View source for Birefringence

{{short description|Optical phenomenon}}
[[File:Crystal on graph paper.jpg|right|thumb|A [[calcite]] crystal laid upon a graph paper with blue lines showing the double refraction]]
[[File:Calcite and polarizing filter.gif|right|thumb|200px|Doubly refracted image as seen through a calcite crystal, seen through a rotating polarizing filter illustrating the opposite polarization states of the two images.]]

'''Birefringence''' is the [[optics|optical]] property of a material having a [[refractive index]] that depends on the [[Polarization (waves)|polarization]] and propagation direction of [[light]].<ref>{{cite web|url=http://www.olympusmicro.com/primer/lightandcolor/birefringence.html |title = Olympus Microscopy Resource Center |publisher=Olympus America Inc. |accessdate=2011-11-13}}</ref> These optically [[anisotropic]] materials are said to be '''birefringent''' (or '''birefractive'''). The birefringence is often quantified as the maximum difference between refractive indices exhibited by the material. [[Crystal]]s with non-cubic [[crystal structure]]s are often birefringent, as are [[plastic]]s under [[mechanical stress]].

Birefringence is responsible for the phenomenon of '''double refraction''' whereby a [[Ray (optics)|ray]] of light, when incident upon a birefringent material, is split by polarization into two rays taking slightly different paths. This effect was first described by the [[Denmark|Danish]] scientist [[Rasmus Bartholin]] in 1669, who observed it<ref>See:
* Erasmus Bartholin, [https://books.google.com/books?id=F7RAAAAAcAAJ&pg=PP11#v=onepage&q&f=false ''Experimenta crystalli islandici disdiaclastici quibus mira & infolita refractio detegitur''] [Experiments on birefringent Icelandic crystal through which is detected a remarkable and unique refraction] (Copenhagen, Denmark: Daniel Paulli, 1669). 
* Erasmus Bartholin (January 1, 1670) [https://books.google.com/books?id=Zl5FAAAAcAAJ&pg=PA2141#v=onepage&q&f=false "An account of sundry experiments made and communicated by that learn'd mathematician, Dr. Erasmus Bartholin, upon a chrystal-like body, sent to him out of Island,"] ''Philosophical Transactions of the Royal Society of London'', '''5''' : 2041-2048.</ref> in [[calcite]], a crystal having one of the strongest birefringences. However it was not until the 19th century that [[Augustin-Jean Fresnel]] described the phenomenon in terms of polarization, understanding light as a wave with field components in transverse polarizations (perpendicular to the direction of the wave vector).

==Explanation==
[[File:Positively birefringent material.svg|right|thumb|Incoming light in the parallel ({{mvar|p}}) polarization sees a different effective [[index of refraction]] than light in the perpendicular ({{mvar|s}}) polarization, and is thus [[Snell's law|refracted]] at a different angle.|alt=Incoming light in the perpendicular ({{mvar|s}}) polarization sees a different effective index of refraction than light in the parallel ({{mvar|p}}) polarization, and is thus refracted at a different angle.]]
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 whereas 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 behavior. 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 would be 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>}}. However, 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 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:
:<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. However 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>}}). However 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.

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, however, 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, the polarizations corresponding to the ordinary and extraordinary rays generally take somewhat different paths. [[Polarization (waves)#Unpolarized and partially polarized light|Unpolarized light]] consists of equal amounts of energy in any two orthogonal polarizations, and even polarized light (except in special cases) will have some energy in each of these polarizations. According to [[Snell's law]] of refraction, the angle of refraction will be governed by the effective [[refractive index]] which is different between these two polarizations. This is clearly seen, for instance, in the [[Wollaston prism]] which is designed to separate incoming light into two linear polarizations using 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 the 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" with a polarization component along the optic axis). In addition, a distinct form of double refraction occurs in cases where the optic axis is not along the refracting surface (nor exactly normal to it); in this case the di[[electric 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 the angle of refraction is zero (according to Snell's law, regardless of effective index of refraction), the energy of the extraordinary ray may be propagated at an angle. This is commonly observed using a piece of calcite cut appropriately with respect to its optic axis, placed above a paper with writing, as in the above two photographs.

==Terminology==
Much of the work involving polarization preceded the understanding of light as a transverse [[electromagnetic wave]], and this has affected some terminology in use. Isotropic materials have symmetry in all directions and the refractive index is the same for any polarization direction. An anisotropic material is called "birefringent" because it will generally refract a single incoming ray in two directions, which we now understand correspond to the two different polarizations. This is true of either a uniaxial or biaxial material.

In a uniaxial material, one ray behaves according to the normal law of refraction (corresponding to the ordinary refractive index), so an incoming ray at normal incidence remains normal to the refracting surface. However, as explained above, the other polarization can be deviated from normal incidence, which cannot be described using the law of refraction. This thus became known as the ''extraordinary ray''. The terms "ordinary" and "extraordinary" are still applied to the polarization components perpendicular to and not perpendicular to the optic axis respectively, even in cases where no double refraction is involved.

A material is termed ''uniaxial'' when it has a single direction of symmetry in its optical behavior, which we term the optic axis. It also happens to be the axis of symmetry of the index ellipsoid (a spheroid in this case). The index ellipsoid could still be described according to the refractive indices, {{math|''n''<sub>α</sub>}}, {{math|''n''<sub>β</sub>}} and {{math|''n''<sub>γ</sub>}}, along three coordinate axes, however in this case two are equal. So if {{math|''n''<sub>α</sub> {{=}} ''n''<sub>β</sub>}} corresponding to the {{mvar|x}} and {{mvar|y}} axes, then the extraordinary index is {{math|''n''<sub>γ</sub>}} corresponding to the {{mvar|z}} axis, which is also called the ''optic axis'' in this case.

However materials in which all three refractive indices are different are termed ''biaxial'' and the origin of this term is more complicated and frequently misunderstood. In a uniaxial crystal, different polarization components of a beam will travel at different phase velocities, ''except'' for rays in the direction of what we call the optic axis. Thus the optic axis has the particular property that rays in that direction do ''not'' exhibit birefringence, with all polarizations in such a beam experiencing the same index of refraction. It is very different when the three principal refractive indices are all different; then an incoming ray in any of those principle directions will still encounter two different refractive indices. But it turns out that there are two special directions (at an angle to all of the 3 axes) where the refractive indices for different polarizations are again equal. For this reason, these crystals were designated as ''biaxial'', with the two "axes" in this case referring to ray directions in which propagation does not experience birefringence.

=== Fast and slow rays ===
In a birefringent material, a wave consists of two polarization components which generally are governed by different effective refractive indices. The so-called ''slow ray'' is the component for which the material has the higher effective refractive index (slower phase velocity), while the ''fast ray'' is the one with a lower effective refractive index. When a beam is incident on such a material from air (or any material with a lower refractive index), the slow ray is thus refracted more towards the normal than the fast ray. In the figure at the top of the page, it can be seen that refracted ray with ''s'' polarization (with its electric vibration in the direction of the optic axis, thus the extraordinary ray<ref>Born & Wolf, 2002, pp.{{nnbsp}}807–8. (In 19th-century terminology, the ordinary ray is said to be polarized in the plane of the optic axis; but this "[[plane of polarization]]" is the plane ''perpendicular'' to the vibration; cf.&nbsp;Fresnel, 1827, tr.&nbsp;Hobson, p.{{nnbsp}}318.)</ref>) is the slow ray in this case.

Using a thin slab of that material at normal incidence, one would implement a [[waveplate]]. In this case there is essentially no spatial separation between the polarizations, however the phase of the wave in the parallel polarization (the slow ray) will be retarded with respect to the perpendicular polarization. These directions are thus known as the slow axis and fast axis of the waveplate.

===Positive or negative===
Uniaxial birefringence is classified as positive when the extraordinary index of refraction {{math|''n''<sub>e</sub>}} is greater than the ordinary index {{math|''n''<sub>o</sub>}}. Negative birefringence means that {{math|Δ''n'' {{=}} ''n''<sub>e</sub> − ''n''<sub>o</sub>}} is less than zero.<ref>''Brad Amos''. [http://homepage.ntlworld.com/w.amos2/Brad%20Amos%27s%20Website/BR1.pdf Birefringence for facetors I: what is birefringence?] {{webarchive |url=https://web.archive.org/web/20131214030317/http://homepage.ntlworld.com/w.amos2/Brad%20Amos%27s%20Website/BR1.pdf |date=December 14, 2013 }} First published in StoneChat, the Journal of the UK Facet Cutter's Guild. January–March. edition 2005.</ref> In other words, the polarization of the fast (or slow) wave is perpendicular to the optic axis when the birefringence of the crystal is positive (or negative, respectively). In the case of biaxial crystals, all three of the principal axes have different refractive indices, so this designation does not apply. But for any defined ray direction one can just as well designate the fast and slow ray polarizations.

== Sources of optical birefringence ==
While birefringence is usually obtained using an anisotropic crystal, it can result from an [[optical isotropy|optically isotropic]] material in a few ways:
* ''Stress birefringence'' results when isotropic materials are stressed or deformed (i.e., stretched or bent) causing a loss of physical isotropy and consequently a loss of isotropy in the material's permittivity tensor.
* ''[[Optical rotation|Circular birefringence]]'' in liquids where there is an [[enantiomeric excess]] in a solution containing a molecule which has [[Stereoisomerism|stereo isomers]].
* Form birefringence, whereby structure elements such as rods, having one refractive index, are suspended in a medium with a different refractive index. When the lattice spacing is much smaller than a wavelength, such a structure is described as a [[metamaterial]].
* By the [[Kerr effect]], whereby an applied electric field induces birefringence at optical frequencies through the effect of [[nonlinear optics]];
* By the [[Faraday effect]], where a magnetic field causes some materials to become ''[[Optical rotation|circularly birefringent]]'' (having slightly different indices of refraction for left- and right-handed [[circular polarization]]s), making the material ''optically active'' until the field is removed;
* By the self or forced alignment into thin films of [[amphiphile|amphiphilic]] molecules such as [[lipids]], some [[surfactants]] or [[liquid crystals]]

== Common birefringent materials ==
[[File:Food Polarization-Dierking.jpg|right|thumb|Light polarization shown on clear polystyrene cutlery between crossed polarizers]]

The best characterized birefringent materials are [[crystals]]. Due to their specific [[crystal structures]] their refractive indices are well defined. Depending on the symmetry of a crystal structure (as determined by one of the 32 possible crystallographic point groups), crystals in that group may be forced to be isotropic (not birefringent), to have uniaxial symmetry, or neither in which case it is a biaxial crystal. The crystal structures permitting uniaxial and biaxial birefringence are noted in the two tables, below, listing the two or three principal refractive indices (at wavelength 590&nbsp;nm) of some better known crystals.<ref name=hypertextbook/>

Many [[plastic]]s are birefringent because their molecules are "frozen" in a stretched conformation when the plastic is molded or extruded.<ref>{{cite journal|doi=10.1002/pen.10347 | volume=38 | issue=10 | title=The use of birefringence for predicting the stiffness of injection molded polycarbonate discs | year=1998 | journal=Polymer Engineering & Science | pages=1770–1777 | last1 = Neves | first1 = N. M.}}</ref> For example, ordinary [[cellophane]] is birefringent. [[Polarizer]]s are routinely used to detect stress in plastics such as [[polystyrene]] and [[polycarbonate]].

[[Cotton]] fiber is birefringent because of high levels of cellulosic material in the fiber's secondary cell wall.

Polarized light microscopy is commonly used in biological tissue, as many biological materials are birefringent. Collagen, found in cartilage, tendon, bone, corneas, and several other areas in the body, is birefringent and commonly studied with polarized light microscopy.<ref>{{cite journal | last1 = Wolman | first1 = M. | last2 = Kasten | first2 = F. H. | year = 1986 | title = Polarized light microscopy in the study of the molecular structure of collagen and reticulin | url = | journal = Histochemistry | volume = 85 | issue = | pages = 41–49 | doi=10.1007/bf00508652}}</ref> Some proteins are also birefringent, exhibiting form birefringence.<ref>{{cite journal | last1 = Sano | first1 = Y | year = 1988 | title = Optical anistropy of bovine serum albumin | url = | journal = J. Colloid Interface Sci. | volume = 124 | issue = | pages = 403–407 | doi=10.1016/0021-9797(88)90178-6|bibcode = 1988JCIS..124..403S }}</ref>

Inevitable manufacturing imperfections in [[optical fiber]] leads to birefringence, which is one cause of [[polarization mode dispersion|pulse broadening]] in [[fiber-optic communication]]s. Such imperfections can be geometrical (lack of circular symmetry), due to stress applied to the optical fiber and/or due to bending of the fiber. Birefringence is ''intentionally'' introduced (for instance, by making the cross-section elliptical) in order to produce [[polarization-maintaining optical fiber]]s.

In addition to anisotropy in the electric polarizability ([[electric susceptibility]]), anisotropy in the magnetic polarizability ([[Permeability (electromagnetism)|magnetic permeability]]) can also cause birefringence. However, at optical frequencies, values of magnetic permeability for natural materials are not measurably different from [[Vacuum permeability|{{math|''µ''<sub>0</sub>}}]], so this is not a source of optical birefringence in practice.
{{col-begin}}
{{col-2}}
{| class="wikitable sortable" style="margin: 0em 0em 1em 1em;"
|+ Uniaxial crystals, at 590&nbsp;nm<ref name=hypertextbook>{{cite web|last=Elert|first=Glenn|title=Refraction|work=The Physics Hypertextbook|url=http://physics.info/refraction/}}</ref>
|-
! Material || Crystal system || {{math|''n''<sub>o</sub>}} || {{math|''n''<sub>e</sub>}} || {{math|Δ''n''}}
|-
| [[barium borate]] BaB<sub>2</sub>O<sub>4</sub>|| [[Trigonal]] ||1.6776 ||1.5534 ||−0.1242
|-
| [[beryl]] Be<sub>3</sub>Al<sub>2</sub>(SiO<sub>3</sub>)<sub>6</sub>|| [[Hexagonal lattice system|Hexagonal]] ||1.602 ||1.557 ||−0.045
|-
| [[calcite]] CaCO<sub>3</sub> || [[Trigonal]] || 1.658 || 1.486 || −0.172
|-
| [[ice]] H<sub>2</sub>O || [[Hexagonal lattice system|Hexagonal]] || 1.309 || 1.313 || +0.004
|-
| [[lithium niobate]] LiNbO<sub>3</sub>|| [[Trigonal]] || 2.272|| 2.187|| −0.085
|-
| [[magnesium fluoride]] MgF<sub>2</sub>|| [[Tetragonal]] || 1.380|| 1.385|| +0.006
|-
| [[quartz]] SiO<sub>2</sub>|| [[Trigonal]] || 1.544|| 1.553|| +0.009
|-
| [[ruby]] Al<sub>2</sub>O<sub>3</sub>|| [[Trigonal]] || 1.770|| 1.762|| −0.008
|-
| [[rutile]] TiO<sub>2</sub>|| [[Tetragonal]] || 2.616|| 2.903|| +0.287
|-
| [[sapphire]] Al<sub>2</sub>O<sub>3</sub>|| [[Trigonal]] || 1.768|| 1.760|| −0.008
|-
| [[silicon carbide]] SiC|| [[Hexagonal lattice system|Hexagonal]] || 2.647|| 2.693|| +0.046
|-
| [[tourmaline]] (complex silicate)|| [[Trigonal]] || 1.669|| 1.638|| −0.031
|-
| [[zircon]], high ZrSiO<sub>4</sub>|| [[Tetragonal]] || 1.960|| 2.015|| +0.055
|-
| zircon, low ZrSiO<sub>4</sub>|| [[Tetragonal]] || 1.920|| 1.967|| +0.047
|}
{{col-2}}
{| class="wikitable sortable" style="margin: 0em 0em 1em 1em;"
|+ Biaxial crystals, at 590&nbsp;nm<ref name=hypertextbook/>
|-
! Material ||Crystal system || {{math|''n''<sub>α</sub>}} || {{math|''n''<sub>β</sub>}} || {{math|''n''<sub>γ</sub>}}
|-
|[[borax]] Na<sub>2</sub>(B<sub>4</sub>O<sub>5</sub>)(OH)<sub>4</sub>·8H<sub>2</sub>O ||[[Monoclinic]]||1.447 ||1.469 ||1.472
|-
|[[Magnesium sulfate|epsom salt]] MgSO<sub>4</sub>·7H<sub>2</sub>O || [[Monoclinic]]||1.433 ||1.455 ||1.461
|-
|[[mica]], [[biotite]] K(Mg,Fe)<sub>3</sub>(AlSi<sub>3</sub>O<sub>10</sub>)(F,OH)<sub>2</sub> ||[[Monoclinic]] ||1.595 ||1.640 ||1.640
|-
|mica, [[muscovite]] KAl<sub>2</sub>(AlSi<sub>3</sub>O<sub>10</sub>)(F,OH)<sub>2</sub> ||[[Monoclinic]] ||1.563 ||1.596 ||1.601
|-
|[[olivine]] (Mg,Fe)<sub>2</sub>SiO<sub>4</sub>||[[Orthorhombic]]||1.640 ||1.660 ||1.680
|-
|[[perovskite]] CaTiO<sub>3</sub> ||[[Orthorhombic]]||2.300 ||2.340 ||2.380
|-
|[[topaz]] Al<sub>2</sub>SiO<sub>4</sub>(F,OH)<sub>2</sub> ||[[Orthorhombic]]||1.618 ||1.620 ||1.627
|-
|[[ulexite]] NaCaB<sub>5</sub>O<sub>6</sub>(OH)<sub>6</sub>·5H<sub>2</sub>O ||[[Triclinic]]||1.490 ||1.510 ||1.520
|}
{{col-end}}

== Measurement ==
Birefringence and other polarization-based optical effects (such as [[optical rotation]] and linear or [[circular dichroism]]) can be measured by measuring the changes in the polarization of light passing through the material. These measurements are known as [[polarimetry]]. Polarized light microscopes, which contain two polarizers that are at 90° to each other on either side of the sample, are used to visualize birefringence. The addition of quarter-wave plates permit examination of circularly polarized light. Birefringence measurements have been made with phase-modulated systems for examining the transient flow behavior of fluids.<ref>Frattini, P., Fuller, G., "A note on phase-modulated flow birefringence: a promising rheo-optical method", J. Rheol., 28: 61 (1984).</ref><ref>Doyle, P., Shaqfeh, E. S. G., Spiegelberg, S. H., McKinley, G. H., "Relaxation of dilute polymer solutions following extensional flow", J. Non-Newtonian Fluid Mech., 86:79–110 (1998).</ref>

Birefringence of [[lipid bilayers]] can be measured using [[dual polarization interferometry]]. This provides a measure of the degree of order within these fluid layers and how this order is disrupted when the layer interacts with other biomolecules.
{{clear}}

== Applications ==
[[Image:LCD layers.svg|left|thumb|250px| Reflective twisted-nematic [[liquid-crystal display]]. Light reflected by surface (6) (or coming from a [[backlight]]) is horizontally polarized (5) and passes through the liquid-crystal modulator (3) sandwiched in between transparent layers (2, 4) containing electrodes. Horizontally polarized light is blocked by the vertically oriented polarizer (1), except where its polarization has been rotated by the liquid crystal (3), appearing bright to the viewer.]]
Birefringence is used in many optical devices. [[Liquid-crystal display]]s, the most common sort of [[flat panel display]], cause their pixels to become lighter or darker through rotation of the polarization (circular birefringence) of linearly polarized light as viewed through a sheet polarizer at the screen's surface. Similarly, [[electro-optic modulator|light modulators]] modulate the intensity of light through [[Pockels cell|electrically induced birefringence]] of polarized light followed by a polarizer. The [[Lyot filter]] is a specialized narrowband spectral filter employing the wavelength dependence of birefringence. [[Wave plate]]s are thin birefringent sheets widely used in certain optical equipment for modifying the polarization state of light passing through it.

Birefringence also plays an important role in [[second-harmonic generation]] and other [[Nonlinear optics|nonlinear optical components]], as the crystals used for this purpose are almost always birefringent. By adjusting the angle of incidence, the effective refractive index of the extraordinary ray can be tuned in order to achieve [[Nonlinear optics#Phase matching|phase matching]], which is required for efficient operation of these devices.

===Medicine===
Birefringence is utilized in medical diagnostics. One powerful accessory used with optical microscopes is a pair of crossed [[polarizer|polarizing]] filters. Light from the source is polarized in the {{mvar|x}} direction after passing through the first polarizer, but above the specimen is a polarizer (a so-called ''analyzer'') oriented in the {{mvar|y}} direction. Therefore, no light from the source will be accepted by the analyzer, and the field will appear dark. However areas of the sample possessing birefringence will generally couple some of the {{mvar|x}}-polarized light into the {{mvar|y}} polarization; these areas will then appear bright against the dark background. Modifications to this basic principle can differentiate between positive and negative birefringence.

[[File:Fluorescent uric acid.JPG|right|thumb|Urate crystals, with the crystals' long axis seen as horizontal in this view being parallel to that of a red compensator filter. These appear as yellow, and are thereby of negative birefringence.]]
For instance, needle aspiration of fluid from a [[gout]]y joint will reveal negatively birefringent monosodium [[urate]] crystals. [[Calcium pyrophosphate]] crystals, in contrast, show weak positive birefringence.<ref>{{cite journal |vauthors=Hardy RH, Nation B |title=Acute gout and the accident and emergency department |journal=Arch Emerg Med |volume=1 |issue=2 |pages=89–95 |date=June 1984 |pmid=6536274 |pmc=1285204 |doi= 10.1136/emj.1.2.89|url=}}</ref> Urate crystals appear yellow, and calcium pyrophosphate crystals appear blue when their long axes are aligned parallel to that of a red compensator filter,<ref>[http://emedicine.medscape.com/article/336054-workup#a0721 The Approach to the Painful Joint Workup] Author: Alan N. Baer; Chief Editor: Herbert S. Diamond. Updated: Nov 22, 2010.</ref> or a crystal of known birefringence is added to the sample for comparison.

Birefringence can be observed in [[amyloid]] plaques such as are found in the brains of [[Alzheimer's disease|Alzheimer's]] patients when stained with a dye such as Congo Red. Modified proteins such as [[immunoglobulin]] light chains abnormally accumulate between cells, forming fibrils. Multiple folds of these fibers line up and take on a beta-pleated sheet [[Chemical structure|conformation]]. [[Congo red]] dye [[intercalation (biochemistry)|intercalates]] between the folds and, when observed under polarized light, causes birefringence.

In [[ophthalmology]], binocular [[retinal birefringence screening]] of the ''Henle fibers'' (photoreceptor axons that go radially outward from the fovea) provides a reliable detection of [[strabismus]] and possibly also of [[Amblyopia#Refractive or anisometropic amblyopia|anisometropic amblyopia]].<ref>{{cite journal|author1=Reed M. Jost|author2=Joost Felius|author3=Eileen E. Birch|title=High sensitivity of binocular retinal birefringence screening for anisometropic amblyopia without strabismus|journal=Journal of American Association for Pediatric Ophthalmology and Strabismus (JAAPOS)|volume=18|number=4|pages=e5–e6|date=August 2014|doi=10.1016/j.jaapos.2014.07.017}}</ref> Furthermore, [[scanning laser polarimetry]] utilises the birefringence of the [[optic nerve]] fibre layer to indirectly quantify its thickness, which is of use in the assessment and monitoring of [[glaucoma]].

Birefringence characteristics in [[sperm head]]s allow the selection of spermatozoa for [[intracytoplasmic sperm injection]].<ref>{{cite journal |author=Gianaroli L. |author2=Magli M. C. |author3=Ferraretti A. P. |title=Birefringence characteristics in sperm heads allow for the selection of reacted spermatozoa for intracytoplasmic sperm injection |journal=Fertil. Steril. |volume= 93|issue= 3|pages= 807–13|date=December 2008 |pmid=19064263 |doi=10.1016/j.fertnstert.2008.10.024 |url=|display-authors=etal}}</ref> Likewise, ''[[zona imaging]]'' uses birefringence on [[oocyte]]s to select the ones with highest chances of successful pregnancy.<ref>{{cite journal |author=Ebner T. |author2=Balaban B. |author3=Moser M. |title=Automatic user-independent zona pellucida imaging at the oocyte stage allows for the prediction of preimplantation development |journal=Fertil. Steril. |volume= 94|issue= 3|pages= 913–920|date=May 2009 |pmid=19439291 |doi=10.1016/j.fertnstert.2009.03.106 |url=|display-authors=etal}}</ref> Birefringence of particles biopsied from pulmonary nodules indicates [[silicosis]].

Dermatologists use dermatascopes to view skin lesions. Dermatascopes use polarized light, allowing the user to view crystalline structures corresponding to dermal collagen in the skin. These structures may appear as shiny white lines or rosette shapes and are only visible under polarized [[dermoscopy]].

===Stress-induced birefringence===
[[Image:Birefringence Stress Plastic.JPG|left|thumb|250px|Color pattern of a plastic box with "frozen in" [[mechanical stress]] placed between two crossed [[polarizer]]s]]
[[Isotropic]] solids do not exhibit birefringence. However, when they are under [[mechanical stress]], birefringence results. The stress can be applied externally or is "frozen in" after a birefringent plastic ware is cooled after it is manufactured using [[injection molding]]. When such a sample is placed between two crossed polarizers, colour patterns can be observed, because polarization of a light ray is rotated after passing through a birefringent material and the amount of rotation is dependent on wavelength. The experimental method called [[photoelasticity]] used for analyzing stress distribution in solids is based on the same principle. There has been recent research on using stress induced birefringence in a glass plate to generate an [[Optical vortex]] and full Poincare beams (optical beams that have every possible polarization states across its cross-section).<ref>{{Cite journal|last=Beckley|first=Amber M.|last2=Brown|first2=Thomas G.|last3=Alonso|first3=Miguel A.|date=2010-05-10|title=Full Poincaré beams|url=https://www.osapublishing.org/abstract.cfm?uri=oe-18-10-10777|journal=Optics Express|language=EN|volume=18|issue=10|pages=10777–10785|doi=10.1364/OE.18.010777|issn=1094-4087|bibcode=2010OExpr..1810777B}}</ref>

===Other cases of birefringence===
[[File:Rutile birefringence.gif|right|thumb|Birefringent [[rutile]] observed in different polarizations using a rotating polarizer (or ''analyzer'')]]
Birefringence is observed in anisotropic [[elastic deformation|elastic]] materials. In these materials, the two polarizations split according to their effective refractive indices, which are also sensitive to stress.

The study of birefringence in [[shear wave]]s traveling through the solid Earth (the Earth's liquid core does not support shear waves) is widely used in [[seismology]].

Birefringence is widely used in mineralogy to identify rocks, minerals, and gemstones.
{{clear}}

==Theory==
[[File:Birifrangence k-surface.gif|400px|thumb|right|Surface of the allowed k-vectors for a fixed frequency for a biaxial crystal (see {{EquationNote|7|eq. 7}}).]]
In an isotropic medium (including free space) the so-called [[electric displacement]] ({{math|'''D'''}}) is just proportional to the electric field ({{math|'''E'''}}) according to {{math|'''D''' {{=}} ''ɛ'''''E'''}} where the material's [[permittivity]] {{math|''ε''}} is just a [[Scalar (physics)|scalar]] (and equal to {{math|''n''<sup>2</sup>[[permittivity of free space|''ε''<sub>0</sub>]]}} where {{math|''n''}} is the [[index of refraction]]). However, in an anisotropic material exhibiting birefringence, the relationship between {{math|'''D'''}} and {{math|'''E'''}} must now be described using a [[tensor]] equation:

{{NumBlk|:|<math>\mathbf D = \boldsymbol\varepsilon \mathbf E </math>|{{EquationRef|1}}}}

where {{math|'''ε'''}} is now a 3&nbsp;×&nbsp;3 permittivity tensor. We assume linearity and no [[magnetic permeability]] in the medium: {{math|''μ'' {{=}} [[permeability of free space|''μ''<sub>0</sub>]]}}. The electric field of a plane wave of angular frequency {{math|''ω''}} can be written in the general form:

{{NumBlk|:|<math>\mathbf{E}=\mathbf{E}_0 e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)}</math>|{{EquationRef|2}}}}

where {{math|'''r'''}} is the position vector, {{math|''t''}} is time, and {{math|'''E'''<sub>0</sub>}} is a vector describing the electric field at {{math|'''r''' {{=}} 0}}, {{math|''t'' {{=}} 0}}. Then we shall find the possible [[wave vector]]s {{math|'''k'''}}. By combining [[Maxwell's equations]] for {{math|∇ × '''E'''}} and {{math|∇ × '''H'''}}, we can eliminate {{math|'''H''' {{=}} {{sfrac|1|''μ''<sub>0</sub>}}'''B'''}} to obtain:

{{NumBlk|:|<math>-\nabla \times \nabla \times \mathbf{E}= \mu_0\frac{\partial^2 }{\partial t^2} \mathbf{D}</math> |{{EquationRef|3a}}}}

With no free charges, Maxwell's equation for the divergence of {{math|'''D'''}} vanishes:
{{NumBlk|:|<math>\nabla \cdot \mathbf{D}=0 </math>|{{EquationRef|3b}}}}

We can apply the vector identity [[Vector calculus identities#Curl of the curl|{{math|∇ × (∇ × '''A''') {{=}} ∇(∇ ⋅ '''A''') − ∇<sup>2</sup>'''A'''}}]] to the left hand side of {{EquationNote|3a|eq. 3a}}, and use the spatial dependence in which each differentiation in {{math|''x''}} (for instance) results in multiplication by {{math|''ik<sub>x</sub>''}} to find:
{{NumBlk|:|<math> - \nabla \times \nabla \times \mathbf{E} =(\mathbf{k} \cdot \mathbf{E}) \mathbf{k} - (\mathbf{k} \cdot \mathbf{k}) \mathbf{E}</math>|{{EquationRef|3c}}}}

The right hand side of {{EquationNote|3a|eq. 3a}} can be expressed in terms of {{math|'''E'''}} through application of the permittivity tensor {{math|'''ε'''}} and noting that differentiation in time results in multiplication by {{math|−''iω''}}, {{EquationNote|3a|eq. 3a}} then becomes:
{{NumBlk|:|<math>(-\mathbf{k}\cdot\mathbf{k})\mathbf{E} + (\mathbf{k} \cdot \mathbf{E}) \mathbf{k}= -\mu_0 \omega^2 (\boldsymbol{\varepsilon} \mathbf{E})</math>|{{EquationRef|4a}}}}

Applying the differentiation rule to {{EquationNote|3b|eq. 3b}} we find:
{{NumBlk|:|<math>\mathbf{k} \cdot \mathbf{D} =0</math>|{{EquationRef|4b}}}}

{{EquationNote|4b|Eq. 4b}} indicates that {{math|'''D'''}} is orthogonal to the direction of the wavevector {{math|'''k'''}}, even though that is no longer generally true for {{math|'''E'''}} as would be the case in an isotropic medium. {{EquationNote|4b|Eq. 4b}} will not be needed for the further steps in the following derivation.

Finding the allowed values of {{math|'''k'''}} for a given {{math|''ω''}} is easiest done by using [[Cartesian coordinates]] with the {{math|''x''}}, {{math|''y''}} and {{math|''z''}} axes chosen in the directions of the symmetry axes of the crystal (or simply choosing {{math|''z''}} in the direction of the optic axis of a uniaxial crystal), resulting in a diagonal matrix for the permittivity tensor {{math|'''ε'''}}:
{{NumBlk|:|<math>\mathbf{\varepsilon}= \varepsilon_0 \begin{bmatrix} n_x^2 & 0 & 0 \\ 0& n_y^2 & 0 \\ 0& 0& n_z^2 \end{bmatrix} </math>|{{EquationRef|4c}}}}

where the diagonal values are squares of the refractive indices for polarizations along the three principal axes {{math|''x''}}, {{math|''y''}} and {{math|''z''}}. With {{math|'''ε'''}} in this form, and substituting in the speed of light {{math|''c''}} using {{math|''c''<sup>2</sup> {{=}} {{sfrac|1|''μ''<sub>0</sub>''ε''<sub>0</sub>}}}}, {{EquationNote|4a|eq. 4a}} becomes
{{NumBlk|:|<math>\left(-k_x^2-k_y^2-k_z^2 \right)E_x + k_x^2 E_x + k_xk_yE_y + k_xk_zE_z = -\frac{\omega^2n_x^2}{c^2} E_x</math>|{{EquationRef|5a}}}}

where {{math|''E<sub>x</sub>''}}, {{math|''E<sub>y</sub>''}}, {{math|''E<sub>z</sub>''}} are the components of {{math|'''E'''}} (at any given position in space and time) and {{math|''k<sub>x</sub>''}}, {{math|''k<sub>y</sub>''}}, {{math|''k<sub>z</sub>''}} are the components of {{math|'''k'''}}. Rearranging, we can write (and similarly for the {{math|''y''}} and {{math|''z''}} components of {{EquationNote|4a|eq. 4a}})
{{NumBlk|:|<math>\left(-k_y^2-k_z^2+\frac{\omega^2n_x^2}{c^2}\right)E_x + k_xk_yE_y + k_xk_zE_z =0</math>|{{EquationRef|5b}}}}
{{NumBlk|:|<math>k_xk_yE_x + \left(-k_x^2-k_z^2+\frac{\omega^2n_y^2}{c^2}\right)E_y + k_yk_zE_z =0</math>|{{EquationRef|5c}}}}
{{NumBlk|:|<math>k_xk_zE_x + k_yk_zE_y + \left(-k_x^2-k_y^2+\frac{\omega^2n_z^2}{c^2}\right)E_z =0</math>|{{EquationRef|5d}}}}

This is a set of linear equations in {{math|''E<sub>x</sub>''}}, {{math|''E<sub>y</sub>''}}, {{math|''E<sub>z</sub>''}}, so it can have a nontrivial solution (that is, one other than {{math|'''E''' {{=}} 0}}) as long as the following [[determinant]] is zero:
{{NumBlk|:|<math>\begin{vmatrix}
\left(-k_y^2-k_z^2+\frac{\omega^2n_x^2}{c^2}\right) & k_xk_y & k_xk_z \\
k_xk_y & \left(-k_x^2-k_z^2+\frac{\omega^2n_y^2}{c^2}\right) & k_yk_z \\
k_xk_z & k_yk_z & \left(-k_x^2-k_y^2+\frac{\omega^2n_z^2}{c^2}\right) \end{vmatrix} =0</math>|{{EquationRef|6}}}}

Evaluating the determinant of {{EquationNote|6|eq. 6}}, and rearranging the terms, we obtain

{{NumBlk|:|<math> \frac{\omega^4}{c^4} - \frac{\omega^2}{c^2}\left(\frac{k_x^2+k_y^2}{n_z^2}+\frac{k_x^2+k_z^2}{n_y^2}+\frac{k_y^2+k_z^2}{n_x^2}\right) + \left(\frac{k_x^2}{n_y^2n_z^2}+\frac{k_y^2}{n_x^2n_z^2}+\frac{k_z^2}{n_x^2n_y^2}\right)\left(k_x^2+k_y^2+k_z^2\right)=0 </math>|{{EquationRef|7}}}}
 
In the case of a uniaxial material, choosing the optic axis to be in the {{math|''z''}} direction so that {{math|''n<sub>x</sub>'' {{=}} ''n<sub>y</sub>'' {{=}} ''n''<sub>o</sub>}} and {{math|''n<sub>z</sub>'' {{=}} ''n''<sub>e</sub>}}, this expression can be factored into

{{NumBlk|:|<math>\left(\frac{k_x^2}{n_\mathrm{o}^2}+\frac{k_y^2}{n_\mathrm{o}^2}+\frac{k_z^2}{n_\mathrm{o}^2} -\frac{\omega^2}{c^2}\right)\left(\frac{k_x^2}{n_\mathrm{e}^2}+\frac{k_y^2}{n_\mathrm{e}^2}+\frac{k_z^2}{n_\mathrm{o}^2} -\frac{\omega^2}{c^2}\right)=0</math>|{{EquationRef|8}}}}

Setting either of the factors in {{EquationNote|8|eq. 8}} to zero will define an [[ellipsoid]]al surface<ref group=note>Although related, note that this is not the same as the [[index ellipsoid]].</ref> in the space of wavevectors {{math|'''k'''}} that are allowed for a given {{math|''ω''}}. The first factor being zero defines a sphere; this is the solution for so-called ordinary rays, in which the effective refractive index is exactly {{math|''n''<sub>o</sub>}} regardless of the direction of {{math|'''k'''}}. The second defines a [[spheroid]] symmetric about the {{math|''z''}} axis. This solution corresponds to the so-called extraordinary rays in which the effective refractive index is in between {{math|''n''<sub>o</sub>}} and {{math|''n''<sub>e</sub>}}, depending on the direction of {{math|'''k'''}}. Therefore, for any arbitrary direction of propagation (other than in the direction of the optic axis), two distinct wavevectors {{math|'''k'''}} are allowed corresponding to the polarizations of the ordinary and extraordinary rays.

For a biaxial material a similar but more complicated condition on the two waves can be described;<ref>Born & Wolf, 2002, §15.3.3</ref> the locus of allowed {{math|'''k'''}} vectors (the ''wavevector surface'') is a 4th-degree two-sheeted surface, so that in a given direction there are generally two permitted {{math|'''k'''}} vectors (and their opposites).<ref name=berry-jeffrey-2007>M.V. Berry and M.R.&nbsp;Jeffrey, [https://michaelberryphysics.files.wordpress.com/2013/07/berry400.pdf "Conical diffraction: Hamilton's diabolical point at the heart of crystal optics"], in E.&nbsp;Wolf&nbsp;(ed.), ''Progress in Optics'', vol.{{nnbsp}}50, Amsterdam: Elsevier, 2007, {{nowrap|pp.{{tsp}}13–50}}, {{doi|10.1016/S0079-6638(07)50002-8}}, at {{nowrap|pp.{{tsp}}20–21}}.</ref> By inspection one can see that {{EquationNote|6|eq. 6}} is generally satisfied for two positive values of {{math|''ω''}}. Or, for a specified optical frequency {{math|''ω''}} and direction normal to the wavefronts {{math|{{sfrac|'''k'''|{{abs|'''k'''}}}}}}, it is satisfied for two [[wavenumbers]] (or propagation constants) {{math|{{abs|'''k'''}}}} (and thus effective refractive indices) corresponding to the propagation of two linear polarizations in that direction.

When those two propagation constants are equal then the effective refractive index is independent of polarization, and there is consequently no birefringence encountered by a wave traveling in that particular direction. For a uniaxial crystal, this is the optic axis, the ±''z'' direction according to the above construction. But when all three refractive indices (or permittivities), {{math|''n<sub>x</sub>''}}, {{math|''n<sub>y</sub>''}} and {{math|''n<sub>z</sub>''}} are distinct, it can be shown that there are exactly two such directions, where the two sheets of the wave-vector surface touch;{{r|berry-jeffrey-2007}} these directions are not at all obvious and do not lie along any of the three principal axes ({{math|''x''}}, {{math|''y''}}, {{math|''z''}} according to the above convention). Historically that accounts for the use of the term "biaxial" for such crystals, as the existence of exactly two such special directions (considered "axes") was discovered well before polarization and birefringence were understood physically. However these two special directions are generally not of particular interest; biaxial crystals are rather specified by their three refractive indices corresponding to the three axes of symmetry.

A general state of polarization launched into the medium can always be decomposed into two waves, one in each of those two polarizations, which will then propagate with different wavenumbers {{math|{{abs|'''k'''}}}}. Applying the different phase of propagation to those two waves over a specified propagation distance will result in a generally ''different'' net polarization state at that point; this is the principle of the [[waveplate]] for instance. However with a waveplate, there is no spatial displacement between the two rays as their {{math|'''k'''}}  vectors are still in the same direction. That is true when each of the two polarizations is either normal  to the optic axis (the ordinary ray) or parallel to it (the extraordinary ray).

In the more general case, there ''is'' a difference not only in the magnitude but the direction of the two rays. For instance, the photograph through a calcite crystal (top of page) shows a shifted image in the two polarizations; this is due to the optic axis being neither parallel nor normal to the crystal surface. And even when the optic axis ''is'' parallel to the surface, this will occur for waves launched at non-normal incidence (as depicted in the explanatory figure). In these case the two {{math|'''k'''}} vectors can be found by solving {{EquationNote|6|eq. 6}} constrained by the boundary condition which requires that the components of the two transmitted waves' {{math|'''k'''}} vectors, and the {{math|'''k'''}} vector of the incident wave, as projected onto the surface of the interface, must all be identical. For a uniaxial crystal it will be found that there is ''not'' a spatial shift for the ordinary ray (hence its name) which will refract as if the material were non-birefringent with an index the same as the two axes which are not the optic axis. For a biaxial crystal neither ray is deemed "ordinary" nor would generally be refracted according to a refractive index equal to one of the principal axes.

== See also ==
* [[Cotton–Mouton effect]]
* [[Crystal optics]]
* [[Dichroism]]
* [[Iceland spar]]
* [[John Kerr (physicist)|John Kerr]]
* [[Periodic poling]]
* [[Pleochroism]]

==Notes==
{{reflist|group=note}}

==References==
{{Reflist}}

==Bibliography==
* <span id="BW02"></span>M. Born and E. Wolf, 2002, ''Principles of Optics'', 7th&nbsp;Ed., Cambridge University Press, 1999 (reprinted with corrections, 2002).
* <span id="SecMem"></span>A. Fresnel, 1827, "Mémoire sur la double réfraction", ''Mémoires de l'Académie Royale des Sciences de l'Institut de France'', vol.{{nnbsp}}{{serif|VII}} (for 1824, printed 1827), [https://archive.org/details/mmoiresdelacad07memo/page/44 pp.{{nnbsp}}45–176]; reprinted as "Second mémoire…" in Fresnel, 1868, [https://books.google.com/books?id=g6tzUG7JmoQC&pg=PA479 pp.{{nnbsp}}479–596]; translated by A.W.&nbsp;Hobson as [https://archive.org/details/scientificmemoir05memo/page/238 "Memoir on double refraction"], in R.{{nnbsp}}Taylor (ed.), ''Scientific Memoirs'', vol.{{nnbsp}}{{serif|V}} (London: Taylor & Francis, 1852), pp.{{nnbsp}}238–333. (Cited page numbers are from the translation.)
* <span id="OC2"></span>A. Fresnel (ed.{{tsp}} H. de Senarmont, E.&nbsp;Verdet, and L.&nbsp;Fresnel), 1868, ''Oeuvres complètes d'Augustin Fresnel'', Paris: Imprimerie Impériale (3&nbsp;vols., 1866–70), [https://books.google.com/books?id=g6tzUG7JmoQC vol.{{nnbsp}}2&nbsp;(1868)].

==External links==
{{Commons|Birefringence}}
* [http://www.photostress.com Stress Analysis Apparatus (based on Birefringence theory)]
* http://www.olympusmicro.com/primer/lightandcolor/birefringence.html
* [https://www.youtube.com/watch?v=BEClYQbuG7U Video of stress birefringence in Polymethylmethacrylate (PMMA or Plexiglas).]
* [http://www.austine.com Artist Austine Wood Comarow employs birefringence to create kinetic figurative images.]
* {{cite web|last=Merrifield|first=Michael|title=Birefringence|url=http://www.sixtysymbols.com/videos/birefringence.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]}}
* [http://iowatom.weebly.com The Birefringence of Thin Ice (Tom Wagner, photographer)]

{{Authority control}}

[[Category:Polarization (waves)]]
[[Category:Optical mineralogy]]