Birefringence
Birefringence, or double refraction, is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals or boron nitride, depending on the polarization of the light.  This effect can occur only if the structure of the material is anisotropic (directionally dependent). If the material has a single axis of anisotropy or optical axis, (i.e. it is uniaxial) birefringence can be formalized by assigning two different refractive indices to the material for different polarizations.  The birefringence magnitude is then defined by
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.

# Creating birefringence
While birefringence is often found naturally (especially in crystals), there are several ways to create it in optically isotropic materials.
- Birefringence results when isotropic materials are deformed such that the isotropy is lost in one direction (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).

# Examples of birefringent materials
Many plastics are birefringent, because their molecules are 'frozen' in a stretched conformation when the plastic is moulded or extruded.
For example, cellophane is a cheap birefringent material. Birefringent materials are used in many devices which manipulate the polarization of light, such as wave plates, polarizing prisms, and Lyot filters.
There are many birefringent crystals: birefringence was first described in calcite crystals by the Danish scientist Rasmus Bartholin in 1669.
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.
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.
Silicon carbide, also known as Moissanite, is strongly birefringent.
The refractive indices of several (uniaxial) birefringent materials are listed below (at wavelength ~ 590 nm)

# Biaxial birefringence
Biaxial birefringence, also known as trirefringence, describes an anisotropic material that has more than one axis of anisotropy. For such a material, the refractive index tensor n, will in general have three distinct eigenvalues that can be labeled n, n and n.

# 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.
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.

# Applications of birefringence
Birefringence is widely used in optical devices, such as liquid crystal displays, light modulators, color filters, wave plates, optical axis gratings, etc. It also plays an important role in second harmonic generation and many other nonlinear processes.  It is also utilized in medical diagnostics: needle aspiration of fluid from a gouty joint will reveal negatively birefringent urate crystals.  Some artists also work with birefringence, the most notable being contemporary American artist Austine Wood Comarow who coined the term "Polage" to describe her polarized light collages.  The artist works by cutting hundreds of small pieces of cellophane and other birefringent films and laminating them between plane polarizing filters.  Comarow's Polage art is exhibited at the Museum of Science, Boston, the New Mexico Museum of Natural History and Science in Albuquerque, NM, and la Cite des Sciences et de l'Industrie (the City of Science and Industry) in Paris.

# 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.

# Electromagnetic waves in an anisotropic material
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 , , and , 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 various continuously between n_o and n_e, depending on the angle with the z axis. The effective refractive index can be constructed from the Index ellipsoid.

## 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 , where the refractive index n, is defined by n\cdot n = \epsilon. If the wave has an electric vector of the form:
where r is the position vector and t is time, then the wave vector k and the angular frequency  must satisfy Maxwell's equations in the medium, leading to the equations:
where c is the speed of light in a vacuum. Substituting eqn. 2 in eqns. 3a-b leads to the conditions:
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 , so that
Hence eqn 4a becomes
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:
Multiplying out eqn (6), and rearranging the terms, we obtain
In the case of a uniaxial material, where nx=ny=no and nz=ne say, eqn 7 can be factorised into
Each of the factors in eqn 8 defines a surface in the space of vectors k  the surface of wave normals. The first factor defines a sphere and the second defines an ellipsoid. Therefore, for each direction of the wave normal, two wavevectors k are allowed. Values of k on the sphere correspond to the ordinary rays while values on the ellipsoid correspond to the extraordinary rays.
For a biaxial material, eqn (7) cannot be factorized in the same way, and describes a more complicated pair of wave-normal surfaces.
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:
where R() is the rotation matrix through an angle . Rather than specifying the complete tensor n, we may now simply specify the magnitude of the birefringence n, and extinction angle , where n = n1  n2.