Birefringence

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$A calcite crystal laid upon a paper with some letters showing the double refraction$

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

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:

$\Delta n=n_e-n_o\,$

(1)

where n_o and n_e 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.

1 Electromagnetic waves in an anisotropic material
2 Creating birefringence
3 Examples of birefringent materials
4 Biaxial birefringence
5 Measuring birefringence
6 Applications of birefringence
7 Elastic birefringence
8 See also
9 References
10 External links

[edit] 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:

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

(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:

$-\nabla \times \nabla \times \mathbf{E}=\frac{1}{c^2}\mathbf{\epsilon} \cdot \frac{\part^2 \mathbf{E}}{\partial t^2}$

(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:

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

(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

$\mathbf{\epsilon}=\begin{bmatrix} n_x^2 & 0 & 0 \\ 0& n_y^2 & 0 \\ 0& 0& n_z^2 \end{bmatrix} \,$

(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 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 non-trivial solution if their determinant is zero:

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

(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 n_x=n_y=n_o and n_z=n_e 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 a biaxial material, eqn (7) cannot be factorised in the same way, and describes a more complicated pair of wave-normal surfaces.^[1]

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 diagonalised 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.

[edit] 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).

[edit] 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), from [1].

Material	n_o	n_e	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

[edit] 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 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	n_α	n_β	n_γ
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

[edit] 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.

[edit] 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 important role in second harmonic generation and many other nonlinear processes. It is also utilized in medical diagnostics. Needle biopsy of suspected gouty joints will be negatively birefringent if urate crystals are present.

[edit] 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.