Encoding information: spin, orbital angular momentum, and channel capacity

Polarisation state (spin) offers two orthogonal modes — LCP and RCP — encoding one classical bit or one quantum bit (polarisation qubit) per photon. Polarisation qubits are among the most practical we have: photons travel without decoherence over hundreds of kilometres of optical fibre and are detected with high efficiency.

But light can carry a second, entirely independent form of angular momentum: orbital angular momentum (OAM). Where spin arises from the polarisation state — the direction \(\mathbf{E}\) oscillates — OAM arises from the spatial structure of the beam as a whole. Understanding it requires looking at what wavefronts actually do.

Flat wavefronts vs helical wavefronts

Before writing any formula, it is worth being precise about what kind of object the electric field is. \(\mathbf{E}(r, \varphi, z, t)\) is a vector — at every point in space and at every instant in time, it is a three-dimensional arrow pointing in some direction with some magnitude. For a beam propagating along \(\hat{z}\), the field is transverse (it lies in the \(xy\)-plane at each point), so the full electric field vector is:

\[ \mathbf{E}(r, \varphi, z, t) = E_0\,\operatorname{Re}\!\Bigl[ \underbrace{(\varepsilon_x\,\hat{x} + \varepsilon_y\,\hat{y})}_{\text{Jones vector (polarisation)}} \;\cdot\; \underbrace{u_p(r, z)\, e^{i\ell\varphi}\, e^{i(kz - \omega t)}}_{\text{scalar complex amplitude } \mathcal{E}(r, \varphi, z, t)} \Bigr], \]

where \((\varepsilon_x, \varepsilon_y)^T\) is the Jones vector from the polarisation section. This \(\mathbf{E}\) is not energy. It is the electric field — the quantity that exerts a force \(q\mathbf{E}\) on a charge. The energy flux (power per unit area) is the Poynting vector \(\mathbf{S} = \frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}\), which requires taking the cross product of the electric and magnetic field vectors — a separate calculation we come to below.

The formula above factors the field into two independent pieces:

  • The Jones vector \((\varepsilon_x\hat{x} + \varepsilon_y\hat{y})\) sets the direction in which \(\mathbf{E}\) points (and rotates, for circular polarisation). It carries the spin angular momentum and is the same at every point in the beam cross-section. This is the SAM piece we studied in the polarisation section.
  • The scalar complex amplitude \(\mathcal{E}(r, \varphi, z, t) = u_p(r,z)\,e^{i\ell\varphi}\,e^{i(kz-\omega t)}\) sets the magnitude of the field and the phase at each point. It is a complex number — one scalar per spacetime point — not a vector. Its modulus-squared \(|\mathcal{E}|^2 \propto |u_p|^2\) gives the intensity profile; its argument \(\arg(\mathcal{E}) = \ell\varphi + kz - \omega t\) determines where the wavefronts are.

A wavefront is a surface of constant phase of \(\mathcal{E}\). For an ordinary plane wave (\(\ell = 0\), \(u_p = \text{const}\)), the phase is \(kz - \omega t\) — constant wherever \(z\) is constant — so wavefronts are flat planes sweeping forward. For a Laguerre–Gaussian (LG) beam, the scalar amplitude carries an extra azimuthal winding:[10]

\[ \mathcal{E}(r, \varphi, z, t) \;\propto\; u_p(r, z)\; e^{i\ell\varphi}\; e^{i(kz - \omega t)}, \]

where \(\ell\) is any integer (the topological charge or OAM index) and \(u_p(r, z)\) is a real radial envelope. The total phase is:

\[ \Phi = \ell\varphi + kz - \omega t. \]

A wavefront (surface of constant \(\Phi\) at fixed \(t\)) satisfies \(\ell\varphi + kz = \text{const}\), or equivalently:

\[ z(\varphi) = z_0 - \frac{\ell}{k}\,\varphi = z_0 - \frac{\ell\lambda}{2\pi}\,\varphi. \]

This is the equation of a helix wound around the \(z\)-axis. As \(\varphi\) completes one full revolution (\(2\pi\)), \(z\) shifts by \(-\ell\lambda\). For \(\ell = 1\): a single-thread corkscrew advancing one wavelength per turn. For \(\ell = 2\): a double-thread corkscrew advancing two wavelengths per turn, with two interleaved helical sheets. For \(\ell = -1\): the same structure wound in the opposite sense.

Phase in the transverse (\(xy\)) plane at a fixed \(z\) and fixed time, for \(\ell = -2\) through \(3\). Hue encodes the phase \(\ell\varphi \bmod 2\pi\) — one complete colour cycle (red → rainbow → red) equals one full \(2\pi\) phase winding. Brightness encodes the LG beam intensity \(|u_{0,\ell}|^2 \propto r^{2|\ell|}e^{-2r^2/w^2}\): note the dark on-axis hole for \(\ell \ne 0\) (doughnut) and the bright Gaussian centre for \(\ell = 0\). For \(\ell = 2\) two full hue cycles complete per revolution; for \(\ell = -1\) the winding runs in the opposite sense.
ℓ =
One wavefront surface (\(\ell\varphi + kz = 0\) at a fixed time), coloured by phase (same wheel as above). For \(\ell = 0\): a flat disc — ordinary plane wave. For \(\ell \ne 0\): a helical sheet that descends \(\ell\lambda\) per revolution around the beam axis (blue cylinder). The dark hole at the centre is the phase singularity where the LG mode vanishes. Drag to rotate.

OAM per photon: why the helical wavefront carries angular momentum

Recall that the Poynting vector \(\mathbf{S} = \frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}\) is the electromagnetic energy flux — it is a vector field giving the power (in W/m²) flowing through a unit surface, and it points in the direction energy travels. For an ordinary plane wave propagating in \(+z\), \(\mathbf{S}\) points purely along \(\hat{z}\): energy marches straight forward, no sideways component.

For an LG beam with \(\ell \ne 0\), the helical wavefront changes this. The local direction of propagation at any point is perpendicular to the local wavefront — and because the wavefront tilts azimuthally (it is a helix, not a flat plane), the local propagation direction also tilts slightly away from \(\hat{z}\). Concretely: the phase gradient of the scalar amplitude \(\mathcal{E}\) in cylindrical coordinates is

\[ \nabla\Phi = \nabla(\ell\varphi + kz - \omega t) = \frac{\ell}{r}\,\hat{\varphi} + k\,\hat{z}. \]

The wave vector \(\mathbf{k}_\text{local} = \nabla\Phi\) has an azimuthal component \(\ell/r\) in addition to the forward component \(k\). Since \(\mathbf{S}\) points in the direction of \(\mathbf{k}_\text{local}\), it too acquires a \(\hat{\varphi}\) component — energy flows in a slight corkscrew around the beam axis rather than purely forward. The angular momentum per unit cross-sectional area is \(r \times S_\varphi / c^2 \propto r\cdot(\ell/r)/c^2 = \ell/c^2\) — independent of \(r\). Integrating over the beam and normalising per photon yields:

\[ L_z = \ell\hbar \quad \text{per photon,} \]

for any integer \(\ell \in \{\ldots, -2, -1, 0, 1, 2, \ldots\}\). The total angular momentum per photon is spin plus orbital:

\[ J_z = \underbrace{(\pm 1)}_{\text{spin (SAM)}}\hbar + \underbrace{\ell}_{\text{orbital (OAM)}}\hbar = (\pm 1 + \ell)\hbar. \]

The doughnut profile

The intensity at a point is proportional to the electric energy density there, which is proportional to \(|\mathbf{E}|^2 = E_0^2\,|\mathcal{E}|^2 = E_0^2\,|u_p(r,z)|^2\) (the Jones vector factor is a unit vector and drops out). For \(\ell \ne 0\), the radial envelope \(u_p(r, z)\) vanishes on the beam axis: the phase factor \(e^{i\ell\varphi}\) is undefined at \(r = 0\) because there is no well-defined azimuthal angle there, so the field is forced to zero on-axis. The intensity therefore has a dark hole at the centre surrounded by a bright annular ring — the doughnut profile characteristic of all OAM beams. For \(\ell = 0\), \(u_p\) is non-zero at \(r=0\) and the standard Gaussian profile with a bright centre is recovered.

Do not confuse OAM with circular polarisation

Both involve something winding around an axis, and the confusion is common. They are completely different physical things:

  • Spin angular momentum (circular polarisation). At a fixed point in space, the electric field vector \(\mathbf{E}(t)\) rotates in the transverse plane as time advances. This is what the Jones vector \(\boldsymbol{\varepsilon} = (1, \pm i)^T/\!\sqrt{2}\) encodes: temporal rotation at a single location. Moving to a neighbouring point along the beam axis, the same rotation continues — every point on the axis sees the same circular dance. This rotation exists even for a plane wave (\(\ell = 0\)) with perfectly flat wavefronts.
  • Orbital angular momentum. At a fixed time and fixed \(z\), the phase of the field changes as you move azimuthally around the beam axis. Walking a full circle of circumference \(2\pi r\) around the axis, the phase \(\ell\varphi\) advances by \(2\pi\ell\). This is a spatial pattern frozen in the beam's cross-section — it has nothing to do with how \(\mathbf{E}\) evolves at any single point over time.

The Jones vector captures only the spin part; the OAM is encoded entirely in the spatial mode \(u_p(r, z)\,e^{i\ell\varphi}\). A laser beam can independently carry any combination:

  • Linear polarisation + \(\ell = 0\): no SAM, no OAM.
  • Linear polarisation + \(\ell = 3\): OAM = \(3\hbar\), no SAM.
  • Circular polarisation (LCP) + \(\ell = 0\): SAM = \(+\hbar\), no OAM.
  • Circular polarisation (LCP) + \(\ell = 2\): SAM = \(+\hbar\), OAM = \(2\hbar\), total = \(3\hbar\).

OAM multiplexing and channel capacity

Different values of \(\ell\) produce spatially orthogonal beam modes that do not interfere with each other — the integral \(\int_0^{2\pi} e^{i\ell\varphi} e^{-i\ell'\varphi}\,d\varphi = 2\pi\,\delta_{\ell\ell'}\) is zero whenever \(\ell \ne \ell'\). Distinct corkscrew modes can therefore be transmitted simultaneously on the same fibre or free-space link, each carrying an independent data stream. This OAM multiplexing is being actively developed to increase optical communication capacity. Since \(\ell\) can be any integer, there is no fundamental upper limit on the number of independent OAM channels; the constraint is engineering, not physics.

So the question "which direction does \(\mathbf{E}\) point?" — the one that opened the previous article — pulls a thread that unravels an unexpectedly rich structure: a complex unit vector in \(\mathbb{C}^2\) encoding both field orientation and spin; a Lorentz-invariant quantum number that makes photons unlike all massive particles; a mechanical torque on matter; a probe of molecular handedness; helical wavefronts carrying unbounded orbital angular momentum; and one of the most promising handles for increasing the information capacity of optical networks.

Do photons have a position?

For the truly curious: physicists have established that photons can't be said to have a position in the sense that a billiard ball does.[8] In quantum field theory, the fundamental observables of the electromagnetic field are the quantized field operators — not position coordinates. When we say "a photon has a 90% probability of being detected in this region," we're really talking about the electromagnetic energy density in that region. There's no position operator for photons analogous to the one for massive particles: the mathematics of a massless, relativistic quantum field simply doesn't support strict spatial localization. Photons can interact at a precise location — as they do when they register on a screen — but that's a detection event, not a pre-existing position being revealed.

The straight line of sight you rely on every waking moment is real, reliable, and extremely useful. It's just a statistical summary of a stranger story — one in which a single photon simultaneously explores every possible path, carries its own intrinsic spin compass, and only the paths that constructively interfere survive to dominate what we observe.

Stay tuned for the next article: Electricity does not work the way you think it does.

References

  1. Physics Stack Exchange. "What's the physical meaning of the statement that 'photons don't have positions'?" physics.stackexchange.com/questions/492711
  2. Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C., & Woerdman, J. P. "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes." Physical Review A 45, 8185 (1992). doi.org/10.1103/PhysRevA.45.8185