Light refraction from a uniaxial crystal with arbitrary optic axis orientation using matrix representation

We have studied theoretically the refraction of a light beam in a uniaxial crystal plate by observing the electric field vector components, for an arbitrary orientation of optic axis with respect to plane of incidence and plane of interface. The electric field components for ordinary and extraordinary light beams are studied with varying the angle of optic axis with respect to plane of incidence and crystal surface and also with angle of incidence, in order to find the optimized position of optic axis orientation for various application prospects. We have used the matrix formulation to transform and simplify the electric field components from principal coordinate of crystal to the laboratory coordinate frame of crystal plate. It was observed that when optic axis is in the plane of incidence and optic axis is in the surface of crystal plate, ordinary and extraordinary beams have nonzero x and y components. While in other cases when optic axis is perpendicular to the plane of incidence and perpendicular to the surface of crystal plate, ordinary beam and extraordinary beams have only y and x components, respectively.