Refractive index#Momentum Paradox

The relative refractive index of an optical medium 2 with respect to another reference medium 1 ({{math|n₂₁}}) is given by the ratio of speed of light in medium 1 to that in medium 2. This can be expressed as follows:

$$n\_\{21\}=\backslash frac\{v\_1\}\{v\_2\}.$$

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|n₂}} 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,

$$n=\backslash frac\{\backslash mathrm\{c\}\}\{v\}.$$

Since {{math|c}} is constant, {{mvar|n}} is inversely proportional to {{mvar|v}}:

$$n\backslash propto\backslash frac\{1\}\{v\}.$$

The phase velocity is the speed at which the crests or the phase of the wave moves, which may be different from the group velocity, the speed at which the pulse of light or the envelope of the wave moves. Historically air at a standardized pressure and temperature has been common as a reference medium.

File:Thomas Young (scientist).jpg coined the term index of refraction in 1807.]]

Thomas Young was presumably the person who first used, and invented, the name "index of refraction", in 1807.{{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}}

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. 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).{{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}} Hauksbee, who called it the "ratio of refraction", wrote it as a ratio with a fixed numerator, like "10000 to 7451.9" (for urine).{{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 }} Hutton wrote it as a ratio with a fixed denominator, like 1.3358 to 1 (water).{{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}}

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}}.{{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{{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}}{{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 }} The symbol {{mvar|n}} gradually prevailed.

{{See also|List of refractive indices}}

File:Brillanten.jpgs 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

$$n(\backslash lambda)\; =\; A\; +\; \backslash frac\; \{B\}\{\backslash lambda^2\}\; +\; \backslash frac\{C\}\{\backslash lambda^4\}\; +\; \backslash cdots,$$

where {{mvar|n}} is the refractive index, {{mvar|λ}} is the wavelength, and {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, etc., are coefficients 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 micrometres.

Usually, it is sufficient to use a two-term form of the equation:

$$n(\backslash lambda)\; =\; A\; +\; \backslash frac\{B\}\{\backslash lambda^2\},$$

where the coefficients {{mvar|A}} and {{mvar|B}} are determined specifically for this form of the equation.

For visible light most 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 D-line of sodium, with a wavelength of 589 nanometers, as is conventionally done.{{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}} 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.{{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. }} 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 polymers can have values as high as 1.76.{{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}}

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.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 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.{{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}}

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.{{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]}}{{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}}.}} 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 frequencies, for absorbing media, in plasmas, and for X-rays. In the X-ray regime the refractive indices are lower than but very {{nobr|close to 1}} (exceptions close to some resonance frequencies).{{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 }}

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).

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), 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.{{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}}

{{See also|Negative index metamaterials}}

File:Split-ring resonator array 10K sq nm.jpg 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 permeability have simultaneous negative values.{{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= Soviet Physics Uspekhi |date= 1968 |volume= 10 |issue= 4 |pages= 509–514 |doi= 10.1070/PU2003v046n07ABEH001614 |bibcode= 1968SvPhU..10..509V |s2cid= 250862458 }} 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 metamaterials.{{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}}{{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 }}

File:Thin section scan crossed polarizers Siilinjärvi R636-105.90.jpg, thin sections are used to study rocks. The method is based on the distinct refractive indices of different minerals.]]

{{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 electrons) 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.{{rp|67}} The charges thus radiate their own electromagnetic wave that is at the same frequency, but usually with a phase delay, as the charges may move out of phase with the force driving them (see sinusoidally driven harmonic oscillator). The light wave traveling in the medium is the macroscopic 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.{{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}}{{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-rays 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 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.{{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 light absorption in opaque materials and corresponds to an 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 lasers 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.

File:WhereRainbowRises.jpg.]]

File:Prism-rainbow.svg, dispersion causes different colors to refract at different angles, splitting 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

{{Main|Dispersion (optics)}}

The refractive index of materials varies with the wavelength (and frequency) of light.{{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}} This is called dispersion and causes prisms and rainbows to divide white light into its constituent spectral colors.{{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}} 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 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. 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:

$$V\; =\; \backslash frac\{n\_\backslash mathrm\{yellow\}\; -\; 1\}\{n\_\backslash mathrm\{blue\}\; -\; n\_\backslash mathrm\{red\}\}.$$

For a more accurate description of the wavelength dependence of the refractive index, the Sellmeier equation can be used.{{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}} 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 lines.

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_d}} and {{mvar|V_e}}. The spectral data provided by glass manufacturers is also often more precise for these two wavelengths.{{cite web |author= Schott Company |date= |title= Interactive Abbe Diagram |url= https://www.schott.com/en-pl/interactive-abbe-diagram |access-date= 2023-08-13 |website= Schott.com}}{{cite web |author= Ohara Corporation |date= |title= Optical Properties |url= https://www.oharacorp.com/o2.html |access-date= 2022-08-15 |website= Oharacorp.com }}{{cite web |author= Hoya Group |date= |title= Optical Properties |url= https://www.hoya-opticalworld.com/english/technical/002.html |access-date= 2023-08-13 |website=Hoya Group Optics Division}}{{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 }}

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.{{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 |url-access= subscription }}{{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 }}

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.{{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}}{{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}} Refractometers usually measure refractive index {{mvar|n_D}}, 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,{{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 }} so a proper subscript should be used to avoid ambiguity.

{{See also|Mathematical descriptions of opacity}}

When light passes through a medium, some part of it will always be absorbed. This can be conveniently taken into account by defining a complex refractive index,

$$\backslash underline\{n\}\; =\; n\; -\; i\backslash kappa.$$

The real and imaginary part of this refractive index are not independent, and are connected through the Kramers–Kronig relations, i.e. the complex refractive index is a linear response function, ensuring causality. {{cite book |last= Stenzel|first=Olaf |date=2022 |title=Light–Matter Interaction |series=UNITEXT for Physics |url=https://link.springer.com/book/10.1007/978-3-030-87144-4 |location= |publisher=Springer, Cham |page=386 |doi=10.1007/978-3-030-87144-4 |isbn=978-3-030-87144-4 }} 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{{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

}}{{rp|36}} indicates the amount of attenuation when the electromagnetic wave propagates through the material.{{rp|p=128}} It is related to the absorption coefficient, $\backslash alpha\_\backslash text\{abs\}$, through:{{rp|41}}

$$\backslash alpha\_\backslash text\{abs\}(\backslash omega)\; =\; \backslash frac\{2\backslash omega\backslash kappa(\backslash omega)\}\{c\}$$

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 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:

$$\backslash begin\{align\}$$

\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}

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

$$I(x)=\; I\_0\; e^\{-4\backslash pi\; \backslash kappa\; x/\backslash lambda\_0\}\; .$$

and thus the absorption coefficient is {{math|α {{=}} 4πκ/λ{{sub|0}}}},{{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}}/4πκ}}.

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 lasers, 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 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 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. 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, or ellipsometric parameters, ellipsometry. 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.

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). Far above the atomic resonance frequency delta can be given by

$$\backslash delta\; =\; \backslash frac\{r\_0\; \backslash lambda^2\; n\_\backslash mathrm\{e\}\}\{2\; \backslash pi\}$$

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|$f\; =\; Z\; +\; f\text{'}\; +\; i\; f\text{'}\text{'}$.}} It follows that

$$\backslash begin\{align\}$$

\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}

with {{mvar|δ}} and {{mvar|β}} typically of the order of {{val|e=−5}} and {{val|e=−6}}.

File:Soap bubble sky.jpg 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,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

\backslash text\{OPL\}\; =\; nd.

This is an important concept in optics because it determines the phase of the light and governs interference and diffraction of light as it propagates. According to Fermat's principle, light rays can be characterized as those curves that optimize the optical path length.{{rp|68–69}}

{{Main|Refraction}}

File:Snells law.svg of light at the interface between two media of different refractive indices, with n₂ > n₁. Since the phase velocity is lower in the second medium (v₂ < v₁), the angle of refraction θ₂ is less than the angle of incidence θ₁; 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 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 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: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

$$n\_1\; \backslash sin\; \backslash theta\_1\; =\; n\_2\; \backslash sin\; \backslash theta\_2.$$

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.

{{Main|Total internal reflection}}

File:Total internal reflection of Chelonia mydas.jpg can be seen at the air-water boundary.]]

If there is no angle {{math|θ{{sub|2}}}} fulfilling Snell's law, i.e.,

$$\backslash frac\{n\_1\}\{n\_2\}\; \backslash sin\; \backslash theta\_1\; >\; 1,$$

the light cannot be transmitted and will instead undergo total internal reflection.{{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{{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 }}

$$\backslash theta\_\backslash mathrm\{c\}\; =\; \backslash arcsin\backslash !\backslash left(\backslash frac\{n\_2\}\{n\_1\}\backslash right)\backslash !.$$

Apart from the transmitted light there is also a 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{{rp|44}}

$$R\_0\; =\; \backslash left|\backslash frac\{n\_1\; -\; n\_2\}\{n\_1\; +\; n\_2\}\backslash right|^2\backslash !.$$

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.{{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 |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}} At other incidence angles the reflectivity will also depend on the 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 {{rp|245}}

$$\backslash theta\_\backslash mathsf\{B\}\; =\; \backslash arctan\; \backslash left(\; \backslash frac\{n\_2\}\{n\_1\}\; \backslash right)\; ~.$$

File:Lupa.na.encyklopedii.jpg of a magnifying glass is determined by the shape and refractive index of the lens.]]

The focal length of a lens is determined by its refractive index {{mvar|n}} and the 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:{{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}}

$$\backslash frac\{1\}\{f\}\; =\; (n\; -\; 1)\backslash left[\backslash frac\{1\}\{R\_1\}\; -\; \backslash frac\{1\}\{R\_2\}\backslash right]\backslash \; ,$$

where {{mvar|f}} is the focal length of the lens.

The 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):{{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 }}{{rp|6}}

$$A\_\backslash mathrm\{Num\}\; =\; n\backslash sin\; \backslash theta\; ~.$$

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.{{rp|14}}

The refractive index of electromagnetic radiation equals

$$n\; =\; \backslash sqrt\{\backslash varepsilon\_\backslash mathrm\{r\}\; \backslash mu\_\backslash mathrm\{r\}\},$$

where {{math|ε{{sub|r}}}} is the material's relative permittivity, and {{math|μ{{sub|r}}}} is its relative permeability.{{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 }}{{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}}}}}}.{{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}} 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

$$\backslash underline\{\backslash varepsilon\}\_\backslash mathrm\{r\}\; =\; \backslash varepsilon\_\backslash mathrm\{r\}\; +\; i\backslash tilde\{\backslash varepsilon\}\_\backslash mathrm\{r\}\; =\; \backslash underline\{n\}^2\; =\; (n\; +\; i\backslash kappa)^2,$$

and their components are related by:{{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 }}

$$\backslash begin\{align\}$$

\varepsilon_\mathrm{r} &= n^2 - \kappa^2\,, \\

\tilde{\varepsilon}_\mathrm{r} &= 2n\kappa\,,

\end{align}

and:

$$\backslash begin\{align\}$$

n &= \sqrt{\frac{|\underline{\varepsilon}_\mathrm{r}| + \varepsilon_\mathrm{r}}{2}}, \\

\kappa &= \sqrt{\frac{|\underline{\varepsilon}_\mathrm{r}| - \varepsilon_\mathrm{r}}{2}}.

\end{align}

where $|\backslash underline\{\backslash varepsilon\}\_\backslash mathrm\{r\}|\; =\; \backslash sqrt\{\backslash varepsilon\_\backslash mathrm\{r\}^2\; +\; \backslash tilde\{\backslash varepsilon\}\_\backslash mathrm\{r\}^2\}$ is the complex modulus.

{{see also|Wave impedance}}

The wave impedance of a plane electromagnetic wave in a non-conductive medium is given by

$$\backslash begin\{align\}$$

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}

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 relative permeability.

In non-magnetic media (that is, in materials with {{math|μ{{sub|r}} {{=}} 1}}), $Z\; =\; \{Z\_0\; \backslash over\; n\}$ and $n\; =\; \{Z\_0\; \backslash over\; Z\}\backslash ,.$

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

$$\backslash begin\{align\}$$

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}

File:density-nd.GIF and borosilicate glasses{{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}}]]

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 and magnesium oxide, while the opposite trend is observed with glasses containing lead(II) oxide and barium oxide 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.{{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 }} This means that it is also proportional to the pressure and inversely proportional to the temperature for 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.{{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|url-access=subscription }} 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.{{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|url-access=subscription }} Based on this concept, the refractive indices of organic materials can be calculated.

File:Annotated Eg vs n.png

The optical refractive index of a semiconductor tends to increase as the bandgap energy decreases. Many attempts{{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 |bibcode=2021PhyB..62013246G |issn=0921-4526|url-access=subscription }} have been made to model this relationship beginning with T. S. Moses in 1949.{{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 |bibcode=1950PPSB...63..167M |issn=0370-1301}} Empirical models can match experimental data over a wide range of materials and yet fail for important cases like InSb, PbS, and Ge.{{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}}

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.{{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 |bibcode=1990JOSAB...7..918B |issn=1520-8540|url-access=subscription }} This is the opposite of most materials, where the refractive index decreases with temperature as a result of a decreasing material density.

{{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}}

$$n\_\backslash mathrm\{g\}\; =\; \backslash frac\{\backslash mathrm\{c\}\}\{v\_\backslash mathrm\{g\}\},$$

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 is small, the group velocity can be linked to the phase velocity by the relation{{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 }}{{rp|22}}

$$v\_\backslash mathrm\{g\}\; =\; v\; -\; \backslash lambda\backslash frac\{\backslash mathrm\{d\}v\}\{\backslash mathrm\{d\}\backslash lambda\},$$

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

$$n\_\backslash mathrm\{g\}\; =\; \backslash frac\{n\}\{1\; +\; \backslash frac\{\backslash lambda\}\{n\}\backslash frac\{\backslash mathrm\{d\}n\}\{\backslash mathrm\{d\}\backslash lambda\}\}.$$

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){{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 }}

$$\backslash begin\{align\}$$

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}

where {{math|λ{{sub|0}}}} is the wavelength in vacuum.

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:

$$\backslash begin\{align\}$$

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}

The momentum of photons in a medium of refractive index {{mvar|n}} is a complex and controversial issue with two different values having different physical interpretations.{{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|url-access=subscription }}

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

In atmospheric applications, refractivity is defined as {{math| N {{=}} n – 1}}, often rescaled as either{{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 }} {{math|N {{=}} {{10^|6}} (n – 1)}}{{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 }}{{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 }} or {{math|N {{=}} {{10^|8}} (n – 1)}};{{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 }} 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 of a substance and can be calculated from the refractive index as

$$A\; =\; \backslash frac\{M\}\{\backslash rho\}\; \backslash cdot\; \backslash frac\{n^2\; -\; 1\}\{n^2\; +\; 2\}\backslash \; ,$$

where {{mvar|ρ}} is the density, and {{mvar|M}} is the molar mass.{{rp|93}}

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.

{{Main|Birefringence}}

File:Calcite.jpg crystal laid upon a paper with some letters showing double refraction]]

File:Plastic Protractor Polarized 05375.jpg.]]

In some materials, the refractive index depends on the polarization and propagation direction of the light.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 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 optical axis of the material.{{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}}}}.{{rp|236}} The birefringence of the material is the difference between these indices of refraction, {{math|Δn {{=}} n{{sub|e}} − n{{sub|o}}}}.{{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.{{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 waveplates.{{rp|237}}

Many crystals are naturally birefringent, but isotropic materials such as plastics 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.

{{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.{{rp|502}} If the index varies quadratically with the field (linearly with the intensity), it is called the optical Kerr effect and causes phenomena such as self-focusing and self-phase modulation.{{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.{{rp|265}}

File:Grin-lens.png

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.{{rp|273}} Light traveling through such a medium can be bent or focused, and this effect can be exploited to produce lenses, some optical fibers, 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.{{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.{{rp|203}} Some common mirages are caused by a spatially varying refractive index of air.

{{Main|Refractometry|Refractometer}}

File:Pulfrich refraktometer en.png

The refractive index of liquids or solids can be measured with refractometers. They typically measure some angle of refraction or the critical angle for total internal reflection. The first laboratory refractometers sold commercially were developed by Ernst Abbe in the late 19th century.{{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 }}

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 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.{{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 }}

File:Refractometer.jpg

This type of device is commonly used in chemical laboratories for identification of substances and for quality control. Handheld variants are used in agriculture by, e.g., wine makers to determine sugar content in grape juice, and inline process refractometers are used in, e.g., 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.{{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 }}

{{Main|Phase-contrast imaging}}

File:S cerevisiae under DIC microscopy.jpg 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 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 by 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 image with a phase-shifting annulus in the 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. In interferometry the illumination is split up into two beams by a 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.{{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 }}

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 lens coatings,{{Cite book |last=Willey |first=Ronald R. |url=https://www.spiedigitallibrary.org/ebooks/FG/Field-Guide-to-Optical-Thin-Films/eISBN-9780819478221/10.1117/3.668269 |title=Field Guide to Optical Thin Films |date=2006-01-27 |publisher=SPIE |isbn=978-0-8194-7822-1 |doi=10.1117/3.668269}} and the light-guiding nature of optical fiber.{{Cite journal |last1=Takeo |first1=Takashi |last2=Hattori |first2=Hajime |date=1982-10-01 |title=Optical Fiber Sensor for Measuring Refractive Index |url=https://iopscience.iop.org/article/10.1143/JJAP.21.1509 |journal=Japanese Journal of Applied Physics |volume=21 |issue=10R |pages=1509 |doi=10.1143/JJAP.21.1509 |bibcode=1982JaJAP..21.1509T |issn=0021-4922|url-access=subscription }} 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. It can be used, for example, to measure the concentration of a solute in an aqueous solution.{{Cite journal |last1=Warren |first1=John R. |last2=Gordon |first2=Julius A. |date=January 1966 |title=On the Refractive Indices of Aqueous Solutions of Urea |url=https://pubs.acs.org/doi/abs/10.1021/j100873a507 |journal=The Journal of Physical Chemistry |language=en |volume=70 |issue=1 |pages=297–300 |doi=10.1021/j100873a507 |issn=0022-3654|url-access=subscription }} It can also be used as a useful tool to differentiate between different types of gemstone, due to the unique 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).

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• 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}}

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{{notelist}}

{{reflist|25em}}

{{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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