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When dealing with linear birefringence we can see the following effects...
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When light undergoes circular polarisation it is instead split into two constantly rotating components.

They are referred to as left-handed circularly polarised light (rotating clockwise) and right-handed circularly polarised light (rotating anticlockwise).
When circularly polarised light is passed through a chiral nematic, only one of the two rotating polarisation directions will have the same 'handedness' as the chiral molecule.

This results in a difference in the liquid crystal's refractive index for the two components - so one will travel faster than the other.

Once again an optical path difference will be created, and the polarisation of the light will have rotated upon exiting the sample.