Birefringence

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Displacement of light rays with perpendicular polarization through a birefringent material..

Calcite crystal put on a square paper with blue lines showing double refractive.

Double refracted image as seen through a calcite crystal, seen through a rotating polarizer filter that illustrates the opposite polarization states of the two images.

Birrefringence in a calcite crystal.

Birrefringence or double refraction is an optical property of certain bodies, especially Iceland spar, which consists of splitting an incident light ray into two linearly polarized rays perpendicular to each other as if the material had two different refractive indices: the first of the two directions follows the normal laws of refraction and is called ordinary ray; the other has a variable speed and refractive index and is called extraordinary ray. Both waves are polarized perpendicular to each other. This phenomenon can only occur if the structure of the material is anisotropic. If the material has only one axis of anisotropy, (ie it is uniaxial), the birefringence can be described by assigning two different refractive indices to the material for the different polarizations.

This effect was first described by the Danish scientist Rasmus Bartholin in 1669, who observed it in calcite, a crystal that has a strong birefringence. However, it was not until the 19th century that the phenomenon was correctly described in terms of polarization, with the understanding of light as a wave, which Augustin-Jean Fresnel did.

Birrefringence is quantified by the relationship:

$Delta n=n_e-n_o ,$

where n_o and n_e are the refractive indices for perpendicular polarizations (ray ordinary) and parallel to the anisotropy axis (ray extraordinary), respectively.

Birrefringence can also appear in magnetic materials, but substantial variations in the magnetic permeability of materials are rare at optical frequencies.

Cellophane is a common birefringent material.

This phenomenon can be seen in potato starch, that is, it is birefringent.

In biological materials, it indicates an arrangement of the molecules, for example oriented towards each other, as in a crystal.

Birrefringence of flow or current is the one seen only when the substance is found in solution of large molecules, such as nucleoproteins.
The crystalline birrefringence or intrinsic is the one that occurs in systems in which the links between molecules or ions present a symmetrical regular layout; it is independent of the media refraction index.
Birrefringence in form is that originated by the regular orientation of asymmetric submicroscopic particles in a substance or object, differing from the refraction index of the surrounding environment; it is the most common form found in living beings.
The birefringence of tension is observed occasionally in isotropic structures when they are subjected to tension or pressure; it occurs in muscle and embryonic tissues, in translucent materials and explains the photoelastic effect.

Theory

Birrefringence in polystyrene

More generally, birrefringence can be defined by a dielectric permit and a tensoral refractive index. Consider a flat wave that spreads in an anisotropic medium, with a tensor of permitivity εwith a tense refractive index n defined by ${displaystyle ncdot n=epsilon }$ . If the wave has a vectorial electric field of the form:

${displaystyle mathbf {E=E_{0}} exp left[i(mathbf {kcdot r} -omega t)right],}$

where r is the position vector and t is time, the wave vector k and the angular frequency must satisfy the equations of Maxwell in the middle, leading to the equation:

${displaystyle -nabla times nabla times mathbf {E} ={frac {1}{c^{2}}}(mathbf {epsilon } cdot {frac {partial ^{2}mathbf {E} }{partial t^{2}}})}$

where c is the speed of light in a vacuum. Substituting the electric field into this equation we get:

${displaystyle |mathbf {k} |^{2}mathbf {E_{0}} -mathbf {(kcdot E_{0})k} ={frac {omega ^{2}}{c^{2}}}(mathbf {epsilon } cdot mathbf {E_{0}})}$

It is common to use the name vector of dielectric displacement for the matrix product ${displaystyle mathbf {D} =(epsilon cdot mathbf {E})}$ . Thus birrefringence deals with the general linear relationships between these two vectors in anisotropic media.

To find the allowed values of k, we can isolate E₀ from the last equation. One way is to write the latter in Cartesian coordinates, with the Cartesian axes in the direction of the eigenvectors of ε, so:

${displaystyle mathbf {epsilon } ={begin{bmatrix}n_{x}^{2}&0&0\0&n_{y}^{2}&0\0&0&n_{z}^{2}end{bmatrix}},}$

In this way, the equation becomes:

${displaystyle (-k_{y}^{2}-k_{z}^{2}+{frac {omega ^{2}n_{x}^{2}}{c^{2}}})E_{x}+k_{x}k_{y}E_{y}+k_{x}k_{z}E_{z}=0}$

${displaystyle k_{x}k_{y}E_{x}+(-k_{x}^{2}-k_{z}^{2}+{frac {omega ^{2}n_{y}^{2}}{c^{2}}})E_{y}+k_{y}k_{z}E_{z}=0}$

${displaystyle k_{x}k_{z}E_{x}+k_{y}k_{z}E_{y}+(-k_{x}^{2}-k_{y}^{2}+{frac {omega ^{2}n_{z}^{2}}{c^{2}}})E_{z}=0}$

where E_x, E_y, E_z, k_x, k_y, and k_z are the Cartesian components of E₀ and k respectively. This is a homogeneous system of linear equations in E_x, E_y and E_z than just can have a non-trivial solution if the associated determinant is zero:

${displaystyle det {begin{bmatrix}(-k_{y}^{2}-k_{z}^{2}+{frac {omega ^{2}n_{x}^{2}}{c^{2}}})&k_{x}k_{y}&k_{x}k_{z}\k_{x}k_{y}&(-k_{x}^{2}-k_{z}^{2}+{frac {omega ^{2}n_{y}^{2}}{c^{2}}})&k_{y}k_{z}\k_{x}k_{z}&k_{y}k_{z}&(-k_{x}^{2}-k_{y}^{2}+{frac {omega ^{2}n_{z}^{2}}{c^{2}}})end{bmatrix}}=0,}$

Expanding the determinant and regrouping it is possible to obtain:

${displaystyle {frac {omega ^{4}}{c^{4}}}-{frac {omega ^{2}}{c^{2}}}left({frac {k_{x}^{2}+k_{y}^{2}}{n_{z}^{2}}}+{frac {k_{x}^{2}+k_{z}^{2}}{n_{y}^{2}}}+{frac {k_{y}^{2}+k_{z}^{2}}{n_{x}^{2}}}right)+left({frac {k_{x}^{2}}{n_{y}^{2}n_{z}^{2}}}+{frac {k_{y}^{2}}{n_{x}^{2}n_{z}^{2}}}+{frac {k_{z}^{2}}{n_{x}^{2}n_{y}^{2}}}right)(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})=0,}$

In a uniaxial material, two of the refractive indices coincide; for example: n_x=n_y=n_o y n_z=n_e. In this case the previous equation is simplified:

${displaystyle left({frac {k_{x}^{2}}{n_{o}^{2}}}+{frac {k_{y}^{2}}{n_{o}^{2}}}+{frac {k_{z}^{2}}{n_{o}^{2}}}-{frac {omega ^{2}}{c^{2}}}right)left({frac {k_{x}^{2}}{n_{e}^{2}}}+{frac {k_{y}^{2}}{n_{e}^{2}}}+{frac {k_{z}^{2}}{n_{o}^{2}}}-{frac {omega ^{2}}{c^{2}}}right)=0,.}$

Each of the two factors in this equation defines a surface in the space of vectors k — the wave vector surface. The first defines a sphere and the second an ellipsoid of revolution. Therefore, for each direction of the wave vector there are two possible wave vectors. The values of k on the sphere correspond to ordinary rays, while the values of the ellipsoid correspond to extraordinary rays.

For a biaxial material the equation cannot be simplified in this way, and the two wave vector surfaces are more complicated.

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