Optical phase singularities going to and coming from infinity with a higher-than-light speed

In this paper, we summarize a remarkable result obtained by Soskin et al. in Phys Rev A 56, 4064 (1997).We show that for an on-axis superposition of two different-waist Laguerre-Gauss beams with numbers (0, n) and (0, m), the topological charge equals TC= m up to a plane where the waist radii become the same, given that the beam (0, m) has a greater waist radius, changing to TC= n after this plane. This occurs because in the initial plane the superposition has an on-axis optical vortex with TC= m and on different axis-centered circles there are ( n - m ) vortices with TC= +1 and ( n - m ) vortices with TC= -1. On approaching the above-specified plane, the vortices with TC= -1 "depart" to infinity with a higher-than-light speed, with the TC of the total beam becoming equal to TC= n. If, on the contrary, the beam (0, m ) has a smaller waist, then the total TC equals n on a path from the initial plane up to a plane where the waist radii become the same, changing to TC= m after the said plane. This occurs because after the said plane, n - m vortices with TC= -1 "arrive" from infinity with a higher-than-light speed.