Editing Birefringence (section)

==Explanation==
[[File:Calcite and polarizing filter.gif|right|thumb|Doubly refracted image as seen through a calcite crystal, seen through a rotating polarizing filter illustrating the opposite polarization states of the two images.]]

A mathematical description of wave propagation in a birefringent medium is presented [[#Theory|below]]. Following is a qualitative explanation of the phenomenon.

===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.

===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.

===Double refraction===
When an arbitrary beam of light strikes the surface of a birefringent material at non-normal incidence, the polarization component normal to the optic axis (ordinary ray) and the other linear polarization (extraordinary ray) will be refracted toward somewhat different paths. Natural light, so-called [[Polarization (waves)#Unpolarized and partially polarized light|unpolarized light]], consists of equal amounts of energy in any two orthogonal polarizations. Even linearly polarized light has some energy in both polarizations, unless aligned along one of the two axes of birefringence. According to [[Snell's law]] of refraction, the two angles of refraction are governed by the effective [[refractive index]] of each of these two polarizations. This is clearly seen, for instance, in the [[Wollaston prism]] which separates incoming light into two linear polarizations using prisms composed of a birefringent material such as [[calcite]].

The different angles of refraction for the two polarization components are shown in the figure at the top of this page, with the optic axis along the surface (and perpendicular to the [[plane of incidence]]), so that the angle of refraction is different for the {{mvar|p}} polarization (the "ordinary ray" in this case, having its electric vector perpendicular to the optic axis) and the {{mvar|s}} polarization (the "extraordinary ray" in this case, whose electric field polarization includes a component in the direction of the optic axis). In addition, a distinct form of double refraction occurs, even with normal incidence, in cases where the optic axis is not along the refracting surface (nor exactly normal to it); in this case, the [[dielectric polarization]] of the birefringent material is not exactly in the direction of the wave's [[electric field]] for the extraordinary ray. The direction of power flow (given by the [[Poynting vector]]) for this [[inhomogenous wave]] is at a finite angle from the direction of the [[wave vector]] resulting in an additional separation between these beams. So even in the case of normal incidence, where one would compute the angle of refraction as zero (according to Snell's law, regardless of the effective index of refraction), the energy of the extraordinary ray is propagated at an angle. If exiting the crystal through a face parallel to the incoming face, the direction of both rays will be restored, but leaving a ''shift'' between the two beams. This is commonly observed using a piece of calcite cut along its natural cleavage, placed above a paper with writing, as in the above photographs. On the contrary, [[waveplate]]s specifically have their optic axis ''along'' the surface of the plate, so that with (approximately) normal incidence there will be no shift in the image from light of either polarization, simply a relative [[phase shift]] between the two light waves.