View source for Refractive index

{{Short description|Property in optics}}
[[File:Refraction photo.png|thumb|A [[ray (optics)|ray]] of light being [[refraction|refracted]] through a glass slab|alt=refer to caption]]
[[File:Refraction at interface.svg|thumb|170px|Refraction of a light ray|alt=Illustration of the incidence and refraction angles]]

In [[optics]], the '''refractive index''' (or '''refraction index''') of an [[optical medium]] is the [[ratio]] of the apparent speed of light in the air or vacuum to the speed in the medium. 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, {{math|1=''n''<sub>1</sub> sin ''θ''<sub>1</sub> = ''n''<sub>2</sub> sin ''θ''<sub>2</sub>}}, where {{math|''θ''<sub>1</sub>}} and {{math|''θ''<sub>2</sub>}} are the [[angle of incidence (optics)|angle of incidence]] and angle of refraction, respectively, of a ray crossing the interface between two media with refractive indices {{math|''n''<sub>1</sub>}} and {{math|''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]], their intensity ([[Fresnel equations]]) 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, <math>n</math>, 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 {{math|1=''v'' = c/''n''}}, and similarly the wavelength in that medium is {{math|1=''λ'' = ''λ''<sub>0</sub>/''n''}}, where {{math|''λ''<sub>0</sub>}} is the wavelength of that light in vacuum. This implies that vacuum has a refractive index of 1, and assumes that the [[frequency]] ({{math|1=''f'' = ''v''/''λ''}}) of the wave is not affected by the refractive index.

The refractive index may vary with wavelength. This causes white light to split into constituent colors when refracted. This is called [[dispersion (optics)|dispersion]]. This effect can be observed in [[Prism (optics)|prisms]] and [[rainbow]]s, and as [[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|publisher=Cambridge University Press }}</ref> The [[Imaginary number|imaginary]] part then handles the [[attenuation]], while the [[Real number|real]] part accounts for refraction. For most materials the refractive index changes with wavelength by several percent across the visible spectrum. Consequently, refractive indices for materials reported using a single value for {{mvar|n}} must specify the wavelength used in the measurement.

The concept of refractive index applies across 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 | last = Kinsler | first = Lawrence E. | title = Fundamentals of Acoustics | url = https://archive.org/details/fundamentalsacou00kins_265 | url-access = limited | publisher = John Wiley | year = 2000 | isbn = 978-0-471-84789-2 | page = [https://archive.org/details/fundamentalsacou00kins_265/page/n151 136]}}</ref> Refraction also occurs in oceans when light passes into the [[halocline]] where salinity has impacted the density of the water column.

For [[lens]]es (such as [[eye glasses]]), a lens made from a high refractive index material will be thinner, and hence lighter, than a conventional lens with a lower refractive index.  Such lenses are generally more expensive to manufacture than conventional ones.

==Definition==
The '''relative refractive index''' of an optical medium 2 with respect to another reference medium 1 ({{math|''n''<sub>21</sub>}}) is given by the ratio of speed of light in medium 1 to that in medium 2. This can be expressed as follows:
<math display="block">n_{21}=\frac{v_1}{v_2}.</math>
If the reference medium 1 is [[vacuum]], then the refractive index of medium 2 is considered with respect to vacuum. It is simply represented as {{math|''n''<sub>2</sub>}} and is called the '''absolute refractive index''' of medium 2.

The '''absolute refractive index''' ''n'' of an optical medium is defined as the ratio of the [[speed of light]] in vacuum, {{math|1=c = {{val|299792458|u=m/s}}}}, and the [[phase velocity]] {{mvar|v}} of light in the medium,
<math display="block">n=\frac{\mathrm{c}}{v}.</math>
Since {{math|c}} is constant, {{mvar|n}} is inversely proportional to {{mvar|v}}:
<math display="block">n\propto\frac{1}{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.<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'' in 1807.]]

[[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 |year= 1807 |title= A course of lectures on natural philosophy and the mechanical arts |publisher= J. Johnson |page= [https://archive.org/details/lecturescourseof01younrich/page/413 413] |url= https://archive.org/details/lecturescourseof01younrich}}</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 |date= 1730 |title= Opticks: Or, A Treatise of the Reflections, Refractions, Inflections and Colours of Light |publisher= William Innys at the West-End of St. Paul's |page= [https://archive.org/details/opticksoratreat00newtgoog/page/n273 247] |url= https://archive.org/details/opticksoratreat00newtgoog}}</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 |last= Hauksbee |first= Francis |date= 1710 |title= A Description of the Apparatus for Making Experiments on the Refractions of Fluids |page= 207 |journal= Philosophical Transactions of the Royal Society of London |volume= 27 |issue= 325–336 |s2cid= 186208526 |doi= 10.1098/rstl.1710.0015 }}</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 |year= 1795 |title= Philosophical and mathematical dictionary |page= 299 |url= https://books.google.com/books?id=lsdJAAAAMAAJ&pg=PA299 |url-status= live |archive-url= https://web.archive.org/web/20170222031446/https://books.google.com/books?id=lsdJAAAAMAAJ&pg=PA299 |archive-date= 2017-02-22}}</ref>

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

==Typical values==
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{{See also|List of refractive indices}}
[[File:Brillanten.jpg|left|thumb|alt=Gemstone diamonds|[[Diamond]]s have a very high refractive index of 2.417.]]

Refractive index also varies with wavelength of the light as given by [[Cauchy's equation]].  The most general form of this equation is
<math display="block"> n(\lambda) = A + \frac {B}{\lambda^2} + \frac{C}{\lambda^4} + \cdots,</math>
where {{mvar|n}} is the refractive index, {{mvar|λ}} is the wavelength, and {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, etc.,  are [[coefficient]]s that can be determined for a material by fitting the equation to measured refractive indices at known wavelengths. The coefficients are usually quoted for {{mvar|λ}} as the [[vacuum wavelength]] in [[micrometre]]s.

Usually, it is sufficient to use a two-term form of the equation:
<math display="block"> n(\lambda) = A + \frac{B}{\lambda^2},</math>
where the coefficients {{mvar|A}} and {{mvar|B}} are determined specifically for this form of the equation.

{| class="wikitable plainrowheaders" style="float:right;"
|+ Selected refractive indices at {{nowrap|1= {{mvar|λ}} = {{val|589|u=nm}}}}.
For references, see the extended [[List of refractive indices]].
! scope="col" | Material
! scope="col" | {{mvar|n}}
|-
! scope="row" | [[Vacuum]]
| {{val|1}}
|-
| colspan="2" style="text-align:center;"| [[Gas]]es at [[Standard temperature and pressure|0&nbsp;°C and 1&nbsp;atm]]
|-
! scope="row" | [[Air]]
| {{val|1.000293}}
|-
! scope="row" | [[Helium]]
| {{val|1.000036}}
|-
! scope="row" | [[Hydrogen]]
| {{val|1.000132}}
|-
! scope="row" | [[Carbon dioxide]]
| {{val|1.00045}}
|-
| colspan="2" style="text-align:center;"| [[Liquid]]s at 20&nbsp;°C
|-
! scope="row" | [[Water]]
| 1.333
|-
! scope="row" | [[Ethanol]]
| 1.36
|-
! scope="row" | [[Olive oil]]
| 1.47
|-
| colspan="2" style="text-align:center;"| [[Solid]]s
|-
! scope="row" | [[Ice]]
| 1.31
|-
! scope="row" | [[Fused silica]] (quartz)
| 1.46<ref>{{cite web |url= https://refractiveindex.info/?shelf=glass&book=fused_silica&page=Malitson |author= Malitson |date= 1965 |title= Refractive Index Database |website= refractiveindex.info |access-date= June 20, 2018}}</ref>
|-
! scope="row" | [[Poly(methyl methacrylate)|PMMA]] (acrylic, plexiglas, lucite, perspex)
| 1.49
|-
! scope="row" | [[Soda-lime glass|Window glass]]
| 1.52<ref>{{cite web |last1= Faick |first1= C. A. |last2= Finn |first2= A. N. |date= July 1931 |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 |access-date= 11 December 2016 |archive-url= https://web.archive.org/web/20161230105725/http://nvlpubs.nist.gov/nistpubs/jres/6/jresv6n6p993_A2b.pdf |archive-date= December 30, 2016 |language= en |url-status= live}}</ref>
|-
! scope="row" | [[Polycarbonate]] (Lexan™)
| 1.58<ref>{{cite journal |last1= Sultanova |first1= N. |last2= Kasarova |first2= S. |last3= Nikolov |first3= I. |date= October 2009 |title= Dispersion Properties of Optical Polymers |journal= Acta Physica Polonica A |volume= 116 |issue= 4 |pages= 585–587 |doi= 10.12693/APhysPolA.116.585 |bibcode= 2009AcPPA.116..585S |doi-access= free}}</ref>
|-
! scope="row" | [[Flint glass]] (typical)
| 1.69
|-
! scope="row" | [[Sapphire]]
| 1.77<ref>{{cite journal |last1= Tapping |first1= J. |last2= Reilly |first2= M. L. |date= 1 May 1986 |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 |volume= 3 |issue= 5 |page= 610 |doi= 10.1364/JOSAA.3.000610 |bibcode= 1986JOSAA...3..610T |url= https://zenodo.org/record/1235646}}</ref>
|-
! scope="row" | [[Cubic zirconia]]
| 2.15
|-
! scope="row" | [[Diamond]]
| 2.417
|-
! scope="row" | [[Moissanite]]
| 2.65
|}

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">{{cite web |title= Forensic Science Communications, Glass Refractive Index Determination |url= https://www.fbi.gov/about-us/lab/forensic-science-communications/fsc/jan2005/index.htm/standards/2005standards9.htm |url-status= dead |archive-url= https://web.archive.org/web/20140910195815/http://www.fbi.gov/about-us/lab/forensic-science-communications/fsc/jan2005/index.htm/standards/2005standards9.htm |archive-date= 2014-09-10 |access-date= 2014-09-08 |publisher= FBI Laboratory Services}}</ref> 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 conference |last1= Tabata |first1= M. |date= 2005 |conference= IEEE Nuclear Science Symposium Conference Record |title= Development of Silica Aerogel with Any Density |volume= 2 |pages= 816–818 |url= http://www.ppl.phys.chiba-u.jp/~makoto/publication/N14-191.pdf |display-authors= 1 |url-status= dead |archive-url= https://web.archive.org/web/20130518075319/http://www.ppl.phys.chiba-u.jp/~makoto/publication/N14-191.pdf |archive-date= 2013-05-18 |doi= 10.1109/NSSMIC.2005.1596380 |isbn= 978-0-7803-9221-2 |s2cid= 18187536 |last2= Tabata |first2= M. |last3= Adachi |first3= I. |last4= Fukushima |first4= T. |last5= Kawai |first5= H. |last6= Kishimoto |first6= H. |last7= Kuratani |first7= A. |last8= Nakayama |first8= H. |last9= Nishida |first9= S. |last10= Noguchi |first10= T. |last11= Okudaira |first11= K. |last12= Tajima |first12= Y. |last13= Yano |first13= H. |last14= Yokogawa |first14= H. |last15= Yoshida |first15= H. }}</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>{{cite web |last1= Sadayori |first1= Naoki |last2= Hotta |first2= Yuji |date= 2004 |title= Polycarbodiimide having high index of refraction and production method thereof |url= https://patents.google.com/patent/US20040158021 |id= US patent 2004/0158021 A1 |publisher= US Patent Office |via= Google Patents}}</ref>

For [[infrared]] light refractive indices can be considerably higher. [[Germanium]] is transparent in the wavelength region from {{val|2|to|14|u=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 termed "[[topological insulators]]", was recently found which have high refractive index of up to 6 in the near to mid infrared frequency range. Moreover, topological insulators are transparent when they have nanoscale thickness. These properties are potentially important for applications in infrared optics.<ref>{{Cite journal|last1=Yue|first1=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}}</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 |last1=Als-Nielsen |first1=J. |last2=McMorrow |first2=D. |date=2011 |title=Elements of Modern X-ray Physics |publisher=Wiley-VCH |isbn=978-0-470-97395-0 |url=https://archive.org/details/elementsmodernxr00alsn |url-access=limited |page=[https://archive.org/details/elementsmodernxr00alsn/page/n38 25]}}</ref>{{efn|
One consequence of the real part of {{mvar|n}} being less than unity is that it implies that the phase velocity inside the material, {{sfrac|{{math|c}}|{{mvar|n}}}}, is larger than the velocity of light, {{math|c}}. This does not, however, violate the law of relativity, which requires that only signals carrying information do not travel faster than {{math|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 {{math|c}}.<ref name=Als-Nielsen2011/>}} 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 {{nobr|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 {{nobr|close to 1}} (exceptions close to some resonance frequencies).<ref name=CXRO>{{cite web |last= Gullikson |first= Eric |title= X-Ray interactions with matter |department=Optical constants |website= The Center for X-Ray Optics |publisher= [[Lawrence Berkeley Laboratory]] |url = http://henke.lbl.gov/optical_constants/ |access-date= 2011-08-30 |url-status= live |archive-url= https://web.archive.org/web/20110827214322/http://henke.lbl.gov/optical_constants/ |archive-date= 2011-08-27 }}</ref>
As an example, water has a refractive index of {{nobr|{{val|0.99999974}} {{=}}}} {{nobr|1 − {{val|2.6|e=-7}}}} for X-ray radiation at a photon energy of {{val|30|ul=keV}} ({{val|0.04|u=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 |date=1967 |title=High Frequency Radio Communications with Emphasis on Polar Problems |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. |author-link= Victor Veselago |title= The electrodynamics of substances with simultaneously negative values of ε and μ |journal= [[Physics-Uspekhi|Soviet Physics Uspekhi]] |date= 1968 |volume= 10 |issue= 4 |pages= 509–514 |doi= 10.1070/PU2003v046n07ABEH001614 |bibcode= 1968SvPhU..10..509V |s2cid= 250862458 }}</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 new phenomena to be actively developed by means of [[negative index metamaterials|metamaterials]].<ref>{{cite web |last1= Pendry |first1= J. B. |last2= Schurig |first2= D. |last3= Smith |first3= D. R. |title= Electromagnetic compression apparatus, methods and systems |url= https://patents.google.com/patent/US7629941 |id= US Patent 7629941 |date= Dec 8, 2009 |publisher= US Patent Office |via= Google Patents}}</ref><ref name=shalaev2007>{{cite journal |last= Shalaev |first= V. M. |author-link= Vladimir Shalaev |title= Optical negative-index metamaterials |journal= [[Nature Photonics]] |volume= 1 |issue= 1 |pages= 41–48 |date= 2007 |doi= 10.1038/nphoton.2006.49 |bibcode= 2007NaPho...1...41S |s2cid= 170678 }}</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 indices 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 |last= Feynman |first= Richard P. |title= Mainly Mechanics, Radiation, and Heat |series= Feynman Lectures on Physics |volume= 1 |edition= The New Millenium |publisher= Basic Books |date= 2011 |isbn= 978-0-465-02493-3}}</ref>{{page needed|date= August 2023}}
* 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]] {{math|c}} (note that the [[signal velocity]] is still less than {{math|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"/>{{page needed|date= August 2023}}
* 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.svg|thumb|left|alt=A white beam of light dispersed into different colors when passing through a triangular prism|In a triangular [[Prism (optics)|prism]], [[Dispersion (optics)|dispersion]] causes different colors to refract at different angles, splitting [[White|white light]] into a [[rainbow]] of colors. The blue color is more deviated (refracted) than the red color because the refractive index of blue is higher than that of red. ]]
[[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>{{cite encyclopedia |last= Paschotta |first= Rüdiger |title= Chromatic Dispersion |url=https://www.rp-photonics.com/chromatic_dispersion.html |archive-url= https://web.archive.org/web/20150629012047/http://www.rp-photonics.com/chromatic_dispersion.html |archive-date= 2015-06-29 |url-status= live |encyclopedia= [[RP Photonics Encyclopedia]] |access-date= 2023-08-13}}</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>{{cite web |last= Nave |first= Carl R. |date= 2000 |title= Dispersion |url= http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/dispersion.html |archive-url= https://web.archive.org/web/20140924222742/http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/dispersion.html |archive-date= 2014-09-24 |website= [[HyperPhysics]] |publisher= Department of Physics and Astronomy, Georgia State University |url-status= live |access-date= 2023-08-13}}</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 {{em|decrease}} with increasing wavelength, and thus {{em|increase}} with frequency. This is called "normal dispersion", in contrast to "anomalous dispersion", where the refractive index {{em|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 display="block">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>{{cite encyclopedia |last= Paschotta |first= Rüdiger |title= Sellmeier formula |url= https://www.rp-photonics.com/sellmeier_formula.html |archive-url= https://web.archive.org/web/20150319205203/http://www.rp-photonics.com/sellmeier_formula.html |archive-date= 2015-03-19 |url-status= live |encyclopedia= [[RP Photonics Encyclopedia]] |access-date= 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.

===Principal refractive index wavelength ambiguity===
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.

Manufacturers of optical glass in general define principal index of refraction at yellow spectral line of helium ({{val|587.56|u=nm}}) and alternatively at a green spectral line of mercury ({{val|546.07|u=nm}}), called {{mvar|d}} and {{mvar|e}} lines respectively. [[Abbe number]] is defined for both and denoted {{mvar|V<sub>d</sub>}} and {{mvar|V<sub>e</sub>}}. The spectral data provided by glass manufacturers is also often more precise for these two wavelengths.<ref>{{cite web |author= Schott Company |date= <!-- undated --> |title= Interactive Abbe Diagram |url= https://www.schott.com/en-pl/interactive-abbe-diagram |access-date= 2023-08-13 |website= Schott.com}}</ref><ref>{{cite web |author= Ohara Corporation |date= <!-- undated --> |title= Optical Properties |url= https://www.oharacorp.com/o2.html |access-date= 2022-08-15 |website= Oharacorp.com }}</ref><ref>{{cite web |author= Hoya Group |date= <!-- undated --> |title= Optical Properties |url= https://www.hoya-opticalworld.com/english/technical/002.html |access-date= 2023-08-13 |website=Hoya Group Optics Division}}</ref><ref>{{cite book |last1= Lentes |first1= Frank-Thomas |last2= Clement |first2= Marc K. Th. |last3= Neuroth |first3= Norbert |last4= Hoffmann |first4= Hans-Jürgen |last5= Hayden |first5= Yuiko T. |last6= Hayden |first6= Joseph S. |last7= Kolberg |first7= Uwe |last8= Wolff |first8= Silke |editor-last1= Bach |editor-first1= Hans |editor-last2= Neuroth |editor-first2= Norbert |date= 1998 |title=The Properties of Optical Glass |chapter= Optical Properties |series=Schott Series on Glass and Glass Ceramics |page= 30 |language=en |doi= 10.1007/978-3-642-57769-7 |isbn= 978-3-642-63349-2 }}</ref>

Both, {{mvar|d}} and {{mvar|e}} spectral lines are singlets and thus are suitable to perform a very precise measurements, such as spectral goniometric method.<ref>{{cite conference |last1= Krey |first1= Stefan |last2= Off |first2= Dennis |last3= Ruprecht |first3= Aiko |editor-last1= Soskind |editor-first1= Yakov G. |editor-last2= Olson |editor-first2= Craig |date= 2014-03-08 |title= Measuring the Refractive Index with Precision Goniometers: A Comparative Study |url= https://www.spiedigitallibrary.org/conference-proceedings-of-spie/8992/89920D/Measuring-the-refractive-index-with-precision-goniometers--a-comparative/10.1117/12.2041760.full |conference=  SPIE OPTO, 2014 |location= San Francisco, California |book-title= Proc. SPIE 8992, Photonic Instrumentation Engineering |publisher= SPIE |volume= 8992 |pages= 56–65 |doi= 10.1117/12.2041760 |bibcode= 2014SPIE.8992E..0DK |s2cid= 120544352 }}</ref><ref>{{Cite book |last1=Rupp |first1=Fabian |last2=Jedamzik |first2=Ralf |last3=Bartelmess |first3=Lothar |last4=Petzold |first4=Uwe |title=Optical Fabrication, Testing, and Metrology VII |chapter=The modern way of refractive index measurement of optical glass at SCHOTT |journal=Optical Fabrication |editor-first1=Reinhard |editor-first2=Roland |editor-first3=Deitze |editor-last1=Völkel |editor-last2=Geyl |editor-last3=Otaduy |date=2021-09-12 |chapter-url=https://www.spiedigitallibrary.org/conference-proceedings-of-spie/11873/1187308/The-modern-way-of-refractive-index-measurement-of-optical-glass/10.1117/12.2597023.full |publisher=SPIE |volume=11873 |pages=15–22 |doi=10.1117/12.2597023|bibcode=2021SPIE11873E..08R |isbn=9781510645905 |s2cid=240561530 }}</ref>

In practical applications, measurements of refractive index are performed on various refractometers, such as [[Abbe refractometer]]. Measurement accuracy of such typical commercial devices is in the order of 0.0002.<ref>{{Cite web |title=Abbe Refractometer{{!}} ATAGO CO., LTD. |url=https://www.atago.net/en/products-abbe-top.php |access-date=2022-08-15 |website=www.atago.net}}</ref><ref>{{Cite web |title=Abbe Multi-Wavelength Refractometer |url=https://www.novatech-usa.com/1412-DR-M2-1550_2 |access-date=2022-08-15 |website=Nova-Tech International |language=en-US}}</ref> Refractometers usually measure refractive index {{mvar|n<sub>D</sub>}}, defined for sodium doublet {{mvar|D}} ({{val|589.29|u=nm}}), which is actually a midpoint between two adjacent yellow spectral lines of sodium. Yellow spectral lines of helium ({{mvar|d}}) and sodium ({{mvar|D}}) are {{val|1.73|u=nm}} apart, which can be considered negligible for typical refractometers, but can cause confusion and lead to errors if accuracy is critical.

All three typical principle refractive indices definitions can be found depending on application and region,<ref>{{Cite book |url=https://link.springer.com/book/10.1007/978-3-642-57769-7 |title=The Properties of Optical Glass |series=Schott Series on Glass and Glass Ceramics |year=1998 |pages=267 |language=en |doi=10.1007/978-3-642-57769-7|isbn=978-3-642-63349-2 |editor1=Bach, Hans |editor2=Neuroth, Norbert }}</ref> so a proper subscript should be used to avoid ambiguity.

==Complex refractive index==
{{See also|Mathematical descriptions of opacity}}

When light passes through a medium, some part of it will always be [[attenuation|absorbed]]. This can be conveniently taken into account by defining a complex refractive index,
<math display="block">\underline{n} = n - i\kappa.</math>

Here, the real part {{mvar|n}} is the refractive index and indicates the [[phase velocity]], while the imaginary part {{mvar|κ}} is called the '''extinction coefficient'''<ref name=DresselhausMITCourse>{{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       = Mildred S.
 |author-link = Mildred Dresselhaus
 |date        = 1999
 |website     = Course 6.732 Solid State Physics
 |publisher   = MIT
 |access-date = 2015-01-05
 |url-status   = live
 |archive-url  = https://web.archive.org/web/20150724051216/http://web.mit.edu/course/6/6.732/www/6.732-pt2.pdf
 |archive-date = 2015-07-24
}}</ref>{{rp|36}} indicates the amount of attenuation when the electromagnetic wave propagates through the material.<ref name="Hecht"/>{{rp|p=128}} It is related to the <!-- do not wikilink, the article by that name covers thermal absorption-->'''absorption coefficient''', <math>\alpha_\text{abs}</math>, through:<ref name=DresselhausMITCourse/>{{rp|41}}
<math display="block">\alpha_\text{abs}(\omega) = \frac{2\omega\kappa(\omega)}{c}</math>
These values depend upon the frequency of the light used in the measurement.

That {{mvar|κ}} corresponds to absorption can be seen by inserting this refractive index into the expression for [[electric field]] of a [[plane wave|plane]] electromagnetic wave traveling in the {{mvar|x}}-direction. This can be done by relating the complex [[wave number]] {{mvar|{{uu|k}}}} to the complex refractive index {{mvar|{{uu|n}}}} through {{math|{{uu|''k''}} {{=}} 2π{{uu|''n''}}/''λ''{{sub|0}}}}, with {{math|''λ''{{sub|0}}}} being the vacuum wavelength; this can be inserted into the plane wave expression for a wave travelling in the {{mvar|x}}-direction as:
<math display="block">\begin{align}
\mathbf{E}(x, t)
  &= \operatorname{Re}\! \left[\mathbf{E}_0 e^{i(\underline{k}x - \omega t)}\right] \\
  &= \operatorname{Re}\! \left[\mathbf{E}_0 e^{i(2\pi(n + i\kappa)x/\lambda_0 - \omega t)}\right] \\
  &= e^{-2\pi \kappa x/\lambda_0} \operatorname{Re}\! \left[\mathbf{E}_0 e^{i(kx - \omega t)}\right].
\end{align}</math>

Here we see that {{mvar|κ}} gives an exponential decay, as expected from the [[Beer–Lambert law]]. Since intensity is proportional to the square of the electric field, intensity will depend on the depth into the material as

<math display="block">I(x)= I_0 e^{-4\pi \kappa x/\lambda_0} .</math>

and thus the [[Attenuation coefficient#Absorption and scattering coefficients|absorption coefficient]] is {{math|''α'' {{=}} 4π''κ''/''λ''{{sub|0}}}},<ref name="Hecht"/>{{rp|p=128}} and the [[penetration depth]] (the distance after which the intensity is reduced by a factor of {{math|1/''e''}}) is {{math|''δ''{{sub|p}} {{=}} 1/''α'' {{=}} ''λ''{{sub|0}}/4π''κ''}}.

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

An alternative convention uses {{math|{{uu|''n''}} {{=}} ''n'' + ''iκ''}} instead of {{math|{{uu|''n''}} {{=}} ''n'' − ''iκ''}}, but where {{math|''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 {{math|Re[exp(−''iωt'')]}} versus {{math|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 {{mvar|n}}, and imaginary {{mvar|κ}}, parts of the complex refractive index are related through the [[Kramers–Kronig relations]]. In 1986, A.R. Forouhi and I. Bloomer deduced an [[Forouhi–Bloomer model|equation]] describing {{mvar|κ}} as a function of photon energy, {{mvar|E}}, applicable to amorphous materials. Forouhi and Bloomer then applied the Kramers–Kronig relation to derive the corresponding equation for [[Forouhi–Bloomer model|{{mvar|n}} as a function of {{mvar|E}}]]. The same formalism was applied to crystalline materials by Forouhi and Bloomer in 1988.

The refractive index and extinction coefficient, {{mvar|n}} and {{mvar|κ}}, are typically measured from quantities that depend on them, such as [[Fresnel equations|reflectance, {{mvar|R}}, or transmittance, {{mvar|T}}]], or ellipsometric parameters, [[ellipsometry|{{mvar|ψ}} and {{mvar|δ}}]]. The determination of {{mvar|n}} and {{mvar|κ}} from such measured quantities will involve developing a theoretical expression for {{mvar|R}} or {{mvar|T}}, or {{mvar|ψ}} and {{mvar|δ}} in terms of a valid physical model for {{mvar|n}} and {{mvar|κ}}. By fitting the theoretical model to the measured {{mvar|R}} or {{mvar|T}}, or {{mvar|ψ}} and {{mvar|δ}} using regression analysis, {{mvar|n}} and {{mvar|κ}} can be deduced.

===X-ray and extreme UV===
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 {{math|{{uu|''n''}} {{=}} 1 − ''δ'' + ''iβ''}} (or {{math|{{uu|''n''}} {{=}} 1 − ''δ'' − ''iβ''}} with the alternative convention mentioned above).<ref name=Attwood/> Far above the atomic resonance frequency delta can be given by
<math display="block"> \delta = \frac{r_0 \lambda^2 n_\mathrm{e}}{2 \pi} </math>
where {{math|''r''{{sub|0}}}} is the [[classical electron radius]], {{mvar|λ}} is the X-ray wavelength, and {{math|''n''{{sub|e}}}} is the electron density. One may assume the electron density is simply the number of electrons per atom {{mvar|Z}} multiplied by the atomic density, but more accurate calculation of the refractive index requires replacing {{mvar|Z}} with the complex [[atomic form factor]] {{nowrap|<math> f = Z + f' + i f'' </math>.}} It follows that
<math display="block">\begin{align}
\delta &= \frac{r_0 \lambda^2}{2 \pi} (Z + f') n_\text{atom} \\
\beta  &= \frac{r_0 \lambda^2}{2 \pi} f'' n_\text{atom}
\end{align}</math>
with {{mvar|δ}} and {{mvar|β}} typically of the order of {{val|e=−5}} and {{val|e=−6}}.

==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 {{mvar|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 display="text">\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 {{math|''n''{{sub|1}}}} to one with refractive index {{math|''n''{{sub|2}}}}, with an [[angle of incidence (optics)|incidence angle]] to the [[surface normal]] of {{math|''θ''{{sub|1}}}}, the refraction angle {{math|''θ''{{sub|2}}}} 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 display="block">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 {{math|''θ''{{sub|2}}}} fulfilling Snell's law, i.e.,
<math display="block">\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 {{math|''θ''{{sub|1}}}} 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|encyclopedia=RP Photonics Encyclopedia |access-date=2015-08-16 |url-status=live |archive-url=https://web.archive.org/web/20150628175307/https://www.rp-photonics.com/total_internal_reflection.html |archive-date=2015-06-28 }}</ref>
<math display="block">\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 display="block">R_0 = \left|\frac{n_1 - n_2}{n_1 + n_2}\right|^2\!.</math>

For common glass in air, {{math|''n''{{sub|1}} {{=}} 1}} and {{math|''n''{{sub|2}} {{=}} 1.5}}, and thus about 4% of the incident power is reflected.<ref name=ri-min>{{cite web |last=Swenson |first=Jim |quote=Incorporates Public Domain material from the [[U.S. Department of Energy]] |title=Refractive Index of Minerals |series=Newton BBS / Argonne National Laboratory |publisher=US DOE |date=November 10, 2009 <!--[http://www.newton.dep.anl.gov/]--> |url=http://www.newton.dep.anl.gov/askasci/env99/env234.htm|access-date=2010-07-28 |url-status=live |archive-url=https://web.archive.org/web/20100528092315/http://www.newton.dep.anl.gov/askasci/env99/env234.htm |archive-date=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 display="block"> \theta_\mathsf{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 {{mvar|n}} and the [[Radius of curvature (optics)|radii of curvature]] {{math|''R''{{sub|1}}}} and {{math|''R''{{sub|2}}}} of its surfaces. The power of a [[thin lens]] in air is given by the simplified version of the [[Lensmaker's formula]]:<ref>{{cite web |first=Carl R. |last=Nave |title=Lens-makers' formula |website=HyperPhysics |series=Department of Physics and Astronomy |publisher=Georgia State University |url=http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenmak.html |access-date=2014-09-08 |archive-url=https://web.archive.org/web/20140926153405/http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenmak.html |archive-date=2014-09-26}}</ref>
<math display="block">\frac{1}{f} = (n - 1)\left[\frac{1}{R_1} - \frac{1}{R_2}\right]\ ,</math>
where {{mvar|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]] ({{math|''A''{{sub|Num}}}}) of its [[objective lens]]. The numerical aperture in turn is determined by the refractive index {{mvar|n}} of the medium filling the space between the sample and the lens and the half collection angle of light {{mvar|θ}} according to Carlsson (2007):<ref name=Carlsson>{{cite report |first = Kjell |last = Carlsson |year = 2007 |title = Light microscopy |url = https://www.kth.se/social/files/542d1251f276544bf2492088/Compendium.Light.Microscopy.pdf |access-date = 2015-01-02 |url-status = live |archive-url = https://web.archive.org/web/20150402122840/https://www.kth.se/social/files/542d1251f276544bf2492088/Compendium.Light.Microscopy.pdf |archive-date = 2015-04-02 }}</ref>{{rp|6}}
<math display="block"> A_\mathrm{Num} = 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 display="block">n = \sqrt{\varepsilon_\mathrm{r} \mu_\mathrm{r}},</math>
where {{math|''ε''{{sub|r}}}} is the material's [[relative permittivity]], and {{math|''μ''{{sub|r}}}} is its [[Permeability (electromagnetism)|relative permeability]].<ref name = bleaney>{{cite book | last1 = Bleaney| first1 = B.| author-link1 = 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 {{math|''μ''{{sub|r}}}} is very close to 1, therefore {{mvar|n}} is approximately {{math|{{sqrt|''ε''{{sub|r}}}}}}.<ref>{{cite book |last1=Andrews |first1=David L. |title=Photonics, Volume 2: Nanophotonic Structures and Materials |date=24 February 2015 |publisher=John Wiley & Sons |isbn=978-1-118-22551-6 |page=54 |url=https://books.google.com/books?id=EYRVBgAAQBAJ |language=en}}</ref> In this particular case, the complex relative permittivity {{math|{{uu|''ε''}}{{sub|r}}}}, with real and imaginary parts {{math|''ε''{{sub|r}}}} and {{math|''ɛ̃''{{sub|r}}}}, and the complex refractive index {{math|{{uu|''n''}}}}, with real and imaginary parts {{mvar|n}} and {{mvar|κ}} (the latter called the "extinction coefficient"), follow the relation
<math display="block">\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 display="block">\begin{align}
\varepsilon_\mathrm{r} &= n^2 - \kappa^2\,, \\
\tilde{\varepsilon}_\mathrm{r} &= 2n\kappa\,,
\end{align}</math>

and:

<math display="block">\begin{align}
n &= \sqrt{\frac{|\underline{\varepsilon}_\mathrm{r}| + \varepsilon_\mathrm{r}}{2}}, \\
\kappa &= \sqrt{\frac{|\underline{\varepsilon}_\mathrm{r}| - \varepsilon_\mathrm{r}}{2}}.
\end{align}</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 display="block">\begin{align}
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}
\end{align}</math>

where {{math|''Z''{{sub|0}}}} is the vacuum wave impedance, {{mvar|μ}} and {{mvar|ε}} are the absolute permeability and permittivity of the medium, {{math|''ε''{{sub|r}}}} is the material's [[relative permittivity]], and {{math|''μ''{{sub|r}}}} is its [[Permeability (electromagnetism)|relative permeability]].

In non-magnetic media (that is, in materials with {{math|''μ''{{sub|r}} {{=}} 1}}), <math>Z = {Z_0 \over n}</math> and <math>n = {Z_0 \over 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''{{sub|0}}}} between two media can thus be expressed both by the wave impedances and the refractive indices as
<math display="block">\begin{align}
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\,.
\end{align}</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|The 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|archive-url=https://web.archive.org/web/20071015124852/http://glassproperties.com/refractive_index/|archive-date=2007-10-15}}</ref>]]
In general, it is assumed that the refractive index of a glass increases with its [[density]]. However, there does not exist an overall linear relationship 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|{{chem2|Li2O}}]] and [[magnesium oxide|{{chem2|MgO}}]], while the opposite trend is observed with glasses containing [[lead(II) oxide|{{chem2|PbO}}]] and [[barium oxide|{{chem2|BaO}}]] as seen in the diagram at the right.

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

For air, {{math|''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 | access-date = 2014-01-11 | url-status = live | archive-url = https://web.archive.org/web/20140111155252/http://emtoolbox.nist.gov/Wavelength/Documentation.asp | archive-date = 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]]. For liquids the same observation can be made as for gases, for instance, the refractive index in alkanes increases nearly perfectly linear with the density. On the other hand, for carboxylic acids, the density decreases with increasing number of C-atoms within the homologeous series. The simple explanation of this finding is that it is not density, but the molar concentration of the chromophore that counts. In homologeous series, this is the excitation of the C-H-bonding. August Beer must have intuitively known that when he gave Hans H. Landolt in 1862 the tip to investigate the refractive index of compounds of homologeous series.<ref>{{Cite journal |last=Landolt |first=H. |date=January 1862 |title=Ueber die Brechungsexponenten flüssiger homologer Verbindungen |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.18621931102 |journal=Annalen der Physik |language=en |volume=193 |issue=11 |pages=353–385 |doi=10.1002/andp.18621931102 |bibcode=1862AnP...193..353L |issn=0003-3804}}</ref> While Landolt did not find this relationship, since, at this time dispersion theory was in its infancy, he had the idea of molar refractivity which can even be assigned to single atoms.<ref>{{Cite journal |last=Landolt |first=H. |date=January 1864 |title=Ueber den Einfluss der atomistischen Zusammensetzung C, H und O-haltiger flüssiger Verbindungen auf die Fortpflanzung des Lichtes |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.18641991206 |journal=Annalen der Physik |language=en |volume=199 |issue=12 |pages=595–628 |doi=10.1002/andp.18641991206 |bibcode=1864AnP...199..595L |issn=0003-3804}}</ref> Based on this concept, the refractive indices of organic materials can be calculated.

=== Bandgap ===
[[File:Annotated Eg vs n.png|thumb|A scatter plot of bandgap energy versus optical refractive index for many common IV, III-V, and II-VI semiconducting elements / compounds. ]]
The optical refractive index of a semiconductor tends to increase as the [[Band gap|bandgap energy]] decreases. Many attempts<ref>{{Cite journal |last1=Gomaa |first1=Hosam M. |last2=Yahia |first2=I. S. |last3=Zahran |first3=H. Y. |date=2021-11-01 |title=Correlation between the static refractive index and the optical bandgap: Review and new empirical approach |url=https://www.sciencedirect.com/science/article/abs/pii/S0921452621004208 |journal=Physica B: Condensed Matter |volume=620 |pages=413246 |doi=10.1016/j.physb.2021.413246 |issn=0921-4526}}</ref> have been made to model this relationship beginning with T. S. Moses in 1949.<ref>{{Cite journal |last=Moss |first=T S |date=1950-03-01 |title=A Relationship between the Refractive Index and the Infra-Red Threshold of Sensitivity for Photoconductors |url= |journal=Proceedings of the Physical Society. Section B |volume=63 |issue=3 |pages=167–176 |doi=10.1088/0370-1301/63/3/302 |issn=0370-1301}}</ref> Empirical models can match experimental data over a wide range of materials and yet fail for important cases like InSb, PbS, and Ge.<ref>{{Cite book |last=Moss |first=T. S. |title=October 1 |chapter-url=https://www.degruyter.com/document/doi/10.1515/9783112495384-003/html |chapter=Relations between the Refractive Index and Energy Gap oi Semiconductors |date=1985-12-31 |publisher=De Gruyter |isbn=978-3-11-249538-4 |pages=415–428 |doi=10.1515/9783112495384-003}}</ref>  

This negative correlation between refractive index and bandgap energy, along with a negative correlation between bandgap and temperature, means that many semiconductors exhibit a positive correlation between refractive index and temperature.<ref>{{Cite journal |last1=Bertolotti |first1=Mario |last2=Bogdanov |first2=Victor |last3=Ferrari |first3=Aldo |last4=Jascow |first4=Andrei |last5=Nazorova |first5=Natalia |last6=Pikhtin |first6=Alexander |last7=Schirone |first7=Luigi |date=1990-06-01 |title=Temperature dependence of the refractive index in semiconductors |url=https://opg.optica.org/josab/abstract.cfm?uri=josab-7-6-918 |journal=JOSA B |language=EN |volume=7 |issue=6 |pages=918–922 |doi=10.1364/JOSAB.7.000918 |issn=1520-8540}}</ref> This is the opposite of most materials, where the refractive index decreases with temperature as a result of a decreasing material density.

===Group index===
{{Redirect distinguish|Group index|Index of a subgroup}}
Sometimes, a "group velocity refractive index", usually called the ''group index'' is defined:{{citation needed|date=June 2015}}
<math display="block">n_\mathrm{g} = \frac{\mathrm{c}}{v_\mathrm{g}},</math>
where {{math|''v''{{sub|g}}}} is the [[group velocity]]. This value should not be confused with {{mvar|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]] | publisher=CUP Archive | edition=7th expanded | last1=Born | first1=Max | author-link1=Max Born | last2=Wolf | first2=Emil | author-link2=Emil Wolf | page=[https://archive.org/details/principlesofopti0006born/page/22 22] | isbn=978-0-521-78449-8 | date=1999 }}</ref>{{rp|22}}
<math display="block">v_\mathrm{g} = v - \lambda\frac{\mathrm{d}v}{\mathrm{d}\lambda},</math>
where {{mvar|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 display="block">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 |last1= Bor |first1= Z. |last2=  Osvay |first2= K. |last3= Rácz |first3= B. |last4= Szabó |first4= G. |date= 1990 |title= Group refractive index measurement by Michelson interferometer |journal= Optics Communications |pages= 109–112 |volume= 78 |doi= 10.1016/0030-4018(90)90104-2 |bibcode= 1990OptCo..78..109B |issue= 2 }}</ref>
<math display="block">\begin{align}
v_\mathrm{g} &= \mathrm{c}\!\left(n - \lambda_0 \frac{\mathrm{d}n}{\mathrm{d}\lambda_0}\right)^{-1}\!, \\
n_\mathrm{g} &= n - \lambda_0 \frac{\mathrm{d}n}{\mathrm{d}\lambda_0},
\end{align}</math>
where {{math|''λ''{{sub|0}}}} is the wavelength in vacuum.

===Velocity, momentum, and polarizability===
As shown in the [[Fizeau experiment]], when light is transmitted through a moving medium, its speed relative to an observer traveling with speed {{mvar|v}} in the same direction as the light is:
<math display="block">\begin{align}
V &= \frac{\mathrm{c}}{n} + \frac{v \left(1 - \frac{1}{n^2} \right)}{1 + \frac{v}{c n}} \\
  &\approx \frac{\mathrm{c}}{n} + v \left(1 - \frac{1}{n^2} \right)\,.
\end{align}</math>

The momentum of photons in a medium of refractive index {{mvar|n}} is a complex and [[Abraham–Minkowski controversy|controversial]] issue with two different values having different physical interpretations.<ref>{{Cite journal |last1=Milonni |first1=Peter W. |last2=Boyd |first2=Robert W. |date=2010-12-31 |title=Momentum of Light in a Dielectric Medium |url=https://opg.optica.org/aop/abstract.cfm?uri=aop-2-4-519 |journal=Advances in Optics and Photonics |language=en |volume=2 |issue=4 |pages=519 |doi=10.1364/AOP.2.000519 |bibcode=2010AdOP....2..519M |issn=1943-8206}}</ref>

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, '''refractivity''' is defined as {{math| ''N'' {{=}} ''n'' – 1}}, often rescaled as either<ref>{{cite web | last = Young | first = A.T. | year = 2011 | title = Refractivity of Air | url = http://mintaka.sdsu.edu/GF/explain/atmos_refr/air_refr.html | access-date = 31 July 2014 | url-status = live | archive-url = https://web.archive.org/web/20150110053602/http://mintaka.sdsu.edu/GF/explain/atmos_refr/air_refr.html | archive-date = 10 January 2015 }}</ref> {{math|''N'' {{=}} {{10^|6}} (''n'' – 1)}}<ref>{{cite journal | last1 = Barrell | first1 = H. | last2 = Sears | first2 = J.E. | year = 1939 | 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 | jstor = 91351 | doi=10.1098/rsta.1939.0004 | doi-access = free |bibcode = 1939RSPTA.238....1B }}</ref><ref>{{cite journal |last1=Aparicio |first1=Josep M. |last2=Laroche |first2=Stéphane |date=2011-06-02 |df=dmy-all |title=An evaluation of the expression of the atmospheric refractivity for GPS signals |journal=[[Journal of Geophysical Research]] |volume=116 |issue=D11 |page=D11104 |doi=10.1029/2010JD015214 |doi-access=free |bibcode=2011JGRD..11611104A }}</ref> or {{math|''N'' {{=}} {{10^|8}} (''n'' – 1)}};<ref>{{cite journal | last = Ciddor | first = P.E. | year = 1996 | title = Refractive index of air: New equations for the visible and near infrared | journal = Applied Optics | volume = 35 | issue = 9 | pages = 1566–1573 | doi=10.1364/ao.35.001566 | pmid = 21085275 |bibcode = 1996ApOpt..35.1566C }}</ref> the multiplication factors are used because the refractive index for air, {{mvar|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 display="block">A = \frac{M}{\rho} \cdot \frac{n^2 - 1}{n^2 + 2}\ ,</math>
where {{mvar|ρ}} is the [[density]], and {{mvar|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 {{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]].

==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 |access-date = 2011-09-03 |url-status = live |archive-url = https://web.archive.org/web/20110612000645/http://www.humboldt.edu/scimus/Essays/EvolAbbeRef/EvolAbbeRef.htm |archive-date = 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 {{mvar|n}} of the liquid can then be calculated from the maximum transmission angle {{mvar|θ}} as {{math|1=''n'' = ''n''{{sub|G}} sin ''θ''}}, where {{math|''n''{{sub|G}}}} is the refractive index of the prism.<ref>{{cite web | url = http://www.refractometer.pl/ | title = Refractometers and refractometry | publisher = Refractometer.pl | year = 2011 | access-date = 2011-09-03 | url-status = live | archive-url = https://web.archive.org/web/20111020123455/http://www.refractometer.pl/ | archive-date = 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 the sugar content of fruits]]

This type of device is 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 the 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 |access-date = 2011-09-03 |url-status = live |archive-url = https://web.archive.org/web/20110910082406/http://www.gemologyproject.com/wiki/index.php?title=Refractometer |archive-date = 2011-09-10 }}</ref>

===Refractive index variations===
{{Main|Phase-contrast imaging}}
[[File:S cerevisiae under DIC microscopy.jpg|thumb|alt=Budding yeast cells with dark borders to the upper left and bright borders to lower right|A [[differential interference contrast microscopy]] image of [[budding 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 constitute 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 the 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 {{abbr|DIC|differential interference contrast microscopy}} 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 the 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 | s2cid = 121322301 | doi-access = free }}</ref>

==Applications==
{{unreferenced section|date=September 2014}}
The refractive index is an 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 the 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. The 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 the refractive index. For a solution of sugar, the refractive index can be used to determine the sugar content (see [[Brix]]).

==See also==
{{div col begin|colwidth=18em}}
* [[Calculation of glass properties]]
* [[Clausius–Mossotti relation]]
* [[Ellipsometry]]
* [[Fermat's principle]]
* [[Index ellipsoid]]
* [[Index-matching material]]
* [[Laser Schlieren deflectometry]]
* [[Optical properties of water and ice]]
* [[Phase-contrast X-ray imaging]]
* [[Prism-coupling refractometry]]
* [[Velocity factor]]
{{div col end}}
{{clear}}

==Footnotes==
{{notelist}}

==References==
{{reflist|25em}}

==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://luxpop.com/ LUXPOP] {{Webarchive|url=https://web.archive.org/web/20130907062841/http://www.luxpop.com/ |date=2013-09-07 }} Thin film and bulk index of refraction and photonics calculations
* [https://www.feynmanlectures.caltech.edu/II_32.html The Feynman Lectures on Physics Vol. II Ch. 32: Refractive Index of Dense Materials]
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[[Category:Dimensionless quantities]]
[[Category:Refraction]]
[[Category:Optical quantities]]