Editing Refractive index (section)

==Nonscalar, nonlinear, or nonhomogeneous refraction==
So far, we have assumed that refraction is given by linear equations involving a spatially constant, scalar refractive index. These assumptions can break down in different ways, to be described in the following subsections.

===Birefringence===
{{Main|Birefringence}}

[[File:Calcite.jpg|thumb|alt=A crystal giving a double image of the text behind it|A [[calcite]] crystal laid upon a paper with some letters showing [[double refraction]]]]
[[File:Plastic Protractor Polarized 05375.jpg|thumb|alt=A transparent plastic protractor with smoothly varying bright colors| Birefringent materials can give rise to colors when placed between crossed polarizers. This is the basis for [[photoelasticity]].]]

In some materials, the refractive index depends on the [[Polarization (waves)|polarization]] and propagation direction of the light.<ref>R. Paschotta, article on [https://www.rp-photonics.com/birefringence.html birefringence] {{webarchive|url=https://web.archive.org/web/20150703221334/http://www.rp-photonics.com/birefringence.html |date=2015-07-03 }} in the [https://www.rp-photonics.com/encyclopedia.html Encyclopedia of Laser Physics and Technology] {{webarchive|url=https://web.archive.org/web/20150813044135/http://www.rp-photonics.com/encyclopedia.html |date=2015-08-13 }}, accessed on 2014-09-09</ref> This is called [[birefringence]] or optical [[anisotropy]].

In the simplest form, uniaxial birefringence, there is only one special direction in the material. This axis is known as the [[Optic axis of a crystal|optical axis]] of the material.<ref name=Hecht/>{{rp|230}} Light with linear polarization perpendicular to this axis will experience an ''ordinary'' refractive index {{math|''n''{{sub|o}}}} while light polarized in parallel will experience an ''extraordinary'' refractive index {{math|''n''{{sub|e}}}}.<ref name=Hecht/>{{rp|236}} The birefringence of the material is the difference between these indices of refraction, {{math|Δ''n'' {{=}} ''n''{{sub|e}} − ''n''{{sub|o}}}}.<ref name=Hecht/>{{rp|237}} Light propagating in the direction of the optical axis will not be affected by the birefringence since the refractive index will be {{math|''n''{{sub|o}}}} independent of polarization. For other propagation directions the light will split into two linearly polarized beams. For light traveling perpendicularly to the optical axis the beams will have the same direction.<ref name=Hecht/>{{rp|233}} This can be used to change the polarization direction of linearly polarized light or to convert between linear, circular, and elliptical polarizations with [[waveplate]]s.<ref name=Hecht/>{{rp|237}}

Many [[crystal]]s are naturally birefringent, but [[isotropic]] materials such as [[plastic]]s and [[glass]] can also often be made birefringent by introducing a preferred direction through, e.g., an external force or electric field. This effect is called [[photoelasticity]], and can be used to reveal stresses in structures. The birefringent material is placed between crossed [[polarizers]]. A change in birefringence alters the polarization and thereby the fraction of light that is transmitted through the second polarizer.

In the more general case of trirefringent materials described by the field of [[crystal optics]], the ''dielectric constant'' is a rank-2 [[tensor]] (a 3 by 3 matrix). In this case the propagation of light cannot simply be described by refractive indices except for polarizations along principal axes.

===Nonlinearity===
{{Main|Nonlinear optics}}
The strong [[electric field]] of high intensity light (such as the output of a [[laser]]) may cause a medium's refractive index to vary as the light passes through it, giving rise to [[nonlinear optics]].<ref name=Hecht/>{{rp|502}} If the index varies quadratically with the field (linearly with the intensity), it is called the [[Kerr effect|optical Kerr effect]] and causes phenomena such as [[self-focusing]] and [[self-phase modulation]].<ref name=Hecht/>{{rp|264}} If the index varies linearly with the field (a nontrivial linear coefficient is only possible in materials that do not possess [[inversion symmetry]]), it is known as the [[Pockels effect]].<ref name=Hecht/>{{rp|265}}

===Inhomogeneity===
[[File:Grin-lens.png|thumb|alt=Illustration with gradually bending rays of light in a thick slab of glass|A gradient-index lens with a parabolic variation of refractive index ({{mvar|n}}) with radial distance ({{mvar|x}}). The lens focuses light in the same way as a conventional lens.]]

If the refractive index of a medium is not constant but varies gradually with the position, the material is known as a gradient-index (GRIN) medium and is described by [[gradient index optics]].<ref name="Hecht"/>{{rp|273}} Light traveling through such a medium can be bent or focused, and this effect can be exploited to produce [[lens (optics)|lenses]], some [[optical fiber]]s, and other devices. Introducing {{abbr|GRIN|gradient-index}} elements in the design of an optical system can greatly simplify the system, reducing the number of elements by as much as a third while maintaining overall performance.<ref name="Hecht"/>{{rp|276}} The crystalline lens of the human eye is an example of a {{abbr|GRIN|gradient-index}} lens with a refractive index varying from about 1.406 in the inner core to approximately 1.386 at the less dense cortex.<ref name="Hecht"/>{{rp|203}} Some common [[mirage]]s are caused by a spatially varying refractive index of [[Earth's atmosphere|air]].