Editing Birefringence (section)

==Terminology==
[[File:Comparison_of_positive_and_negative_birefringence.svg|thumb|Comparison of positive and negative birefringence : In positive birefringence (figure 1), the ordinary ray (p-polarisation in this case w.r.t. magenta-coloured plane of incidence), perpendicular to optic axis A is the fast ray (F) while the extraordinary ray (s-polarisation in this case and parallel to optic axis A) is the slow ray (S). In negative birefringence (figure 2), it is the reverse.]]
Much of the work involving polarization preceded the understanding of light as a transverse [[electromagnetic wave]], and this has affected some terminology in use. Isotropic materials have symmetry in all directions and the refractive index is the same for any polarization direction. An anisotropic material is called "birefringent" because it will generally refract a single incoming ray in two directions, which we now understand correspond to the two different polarizations. This is true of either a uniaxial or biaxial material.

In a uniaxial material, one ray behaves according to the normal law of refraction (corresponding to the ordinary refractive index), so an incoming ray at normal incidence remains normal to the refracting surface. As explained above, the other polarization can deviate from normal incidence, which cannot be described using the law of refraction. This thus became known as the ''extraordinary ray''. The terms "ordinary" and "extraordinary" are still applied to the polarization components perpendicular to and not perpendicular to the optic axis respectively, even in cases where no double refraction is involved.

A material is termed ''uniaxial'' when it has a single direction of symmetry in its optical behavior, which we term the optic axis. It also happens to be the axis of symmetry of the index ellipsoid (a spheroid in this case). The index ellipsoid could still be described according to the refractive indices, {{math|''n''<sub>α</sub>}}, {{math|''n''<sub>β</sub>}} and {{math|''n''<sub>γ</sub>}}, along three coordinate axes; in this case two are equal. So if {{math|''n''<sub>α</sub> {{=}} ''n''<sub>β</sub>}} corresponding to the {{mvar|x}} and {{mvar|y}} axes, then the extraordinary index is {{math|''n''<sub>γ</sub>}} corresponding to the {{mvar|z}} axis, which is also called the ''optic axis'' in this case.

Materials in which all three refractive indices are different are termed ''biaxial'' and the origin of this term is more complicated and frequently misunderstood. In a uniaxial crystal, different polarization components of a beam will travel at different phase velocities, ''except'' for rays in the direction of what we call the optic axis. Thus the optic axis has the particular property that rays in that direction do ''not'' exhibit birefringence, with all polarizations in such a beam experiencing the same index of refraction. It is very different when the three principal refractive indices are all different; then an incoming ray in any of those principal directions will still encounter two different refractive indices. But it turns out that there are two special directions (at an angle to all of the 3 axes) where the refractive indices for different polarizations are again equal. For this reason, these crystals were designated as ''biaxial'', with the two "axes" in this case referring to ray directions in which propagation does not experience birefringence.

=== Fast and slow rays ===
In a birefringent material, a wave consists of two polarization components which generally are governed by different effective refractive indices. The so-called ''slow ray'' is the component for which the material has the higher effective refractive index (slower phase velocity), while the ''fast ray'' is the one with a lower effective refractive index. When a beam is incident on such a material from air (or any material with a lower refractive index), the slow ray is thus refracted more towards the normal than the fast ray. In the example figure at top of this page, it can be seen that refracted ray with ''s'' polarization (with its electric vibration along the direction of the optic axis, thus called the extraordinary ray<ref>Born & Wolf, 2002, pp.{{nnbsp}}807–808. (In 19th-century terminology, the ordinary ray is said to be polarized in the plane of the optic axis; but this "[[plane of polarization]]" is the plane ''perpendicular'' to the vibration; cf.&nbsp;Fresnel, 1827, tr.&nbsp;Hobson, p.{{nnbsp}}318.)</ref>) is the slow ray in given scenario.

Using a thin slab of that material at normal incidence, one would implement a [[waveplate]]. In this case, there is essentially no spatial separation between the polarizations, the phase of the wave in the parallel polarization (the slow ray) will be retarded with respect to the perpendicular polarization. These directions are thus known as the slow axis and fast axis of the waveplate.

===Positive or negative===
Uniaxial birefringence is classified as positive when the extraordinary index of refraction {{math|''n''<sub>e</sub>}} is greater than the ordinary index {{math|''n''<sub>o</sub>}}. Negative birefringence means that {{math|Δ''n'' {{=}} ''n''<sub>e</sub> − ''n''<sub>o</sub>}} is less than zero.<ref>''Brad Amos''. [http://homepage.ntlworld.com/w.amos2/Brad%20Amos%27s%20Website/BR1.pdf Birefringence for facetors I: what is birefringence?] {{webarchive |url=https://web.archive.org/web/20131214030317/http://homepage.ntlworld.com/w.amos2/Brad%20Amos%27s%20Website/BR1.pdf |date=December 14, 2013 }} First published in StoneChat, the Journal of the UK Facet Cutter's Guild. January–March. edition 2005.</ref> In other words, the polarization of the fast (or slow) wave is perpendicular to the optic axis when the birefringence of the crystal is positive (or negative, respectively). In the case of biaxial crystals, all three of the principal axes have different refractive indices, so this designation does not apply. But for any defined ray direction one can just as well designate the fast and slow ray polarizations.