Chiral nematic liquid crystals also exhibit birefringence – however due to their chirality the manner in which they split light into components is slightly different.
When light is travelling along the helical axis of a chiral nematic it does not undergo regular (‘linear’) birefringence. This is because as the director vector rotates the two components rotate along with it, and having travelled through one 360° pitch the components experience exactly the same overall refractive index. The result is that one component does not end up travelling faster than the other and so we see no optical path difference.
However, in a chiral material light can become circularly polarised. In this case the light is split not into two perpendicular components, but instead into two components that are constantly rotating in opposite directions. The difference between linear polarisation (as in a nematic) and circular polarisation (as in a chiral nematic) is illustrated in the demonstration below: