birefringence

Birefringence, also known as double refraction, is an optical property of anisotropic materials in which a light beam entering the material is split into two polarized beams that travel at different speeds and follow different paths. This occurs because the refractive index that the material presents to light depends on the light's polarization and propagation direction.

In transparent crystals, the difference arises from the crystal's internal structure. In uniaxial crystals, such as calcite and quartz, there is a single optic axis along which the optical properties differ. There are two principal refractive indices: the ordinary index n_o, for the ray that experiences ordinary refraction, and the extraordinary index n_e, for the ray whose velocity depends on its propagation angle relative to the optic axis. In biaxial crystals, two optic axes exist and the behavior is more complex.

When a ray enters a birefringent material, it splits into an o-ray and an e-ray, each polarized perpendicularly. In many experiments, placing the material between crossed polarizers reveals interference colors or doubled images. The phase difference between the two rays after traversing a thickness t is Δφ = (2π/λ) Δn t, where Δn = n_e − n_o. This retardation underpins devices like waveplates.

Birefringence is quantified by Δn. Materials with large Δn are highly birefringent. It is widely used in optics and materials science: waveplates and retarders control polarization, polarizers exploit birefringence, and in geology optical mineralogy, birefringence helps identify crystals such as calcite.

Liquid crystals also exhibit birefringence, which is central to their electro-optic behavior.