The Poincaré sphere represents polarisation states of light of fixed intensity by…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Manifolds, charts and tangent spaces

Problem

The Poincaré sphere represents polarisation states of light of fixed intensity by points on the unit sphere with Cartesian coordinates (S1,S2,S3)=(cos⁡2χcos⁡2ψ, cos⁡2χsin⁡2ψ, sin⁡2χ),(S_1, S_2, S_3) = (\cos2\chi\cos2\psi,\ \cos2\chi\sin2\psi,\ \sin2\chi), where ψ\psi is the orientation angle of the polarisation ellipse and χ\chi is its ellipticity angle. A stereographic chart on the sphere, projecting from the north pole, is (u,v)=(S1,S2)1−S3.(u, v) = \frac{(S_1, S_2)}{1 - S_3}. Find the chart coordinates (u,v)(u, v) of the polarisation state with ψ=30∘\psi = 30^\circ and χ=15∘\chi = 15^\circ.

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