Math 2.0

Optics, electromagnetic theory Polarization and crystal optics Object

Poincaré sphere

Builds on Stokes parametersSphereRiemann sphere

Description

Let (S0,S1,S2,S3)(S_0, S_1, S_2, S_3) be the Stokes parameters of fully polarized light, so that S02=S12+S22+S32S_0^2 = S_1^2 + S_2^2 + S_3^2 and S0>0S_0 > 0. The point (S1,S2,S3)/S0(S_1, S_2, S_3)/S_0 lies on the unit sphere of ℝ3\R^3. The Poincaré sphere is this unit sphere, seen as the set of all states of full polarization. Let ψ\psi be the orientation angle of the polarization ellipse, and let tan⁡χ\tan \chi be its ratio of axes, with a sign for the handedness. In these spherical coordinates, S1=S0cos⁡2χcos⁡2ψS_1 = S_0 \cos 2\chi \cos 2\psi, S2=S0cos⁡2χsin⁡2ψS_2 = S_0 \cos 2\chi \sin 2\psi and S3=S0sin⁡2χS_3 = S_0 \sin 2\chi.

Remarks

Linear polarizations lie on the equator, and the two circular polarizations are the poles. Horizontal polarization is the point (1,0,0)(1, 0, 0) and vertical polarization is (−1,0,0)(-1, 0, 0). Two orthogonal polarization states are always antipodal points. The angles are doubled: turning a linear polarizer by 90∘90^\circ moves the point half way around the equator. A lossless wave plate acts on the sphere as a rotation. Partially polarized light gives a point inside the ball, at distance equal to the degree of polarization from the centre. The sphere is also the complex projective line of the ratio Ey/ExE_y/E_x of the field amplitudes, that is, a copy of the Riemann sphere.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Riemann sphere, concept, no evidence levelRiemann sphereconceptSphere, concept, no evidence levelSphereconceptStokes parameters, concept, no evidence levelStokes parametersconceptPoincaré sphere, object, no evidence levelPoincaré sphereobject
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  1. Riemann sphere concept
  2. Sphere concept
  3. Stokes parameters concept
  4. Poincaré sphere object

Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Stokes_parameters
About this page
TagJ1T8NZCVTB
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WritingW0 stub
MSC78A10
Instance ofSphere
UsesStokes parameters, Sphere, Riemann sphere
Changed3 November 2026

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