Birefringence

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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:

<math>\Delta n=n_e-n_o</math> (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.

Contents 1 Electromagnetic waves in an anisotropic material 2 Examples of birefringent materials 3 Biaxial birefringence 4 Measuring birefringence 5 Birefringence in fiber optics 6 See also

Electromagnetic waves in an anisotropic material

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:

<math>\mathbf{E=E_0}\exp i(\mathbf{k \cdot r}-\omega t) \,,</math> (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:

<math>-\nabla \times \nabla \times \mathbf{E}=\frac{1}{c^2}\mathbf{\epsilon} \cdot \frac{\part^2 \mathbf{E}}{\partial t^2} </math> (3a)

<math> \nabla \cdot \mathbf{\epsilon} \cdot \mathbf{E} =0 </math> (3b)

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

<math>|\mathbf{k}|^2\mathbf{E_0}-\mathbf{(k.E_0) k}= \frac{\omega^2}{c^2} \mathbf{\epsilon} \cdot \mathbf{E_0} </math> (4a)

<math>\mathbf{k} \cdot \mathbf{\epsilon} \cdot \mathbf{E_0} =0 </math> (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

<math>\mathbf{\epsilon}=\begin{bmatrix} n_x^2 & 0 & 0 \\ 0& n_y^2 & 0 \\ 0& 0& n_z^2 \end{bmatrix} \,.</math>

Hence eqn 4a becomes

<math>(-k_y^2-k_z^2+\frac{\omega^2n_x^2}{c^2})E_x + k_xk_yE_y + k_xk_zE_z =0</math> (5a)

<math>k_xk_yE_x + (-k_x^2-k_z^2+\frac{\omega^2n_y^2}{c^2})E_y + k_yk_zE_z =0</math> (5b)

<math>k_xk_zE_x + k_yk_zE_y + (-k_x^2-k_y^2+\frac{\omega^2n_z^2}{c^2})E_z =0</math> (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:

<math>\det\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\,.</math> (6)

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

<math>\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\,. </math> (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

<math>\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\,.</math> (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 (see Template:Ref).

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:

<math>\mathbf{n} = \mathbf{R(\chi)} \cdot \begin{bmatrix} n_1 & 0 \\ 0 & n_2 \end{bmatrix} \cdot \mathbf{R(\chi)}^\textrm{T} </math> (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.

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.

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

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

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 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 [2].

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

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.

Birefringence in fiber optics

In modern fiber optic cables, the coupled nonlinear Schrödinger equation models birefringence. The effect of birefringence in fiber optic cables is to cause differently polarized light to travel at different velocities down the cable. Using the Inverse Scattering Transform on the coupled nonlinear Schrödinger equation, as in the non-birefringent case (modeled by the plain nonlinear Schrödinger equation), produces a system of linear ordinary differential equations, known as the Manakov system.

See also

Template:Commons

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

1. Template:Note Born M, and Wolf E, Principles of Optics, 7th Ed. 1999 (Cambridge University Press), §15.3.3de:Doppelbrechung

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