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Encyclopedia > Birefringence
A calcite crystal laid upon a paper with some letters showing the double refraction

$Delta n=n_e-n_o,$

where no and ne are the refractive indices for polarizations perpendicular (ordinary) and parallel (extraordinary) to the axis of anisotropy respectively.

Birefringence can also arise in magnetic, not dielectric, materials, but substantial variations in magnetic permeability of materials are rare at optical frequencies. For other senses of this word, see magnetism (disambiguation). ... A dielectric is a nonconducting substance, i. ... In electromagnetism, permeability is the degree of magnetization of a material that responds linearly to an applied magnetic field. ...

While birefringence is often found naturally (especially in crystals), there are several ways to create it in optically isotropic materials. Look up Isotropy in Wiktionary, the free dictionary. ...

• Birefringence results when isotropic materials are deformed such that the isotropy is lost in one direction (ie, stretched or bent). Example
• Applying an electric field can induce molecules to line up or behave asymmetrically, introducing anisotropy and resulting in birefringence. (see Pockels effect)
• Applying a magnetic field can cause a material to be circularly birefringent, with different indices of refraction for oppositely-handed circular polarizations (see Faraday effect).

The Pockels effect, or Pockels electro-optic effect, produces birefringence in an optical medium induced by a constant or varying electric field. ... In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. ... In physics, the Faraday effect or Faraday rotation is a magneto-optical phenomenon, or an interaction between light and a magnetic field. ...

## Examples of birefringent materials

Uniaxial materials, at 590 nm[1]
Material no ne Δn
beryl Be3Al2(SiO3)6 1.602 1.557 -0.045
calcite CaCO3 1.658 1.486 -0.172
calomel Hg2Cl2 1.973 2.656 +0.683
ice H2O 1.309 1.313 +0.014
lithium niobate LiNbO3 2.272 2.187 -0.085
magnesium fluoride MgF2 1.380 1.385 +0.006
quartz SiO2 1.544 1.553 +0.009
ruby Al2O3 1.770 1.762 -0.008
rutile TiO2 2.616 2.903 +0.287
peridot (Mg, Fe)2SiO4 1.690 1.654 -0.036
sapphire Al2O3 1.768 1.760 -0.008
sodium nitrate NaNO3 1.587 1.336 -0.251
tourmaline (complex silicate ) 1.669 1.638 -0.031
zircon, high ZrSiO4 1.960 2.015 +0.055
zircon, low ZrSiO4 1.920 1.967 +0.047

There are many birefringent crystals: birefringence was first described in calcite crystals by the Danish scientist Rasmus Bartholin in 1669. Rasmus Bartholin (Latinized Erasmus Bartholinus; August 13, 1625 - November 4, 1698) was a Danish scientist and physician. ... // Events Samuel Pepys stopped writing his diary. ...

Birefringence can be observed in amyloid plaque deposits such as are found in the brains of Alzheimer's victims. 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. For other uses, see Amyloid (disambiguation). ... Alzheimers disease (AD), also known simply as Alzheimers, is a neurodegenerative disease that, in its most common form, is found in people over age 65. ... Schematic of antibody binding to an antigen An antibody is a protein complex used by the immune system to identify and neutralize foreign objects like bacteria and viruses. ... Conformation generally means structural arrangement. ... Chemical structure of congo red Congo red is the sodium salt of benzidinedflandersiazo-bis-1-naphtylamine-4-sulfonic acid (formula: C32H22N6Na2O6S2; molecular weight: 696. ... Intercalation induces structural distortions. ...

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

Slight imperfections in optical fiber can cause birefringence, which can cause distortion in fiber-optic communication; see polarization mode dispersion. Optical fibers An optical fiber (or fibre) is a glass or plastic fiber designed to guide light along its length. ... Fiber-optic communication is a method of transmitting information from one place to another by sending light through an optical fiber. ... Polarization mode dispersion (PMD) 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. ...

Silicon carbide, also known as Moissanite, is strongly birefringent. Except where noted otherwise, data are given for materials in their standard state (at 25 Â°C, 100 kPa) Infobox disclaimer and references Silicon carbide (SiC) is a ceramic compound of silicon and carbon that is manufactured on a large scale for use mainly as an abrasive but also occurs in...

The refractive indices of several (uniaxial) birefringent materials are listed below (at wavelength ~ 590 nm)[1]

## Biaxial birefringence

Biaxial materials, at 590 nm[1]
Material nα nβ nγ
borax 1.447 1.469 1.472
epsom salt MgSO4·7(H2O) 1.433 1.455 1.461
mica, biotite 1.595 1.640 1.640
mica, muscovite 1.563 1.596 1.601
olivine (Mg, Fe)2SiO4 1.640 1.660 1.680
perovskite CaTiO3 2.300 2.340 2.380
topaz 1.618 1.620 1.627
ulexite 1.490 1.510 1.520

## Measuring birefringence

Birefringence and related optical effects (such as optical rotation and linear or circular dichroism) can be measured by measuring the changes in the polarization of light passing through the material. These measurements are known as polarimetry. When polarized light is passed through a substance containing chiral molecules (or nonchiral molecules arranged asymmetrically), the direction of polarization can be changed. ... Circular dichroism (CD) is a form of spectroscopy based on the differential absorption of left- and right-handed circularly polarized light. ... Polarimetry is the measurement of the polarisation of light; a polarimeter is the scientific instrument used to make these measurements. ...

A common feature of optical microscopes is a pair of crossed polarizing filters. Between the crossed polarizers, a birefringent sample will appear bright against a dark (isotropic) background. A polarizer is a device that converts an unpolarized or mixed-polarization beam of electromagnetic waves (e. ...

## Elastic birefringence

Another form of birefringence is observed in anisotropic elastic materials. In these materials, shear waves split according to similar principles as the light waves discussed above. The study of birefringent shear waves in the earth is a part of seismology. Birefringence is also used in optical mineralogy to determine the chemical composition, and history of minerals and rocks. In engineering mechanics, deformation is a change in shape due to an applied force. ... A type of seismic wave, the S-wave moves in a shear or transverse wave, so motion is perpendicular to the direction of wave propagation. ... Seismology (from the Greek seismos = earthquake and logos = word) is the scientific study of earthquakes and the propagation of elastic waves through the Earth. ...

## Electromagnetic waves in an anisotropic material

Effective refractive indices in uniaxial materials
Propagation
direction
Ordinary ray Extraordinary ray
Polarization neff Polarization neff
z xy-plane no n/a n/a
xy-plane xy-plane no z ne
xz-plane y no xz-plane ne < n < no
other analogous to xz-plane

The behavior of a light ray that propagates through an anisotropic material is dependent on its polarization. For a given propagation direction, there are generally two perpendicular polarizations for which the medium behaves as if it had a single effective refractive index. In a uniaxial material, rays with these polarizations are called the extraordinary and the ordinary ray (e and o rays), corresponding to the extraordinary and ordinary refractive indices. In a biaxial material, there are three refractive indices α, β, and γ, yet only two rays, which are called the fast and the slow ray. The slow ray is the ray that has the highest effective refractive index.

For a uniaxial material with the z axis defined to be the optical axis, the effective refractive indices are as in the table on the right. For rays propagating in the xz plane, the effective refractive index of the e polarization varies continuously between no and ne, depending on the angle with the z axis. The effective refractive index can be constructed from the Index ellipsoid. The index ellipsoid is a diagram of an ellipsoid that depictes the orientation and relative magnitude of refractive indices in a crystal. ...

### Mathematical description

More generally, birefringence can be defined by considering a dielectric permittivity and a refractive index that are tensors. Consider a plane wave propagating in an anisotropic medium, with a relative permittivity tensor ε, where the refractive index n, is defined by $ncdot n = epsilon$. If the wave has an electric vector of the form: Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In the physics of wave propagation (especially electromagnetic waves), a plane wave (also spelled planewave) is a constant-frequency wave whose wavefronts (surfaces of constant amplitude and phase) are infinite parallel planes normal to the propagation direction. ... This article is about vectors that have a particular relation to the spatial coordinates. ...

$mathbf{E=E_0}exp i(mathbf{k cdot r}-omega t) ,$ (2)

where r is the position vector and t is time, then the wave vector k and the angular frequency ω must satisfy Maxwell's equations in the medium, leading to the equations: A wave vector is a vector that represents two properties of a wave: the magnitude of the vector represents wavenumber (inversely related to wavelength), and the vector points in the direction of wave propagation. ... For thermodynamic relations, see Maxwell relations. ...

$-nabla times nabla times mathbf{E}=frac{1}{c^2}mathbf{epsilon} cdot frac{part^2 mathbf{E} }{partial t^2}$ (3a)

$nabla cdot mathbf{epsilon} cdot mathbf{E} =0$ (3b)

where c is the speed of light in a vacuum. Substituting eqn. 2 in eqns. 3a-b leads to the conditions: The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ...

$|mathbf{k}|^2mathbf{E_0}-mathbf{(k cdot E_0) k}= frac{omega^2}{c^2} mathbf{epsilon} cdot mathbf{E_0}$ (4a)

$mathbf{k} cdot mathbf{epsilon} cdot mathbf{E_0} =0$ (4b)

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 Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

$mathbf{epsilon}=begin{bmatrix} n_x^2 & 0 & 0 0& n_y^2 & 0 0& 0& n_z^2 end{bmatrix} ,$ (4c)

Hence eqn 4a becomes

$(-k_y^2-k_z^2+frac{omega^2n_x^2}{c^2})E_x + k_xk_yE_y + k_xk_zE_z =0$ (5a)

$k_xk_yE_x + (-k_x^2-k_z^2+frac{omega^2n_y^2}{c^2})E_y + k_yk_zE_z =0$ (5b)

$k_xk_zE_x + k_yk_zE_y + (-k_x^2-k_y^2+frac{omega^2n_z^2}{c^2})E_z =0$ (5c)

where Ex, Ey, Ez, kx, ky and kz are the components of E0 and k. This is a set of linear equations in Ex, Ey, Ez, and they have a non-trivial solution if their determinant is zero: In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

$detbegin{bmatrix} (-k_y^2-k_z^2+frac{omega^2n_x^2}{c^2}) & k_xk_y & k_xk_z k_xk_y & (-k_x^2-k_z^2+frac{omega^2n_y^2}{c^2}) & k_yk_z k_xk_z & k_yk_z & (-k_x^2-k_y^2+frac{omega^2n_z^2}{c^2}) end{bmatrix} =0,$ (6)

Multiplying out eqn (6), and rearranging the terms, we obtain

$frac{omega^4}{c^4} - frac{omega^2}{c^2}left(frac{k_x^2+k_y^2}{n_z^2}+frac{k_x^2+k_z^2}{n_y^2}+frac{k_y^2+k_z^2}{n_x^2}right) + left(frac{k_x^2}{n_y^2n_z^2}+frac{k_y^2}{n_x^2n_z^2}+frac{k_z^2}{n_x^2n_y^2}right)(k_x^2+k_y^2+k_z^2)=0,$ (7)

In the case of a uniaxial material, where nx=ny=no and nz=ne say, eqn 7 can be factorised into

$left(frac{k_x^2}{n_o^2}+frac{k_y^2}{n_o^2}+frac{k_z^2}{n_o^2} -frac{omega^2}{c^2}right)left(frac{k_x^2}{n_e^2}+frac{k_y^2}{n_e^2}+frac{k_z^2}{n_o^2} -frac{omega^2}{c^2}right)=0,.$ (8)

Each of the factors in eqn 8 defines a surface in the space of vectors k — the surface of wave normals. The first factor defines a sphere and the second defines an ellipsoid. Therefore, for each direction of the wave normal, two wavevectors k are allowed. Values of k on the sphere correspond to the ordinary rays while values on the ellipsoid correspond to the extraordinary rays. For other uses, see Sphere (disambiguation). ... 3D rendering of an ellipsoid In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ...

For a biaxial material, eqn (7) cannot be factorized in the same way, and describes a more complicated pair of wave-normal surfaces.[3]

Birefringence is often measured for rays propagating along one of the optical axes (or measured in a two-dimensional material). In this case, n has two eigenvalues which can be labeled n1 and n2. n can be diagonalized by:

$mathbf{n} = mathbf{R(chi)} cdot begin{bmatrix} n_1 & 0 0 & n_2 end{bmatrix} cdot mathbf{R(chi)}^textrm{T}$ (9)

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 χ, where Δn = n1 − n2.

Wikimedia Commons has media related to:
Birefringence

Image File history File links Commons-logo. ... Crystal optics is the branch of optics that describes the behaviour of light in anisotropic media, that is, media (such as crystals) in which light behaves differently depending on which direction the light is propagating. ... Rev. ... Periodic poling is formation of layers with alternate orientation in a birefringent material. ... In optics, the term dichroic has two related but distinct meanings. ...

## References

1. ^ a b c Elert, Glenn. Refraction. The Physics Hypertextbook.
2. ^ The Use of Birefringence for Predicting the Stiffness of Injection Moulded Polycarbonate Discs
3. ^ Born M, and Wolf E, Principles of Optics, 7th Ed. 1999 (Cambridge University Press), §15.3.3

Results from FactBites:

 Birefringence (562 words) Birefringent materials are used widely in optics to produce polarizing prisms and retarder plates such as the quarter-wave plate. Its birefringence is extremely large, with indices of refraction for the o- and e-rays of 1.6584 and 1.4864 respectively. The property called birefringence has to do with anisotropy in the binding forces between the atoms forming a crystal, so it can be visualized as the atoms having stronger "springs" holding them together in some crystalline directions.
 Olympus Microscopy Resource Center: Light and Color - Optical Birefringence (4733 words) Birefringence is formally defined as the double refraction of light in a transparent, molecularly ordered material, which is manifested by the existence of orientation-dependent differences in refractive index. In contrast, birefringence refers to the physical origin of the separation, which is the existence of a variation in refractive index that is sensitive to direction in a geometrically ordered material. The maximum brightness for the birefringent material is observed when the long (optical) axis of the crystal is oriented at a 45 degree angle with respect to both the polarizer and analyzer, as illustrated in Figure 8(c).
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