The life-cycle of Riemann–Silberstein electromagnetic vortices

Abstract

To study the singularities of a monochromatic electromagnetic wave field in free space, it is desirable to use a quantity that combines both the electric field E and the magnetic field B in equal measure. The Riemann–Silberstein (R–S) field is a way of doing this. It is based on the real physical E and B and one constructs from them the complex vector field Then, one constructs and studies the optical vortices of this R–S complex scalar field. Unlike the better-known and much studied optical vortices of a monochromatic complex scalar field, which are stationary, these vortices are normally in continual motion; they oscillate at the optical frequency. We study their life cycle in the simplest model that is sufficiently generic, namely, fields generated by the interference of four randomly chosen plane elliptically polarised waves. The topological events in the life cycle do not repeat on a 3D space lattice in a stationary laboratory frame. In space–time, however, the R–S vortices are invariant under any Lorentz transformation, and because of this and the inherent time repetition there is a particular moving frame in space–time, reached by a Lorentz transformation, where there exists a repeating pattern of events in space. Its 4D unit cell constitutes, in effect, a description of the whole infinite pattern. Just because they are in constant motion, it is not surprising that the R–S vortex lines in the model make reconnections and appear as rings that either shrink to nothing or appear from nothing. However, these processes occur in groups of four, reflecting the fact that the unit cell is face-centred. What distinguishes the R–S field from the other complex scalar fields containing vortices is the existence of this face-centred repeating cell.