A plate of an unidentified transparent crystal

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Polarisation and anisotropic media · Polarisation and anisotropic media

Optics and Photonics, Ch Birefringence and crystal optics

Problem

A plate of an unidentified transparent crystal, thickness d=4.00 mmd = 4.00\ \text{mm}, is suspected to be optically active. It is set up between a polariser and an analyser with light of vacuum wavelength λ=546 nm\lambda = 546\ \text{nm} (the mercury green line) travelling along a direction for which the crystal passes right- and left-circular light unchanged in form, with indices nRn_R and nLn_L. With the input polariser set at azimuth 0∘0^\circ and the analyser crossed with it at 90∘90^\circ, light comes through; turning the analyser to azimuth 57.6∘57.6^\circ restores extinction. With the input polariser reset to 50.0∘50.0^\circ, extinction requires the analyser at azimuth 107.6∘107.6^\circ. Azimuths are measured anticlockwise as seen looking towards the source (CV6), and the plate is thin enough that the smaller of the two possible turns is the real one. ((a)) Starting from the decomposition of linear light into right- and left-circular Jones vectors (Section 9.7), and letting each component pick up its own propagation phase over the thickness dd from its own index, show that if the plate passes such circular states with indices nR≠nLn_R \ne n_L, the emerging light is linear at every input azimuth, turned by the same fixed angle θrot=π(nL−nR)d/λ\theta_{\mathrm{rot}} = \pi(n_L-n_R)d/\lambda. Explain briefly why the two measurements quoted above are exactly what this predicts, and why they would not be consistent with the plate being a simple retarder. ((b)) Find the crystal's circular index difference nL−nRn_L - n_R.

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