Birefringence

Birefringence, or double refraction, is the division of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals, depending on the polarization of the light. This is explained by assigning two different refractive indices to the material for different polarizations. The birefringence is quantified by:

where is the refractive index for the ordinary ray and is the refractive index for the extraordinary ray.

More generally, an anisotropic dielectric material has a dielectric constant that is a rank-2 tensor (3 by 3 matrix). A birefringent material corresponds to the special cases of a real-symmetric dielectric tensor ε with eigenvalues of , , and along the three orthogonal principle axes of polarization. (Or, sometimes, only two axes are considered, corresponding to a single propagation direction.)

(In principle, birefringence could also arise in magnetic, not dielectric, materials, but substantial variations in magnetic permeability are rare at optical frequencies.)