evolution of the self-focusing beam; they naturally lead to the application of the concept of geometric phase to the self-focusing beam presented in the paper.
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