A conformally invariant derivation of average electromagnetic helicity

TL;DR

This paper derives a conformally invariant formula for the average electromagnetic helicity, providing multiple equivalent integral expressions that can be directly computed from standard Maxwell solver outputs.

Contribution

It introduces a new conformally invariant derivation of electromagnetic helicity with versatile integral forms suitable for numerical evaluation.

Findings

Derived conformally invariant expressions for average helicity.

Established equivalence with traditional volume integral formulations.

Provided formulas compatible with common Maxwell simulation outputs.

Abstract

The average helicity of a given electromagnetic field measures the difference between the number of left- and right-handed photons contained in the field. In here, the average helicity is derived using the conformally-invariant inner-product for Maxwell fields. Several equivalent integral expressions in momentum space, in $(\mathbf{r},t)$ space, and in the time-harmonic $(\mathbf{r},\omega)$ space are obtained, featuring Riemann-Silberstein-like fields and potentials. The time-harmonic expressions can be directly evaluated using the outputs of common numerical solvers of Maxwell equations. The results are shown to be equivalent to the well-known volume integral for the average helicity, featuring the electric and magnetic fields and potentials.