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Double Refraction by a Uniaxial Crystal

Crystalline materials can have different indices of refraction in different crystallographic directions. Crystals belonging to the hexagonal, tetragonal, or rhombohedral classes are uniaxial, in that they possess a unique optical axis, most often coincident with the crystallographic axis. Light traveling through such an anisotropic medium can exhibit double refraction or birefringence, in which an unpolarized incident light ray splits into two polarized rays with mutually perpendicular planes of vibration. The ray with its electric field vibrating perpendicular to the optical axis is called the ordinary ray, and is characterized by an index of refraction . The ray that vibrates parallel to the optical axis is called the extraordinary ray, with its index of refraction designated . The birefringence is the difference Δ, which can be positive or negative.
The best-known birefringent crystal is the mineral calcite (Iceland spar), the colorless, transparent rhombohedral salt calcium carbonate, . The optical axis coincides with the three-fold axes of the equilateral groups, along which the ions are also situated. For 590 nm light, the yellow sodium-D lines, the indices for calcite are =1.658, =1.486, =0.172. Some other common uniaxial minerals are: tourmaline, beryl, quartz, ruby, sapphire and zircon. Birefringent materials find several applications in optics, for example Nicol prisms and quarter-wave plates.
In this Demonstration, the angle of incidence of a light ray entering a crystal can be varied, as can the two refractive indices (the default values are those of calcite). The direction of the optical axis is marked with a gray arrow. The ordinary ray is shown in orange, the extraordinary ray in blue. After refraction by the crystal, the light emerges in two parallel rays. You also have the option to display the polarizations of the various rays.

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Snapshot 1: conventional refraction when and are equal
Snapshot 2: double refraction by a hypothetical crystal with large birefringence
Snapshot 3: same crystal with polarizations shown

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