Lensless focusing of hypergeometric laser beams

Using the weak nonparaxial approximation, we derive explicit analytical relationships for three projections of the electric field vector of a hypergeometric (HyG) laser beam. For the HyG beams with topological charge n = 0, 1 we derive explicit formulae for the axial intensity, which suggest that the waist position is related to the parameters of the HyG beam and shifted with respect to the plane z = 0. Such a displacement of the axial intensity peak has been given the name 'HyG beam lensless focusing' and formulae for calculating the said shift magnitude (focus distance) are deduced. The propagation of the HyG beam at n = 0 is simulated using the finite-difference time-domain method. The waist shift and the axial intensity magnitudes are shown to agree with those predicted by the approximate relations.