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🌀 Optical Vortex Beams

Interactive Laguerre–Gauss vortex beam simulator: render LG mode intensity, helical phase and spiral/fork interference. See orbital angular momentum and topological charge live.

Optics & Light3DAdvanced60 FPS

About this simulation

This renderer evaluates the exact Laguerre-Gauss mode LG_p,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.

🔬 What it shows

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).

🎮 How to use

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.

💡 Did you know?

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.

Frequently asked questions

What is topological charge ℓ?

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.

Why is there a dark spot at the centre of the beam?

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.

What does the radial index p control?

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.

Why do the spiral and fork interference patterns reveal ℓ?

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|.

What is orbital angular momentum (OAM) used for in real optics?

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.

⚙ Under the hood

Laguerre–Gauss beams carry orbital angular momentum: a helical phase exp(iℓφ) makes a doughnut intensity with a dark core. Interfere them to see the spiral and fork patterns that measure OAM.

Canvas 2DOpticsLaguerre-GaussOAMVortex

3D · Three.js / WebGL renderer · 60 FPS target · runs fully client-side, no install