Topological events on wave dislocation lines: birth and death of loops, and reconnection

In three-dimensional space, a wave dislocation, that is, a quantized (optical) vortex or phase singularity, is a line zero of a complex scalar wavefunction. As a 'time' parameter varies, the topology of the vortex can change by encounter with a line of vanishing vorticity (curl of the current associated with the wavefunction). An isolated critical point of the field intensity, sliding along the zero-vorticity line like a bead on a wire, meets the vortex as it encounters the line, and so participates in the singular event. Local expansion and gauge and coordinate transformations show that the vortex topology can change generically by the appearance or disappearance of a loop, or by the reconnection of branches of a pair of hyperbolas.