The Poincaré sphere and its stereographic projection
Every polarization state of light is a point on a sphere. Project that sphere from its north pole onto a plane and each point becomes a polarization ellipse: lines on the equator, circles at the poles, and ellipses turning opposite ways in the two hemispheres. Follow the six short steps, or just drag the point and explore.
Based on an interactive 3D figure by Miguel A. Alonso, presented at the POLYTOPO Heraklion Workshop 2026 (acknowledgement).
View σ
drag the point or the ellipse · drag elsewhere to rotate
latitude θlongitude φprojection lineimage of the equatoranticlockwiseclockwise
The ellipse in the beam
Landmark states
H, V, D, A lie on the equator. C+ and C− are the two circular states at the poles.
The state in numbers
Jones vector
Projected point ζ
Axis ratio b/a
Type any value and press Enter: the sphere follows. Stokes values are rescaled onto the sphere.
Linear
Conventions used here
Light travels along +z with time dependence e−iωt. The Jones vector (Ex, Ey) gives S₁ = |Ex|² − |Ey|², S₂ = 2 Re(Ex* Ey), S₃ = 2 Im(Ex* Ey), normalised to |S| = 1.
Facing the source (σ = +1) shows x to the right and y up. Then S₃ > 0 turns anticlockwise. From behind (σ = −1) is the mirror image, so every sense reverses.
Textbooks disagree on whether S₃ > 0 is called right- or left-handed (optics and engineering conventions differ). That is why the poles are labelled C+ and C− here, by the sign of S₃.
How to use
Drag the white point on the sphere, or drag the ellipse on the plane. Drag anywhere else to turn the scene.
The six tour steps each end with a small task. The page checks it for you and ticks it off.
▶ Field animates the electric field. ⧉ Link copies a link to the current state, handy for teaching.
Step 6 is a game: find the mystery state from its ellipse alone.
Reading the colours
Blue: on the equator. The ellipse collapses to a line and the field only oscillates (linear polarization).
Red: the field turns anticlockwise in the current view. Green: clockwise. The two hemispheres turn opposite ways, and switching the view swaps them.
Why a sphere, and why project it?
A polarization state is a Jones vector up to an overall phase, i.e. a point of the complex projective line ℂP¹. Topologically this is the 2-sphere: the Poincaré sphere.
Stereographic projection from the north pole identifies the sphere with the extended plane ℂ ∪ {∞}. It is conformal and sends circles to circles. The equator lands on the circle of radius 2, and the north pole goes off to infinity.
The projected point is a complex number ζ whose size is twice the ratio of the two circular components, |ζ| = 2|E₊/E₋|, and whose angle is twice the tilt of the ellipse.
The doubled angles (φ = 2ψ, θ = 2χ) are the same doubling that links spinors to rotations (SU(2) → SO(3)), as in Dirac's belt trick in the Outreach section.
The formulas behind the picture
Point on the sphere (Stokes vector): S = (cos θ cos φ, cos θ sin φ, sin θ), with ψ = φ/2, χ = θ/2.
Jones vector, up to a global phase: E = (cos ψ cos χ − i sin ψ sin χ, sin ψ cos χ + i cos ψ sin χ)
Field in the beam, facing the source: x(t) = cos t cos χ cos ψ − sin t sin χ sin ψy(t) = sin t sin χ cos ψ + cos t cos χ sin ψ. From behind, x changes sign. Axis ratio tan |χ|.
Projection from N = (0, 0, 1) onto the plane z = −1: ζ = R eiφ, R = 2 tan(π/4 + χ) = 2 |E₊ / E₋|.
Based on
This simulation is adapted from an interactive 3D figure made in Mathematica by Prof. Miguel A. Alonso (Institut Fresnel, Centrale Méditerranée & University of Rochester). 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.
Web adaptation
Yorgis Androulakis, IACM / FORTH
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