Optical Vortex Beams & OAM

A beam with a hole burned in the middle

A plain laser beam has flat wavefronts and a bright Gaussian spot. An optical vortex beam looks completely different: a dark, perfectly black point on the axis, surrounded by one or more bright rings. The darkness is not a filter or a mask — it is a mathematical necessity. The beam's phase winds around the axis by 2πℓ for every full loop, and at the exact centre every possible phase value converges on one point, which is impossible for a well-defined wave. The amplitude is forced to zero there, leaving a phase singularity threaded through the beam like the eye of a hurricane.

The cleanest mathematical description is the Laguerre-Gauss (LG) mode, a solution of the paraxial wave equation in cylindrical coordinates. Its field carries an explicit helical phase term:

E(r, φ, z) ∝ r^|ℓ| · exp(-r²/w²) · exp(i·ℓ·φ) · exp(i·k·z)
                                    ↑
                       azimuthal phase — winds ℓ times around the axis

Orbital angular momentum: ℓħ per photon

In 1992 Les Allen and colleagues showed that a beam with this exp(iℓφ) term carries a well-defined orbital angular momentum (OAM) of ℓħ per photon, on top of whatever spin angular momentum its polarization carries. The integer ℓ is the beam's topological charge — it can be any positive or negative whole number, and it is what makes vortex beams fundamentally different from ordinary polarized light, whose spin only ever takes two values.

live demo · a helical phase front winding around a dark core● LIVE

How you make one: spiral plates and forked holograms

A spiral phase plate is the most direct route: a transparent disc machined so its thickness increases smoothly around the axis, adding exactly the right extra path length to imprint the exp(iℓφ) twist onto a plane wave passing through. More common in the lab is a computer-generated hologram — a diffraction grating with a fork-shaped dislocation at its centre, displayed on a spatial light modulator (SLM) or etched into glass. Diffracting a Gaussian beam off the fork sends different diffraction orders out with different, well-defined values of ℓ, letting one device generate a whole family of vortex beams just by changing the pattern shown on the SLM. A third route, cylindrical-lens mode converters, reshapes an ordinary higher-order Hermite-Gauss laser mode into an LG mode using two tilted cylindrical lenses.

Reading the charge back out

You cannot see ℓ directly in the plain intensity ring — a charge of +3 and -3 look identical as a doughnut. To measure it, you interfere the vortex beam with a reference wave. Against a tilted plane wave, the interference pattern is a spiral with exactly |ℓ| arms; against an ordinary Gaussian beam, it produces a fork pattern whose number of extra tines again equals |ℓ|, and the direction the spiral turns (or the way the fork opens) reveals the sign.

Why anyone builds these

Optical tweezers built from vortex beams transfer their orbital angular momentum to trapped microscopic particles, spinning them without any mechanical contact — useful for micro-rotors and for probing the mechanics of cells. Because ℓ can in principle take any integer value, a single photon can encode far more than one bit, which is the basis of proposed high-capacity free-space and fibre communication links that multiplex many OAM channels on the same wavelength, and of high-dimensional quantum key distribution protocols that use OAM states instead of, or alongside, polarization.

Frequently asked questions

Why is the centre of a vortex beam completely dark?

Because the phase is undefined exactly on the beam axis — every azimuthal angle meets there at once. A wave cannot have an undefined phase, so the amplitude is forced to zero at that point, leaving a dark, singular core surrounded by a bright ring.

How is orbital angular momentum different from spin angular momentum in light?

Spin angular momentum comes from circular polarization and only takes two values, +ħ or -ħ per photon, regardless of the beam's shape. Orbital angular momentum comes from a helically twisted wavefront and can take any integer multiple ℓħ of ħ, giving an essentially unbounded set of distinguishable states for a single photon.

How do you tell the topological charge of a vortex beam just by looking at it?

Interfere it with a plane reference wave and count the spiral arms in the resulting fringe pattern, or interfere it with an ordinary Gaussian beam and count the forks in the fringes — both counts equal |ℓ|, and the direction of the spiral or the way the forks open tells you the sign.

Try it live

Everything above runs in your browser — open Optical Vortex Beams and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

▶ Open Optical Vortex Beams simulation

Optical Vortex Beams: Light With a Twist