Helicity, polarization, and Riemann-Silberstein vortices

TL;DR

This paper demonstrates that Riemann-Silberstein vortices, which are null surfaces in electromagnetic fields, become steady and coincide with polarization C-lines when the field has definite helicity, linking relativistic invariance with observable polarization features.

Contribution

It shows that under the assumption of definite helicity, RS vortices are steady and coincide with polarization C-lines, unifying these concepts for waves of definite frequency and helicity.

Findings

RS vortices are steady with definite helicity.

RS vortices coincide with electric and magnetic C-lines.

The concept generalizes C-lines for wideband waves of definite helicity.

Abstract

Riemann-Silberstein (RS) vortices have been defined as surfaces in spacetime where the complex form of a free electromagnetic field given by F=E+iB is null (F.F=0), and they can indeed be interpreted as the collective history swept out by moving vortex lines of the field. Formally, the nullity condition is similar to the definition of "C-lines" associated with a monochromatic electric or magnetic field, which are curves in space where the polarization ellipses degenerate to circles. However, it was noted that RS vortices of monochromatic fields generally oscillate at optical frequencies and are therefore unobservable while electric and magnetic C-lines are steady. Here I show that under the additional assumption of having definite helicity, RS vortices are not only steady but they coincide with both sets of C-lines, electric and magnetic. The two concepts therefore become one for waves of definite frequency and helicity. Since the definition of RS vortices is relativistically invariant while that of C-lines is not, it may be useful to regard the vortices as a wideband generalization of C-lines for waves of definite helicity.