Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light.[1] 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. Crystals with non-cubic crystal structures are often birefringent, as are plastics under mechanical stress.

Birefringence is responsible for the phenomenon of double refraction whereby a 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 Danish scientist Rasmus Bartholin in 1669, who observed it[2] in calcite, a crystal having one of the strongest birefringences. However it was not until the 19th century that Augustin-Jean Fresnel described the phenomenon in terms of polarization, understanding light as a wave with field components in transverse polarizations (perpendicular to the direction of the wave vector).

Crystal on graph paper
A calcite crystal laid upon a graph paper with blue lines showing the double refraction
Calcite and polarizing filter
Doubly refracted image as seen through a calcite crystal, seen through a rotating polarizing filter illustrating the opposite polarization states of the two images.


Positively birefringent material
Incoming light in the parallel (p) polarization sees a different effective index of refraction than light in the perpendicular (s) polarization, and is thus refracted at a different angle.

A mathematical description of wave propagation in a birefringent medium is presented below. Following is a qualitative explanation of the phenomenon.

Uniaxial materials

The simplest type of birefringence is described as uniaxial, meaning that there is a single direction governing the optical anisotropy whereas all directions perpendicular to it (or at a given angle to it) are optically equivalent. Thus rotating the material around this axis does not change its optical behavior. This special direction is known as the optic axis of the material. Light whose polarization is perpendicular to the optic axis is governed by a refractive index no (for "ordinary"). Light whose polarization is in the direction of the optic axis sees an optical index ne (for "extraordinary"). For any ray direction there is a linear polarization direction perpendicular to the optic axis, and this is called an ordinary ray. However, for ray directions not parallel to the optic axis, the polarization direction perpendicular to the ordinary ray's polarization will be partly in the direction of the optic axis, and this is called an extraordinary ray. I.e., when unpolarized light enters an uniaxial birefringent material it is split into two beams travelling different directions; the ordinary ray doesn't change direction while the extraordinary ray bends (is refracted) as it travels through the material. The ordinary ray will always experience a refractive index of no, whereas the refractive index of the extraordinary ray will be in between no and ne, depending on the ray direction as described by the index ellipsoid. The magnitude of the difference is quantified by the birefringence:

The propagation (as well as reflection coefficient) of the ordinary ray is simply described by no as if there were no birefringence involved. However the extraordinary ray, as its name suggests, propagates unlike any wave in a homogenous optical material. Its refraction (and reflection) at a surface can be understood using the effective refractive index (a value in between no and ne). However it is in fact an inhomogeneous wave whose power flow (given by the Poynting vector) is not exactly in the direction of the wave vector. This causes an additional shift in that beam, even when launched at normal incidence, as is popularly observed using a crystal of calcite as photographed above. Rotating the calcite crystal will cause one of the two images, that of the extraordinary ray, to rotate slightly around that of the ordinary ray, which remains fixed.

When the light propagates either along or orthogonal to the optic axis, such a lateral shift does not occur. In the first case, both polarizations 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 ne) 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 quarter-wave plate is commonly used to create circular polarization from a linearly polarized source.

Biaxial materials

The case of so-called biaxial crystals is substantially more complex.[3] These are characterized by three refractive indices corresponding to three principal axes of the crystal. For most ray directions, both polarizations would be classified as extraordinary rays but with different effective refractive indices. Being extraordinary waves, however, the direction of power flow is not identical to the direction of the wave vector in either case.

The two refractive indices can be determined using the index ellipsoids for given directions of the polarization. Note that for biaxial crystals the index ellipsoid will not be an ellipsoid of revolution ("spheroid") but is described by three unequal principle refractive indices nα, nβ and nγ. Thus there is no axis around which a rotation leaves the optical properties invariant (as there is with uniaxial crystals whose index ellipsoid is a spheroid).

Although there is no axis of symmetry, there are two optical axes or binormals which are defined as directions along which light may propagate without birefringence, i.e., directions along which the wavelength is independent of polarization.[3] For this reason, birefringent materials with three distinct refractive indices are called biaxial. Additionally, there are two distinct axes known as optical ray axes or biradials along which the group velocity of the light is independent of polarization.

Double refraction

When an arbitrary beam of light strikes the surface of a birefringent material, the polarizations corresponding to the ordinary and extraordinary rays generally take somewhat different paths. Unpolarized light consists of equal amounts of energy in any two orthogonal polarizations, and even polarized light (except in special cases) will have some energy in each of these polarizations. According to Snell's law of refraction, the angle of refraction will be governed by the effective refractive index which is different between these two polarizations. This is clearly seen, for instance, in the Wollaston prism which is designed to separate incoming light into two linear polarizations using a birefringent material such as calcite.

The different angles of refraction for the two polarization components are shown in the figure at the top of the page, with the optic axis along the surface (and perpendicular to the plane of incidence), so that the angle of refraction is different for the p polarization (the "ordinary ray" in this case, having its polarization perpendicular to the optic axis) and the s polarization (the "extraordinary ray" with a polarization component along the optic axis). In addition, a distinct form of double refraction occurs in cases where the optic axis is not along the refracting surface (nor exactly normal to it); in this case the electric polarization of the birefringent material is not exactly in the direction of the wave's electric field for the extraordinary ray. The direction of power flow (given by the Poynting vector) for this inhomogenous wave is at a finite angle from the direction of the wave vector resulting in an additional separation between these beams. So even in the case of normal incidence, where the angle of refraction is zero (according to Snell's law, regardless of effective index of refraction), the energy of the extraordinary ray may be propagated at an angle. This is commonly observed using a piece of calcite cut appropriately with respect to its optic axis, placed above a paper with writing, as in the above two photographs.


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

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

A material is termed uniaxial when it has a single direction of symmetry in its optical behavior, which we term the optic axis. It also happens to be the axis of symmetry of the index ellipsoid (a spheroid in this case). The index ellipsoid could still be described according to the refractive indices, nα, nβ and nγ, along three coordinate axes, however in this case two are equal. So if nα = nβ corresponding to the x and y axes, then the extraordinary index is nγ corresponding to the z axis, which is also called the optic axis in this case.

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

Fast and slow rays

In a birefringent material, a wave consists of two polarization components which generally are governed by different effective refractive indices. The so-called slow ray is the component for which the material has the higher effective refractive index (slower phase velocity), while the fast ray is the one with a lower effective refractive index. When a beam is incident on such a material from air (or any material with a lower refractive index), the slow ray is thus refracted more towards the normal than the fast ray. In the figure at the top of the page, it can be seen that refracted ray with s polarization in the direction of the optic axis (thus the extraordinary ray) is the slow ray in this case.

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

Positive or negative

Uniaxial birefringence is classified as positive when the extraordinary index of refraction ne is greater than the ordinary index no. Negative birefringence means that Δn = neno is less than zero.[4] In other words, the polarization of the fast (or slow) wave is perpendicular to the optic axis when the birefringence of the crystal is positive (or negative, respectively). In the case of biaxial crystals, all three of the principal axes have different refractive indices, so this designation does not apply. But for any defined ray direction one can just as well designate the fast and slow ray polarizations.

Sources of optical birefringence

While birefringence is usually obtained using an anisotropic crystal, it can result from an optically isotropic material in a few ways:

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

Common birefringent materials

Food Polarization-Dierking
Light polarization shown on clear polystyrene cutlery between crossed polarizers

The best characterized birefringent materials are crystals. Due to their specific crystal structures their refractive indices are well defined. Depending on the symmetry of a crystal structure (as determined by one of the 32 possible crystallographic point groups), crystals in that group may be forced to be isotropic (not birefringent), to have uniaxial symmetry, or neither in which case it is a biaxial crystal. The crystal structures permitting uniaxial and biaxial birefringence are noted in the two tables, below, listing the two or three principal refractive indices (at wavelength 590 nm) of some better known crystals.[5]

Many plastics are birefringent because their molecules are "frozen" in a stretched conformation when the plastic is molded or extruded.[6] For example, ordinary cellophane is birefringent. Polarizers are routinely used to detect stress in plastics such as polystyrene and polycarbonate.

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

Polarized light microscopy is commonly used in biological tissue, as many biological materials are birefringent. Collagen, found in cartilage, tendon, bone, corneas, and several other areas in the body, is birefringent and commonly studied with polarized light microscopy.[7] Some proteins are also birefringent, exhibiting form birefringence.[8]

Inevitable manufacturing imperfections in optical fiber leads to birefringence, which is one cause of pulse broadening in fiber-optic communications. 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 fibers.

In addition to anisotropy in the electric polarizability (electric susceptibility), anisotropy in the magnetic polarizability (magnetic permeability) can also cause birefringence. However, at optical frequencies, values of magnetic permeability for natural materials are not measurably different from µ0, so this is not a source of optical birefringence in practice.

Uniaxial crystals, at 590 nm[5]
Material Crystal system no ne Δn
barium borate BaB2O4 Trigonal 1.6776 1.5534 −0.1242
beryl Be3Al2(SiO3)6 Hexagonal 1.602 1.557 −0.045
calcite CaCO3 Trigonal 1.658 1.486 −0.172
ice H2O Hexagonal 1.309 1.313 +0.004
lithium niobate LiNbO3 Trigonal 2.272 2.187 −0.085
magnesium fluoride MgF2 Tetragonal 1.380 1.385 +0.006
quartz SiO2 Trigonal 1.544 1.553 +0.009
ruby Al2O3 Trigonal 1.770 1.762 −0.008
rutile TiO2 Tetragonal 2.616 2.903 +0.287
sapphire Al2O3 Trigonal 1.768 1.760 −0.008
silicon carbide SiC Hexagonal 2.647 2.693 +0.046
tourmaline (complex silicate) Trigonal 1.669 1.638 −0.031
zircon, high ZrSiO4 Tetragonal 1.960 2.015 +0.055
zircon, low ZrSiO4 Tetragonal 1.920 1.967 +0.047
Biaxial crystals, at 590 nm[5]
Material Crystal system nα nβ nγ
borax Na2(B4O5)(OH)4·8H2O Monoclinic 1.447 1.469 1.472
epsom salt MgSO4·7H2O Monoclinic 1.433 1.455 1.461
mica, biotite K(Mg,Fe)3(AlSi3O10)(F,OH)2 Monoclinic 1.595 1.640 1.640
mica, muscovite KAl2(AlSi3O10)(F,OH)2 Monoclinic 1.563 1.596 1.601
olivine (Mg,Fe)2SiO4 Orthorhombic 1.640 1.660 1.680
perovskite CaTiO3 Orthorhombic 2.300 2.340 2.380
topaz Al2SiO4(F,OH)2 Orthorhombic 1.618 1.620 1.627
ulexite NaCaB5O6(OH)6·5H2O Triclinic 1.490 1.510 1.520


Birefringence and other polarization-based optical effects (such as optical rotation and linear or circular dichroism) can be measured by measuring the changes in the polarization of light passing through the material. These measurements are known as polarimetry. Polarized light microscopes, which contain two polarizers that are at 90° to each other on either side of the sample, are used to visualize birefringence. The addition of quarter-wave plates permit examination of circularly polarized light. Birefringence measurements have been made with phase-modulated systems for examining the transient flow behavior of fluids.[9][10]

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


LCD layers
Reflective twisted-nematic liquid-crystal display. Light reflected by surface (6) (or coming from a backlight) is horizontally polarized (5) and passes through the liquid-crystal modulator (3) sandwiched in between transparent layers (2, 4) containing electrodes. Horizontally polarized light is blocked by the vertically oriented polarizer (1), except where its polarization has been rotated by the liquid crystal (3), appearing bright to the viewer.

Birefringence is used in many optical devices. Liquid-crystal displays, 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, light modulators modulate the intensity of light through 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 plates are thin birefringent sheets widely used in certain optical equipment for modifying the polarization state of light passing through it.

Birefringence also plays an important role in second-harmonic generation and other nonlinear 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 phase matching, which is required for efficient operation of these devices.


Birefringence is utilized in medical diagnostics. One powerful accessory used with optical microscopes is a pair of crossed 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.

Fluorescent uric acid
Urate crystals, with the crystals' long axis seen as horizontal in this view being parallel to that of a red compensator filter. These appear as yellow, and are thereby of negative birefringence.

For instance, needle aspiration of fluid from a gouty joint will reveal negatively birefringent monosodium urate crystals. Calcium pyrophosphate crystals, in contrast, show weak positive birefringence.[11] Urate crystals appear yellow, and calcium pyrophosphate crystals appear blue when their long axes are aligned parallel to that of a red compensator filter,[12] or a crystal of known birefringence is added to the sample for comparison.

Birefringence can be observed in amyloid plaques such as are found in the brains of Alzheimer's 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 conformation. Congo red dye intercalates between the folds and, when observed under polarized light, causes birefringence.

In ophthalmology, binocular retinal birefringence screening of the Henle fibers (photoreceptor axons that go radially outward from the fovea) provides a reliable detection of strabismus and possibly also of anisometropic amblyopia.[13] Furthermore, scanning laser polarimetry utilises the birefringence of the optic nerve fibre layer to indirectly quantify its thickness, which is of use in the assessment and monitoring of glaucoma.

Birefringence characteristics in sperm heads allow the selection of spermatozoa for intracytoplasmic sperm injection.[14] Likewise, zona imaging uses birefringence on oocytes to select the ones with highest chances of successful pregnancy.[15] Birefringence of particles biopsied from pulmonary nodules indicates silicosis.

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

Stress-induced birefringence

Birefringence Stress Plastic
Color pattern of a plastic box with "frozen in" mechanical stress 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, because polarization of a light ray is rotated after passing through a birefringent material and the amount of rotation is dependent on wavelength. The experimental method called photoelasticity used for analyzing stress distribution in solids is based on the same principle. There has been recent research on using stress induced birefringence in a glass plate to generate an Optical vortex and full Poincare beams (optical beams that have every possible polarization states across its cross-section).[16]

Other cases of birefringence

Rutile birefringence
Birefringent rutile observed in different polarizations using a rotating polarizer (or analyzer)

Birefringence is observed in anisotropic 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.


In an isotropic medium (including free space) the so-called electric displacement (D) is just proportional to the electric field (E) according to D = ɛE where the material's permittivity ε is just a scalar (and equal to n2ε0 where n is the index of refraction). However, in an anisotropic material exhibiting birefringence, the relationship between D and E must now be described using a tensor equation:


where ε is now a 3 × 3 permittivity tensor. We assume linearity and no magnetic permeability in the medium: μ = μ0. The electric field of a plane wave of angular frequency ω can be written in the general form:


where r is the position vector, t is time, and E0 is a vector describing the electric field at r = 0, t = 0. Then we shall find the possible wave vectors k. By combining Maxwell's equations for ∇ × E and ∇ × H, we can eliminate H = 1/μ0B to obtain:


With no free charges, Maxwell's equation for the divergence of D vanishes:


We can apply the vector identity ∇ × (∇ × A) = ∇(∇ ⋅ A) − ∇2A to the left hand side of eq. 3a, and use the spatial dependence in which each differentiation in x (for instance) results in multiplication by ikx to find:


The right hand side of eq. 3a can be expressed in terms of E through application of the permittivity tensor ε and noting that differentiation in time results in multiplication by , eq. 3a then becomes:


Applying the differentiation rule to eq. 3b we find:


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

Finding the allowed values of k for a given ω is easiest done by using Cartesian coordinates with the x, y and z axes chosen in the directions of the symmetry axes of the crystal (or simply choosing z in the direction of the optic axis of a uniaxial crystal), resulting in a diagonal matrix for the permittivity tensor ε:


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 substituting in the speed of light c using c2 = 1/μ0ε0, eq. 4a becomes


where Ex, Ey, Ez are the components of E (at any given position in space and time) and kx, ky, kz are the components of k. Rearranging, we can write (and similarly for the y and z components of eq. 4a)




This is a set of linear equations in Ex, Ey, Ez, so it can have a nontrivial solution (that is, one other than E = 0) as long as the following determinant is zero:


Evaluating the determinant of eq. 6, and rearranging the terms, we obtain


In the case of a uniaxial material, choosing the optic axis to be in the z direction so that nx = ny = no and nz = ne, this expression can be factored into


Setting either of the factors in eq. 8 to zero will define an ellipsoidal surface[note 1] in the space of wavevectors k that are allowed for a given ω. The first factor being zero defines a sphere; this is the solution for so-called ordinary rays, in which the effective refractive index is exactly no regardless of the direction of k. The second defines a spheroid symmetric about the z axis. This solution corresponds to the so-called extraordinary rays in which the effective refractive index is in between no and ne, depending on the direction of k. Therefore, for any arbitrary direction of propagation (other than in the direction of the optic axis), two distinct wavevectors k are allowed corresponding to the polarizations of the ordinary and extraordinary rays.

For a biaxial material a similar but more complicated condition on the two waves can be described;[17] the allowed k vectors in a specified direction now lie on one of two ellipsoids. By inspection one can see that eq. 6 is generally satisfied for two positive values of ω. Or, for a specified optical frequency ω and direction normal to the wavefronts k/|k|, it is satisfied for two wavenumbers (or propagation constants) |k| (and thus effective refractive indices) corresponding to the propagation of two linear polarizations in that direction.

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

A general state of polarization launched into the medium can always be decomposed into two waves, one in each of those two polarizations, which will then propagate with different wavenumbers |k|. Applying the different phase of propagation to those two waves over a specified propagation distance will result in a generally different net polarization state at that point; this is the principle of the waveplate for instance. However, when you have a wave launched into a birefringent material at non-normal incidence, the problem is yet more complicated since the two polarization components will now not only have distinct wavenumbers but the k vectors will not even be in exactly the same direction (see figure at the top of the page). In this case the two k vectors are rather solutions of eq. 6 constrained by the boundary condition which requires that the components of the two transmitted waves' k vectors, and the k vector of the incident wave, as projected onto the surface of the interface, must all be identical.

See also


  1. ^ Although related, note that this is not the same as the index ellipsoid.


  1. ^ "Olympus Microscopy Resource Center". Olympus America Inc. Retrieved 2011-11-13.
  2. ^ See:
  3. ^ a b Landau, L. D., and Lifshitz, E. M., Electrodynamics of Continuous Media, Vol. 8 of the Course of Theoretical Physics 1960 (Pergamon Press), §79
  4. ^ Brad Amos. Birefringence for facetors I: what is birefringence? Archived December 14, 2013, at the Wayback Machine. First published in StoneChat, the Journal of the UK Facet Cutter's Guild. January–March. edition 2005.
  5. ^ a b c Elert, Glenn. "Refraction". The Physics Hypertextbook.
  6. ^ Neves, N. M. (1998). "The use of birefringence for predicting the stiffness of injection molded polycarbonate discs". Polymer Engineering & Science. 38 (10): 1770–1777. doi:10.1002/pen.10347.
  7. ^ Wolman, M.; Kasten, F. H. (1986). "Polarized light microscopy in the study of the molecular structure of collagen and reticulin". Histochemistry. 85: 41–49. doi:10.1007/bf00508652.
  8. ^ Sano, Y (1988). "Optical anistropy of bovine serum albumin". J. Colloid Interface Sci. 124: 403–407. Bibcode:1988JCIS..124..403S. doi:10.1016/0021-9797(88)90178-6.
  9. ^ Frattini, P., Fuller, G., "A note on phase-modulated flow birefringence: a promising rheo-optical method", J. Rheol., 28: 61 (1984).
  10. ^ Doyle, P., Shaqfeh, E. S. G., Spiegelberg, S. H., McKinley, G. H., "Relaxation of dilute polymer solutions following extensional flow", J. Non-Newtonian Fluid Mech., 86:79–110 (1998).
  11. ^ Hardy RH, Nation B (June 1984). "Acute gout and the accident and emergency department". Arch Emerg Med. 1 (2): 89–95. doi:10.1136/emj.1.2.89. PMC 1285204. PMID 6536274.
  12. ^ The Approach to the Painful Joint Workup Author: Alan N. Baer; Chief Editor: Herbert S. Diamond. Updated: Nov 22, 2010.
  13. ^ Reed M. Jost; Joost Felius; Eileen E. Birch (August 2014). "High sensitivity of binocular retinal birefringence screening for anisometropic amblyopia without strabismus". Journal of American Association for Pediatric Ophthalmology and Strabismus (JAAPOS). 18 (4): e5–e6. doi:10.1016/j.jaapos.2014.07.017.
  14. ^ Gianaroli L.; Magli M. C.; Ferraretti A. P.; et al. (December 2008). "Birefringence characteristics in sperm heads allow for the selection of reacted spermatozoa for intracytoplasmic sperm injection". Fertil. Steril. 93 (3): 807–13. doi:10.1016/j.fertnstert.2008.10.024. PMID 19064263.
  15. ^ Ebner T.; Balaban B.; Moser M.; et al. (May 2009). "Automatic user-independent zona pellucida imaging at the oocyte stage allows for the prediction of preimplantation development". Fertil. Steril. 94 (3): 913–920. doi:10.1016/j.fertnstert.2009.03.106. PMID 19439291.
  16. ^ Beckley, Amber M.; Brown, Thomas G.; Alonso, Miguel A. (2010-05-10). "Full Poincaré beams". Optics Express. 18 (10): 10777–10785. Bibcode:2010OExpr..1810777B. doi:10.1364/OE.18.010777. ISSN 1094-4087.
  17. ^ Born M, and Wolf E, Principles of Optics, 7th Ed. 1999 (Cambridge University Press), §15.3.3

External links


Cinnabar () and cinnabarite (), likely deriving from the Ancient Greek: κιννάβαρι (kinnabari), refer to the common bright scarlet to brick-red form of mercury(II) sulfide (HgS) that is the most common source ore for refining elemental mercury, and is the historic source for the brilliant red or scarlet pigment termed vermilion and associated red mercury pigments.

Cinnabar generally occurs as a vein-filling mineral associated with recent volcanic activity and alkaline hot springs. The mineral resembles quartz in symmetry and in its exhibiting birefringence; cinnabar has a mean refractive index of approximately 3.2, a hardness between 2.0 and 2.5, and a specific gravity of approximately 8.1. The color and properties derive from a structure that is a hexagonal crystalline lattice belonging to the trigonal crystal system, crystals that sometimes exhibit twinning.

Cinnabar has been used for its color since antiquity in the Near East, including as a rouge-type cosmetic, in the New World since the Olmec culture, and in China since as early as the Yangshao culture, where it was used in coloring stoneware.

Associated modern precautions for use and handling of cinnabar arise from the toxicity of the mercury component, which was recognized as early as ancient Rome.

Conoscopic interference pattern

This page is about the geology/optical mineralogy term. For general information about interference, see Interference (wave propagation) or Interference patterns.A conoscopic interference pattern or interference figure is a pattern of birefringent colours crossed by dark bands (or isogyres), which can be produced using a geological petrographic microscope for the purposes of mineral identification and investigation of mineral optical and chemical properties. The figures are produced by optical interference when diverging light rays travel through an optically non-isotropic substance - that is, one in which the substance's refractive index varies in different directions within it. The figure can be thought of as a "map" of how the birefringence of a mineral would vary with viewing angle away from perpendicular to the slide, where the central colour is the birefringence seen looking straight down, and the colours further from the centre equivalent to viewing the mineral at ever increasing angles from perpendicular. The dark bands correspond to positions where optical extinction (apparent isotropy) would be seen. In other words, the interference figure presents all possible birefringence colours for the mineral at once.

Viewing the interference figure is a foolproof way to determine if a mineral is optically uniaxial or biaxial. If the figure is aligned correctly, use of a sensitive tint plate in conjunction with the microscope allows the user to determine mineral optic sign and optic angle.


Electro-optics is a branch of electrical engineering, electronic engineering, materials science, and material physics involving components, devices (e.g. Lasers, LEDs, waveguides etc.) and systems which operate by the propagation and interaction of light with various tailored materials. It is essentially the same as what is popularly described today as photonics. It is not only concerned with the "Electro-Optic effect". Thus it concerns the interaction between the electromagnetic (optical) and the electrical (electronic) states of materials.

Euler–Heisenberg Lagrangian

In physics, the Euler–Heisenberg Lagrangian describes the non-linear dynamics of electromagnetic fields in vacuum. It was first obtained by Werner Heisenberg and Hans Heinrich Euler in 1936. By treating the vacuum as a medium, it predicts rates of quantum electrodynamics (QED) light interaction processes.

Experiments of Rayleigh and Brace

The experiments of Rayleigh and Brace (1902, 1904) were aimed to show whether length contraction leads to birefringence or not. They were some of the first optical experiments measuring the relative motion of Earth and the luminiferous aether which were sufficiently precise to detect magnitudes of second order to v/c. The results were negative, which was of great importance for the development of the Lorentz transformation and consequently of the theory of relativity. See also Tests of special relativity.

Flow birefringence

In biochemistry, flow birefringence is a hydrodynamic technique for measuring the rotational diffusion constants (or, equivalently, the rotational drag coefficients). The birefringence of a solution sandwiched between two concentric cylinders is measured as a function of the difference in rotational speed between the inner and outer cylinders. The flow tends to orient an ellipsoidal particle (typically, a protein, virus, etc.) in one direction, whereas rotational diffusion (tumbling) causes the molecule to become disoriented. The equilibrium between these two processes as a function of the flow provides a measure of the axial ratio of the ellipsoidal particle.

Interference colour chart

In optical mineralogy an interference colour chart, first developed by Auguste Michel-Lévy, is a tool to identify minerals in thin section using a petrographic microscope. With a known thickness of the thin section, minerals have specific and predictable colours in cross-polarized light, and this chart can help identify minerals. The colours are produced by the difference in speed in the fast and slow rays, also known as birefringence.

When using the chart, it is important to remember these tips:

Isotropic and opaque (metallic) minerals cannot be identified this way.

The stage of the microscope should be rotated until maximum colour is found, and therefore, the maximum birefringence.

Each mineral, depending on the orientation, may not exhibit the maximum birefringence. It is important to sample a number of similar minerals in order to get the best value of birefringence.

Uniaxial minerals can look isotropic (always extinct) if the mineral is cut perpendicular to the optic axis (this situation can be revealed with the conoscopic interference pattern).


Muscovite (also known as common mica, isinglass, or potash mica) is a hydrated phyllosilicate mineral of aluminium and potassium with formula KAl2(AlSi3O10)(FOH)2, or (KF)2(Al2O3)3(SiO2)6(H2O). It has a highly perfect basal cleavage yielding remarkably thin laminae (sheets) which are often highly elastic. Sheets of muscovite 5 meters × 3 meters (16.5 feet × 10 feet) have been found in Nellore, India.

Muscovite has a Mohs hardness of 2–2.25 parallel to the [001] face, 4 perpendicular to the [001] and a specific gravity of 2.76–3. It can be colorless or tinted through grays, browns, greens, yellows, or (rarely) violet or red, and can be transparent or translucent. It is anisotropic and has high birefringence. Its crystal system is monoclinic. The green, chromium-rich variety is called fuchsite; mariposite is also a chromium-rich type of muscovite.

Muscovite is the most common mica, found in granites, pegmatites, gneisses, and schists, and as a contact metamorphic rock or as a secondary mineral resulting from the alteration of topaz, feldspar, kyanite, etc. In pegmatites, it is often found in immense sheets that are commercially valuable. Muscovite is in demand for the manufacture of fireproofing and insulating materials and to some extent as a lubricant.

The name muscovite comes from Muscovy-glass, a name given to the mineral in Elizabethan England due to its use in medieval Russia as a cheaper alternative to glass in windows. This usage became widely known in England during the sixteenth century with its first mention appearing in letters by George Turberville, the secretary of England's ambassador to the Russian tsar Ivan the Terrible, in 1568.

Optic axis of a crystal

An optic axis of a crystal is a direction in which a ray of transmitted light suffers no birefringence (double refraction). An optical axis is a direction rather than a single line: all rays that are parallel to that direction exhibit the same lack of birefringence.Crystals may have a single optic axis, in which case they are uniaxial, or two different optic axes, in which case they are biaxial. Non-crystalline materials generally have no birefringence and thus, no optic axis. A uniaxial crystal (e.g. calcite, quartz) is isotropic within the plane orthogonal to the optic axis of the crystal.

Optical rotation

Optical rotation or optical activity (sometimes referred to as rotary polarization) is the rotation of the plane of polarization of linearly polarized light as it travels through certain materials. Optical activity occurs only in chiral materials, those lacking microscopic mirror symmetry. Unlike other sources of birefringence which alter a beam's state of polarization, optical activity can be observed in fluids. This can include gases or solutions of chiral molecules such as sugars, molecules with helical secondary structure such as some proteins, and also chiral liquid crystals. It can also be observed in chiral solids such as certain crystals with a rotation between adjacent crystal planes (such as quartz) or metamaterials. Rotation of light's plane of polarization may also occur through the Faraday effect which involves a static magnetic field, however this is a distinct phenomenon that is not usually classified under "optical activity."

The rotation of the plane of polarization may be either clockwise, to the right (dextrorotary — d-rotary), or to the left (levorotary — l-rotary) depending on which stereoisomer is present (or dominant). For instance, sucrose and camphor are d-rotary whereas cholesterol is l-rotary. For a given substance, the angle by which the polarization of light of a specified wavelength is rotated is proportional to the path length through the material and (for a solution) proportional to its concentration. The rotation is not dependent on the direction of propagation, unlike the Faraday effect where the rotation is dependent on the relative direction of the applied magnetic field.

Optical activity is measured using a polarized source and polarimeter. This is a tool particularly used in the sugar industry to measure the sugar concentration of syrup, and generally in chemistry to measure the concentration or enantiomeric ratio of chiral molecules in solution. Modulation of a liquid crystal's optical activity, viewed between two sheet polarizers, is the principle of operation of liquid-crystal displays (used in most modern televisions and computer monitors).


PVLAS (Polarizzazione del Vuoto con LASer, "polarization of the vacuum with laser") aims to carry out a test of quantum electrodynamics and possibly detect dark matter at the Department of Physics and National Institute of Nuclear Physics in Ferrara, Italy. It searches for vacuum polarization causing nonlinear optical behavior in magnetic fields. Experiments began in 2001 at the INFN Laboratory in Legnaro (Padua, Italy) and continue today with new equipment.

Photoelastic modulator

A photoelastic modulator (PEM) is an optical device used to modulate the polarization of a light source. The photoelastic effect is used to change the birefringence of the optical element in the photoelastic modulator.

PEM was first invented by J. Badoz in the 1960s and originally called a "birefringence modulator." It was initially developed for physical measurements including optical rotary dispersion and Faraday rotation, polarimetry of astronomical objects, strain-induced birefringence, and ellipsometry. Later developers of the photoelastic modulator include J.C Kemp, S.N Jasperson and S.E Schnatterly.


Photoelasticity describes changes in the optical properties of a material under mechanical deformation. It is a property of all dielectric media and is often used to experimentally determine the stress distribution in a material, where it gives a picture of stress distributions around discontinuities in materials. Photoelastic experiments (also informally referred to as photoelasticity) are an important tool for determining critical stress points in a material, and are used for determining stress concentration in irregular geometries.

Pockels effect

The Pockels effect (after Friedrich Carl Alwin Pockels who studied the effect in 1893), or Pockels electro-optic effect, changes or produces birefringence in an optical medium induced by an electric field. In the Pockels effect, also known as the linear electro-optic effect, the birefringence is proportional to the electric field. In the Kerr effect, the refractive index change (birefringence) is proportional to the square of the field. The Pockels effect occurs only in crystals that lack inversion symmetry, such as lithium niobate, and in other noncentrosymmetric media such as electric-field poled polymers or glasses.


Polarimetry is the measurement and interpretation of the polarization of transverse waves, most notably electromagnetic waves, such as radio or light waves. Typically polarimetry is done on electromagnetic waves that have traveled through or have been reflected, refracted or diffracted by some material in order to characterize that object.

Polarization (waves)

Polarization (also polarisation) is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of the wave. A simple example of a polarized transverse wave is vibrations traveling along a taut string (see image); for example, in a musical instrument like a guitar string. Depending on how the string is plucked, the vibrations can be in a vertical direction, horizontal direction, or at any angle perpendicular to the string. In contrast, in longitudinal waves, such as sound waves in a liquid or gas, the displacement of the particles in the oscillation is always in the direction of propagation, so these waves do not exhibit polarization. Transverse waves that exhibit polarization include electromagnetic waves such as light and radio waves, gravitational waves, and transverse sound waves (shear waves) in solids. In some types of transverse waves, the wave displacement is limited to a single direction, so these also do not exhibit polarization; for example, in surface waves in liquids (gravity waves), the wave displacement of the particles is always in a vertical plane.

An electromagnetic wave such as light consists of a coupled oscillating electric field and magnetic field which are always perpendicular; by convention, the "polarization" of electromagnetic waves refers to the direction of the electric field. In linear polarization, the fields oscillate in a single direction. In circular or elliptical polarization, the fields rotate at a constant rate in a plane as the wave travels. The rotation can have two possible directions; if the fields rotate in a right hand sense with respect to the direction of wave travel, it is called right circular polarization, or, if the fields rotate in a left hand sense, it is called left circular polarization.

Light or other electromagnetic radiation from many sources, such as the sun, flames, and incandescent lamps, consists of short wave trains with an equal mixture of polarizations; this is called unpolarized light. Polarized light can be produced by passing unpolarized light through a polarizer, which allows waves of only one polarization to pass through. The most common optical materials (such as glass) are isotropic and do not affect the polarization of light passing through them; however, some materials—those that exhibit birefringence, dichroism, or optical activity—can change the polarization of light. Some of these are used to make polarizing filters. Light is also partially polarized when it reflects from a surface.

According to quantum mechanics, electromagnetic waves can also be viewed as streams of particles called photons. When viewed in this way, the polarization of an electromagnetic wave is determined by a quantum mechanical property of photons called their spin. A photon has one of two possible spins: it can either spin in a right hand sense or a left hand sense about its direction of travel. Circularly polarized electromagnetic waves are composed of photons with only one type of spin, either right- or left-hand. Linearly polarized waves consist of equal numbers of right and left hand spinning photons, with their phase synchronized so they superpose to give oscillation in a plane.

Polarization is an important parameter in areas of science dealing with transverse waves, such as optics, seismology, radio, and microwaves. Especially impacted are technologies such as lasers, wireless and optical fiber telecommunications, and radar.

Retinal birefringence scanning

Retinal birefringence scanning (RBS) is a method for detection the central fixation of the eye. The method can be used in pediatric ophthalmology for screening purposes. By simultaneously measuring the central fixation of both eyes, small- and large-angle strabismus can be detected. The method is non-invasive and requires little cooperation by the patient, so that it can be used for detecting strabismus in young children. The method provides a reliable detection of strabismus and has also been used for detecting certain kinds of amblyopia.

Retinal birefringence scanning uses the human eye's birefringent properties to identify the position of the fovea and the direction of gaze, and thereby to measure any binocular misalignment.


Rutile is a mineral composed primarily of titanium dioxide (TiO2).

Rutile is the most common natural form of TiO2. Other rarer polymorphs of TiO2 are known including anatase, and brookite.

Rutile has one of the highest refractive indices at visible wavelengths of any known crystal and also exhibits a particularly large birefringence and high dispersion. Owing to these properties, it is useful for the manufacture of certain optical elements, especially polarization optics, for longer visible and infrared wavelengths up to about 4.5 μm.

Natural rutile may contain up to 10% iron and significant amounts of niobium and tantalum. Rutile derives its name from the Latin rutilus, red, in reference to the deep red color observed in some specimens when viewed by transmitted light. Rutile was first described in 1803 by Abraham Gottlob Werner.

Vector soliton

In physical optics or wave optics, a vector soliton is a solitary wave with multiple components coupled together that maintains its shape during propagation. Ordinary solitons maintain their shape but have effectively only one (scalar) polarization component, while vector solitons have two distinct polarization components. Among all the types of solitons, optical vector solitons draw the most attention due to their wide range of applications, particularly in generating ultrafast pulses and light control technology. Optical vector solitons can be classified into temporal vector solitons and spatial vector solitons. During the propagation of both temporal solitons and spatial solitons, despite being in a medium with birefringence, the orthogonal polarizations can copropagate as one unit without splitting due to the strong cross-phase modulation and coherent energy exchange between the two polarizations of the vector soliton which may induce intensity differences between these two polarizations. Thus vector solitons are no longer linearly polarized but rather elliptically polarized.

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