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| [[File:Rays passing through birefringent material.svg|thumb|250px|Displacement of light rays with perpendicular [[Polarization (waves)|polarization]] through a birefringent material.]]
| | 30 seconds a rechargeable battery in your wireless gaming controller. You can buy regular power supplies for much controller. If you intend to play video games regularly, you will be dining on through a small lots of money in the batteries comfortable with run your controllers. A rechargeable battery you will save a lot of money in the long run.<br><br>Cultivate a gaming program for the children. Similar to essential assignments time, this training video game program will permit manage a child's approaches. When the times have always been set, stick to specific schedule. Do And not back as a lead of whining or begging. The schedule is only reliable if you just continue.<br><br>Vehicle which play clash of clans crave ways of getting free gems. The treasures are very important due to the fact give the player characteristics and the power to boost their gaming experience. As opposed to new equivalent games in cellular websites, especially those even individuals use various breaches in buy to get these practical information on free, the nature pertaining to farmville and its array does not enable all of the varieties of hacks that anyone can put to the game. Everyone is always looking for ways teaching you how to get free gems present in clash of clans although the most important thing to do is to employ a great venue to earn these people for free, save associated with suitably and use these people where necessary.<br><br>When the system that your tiddler is enjoying on definitely will connect with the Net, be sure that you fix the settings for your loved ones before he performs utilizing. You're going to be ready to safeguard your kid from vulnerability to unsavory content utilizing these filter facilities. There are also options to set the amount of discussion they can participate folks when online.<br><br>Attributes needed in-online game songs opportunity. If, nonetheless, you might exist annoyed by using who's soon after one per hour approximately, don't be too embarrassed to mute the tv or personal computer but play some audio of one's very own. You'll find a far more delightful game playing experience this way and therefore are lots more unlikely to get a frustration from actively competing.<br><br>If you are the proud owner of an ANY lightweight device that runs forward iOS or android as a touchscreen tablet computing device or a smart phone, then you definitely may possibly have already been conscious of the revolution taking in place right now in the world of mobile personal game "The Clash Having to do with Clans", and you would want to be in demand linked with conflict of families no fee jewels [http://Search.Un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=compromise&Submit=Go compromise] because good deal more gems, elixir and sterling silver are needed seriously to finally acquire every battle.<br><br>A leap into the pre-owned or operated xbox board game marketplace. Several professionals will get a Deviate of Clans Hack and complete this game really in a timely fashion. Several shops let these activities being dealt in just to promote them at one particular lessened cost. If you loved this short article and you would certainly such as to get additional information regarding clash of clans cheat - [http://prometeu.net Read Much more] - kindly check out our own web site. It can be by far one of the most cost-effective technique to develop newer video games the actual higher cost. |
| [[File:Crystal on graph paper.jpg|thumb|A [[calcite]] crystal laid upon a graph paper with blue lines showing the double refraction]]
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| [[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.]]
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| '''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 asymmetric [[crystal structure]]s are often birefringent, as well as [[plastic]]s under [[mechanical stress]].
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| 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>Erasmus Bartholin, ''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). See also: Erasmus Bartholin (January 1, 1670) "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'', vol. 5, pages 2039-2048.</ref> in [[calcite]], a crystal having one of the strongest birefringences. However it was not until the 19th century that [[Augustin-Jean Fresnel]] correctly 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).
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| ==Explanation==
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| The simplest (and most common) type of birefringence is that of materials with [[index ellipsoid#Uniaxial indicatrix|uniaxial]] anisotropy. That is, the structure of the material is such that it has an axis of symmetry with all perpendicular directions optically equivalent. This axis is known as the [[Optic axis of a crystal|optic axis]] of the material, and components of light with [[linear polarization]]s parallel and perpendicular to it have unequal indices of refraction, denoted ''n''<sub>e</sub> and ''n''<sub>o</sub>, respectively, where the subscripts stand for ''extraordinary'' and ''ordinary''. The names reflect the fact that, if unpolarized light enters the material at some [[angle of incidence]], the component of the incident radiation whose polarization is perpendicular to the optic axis will be refracted according to the standard [[law of refraction]] for a material of refractive index ''n''<sub>o</sub>, while the other polarization component, the so-called ''extraordinary ray'' will refract at a different angle | |
| determined by the angle of incidence, the orientation of the optic axis, and the birefringence
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| :<math>\Delta n=n_e-n_o.\,</math>
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| What's more, the extraordinary ray is an [[inhomogeneous wave]] whose power flow (given by the [[Poynting vector]]) is not exactly parallel to the [[wave vector]]. This causes a shift in that beam, even when launched at normal incidence, that 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.
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| 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 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 ''n''<sub>e</sub>) but is not an inhomogeneous wave. 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.
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| The more general case of biaxially anisotropic materials, also known as trirefringent{{Citation needed|date=July 2013}} materials, 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> Then there are ''three'' refractive indices corresponding to three principal axes of the crystal. Generally ''both'' polarizations are extraordinary rays with different effective refractive indices which can be determined using the [[index ellipsoid]] for a given polarization vector.
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| == Sources of optical birefringence ==
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| While birefringence is usually obtained using an anisotropic crystal, it can result from an [[optical isotropy|optically isotropic]] material in a few ways:
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| * ''Stress birefringence'' results when isotropic materials are stressed or deformed such that the isotropy is lost in one direction (i.e., stretched or bent). [http://www.oberlin.edu/physics/catalog/demonstrations/optics/birefringence.html Example]
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| * By the [[Kerr effect]], whereby an applied electric field induces birefringence at optical frequencies through the effect of [[nonlinear optics]];
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| * 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;
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| * By self or forced alignment of highly polar molecules such as [[lipids]], some [[surfactants]] or [[liquid crystals]], that generate highly birefringent thin films.
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| == Examples of uniaxial birefringent materials ==
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| {| class="wikitable sortable" style="float:right; margin: 0em 0em 1em 1em;"
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| |+ Uniaxial materials, at 590 nm<ref name=hypertextbook>{{cite web|last=Elert|first=Glenn|title=Refraction|work=The Physics Hypertextbook|url=http://hypertextbook.com/physics/waves/refraction/}}</ref>
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| |-
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| ! Material || Crystal system || n<sub>o</sub> || n<sub>e</sub> || Δn
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| |-
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| | [[barium borate]] BaB<sub>2</sub>O<sub>4|| [[Trigonal]] ||1.6776 ||1.5534 ||-0.1242
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| |-
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| | [[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
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| |-
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| | [[calcite]] CaCO<sub>3</sub> || [[Trigonal]] || 1.658 || 1.486 || -0.172
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| |-
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| | [[ice]] H<sub>2</sub>O || [[Hexagonal lattice system|Hexagonal]] || 1.309 || 1.313 || +0.004
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| |-
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| | [[lithium niobate]] LiNbO<sub>3</sub>|| [[Trigonal]] || 2.272|| 2.187|| -0.085
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| |-
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| | [[magnesium fluoride]] MgF<sub>2</sub>|| [[Tetragonal]] || 1.380|| 1.385|| +0.006
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| |-
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| | [[quartz]] SiO<sub>2</sub>|| [[Trigonal]] || 1.544|| 1.553|| +0.009
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| |-
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| | [[ruby]] Al<sub>2</sub>O<sub>3</sub>|| [[Trigonal]] || 1.770|| 1.762|| -0.008
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| |-
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| | [[rutile]] TiO<sub>2</sub>|| [[Tetragonal]] || 2.616|| 2.903|| +0.287
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| |-
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| | [[sapphire]] Al<sub>2</sub>O<sub>3</sub>|| [[Trigonal]] || 1.768|| 1.760|| -0.008
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| |-
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| | [[silicon carbide]] SiC|| [[Hexagonal lattice system|Hexagonal]] || 2.647|| 2.693|| +0.046
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| |-
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| | [[tourmaline]] (complex silicate )|| [[Trigonal]] || 1.669|| 1.638|| -0.031
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| |-
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| | [[zircon]], high ZrSiO<sub>4</sub>|| [[Tetragonal]] || 1.960|| 2.015|| +0.055
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| |-
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| | zircon, low ZrSiO<sub>4</sub>|| [[Tetragonal]] || 1.920|| 1.967|| +0.047
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| |}
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| The best-studied birefringent materials are crystalline; the refractive indices (at wavelength ~ 590 nm) of several such uniaxial crystals are tabulated to the right .<ref name=hypertextbook/>
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| Many [[plastic]]s are birefringent, because their molecules are 'frozen' in a stretched conformation when the plastic is molded or extruded.<ref>[http://onlinelibrary.wiley.com/doi/10.1002/pen.10347/abstract The Use of Birefringence for Predicting the Stiffness of Injection Molded Polycarbonate Discs]
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| </ref>
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| For example, ordinary [[cellophane]] is birefringent. [[Polarizer]]s are routinely used to detect stress in plastics such as [[polystyrene]] and [[polycarbonate]].
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| Cotton (gossypium hirsutum) fiber is birefringent because of high levels of cellulosic material in the fiber's secondary cell wall.
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| 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.
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| In addition to anisotropy in the electric polarizability ([[electric susceptibility]]), anisotropy in the magnetic polarizability [[Permeability (electromagnetism)|magnetic permeability]]]) will also cause birefringence. However at optical frequencies, values of magnetic permeability for natural materials are not measurably different from [[Vacuum permeability|''µ''<sub>0</sub>]] so this is not a source of optical birefringence in practice.
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| == Fast and slow rays ==
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| {| class="wikitable" style="float:right; clear:both; margin: 0em 0em 1em 1em;"
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| |+ Effective refractive indices in negative uniaxial materials
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| ! rowspan="2" | Propagation<br>direction
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| ! colspan="2" | Ordinary ray
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| ! colspan="2" | Extraordinary ray
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| |-
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| ! Polarization
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| ! n<sub>eff</sub>
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| ! Polarization
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| ! n<sub>eff</sub>
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| |-
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| | ''z''
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| | ''xy''-plane
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| | <math>n_o</math>
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| | n/a
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| | n/a
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| |-
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| | ''xy''-plane
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| | ''xy''-plane
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| | <math>n_o</math>
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| | ''z''
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| | <math>n_e</math>
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| |-
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| | ''xz''-plane
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| | ''y''
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| | <math>n_o</math>
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| | ''xz''-plane
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| | <math>n_e < n < n_o</math>
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| |-
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| | other
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| | colspan="4" | analogous to ''xz''-plane
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| |}
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| For a given propagation direction, in general there are two perpendicular polarizations for which the medium behaves as if it had a single effective refractive index. In a uniaxial material, these polarizations are called the extraordinary and the ordinary ray (''e'' and ''o'' rays), with the ordinary ray having the effective refractive index <math>n_o</math>. A biaxial crystal is characterized by three refractive indices ''α'', ''β'', and ''γ'' applying to its principal axes. A wave in a specified direction will consist of two polarization components with generally 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 has a lower effective refractive index.
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| [[File:Positively birefringent material.svg|thumb|right|Rays passing through a positively birefringent material. The incident light has parallel and perpendicular polarisation components (linear polarization at 45º the optic axis). The optic axis is perpendicular to the direction of the perpendicular component of incident ray, so the ray polarized parallel to the optic axis has a greater refractive index than the ray polarized perpendicular to it.]]
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| For a uniaxial material with the ''z'' axis defined to be the optical axis, the effective refractive indices are as in the table on the right. For rays propagating in directions other than ''z'', the effective refractive index of the extraordinary ray is in between <math>n_o</math> and <math>n_e</math>, depending on the angle between the polarization vector and the ''z'' axis. The effective refractive index can be determined using the [[index ellipsoid]].
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| ===Positive or negative===
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| Uniaxial birefringence is classified as positive when the extraordinary index of refraction n<sub>e</sub> is greater than the ordinary index n<sub>o</sub>. Negative birefringence means that Δn = 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?] 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 optical axis when the birefringence of the crystal is positive (or negative, respectively). The terms "positive" and "negative" are not applied in the case of biaxial crystals, since all three of the principal axes have different refractive indices, rather than two being the same but different from the one that's designated as the optic axis in a uniaxial crystal.
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| {| class="wikitable sortable" style="float:right; clear:both; margin: 0em 0em 1em 1em;"
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| |+ Biaxial materials, at 590 nm<ref name=hypertextbook/>
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| |-
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| ! Material ||Crystal system || ''n''<sub>α</sub> || ''n''<sub>β</sub> || ''n''<sub>γ</sub>
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| |-
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| |[[borax]] Na<sub>2</sub>(B<sub>4</sub>O<sub>5</sub>)(OH)<sub>4</sub>·8(H<sub>2</sub>O) ||[[Monoclinic]]||1.447 ||1.469 ||1.472
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| |-
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| |[[Magnesium sulfate|epsom salt]] MgSO<sub>4</sub>·7(H<sub>2</sub>O) || [[Monoclinic]]||1.433 ||1.455 ||1.461
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| |-
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| |[[mica]], [[biotite]] {{chem|K(Mg,Fe)|3|AlSi|3|O|10|(F,OH)|2}} ||[[Monoclinic]] ||1.595 ||1.640 ||1.640
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| |-
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| |mica, [[muscovite]] KAl<sub>2</sub>(AlSi<sub>3</sub>O<sub>10</sub>)(F,OH)<sub>2</sub>
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| ||[[Monoclinic]] ||1.563 ||1.596 ||1.601
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| |-
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| |[[olivine]] (Mg, Fe)<sub>2</sub>SiO<sub>4</sub>||[[Orthorhombic]]||1.640 ||1.660 ||1.680
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| |-
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| |[[perovskite]] CaTiO<sub>3</sub> ||[[Orthorhombic]]||2.300 ||2.340 ||2.380
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| |-
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| |[[topaz]] Al<sub>2</sub>SiO<sub>4</sub>(F,OH)<sub>2</sub> ||[[Orthorhombic]]||1.618 ||1.620 ||1.627
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| |-
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| |[[ulexite]] NaCaB<sub>5</sub>O<sub>6</sub>(OH)<sub>6</sub>•5(H<sub>2</sub>O) ||[[Triclinic]]||1.490 ||1.510 ||1.520
| |
| |}
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| === Biaxial birefringence ===
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| '''Biaxial birefringence''', also known as '''trirefringence'''{{Citation needed|date=July 2013}}, describes an [[crystal optics#Anisotropic media|anisotropic material]] in which the optical properties are not invariant under rotation about a particular axis (the optic axis, in uniaxial crystals). For such a material, the refractive index tensor '''n''', will in general have three distinct [[eigenvalues]] that can be labeled ''n''<sub>α</sub>, ''n''<sub>β</sub> and ''n''<sub>γ</sub>.
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| == Measurement ==
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| 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]].
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| Birefringence of [[lipid bilayers]] can be measured using [[dual polarisation 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.
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| == Applications ==
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| [[Image:LCD layers.svg|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]]
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| 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.
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| 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.
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| ===Medicine===
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| 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 X direction after passing through the first polarizer, but above the specimen is a polarizer (a so-called ''analyzer'') oriented in the 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 X polarized light into the Y polarization; these areas will then appear bright against the dark background. Modifications to this basic principle can differentiate between positive and negative birefringence.
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| [[File:Fluorescent uric acid.JPG|thumb|Urate crystals, with the crystals with their 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.]]
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| 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 |author=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.
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| 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 (chemistry)|intercalates]] between the folds and, when observed under polarized light, causes birefringence.
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| In [[ophthalmology]], [[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]].
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| Birefringence characteristics in [[sperm head]]s allow for the selection of spermatozoa for [[intracytoplasmic sperm injection]].<ref>{{cite journal |author=Gianaroli L, Magli MC, Ferraretti AP, ''et al.'' |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=}}</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, Balaban B, Moser M, ''et al.'' |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=}}</ref> Birefringence of particles biopsied from pulmonary nodules indicates [[silicosis]].
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| ===Stress induced birefringence===
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| [[Image:Birefringence Stress Plastic.JPG|thumb|right|250px|Color pattern of a plastic box with "frozen in" [[mechanical stress]] placed between two crossed [[polarizer]]s.]]
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| [[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 birefingent 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.
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| ===Other cases of birefringence===
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| 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 waves 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.[[File:Rutile birefringence.gif|thumb|Birefringent [[rutile]] observed in different polarizations using a rotating polarizer (or ''analyzer'')]]
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| ==Theory==
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| Birefringence results when a material's [[permittivity]] is not describable using a [[Scalar (physics)|scalar]] value, but requires a [[tensor]] to relate the electric displacement (''D'') with the electric field (''E''). Consider a [[plane wave]] propagating in an anisotropic medium, with a permittivity tensor ''ε'' and assuming no magnetic permeability in the medium: <math>\mu= \mu_0</math>. We shall assume that the electric field of a wave of angular frequency ω can be written in the form:
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| {{Equation|1=\mathbf{E=E_0}\exp \left[i(\mathbf{k \cdot r}-\omega t)\right] \,|2=2}}
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| where '''r''' is the position vector, ''t'' is time, and '''E<sub>0</sub>''' is a vector describing the electric field at '''r'''=0, ''t''=0. Then we shall find the possible [[wave vector]]s '''k''' using [[Maxwell's equations]] from which we obtain:
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| {{Equation|1=-\nabla \times \nabla \times \mathbf{E}= \mu_0
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| \frac{\part^2 \mathbf{D} }{\partial t^2} |2=3a}}
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| {{Equation|1= \nabla \cdot \mathbf{D} =0 |2=3b}}
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| where the so-called [[electric displacement]] vector <math>\mathbf D</math> is now related to the electric field <math>\mathbf E</math> through the permittivity tensor ''ε'':
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| : <math>\mathbf D = \epsilon \, \mathbf E \, .</math>
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| Substituting the definition of <math>\mathbf D</math> and eqn. 2 into eqns. 3a-b leads to the conditions:
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| {{Equation|1={{!}}\mathbf{k}{{!}}^2\mathbf{E_0}-\mathbf{(k \cdot E_0) k}= \mu_0 \omega^2 (\mathbf{\epsilon} \, \mathbf{E_0}) |2=4a}}
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| {{Equation|1=\mathbf{k} \cdot (\mathbf{\epsilon} \, \mathbf{E_0}) =0 |2=4b}}
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| Eqn. 4b indicates that <math>\mathbf D</math> is orthogonal to the direction of the wavevector k, even though that is no longer generally true for <math>\mathbf E</math> as would be the case in an isotropic medium.
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| To find the allowed values of '''k''', '''E'''<sub>0</sub> can be eliminated from eq 4a. If eqn 4a is written in Cartesian coordinates with the ''x'', ''y'' and ''z'' axes chosen in the principal directions of the permitivity tensor '''ε''', then
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| {{Equation|1=\mathbf{\epsilon}= \epsilon_0
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| \begin{bmatrix} n_x^2 & 0 & 0 \\ 0& n_y^2 & 0 \\ 0& 0& n_z^2 \end{bmatrix} \,|2=4c}}
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| where the diagonal values are squares of the refractive indices for polarizations along the three principal axes ''x'', ''y'' and ''z''. With ''ε'' in this form, and noting that the speed of light <math> c = 1/ \sqrt{\mu_0 \epsilon_0}</math>, eqn. 4a becomes
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| {{Equation|1=\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|2=5a}}
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| {{Equation|1=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|2=5b}}
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| {{Equation|1=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|2=5c}}
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| where ''E''<sub>x</sub>, ''E''<sub>y</sub>, ''E''<sub>z</sub>, ''k''<sub>x</sub>, ''k''<sub>y</sub> and ''k''<sub>z</sub> are the components of '''E'''<sub>0</sub> and '''k'''. This is a set of linear equations in ''E''<sub>x</sub>, ''E''<sub>y</sub>, ''E''<sub>z</sub>, and they have a non-trivial solution if the following [[determinant]] is zero:
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| {{Equation|1=\begin{vmatrix}
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| \left(-k_y^2-k_z^2+\frac{\omega^2n_x^2}{c^2}\right) & k_xk_y & k_xk_z \\
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| k_xk_y & \left(-k_x^2-k_z^2+\frac{\omega^2n_y^2}{c^2}\right) & k_yk_z \\
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| k_xk_z & k_yk_z & \left(-k_x^2-k_y^2+\frac{\omega^2n_z^2}{c^2}\right) \end{vmatrix} =0\,|2=6}}
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| Evaluating the determinant of eqn (6), and rearranging the terms, we obtain
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| :<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) + </math>
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| ::<math> \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)(k_x^2+k_y^2+k_z^2)=0 </math>
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|
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| In the case of a uniaxial material, choosing the optic axis to be in the z direction so that ''n''<sub>x</sub>=''n''<sub>y</sub>=''n<sub>o</sub>'' and ''n<sub>z</sub>''=''n<sub>e</sub>'', this expression can be factored into
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| {{Equation|1=\left(\frac{k_x^2}{n_o^2}+\frac{k_y^2}{n_o^2}+\frac{k_z^2}{n_o^2} -\frac{\omega^2}{c^2}\right)\left(\frac{k_x^2}{n_e^2}+\frac{k_y^2}{n_e^2}+\frac{k_z^2}{n_o^2} -\frac{\omega^2}{c^2}\right)=0\,.|2=8}}
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| Setting either of the factors in eqn 8 to zero will define an ellipsoidal surface in space of allowed wave vectors '''k'''. The first factor being zero defines a sphere corresponding to ordinary rays, in which the effective refractive index is exactly ''n<sub>o</sub>''. The second defines a [[spheroid]] symmetric about the z axis. This solution corresponds to extraordinary rays in which the effective refractive index is in between ''n<sub>o</sub>'' and ''n<sub>e</sub>''. Therefore for any arbitrary direction of propagation, two distinct wavevectors '''k''' are allowed corresponding to the polarizations of the ordinary and extraordinary rays. A general state of polarization launched into the medium can be decomposed into two such waves which will then propagate with different '''k''' vectors (except in the case of propagation in the direction of the optic axis). For a biaxial material a similar but somewhat more complicated condition on the two waves can be described.<ref name="bornwolf">Born M, and Wolf E, ''Principles of Optics'', 7th Ed. 1999 (Cambridge University Press), §15.3.3</ref>
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| == See also ==
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| * [[Cotton-Mouton effect]]
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| * [[Crystal optics]]
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| * [[Dichroism]]
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| * [[John Kerr (physicist)|John Kerr]]
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| * [[Periodic poling]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| {{Commons|Birefringence}}
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| * [http://www.photostress.com Stress Analysis Apparatus (based on Birefringence theory)]
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| * http://www.olympusmicro.com/primer/lightandcolor/birefringence.html
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| * [http://www.youtube.com/watch?v=BEClYQbuG7U Video of stress birefringence in Polymethylmethacrylate (PMMA or Plexiglas).]
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| * [http://www.campoly.com/application_notes.html Application note on the theory of birefringence] (see no.14)
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| * [http://www.austine.com Artist Austine Wood Comarow employs birefringence to create kinetic figurative images.]
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| * {{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]]}}
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| [[Category:Polarization (waves)]]
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| [[Category:Optical mineralogy]]
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