Birefringence

Birefringence, or double refraction, is the optical property of an anisotropic material to exhibit two different refractive indices for two different polarisations of light propagating through it. An unpolarised beam entering a birefringent crystal (calcite, quartz, many plastics under stress) splits into two linearly polarised beams — the ordinary ray, which obeys Snell's law with an isotropic index n_o, and the extraordinary ray, whose index n_e depends on direction of propagation relative to the crystal's optic axis. The two emerge separated in space and typically with a phase difference, which is what allows wave plates, polarising beam splitters, and optical modulators to be built.

The microscopic origin is crystal anisotropy: the dielectric tensor ε is not scalar but a 3×3 tensor, and the refractive index depends on the angle between the wave's E-field and the crystal's principal axes. Uniaxial crystals (calcite, quartz, ice) have two of the three principal dielectric values equal and one different, so there is a special "optic axis" along which the anisotropy vanishes. Biaxial crystals (mica, topaz) have all three values different. Birefringence can also be induced in normally isotropic materials by stress (stress-birefringence, used for photoelastic analysis of mechanical stress), by an electric field (Kerr effect, used for fast optical shutters), or by a magnetic field (Cotton–Mouton effect). Every liquid-crystal display on every laptop and phone screen uses controlled electrically-induced birefringence as its fundamental switching mechanism.