Conservation of the spin and orbital angular momenta in electromagnetism

TL;DR

This paper develops physically measurable, locally conserved densities and fluxes for the spin and orbital angular momentum of electromagnetic fields, clarifying their physical meaning and interaction effects in optics.

Contribution

It constructs the missing locally conserved spin and orbital angular momentum densities and fluxes for electromagnetic fields, including spin-orbit interaction effects, in both quantum and classical frameworks.

Findings

Derived physically measurable and conserved spin and orbital AM densities and fluxes.

Included a spin-orbit interaction term in the fluxes for nonparaxial fields.

Validated the theory using nonparaxial optical vortex beams.

Abstract

We review and re-examine the description and separation of the spin and orbital angular momenta (AM) of an electromagnetic field in free space. While the spin and orbital AM of light are not separately-meaningful physical quantities in orthodox quantum mechanics or classical field theory, these quantities are routinely measured and used for applications in optics. A meaningful quantum description of the spin and orbital AM of light was recently provided by several authors, which describes separately conserved and measurable integral values of these quantities. However, the electromagnetic field theory still lacks corresponding locally-conserved spin and orbital AM currents. In this paper, we construct these missing spin and orbital AM densities and fluxes that satisfy the proper continuity equations. We show that these are physically measurable and conserved quantities. These are, however, not Lorentz-covariant, so only make sense in the single laboratory reference frame of the measurement probe. The fluxes we derive improve the canonical (non-conserved) spin and orbital AM fluxes, and include a `spin-orbit' term that describes the spin-orbit interaction effects observed in nonparaxial optical fields. We also consider both standard and dual-symmetric versions of the electromagnetic field theory. Applying the general theory to nonparaxial optical vortex beams validates our results and allows us to discriminate between earlier approaches to the problem. Our treatment yields the complete and consistent description of the spin and orbital AM of free Maxwell fields in both quantum-mechanical and field-theory approaches.