**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 (assuming a relative magnetic permeability ). If the wave has an electric vector of the form:

*(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:

*(3a)*

*(3b)*

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

*(4a)*

*(4b)*

For the matrix product often a separate name is used, the *dielectric displacement vector* . So essentially birefringence concerns the general theory of linear relationships between these two vectors in anisotropic media.

To find the allowed values of **k**, **E**_{0} 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

*(4c)*

Hence eqn 4a becomes

*(5a)*

*(5b)*

*(5c)*

where *E*_{x}, *E*_{y}, *E*_{z}, *k*_{x}, *k*_{y} and *k*_{z} are the components of **E**_{0} 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:

*(6)*

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

*(7)*

In the case of a uniaxial material, where *n*_{x}=*n*_{y}=*n _{o}* and

*n*=

_{z}*n*say, eqn 7 can be factorised into

_{e} *(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 *n*_{1} and *n*_{2}. **n** can be diagonalized by:

*(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 Δ*n*, and *extinction angle* χ, where Δ*n* = *n*_{1} − *n*_{2}.

Read more about this topic: Birefringence

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### Famous quotes containing the word theory:

“Won’t this whole instinct matter bear revision?

Won’t almost any *theory* bear revision?

To err is human, not to, animal.”

—Robert Frost (1874–1963)

“every subjective phenomenon is essentially connected with a single point of view, and it seems inevitable that an objective, physical *theory* will abandon that point of view.”

—Thomas Nagel (b. 1938)

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