Optical vortices evolving from helicoidal integer and fractional phase steps
Abstract
The evolution of a wave starting at z = 0 as exp(i agrphgr) (0 \leqslant \phi<2\pi ), i.e. with unit amplitude and a phase step 2pgragr on the positive x axis, is studied exactly and paraxially. For integer steps (agr = 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 (agrphgr+kz)} anda 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 agr, no fractionalstrength vortices can propagate; instead, the interference between an additional diffracted wave, from the phase step discontinuity, with exp{i (agrphgr+kz)} andthe wave scattered from r = 0, generates a pattern of strength1 vortex lines, whose total (signed) strength S_{agr} is the nearest integer to agr. For small agrn, these lines are close to the z axis. As agr passes n+1/2, S_{agr} jumps by unity, so a vortex is born. The mechanism involves an infinite chain of alternatingstrength vortices close to the positive x axis for agr = n+1/2, which annihilate in pairs differently when agr>n+1/2 and when agr<n+1/2. There is a partial analogy between agr and the quantum flux in the AharonovBohm effect.
 Publication:

Journal of Optics A: Pure and Applied Optics
 Pub Date:
 February 2004
 DOI:
 10.1088/14644258/6/2/018
 Bibcode:
 2004JOptA...6..259B
 Keywords:

 vorticessingularitiesasymptoticsphase