Editing Birefringence (section)

===Uniaxial materials===
The simplest type of birefringence is described as ''uniaxial'', meaning that there is a single direction governing the optical anisotropy whereby all directions perpendicular to it (or at a given angle to it) are optically equivalent. Thus rotating the material around this axis does not change its optical behaviour. This special direction is known as the [[Optic axis of a crystal|optic axis]] of the material. Light propagating parallel to the optic axis (whose polarization is always ''perpendicular'' to the optic axis) is governed by a refractive index {{math|''n''<sub>o</sub>}} (for "ordinary") regardless of its specific polarization. For rays with any other propagation direction, there is one linear polarization that is perpendicular to the optic axis, and a ray with that polarization is called an ''ordinary ray'' and is governed by the same refractive index value {{math|''n''<sub>o</sub>}}. For a ray propagating in the same direction but with a polarization perpendicular to that of the ordinary ray, the polarization direction will be partly in the direction of (parallel to) the optic axis, and this ''extraordinary ray'' will be governed by a different, ''direction-dependent'' refractive index. Because the index of refraction depends on the polarization when unpolarized light enters a uniaxial birefringent material, it is split into two beams travelling in different directions, one having the polarization of the ordinary ray and the other the polarization of the extraordinary ray.
The ordinary ray will always experience a refractive index of {{math|''n''<sub>o</sub>}}, whereas the refractive index of the extraordinary ray will be in between {{math|''n''<sub>o</sub>}} and {{math|''n''<sub>e</sub>}}, depending on the ray direction as described by the [[index ellipsoid]]. The magnitude of the difference is quantified by the birefringence<ref>{{Cite book |last=Ehlers |first=Ernest G. |title=Optical Mineralogy: Theory and Technique |publisher=Blackwell Scientific Publications |year=1987 |isbn=0-86542-323-7 |volume=1 |location=Palo Alto |pages=28}}</ref>

:<math>\Delta n=n_\mathrm{e}-n_\mathrm{o}\,.</math>

The propagation (as well as [[Fresnel equations|reflection coefficient]]) of the ordinary ray is simply described by {{math|''n''<sub>o</sub>}} as if there were no birefringence involved. The extraordinary ray, as its name suggests, propagates unlike any wave in an isotropic optical material. Its refraction (and reflection) at a surface can be understood using the effective refractive index (a value in between {{math|''n''<sub>o</sub>}} and {{math|''n''<sub>e</sub>}}). Its power flow (given by the [[Poynting vector]]) is not exactly in the direction of the [[wave vector]]. This causes an additional shift in that beam, even when launched at normal incidence, as is popularly observed using a crystal of [[calcite]] as photographed above. Rotating the calcite crystal will cause one of the two images, that of the extraordinary ray, to rotate slightly around that of the ordinary ray, which remains fixed.{{Verify source|date=February 2020}}

When the light propagates either along or orthogonal to the optic axis, such a lateral shift does not occur. In the first case, both polarizations are perpendicular to the optic axis and see the same effective refractive index, so there is no extraordinary ray. In the second case the extraordinary ray propagates at a different phase velocity (corresponding to {{math|''n''<sub>e</sub>}}) but still has the power flow in the direction of the [[wave vector]]. A crystal with its optic axis in this orientation, parallel to the optical surface, may be used to create a [[waveplate]], in which there is no distortion of the image but an intentional modification of the state of polarization of the incident wave. For instance, a [[Waveplate#Quarter-wave plate|quarter-wave plate]] is commonly used to create [[circular polarization]] from a linearly polarized source.