User:Ddcampayo/Birefrigence

Theory

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 $n\cdot n=\epsilon$. If the wave has an electric vector of the form:

$\mathbf {E=E_{0}} \exp \left[i(\mathbf {k\cdot r} -\omega t)\right]\,$ (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:

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

For the matrix product $(\epsilon \cdot \mathbf {E} )$ often a separate name is used, the dielectric displacement vector $\mathbf {D}$. 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

$\mathbf {\epsilon } ={\begin{bmatrix}n_{x}^{2}&0&0\\0&n_{y}^{2}&0\\0&0&n_{z}^{2}\end{bmatrix}}\,$ (4c)

Hence eqn 4a becomes

$(-k_{y}^{2}-k_{z}^{2}+{\frac {\omega ^{2}n_{x}^{2}}{c^{2}}})E_{x}+k_{x}k_{y}E_{y}+k_{x}k_{z}E_{z}=0$ (5a)

$k_{x}k_{y}E_{x}+(-k_{x}^{2}-k_{z}^{2}+{\frac {\omega ^{2}n_{y}^{2}}{c^{2}}})E_{y}+k_{y}k_{z}E_{z}=0$ (5b)

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

$\det {\begin{bmatrix}(-k_{y}^{2}-k_{z}^{2}+{\frac {\omega ^{2}n_{x}^{2}}{c^{2}}})&k_{x}k_{y}&k_{x}k_{z}\\k_{x}k_{y}&(-k_{x}^{2}-k_{z}^{2}+{\frac {\omega ^{2}n_{y}^{2}}{c^{2}}})&k_{y}k_{z}\\k_{x}k_{z}&k_{y}k_{z}&(-k_{x}^{2}-k_{y}^{2}+{\frac {\omega ^{2}n_{z}^{2}}{c^{2}}})\end{bmatrix}}=0\,$ (6)

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

${\frac {\omega ^{4}}{c^{4}}}-{\frac {\omega ^{2}}{c^{2}}}\left({\frac {k_{x}^{2}+k_{y}^{2}}{n_{z}^{2}}}+{\frac {k_{x}^{2}+k_{z}^{2}}{n_{y}^{2}}}+{\frac {k_{y}^{2}+k_{z}^{2}}{n_{x}^{2}}}\right)+\left({\frac {k_{x}^{2}}{n_{y}^{2}n_{z}^{2}}}+{\frac {k_{y}^{2}}{n_{x}^{2}n_{z}^{2}}}+{\frac {k_{z}^{2}}{n_{x}^{2}n_{y}^{2}}}\right)(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})=0\,$ (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 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 that can be labeled n₁ and n₂. n can be diagonalized 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.

A crystal of rutile exhibiting birefringence for rotating polarizer