# Spin angular momentum of light

The spin angular momentum of light (SAM) is the component of angular momentum of a light beam that can be associated with its circular or elliptical polarization.

## Introduction

Left and right handed circular polarization, and their associate angular momenta.

An electromagnetic wave is said to have circular polarization when its electric and magnetic fields rotate continuously around the beam axis during the propagation. The circular polarization is left ($L$) or right ($R$) depending on the field rotation direction (but be careful that both conventions are used in science, depending on the subfield).

When a light beam is circularly polarized, each of its photons carries a spin angular momentum of $\pm\hbar$, where $\hbar$ is the reduced Planck constant and the $\pm$ sign is positive for Left and negative for Right circular polarizations (this is adopting the convention most commonly used in optics). This SAM is directed along the beam axis (parallel if positive, antiparallel if negative). The above figure shows the instantaneous structure of the electric field of left ($L$) and right ($R$) circularly polarized light in space. The green arrows indicate the propagation direction.

The mathematical expressions reported under the figures give the three electric field components of circularly polarized plane wave propagating in the $z$-direction, in complex notation.

## Mathematical expression

In the following the main formula used for the spin angular momentum of light are given: General expression (paraxial limit only):[1]

$\mathbf{S}=\epsilon_0\int \left(\mathbf{E}\times\mathbf{A}\right)d^{3}\mathbf{r} ,$

where $\mathbf{E}$ and $\mathbf{A}$ are the electric field and vector potential, respectively, $\epsilon_0$ is the vacuum permittivity and we are using SI units.

Monochromatic wave case:[2]

$\mathbf{S}= \frac{\epsilon_0}{2i\omega}\int \left(\mathbf{E}^\ast\times\mathbf{E}\right)d^{3}\mathbf{r} .$

In particular, this expression shows that the SAM is nonzero when the light polarization is elliptical or circular, while it vanishes if the light polarization is linear. In the quantum theory of the electromagnetic field, the SAM is a quantum observable, described by a corresponding operator.

$\mathbf{S}=\sum_{\mathbf{k}}\hbar\mathbf{u}_{\mathbf{k}}\left(\hat{a}^{\dagger}_{\mathbf{k},L}\hat{a}_{\mathbf{k},L}-\hat{a}^{\dagger}_{\mathbf{k},R}\hat{a}_{\mathbf{k},R}\right),$

where $\mathbf{u}_{\mathbf{k}}$ is the unit vector in the propagation direction, $\hat{a}^{\dagger}_{\mathbf{k},\pi}$ and $\hat{a}_{\mathbf{k},\pi}$ are the creation and annihilation operators for photons in the mode k and polarization state $\pi$, respectively.

In this case, for a single photon the SAM can only have two values (eigenvalues of the SAM operator):

$\mathbf{S}_z=\pm\hbar .$

The corresponding eigenfunctions describing photons with well defined values of SAM are described as circularly polarized waves:

$|\pm\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[l]{c} 1\\ \pm i \end{array}\right).$