“…While the existence of this shelf of radiation has been shown from perturbation theory [18,19], the reason for this radiation can most easily be seen from the dispersion relation for the linear electric field equation, ω = k 2 /2, where k is the wave number. The group velocity is then c g = k, so that low-wave-number radiation has a low velocity and thus accumulates under the vortex, resulting in a shelf of radiation, or a pedestal, as this radiation effect is referred to in optical fibers [20,21]. Also, in analogy with previous studies of nonlocal vortices [14][15][16], the parameters a, w, g, and σ should depend not only on the space variable z, but also on the azimuthal angle θ .…”