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Normal glasses are homogeneous and isotropic, that is, they have the same refractive index no matter what direction light travels through them. Uniaxial materials, such as Calcite, have a crystal axis which defines an axis of symmetry. These materials refract rays differently, depending on the polarization state of the ray and the angle the ray makes with respect to the crystal axis. Therefore there are two possible refraction angles for any ray, representing two orthogoanl polarization states. This phenomenon is known as double refraction or birefringence.

Birefringent materials always bend rays according to Snell's law, but the effective index of refraction in the media depends on the input polarization state of the ray and the angle the ray makes to the crystal axis. "Ordinary" rays are refracted by:

nsinq = n_osinq'

where n_o is the ordinary refractive index. which is just Snell's law. "Extraordinary" rays are refracted by:

nsinq = n(q_w)sinq'

which is also Snell's law, but note that the refractive index is now a function of the angle  q_w, which is the angle between the crystal axis vector a and the refracted wavevector k.

Now here is the hard part. The ray vector S, which points in the direction of energy flow, does not follow the wave vector k but instead makes a small angle with respect to it. In normal glasses k and S are the same vector and we just keep track of k. In birefringent media we must consider the ray and wave vectors as being different. The vectors k and S both lie in the same plane as the crystal axis vector a, and

cosq_w = k•a

The effective refractive index seen by the extraordinary ray is defined by:

(1/n(q_w))² =  (cosq_w /n_o)² + (sinq_w /n_e)²

where n_e is the extraordinary index of refraction.