Editing Birefringence (section)

===Biaxial materials===
The case of so-called biaxial crystals is substantially more complex.<ref name="landaulifshitz">Landau, L. D., and Lifshitz, E. M., ''Electrodynamics of Continuous Media'', Vol. 8 of the ''Course of Theoretical Physics'' 1960 (Pergamon Press), §79</ref> These are characterized by ''three'' refractive indices corresponding to three principal axes of the crystal. For most ray directions, ''both'' polarizations would be classified as extraordinary rays but with different effective refractive indices. Being extraordinary waves, the direction of power flow is not identical to the direction of the wave vector in either case.

The two refractive indices can be determined using the [[index ellipsoid]]s for given directions of the polarization. Note that for biaxial crystals the index ellipsoid will ''not'' be an ellipsoid of revolution ("[[spheroid]]") but is described by three unequal principle refractive indices {{math|''n''<sub>α</sub>}}, {{math|''n''<sub>β</sub>}} and {{math|''n''<sub>γ</sub>}}. Thus there is no axis around which a rotation leaves the optical properties invariant (as there is with uniaxial crystals whose index ellipsoid ''is'' a spheroid).

Although there is no axis of symmetry, there are ''two'' optical axes or ''binormals'' which are defined as directions along which light may propagate without birefringence, i.e., directions along which the wavelength is independent of polarization.<ref name="landaulifshitz"/> For this reason, birefringent materials with three distinct refractive indices are called ''biaxial''. Additionally, there are two distinct axes known as ''optical ray axes'' or ''biradials'' along which the group velocity of the light is independent of polarization.