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Polarisation & Topology

Three interactive simulations, from a single point on a sphere to beams that wrap around it
POLYTOPO · Interactive Simulations

Polarisation & Topology

The polarisation of light has a geometry of its own. Every polarisation state is a point on a sphere, every optical element moves that point in a definite way, and a structured beam can wrap its whole cross-section around the sphere a whole number of times. That number is a topological invariant, the same idea that protects skyrmions in magnets. These three interactive simulations build the picture step by step, each one starting where the previous one ends.

Series · 3 simulations
Part 1 · One state, one point
The Poincaré Sphere

Drag a point over the sphere and watch its polarisation ellipse change: lines on the equator, circles at the poles. Then project the sphere onto a plane and see every state laid out at once.

Open Part 1 → Part 2 · Moving the point
The Optics Bench

Build a bench from wave plates, polarisers and depolarisers. Wave plates turn the sphere, polarisers project onto it, depolarisers pull the point inside, and a closed loop leaves behind a phase that comes from geometry alone.

Open Part 2 → Part 3 · Covering the sphere
Polarisation Skyrmions

Give every point of a beam its own polarisation and the beam becomes a map onto the sphere. Some beams cover it completely, once or several times: the number of coverings is the skyrmion number.

Open Part 3 →

New to the subject? Start with Part 1 and follow the parts in order.

Based on

The series grew out of an original Mathematica notebook by Prof. Miguel A. Alonso (Institut Fresnel, Centrale Méditerranée & University of Rochester), which Part 1 adapts for the web. He presented it in his talk “Covering spheres and higher order compact manifolds with optical fields” at the POLYTOPO Workshop on Emerging Applications of Topological Textures, Heraklion, Crete, 16–19 September 2026. We thank him for sharing it.

Created by

Yorgis Androulakis, IACM / FORTH: web adaptation of Part 1; concept, design & development of Parts 2 and 3.

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Polarisation & Topology