Optical vortices evolving from helicoidal integer and fractional phase steps

The evolution of a wave starting at z = 0 as exp(ialphaphi) (0 less than or equal to phi < 2pi), i.e. with unit amplitude and a phase step 2pialpha on the positive x axis, is studied exactly and paraxially. For integer steps (alpha = n), the singularity at the origin r = 0 becomes for z > 0 a strength n optical vortex, whose neighbourhood is described in detail. Far from the axis, the wave is the sum of exp{i(alphaphi + kz)} and a diffracted wave from r = 0. The paraxial wave and the wave far from the vortex are incorporated into a uniform approximation that describes the wave with high accuracy, even well into the evanescent zone. For fractional a, no fractional-strength vortices can propagate; instead, the interference between an additional diffracted wave, from the phase step discontinuity, with exp{i(alphaphi + kz)} and the wave scattered from r = 0, generates a pattern of strength-1 vortex lines, whose total (signed) strength S-alpha is the nearest integer to alpha. For small \alpha-n\, these lines are close to the z axis. As alpha passes n + 1/2, S-alpha jumps by unity, so a vortex is born. The mechanism involves an infinite chain of alternating-strength vortices close to the positive x axis for alpha = n + 1/2, which annihilate in pairs differently when alpha > n + 1/2 and when alpha < n + 1/2. There is a partial analogy between a and the quantum flux in the Aharonov-Bohm effect.