Optical helicity and Hertz vectors

TL;DR

This paper investigates optical helicity, a conserved quantity in Maxwell's equations, using Hertz vectors, explores its anomaly in charged systems, and discusses its role in topological effects and dual symmetric formulations.

Contribution

It introduces a Hertz vector-based approach to optical helicity, derives a dual symmetric Hertz Lagrangian, and connects helicity evolution to chiral anomalies and topological phenomena.

Findings

Optical helicity is linked to the dual symmetry of Maxwell's equations.

An alternative expression for optical helicity is derived.

A dual symmetric Hertz Lagrangian with conserved charge is constructed.

Abstract

We study the conserved quantity associated with the dual symmetry of the Maxwell equations, called the optical helicity, by means of transverse Hertz vectors. In the presence of charges, its evolution yields the integral of $\bm{E}\cdot\bm{B}$ which is the anomaly term for chiral fermions. We also discuss the helicity change in condensed matter systems where topological magnetoelectric effect emerges. An alternative expression of the optical helicity is also found. Lastly, a dual symmetric Hertz Lagrangian is constructed and its conserved charge is derived.