Editing Birefringence (section)

==Theory==
[[File:Birifrangence k-surface.gif|400px|thumb|right|Surface of the allowed '''k''' vectors for a fixed frequency for a biaxial crystal (see {{EquationNote|7|eq. 7}}).]]
In an isotropic medium (including free space) the so-called [[electric displacement]] ({{math|'''D'''}}) is just proportional to the electric field ({{math|'''E'''}}) according to {{math|'''D''' {{=}} ''ɛ'''''E'''}} where the material's [[permittivity]] {{math|''ε''}} is just a [[Scalar (physics)|scalar]] (and equal to {{math|''n''<sup>2</sup>[[permittivity of free space|''ε''<sub>0</sub>]]}} where {{math|''n''}} is the [[index of refraction]]). In an anisotropic material exhibiting birefringence, the relationship between {{math|'''D'''}} and {{math|'''E'''}} must now be described using a [[tensor]] equation:

{{NumBlk|:|<math>\mathbf D = \boldsymbol\varepsilon \mathbf E </math>|{{EquationRef|1}}}}

where {{math|'''ε'''}} is now a 3&nbsp;×&nbsp;3 permittivity tensor. We assume linearity and no [[magnetic permeability]] in the medium: {{math|''μ'' {{=}} [[permeability of free space|''μ''<sub>0</sub>]]}}. The electric field of a plane wave of angular frequency {{math|''ω''}} can be written in the general form:

{{NumBlk|:|<math>\mathbf{E}=\mathbf{E}_0 e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)}</math>|{{EquationRef|2}}}}

where {{math|'''r'''}} is the position vector, {{math|''t''}} is time, and {{math|'''E'''<sub>0</sub>}} is a vector describing the electric field at {{math|'''r''' {{=}} 0}}, {{math|''t'' {{=}} 0}}. Then we shall find the possible [[wave vector]]s {{math|'''k'''}}. By combining [[Maxwell's equations]] for {{math|∇ × '''E'''}} and {{math|∇ × '''H'''}}, we can eliminate {{math|'''H''' {{=}} {{sfrac|1|''μ''<sub>0</sub>}}'''B'''}} to obtain:

{{NumBlk|:|<math>-\nabla \times \nabla \times \mathbf{E}= \mu_0\frac{\partial^2 }{\partial t^2} \mathbf{D}</math> |{{EquationRef|3a}}}}

With no free charges, Maxwell's equation for the divergence of {{math|'''D'''}} vanishes:
{{NumBlk|:|<math>\nabla \cdot \mathbf{D}=0 </math>|{{EquationRef|3b}}}}

We can apply the vector identity {{math|[[Vector calculus identities#Curl of curl|∇ × (∇ × '''A''') {{=}} ∇(∇ ⋅ '''A''') − ∇<sup>2</sup>'''A''']]}} to the left hand side of {{EquationNote|3a|eq. 3a}}, and use the spatial dependence in which each differentiation in {{math|''x''}} (for instance) results in multiplication by {{math|''ik<sub>x</sub>''}} to find:
{{NumBlk|:|<math> - \nabla \times \nabla \times \mathbf{E} =(\mathbf{k} \cdot \mathbf{E}) \mathbf{k} - (\mathbf{k} \cdot \mathbf{k}) \mathbf{E}</math>|{{EquationRef|3c}}}}

The right hand side of {{EquationNote|3a|eq. 3a}} can be expressed in terms of {{math|'''E'''}} through application of the permittivity tensor {{math|'''ε'''}} and noting that differentiation in time results in multiplication by {{math|−''iω''}}, {{EquationNote|3a|eq. 3a}} then becomes:
{{NumBlk|:|<math>(-\mathbf{k}\cdot\mathbf{k})\mathbf{E} + (\mathbf{k} \cdot \mathbf{E}) \mathbf{k}= -\mu_0 \omega^2 (\boldsymbol{\varepsilon} \mathbf{E})</math>|{{EquationRef|4a}}}}

Applying the differentiation rule to {{EquationNote|3b|eq. 3b}} we find:
{{NumBlk|:|<math>\mathbf{k} \cdot \mathbf{D} =0</math>|{{EquationRef|4b}}}}

{{EquationNote|4b|Eq. 4b}} indicates that {{math|'''D'''}} is orthogonal to the direction of the wavevector {{math|'''k'''}}, even though that is no longer generally true for {{math|'''E'''}} as would be the case in an isotropic medium. {{EquationNote|4b|Eq. 4b}} will not be needed for the further steps in the following derivation.

Finding the allowed values of {{math|'''k'''}} for a given {{math|''ω''}} is easiest done by using [[Cartesian coordinates]] with the {{math|''x''}}, {{math|''y''}} and {{math|''z''}} axes chosen in the directions of the symmetry axes of the crystal (or simply choosing {{math|''z''}} in the direction of the optic axis of a uniaxial crystal), resulting in a diagonal matrix for the permittivity tensor {{math|'''ε'''}}:
{{NumBlk|:|<math>\mathbf{\varepsilon}= \varepsilon_0 \begin{bmatrix} n_x^2 & 0 & 0 \\ 0& n_y^2 & 0 \\ 0& 0& n_z^2 \end{bmatrix} </math>|{{EquationRef|4c}}}}

where the diagonal values are squares of the refractive indices for polarizations along the three principal axes {{math|''x''}}, {{math|''y''}} and {{math|''z''}}. With {{math|'''ε'''}} in this form, and substituting in the speed of light {{math|''c''}} using {{math|''c''<sup>2</sup> {{=}} {{sfrac|1|''μ''<sub>0</sub>''ε''<sub>0</sub>}}}}, the {{math|''x''}} component of the vector equation {{EquationNote|4a|eq. 4a}} becomes
{{NumBlk|:|<math>\left(-k_x^2-k_y^2-k_z^2 \right)E_x + k_x^2 E_x + k_xk_yE_y + k_xk_zE_z = -\frac{\omega^2n_x^2}{c^2} E_x</math>|{{EquationRef|5a}}}}

where {{math|''E<sub>x</sub>''}}, {{math|''E<sub>y</sub>''}}, {{math|''E<sub>z</sub>''}} are the components of {{math|'''E'''}} (at any given position in space and time) and {{math|''k<sub>x</sub>''}}, {{math|''k<sub>y</sub>''}}, {{math|''k<sub>z</sub>''}} are the components of {{math|'''k'''}}. Rearranging, we can write (and similarly for the {{math|''y''}} and {{math|''z''}} components of {{EquationNote|4a|eq. 4a}})
{{NumBlk|:|<math>\left(-k_y^2-k_z^2+\frac{\omega^2n_x^2}{c^2}\right)E_x + k_xk_yE_y + k_xk_zE_z =0</math>|{{EquationRef|5b}}}}
{{NumBlk|:|<math>k_xk_yE_x + \left(-k_x^2-k_z^2+\frac{\omega^2n_y^2}{c^2}\right)E_y + k_yk_zE_z =0</math>|{{EquationRef|5c}}}}
{{NumBlk|:|<math>k_xk_zE_x + k_yk_zE_y + \left(-k_x^2-k_y^2+\frac{\omega^2n_z^2}{c^2}\right)E_z =0</math>|{{EquationRef|5d}}}}

This is a set of linear equations in {{math|''E<sub>x</sub>''}}, {{math|''E<sub>y</sub>''}}, {{math|''E<sub>z</sub>''}}, so it can have a nontrivial solution (that is, one other than {{math|'''E''' {{=}} 0}}) as long as the following [[determinant]] is zero:
{{NumBlk|:|<math>\begin{vmatrix}
\left(-k_y^2-k_z^2+\frac{\omega^2n_x^2}{c^2}\right) & k_xk_y & k_xk_z \\
k_xk_y & \left(-k_x^2-k_z^2+\frac{\omega^2n_y^2}{c^2}\right) & k_yk_z \\
k_xk_z & k_yk_z & \left(-k_x^2-k_y^2+\frac{\omega^2n_z^2}{c^2}\right) \end{vmatrix} =0</math>|{{EquationRef|6}}}}

Evaluating the determinant of {{EquationNote|6|eq. 6}}, and rearranging the terms according to the powers of <math>\frac{\omega^2}{c^2}</math>, the constant terms cancel. After eliminating the common factor <math>\frac{\omega^2}{c^2}</math> from the remaining terms, we obtain

{{NumBlk|:|<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)\left(k_x^2+k_y^2+k_z^2\right)=0 </math>|{{EquationRef|7}}}}
 
In the case of a uniaxial material, choosing the optic axis to be in the {{math|''z''}} direction so that {{math|''n<sub>x</sub>'' {{=}} ''n<sub>y</sub>'' {{=}} ''n''<sub>o</sub>}} and {{math|''n<sub>z</sub>'' {{=}} ''n''<sub>e</sub>}}, this expression can be factored into

{{NumBlk|:|<math>\left(\frac{k_x^2}{n_\mathrm{o}^2}+\frac{k_y^2}{n_\mathrm{o}^2}+\frac{k_z^2}{n_\mathrm{o}^2} -\frac{\omega^2}{c^2}\right)\left(\frac{k_x^2}{n_\mathrm{e}^2}+\frac{k_y^2}{n_\mathrm{e}^2}+\frac{k_z^2}{n_\mathrm{o}^2} -\frac{\omega^2}{c^2}\right)=0</math>|{{EquationRef|8}}}}

Setting either of the factors in {{EquationNote|8|eq. 8}} to zero will define an [[ellipsoid]]al surface<ref group=note>Although related, note that this is not the same as the [[index ellipsoid]].</ref> in the space of wavevectors {{math|'''k'''}} that are allowed for a given {{math|''ω''}}. The first factor being zero defines a sphere; this is the solution for so-called ordinary rays, in which the effective refractive index is exactly {{math|''n''<sub>o</sub>}} regardless of the direction of {{math|'''k'''}}. The second defines a [[spheroid]] symmetric about the {{math|''z''}} axis. This solution corresponds to the so-called extraordinary rays in which the effective refractive index is in between {{math|''n''<sub>o</sub>}} and {{math|''n''<sub>e</sub>}}, depending on the direction of {{math|'''k'''}}. Therefore, for any arbitrary direction of propagation (other than in the direction of the optic axis), two distinct wavevectors {{math|'''k'''}} are allowed corresponding to the polarizations of the ordinary and extraordinary rays.

For a biaxial material a similar but more complicated condition on the two waves can be described;<ref>Born & Wolf, 2002, §15.3.3</ref> the locus of allowed {{math|'''k'''}} vectors (the ''wavevector surface'') is a 4th-degree two-sheeted surface, so that in a given direction there are generally two permitted {{math|'''k'''}} vectors (and their opposites).<ref name=berry-jeffrey-2007>M.V. Berry and M.R.&nbsp;Jeffrey, "Conical diffraction: Hamilton's diabolical point at the heart of crystal optics", in E.&nbsp;Wolf&nbsp;(ed.), ''Progress in Optics'', vol.{{nnbsp}}50, Amsterdam: Elsevier, 2007, {{nowrap|pp.{{tsp}}13–50}}, {{doi|10.1016/S0079-6638(07)50002-8}}, at {{nowrap|pp.{{tsp}}20–21}}.</ref> By inspection one can see that {{EquationNote|6|eq. 6}} is generally satisfied for two positive values of {{math|''ω''}}. Or, for a specified optical frequency {{math|''ω''}} and direction normal to the wavefronts {{math|{{sfrac|'''k'''|{{abs|'''k'''}}}}}}, it is satisfied for two [[wavenumbers]] (or propagation constants) {{math|{{abs|'''k'''}}}} (and thus effective refractive indices) corresponding to the propagation of two linear polarizations in that direction.

When those two propagation constants are equal then the effective refractive index is independent of polarization, and there is consequently no birefringence encountered by a wave traveling in that particular direction. For a uniaxial crystal, this is the optic axis, the ±''z'' direction according to the above construction. But when all three refractive indices (or permittivities), {{math|''n<sub>x</sub>''}}, {{math|''n<sub>y</sub>''}} and {{math|''n<sub>z</sub>''}} are distinct, it can be shown that there are exactly two such directions, where the two sheets of the wave-vector surface touch;{{r|berry-jeffrey-2007}} these directions are not at all obvious and do not lie along any of the three principal axes ({{math|''x''}}, {{math|''y''}}, {{math|''z''}} according to the above convention). Historically that accounts for the use of the term "biaxial" for such crystals, as the existence of exactly two such special directions (considered "axes") was discovered well before polarization and birefringence were understood physically. These two special directions are generally not of particular interest; biaxial crystals are rather specified by their three refractive indices corresponding to the three axes of symmetry.

A general state of polarization launched into the medium can always be decomposed into two waves, one in each of those two polarizations, which will then propagate with different wavenumbers {{math|{{abs|'''k'''}}}}. Applying the different phase of propagation to those two waves over a specified propagation distance will result in a generally ''different'' net polarization state at that point; this is the principle of the [[waveplate]] for instance. With a waveplate, there is no spatial displacement between the two rays as their {{math|'''k'''}}  vectors are still in the same direction. That is true when each of the two polarizations is either normal  to the optic axis (the ordinary ray) or parallel to it (the extraordinary ray).

In the more general case, there ''is'' a difference not only in the magnitude but the direction of the two rays. For instance, the photograph through a calcite crystal (top of page) shows a shifted image in the two polarizations; this is due to the optic axis being neither parallel nor normal to the crystal surface. And even when the optic axis ''is'' parallel to the surface, this will occur for waves launched at non-normal incidence (as depicted in the explanatory figure). In these cases the two {{math|'''k'''}} vectors can be found by solving {{EquationNote|6|eq. 6}} constrained by the boundary condition which requires that the components of the two transmitted waves' {{math|'''k'''}} vectors, and the {{math|'''k'''}} vector of the incident wave, as projected onto the surface of the interface, must all be identical. For a uniaxial crystal it will be found that there is ''not'' a spatial shift for the ordinary ray (hence its name) which will refract as if the material were non-birefringent with an index the same as the two axes which are not the optic axis. For a biaxial crystal neither ray is deemed "ordinary" nor would generally be refracted according to a refractive index equal to one of the principal axes.