Refractive index: Difference between revisions

Revision as of 19:44, 15 July 2025

Revision as of 23:49, 22 July 2025

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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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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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=== 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 |pagesarticle-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.