On quantifying the spin angular momentum density of light

TL;DR

The paper investigates how to quantify spin angular momentum (SAM) density in light and resolves longstanding debates about SAM in plane waves. It offers a purely classical derivation using Maxwell's equations and the Coulomb force, avoiding quantum or field-theoretic frameworks. The main contributions include a direct SAM current density for plane waves and a torque identity that links areal torque to SAM, demonstrated through radiation pressure and waveplate analysis with internal reflections. The work clarifies that elliptically polarized plane waves possess SAM and provides an accessible, Maxwell-based framework for light-matter torque phenomena with implications for optomechanics and photonic manipulation.

Abstract

In addition to energy, light carries linear and angular momentum. These are key quantities in rapidly developing optics research and in technologies focusing on light induced forces and torques on materials. Spin angular momentum (SAM) density is of particular interest, since unlike orbital angular momentum, it is uncoupled from linear momentum. The SAM density of light was first estimated in 1909 by Poynting, using a mechanical analogy. Exact expressions, based on results from quantum mechanics and field theory were subsequently developed, and are in common use today. In this paper, we show that the SAM density of light can be obtained directly from the Coulomb force and Maxwell's equations, without reliance on quantum mechanics or field theories; it could have been calculated by Maxwell and his contemporaries. Besides its historical significance, the simple derivation of our result makes it readily accessible to non-experts in the field.

Figures (4)

• Figure 1: Illustration of the sample and fields used in the calculations. The magnetic fields associated with the electric fields are not shown, they are in the $\mathbf{\hat{k}\times E}$ direction.
• Figure 2: Radiation pressure $P$, normalized by $\varepsilon_{0}|\mathbf{E}_{i}|^2$, as a function of thickness $d/\lambda_{0}$. Left: isotropic slab with index $n=3$, described by the Airy function. Right: waveplate with indices $n_{1}=1.7$ and $n_{2}=1.2$. The radiation pressure $P$ in general is quasiperioedic; here the period is $5$.
• Figure 3: Illustration of the sample and fields used in the calculations. The magnetic fields associated with the electric fields are not shown, they are in the $\mathbf{\hat{k}\times E}$ direction. The optic axis is indicated with the double arrow.
• Figure 4: Areal torque density $\mathbf{\tau }_{A}/\tau _{0}$, on a wave plate with indices $n_{1}=1.7$ and $n_{2}=1.2$ as function of thickness $d/\lambda _{0}$ for (left) left-circular, (middle) linear and (right) right-circular polarizations, from the point of view of the source. The areal torque density in general is quasiperiodic; here the period is $10$.