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When dealing with linear birefringence we can see the following effects...
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.