Editing Refractive index (section)

==Relations to other quantities==

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

[[Optical path length]] (OPL) is the product of the geometric length {{mvar|d}} of the path light follows through a system, and the index of refraction of the medium through which it propagates,<ref>R. Paschotta, article on [https://www.rp-photonics.com/optical_thickness.html optical thickness] {{webarchive|url=https://web.archive.org/web/20150322115346/http://www.rp-photonics.com/optical_thickness.html |date=2015-03-22 }} in the [https://www.rp-photonics.com/encyclopedia.html Encyclopedia of Laser Physics and Technology] {{webarchive|url=https://web.archive.org/web/20150813044135/http://www.rp-photonics.com/encyclopedia.html |date=2015-08-13 }}, accessed on 2014-09-08</ref>
<math display="text">\text{OPL} = nd.</math>
This is an important concept in optics because it determines the [[phase (waves)|phase]] of the light and governs [[interference (wave propagation)|interference]] and [[diffraction]] of light as it propagates. According to [[Fermat's principle]], light rays can be characterized as those curves that [[Mathematical optimization|optimize]] the optical path length.<ref name=Hecht/>{{rp|68–69}}

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

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

When light moves from one medium to another, it changes direction, i.e. it is [[Refraction|refracted]]. If it moves from a medium with refractive index {{math|''n''{{sub|1}}}} to one with refractive index {{math|''n''{{sub|2}}}}, with an [[angle of incidence (optics)|incidence angle]] to the [[surface normal]] of {{math|''θ''{{sub|1}}}}, the refraction angle {{math|''θ''{{sub|2}}}} can be calculated from [[Snell's law]]:<ref>R. Paschotta, article on [https://www.rp-photonics.com/refraction.html refraction] {{webarchive|url=https://web.archive.org/web/20150628174941/https://www.rp-photonics.com/refraction.html |date=2015-06-28 }} in the [https://www.rp-photonics.com/encyclopedia.html Encyclopedia of Laser Physics and Technology] {{webarchive|url=https://web.archive.org/web/20150813044135/http://www.rp-photonics.com/encyclopedia.html |date=2015-08-13 }}, accessed on 2014-09-08</ref>
<math display="block">n_1 \sin \theta_1 = n_2 \sin \theta_2.</math>

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

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

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

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

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

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

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

The [[focal length]] of a [[lens (optics)|lens]] is determined by its refractive index {{mvar|n}} and the [[Radius of curvature (optics)|radii of curvature]] {{math|''R''{{sub|1}}}} and {{math|''R''{{sub|2}}}} of its surfaces. The power of a [[thin lens]] in air is given by the simplified version of the [[Lensmaker's formula]]:<ref>{{cite web |first=Carl R. |last=Nave |title=Lens-makers' formula |website=HyperPhysics |series=Department of Physics and Astronomy |publisher=Georgia State University |url=http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenmak.html |access-date=2014-09-08 |archive-url=https://web.archive.org/web/20140926153405/http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenmak.html |archive-date=2014-09-26}}</ref>
<math display="block">\frac{1}{f} = (n - 1)\left[\frac{1}{R_1} - \frac{1}{R_2}\right]\ ,</math>
where {{mvar|f}} is the focal length of the lens.

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

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

===Relative permittivity and permeability===
The refractive index of electromagnetic radiation equals
<math display="block">n = \sqrt{\varepsilon_\mathrm{r} \mu_\mathrm{r}},</math>
where {{math|''ε''{{sub|r}}}} is the material's [[relative permittivity]], and {{math|''μ''{{sub|r}}}} is its [[Permeability (electromagnetism)|relative permeability]].<ref name = bleaney>{{cite book | last1 = Bleaney| first1 = B.| author-link1 = Brebis Bleaney |last2 = Bleaney |first2 = B.I. | title = Electricity and Magnetism | publisher = [[Oxford University Press]] | edition = Third | date = 1976 | isbn = 978-0-19-851141-0 }}</ref>{{rp|229}} The refractive index is used for optics in [[Fresnel equations]] and [[Snell's law]]; while the relative permittivity and permeability are used in [[Maxwell's equations]] and electronics. Most naturally occurring materials are non-magnetic at optical frequencies, that is {{math|''μ''{{sub|r}}}} is very close to 1, therefore {{mvar|n}} is approximately {{math|{{sqrt|''ε''{{sub|r}}}}}}.<ref>{{cite book |last1=Andrews |first1=David L. |title=Photonics, Volume 2: Nanophotonic Structures and Materials |date=24 February 2015 |publisher=John Wiley & Sons |isbn=978-1-118-22551-6 |page=54 |url=https://books.google.com/books?id=EYRVBgAAQBAJ |language=en}}</ref> In this particular case, the complex relative permittivity {{math|{{uu|''ε''}}{{sub|r}}}}, with real and imaginary parts {{math|''ε''{{sub|r}}}} and {{math|''ɛ̃''{{sub|r}}}}, and the complex refractive index {{math|{{uu|''n''}}}}, with real and imaginary parts {{mvar|n}} and {{mvar|κ}} (the latter called the "extinction coefficient"), follow the relation
<math display="block">\underline{\varepsilon}_\mathrm{r} = \varepsilon_\mathrm{r} + i\tilde{\varepsilon}_\mathrm{r} = \underline{n}^2 = (n + i\kappa)^2,</math>

and their components are related by:<ref>{{cite book|first=Frederick|last=Wooten|title=Optical Properties of Solids|page=49|publisher=[[Academic Press]]|location=New York City|year= 1972|isbn=978-0-12-763450-0}}[http://www.lrsm.upenn.edu/~frenchrh/download/0208fwootenopticalpropertiesofsolids.pdf  (online pdf)] {{webarchive|url=https://web.archive.org/web/20111003034948/http://www.lrsm.upenn.edu/~frenchrh/download/0208fwootenopticalpropertiesofsolids.pdf |date=2011-10-03 }}</ref>
<math display="block">\begin{align}
\varepsilon_\mathrm{r} &= n^2 - \kappa^2\,, \\
\tilde{\varepsilon}_\mathrm{r} &= 2n\kappa\,,
\end{align}</math>

and:

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

where <math>|\underline{\varepsilon}_\mathrm{r}| = \sqrt{\varepsilon_\mathrm{r}^2 + \tilde{\varepsilon}_\mathrm{r}^2}</math> is the [[modulus of complex number|complex modulus]].

===Wave impedance===
{{see also|Wave impedance}}
The wave impedance of a plane electromagnetic wave in a non-conductive medium is given by
<math display="block">\begin{align}
Z &= \sqrt{\frac{\mu}{\varepsilon}} = \sqrt{\frac{\mu_\mathrm{0}\mu_\mathrm{r}}{\varepsilon_\mathrm{0}\varepsilon_\mathrm{r}}} = \sqrt{\frac{\mu_\mathrm{0}}{\varepsilon_\mathrm{0}}}\sqrt{\frac{\mu_\mathrm{r}}{\varepsilon_\mathrm{r}}} \\
  &= Z_0 \sqrt{\frac{\mu_\mathrm{r}}{\varepsilon_\mathrm{r}}} \\
  &= Z_0 \frac{\mu_\mathrm{r}}{n}
\end{align}</math>

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

In non-magnetic media (that is, in materials with {{math|''μ''{{sub|r}} {{=}} 1}}), <math>Z = {Z_0 \over n}</math> and <math>n = {Z_0 \over Z}\,.</math>

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

The reflectivity {{math|''R''{{sub|0}}}} between two media can thus be expressed both by the wave impedances and the refractive indices as
<math display="block">\begin{align}
R_0 &= \left| \frac{n_1 - n_2}{n_1 + n_2} \right|^2 \\
    &= \left| \frac{Z_2 - Z_1}{Z_2 + Z_1} \right|^2\,.
\end{align}</math>

===Density===
[[File:density-nd.GIF|thumb|upright=1.7|alt=A scatter plot showing a strong correlation between glass density and refractive index for different glasses|The relation between the refractive index and the density of [[silicate glass|silicate]] and [[borosilicate glass]]es<ref>{{cite web|url=http://www.glassproperties.com/refractive_index/|title=Calculation of the Refractive Index of Glasses|work=Statistical Calculation and Development of Glass Properties|url-status=live|archive-url=https://web.archive.org/web/20071015124852/http://glassproperties.com/refractive_index/|archive-date=2007-10-15}}</ref>]]
In general, it is assumed that the refractive index of a glass increases with its [[density]]. However, there does not exist an overall linear relationship between the refractive index and the density for all silicate and borosilicate glasses. A relatively high refractive index and low density can be obtained with glasses containing light metal oxides such as [[lithium oxide|{{chem2|Li2O}}]] and [[magnesium oxide|{{chem2|MgO}}]], while the opposite trend is observed with glasses containing [[lead(II) oxide|{{chem2|PbO}}]] and [[barium oxide|{{chem2|BaO}}]] as seen in the diagram at the right.

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

For air, {{math|''n'' - 1}} is proportional to the density of the gas as long as the chemical composition does not change.<ref>{{cite web | url = http://emtoolbox.nist.gov/Wavelength/Documentation.asp | first1 = Jack A. | last1 = Stone | first2 = Jay H. | last2 = Zimmerman | date = 2011-12-28 | website = Engineering metrology toolbox | publisher = National Institute of Standards and Technology (NIST) | title = Index of refraction of air | access-date = 2014-01-11 | url-status = live | archive-url = https://web.archive.org/web/20140111155252/http://emtoolbox.nist.gov/Wavelength/Documentation.asp | archive-date = 2014-01-11 }}</ref> This means that it is also proportional to the pressure and inversely proportional to the temperature for [[ideal gas law|ideal gases]]. For liquids the same observation can be made as for gases, for instance, the refractive index in alkanes increases nearly perfectly linear with the density. On the other hand, for carboxylic acids, the density decreases with increasing number of C-atoms within the homologeous series. The simple explanation of this finding is that it is not density, but the molar concentration of the chromophore that counts. In homologeous series, this is the excitation of the C-H-bonding. August Beer must have intuitively known that when he gave Hans H. Landolt in 1862 the tip to investigate the refractive index of compounds of homologeous series.<ref>{{Cite journal |last=Landolt |first=H. |date=January 1862 |title=Ueber die Brechungsexponenten flüssiger homologer Verbindungen |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.18621931102 |journal=Annalen der Physik |language=en |volume=193 |issue=11 |pages=353–385 |doi=10.1002/andp.18621931102 |bibcode=1862AnP...193..353L |issn=0003-3804|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 |pages=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>
where {{math|''v''{{sub|g}}}} is the [[group velocity]]. This value should not be confused with {{mvar|n}}, which is always defined with respect to the [[phase velocity]]. When the [[dispersion (optics)|dispersion]] is small, the group velocity can be linked to the phase velocity by the relation<ref name=bornwolf>{{cite book | title=[[Principles of Optics]] | publisher=CUP Archive | edition=7th expanded | last1=Born | first1=Max | author-link1=Max Born | last2=Wolf | first2=Emil | author-link2=Emil Wolf | page=[https://archive.org/details/principlesofopti0006born/page/22 22] | isbn=978-0-521-78449-8 | date=1999 }}</ref>{{rp|22}}
<math display="block">v_\mathrm{g} = v - \lambda\frac{\mathrm{d}v}{\mathrm{d}\lambda},</math>
where {{mvar|λ}} is the wavelength in the medium. In this case the group index can thus be written in terms of the wavelength dependence of the refractive index as
<math display="block">n_\mathrm{g} = \frac{n}{1 + \frac{\lambda}{n}\frac{\mathrm{d}n}{\mathrm{d}\lambda}}.</math>

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

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

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

===Refractivity===
In atmospheric applications, '''refractivity''' is defined as {{math| ''N'' {{=}} ''n'' – 1}}, often rescaled as either<ref>{{cite web | last = Young | first = A.T. | year = 2011 | title = Refractivity of Air | url = http://mintaka.sdsu.edu/GF/explain/atmos_refr/air_refr.html | access-date = 31 July 2014 | url-status = live | archive-url = https://web.archive.org/web/20150110053602/http://mintaka.sdsu.edu/GF/explain/atmos_refr/air_refr.html | archive-date = 10 January 2015 }}</ref> {{math|''N'' {{=}} {{10^|6}} (''n'' – 1)}}<ref>{{cite journal | last1 = Barrell | first1 = H. | last2 = Sears | first2 = J.E. | year = 1939 | title = The refraction and dispersion of air for the visible spectrum | journal = [[Philosophical Transactions of the Royal Society of London]] | series =  A, Mathematical and Physical Sciences | volume = 238 | issue = 786 | pages = 1–64 | jstor = 91351 | doi=10.1098/rsta.1939.0004 | doi-access = free |bibcode = 1939RSPTA.238....1B }}</ref><ref>{{cite journal |last1=Aparicio |first1=Josep M. |last2=Laroche |first2=Stéphane |date=2011-06-02 |df=dmy-all |title=An evaluation of the expression of the atmospheric refractivity for GPS signals |journal=[[Journal of Geophysical Research]] |volume=116 |issue=D11 |page=D11104 |doi=10.1029/2010JD015214 |doi-access=free |bibcode=2011JGRD..11611104A }}</ref> or {{math|''N'' {{=}} {{10^|8}} (''n'' – 1)}};<ref>{{cite journal | last = Ciddor | first = P.E. | year = 1996 | title = Refractive index of air: New equations for the visible and near infrared | journal = Applied Optics | volume = 35 | issue = 9 | pages = 1566–1573 | doi=10.1364/ao.35.001566 | pmid = 21085275 |bibcode = 1996ApOpt..35.1566C }}</ref> the multiplication factors are used because the refractive index for air, {{mvar|n}} deviates from unity by at most a few parts per ten thousand.

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