Birefringence

A calcite crystal laid upon a paper with some letters showing the double refraction

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:

Δn = n_e - n_o

where n_o is the refractive index for the ordinary ray and n_e 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 n_o², n_o², and n_e² along the three orthogonal principle axes of polarization. (Or, sometimes, only two axes are considered, corresponding to a single propagation direction.)

Cellophane paper is a cheap birefringent material.

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

See also: crystal optics, John Kerr.