A paraxial formulation of steady-state self-focusing is presented, tak
ing due care of the correct paraxial approximation of the plasma frequ
ency and the resultant nonlinear refractive index of the plasma in the
presence of the electromagnetic field of a high-powered laser beam. F
or 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 con
ditions 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 (Gau
ssian) beam. These laws define a one-complex-parameter group of transf
ormations [the SU(2) group for the absorptionless case and the restric
ted Lorentz group in general] that describe the evolution of the self-
focusing beam; they naturally lead to the application of the concept o
f geometric phase to the self-focusing beam presented in the paper. (C
) 1998 American Institute of Physics. [S1070-664X(98)03308-4]