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, depending on the polarization of the light. This effect can occur only if the structure of the material is anisotropic. If the material has a single axis of anisotropy, (i.e. it is uniaxial,) birefringence can be formalised by assigning two different refractive indices to the material for different polarizations. The birefringence magnitude is then defined by:

$\backslash Delta\; n=n\_e-n\_o$ (1)

where n_o and n_e are the refractive indices for polarisations perpendicular and parallel to the axis of anisotropy respectively. (These rays are labelled 'ordinary' and 'extraordinary' respectively.)

Birefringence can also arise in magnetic, not dielectric, materials, but substantial variations in magnetic permeability of materials are rare at optical frequencies.

Electromagnetic waves in an anisotropic material

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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 n.n = ε. If the wave has an electric vector of the form:

$\backslash mathbf\{E=E\_0\}\backslash exp\; i(\backslash mathbf\{k\; \backslash cdot\; r\}-\backslash omega\; t)\; \backslash ,,$ (2)

where r is the position vector and t is time, then the wavevector k and the angular frequency ω must satisfy Maxwell's equations in the medium, leading to the equations:

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

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

where c is the speed of light in a vacuum. Substituting eqn. 2 in eqns. 3a-b leads to the conditions:

$|\backslash mathbf\{k\}|^2\backslash mathbf\{E\_0\}-\backslash mathbf\{(k\; \backslash cdot\; E\_0)\; k\}=\; \backslash frac\{\backslash omega^2\}\{c^2\}\; \backslash mathbf\{\backslash epsilon\}\; \backslash cdot\; \backslash mathbf\{E\_0\}$ (4a)

$\backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{\backslash epsilon\}\; \backslash cdot\; \backslash mathbf\{E\_0\}\; =0$ (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

Hence eqn 4a becomes

$(-k\_y^2-k\_z^2+\backslash frac\{\backslash omega^2n\_x^2\}\{c^2\})E\_x\; +\; k\_xk\_yE\_y\; +\; k\_xk\_zE\_z\; =0$ (5a)

$k\_xk\_yE\_x\; +\; (-k\_x^2-k\_z^2+\backslash frac\{\backslash omega^2n\_y^2\}\{c^2\})E\_y\; +\; k\_yk\_zE\_z\; =0$ (5b)

$k\_xk\_zE\_x\; +\; k\_yk\_zE\_y\; +\; (-k\_x^2-k\_y^2+\backslash frac\{\backslash omega^2n\_z^2\}\{c^2\})E\_z\; =0$ (5c)

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 solution if their determinant is zero:

$\backslash det\backslash begin\{bmatrix\}$

(-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

$\backslash frac\{\backslash omega^4\}\{c^4\}\; +\; \backslash frac\{\backslash omega^2\}\{c^2\}\backslash left(\backslash frac\{k\_x^2+k\_y^2\}\{n\_z^2\}+\backslash frac\{k\_x^2+k\_z^2\}\{n\_y^2\}+\backslash frac\{k\_y^2+k\_z^2\}\{n\_x^2\}\backslash right)\; +\; \backslash left(\backslash frac\{k\_x^2\}\{n\_y^2n\_z^2\}+\backslash frac\{k\_y^2\}\{n\_x^2n\_z^2\}+\backslash frac\{k\_z^2\}\{n\_x^2n\_y^2\}\backslash right)(k\_x^2+k\_y^2+k\_z^2)=0\backslash ,.$ (7)

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

$\backslash left(\backslash frac\{k\_x^2\}\{n\_o^2\}+\backslash frac\{k\_y^2\}\{n\_o^2\}+\backslash frac\{k\_z^2\}\{n\_o^2\}\; -\backslash frac\{\backslash omega^2\}\{c^2\}\backslash right)\backslash left(\backslash frac\{k\_x^2\}\{n\_e^2\}+\backslash frac\{k\_y^2\}\{n\_e^2\}+\backslash frac\{k\_z^2\}\{n\_o^2\}\; -\backslash frac\{\backslash omega^2\}\{c^2\}\backslash right)=0\backslash ,.$ (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 a biaxial material, eqn (7) cannot be factorised in the same way, and describes a more complicated pair of wave-normal surfaces.Born M, and Wolf E, Principles of Optics, 7th Ed. 1999 (Cambridge University Press), §15.3.3

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 labelled n₁ and n₂. n can be diagonalised by:

(8)

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.

Stress birefringence

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For isotropic materials, when they are squeezed or bent to become anisotropic, birefringence results.

Example: http://www.oberlin.edu/physics/catalog/demonstrations/optics/birefringence.html

Examples of birefringent materials

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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.

The refractive indices of several (uniaxial) birefringent materials are listed below (at wavelength ~ 590 nm), from ^*.

Material	no	ne	Δn
beryl	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	1.690	1.654	-0.036
sapphire Al2O3	1.768	1.760	-0.008
sodium nitrate NaNO3	1.587	1.336	-0.251
tourmaline	1.669	1.638	-0.031
zircon, high ZrSiO4	1.960	2.015	+0.055
zircon, low ZrSiO4	1.920	1.967	+0.047

Biaxial birefringence

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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 labelled n_α, n_β and n_γ.

The refractive indices of some trirefringent materials are listed below (at wavelength ~ 590 nm), from ^*.

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)2SiO	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

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Birefringence and related optical effects (such as optical rotation and linear or circular dichroism) can be measured by measuring the changes in the polarisation of light passing through the material. These measurements are known as polarimetry.

A common feature of optical microscopes is a pair of crossed polarising filters. Between the crossed polarisers, a birefringent sample will appear bright against a dark (isotropic) background.

Applications of birefringence

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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 important role in second harmonic generation and many other nonlinear processes.

Elastic birefringence

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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.

See also

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• Crystal optics
• John Kerr

References

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External links

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• "Refraction" in The Physics Hypertextbook by Glenn Elert, lists several birefringent/trirefringent materials.

Polarization

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