View source for Refractive index

[[File:Refraction photo.png|thumb|A [[ray (optics)|ray]] of light being [[refraction|refracted]] in a plastic block|alt=refer to caption]]

In [[optics]], the '''refractive index''' or '''index of refraction''' of a [[optical medium|material]] is a [[dimensionless number]] that describes how fast [[EM radiation|light]]  travels through the material. It is defined as
:<math>n = \frac{c}{v},</math>
where ''c'' is the [[speed of light]] in [[vacuum]] and ''v'' is the [[phase velocity]] of light in the medium. For example, the refractive index of water is 1.333, meaning that light travels 1.333 times as fast in vacuum as in water.

[[File:Refraction at interface.svg|thumb|170px|Refraction of a light ray|alt=Illustration of the incidence and refraction angles]]

The refractive index determines how much the path of light is bent, or [[refraction|refracted]], when entering a material. This is described by [[Snell's law]] of refraction, ''n''<sub>1</sub>&nbsp;sin''θ''<sub>1</sub>&nbsp;=&nbsp;''n''<sub>2</sub>&nbsp;sin''θ''<sub>2</sub>,
where ''θ''<sub>1</sub> and ''θ''<sub>2</sub> are the [[angle of incidence (optics)|angles of incidence]] and refraction, respectively, of a ray crossing the interface between two media with refractive indices ''n''<sub>1</sub> and ''n''<sub>2</sub>. The refractive indices also determine the amount of light that is [[reflectivity|reflected]] when reaching the interface, as well as the critical angle for [[total internal reflection]] and [[Brewster's angle]].<ref name="Hecht">{{cite book  | author = Hecht, Eugene  | title = Optics | publisher = Addison-Wesley | year = 2002 | isbn = 978-0-321-18878-6}}</ref>

The refractive index can be seen as the factor by which the speed and the [[wavelength]] of the radiation are reduced with respect to their vacuum values: the speed of light in a medium is ''v'' = ''c''/''n'', and similarly the wavelength in that medium is ''λ'' = ''λ''<sub>0</sub>/''n'', where ''λ''<sub>0</sub> is the wavelength of that light in vacuum. This implies that vacuum has a refractive index of 1, and that the [[frequency]] (''f'' = ''v''/''λ'') of the wave is not affected by the refractive index. As a result, the perceived color of the refracted light to a human eye which depends on the frequency is not affected by the refraction or the refractive index of the medium.

While the refractive index affects wavelength, it depends on frequency, color and energy so the resulting difference in the bending angles causes white light to split into its constituent colors. This is called [[dispersion (optics)|dispersion]]. It can be observed in [[Prism (optics)|prisms]] and [[rainbow]]s, and [[chromatic aberration]] in lenses. Light propagation in [[Absorption (electromagnetic radiation)|absorbing]] materials can be described using a [[complex number|complex]]-valued refractive index.<ref name="Attwood">{{cite book|title=Soft X-rays and extreme ultraviolet radiation: principles and applications|author=Attwood, David |page=60|isbn=978-0-521-02997-1|year=1999}}</ref> The [[Imaginary number|imaginary]] part then handles the [[attenuation]], while the [[Real number|real]] part accounts for refraction.

The concept of refractive index applies within the full [[electromagnetic spectrum]], from [[X-ray]]s to [[radio wave]]s. It can also be applied to [[wave]] phenomena such as [[sound]]. In this case the speed of sound is used instead of that of light, and a reference medium other than vacuum must be chosen.<ref name=Kinsler>{{cite book | author = Kinsler, Lawrence E. | title = Fundamentals of Acoustics | publisher = John Wiley | year = 2000 | isbn = 978-0-471-84789-2 | page = 136}}</ref>

==Definition==
The refractive index ''n'' of an optical medium is defined as the ratio of the speed of light in vacuum, ''c'' = {{val|299792458|u=m/s}}, and the [[phase velocity]] ''v'' of light in the medium,<ref name=Hecht/>
:<math>n=\frac{c}{v}.</math>
The phase velocity is the speed at which the crests or the [[phase (waves)|phase]] of the [[wave]] moves, which may be different from the [[group velocity]], the speed at which the pulse of light or the [[Envelope (waves)|envelope]] of the wave moves.

The definition above is sometimes referred to as the '''absolute refractive index''' or the '''absolute index of refraction''' to distinguish it from definitions where the speed of light in other reference media than vacuum is used.<ref name=Hecht/> Historically [[air]] at a standardized [[pressure]] and [[temperature]] has been common as a reference medium.

==History==
[[File:Thomas Young (scientist).jpg|thumb|120px|alt=Stipple engraving of Thomas Young|[[Thomas Young (scientist)|Thomas Young]] coined the term ''index of refraction''.]]

[[Thomas Young (scientist)|Thomas Young]] was presumably the person who first used, and invented, the name "index of refraction", in 1807.<ref>{{cite book|last=Young|first=Thomas|title=A course of lectures on natural philosophy and the mechanical arts|year=1807|page=413|url=https://books.google.com/books?id=YPRZAAAAYAAJ&pg=PA413|url-status=live|archiveurl=https://web.archive.org/web/20170222083944/https://books.google.com/books?id=YPRZAAAAYAAJ&pg=PA413|archivedate=2017-02-22}}</ref>
At the same time he changed this value of refractive power into a single number, instead of the traditional ratio of two numbers. The ratio had the disadvantage of different appearances. [[Isaac Newton|Newton]], who called it the "proportion of the sines of incidence and refraction", wrote it as a ratio of two numbers, like "529 to 396" (or "nearly 4 to 3"; for water).<ref name=Newton>{{cite book|last=Newton|first=Isaac|title=Opticks: Or, A Treatise of the Reflections, Refractions, Inflections and Colours of Light|year=1730|page=247|url=https://books.google.com/books?id=GnAFAAAAQAAJ&printsec=frontcover|url-status=live|archiveurl=https://web.archive.org/web/20151128124054/https://books.google.com/books?id=GnAFAAAAQAAJ&printsec=frontcover|archivedate=2015-11-28}}</ref>  [[Francis Hauksbee|Hauksbee]], who called it the "ratio of refraction", wrote it as a ratio with a fixed numerator, like "10000 to 7451.9" (for urine).<ref name=Hauksbee>{{cite journal|doi=10.1098/rstl.1710.0015|last=Hauksbee|first=Francis|title=A Description of the Apparatus for Making Experiments on the Refractions of Fluids|year=1710|page=207|journal=Philosophical Transactions of the Royal Society of London|volume=27|issue=325–336}}</ref>  [[Charles Hutton|Hutton]] wrote it as a ratio with a fixed denominator, like 1.3358 to 1 (water).<ref name=Hutton>{{cite book|last=Hutton|first=Charles|title=Philosophical and mathematical dictionary|year=1795|page=299|url=https://books.google.com/books?id=lsdJAAAAMAAJ&pg=PA299|url-status=live|archiveurl=https://web.archive.org/web/20170222031446/https://books.google.com/books?id=lsdJAAAAMAAJ&pg=PA299|archivedate=2017-02-22}}</ref>

Young did not use a symbol for the index of refraction, in 1807. In the next years, others started using different symbols:
n, m, and µ.<ref name=Fraunhofer>{{cite journal|last=[[Joseph von Fraunhofer|von Fraunhofer]]|first=Joseph|title=Bestimmung des Brechungs und Farbenzerstreuungs Vermogens verschiedener Glasarten|journal=Denkschriften der Königlichen Akademie der Wissenschaften zu München|year=1817|volume=5|page=208|url=https://books.google.com/books?id=lMRSAAAAcAAJ&pg=PA208|url-status=live|archiveurl=https://web.archive.org/web/20170222074213/https://books.google.com/books?id=lMRSAAAAcAAJ&pg=PA208|archivedate=2017-02-22}} Exponent des Brechungsverhältnisses is index of refraction</ref><ref name=Brewster>{{cite journal|last=[[David Brewster|Brewster]]|first=David|title=On the structure of doubly refracting crystals|journal=Philosophical Magazine|year=1815|volume=45|issue=202|page=126|url=https://books.google.com/books?id=GhpRAAAAYAAJ&pg=PA124|doi=10.1080/14786441508638398|url-status=live|archiveurl=https://web.archive.org/web/20170222103726/https://books.google.com/books?id=GhpRAAAAYAAJ&pg=PA124|archivedate=2017-02-22}}</ref><ref name=Herschel>{{cite book|last=[[John Herschel|Herschel]]|first=John F.W.|title=On the Theory of Light|year=1828|page=368|url=https://books.google.com/books?id=Lo4_AAAAcAAJ&printsec=frontcover|url-status=live|archiveurl=https://web.archive.org/web/20151124000212/https://books.google.com/books?id=Lo4_AAAAcAAJ&printsec=frontcover|archivedate=2015-11-24}}</ref> The symbol n gradually prevailed.

==Typical values==
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[[File:Brillanten.jpg|left|thumb|alt=Gemstone diamonds|[[Diamond]]s have a very high refractive index of 2.42.]]

{| style="float:right;"  class="wikitable"
|+Selected refractive indices at λ=589&nbsp;nm.
For references, see the extended [[List of refractive indices]].
!Material||''n''
|-
|[[Vacuum]] || {{val|1}}
|-
| colspan="2" style="text-align:center;"| [[Gas]]es at [[Standard temperature and pressure|0&nbsp;°C and 1&nbsp;atm]]
|-
|[[Air]] || {{val|1.000293}}
|-
|[[Helium]] || {{val|1.000036}}
|-
|[[Hydrogen]] || {{val|1.000132}}
|-
|[[Carbon dioxide]] || {{val|1.00045}}
|-
| colspan="2" style="text-align:center;"| [[Liquid]]s at 20&nbsp;°C
|-
|[[Water]] || 1.333
|-
|[[Ethanol]] || 1.36
|-
|[[Olive oil]] || 1.47
|-
| colspan="2" style="text-align:center;"| [[Solid]]s
|-
|[[Ice]] || 1.31
|-
|[[Fused silica]] (quartz) || 1.46<ref>{{cite web|url=https://refractiveindex.info/?shelf=glass&book=fused_silica&page=Malitson|author=Malitson|year=1965|title=Refractive Index Database|website=refractiveindex.info|accessdate=June 20, 2018}}</ref>
|-
|[[Poly(methyl methacrylate)|PMMA]] (acrylic, plexiglas, lucite, perspex) || 1.49
|-
|[[Soda-lime glass|Window glass]] || 1.52<ref>{{cite web|author1=Faick, C.A.|author2=Finn, A.N.|title=The Index of Refraction of Some Soda-Lime-Silica Glasses as a Function of the Composition|url=http://nvlpubs.nist.gov/nistpubs/jres/6/jresv6n6p993_A2b.pdf|publisher=National Institute of Standards and Technology|accessdate=11 December 2016|archiveurl=https://web.archive.org/web/20161230105725/http://nvlpubs.nist.gov/nistpubs/jres/6/jresv6n6p993_A2b.pdf|archivedate=December 30, 2016|language=English|format=.pdf|date=July 1931|url-status=live}}</ref>
|-
|[[Polycarbonate]] (Lexan™) || 1.58<ref>{{cite journal|last1=Sultanova|first1=N.|last2=Kasarova|first2=S.|last3=Nikolov|first3=I.|title=Dispersion Properties of Optical Polymers|journal=Acta Physica Polonica A|date=October 2009|volume=116|issue=4|pages=585–587|doi=10.12693/APhysPolA.116.585}}</ref>
|-
|[[Flint glass]] (typical) || 1.62
|-
|[[Sapphire]] || 1.77<ref>{{cite journal|last1=Tapping|first1=J.|last2=Reilly|first2=M. L.|title=Index of refraction of sapphire between 24 and 1060°C for wavelengths of 633 and 799 nm|journal=Journal of the Optical Society of America A|date=1 May 1986|volume=3|issue=5|pages=610|doi=10.1364/JOSAA.3.000610|bibcode=1986JOSAA...3..610T|df=}}</ref>
|-
|[[Cubic zirconia]] || 2.15
|-
|[[Diamond]] || 2.42
|-
|[[Moissanite]] || 2.65
|}

{{See also|List of refractive indices}}
For [[visible light]] most [[transparency and translucency|transparent]] media have refractive indices between 1 and 2. A few examples are given in the adjacent table. These values are measured at the yellow doublet [[D2 line|D-line]] of [[sodium]], with a wavelength of 589 [[nanometers]], as is conventionally done.<ref name=FBI/> Gases at atmospheric pressure have refractive indices close to 1 because of their low density. Almost all solids and liquids have refractive indices above 1.3, with [[aerogel]] as the clear exception. Aerogel is a very low density solid that can be produced with refractive index in the range from 1.002 to 1.265.<ref>{{cite journal| author=Tabata, M.| title=Development of Silica Aerogel with Any Density| journal=2005 IEEE Nuclear Science Symposium Conference Record| volume=2| pages=816–818| year=2005| url=http://www.ppl.phys.chiba-u.jp/~makoto/publication/N14-191.pdf| display-authors=etal| url-status=live| archiveurl=https://web.archive.org/web/20130518075319/http://www.ppl.phys.chiba-u.jp/~makoto/publication/N14-191.pdf| archivedate=2013-05-18| doi=10.1109/NSSMIC.2005.1596380| isbn=978-0-7803-9221-2}}</ref> [[Moissanite]] lies at the other end of the range with a refractive index as high as 2.65. Most plastics have refractive indices in the range from 1.3 to 1.7, but some [[high-refractive-index polymer]]s can have values as high as 1.76.<ref>Naoki Sadayori and Yuji Hotta "Polycarbodiimide having high index of refraction and production method thereof" [http://www.google.com/patents?vid=va2WAAAAEBAJ US patent 2004/0158021 A1] (2004)</ref>

For [[infrared]] light refractive indices can be considerably higher. [[Germanium]] is transparent in the wavelength region from 2 to 14&nbsp;µm and has a refractive index of about 4.<ref>Tosi, Jeffrey L., article on [http://www.photonics.com/EDU/Handbook.aspx?AID=25495 Common Infrared Optical Materials] in the Photonics Handbook, accessed on 2014-09-10</ref> A type of new materials, called topological insulator, was recently found holding higher refractive index of up to 6 in near to mid infrared frequency range. Moreover, topological insulator material are transparent when they have nanoscale thickness. These excellent properties make them a type of significant materials for infrared optics.<ref>{{Cite journal|last=Yue|first=Zengji|last2=Cai|first2=Boyuan|last3=Wang|first3=Lan|last4=Wang|first4=Xiaolin|last5=Gu|first5=Min|date=2016-03-01|title=Intrinsically core-shell plasmonic dielectric nanostructures with ultrahigh refractive index|journal=Science Advances|language=en|volume=2|issue=3|pages=e1501536|doi=10.1126/sciadv.1501536|issn=2375-2548|pmc=4820380|pmid=27051869|bibcode=2016SciA....2E1536Y|df=}}</ref>

===Refractive index below unity===
According to the [[theory of relativity]], no information can travel faster than the speed of light in vacuum, but this does not mean that the refractive index cannot be less than 1. The refractive index measures the [[phase velocity]] of light, which does not carry [[information]].<ref name=Als-Nielsen2011>{{cite book|last=Als-Nielsen|first=J.; McMorrow, D.|title=Elements of Modern X-ray Physics|year=2011|publisher=Wiley-VCH|isbn=978-0-470-97395-0|page=25|quote=One consequence of the real part of ''n'' being less than unity is that it implies that the phase velocity inside the material, ''c''/''n'', is larger than the velocity of light, ''c''. This does not, however, violate the law of relativity, which requires that only signals carrying information do not travel faster than ''c''. Such signals move with the group velocity, not with the phase velocity, and it can be shown that the group velocity is in fact less than ''c''.}}</ref> The phase velocity is the speed at which the crests of the wave move and can be faster than the speed of light in vacuum, and thereby give a refractive index below 1. This can occur close to [[resonance frequency|resonance frequencies]], for absorbing media, in [[plasma (physics)|plasmas]], and for [[X-ray]]s. In the X-ray regime the refractive indices are lower than but very close to 1 (exceptions close to some resonance frequencies).<ref name=CXRO>{{cite web |url = http://henke.lbl.gov/optical_constants/ |title = X-Ray Interactions With Matter |publisher = The Center for X-Ray Optics |accessdate = 2011-08-30 |url-status = live |archiveurl = https://web.archive.org/web/20110827214322/http://henke.lbl.gov/optical_constants/ |archivedate = 2011-08-27 }}</ref>
As an example, water has a refractive index of {{val|0.99999974}} = 1 − {{val|2.6|e=-7}} for X-ray radiation at a photon energy of {{val|30|ul=keV}} (0.04&nbsp;nm wavelength).<ref name=CXRO/>

An example of a plasma with an index of refraction less than unity is Earth's [[ionosphere]].  Since the refractive index of the ionosphere (a [[Plasma (physics)|plasma]]), is less than unity, electromagnetic waves propagating through the plasma are bent "away from the normal" (see [[Geometric optics]]) allowing the radio wave to be refracted back toward earth, thus enabling long-distance radio communications.  See also [[Radio Propagation]] and [[Skywave]].<ref>{{cite book|last1=Lied|first1=Finn|title=High Frequency Radio Communications with Emphasis on Polar Problems|date=1967|publisher=The Advisory Group for Aerospace Research and Development|pages=1–7}}</ref>

===Negative refractive index===
{{See also|Negative index metamaterials}}
[[File:Split-ring resonator array 10K sq nm.jpg|thumb|250px|alt=A 3D grid of open copper rings made from interlocking standing sheets of fiberglass circuit boards|A [[split-ring resonator]] array arranged to produce a negative index of refraction for [[microwaves]]]]

Recent research has also demonstrated the existence of materials with a negative refractive index, which can occur if [[permittivity]] and [[magnetic permeability|permeability]] have simultaneous negative values.<ref name=veselago1968>{{cite journal|last=Veselago|first=V. G.| authorlink = Victor Veselago |title=The electrodynamics of substances with simultaneously negative values of ε and μ|journal=[[Physics-Uspekhi|Soviet Physics Uspekhi]]|year=1968|volume=10|issue=4|pages=509–514|doi=10.1070/PU1968v010n04ABEH003699|bibcode = 1968SvPhU..10..509V }}</ref> This can be achieved with periodically constructed [[metamaterials]]. The resulting [[negative refraction]] (i.e., a reversal of [[Snell's law]]) offers the possibility of the [[superlens]] and other exotic phenomena.<ref name=shalaev2007>{{cite journal | last = Shalaev | first = V. M. | authorlink = Vladimir Shalaev | title = Optical negative-index metamaterials | journal = [[Nature Photonics]] | volume = 1 | issue =  | pages = 41–48 | date = 2007 | jstor =  | issn =  | doi = 10.1038/nphoton.2006.49 | id =  | mr =  | zbl =  | jfm =  | bibcode = 2007NaPho...1...41S | df =  }}</ref>

==Microscopic explanation==
[[File:Thin section scan crossed polarizers Siilinjärvi R636-105.90.jpg|thumb|In [[optical mineralogy]], [[thin section]]s are used to study rocks. The method is based on the distinct refractive indexes of different [[mineral]]s.]]
{{main|Ewald–Oseen extinction theorem}}
At the atomic scale, an electromagnetic wave's phase velocity is slowed in a material because the [[electric field]] creates a disturbance in the charges of each atom (primarily the [[electron]]s) proportional to the [[electric susceptibility]] of the medium. (Similarly, the [[magnetic field]] creates a disturbance proportional to the [[magnetic susceptibility]].) As the electromagnetic fields oscillate in the wave, the charges in the material will be "shaken" back and forth at the same frequency.<ref name = Hecht />{{rp|67}} The charges thus radiate their own electromagnetic wave that is at the same frequency, but usually with a [[phase (waves)|phase delay]], as the charges may move out of phase with the force driving them (see [[Harmonic oscillator#Sinusoidal driving force|sinusoidally driven harmonic oscillator]]). The light wave traveling in the medium is the macroscopic [[superposition principle|superposition (sum)]] of all such contributions in the material: the original wave plus the waves radiated by all the moving charges. This wave is typically a wave with the same frequency but shorter wavelength than the original, leading to a slowing of the wave's phase velocity. Most of the radiation from oscillating material charges will modify the incoming wave, changing its velocity. However, some net energy will be radiated in other directions or even at other frequencies (see [[scattering]]).

Depending on the relative phase of the original driving wave and the waves radiated by the charge motion, there are several possibilities:
* If the electrons emit a light wave which is 90° out of phase with the light wave shaking them, it will cause the total light wave to travel slower. This is the normal refraction of transparent materials like glass or water, and corresponds to a refractive index which is real and greater than 1.<ref name="Feynman, Richard P. 2011">{{cite book | author = Feynman, Richard P. | title = Feynman Lectures on Physics 1: Mainly Mechanics, Radiation, and Heat | publisher = Basic Books | year = 2011 | page =  | isbn = 978-0-465-02493-3}}</ref>
* If the electrons emit a light wave which is 270° out of phase with the light wave shaking them, it will cause the wave to travel faster. This is called "anomalous refraction", and is observed close to absorption lines (typically in infrared spectra), with [[X-ray]]s in ordinary materials, and with radio waves in Earth's [[ionosphere]]. It corresponds to a [[permittivity]] less than 1, which causes the refractive index to be also less than unity and the [[phase velocity]] of light greater than the [[speed of light|speed of light in vacuum]] ''c'' (note that the [[signal velocity]] is still less than ''c'', as discussed above). If the response is sufficiently strong and out-of-phase, the result is a negative value of [[permittivity]] and imaginary index of refraction, as observed in metals or plasma.<ref name="Feynman, Richard P. 2011"/>
* If the electrons emit a light wave which is 180° out of phase with the light wave shaking them, it will destructively interfere with the original light to reduce the total light intensity. This is [[absorption (electromagnetic radiation)|light absorption in opaque materials]] and corresponds to an [[imaginary number|imaginary]] refractive index.
* If the electrons emit a light wave which is in phase with the light wave shaking them, it will amplify the light wave. This is rare, but occurs in [[laser]]s due to [[stimulated emission]]. It corresponds to an imaginary index of refraction, with the opposite sign to that of absorption.
For most materials at visible-light frequencies, the phase is somewhere between 90° and 180°, corresponding to a combination of both refraction and absorption.

==Dispersion==
[[File:WhereRainbowRises.jpg|thumb|150px|alt=A rainbow|Light of different colors has slightly different refractive indices in water and therefore shows up at different positions in the [[rainbow]].]]
[[File:Prism rainbow schema.png|thumb|left|alt=A white beam of light dispersed into different colors when passing through a triangular prism|In a prism, dispersion causes different colors to refract at different angles, splitting white light into a rainbow of colors.]]
[[File:Mplwp dispersion curves.svg|right|thumb|320px|alt=A graph showing the decrease in refractive index with increasing wavelength for different types of glass|The variation of refractive index with wavelength for various glasses. The shaded zone indicates the range of visible light.]]
{{Main|Dispersion (optics)}}

The refractive index of materials varies with the wavelength (and [[frequency]]) of light.<ref name=dispersion_ELPT>R. Paschotta, article on [https://www.rp-photonics.com/chromatic_dispersion.html chromatic dispersion] {{webarchive|url=https://web.archive.org/web/20150629012047/http://www.rp-photonics.com/chromatic_dispersion.html |date=2015-06-29 }} 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 2014-09-08</ref> This is called dispersion and causes [[prism (optics)|prisms]] and [[rainbow]]s to divide white light into its constituent spectral [[color]]s.<ref name=hyperphysics_dispersion>Carl R. Nave, page on [http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/dispersion.html Dispersion] {{webarchive|url=https://web.archive.org/web/20140924222742/http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/dispersion.html |date=2014-09-24 }} in [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics] {{webarchive|url=https://web.archive.org/web/20071028155517/http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html |date=2007-10-28 }}, Department of Physics and Astronomy, Georgia State University, accessed on 2014-09-08</ref> As the refractive index varies with wavelength, so will the refraction angle as light goes from one material to another. Dispersion also causes the [[focal length]] of [[Lens (optics)|lenses]] to be wavelength dependent. This is a type of [[chromatic aberration]], which often needs to be corrected for in imaging systems. In regions of the spectrum where the material does not absorb light, the refractive index tends to decrease with increasing wavelength, and thus increase with frequency. This is called "normal dispersion", in contrast to "anomalous dispersion", where the refractive index increases with wavelength.<ref name=dispersion_ELPT/> For visible light normal dispersion means that the refractive index is higher for blue light than for red.

For optics in the visual range, the amount of dispersion of a lens material is often quantified by the [[Abbe number]]:<ref name=hyperphysics_dispersion/>
:<math>V = \frac{n_\mathrm{yellow} - 1}{n_\mathrm{blue} - n_\mathrm{red}}.</math>
For a more accurate description of the wavelength dependence of the refractive index, the [[Sellmeier equation]] can be used.<ref>R. Paschotta, article on [https://www.rp-photonics.com/sellmeier_formula.html Sellmeier formula] {{webarchive|url=https://web.archive.org/web/20150319205203/http://www.rp-photonics.com/sellmeier_formula.html |date=2015-03-19 }} 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 2014-09-08</ref> It is an empirical formula that works well in describing dispersion. ''Sellmeier coefficients'' are often quoted instead of the refractive index in tables.

Because of dispersion, it is usually important to specify the vacuum wavelength of light for which a refractive index is measured. Typically, measurements are done at various well-defined spectral [[emission line]]s; for example, ''n''<sub>D</sub> usually denotes the refractive index at the [[Fraunhofer lines|Fraunhofer]] "D" line, the centre of the yellow [[sodium]] double emission at 589.29 [[nanometre|nm]] wavelength.<ref name=FBI>{{cite web |url = https://www.fbi.gov/about-us/lab/forensic-science-communications/fsc/jan2005/index.htm/standards/2005standards9.htm |title = Forensic Science Communications, Glass Refractive Index Determination |publisher = FBI Laboratory Services |accessdate = 2014-09-08 |url-status = dead |archiveurl = https://web.archive.org/web/20140910195815/http://www.fbi.gov/about-us/lab/forensic-science-communications/fsc/jan2005/index.htm/standards/2005standards9.htm |archivedate = 2014-09-10 }}</ref>

==Complex refractive index==
[[File:Gradndfilter.jpg|thumb|alt=A glass plate, half of which is darkened|A [[graduated neutral density filter]] showing light absorption in the upper half]]
{{see also|Mathematical descriptions of opacity}}

When light passes through a medium, some part of it will always be [[Attenuation|attenuated]]. This can be conveniently taken into account by defining a complex refractive index,
:<math>\underline{n} = n + i\kappa.</math>
Here, the real part ''n'' is the refractive index and indicates the [[phase velocity]], while the imaginary part ''κ'' is called the '''extinction coefficient''' — although ''κ'' can also refer to the [[mass attenuation coefficient]]—<ref>{{cite web
 |url         = http://web.mit.edu/course/6/6.732/www/6.732-pt2.pdf
 |title       = Solid State Physics Part II Optical Properties of Solids
 |last        = Dresselhaus
 |first       = M. S.
 |date        = 1999
 |website     = Course 6.732 Solid State Physics
 |publisher   = MIT
 |accessdate  = 2015-01-05
 |url-status     = live
 |archiveurl  = https://web.archive.org/web/20150724051216/http://web.mit.edu/course/6/6.732/www/6.732-pt2.pdf
 |archivedate = 2015-07-24
}}</ref>{{rp|3}} and indicates the amount of attenuation when the electromagnetic wave propagates through the material.<ref name="Hecht"/>{{rp|128}}

That ''κ'' corresponds to attenuation can be seen by inserting this refractive index into the expression for [[electric field]] of a [[plane wave|plane]] electromagnetic wave traveling in the ''z''-direction. We can do this by relating the complex wave number <u>''k''</u> to the complex refractive index <u>''n''</u> through <u>''k''</u> = 2π<u>''n''</u>/''λ''<sub>0</sub>, with ''λ''<sub>0</sub> being the vacuum wavelength; this can be inserted into the plane wave expression as
:<math>\mathbf{E}(z, t) = \operatorname{Re}\! \left[\mathbf{E}_0 e^{i(\underline{k}z - \omega t)}\right] = \operatorname{Re}\! \left[\mathbf{E}_0 e^{i(2\pi(n + i\kappa)z/\lambda_0 - \omega t)}\right] = e^{-2\pi \kappa z/\lambda_0} \operatorname{Re}\! \left[\mathbf{E}_0 e^{i(kz - \omega t)}\right].</math>
Here we see that ''κ'' gives an exponential decay, as expected from the [[Beer–Lambert law]]. Since intensity is proportional to the square of the electric field, it will depend on the depth into the material as exp(−4π''κz''/''λ''<sub>0</sub>), and the [[attenuation coefficient]] becomes ''α'' = 4π''κ''/''λ''<sub>0</sub>.<ref name="Hecht"/>{{rp|128}} This also relates it to the [[penetration depth]], the distance after which the intensity is reduced by 1/''e'', ''δ''<sub>p</sub> = 1/''α'' = ''λ''<sub>0</sub>/(4π''κ'').

Both ''n'' and ''κ'' are dependent on the frequency. In most circumstances ''κ'' > 0 (light is absorbed) or ''κ'' = 0 (light travels forever without loss). In special situations, especially in the [[gain medium]] of [[laser]]s, it is also possible that ''κ'' < 0, corresponding to an amplification of the light.

An alternative convention uses <u>''n''</u> = ''n'' − ''iκ'' instead of <u>''n''</u> = ''n'' + ''iκ'', but where ''κ'' > 0 still corresponds to loss. Therefore, these two conventions are inconsistent and should not be confused. The difference is related to defining sinusoidal time dependence as Re[exp(−''iωt'')] versus Re[exp(+''iωt'')]. See [[Mathematical descriptions of opacity]].

Dielectric loss and non-zero DC conductivity in materials cause absorption. Good dielectric materials such as glass have extremely low DC conductivity, and at low frequencies the dielectric loss is also negligible, resulting in almost no absorption. However, at higher frequencies (such as visible light), dielectric loss may increase absorption significantly, reducing the material's [[transparency (optics)|transparency]] to these frequencies.

The real, ''n'', and imaginary, ''κ'', parts of the complex refractive index are related through the [[Kramers–Kronig relation]]s. In 1986 A.R. Forouhi and I. Bloomer deduced an [[Refractive index and extinction coefficient of thin film materials|equation]] describing ''κ'' as a function of photon energy, ''E'', applicable to amorphous materials. Forouhi and Bloomer then applied the Kramers–Kronig relation to derive the corresponding equation for [[Refractive index and extinction coefficient of thin film materials|''n'' as a function of ''E'']]. The same formalism was applied to crystalline materials by Forouhi and Bloomer in 1988.

The refractive index and extinction coefficient, ''n'' and ''κ'', cannot be measured directly. They must be determined indirectly from measurable quantities that depend on them, such as [[Refractive index and extinction coefficient of thin film materials|reflectance, ''R'', or transmittance, ''T'']], or ellipsometric parameters, [[ellipsometry|''ψ'' and ''δ'']]. The determination of ''n'' and ''κ'' from such measured quantities will involve developing a theoretical expression for ''R'' or ''T'', or ''ψ'' and ''δ'' in terms of a valid physical model for ''n'' and ''κ''. By fitting the theoretical model to the measured ''R'' or ''T'', or ''ψ'' and ''δ'' using regression analysis, ''n'' and ''κ'' can be deduced.

For [[X-ray]] and [[extreme ultraviolet]] radiation the complex refractive index deviates only slightly from unity and usually has a real part smaller than 1. It is therefore normally written as <u>''n''</u> = 1 − ''δ'' + ''iβ'' (or <u>''n''</u> = 1 − ''δ'' − ''iβ'' with the alternative convention mentioned above).<ref name=Attwood/> Far above the atomic resonance frequency delta can be given by

:<math> \delta = \frac{r_0 \lambda^2 n_e}{2 \pi} </math>

where <math>r_0</math> is the [[classical electron radius]], <math> \lambda </math> is the X-ray wavelength, and <math>n_e</math> is the electron density. One may assume the electron density is simply the number of electrons per atom Z multiplied by the atomic density, but more accurate calculation of the refractive index requires replacing Z with the complex [[atomic form factor]] <math> f = Z + f' + i f'' </math>. It follows that

:<math> \delta = \frac{r_0 \lambda^2}{2 \pi} (Z + f') n_{Atom} </math>
:<math> \beta = \frac{r_0 \lambda^2}{2 \pi} f'' n_{Atom} </math>

with <math>\delta</math> and <math>\beta</math> typically of the order of 10<sup>−5</sup> and 10<sup>−6</sup>.

==Relations to other quantities==

===Optical path length===
[[File:Soap bubble sky.jpg|thumb|alt=Soap bubble|The colors of a [[soap bubble]] are determined by the [[optical path length]] through the thin soap film in a phenomenon called [[thin-film interference]].]]

[[Optical path length]] (OPL) is the product of the geometric length ''d'' of the path light follows through a system, and the index of refraction of the medium through which it propagates,<ref>R. Paschotta, article on [https://www.rp-photonics.com/optical_thickness.html optical thickness] {{webarchive|url=https://web.archive.org/web/20150322115346/http://www.rp-photonics.com/optical_thickness.html |date=2015-03-22 }} 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 2014-09-08</ref>
:<math>\text{OPL} = nd.</math>
This is an important concept in optics because it determines the [[phase (waves)|phase]] of the light and governs [[interference (wave propagation)|interference]] and [[diffraction]] of light as it propagates. According to [[Fermat's principle]], light rays can be characterized as those curves that [[Mathematical optimization|optimize]] the optical path length.<ref name=Hecht/>{{rp|68–69}}

===Refraction===
{{Main|Refraction}}

[[File:Snells law.svg|thumb|alt=refer to caption|[[Refraction]] of light at the interface between two media of different refractive indices, with ''n''<sub>2</sub> > ''n''<sub>1</sub>. Since the [[phase velocity]] is lower in the second medium (''v''<sub>2</sub> < ''v''<sub>1</sub>), the angle of refraction ''θ''<sub>2</sub> is less than the angle of incidence ''θ''<sub>1</sub>; that is, the ray in the higher-index medium is closer to the normal.]]

When light moves from one medium to another, it changes direction, i.e. it is [[Refraction|refracted]]. If it moves from a medium with refractive index ''n''<sub>1</sub> to one with refractive index ''n''<sub>2</sub>, with an [[angle of incidence (optics)|incidence angle]] to the [[surface normal]] of ''θ''<sub>1</sub>, the refraction angle ''θ''<sub>2</sub> can be calculated from [[Snell's law]]:<ref>R. Paschotta, article on [https://www.rp-photonics.com/refraction.html refraction] {{webarchive|url=https://web.archive.org/web/20150628174941/https://www.rp-photonics.com/refraction.html |date=2015-06-28 }} 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 2014-09-08</ref>
:<math>n_1 \sin \theta_1 = n_2 \sin \theta_2.</math>

When light enters a material with higher refractive index, the angle of refraction will be smaller than the angle of incidence and the light will be refracted towards the normal of the surface. The higher the refractive index, the closer to the normal direction the light will travel. When passing into a medium with lower refractive index, the light will instead be refracted away from the normal, towards the surface.

===Total internal reflection===
{{Main|Total internal reflection}}

[[File:Total internal reflection of Chelonia mydas.jpg|thumb|alt=A sea turtle being reflected in the water surface above|[[Total internal reflection]] can be seen at the air-water boundary.]]
If there is no angle ''θ''<sub>2</sub> fulfilling Snell's law, i.e.,
:<math>\frac{n_1}{n_2} \sin \theta_1 > 1,</math>
the light cannot be transmitted and will instead undergo [[total internal reflection]].<ref name = bornwolf />{{rp|49–50}} This occurs only when going to a less optically dense material, i.e., one with lower refractive index. To get total internal reflection the angles of incidence ''θ''<sub>1</sub> must be larger than the critical angle<ref>{{cite encyclopedia |first=R. |last=Paschotta |url=https://www.rp-photonics.com/total_internal_reflection.html|title=Total Internal Reflection|work=RP Photonics Encyclopedia |accessdate=2015-08-16 |url-status=live |archiveurl=https://web.archive.org/web/20150628175307/https://www.rp-photonics.com/total_internal_reflection.html |archivedate=2015-06-28 }}</ref>
:<math>\theta_\mathrm{c} = \arcsin\!\left(\frac{n_2}{n_1}\right)\!.</math>

===Reflectivity===
Apart from the transmitted light there is also a [[reflection (physics)|reflected]] part. The reflection angle is equal to the incidence angle, and the amount of light that is reflected is determined by the [[reflectivity]] of the surface. The reflectivity can be calculated from the refractive index and the incidence angle with the [[Fresnel equations]], which for [[normal incidence]] reduces to<ref name = bornwolf />{{rp|44}}

:<math>R_0 = \left|\frac{n_1 - n_2}{n_1 + n_2}\right|^2\!.</math>

For common glass in air, ''n''<sub>1</sub> = 1 and ''n''<sub>2</sub> = 1.5, and thus about 4% of the incident power is reflected.<ref name=ri-min>{{cite web|last=Swenson|first=Jim|author2=Incorporates Public Domain material from the [[U.S. Department of Energy]]|title=Refractive Index of Minerals|publisher=Newton BBS, Argonne National Laboratory, US DOE|date=November 10, 2009<!--[http://www.newton.dep.anl.gov/]-->|url=http://www.newton.dep.anl.gov/askasci/env99/env234.htm|accessdate=2010-07-28|url-status=live|archiveurl=https://web.archive.org/web/20100528092315/http://www.newton.dep.anl.gov/askasci/env99/env234.htm|archivedate=May 28, 2010}}</ref> At other incidence angles the reflectivity will also depend on the [[polarization (waves)|polarization]] of the incoming light. At a certain angle called [[Brewster's angle]], p-polarized light (light with the electric field in the [[plane of incidence]]) will be totally transmitted. Brewster's angle can be calculated from the two refractive indices of the interface as <ref name=Hecht/>{{rp|245}}
:<math>\theta_\mathrm{B} = \arctan\!\left(\frac{n_2}{n_1}\right)\!.</math>

===Lenses===
[[File:Lupa.na.encyklopedii.jpg|thumb|alt=A magnifying glass|The [[optical power|power]] of a [[magnifying glass]] is determined by the shape and refractive index of the lens.]]

The [[focal length]] of a [[lens (optics)|lens]] is determined by its refractive index ''n'' and the [[Radius of curvature (optics)|radii of curvature]] ''R''<sub>1</sub> and ''R''<sub>2</sub> of its surfaces. The power of a [[thin lens]] in air is given by the [[Lensmaker's formula]]:<ref>Carl R. Nave, page on the [http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenmak.html Lens-Maker's Formula] {{webarchive|url=https://web.archive.org/web/20140926153405/http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenmak.html |date=2014-09-26 }} in [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics] {{webarchive|url=https://web.archive.org/web/20071028155517/http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html |date=2007-10-28 }}, Department of Physics and Astronomy, Georgia State University, accessed on 2014-09-08</ref>
:<math>\frac{1}{f} = (n - 1)\!\left(\frac{1}{R_1} - \frac{1}{R_2}\right)\!,</math>
where ''f'' is the focal length of the lens.

===Microscope resolution===
The [[optical resolution|resolution]] of a good optical [[microscope]] is mainly determined by the [[numerical aperture]] (NA) of its [[objective lens]]. The numerical aperture in turn is determined by the refractive index ''n'' of the medium filling the space between the sample and the lens and the half collection angle of light ''θ'' according to<ref name=Carlsson>{{cite web |first = Kjell |last = Carlsson |url = https://www.kth.se/social/files/542d1251f276544bf2492088/Compendium.Light.Microscopy.pdf |title = Light microscopy |year = 2007 |accessdate = 2015-01-02 |url-status = live |archiveurl = https://web.archive.org/web/20150402122840/https://www.kth.se/social/files/542d1251f276544bf2492088/Compendium.Light.Microscopy.pdf |archivedate = 2015-04-02 }}</ref>{{rp|6}}
:<math>\mathrm{NA} = n\sin \theta.</math>

For this reason [[oil immersion]] is commonly used to obtain high resolution in microscopy. In this technique the objective is dipped into a drop of high refractive index immersion oil on the sample under study.<ref name=Carlsson/>{{rp|14}}

===Relative permittivity and permeability===
The refractive index of electromagnetic radiation equals
:<math>n = \sqrt{\varepsilon_\mathrm{r} \mu_\mathrm{r}},</math>
where ''ε''<sub>r</sub> is the material's [[relative permittivity]], and ''μ''<sub>r</sub> is its [[Permeability (electromagnetism)|relative permeability]].<ref name = bleaney>{{cite book | last1 = Bleaney| first1 = B.| authorlink1 = Brebis Bleaney |last2 = Bleaney |first2 = B.I. | title = Electricity and Magnetism | publisher = [[Oxford University Press]] | edition = Third | date = 1976 | isbn = 978-0-19-851141-0 }}</ref>{{rp|229}} The refractive index is used for optics in [[Fresnel equations]] and [[Snell's law]]; while the relative permittivity and permeability are used in [[Maxwell's equations]] and electronics. Most naturally occurring materials are non-magnetic at optical frequencies, that is ''μ<sub>r</sub>'' is very close to 1,{{Citation needed|date=November 2015}} therefore ''n'' is approximately {{sqrt|''ε''<sub>r</sub>}}. In this particular case, the complex relative permittivity <u>''ε''</u><sub>r</sub>, with real and imaginary parts ''ε''<sub>r</sub> and ''ɛ̃''<sub>r</sub>, and the complex refractive index <u>''n''</u>, with real and imaginary parts ''n'' and ''κ'' (the latter called the "extinction coefficient"), follow the relation
:<math>\underline{\varepsilon}_\mathrm{r} = \varepsilon_\mathrm{r} + i\tilde{\varepsilon}_\mathrm{r} = \underline{n}^2 = (n + i\kappa)^2,</math>
and their components are related by:<ref>{{cite book|first=Frederick|last=Wooten|title=Optical Properties of Solids|page=49|publisher=[[Academic Press]]|location=New York City|year= 1972|isbn=978-0-12-763450-0}}[http://www.lrsm.upenn.edu/~frenchrh/download/0208fwootenopticalpropertiesofsolids.pdf  (online pdf)] {{webarchive|url=https://web.archive.org/web/20111003034948/http://www.lrsm.upenn.edu/~frenchrh/download/0208fwootenopticalpropertiesofsolids.pdf |date=2011-10-03 }}</ref>
:<math>\varepsilon_\mathrm{r} = n^2 - \kappa^2,</math>
:<math>\tilde{\varepsilon}_\mathrm{r} = 2n\kappa,</math>
and:
:<math>n = \sqrt{\frac{|\underline{\varepsilon}_\mathrm{r}| + \varepsilon_\mathrm{r}}{2}},</math>
:<math>\kappa = \sqrt{\frac{|\underline{\varepsilon}_\mathrm{r}| - \varepsilon_\mathrm{r}}{2}}.</math>
where <math>|\underline{\varepsilon}_\mathrm{r}| = \sqrt{\varepsilon_\mathrm{r}^2 + \tilde{\varepsilon}_\mathrm{r}^2}</math> is the [[modulus of complex number|complex modulus]].

===Wave impedance===
{{see also|Wave impedance}}
The wave impedance of a plane electromagnetic wave in a non-conductive medium is given by
:<math>Z = \sqrt{\frac{\mu}{\varepsilon}} = \sqrt{\frac{\mu_\mathrm{0}\mu_\mathrm{r}}{\varepsilon_\mathrm{0}\varepsilon_\mathrm{r}}} = \sqrt{\frac{\mu_\mathrm{0}}{\varepsilon_\mathrm{0}}}\sqrt{\frac{\mu_\mathrm{r}}{\varepsilon_\mathrm{r}}} = Z_0\sqrt{\frac{\mu_\mathrm{r}}{\varepsilon_\mathrm{r}}} = Z_0\frac{\mu_\mathrm{r}}{n}</math>
where <math>Z_0</math> is the vacuum wave impedance, ''μ'' and ''ϵ'' are the absolute permeability and permittivity of the medium, ''ε''<sub>r</sub> is the material's [[relative permittivity]], and ''μ''<sub>r</sub> is its [[Permeability (electromagnetism)|relative permeability]].

In non-magnetic media with <math>\mu_\mathrm{r}=1</math>,
:<math>Z = \frac{Z_0}{n},</math>

:<math>n = \frac{Z_0}{Z}.</math>

Thus refractive index in a non-magnetic media is the ratio of the vacuum wave impedance to the wave impedance of the medium.

The reflectivity <math>R_0</math> between two media can thus be expressed both by the wave impedances and the refractive indices as
:<math>R_0 = \left|\frac{n_1 - n_2}{n_1 + n_2}\right|^2\! = \left|\frac{Z_2 - Z_1}{Z_2 + Z_1}\right|^2\!.</math>

===Density===
[[File:density-nd.GIF|thumb|upright=1.7|alt=A scatter plot showing a strong correlation between glass density and refractive index for different glasses|Relation between the refractive index and the density of [[silicate glass|silicate]] and [[borosilicate glass]]es<ref>{{cite web|url=http://www.glassproperties.com/refractive_index/|title=Calculation of the Refractive Index of Glasses|work=Statistical Calculation and Development of Glass Properties|url-status=live|archiveurl=https://web.archive.org/web/20071015124852/http://glassproperties.com/refractive_index/|archivedate=2007-10-15}}</ref>]]
In general, the refractive index of a glass increases with its [[density]]. However, there does not exist an overall linear relation between the refractive index and the density for all silicate and borosilicate glasses. A relatively high refractive index and low density can be obtained with glasses containing light metal oxides such as [[lithium oxide|Li<sub>2</sub>O]] and [[magnesium oxide|MgO]], while the opposite trend is observed with glasses containing [[lead(II) oxide|PbO]] and [[barium oxide|BaO]] as seen in the diagram at the right.

Many oils (such as [[olive oil]]) and [[ethyl alcohol]] are examples of liquids which are more refractive, but less dense, than water, contrary to the general correlation between density and refractive index.

For air, ''n'' − 1 is proportional to the density of the gas as long as the chemical composition does not change.<ref>{{cite web | url = http://emtoolbox.nist.gov/Wavelength/Documentation.asp | first1 = Jack A. | last1 = Stone | first2 = Jay H. | last2 = Zimmerman | date = 2011-12-28 | website = Engineering metrology toolbox | publisher = National Institute of Standards and Technology (NIST) | title = Index of refraction of air | accessdate = 2014-01-11 | url-status = live | archiveurl = https://web.archive.org/web/20140111155252/http://emtoolbox.nist.gov/Wavelength/Documentation.asp | archivedate = 2014-01-11 }}</ref> This means that it is also proportional to the pressure and inversely proportional to the temperature for [[ideal gas law|ideal gases]].

===Group index===
Sometimes, a "group velocity refractive index", usually called the ''group index'' is defined:{{citation needed|date=June 2015}}
:<math>n_\mathrm{g} = \frac{\mathrm{c}}{v_\mathrm{g}},</math>
where ''v''<sub>g</sub> is the [[group velocity]]. This value should not be confused with ''n'', which is always defined with respect to the [[phase velocity]]. When the [[dispersion (optics)|dispersion]] is small, the group velocity can be linked to the phase velocity by the relation<ref name=bornwolf>{{cite book | title=Principles of Optics | edition=7th expanded | last1=Born | first1=Max | authorlink1=Max Born | last2=Wolf | first2=Emil | authorlink2=Emil Wolf | url=https://books.google.com/books?id=oV80AAAAIAAJ&pg=PA22 | isbn=978-0-521-78449-8 | date=1999 | url-status=live | archiveurl=https://web.archive.org/web/20170222111359/https://books.google.com/books?id=oV80AAAAIAAJ&pg=PA22 | archivedate=2017-02-22 }}</ref>{{rp|22}}
:<math>v_\mathrm{g} = v - \lambda\frac{\mathrm{d}v}{\mathrm{d}\lambda},</math>
where ''λ'' is the wavelength in the medium. In this case the group index can thus be written in terms of the wavelength dependence of the refractive index as
:<math>n_\mathrm{g} = \frac{n}{1 + \frac{\lambda}{n}\frac{\mathrm{d}n}{\mathrm{d}\lambda}}.</math>

When the refractive index of a medium is known as a function of the vacuum wavelength (instead of the wavelength in the medium), the corresponding expressions for the group velocity and index are (for all values of dispersion) <ref>{{Cite journal | doi = 10.1016/0030-4018(90)90104-2 | title = Group refractive index measurement by Michelson interferometer
 | year = 1990 | journal = Optics Communications | pages = 109–112 | volume = 78 | last1 = Bor | first1 =  Z. | last2 =  Osvay | first2 =  K. | last3 =  Rácz | first3 =  B. | last4 =  Szabó | first4 =  G. |bibcode = 1990OptCo..78..109B
 | issue = 2 }}</ref>
:<math>v_\mathrm{g} = \mathrm{c}\!\left(n - \lambda_0 \frac{\mathrm{d}n}{\mathrm{d}\lambda_0}\right)^{-1}\!,</math>
:<math>n_\mathrm{g} = n - \lambda_0 \frac{\mathrm{d}n}{\mathrm{d}\lambda_0},</math>
where ''λ''<sub>0</sub> is the wavelength in vacuum.

===Momentum (Abraham–Minkowski controversy)===
{{Main|Abraham–Minkowski controversy}}

In 1908, [[Hermann Minkowski]] calculated the momentum ''p'' of a refracted ray as follows:<ref>{{cite journal|last=Minkowski|first=Hermann|year=1908|title=Die Grundgleichung für die elektromagnetischen Vorgänge in bewegten Körpern|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse|volume=1908|issue=1|pages=53–111|url=http://www.digizeitschriften.de/resolveppn/GDZPPN00250152X}}</ref>
:<math>p = \frac{nE}{\mathrm{c}},</math>
where ''E'' is the energy of the photon, c is the speed of light in vacuum and ''n'' is the refractive index of the medium. In 1909, [[Max Abraham]] proposed the following formula for this calculation:<ref>{{cite journal|first=Max|last=Abraham|title=Zur Elektrodynamik bewegter Körper |journal=[[Rendiconti del Circolo Matematico di Palermo]]|volume=28|year=1909|issue=1|title-link=:de:s:Zur Elektrodynamik bewegter Körper (Abraham)|url=https://zenodo.org/record/1428462/files/article.pdf}}</ref>
:<math>p = \frac{E}{n\mathrm{c}}.</math>

A 2010 study suggested that ''both'' equations are correct, with the Abraham version being the [[kinetic momentum]] and the Minkowski version being the [[canonical momentum]], and claims to explain the contradicting experimental results using this interpretation.<ref>{{cite journal |doi=10.1103/PhysRevLett.104.070401 |title=Resolution of the Abraham-Minkowski Dilemma |first=Stephen |last=Barnett |journal=Phys. Rev. Lett. |date=2010-02-07 |volume=104 |issue=7 |page=070401 |pmid=20366861|bibcode = 2010PhRvL.104g0401B |url=https://strathprints.strath.ac.uk/26871/5/AbMinPRL.pdf }}</ref>

===Other relations===
As shown in the [[Fizeau experiment]], when light is transmitted through a moving medium, its speed relative to an observer traveling with speed ''v'' in the same direction as the light is:
:<math>V = \frac{\mathrm{c}}{n} + \frac{v\left(1-\frac{1}{n^2}\right)}{1+\frac{v}{cn}}\approx \frac{\mathrm{c}}{n} + v\left(1-\frac{1}{n^2}\right) \ .</math>

The refractive index of a substance can be related to its [[polarizability]] with the [[Lorentz–Lorenz equation]] or to the [[molar refractivity|molar refractivities]] of its constituents by the [[Gladstone–Dale relation]].

===Refractivity===
In atmospheric applications, the ''refractivity'' is taken as ''N'' = ''n'' – 1.  Atmospheric refractivity is often expressed as either<ref>{{Citation | last = Young | first = A. T. | title = Refractivity of Air | year = 2011 | url = http://mintaka.sdsu.edu/GF/explain/atmos_refr/air_refr.html | accessdate = 31 July 2014 | url-status = live | archiveurl = https://web.archive.org/web/20150110053602/http://mintaka.sdsu.edu/GF/explain/atmos_refr/air_refr.html | archivedate = 10 January 2015 }}</ref> ''N'' = {{val|e=6}}(''n'' – 1)<ref>{{Citation | last = Barrell | first = H. | last2 = Sears | first2 = J. E.  | title = The Refraction and Dispersion of Air for the Visible Spectrum | journal = Philosophical Transactions of the Royal Society of London | series =  A, Mathematical and Physical Sciences | volume = 238 | issue = 786 | pages = 1–64 | year = 1939 | url = | jstor = 91351 | doi=10.1098/rsta.1939.0004|bibcode = 1939RSPTA.238....1B }}</ref><ref>{{cite journal |last=Aparicio |first=Josep M. |last2=Laroche |first2=Stéphane |date=2011-06-02 |title=An evaluation of the expression of the atmospheric refractivity for GPS signals |journal=Journal of Geophysical Research |volume=116 |issue=D11 |pages=D11104 |doi=10.1029/2010JD015214 |bibcode=2011JGRD..11611104A |df= }}</ref> or ''N'' = {{val|e=8}}(''n'' – 1)<ref>{{Citation | last = Ciddor | first = P. E. | title = Refractive Index of Air: New Equations for the Visible and Near Infrared | journal = Applied Optics | volume = 35 | issue = 9 | pages = 1566–1573 | year = 1996 | url = | jstor = | doi=10.1364/ao.35.001566| pmid = 21085275 |bibcode = 1996ApOpt..35.1566C }}</ref>  The multiplication factors are used because the refractive index for air, ''n'' deviates from unity by at most a few parts per ten thousand.

[[Molar refractivity]], on the other hand, is a measure of the total [[polarizability]] of a [[mole (unit)|mole]] of a substance and can be calculated from the refractive index as
:<math>A = \frac{M}{\rho} \frac{n^2 - 1}{n^2 + 2},</math>
where ''ρ'' is the [[density]], and ''M'' is the [[molar mass]].<ref name = bornwolf />{{rp|93}}

==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 ''n''<sub>o</sub> while light polarized in parallel will experience an ''extraordinary'' refractive index ''n''<sub>e</sub>.<ref name=Hecht/>{{rp|236}} The birefringence of the material is the difference between these indices of refraction, Δ''n'' = ''n''<sub>e</sub> − ''n''<sub>o</sub>.<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 ''n''<sub>o</sub> 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 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 (''n'') with radial distance (''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 position, the material is known as a gradient-index or 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 GRIN 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 GRIN 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]].

==Refractive index measurement==

===Homogeneous media===
{{Main|Refractometry|Refractometer}}
[[File:Pulfrich refraktometer en.png|thumb|alt=Illustration of a refractometer measuring the refraction angle of light passing from a sample into a prism along the interface|The principle of many refractometers]]

The refractive index of liquids or solids can be measured with [[refractometer]]s. They typically measure some angle of refraction or the critical angle for total internal reflection. The first [[Abbe refractometer|laboratory refractometers]] sold commercially were developed by [[Ernst Abbe]] in the late 19th century.<ref>{{cite web |url = http://www.humboldt.edu/scimus/Essays/EvolAbbeRef/EvolAbbeRef.htm |title = The Evolution of the Abbe Refractometer |publisher = Humboldt State University, Richard A. Paselk |year = 1998 |accessdate = 2011-09-03 |url-status = live |archiveurl = https://web.archive.org/web/20110612000645/http://www.humboldt.edu/scimus/Essays/EvolAbbeRef/EvolAbbeRef.htm |archivedate = 2011-06-12 }}</ref>
The same principles are still used today. In this instrument a thin layer of the liquid to be measured is placed between two prisms. Light is shone through the liquid at incidence angles all the way up to 90°, i.e., light rays [[parallel (geometry)|parallel]] to the surface. The second prism should have an index of refraction higher than that of the liquid, so that light only enters the prism at angles smaller than the critical angle for total reflection. This angle can then be measured either by looking through a [[telescope]],{{clarify|date=June 2017}} or with a digital [[photodetector]] placed in the focal plane of a lens. The refractive index ''n'' of the liquid can then be calculated from the maximum transmission angle ''θ'' as ''n'' = ''n''<sub>G</sub> sin ''θ'', where ''n''<sub>G</sub> is the refractive index of the prism.<ref>{{cite web | url = http://www.refractometer.pl/ | title = Refractometers and refractometry | publisher = Refractometer.pl | year = 2011 | accessdate = 2011-09-03 | url-status = live | archiveurl = https://web.archive.org/web/20111020123455/http://www.refractometer.pl/ | archivedate = 2011-10-20 }}</ref>

[[File:Refractometer.jpg|thumb|alt=A small cylindrical refractometer with a surface for the sample at one end and an eye piece to look into at the other end|A handheld refractometer used to measure sugar content of fruits]]

This type of devices are commonly used in [[chemistry|chemical]] laboratories for identification of [[chemical substance|substances]] and for [[quality control]]. [[Digital handheld refractometer|Handheld variants]] are used in [[agriculture]] by, e.g., [[wine maker]]s to determine [[Brix|sugar content]] in [[grape]] juice, and [[inline process refractometer]]s are used in, e.g., [[chemical industry|chemical]] and [[pharmaceutical industry]] for [[process control]].

In [[gemology]] a different type of refractometer is used to measure index of refraction and birefringence of [[gemstones]]. The gem is placed on a high refractive index prism and illuminated from below. A high refractive index contact liquid is used to achieve optical contact between the gem and the prism. At small incidence angles most of the light will be transmitted into the gem, but at high angles total internal reflection will occur in the prism. The critical angle is normally measured by looking through a telescope.<ref>{{cite web |url = http://gemologyproject.com/wiki/index.php?title=Refractometer |title = Refractometer |publisher = The Gemology Project |accessdate = 2011-09-03 |url-status = live |archiveurl = https://web.archive.org/web/20110910082406/http://www.gemologyproject.com/wiki/index.php?title=Refractometer |archivedate = 2011-09-10 }}</ref>

===Refractive index variations===
{{Main|Phase-contrast imaging}}
[[File:S cerevisiae under DIC microscopy.jpg|thumb|alt=Yeast cells with dark borders to the upper left and bright borders to lower right|A differential interference contrast microscopy image of yeast cells]]

Unstained biological structures appear mostly transparent under [[Bright-field microscopy]] as most cellular structures do not attenuate appreciable quantities of light. Nevertheless, the variation in the materials that constitutes these structures also corresponds to a variation in the refractive index.  The following techniques convert such variation into measurable amplitude differences:

To measure the spatial variation of refractive index in a sample [[phase-contrast imaging]] methods are used. These methods measure the variations in [[phase (waves)|phase]] of the light wave exiting the sample. The phase is proportional to the [[optical path length]] the light ray has traversed, and thus gives a measure of the [[integral]] of the refractive index along the ray path. The phase cannot be measured directly at optical or higher frequencies, and therefore needs to be converted into [[intensity (physics)|intensity]] by [[interference (optics)|interference]] with a reference beam. In the visual spectrum this is done using Zernike [[phase-contrast microscopy]], [[differential interference contrast microscopy]] (DIC) or [[interferometry]].

Zernike phase-contrast microscopy introduces a phase shift to the low [[spatial frequency]] components of the [[Real image|image]] with a phase-shifting [[annulus (geometry)|annulus]] in the [[Fourier optics|Fourier plane]] of the sample, so that high-spatial-frequency parts of the image can interfere with the low-frequency reference beam. In DIC the illumination is split up into two beams that are given different polarizations, are phase shifted differently, and are shifted transversely with slightly different amounts. After the specimen, the two parts are made to interfere, giving an image of the derivative of the optical path length in the direction of the difference in transverse shift.<ref name=Carlsson/> In interferometry the illumination is split up into two beams by a [[Beam splitter|partially reflective mirror]]. One of the beams is let through the sample before they are combined to interfere and give a direct image of the phase shifts. If the optical path length variations are more than a wavelength the image will contain fringes.

There exist several [[phase-contrast X-ray imaging]] techniques to determine 2D or 3D spatial distribution of refractive index of samples in the X-ray regime.<ref>{{Cite journal | first = Richard | last = Fitzgerald | title = Phase‐Sensitive X‐Ray Imaging | journal = Physics Today | volume = 53 | page = 23 | date = July 2000 | doi = 10.1063/1.1292471|bibcode = 2000PhT....53g..23F | issue = 7 }}</ref>

==Applications==
{{unreferenced section|date=September 2014}}
The refractive index is a very important property of the components of any [[optical instrument]]. It determines the focusing power of lenses, the dispersive power of prisms, the reflectivity of [[anti-reflective coating|lens coatings]], and the light-guiding nature of [[optical fiber]]. Since refractive index is a fundamental physical property of a substance, it is often used to identify a particular substance, confirm its purity, or measure its concentration. Refractive index is used to measure solids, liquids, and gases. Most commonly it is used to measure the concentration of a solute in an [[aqueous]] [[solution]]. It can also be used as a useful tool to differentiate between different types of gemstone, due to the unique [[Chatoyancy|chatoyance]] each individual stone displays. A [[refractometer]] is the instrument used to measure refractive index. For a solution of sugar, the refractive index can be used to determine the sugar content (see [[Brix]]).

==See also==
{{Div col}}
* [[Fermat's principle]]
* [[Calculation of glass properties]]
* [[Clausius–Mossotti relation]]
* [[Ellipsometry]]
* [[Index-matching material]]
* [[Index ellipsoid]]
* [[Laser Schlieren Deflectometry]]
* [[Optical properties of water and ice]]
* [[Prism-coupling refractometry]]
* [[Phase-contrast X-ray imaging]]
{{Div col end}}
{{clear}}

==References==
{{Reflist|35em}}

==External links==
{{Commons category|Refraction}}
* [http://emtoolbox.nist.gov/Wavelength/Documentation.asp NIST calculator for determining the refractive index of air]
* [http://www.tf.uni-kiel.de/matwis/amat/elmat_en/index.html Dielectric materials]
* [http://scienceworld.wolfram.com/physics/IndexofRefraction.html Science World]
* [http://www.filmetrics.com/refractive-index-database Filmetrics' online database] Free database of refractive index and absorption coefficient information
* [http://RefractiveIndex.INFO/ RefractiveIndex.INFO] Refractive index database featuring online plotting and parameterisation of data
* [http://www.sopra-sa.com/ sopra-sa.com] Refractive index database as text files (sign-up required)
* [http://luxpop.com/ LUXPOP] Thin film and bulk index of refraction and photonics calculations
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[[Category:Dimensionless numbers]]
[[Category:Physical quantities]]
[[Category:Refraction]]