A linear polariser's transmission axis is a line through its centre

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Manifolds, charts and tangent spaces

Problem

A linear polariser's transmission axis is a line through its centre, not an arrow: rotating the polariser through π\pi radians about its normal leaves it in a physically identical state, so light polarised at angle θ\theta and at θ+π\theta+\pi are transmitted identically. The configuration space of "which axis direction" is therefore the set of angles θ\theta with θ\theta and θ+π\theta+\pi identified. ((a)) Chart AA: m=tan⁡θm=\tan\theta for θ∈(−π/2,π/2)\theta\in(-\pi/2,\pi/2). Show AA is a smooth bijection of this range of directions onto R\mathbb{R}, and say which single direction it excludes. ((b)) Chart BB: m′=tan⁡(θ−π/2)m'=\tan(\theta-\pi/2) for θ∈(0,π)\theta\in(0,\pi). Show m′=−cot⁡θm'=-\cot\theta and that BB excludes the direction θ=0≡π\theta=0\equiv\pi. ((c)) Find m′m' as a smooth function of mm on the overlap of the two domains. ((d)) Explain why {A,B}\{A,B\} makes the space of polariser-axis directions a smooth one-dimensional manifold, and why no chart of the form m=tan⁡(θ−θ0)m=\tan(\theta-\theta_0) can cover every axis direction by itself.

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