Birefringent Materials Optic Axis

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My answer to the question is in the attached word document.

A transparent material like glass allows light to pass through it. The velocity of light inside any transparent material like glass is decided by the refractive index of the material. The more the absolute refractive index, the less is the velocity of light in the medium. But in a material like glass, the refractive index doesn't depend on the direction of passage of light inside the material, or on the plane of polarization of the light with respect to the material. The refractive index is the same in all directions. Or the material is isotropic.
But this is not the situation in many materials, like calcite and quartz. These materials are crystalline in nature. The refractive index of these materials is not a constant one. The refractive index can vary with the direction light make with certain internal symmetry directions of the crystal. In other words the material is anisotropic.
Let us take the example of a calcite crystal. A calcite crystal has the shape of rhombohedra. Each of the six faces of this crystal is a parallelogram with angles 78o 5' and 101o 55'. When light falls on the face of a calcite crystal, it doesn't get transmitted like a single beam as in glass. Instead the incident beam of light gets split up into two beams and these two beams travels in different directions. One of these two beams has a constant velocity in all directions, or this ray of light has an ordinary behavior. This ray is called as the Ordinary ray (O-ray). The other ray of light behaves extra-ordinarily. It has different refractive indices in different directions and therefore travels with different velocities. The velocity of the ordinary ray is determined by the crystal symmetry.
In a uniaxial crystal there is only one optic axis. Optic axis is the direction inside the crystal along which both the rays (O-ray and E-ray) have equal velocities and ...