This renderer evaluates the exact Laguerre-Gauss mode LGp,ℓ — radial term (r√2/w)^|ℓ|, an associated Laguerre polynomial L_p^|ℓ|(2r²/w²) computed by recurrence, and a Gaussian envelope e^(−r²/w²) — combined with the helical phase factor e^(iℓφ). Four views expose different aspects of the same complex field: raw intensity (a doughnut with p+1 rings), the phase itself as an HSV colour wheel, and two interference patterns (against a spherical or tilted plane reference wave) whose spiral arms or fork-grating prongs directly reveal the topological charge ℓ.
Optical vortex beams: light whose wavefront twists helically around a dark central singularity, carrying orbital angular momentum of ℓℏ per photon — a property distinct from ordinary spin angular momentum (polarisation).
Drag Topological charge ℓ, Radial index p, and Beam waist w to reshape the mode; switch between Intensity, Phase, Int. spiral, and Int. fork views to see the doughnut, the HSV phase wheel, or interference patterns; change Colormap for contrast, and watch OAM/photon, phase windings, and ring count update live in Stats.
Because each photon in one of these beams carries ℓℏ of orbital angular momentum, optical tweezers using vortex beams can set trapped microscopic particles spinning — a mechanical effect from a purely optical structure, with no polarisation involved.
It's the integer number of times the beam's phase winds by 2π around the beam axis; it directly sets the beam's orbital angular momentum (ℓℏ per photon) and determines both the size of the dark central core and the number of arms in interference patterns.
At the exact axis the phase is undefined (a singularity), so the field amplitude must go to zero there for continuity — this forces the characteristic doughnut-shaped intensity profile whenever ℓ≠0.
p sets the number of radial nodes in the Laguerre polynomial term, producing p+1 concentric bright rings in the intensity profile instead of the single ring seen when p=0.
Interfering the vortex beam with a reference wave (spherical for the spiral pattern, tilted plane for the fork pattern) converts the invisible helical phase into a visible geometric feature — the number of spiral arms or the number of extra prongs at the fork's centre exactly equals |ℓ|.
Vortex beams carrying OAM are used in optical tweezers to rotate trapped particles, in free-space and fibre optical communications to encode extra data channels via different ℓ values, and in super-resolution microscopy techniques such as STED.