Explanation
The simplest type of birefringence is that of materials with uniaxial anisotropy. That is, the structure of the material is such that it has an axis of symmetry with all perpendicular directions optically equivalent. This axis is known as the optical axis of the material, and light with linear polarizations parallel and perpendicular to it has unequal indices of refraction, denoted ne and no, respectively, where the subscripts stand for extraordinary and ordinary. The names reflect the fact that, if unpolarized light enters the material at a nonzero acute angle to the optical axis, the component with polarization perpendicular to this axis will be refracted as per the standard law of refraction, while the complementary polarization component will refract at a nonstandard angle determined by the angle of entry and the birefringence
The light will therefore split into two linearly polarized beams, known as ordinary and extraordinary. Exceptions arise when the light propagates either along or orthogonal to the optical axis. In the first case, both polarizations and rays are ordinary and are not split. In the second case also, there is no splitting of the light into two separate directions, but the ordinary and extraordinary components travel at different speeds, and the effect is used to interconvert between linear and circular or elliptical polarizations.
The more general case of biaxially anisotropic materials, also known as trirefringent materials, is described by crystal optics which is substantially more complex.
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