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{{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.
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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 |
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.
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{{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]] |date= 17 April 2005 |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]] |date= 20 February 2005 |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===
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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 |url-access= subscription }}</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.
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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
The real and imaginary parts 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. <ref name=lightmatterinteractionbook>{{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 }}</ref> 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 =
|author-link = Mildred Dresselhaus
|date = 1999
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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===
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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 |date=5 April 2013 |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>
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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|url-access=subscription }}</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|url-access=subscription }}</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 |article-number=413246 |doi=10.1016/j.physb.2021.413246 |bibcode=2021PhyB..62013246G |issn=0921-4526|url-access=subscription }}</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 |bibcode=1950PPSB...63..167M |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 |bibcode=1990JOSAB...7..918B |issn=1520-8540|url-access=subscription }}</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>
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\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|url-access=subscription }}</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]].
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===Inhomogeneity===
[[File:Grin-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]].
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==Applications==
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]],<ref>{{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}}</ref> and the light-guiding nature of [[optical fiber]].<ref>{{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 }}</ref> 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.
▲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==
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