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Study of orthogonal polarization approximation for ordinary and extraordinary rays in ray tracing for uniaxial crystals

We study the assumption of orthogonal polarization for ordinary and extraordinary rays inside uniaxial crystals, using a closed-form expression for the angle between the polarizations. We highlight that orthogonality holds only when the crystal axis is rather coplanar to the ordinary and extraordinary wave vectors or orthogonal to the extraordinary wave vector, which are the same conditions for the extraordinary ray to stay in the plane of incidence. We show that in general the deviation from orthogonality is rather small, for it depends on the diﬀerence of the optical indices, and that negative and positive crystals have diﬀerent behaviors. Using the paraxial approximation we derive expressions for the polarizations of rays and point out that even under the paraxial regime orthogonality does not hold. Speciﬁc examples for calcite and quartz are given.