How does a twisted-nematic liquid crystal layer rotate polarized light by 90 degrees?
In nematic arrangements the crystals are randomly positioned with all their long axes pointing in essentially the same direction, given by a vector known as the director. In smectic arrangements the crystals are further constrained to parallel planes; in smectic A the director is perpendicular to these planes, while in smectic C the director is at some other angle to the planes.
In a twisted-nematic (TN) cell, the rotation of polarized light is achieved through the specific structural arrangement of the liquid crystal molecules between two substrate plates.12
The inner surfaces of these plates are coated with a polymer layer that is rubbed in a single direction to align the liquid crystal molecules. By arranging the top and bottom plates so that their alignment directions are perpendicular to each other, the liquid crystal molecules are forced to adopt a smooth, 90-degree twisted configuration across the gap between the plates.12
When light passes through the first polarizer, its polarization is aligned with the molecules at the bottom of the layer. As the light travels through the cell, the twisted structure of the liquid crystal acts as a waveguide, gradually rotating the polarization of the light along with the molecular twist. By the time the light reaches the top of the layer, its polarization has been rotated by 90 degrees, allowing it to pass through the second, crossed polarizer.21