Self-focusing of hypergeometric laser beams

We have obtained explicit nonparaxial expressions for three components of the electric vector of the hypergeometric laser beam (HyG-beam). For HyG-beams with topological charge n = 0, 1 explicit expressions for on-axis intensity have been derived and it follows that position of the beam waist depends on HyG-beam parameters (gamma, m) and is displaced from the input plane z = 0. Such displacement of maximal on-axis intensity has been called self-focusing of the HyG-beam and formulae for this displacement (focusing distance) have been obtained. Numerical simulation of HyG-beam propagation with n = 0 has been done by the FDTD-method and it is shown that displacement of beam waist and on-axis intensity are in good accordance with values predicted by approximate formulae.