polarization through a birefringent material.]]  crystal laid upon a paper with all letters showing the double refraction]] A calcite crystal seen through a polarizing filter  '''Birefringence''', or '''double refraction''', is the decomposition of a ray of light into two rays when it passes through certain anisotropic materials, such as crystals of calcite or boron nitride. The effect was first described by the Danish scientist Rasmus Bartholin in 1669, who saw it in calcite.Erasmus Bartholin, ''Experimenta crystalli islandici disdiaclastici quibus mira &amp; 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) &quot;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,&quot; ''Philosophical Transactions of the Royal Society of London'', vol. 5, pages 2039-2048. The effect is now known to also occur in certain plastics, magnetic materials, various noncrystalline materials, and liquid crystals.The Science of Color, by Steven K. Shevell, Optical Society of America. Published 2003. ISBN 0444512519  The simplest instance of the effect arises in materials with uniaxial anisotropy. That is, the structure of the material is such that it has an axis of symmetry with no equivalent axis in the plane perpendicular to it. (Cubic crystals are thereby ruled out.) This axis is known as the optical axis of the material, and  light with linear polarizations parallel and perpendicular to it has unequal indices of refraction, denoted ''n''e and ''n''o respectively, where the suffixes stand for '''extraordinary''' and '''ordinary'''. The names reflect the fact that if unpolarized light enters the material at a nonzero acute angle to the optical axis, the component with polarization perpendicular to this axis will be refracted as per the standard law of refraction, while the complementary polarization component will refract at a nonstandard angle determined by the angle of entry and the difference between the indices of refraction, :\Delta n=n_e-n_o\, known as the birefringence magnitude. The light will therefore split into two linearly polarized beams, correspondingly known as ordinary and extraordinary.[ Eric Weisstein's World of Physics on Birefringence] Exceptions arise when the light propagates either along or orthogonal to the optical axis. In the first case, both polarizations and rays are ordinary and are not split. In the second case also there is no splitting of the light into two separate directions, but the ordinary and extraordinary components travel at different speeds, and the effect is used to interconvert between linear and circular or elliptical polarizations.   Double refraction also occurs in biaxially anisotropic materials, which are also known as trirefringent, but its description is then substantially more complex.Landau, L. D., and Lifshitz, E. M., ''Electrodynamics of Continuous Media'', Vol. 8 of the ''Course of Theoretical Physics'' 1960 (Pergamon Press), §79  == Creation == While birefringence is often found naturally (especially in crystals), there are several ways to create it in optically isotropic materials. *Birefringence results when isotropic materials are deformed such that the isotropy is lost in one direction (i.e., stretched or bent). [ Example] *Applying an electric field can induce molecules to line up or behave asymmetrically, introducing anisotropy and resulting in birefringence. (''see'' Pockels effect) *Applying a magnetic field can cause a material to be '''circularly birefringent''', with different indices of refraction for oppositely-handed circular polarizations (''see'' Faraday effect) *Self alignment of highly polar molecules such as lipids and some surfactants will generate highly birefringent thin films (see also Liquid crystal)  == Examples of uniaxial birefringent materials == {| class=&quot;wikitable sortable&quot; style=&quot;float:right; margin: 0em 0em 1em 1em;&quot; |+ Uniaxial materials, at 590&amp;nbsp;nm |- ! Material || no || ne || Δn |- | beryl Be3Al2(SiO3)6||1.602 ||1.557 ||-0.045 |- | calcite CaCO3 || 1.658 || 1.486 || -0.172 |- | calomel	Hg2Cl2	|| 1.973	|| 2.656	|| +0.683 |- | ice	H2O	|| 1.309	|| 1.313	|| +0.004 |- | lithium niobate LiNbO3|| 	2.272|| 	2.187|| 	-0.085 |- | magnesium fluoride	MgF2|| 	1.380|| 	1.385|| 	+0.006 |- | quartz	SiO2|| 	1.544|| 	1.553|| 	+0.009 |- | ruby	Al2O3|| 	1.770|| 	1.762|| 	-0.008 |- | rutile	TiO2|| 	2.616|| 	2.903|| 	+0.287 |- | peridot	(Mg, Fe)2SiO4 	|| 1.690|| 	1.654|| 	-0.036 |- | sapphire	Al2O3|| 	1.768|| 	1.760|| 	-0.008 |- | sodium nitrate	NaNO3|| 	1.587|| 	1.336|| 	-0.251 |- | tourmaline (complex silicate )|| 	 	1.669|| 	1.638|| 	-0.031 |- | zircon, high	ZrSiO4|| 	1.960|| 	2.015|| 	+0.055 |- | zircon, low	ZrSiO4|| 	1.920|| 	1.967|| 	+0.047 |} The best studied uniaxial birefringent materials are crystalline, the refractive indices of several of which are tabulated to the right (at wavelength ~ 590&amp;nbsp;nm). Silicon carbide, also known as Moissanite, is strongly birefringent.  Many plastics are birefringent, because their molecules are 'frozen' in a stretched conformation when the plastic is moulded or extruded.[ The Use of Birefringence for Predicting the Stiffness of Injection Moulded Polycarbonate Discs]  For example, cellophane is a cheap birefringent material, and Polaroid sheets are commonly used to examine for orientation in birefringent plastics like polystyrene and polycarbonate. Birefringent materials are used in many devices which manipulate the polarization of light, such as wave plates, polarizing prisms, and Lyot filters.  As stated above, birefringence can also arise in magnetic materials, but substantial variations in magnetic permeability of materials are rare at optical frequencies.  Birefringence can be observed in amyloid plaque deposits such as are found in the brains of Alzheimer's patients.  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 conformation.  Congo red dye intercalates between the folds and, when observed under polarized light, causes birefringence.  Cotton (Gossypium hirsutum) fiber is birefringent because of high levels of cellulosic material in the fiber's secondary cell wall.  Slight imperfections in optical fiber can cause birefringence, which can cause distortion in fiber-optic communication; see polarization mode dispersion.  The imperfections can be geometrically based, or a result of photoelastic effects from loading on the optical fiber.  == Fast and slow rays == {| class=&quot;wikitable&quot; style=&quot;float:right; clear:both; margin: 0em 0em 1em 1em;&quot; |+ Effective refractive indices in negative uniaxial materials ! rowspan=&quot;2&quot; | Propagationdirection ! colspan=&quot;2&quot; | Ordinary ray ! colspan=&quot;2&quot; | Extraordinary ray |- ! Polarization ! neff ! Polarization ! neff |- | ''z'' | ''xy''-plane | n_o | n/a | n/a |- | ''xy''-plane | ''xy''-plane | n_o | ''z'' | n_e |- | ''xz''-plane | ''y'' | n_o | ''xz''-plane | n_e  |- | other | colspan=&quot;4&quot; | analogous to ''xz''-plane |} For a given propagation direction, there are generally two perpendicular polarizations for which the medium behaves as if it had a single effective refractive index. In a uniaxial material, rays with these polarizations are called the extraordinary and the ordinary ray (''e'' and ''o'' rays), corresponding to the extraordinary and ordinary refractive indices. In a biaxial material, there are three refractive indices ''α'', ''β'', and ''γ'', yet only two rays, which are called the fast and the slow ray. The slow ray is the ray that has the highest effective refractive index.  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 the ''xz'' plane, the effective refractive index of the ''e'' polarization varies continuously between n_o and n_e, depending on the angle with the ''z'' axis. The effective refractive index can be constructed from the Index ellipsoid.  == Biaxial birefringence == {| class=&quot;wikitable sortable&quot; style=&quot;float:right; clear:both;  margin: 0em 0em 1em 1em;&quot; |+  Biaxial materials, at 590&amp;nbsp;nm |- ! Material || ''n''α || ''n''β || ''n''γ |- |borax	 	||1.447	||1.469	||1.472 |- |epsom salt	MgSO4·7(H2O)	||1.433	||1.455	||1.461 |- |mica, biotite	 	||1.595	||1.640	||1.640 |- |mica, muscovite	 	||1.563	||1.596	||1.601 |- |olivine	(Mg, Fe)2SiO4	||1.640	||1.660	||1.680 |- |perovskite	CaTiO3	||2.300	||2.340	||2.380 |- |topaz	 	||1.618	||1.620	||1.627 |- |ulexite	 	||1.490	||1.510	||1.520 |} '''Biaxial birefringence''', also known as '''trirefringence''', describes an anisotropic material that has more than one axis of anisotropy. For such a material, the refractive index tensor '''n''', will in general have three distinct eigenvalues that can be labeled ''n''α, ''n''β and ''n''γ.  ==Positive or negative== Rays passing through a positively birefringent material. The incident light is having parallel and perpendicular polarisation components (unpolarised light). The optical 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.  Uniaxial birefringent materials are classified as positively (or negatively) birefringent when, for light (having parallel and perpendicular components) directed to the optic axis, the refractive index of light polarized parallel to the optic axis is greater (or smaller, respectively,) than light polarized perpendicularly to the optic axis.''Brad Amos''.[ Birefringence for facetors I : what is birefringence?] First published in StoneChat, the Journal of the UK Facet Cutter's Guild. January–March. edition 2005 In other words, the polarization of the slow (or fast) wave is parallel to the optical axis when the birefringence of the crystal is positive (or negative, respectively).  Biaxial crystals are defined as positively (or negatively) birefringent when the slow ray (or fast ray, respectively) bisects the acute angle formed by the optical axes.Page 509 in: [  In practice, when using an optical compensator that emits red light, a crystal with positive birefringence appears blue when its long dimension is parallel to the slow axis of the compensator. In contrast, a crystal with negative birefringence appears yellow when its long dimension is parallel to the slow axis of the compensator, and the slow ray of the compensator is oriented perpendicularly to the long axis of the crystal. The reason for these phenomena is that the wavelength of emitted light is shifted higher in positively birefringent crystals, because the slow ray of the crystal is parallel to the slow axis of the compensator, while for negatively birefringent crystals the wavelength of emitted light is shifted lower, because the fast ray of the crystal is parallel to the slow axis of the compensator. The order of colors resulting from the use of compensators in a polarized light system differs from that of a typical spectrum, instead having an order including yellow-orange-red-violet-blue.  == Measurement == Birefringence and related 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.  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.  A common feature of optical microscopes is a pair of crossed polarizing filters. Between the crossed polarizers, a birefringent sample will appear bright against a dark (isotropic) background.  For a fixed composition such as calcium carbonate, a crystal such as calcite or its polymorphs, the index of refraction depends on the direction of light through the crystal structure. The refraction also depends on composition, and can be calculated using the Gladstone-Dale relation.  == Applications ==  Birefringence is widely used in optical devices, such as liquid crystal displays, light modulators, color filters, wave plates, optical axis gratings, etc. It also plays an important role in second harmonic generation and many other nonlinear processes.  Birefringent filters are also used as spatial low-pass filters in electronic cameras, where the thickness of the crystal is controlled to spread the image in one direction, thus increasing the spot-size. This is essential to the proper working of all television and electronic film cameras, to avoid spatial aliasing, the folding back of frequencies higher than can be sustained by the pixel matrix of the camera.  ===Medicine=== Birefringence is utilized in medical diagnostics.   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. Needle aspiration of fluid from a gouty joint will reveal negatively birefringent urate crystals. Calcium pyrophosphate crystals, in contrast, show weak positive birefringence. In practice, urate crystals appear yellow and calcium pyrophosphate crystals appear blue when their long axes are aligned parallel to that of a red compensator filter,[ The Approach to the Painful Joint Workup] Author: Alan N Baer; Chief Editor: Herbert S Diamond. Updated: Nov 22, 2010 or a crystal of known birefringence is added to the sample for comparison.  In ophthalmology, scanning laser polarimetry utilises the birefringence of the retinal nerve fibre layer to indirectly quantify its thickness, which is of use in the assessment and monitoring of glaucoma.   Birefringence characteristics in sperm heads allow for the selection of spermatozoa for intracytoplasmic sperm injection. Likewise, ''zona imaging'' uses birefringence on oocytes to select the ones with highest chances of successful pregnancy. Birefringence of particles biopsied from pulmonary nodules indicates silicosis.  ==Elastic birefringence== Another form of birefringence is observed in anisotropic elastic materials.  In these materials, shear waves split according to similar principles as the light waves discussed above.  The study of birefringent shear waves in the earth is a part of seismology. Birefringence is also used in optical mineralogy to determine the chemical composition, and history of minerals and rocks.  ==Stress induced birefringence==  placed between two crossed polarizers.]]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 due to stress induced birefringence. The reason is that polarization of a light ray is usually rotated after passing through a birefingent material and the amount of rotation is dependent on  wavelength.  ==Theory==  More generally, birefringence can be defined by considering a dielectric permittivity and a refractive index that are tensors. Consider a plane wave propagating in an anisotropic medium, with a relative permittivity tensor '''ε''', where the refractive index '''n''', is defined by n\cdot n = \epsilon. If the wave has an electric vector of the form:    where '''r''' is the position vector and ''t'' is time, then the wave vector '''k''' and the angular frequency ω must satisfy Maxwell's equations in the medium, leading to the equations:      where ''c'' is the speed of light in a vacuum. Substituting eqn. 2 in eqns. 3a-b leads to the conditions:  \mathbf{k}^2\mathbf{E_0}-\mathbf{(k \cdot E_0) k}=    \frac{\omega^2}{c^2} (\mathbf{\epsilon} \cdot \mathbf{E_0}) |2=4a}}    For the matrix product (\epsilon\cdot\mathbf E) often a separate name is used, the ''dielectric displacement vector'' \mathbf D. So essentially birefringence concerns the general theory of linear relationships between these two vectors in anisotropic media.  To find the allowed values of '''k''', '''E'''0 can be eliminated from eq 4a. One way to do this is to write eqn 4a in Cartesian coordinates, where the ''x'', ''y'' and ''z'' axes are chosen in the directions of the eigenvectors of '''ε''', so that    Hence eqn 4a becomes      where ''E''x,  ''E''y,  ''E''z, ''k''x,  ''k''y and  ''k''z are the components of '''E'''0 and  '''k'''. This is a set of linear equations in ''E''x,  ''E''y,  ''E''z, and they have a non-trivial solution if their determinant is zero:    Multiplying out eqn (6), and rearranging the terms, we obtain    In the case of a uniaxial material, where ''n''x=''n''y=''no'' and ''nz''=''ne'' say, eqn 7 can be factorised into    Each of the factors in eqn 8 defines a surface in the space of vectors '''k''' — the '''surface of wave normals'''. The first factor defines a sphere and the second defines an ellipsoid. Therefore, for each direction of the wave normal, two wavevectors '''k''' are allowed. Values of '''k''' on the sphere correspond to the '''ordinary rays''' while values on the ellipsoid correspond to the '''extraordinary rays'''.  For a biaxial material, eqn (7) cannot be factorized in the same way, and describes a more complicated pair of wave-normal surfaces.Born M, and Wolf E, ''Principles of Optics'', 7th Ed. 1999 (Cambridge University Press), §15.3.3  Birefringence is often measured for rays propagating along one of the optical axes (or measured in a two-dimensional material). In this case, '''n''' has two eigenvalues which can be labeled ''n''1 and ''n''2. '''n''' can be diagonalized by:    where '''R'''(χ) is the rotation matrix through an angle χ. Rather than specifying the complete tensor '''n''', we may now simply specify the ''magnitude'' of the birefringence Δ''n'', and ''extinction angle'' χ, where Δ''n'' = ''n''1&amp;nbsp;−&amp;nbsp;''n''2.  