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Birefringence

Overview

Birefringence, or double refraction, is the decomposition of a ray

Ray (optics)

In optics, a ray is an idealized narrow beam of light. Rays are used to model the propagation of light through an optical system, by dividing the real light field up into discrete rays that can be computationally propagated through the system by the techniques of ray tracing. This allows even very...

of light

Light

Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

into two rays when it passes through certain anisotropic materials, such as crystals of calcite

Calcite

Calcite is a carbonate mineral and the most stable polymorph of calcium carbonate . The other polymorphs are the minerals aragonite and vaterite. Aragonite will change to calcite at 380-470°C, and vaterite is even less stable.-Properties:...

or boron nitride

Boron nitride

Boron nitride is a chemical compound with chemical formula BN, consisting of equal numbers of boron and nitrogen atoms. BN is isoelectronic to a similarly structured carbon lattice and thus exists in various crystalline forms...

. The effect was first described by the Danish

Denmark

Denmark is a Scandinavian country in Northern Europe. The countries of Denmark and Greenland, as well as the Faroe Islands, constitute the Kingdom of Denmark . It is the southernmost of the Nordic countries, southwest of Sweden and south of Norway, and bordered to the south by Germany. Denmark...

scientist Rasmus Bartholin

Rasmus Bartholin

Rasmus Bartholin was a Danish scientist and physician. As part of his studies, he travelled in Europe for ten years. Professor at the University of Copenhagen, first in Geometry, later in Medicine...

in 1669, who saw it in calcite. The effect is now known to also occur in certain plastics, magnetic materials, various noncrystalline materials, and liquid crystals.

Discussion

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Unanswered Questions

Considering that birefringence results from a difference in effective propagation velocity of the electromagnetic field through an anisotropic refract...

Is birefringence shown best by polarised light? e mail, chomghall@yahoo.ie

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So this is the birefringence territitory, looking forward to it

Encyclopedia

Birefringence, or double refraction, is the decomposition of a ray

Ray (optics)

of light

Light

into two rays when it passes through certain anisotropic materials, such as crystals of calcite

Calcite

or boron nitride

Boron nitride

. The effect was first described by the Danish

Denmark

scientist Rasmus Bartholin

Rasmus Bartholin

Rasmus Bartholin was a Danish scientist and physician. As part of his studies, he travelled in Europe for ten years. Professor at the University of Copenhagen, first in Geometry, later in Medicine...

in 1669, who saw it in calcite. The effect is now known to also occur in certain plastics, magnetic materials, various noncrystalline materials, and liquid crystals.

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

Optic axis of a crystal

The optic axis of a crystal is the direction in which a ray of transmitted light suffers no birefringence . Due to the internal structure of the crystal , light propagates along the optical axis differently than in other directions...

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,

known as the birefringence magnitude. The light will therefore split into two linearly polarized beams, known as ordinary and extraordinary. 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.

Creation

While birefringence is often found naturally (like anisotropic 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
By the Pockels effect
Pockels effect
The Pockels effect , or Pockels electro-optic effect, produces birefringence in an optical medium induced by a constant or varying electric field. It is distinguished from the Kerr effect by the fact that the birefringence is proportional to the electric field, whereas in the Kerr effect it is...

, where an electric field can induce molecules to line up or behave asymmetrically, inducing anisotropy;
by the Faraday effect
Faraday effect
In physics, the Faraday effect or Faraday rotation is a Magneto-optical phenomenon, that is, an interaction between light and a magnetic field in a medium...

, where a magnetic field can cause a material to be circularly birefringent, with different indices of refraction for oppositely-handed circular polarization
Circular polarization
In electrodynamics, circular polarization of an electromagnetic wave is a polarization in which the electric field of the passing wave does not change strength but only changes direction in a rotary type manner....

s similar to an optically active material;
by self or forced alignment of highly polar molecules such as lipids, some surfactants or liquid crystals, that generate highly birefringent thin films.

Examples of uniaxial birefringent materials

Uniaxial materials, at 590 nm
Material	n_o	n_e	Δn
beryl Beryl The mineral beryl is a beryllium aluminium cyclosilicate with the chemical formula Be3Al26. The hexagonal crystals of beryl may be very small or range to several meters in size. Terminated crystals are relatively rare... Be₃Al₂(SiO₃)₆	1.602	1.557
calcite Calcite Calcite is a carbonate mineral and the most stable polymorph of calcium carbonate . The other polymorphs are the minerals aragonite and vaterite. Aragonite will change to calcite at 380-470°C, and vaterite is even less stable.-Properties:... CaCO₃	1.658	1.486	-0.172
calomel Hg₂Cl₂	1.973	2.656	+0.683
ice Ice Ice is water frozen into the solid state. Usually ice is the phase known as ice Ih, which is the most abundant of the varying solid phases on the Earth's surface. It can appear transparent or opaque bluish-white color, depending on the presence of impurities or air inclusions... H₂O	1.309	1.313	+0.004
lithium niobate Lithium niobate Lithium niobate is a compound of niobium, lithium, and oxygen. Its single crystals are an important material for optical waveguides, mobile phones, optical modulators and various other linear and non-linear optical applications.-Properties:... LiNbO₃	2.272	2.187	-0.085
magnesium fluoride Magnesium fluoride Magnesium fluoride is an inorganic compound with the formula MgF2. The compound is a white crystalline salt and is transparent over a wide range of wavelengths, with commercial uses in optics.-Production and structure:... MgF₂	1.380	1.385	+0.006
quartz Quartz Quartz is the second-most-abundant mineral in the Earth's continental crust, after feldspar. It is made up of a continuous framework of SiO4 silicon–oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall formula SiO2. There are many different varieties of quartz,... SiO₂	1.544	1.553	+0.009
ruby Ruby A ruby is a pink to blood-red colored gemstone, a variety of the mineral corundum . The red color is caused mainly by the presence of the element chromium. Its name comes from ruber, Latin for red. Other varieties of gem-quality corundum are called sapphires... Al₂O₃	1.770	1.762	-0.008
rutile Rutile Rutile is a mineral composed primarily of titanium dioxide, TiO2.Rutile is the most common natural form of TiO2. Two rarer polymorphs of TiO2 are known:... TiO₂	2.616	2.903	+0.287
peridot Peridot -Chemistry:The chemical composition of peridot is 2SiO4, with Mg in greater quantities than Fe.-Etymology:The origin of the name "peridot" is uncertain... (Mg, Fe)₂SiO₄	1.690	1.654	-0.036
sapphire Sapphire Sapphire is a gemstone variety of the mineral corundum, an aluminium oxide , when it is a color other than red or dark pink; in which case the gem would instead be called a ruby, considered to be a different gemstone. Trace amounts of other elements such as iron, titanium, or chromium can give... Al₂O₃	1.768	1.760	-0.008
sodium nitrate Sodium nitrate Sodium nitrate is the chemical compound with the formula NaNO3. This salt, also known as Chile saltpeter or Peru saltpeter to distinguish it from ordinary saltpeter, potassium nitrate, is a white solid which is very soluble in water... NaNO₃	1.587	1.336	-0.251
tourmaline Tourmaline Tourmaline is a crystal boron silicate mineral compounded with elements such as aluminium, iron, magnesium, sodium, lithium, or potassium. Tourmaline is classified as a semi-precious stone and the gem comes in a wide variety of colors... (complex silicate )	1.669	1.638	-0.031
zircon Zircon Zircon is a mineral belonging to the group of nesosilicates. Its chemical name is zirconium silicate and its corresponding chemical formula is ZrSiO4. A common empirical formula showing some of the range of substitution in zircon is 1–x4x–y... , 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 tabulated to the right (at wavelength ~ 590 nm). Silicon carbide

Silicon carbide

Silicon carbide , also known as carborundum, is a compound of silicon and carbon with chemical formula SiC. It occurs in nature as the extremely rare mineral moissanite. Silicon carbide powder has been mass-produced since 1893 for use as an abrasive...

, also known as Moissanite, is strongly birefringent.

Many plastic

Plastic

A plastic material is any of a wide range of synthetic or semi-synthetic organic solids used in the manufacture of industrial products. Plastics are typically polymers of high molecular mass, and may contain other substances to improve performance and/or reduce production costs...

s are birefringent, because their molecules are 'frozen' in a stretched conformation when the plastic is moulded or extruded.
For example, cellophane

Cellophane

Cellophane is a thin, transparent sheet made of regenerated cellulose. Its low permeability to air, oils, greases, bacteria and water makes it useful for food packaging...

is a cheap birefringent material, and Polaroid

Instant film

Instant film is a type of photographic film first introduced by Polaroid that is designed to be used in an instant camera...

sheets are commonly used to examine for orientation in birefringent plastics like polystyrene

Polystyrene

Polystyrene ) also known as Thermocole, abbreviated following ISO Standard PS, is an aromatic polymer made from the monomer styrene, a liquid hydrocarbon that is manufactured from petroleum by the chemical industry...

and polycarbonate

Polycarbonate

PolycarbonatePhysical PropertiesDensity 1.20–1.22 g/cm3Abbe number 34.0Refractive index 1.584–1.586FlammabilityV0-V2Limiting oxygen index25–27%Water absorption – Equilibrium0.16–0.35%Water absorption – over 24 hours0.1%...

. Birefringent materials are used in many devices that manipulate the polarization of light, such as wave plate

Wave plate

A wave plate or retarder is an optical device that alters the polarization state of a light wave travelling through it.- Operation :A wave plate works by shifting the phase between two perpendicular polarization components of the light wave. A typical wave plate is simply a birefringent crystal...

s, polarizing

Polarizer

A polarizer is an optical filter that passes light of a specific polarization and blocks waves of other polarizations. It can convert a beam of light of undefined or mixed polarization into a beam with well-defined polarization. The common types of polarizers are linear polarizers and circular...

prisms

Prism (optics)

In optics, a prism is a transparent optical element with flat, polished surfaces that refract light. The exact angles between the surfaces depend on the application. The traditional geometrical shape is that of a triangular prism with a triangular base and rectangular sides, and in colloquial use...

, and Lyot filter

Lyot filter

A Lyot filter, named for its inventor Bernard Lyot, is a type of optical filter that uses birefringence to produce a narrow passband of transmitted wavelengths...

s.

As stated above, birefringence can also arise in magnetic

Magnetism

Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...

materials, but substantial variations in magnetic permeability

Permeability (electromagnetism)

In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...

of materials are rare at optical frequencies.

Birefringence can be observed in amyloid

Amyloid

Amyloids are insoluble fibrous protein aggregates sharing specific structural traits. Abnormal accumulation of amyloid in organs may lead to amyloidosis, and may play a role in various neurodegenerative diseases.-Definition:...

plaque deposits such as are found in the brains of Alzheimer's

Alzheimer's disease

Alzheimer's disease also known in medical literature as Alzheimer disease is the most common form of dementia. There is no cure for the disease, which worsens as it progresses, and eventually leads to death...

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

Chemical structure

A chemical structure includes molecular geometry, electronic structure and crystal structure of molecules. Molecular geometry refers to the spatial arrangement of atoms in a molecule and the chemical bonds that hold the atoms together. Molecular geometry can range from the very simple, such as...

. Congo red

Congo red

Congo red is the sodium salt of 3,3'-bis. It is a secondary diazo dye...

dye intercalates

Intercalation (chemistry)

In chemistry, intercalation is the reversible inclusion of a molecule between two other molecules . Examples include DNA intercalation and graphite intercalation compounds.- DNA intercalation :...

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

Optical fiber

An optical fiber is a flexible, transparent fiber made of a pure glass not much wider than a human hair. It functions as a waveguide, or "light pipe", to transmit light between the two ends of the fiber. The field of applied science and engineering concerned with the design and application of...

can cause birefringence, which can cause distortion in fiber-optic communication

Fiber-optic communication

Fiber-optic communication is a method of transmitting information from one place to another by sending pulses of light through an optical fiber. The light forms an electromagnetic carrier wave that is modulated to carry information...

; see polarization mode dispersion

Polarization mode dispersion

Polarization mode dispersion is a form of modal dispersion where two different polarizations of light in a waveguide, which normally travel at the same speed, travel at different speeds due to random imperfections and asymmetries, causing random spreading of optical pulses...

. The imperfections can be geometrically based, or a result of photoelastic effects from loading on the optical fiber.

Fast and slow rays

Effective refractive indices in negative uniaxial materials
Propagation direction	Ordinary ray		Extraordinary ray
Propagation direction	Polarization	n_eff	Polarization	n_eff
z	xy-plane		n/a	n/a
xy-plane	xy-plane		z
xz-plane	y		xz-plane
other	analogous to xz-plane

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, 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

and

, depending on the angle with the z axis. The effective refractive index can be constructed from the Index ellipsoid

Index ellipsoid

In optics, an index ellipsoid is a diagram of an ellipsoid that depicts the orientation and relative magnitude of refractive indices in a crystal....

Biaxial birefringence

Biaxial materials, at 590 nm
Material	n_α	n_β	n_γ
borax Borax Borax, also known as sodium borate, sodium tetraborate, or disodium tetraborate, is an important boron compound, a mineral, and a salt of boric acid. It is usually a white powder consisting of soft colorless crystals that dissolve easily in water.Borax has a wide variety of uses...	1.447	1.469	1.472
epsom salt Magnesium sulfate Magnesium sulfate is a chemical compound containing magnesium, sulfur and oxygen, with the formula MgSO4. It is often encountered as the heptahydrate epsomite , commonly called Epsom salt, from the town of Epsom in Surrey, England, where the salt was distilled from the springs that arise where the... MgSO₄·7(H₂O)	1.433	1.455	1.461
mica Mica The mica group of sheet silicate minerals includes several closely related materials having highly perfect basal cleavage. All are monoclinic, with a tendency towards pseudohexagonal crystals, and are similar in chemical composition... , biotite Biotite Biotite is a common phyllosilicate mineral within the mica group, with the approximate chemical formula . More generally, it refers to the dark mica series, primarily a solid-solution series between the iron-endmember annite, and the magnesium-endmember phlogopite; more aluminous endmembers...	1.595	1.640	1.640
mica, muscovite Muscovite Muscovite is a phyllosilicate mineral of aluminium and potassium with formula KAl22, or 236. It has a highly-perfect basal cleavage yielding remarkably-thin laminæ which are often highly elastic...	1.563	1.596	1.601
olivine Olivine The mineral olivine is a magnesium iron silicate with the formula 2SiO4. It is a common mineral in the Earth's subsurface but weathers quickly on the surface.... (Mg, Fe)₂SiO₄	1.640	1.660	1.680
perovskite Perovskite A perovskite structure is any material with the same type of crystal structure as calcium titanium oxide , known as the perovskite structure, or XIIA2+VIB4+X2−3 with the oxygen in the face centers. Perovskites take their name from this compound, which was first discovered in the Ural mountains of... CaTiO₃	2.300	2.340	2.380
topaz Topaz Topaz is a silicate mineral of aluminium and fluorine with the chemical formula Al2SiO42. Topaz crystallizes in the orthorhombic system and its crystals are mostly prismatic terminated by pyramidal and other faces.-Color and varieties:...	1.618	1.620	1.627
ulexite Ulexite Ulexite , sometimes known as TV rock, is a mineral occurring in silky white rounded crystalline masses or in parallel fibers. The natural fibers of ulexite conduct light along their long axes, by internal reflection...	1.490	1.510	1.520

Biaxial birefringence, also known as trirefringence, describes an anisotropic material that has more than one axis of anisotropy

Index ellipsoid

In optics, an index ellipsoid is a diagram of an ellipsoid that depicts the orientation and relative magnitude of refractive indices in a crystal....

. 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

Uniaxial birefringent materials are classified as positively (or negatively) birefringent when, for light (having parallel and perpendicular components) directed to the optic axis

Optic axis of a crystal

, the refractive index

Refractive index

In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

of light polarized parallel to the optic axis is greater (or smaller, respectively) than light polarized perpendicularly to the optic axis. 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.

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

Spectrum

A spectrum is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by...

, instead having an order including yellow-orange-red-violet-blue.

Measurement

Birefringence and related optical effects (such as optical rotation

Optical rotation

Optical rotation is the turning of the plane of linearly polarized light about the direction of motion as the light travels through certain materials. It occurs in solutions of chiral molecules such as sucrose , solids with rotated crystal planes such as quartz, and spin-polarized gases of atoms...

and linear or circular dichroism

Circular dichroism

Circular dichroism refers to the differential absorption of left and right circularly polarized light. This phenomenon was discovered by Jean-Baptiste Biot, Augustin Fresnel, and Aimé Cotton in the first half of the 19th century. It is exhibited in the absorption bands of optically active chiral...

) can be measured by measuring the changes in the polarization of light passing through the material. These measurements are known as polarimetry

Polarimetry

Polarimetry is the measurement and interpretation of the polarization of transverse waves, most notably electromagnetic waves, such as radio or light waves...

.

Birefringence of lipid bilayers can be measured using dual polarisation interferometry

Dual Polarisation Interferometry

Dual polarization interferometry is an analytical technique that can probe molecular scale layers adsorbed to the surface of a waveguide by using the evanescent wave of a laser beam confined to the waveguide...

. 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

Polarizer

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

Gladstone-Dale relation

The Gladstone–Dale relation is used for optical analysis , or to calculate the density of a liquid for use in fluid dynamics .The relation has also been used to calculate refractive index of glass and minerals in optical mineralogy The Gladstone–Dale relation (J. H. Gladstone and T. P. Dale, 1864)...

Applications

Birefringence is widely used in optical devices, such as liquid crystal display

Liquid crystal display

A liquid crystal display is a flat panel display, electronic visual display, or video display that uses the light modulating properties of liquid crystals . LCs do not emit light directly....

s, light modulators

Electro-optic modulator

Electro-optic modulator is an optical device in which a signal-controlled element displaying electro-optic effect is used to modulate a beam of light. The modulation may be imposed on the phase, frequency, amplitude, or polarization of the modulated beam...

, color filters

Lyot filter

A Lyot filter, named for its inventor Bernard Lyot, is a type of optical filter that uses birefringence to produce a narrow passband of transmitted wavelengths...

, wave plate

Wave plate

s, optical axis gratings

Optical axis gratings

Optical axis gratings are gratings of optical axis of a birefringent material. In OAGs, the birefringence of the material is constant, while the direction of optical axis is periodically modulated in a fixed direction...

, etc. It also plays an important role in second harmonic generation

Second harmonic generation

An optical frequency multiplier is a nonlinear optical device, in which photons interacting with a nonlinear material are effectively "combined" to form new photons with greater energy, and thus higher frequency...

and many other nonlinear processes

Nonlinear optics

Nonlinear optics is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light...

.

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.

Needle aspiration of fluid from a gout

Gout

Gout is a medical condition usually characterized by recurrent attacks of acute inflammatory arthritis—a red, tender, hot, swollen joint. The metatarsal-phalangeal joint at the base of the big toe is the most commonly affected . However, it may also present as tophi, kidney stones, or urate...

y joint will reveal negatively birefringent monosodium urate crystals. Calcium pyrophosphate

Calcium pyrophosphate

Calcium pyrophosphate is a chemical compound that can be formed by the reaction of pyrophosphoric acid and a calcium base or by strongly heating calcium phosphate or calcium ammonium phosphate....

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, or a crystal of known birefringence is added to the sample for comparison.

In ophthalmology

Ophthalmology

Ophthalmology is the branch of medicine that deals with the anatomy, physiology and diseases of the eye. An ophthalmologist is a specialist in medical and surgical eye problems...

, scanning laser polarimetry

Scanning laser polarimetry

Scanning laser polarimetry is the use of polarised light to measure the thickness of the retinal nerve fiber layer as part of a glaucoma workup. The GDx-VCC is one example....

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

Glaucoma

Glaucoma is an eye disorder in which the optic nerve suffers damage, permanently damaging vision in the affected eye and progressing to complete blindness if untreated. It is often, but not always, associated with increased pressure of the fluid in the eye...

.

Birefringence characteristics in sperm heads allow for the selection of spermatozoa for intracytoplasmic sperm injection

Intracytoplasmic sperm injection

Intracytoplasmic sperm injection is an in vitro fertilization procedure in which a single sperm is injected directly into an egg.-Indications:...

. Likewise, zona imaging uses birefringence on oocyte

Oocyte

An oocyte, ovocyte, or rarely ocyte, is a female gametocyte or germ cell involved in reproduction. In other words, it is an immature ovum, or egg cell. An oocyte is produced in the ovary during female gametogenesis. The female germ cells produce a primordial germ cell which undergoes a mitotic...

s to select the ones with highest chances of successful pregnancy. Birefringence of particles biopsied from pulmonary nodules indicates silicosis

Silicosis

Silicosis, also known as Potter's rot, is a form of occupational lung disease caused by inhalation of crystalline silica dust, and is marked by inflammation and scarring in forms of nodular lesions in the upper lobes of the lungs...

Elastic birefringence

Another form of birefringence is observed in anisotropic elastic materials. In these materials, shear wave

S-wave

A type of seismic wave, the S-wave, secondary wave, or shear wave is one of the two main types of elastic body waves, so named because they move through the body of an object, unlike surface waves....

s 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

Seismology

Seismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies. The field also includes studies of earthquake effects, such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic,...

. Birefringence is also used in optical mineralogy to determine the chemical composition, and history of minerals and rocks.

Stress induced birefringence

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

Injection molding

Injection molding is a manufacturing process for producing parts from both thermoplastic and thermosetting plastic materials. Material is fed into a heated barrel, mixed, and forced into a mold cavity where it cools and hardens to the configuration of the cavity...

. 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

Photoelasticity

Photoelasticity is an experimental method to determine the stress distribution in a material. The method is mostly used in cases where mathematical methods become quite cumbersome. Unlike the analytical methods of stress determination, photoelasticity gives a fairly accurate picture of stress...

used for analyzing stress distribution in solids is based on the same principle.

Theory

More generally, birefringence can be defined by considering a dielectric permittivity

Permittivity

In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...

and a refractive index that are tensor

Tensor

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

s. Consider a plane wave

Plane wave

In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....

propagating in an anisotropic medium, with a relative permittivity tensor ε, where the refractive index n, is defined by

. If the wave has an electric vector of the form:
where r is the position vector and t is time, then the wave vector

Wave vector

In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave , and its direction is ordinarily the direction of wave propagation In...

k and the angular frequency ω must satisfy Maxwell's equations

Maxwell's equations

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

in the medium, leading to the equations:

where c is the speed of light

Speed of light

The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

in a vacuum. Substituting eqn. 2 in eqns. 3a-b leads to the conditions:

For the matrix product

often a separate name is used, the dielectric displacement vector

. So essentially birefringence concerns the general theory of linear relationships between these two vectors in anisotropic media.

To find the allowed values of k, E₀ 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 E₀ 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

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

is zero:
Multiplying out eqn (6), and rearranging the terms, we obtain
In the case of a uniaxial material, where n_x=n_y=n_o and n_z=n_e 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

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

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.

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 that can be labeled n₁ and n₂. 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 Δn, and extinction angle
Extinction angle
Extinction is a term used in optical mineralogy and petrology, which describes when cross-polarized light dims, as viewed through a thin section of a mineral in a petrographic microscope. An isotropic minerals, opaque minerals, or amophous materials show no light...

χ, where Δn = n₁ − n₂.

Vacuum birefringence

Due to quantum effects, the vacuum can behave like a birefringent medium. In the limit of weak fields and slowly varying fields, the Euler-Heisenberg Lagrangian

Euler-Heisenberg Lagrangian

In physics, the Euler-Heisenberg Lagrangian describes the non-linear dynamics of electromagnetic fields in vacuum. It takes into account vacuum polarization to one loop, and it is valid for electromagnetic fields that change slowly compared to the inverse electron mass...

becomes:

where

is the fine sructure constant and m the electron mass.

The effective electric permittivity tensor and magnetic permeability tensor are then given by :

In the case of an plane wave propagating through a region of constant magnetic field, one finds that there are two eigenmodes. One has the electric component of the plane wave perpendicular to the plane spanned by the wavevector and the magnetic field, the other has the magnetic component perpendicular to this plane. The former has an index of refraction of