Optical vortices evolving from helicoidal integer and fractional phase steps

摘要

The evolution of a wave starting at z = 0 as exp(iαφ) (0 ≤ φ < 2π), i.e. with unit amplitude and a phase step 2πα on the positive x axis, is studied exactly and paraxially. For integer steps (α= 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(αφ+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 , no fractional-strength vortices can propagate; instead, the interference between an additional diffracted wave, from the phase step discontinuity, with exp{i (αφ+kz)} and the wave scattered from r = 0, generates a pattern of strength-1 vortex lines, whose total (signed) strength S_α is the nearest integer to α. For small |α-n|, these lines are close to the z axis. As α passes n + 1/2, S_αjumps by unity, so a vortex is born. The mechanism involves an infinite chain of alternating-strength vortices close to the positive x axis for α= n + 1/2, which annihilate in pairs differently when α> n + 1/2 and when α < n + 1/2. There is a partial analogy between α and the quantum flux in the Aharonov–Bohm effect.