Angular momentum of light

Mathematical expressions for the angular momentum of light

One commonly used expression for the total angular momentum of an electromagnetic field is the following one, in which there is no explicit distinction between the two forms of rotation: $${\displaystyle \mathbf{J}={\epsilon }_0\int \mathbf{r}\times \left(\mathbf{E}\times \mathbf{B}\right)d^3\mathbf{r},}$$ where ${\displaystyle \mathbf{E}}$ and ${\displaystyle \mathbf{B}}$ are the electric and magnetic fields, respectively, ${\displaystyle {ϵ}_0}$ is the vacuum permittivity and we are using SI units.

However, another expression of the angular momentum naturally arising from Noether's theorem is the following one, in which there are two separate terms that may be associated with SAM (${\displaystyle \mathbf{S}}$) and OAM (${\displaystyle \mathbf{L}}$): $${\displaystyle \mathbf{J}={\epsilon }_0\int \left(\mathbf{E}\times \mathbf{A}\right)d^3\mathbf{r}+{\epsilon }_0\sum _{i=x,y,z}\int \left(E^i\left(\mathbf{r}\times \mathbf{\nabla }\right)A^i\right)d^3\mathbf{r}=\mathbf{S}+\mathbf{L},}$$ where ${\displaystyle \mathbf{A}}$ is the vector potential of the magnetic field, and the i-superscripted symbols denote the cartesian components of the corresponding vectors.

These two expressions can be proved to be equivalent to each other for any electromagnetic field that satisfies Maxwell's equations with no source charges and vanishes fast enough outside a finite region of space. The two terms in the second expression however are physically ambiguous, as they are not gauge-invariant. A gauge-invariant version can be obtained by replacing the vector potential A and the electric field E with their "transverse" or radiative component ${\displaystyle {\mathbf{A}}_{\perp }}$ and ${\displaystyle {\mathbf{E}}_{\perp }}$, thus obtaining the following expression: $${\displaystyle {\mathbf{J}}_{\perp }={\epsilon }_0\int \left({\mathbf{E}}_{\perp }\times {\mathbf{A}}_{\perp }\right)d^3\mathbf{r}+{\epsilon }_0\sum _{i=x,y}\int \left({E^i}_{\perp }\left(\mathbf{r}\times \mathbf{\nabla }\right)A_{\perp }^i\right)d^3\mathbf{r}.}$$

A justification for taking this step is yet to be provided. The latter expression has further problems, as it can be shown that the two terms are not true angular momenta as they do not obey the correct quantum commutation rules. Their sum, that is the total angular momentum, instead does.

An equivalent but simpler expression for a monochromatic wave of frequency ω, using the complex notation for the fields, is the following: $${\displaystyle \mathbf{J}=\frac{{\epsilon }_0}{2i\omega }\int \left({\mathbf{E}}^{\ast }\times \mathbf{E}\right)d^3\mathbf{r}+\frac{{\epsilon }_0}{2i\omega }\sum _{i=x,y,z}\int \left({E^i}^{\ast }\left(\mathbf{r}\times \mathbf{\nabla }\right)E^i\right)d^3\mathbf{r}.}$$

Let us now consider the paraxial limit, with the beam axis assumed to coincide with the z axis of the coordinate system. In this limit the only significant component of the angular momentum is the z one, that is the angular momentum measuring the light beam rotation around its own axis, while the other two components are negligible.

$${\displaystyle \mathbf{J}\approx \frac{\hat{z}{ϵ}_0}{2\omega }\int \left({\left|E_{\text{L}}\right|}^2-{\left|E_{\text{R}}\right|}^2\right)d^3\mathbf{r}+\frac{\hat{z}{\epsilon }_0}{2i\omega }\int \sum _{i=x,y,z}\left({E^i}^{\ast }\frac{\mathrm{\partial }}{\mathrm{\partial }\varphi }{E}^i\right)d^3\mathbf{r}.}$$ where ${\displaystyle E_{\text{L}}}$ and ${\displaystyle E_{\text{R}}}$ denote the left and right circular polarization components, respectively.

Exchange of spin and orbital angular momentum with matter

When a light beam carrying nonzero angular momentum impinges on an absorbing particle, its angular momentum can be transferred on the particle, thus setting it in rotational motion. This occurs both with SAM and OAM. However, if the particle is not at the beam center the two angular momenta will give rise to different kinds of rotation of the particle. SAM will give rise to a rotation of the particle around its own center, i.e., to a particle spinning. OAM, instead, will generate a revolution of the particle around the beam axis. These phenomena are schematically illustrated in the figure.

In the case of transparent media, in the paraxial limit, the optical SAM is mainly exchanged with anisotropic systems, for example birefringent crystals. Indeed, thin slabs of birefringent crystals are commonly used to manipulate the light polarization. Whenever the polarization ellipticity is changed, in the process, there is an exchange of SAM between light and the crystal. If the crystal is free to rotate, it will do so. Otherwise, the SAM is finally transferred to the holder and to the Earth.

Spiral phase plate (SPP)

In the paraxial limit, the OAM of a light beam can be exchanged with material media that have a transverse spatial inhomogeneity. For example, a light beam can acquire OAM by crossing a spiral phase plate, with an inhomogeneous thickness (see figure).

Pitch-fork hologram

A more convenient approach for generating OAM is based on using diffraction on a fork-like or pitchfork hologram (see figure). Holograms can be also generated dynamically under the control of a computer by using a spatial light modulator. As a result, this allows one to obtain arbitrary values of the orbital angular momentum.

Q-plate

Another method for generating OAM is based on the SAM-OAM coupling that may occur in a medium which is both anisotropic and inhomogeneous. In particular, the so-called q-plate is a device, currently realized using liquid crystals, polymers or sub-wavelength gratings, which can generate OAM by exploiting a SAM sign-change. In this case, the OAM sign is controlled by the input polarization.

Cylindrical mode converters

OAM can also be generated by converting a Hermite-Gaussian beam into a Laguerre-Gaussian one by using an astigmatic system with two well-aligned cylindrical lenses placed at a specific distance (see figure) in order to introduce a well-defined relative phase between horizontal and vertical Hermite-Gaussian beams.

Possible applications of the orbital angular momentum of light

The applications of the spin angular momentum of light are undistinguishable from the innumerable applications of the light polarization and will not be discussed here. The possible applications of the orbital angular momentum of light are instead currently the subject of research. In particular, the following applications have been already demonstrated in research laboratories, although they have not yet reached the stage of commercialization:

1. Orientational manipulation of particles or particle aggregates in optical tweezers
2. High-bandwidth information encoding in free-space optical communication
3. Higher-dimensional quantum information encoding, for possible future quantum cryptography or quantum computation applications
4. Sensitive optical detection

Controversy: Non-existence of SAM and OAM of light

The decomposition of total angular momentum into spin and orbital parts for massless bosons—such as photons and gluons—has been a longstanding controversy in both classical and quantum field theory. Since the early days of quantum electrodynamics, physicists have debated whether the total angular momentum of light can be meaningfully separated into intrinsic spin and orbital angular momentum components. While such a decomposition is well-defined for massive particles using the Newton–Wigner position operator and related constructions, analogous definitions for massless particles have led to ambiguities, nonlocal expressions, or violations of Lorentz symmetry. In 2025, four rigorous no-go theorems were established that clarify the root of these issues. It was proved that it is mathematically impossible to construct operators that simultaneously satisfy the canonical angular momentum algebra, preserve Poincaré symmetry, and yield a physically meaningful spin–orbit decomposition for massless vector bosons. Any such attempt necessarily fails to commute with the Hamiltonian, violates Lorentz invariance, or leads to frame-dependent nonlocality. These results, grounded in the representation theory of the Poincaré group, explain why a consistent spin–orbit separation is fundamentally obstructed for massless gauge bosons. Nevertheless subsequent work has shown that helicity operators, which is not related to the angular momentum operator, can be used as alternatives.