Birefringence, or double refraction, is the decomposition of a ray of light into two rays when it passes through certain anisotropic materials, such as crystals of calcite or boron nitride. The effect was first described by the Danish scientist Rasmus Bartholin in 1669, who saw it in calcite. 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 of the material, and light with linear polarizations parallel and perpendicular to it has unequal indices of refraction, denoted ne and no 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, correspondingly 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.
As stated above, birefringence can also arise in magnetic materials, but substantial variations in magnetic permeability of materials are rare at optical frequencies.
Birefringence can be observed in amyloid plaque deposits such as are found in the brains of Alzheimer's patients. Modified proteins such as immunoglobulin light chains abnormally accumulate between cells, forming fibrils. Multiple folds of these fibers line up and take on a beta-pleated sheet conformation. Congo red dye intercalates between the folds and, when observed under polarized light, causes birefringence.
Cotton (Gossypium hirsutum) fiber is birefringent because of high levels of cellulosic material in the fiber's secondary cell wall.
Slight imperfections in optical fiber can cause birefringence, which can cause distortion in fiber-optic communication; see polarization mode dispersion. The imperfections can be geometrically based, or a result of photoelastic effects from loading on the optical fiber.
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.
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. In ophthalmology, scanning laser polarimetry utilises the birefringence of the retinal nerve fibre layer to indirectly quantify its thickness, which is of use in the assessment and monitoring of glaucoma. Birefringence characteristics in sperm heads allow for the selection of spermatozoa for intracytoplasmic sperm injection. Likewise, zona imaging uses birefringence on oocytes to select the ones with highest chances of successful pregnancy. Birefringence of particles biopsied from pulmonary nodules indicates silicosis.
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.
where r is the position vector and t is time, then the wave vector k and the angular frequency ω must satisfy Maxwell's equations in the medium, leading to the equations:
where c is the speed of light in a vacuum. Substituting eqn. 2 in eqns. 3a-b leads to the conditions:
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 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(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.
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