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Speed of light
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=== In a medium === {{See also|Refractive index}} In a medium, light usually does not propagate at a speed equal to ''c''; further, different types of light wave will travel at different speeds. The speed at which the individual crests and troughs of a [[plane wave]] (a wave filling the whole space, with only one [[frequency]]) propagate is called the [[phase velocity]] ''v''<sub>p</sub>. A physical signal with a finite extent (a pulse of light) travels at a different speed. The overall [[Envelope (waves)|envelope]] of the pulse travels at the [[group velocity]] ''v''<sub>g</sub>, and its earliest part travels at the [[front velocity]] ''v''<sub>f</sub>.<ref name="Milonni">{{Cite book|author=Milonni|first=Peter W.|url=https://books.google.com/books?id=kE8OUCvt7ecC&pg=PA26|title=Fast light, slow light and left-handed light|publisher=CRC Press|year=2004|isbn=978-0-7503-0926-4|pages=25 ''ff''|authorlink1=Peter W. Milonni}}</ref> [[File:frontgroupphase.gif|thumb|The blue dot moves at the speed of the ripples, the phase velocity; the green dot moves with the speed of the envelope, the group velocity; and the red dot moves with the speed of the foremost part of the pulse, the front velocity.|alt=A modulated wave moves from left to right. There are three points marked with a dot: A blue dot at a node of the carrier wave, a green dot at the maximum of the envelope, and a red dot at the front of the envelope.]] The phase velocity is important in determining how a light wave travels through a material or from one material to another. It is often represented in terms of a ''refractive index''. The refractive index of a material is defined as the ratio of ''c'' to the phase velocity ''v''<sub>p</sub> in the material: larger indices of refraction indicate lower speeds. The refractive index of a material may depend on the light's frequency, intensity, [[polarization (waves)|polarization]], or direction of propagation; in many cases, though, it can be treated as a material-dependent constant. The [[refractive index of air]] is approximately 1.0003.<ref name=Podesta> {{Cite book |last=de Podesta |first=M. |year=2002 |title=Understanding the Properties of Matter |url=https://books.google.com/books?id=h8BNvnR050cC&pg=PA131 |page=131 |publisher=CRC Press |isbn=978-0-415-25788-6 }}</ref> Denser media, such as [[Optical properties of water and ice|water]],<ref> {{Cite web |title=Optical constants of H<sub>2</sub>O, D<sub>2</sub>O (Water, heavy water, ice) |url=https://refractiveindex.info/?shelf=main&book=H2O&page=Hale |publisher=Mikhail Polyanskiy |work=refractiveindex.info |access-date=7 November 2017 }}</ref> [[glass]],<ref> {{Cite web |title=Optical constants of Soda lime glass |url=https://refractiveindex.info/?shelf=glass&book=soda-lime&page=Rubin-clear |publisher=Mikhail Polyanskiy |work=refractiveindex.info |access-date=7 November 2017 }}</ref> and [[Material properties of diamond#Optical properties|diamond]],<!--there must be a way to make it clearer where these links go--><ref> {{Cite web |title=Optical constants of C (Carbon, diamond, graphite) |url=https://refractiveindex.info/?shelf=main&book=C&page=Phillip |publisher=Mikhail Polyanskiy |work=refractiveindex.info |access-date =7 November 2017 }}</ref> have refractive indexes of around 1.3, 1.5 and 2.4, respectively, for visible light. In exotic materials like [[Bose–Einstein condensate]]s near [[absolute zero]], the effective speed of light may be only a few metres per second. However, this represents absorption and re-radiation delay between atoms, as do all slower-than-''c'' speeds in material substances. As an extreme example of light "slowing" in matter, two independent teams of physicists claimed to bring light to a "complete standstill" by passing it through a Bose–Einstein condensate of the element [[rubidium]]. The popular description of light being "stopped" in these experiments refers only to light being stored in the excited states of atoms, then re-emitted at an arbitrarily later time, as stimulated by a second laser pulse. During the time it had "stopped", it had ceased to be light. This type of behaviour is generally microscopically true of all transparent media which "slow" the speed of light.<ref>{{Cite web |last=Cromie |first=William J. |url=http://www.news.harvard.edu/gazette/2001/01.24/01-stoplight.html |title=Researchers now able to stop, restart light |website=Harvard University Gazette |date=24 January 2001 |access-date=8 November 2011 |url-status=dead |archive-url=https://web.archive.org/web/20111028041346/http://www.news.harvard.edu/gazette/2001/01.24/01-stoplight.html |archive-date=28 October 2011 }}</ref> In transparent materials, the refractive index generally is greater than 1, meaning that the phase velocity is less than ''c''. In other materials, it is possible for the refractive index to become smaller than{{nbsp}}1 for some frequencies; in some exotic materials it is even possible for the index of refraction to become negative.<ref> {{Cite book |title=Fast light, slow light and left-handed light |last=Milonni |first=P. W. |author-link1=Peter W. Milonni |url=https://books.google.com/books?id=kE8OUCvt7ecC&pg=PA25 |page=25 |isbn=978-0-7503-0926-4 |year=2004 |publisher=CRC Press }}</ref> The requirement that causality is not violated implies that the [[real and imaginary parts]] of the [[dielectric constant]] of any material, corresponding respectively to the index of refraction and to the [[attenuation coefficient]], are linked by the [[Kramers–Kronig relation]]s.<ref> {{Cite journal |last=Toll |first=J. S. |year=1956 |title=Causality and the Dispersion Relation: Logical Foundations |journal=[[Physical Review]] |volume=104 |issue=6 |pages=1760–1770 |doi=10.1103/PhysRev.104.1760 |bibcode = 1956PhRv..104.1760T }}</ref><ref>{{Cite book|last=Wolf|first=Emil|url=https://www.worldcat.org/oclc/261134839|title=Selected Works of Emil Wolf: with commentary|date=2001|publisher=World Scientific|isbn=978-981-281-187-5|location=River Edge, N.J.|pages=577–584|chapter=Analyticity, Causality and Dispersion Relations|oclc=261134839|author-link=Emil Wolf}}</ref> In practical terms, this means that in a material with refractive index less than 1, the wave will be absorbed quickly.<ref>{{Cite journal |last1=Libbrecht |first1=K. G. |last2=Libbrecht |first2=M. W. |date=December 2006 |title=Interferometric measurement of the resonant absorption and refractive index in rubidium gas |url=https://authors.library.caltech.edu/12639/1/LIBajp06.pdf |journal=American Journal of Physics |language=en |volume=74 |issue=12 |pages=1055–1060 |doi=10.1119/1.2335476 |bibcode=2006AmJPh..74.1055L |issn=0002-9505}}</ref> A pulse with different group and phase velocities (which occurs if the phase velocity is not the same for all the frequencies of the pulse) smears out over time, a process known as [[Dispersion (optics)|dispersion]]. Certain materials have an exceptionally low (or even zero) group velocity for light waves, a phenomenon called [[slow light]].<ref>See, for example: * {{Cite journal |last1=Hau |first1=L. V. |author-link1=Lene Hau |last2=Harris |first2=S. E. |author-link2=Stephen E. Harris |last3=Dutton |first3=Z. |author-link3=Zachary Dutton |last4=Behroozi |first4=C. H. |year=1999 |title=Light speed reduction to 17 metres per second in an ultracold atomic gas |journal=Nature |volume=397 |issue=6720 |pages=594–598 |doi=10.1038/17561 |bibcode = 1999Natur.397..594V |s2cid=4423307 |url=http://www.seas.harvard.edu/haulab/publications/pdf/Slow_Light_1999.pdf }} * {{Cite journal |last1=Liu |first1=C. |last2=Dutton |first2=Z. |author-link2=Zachary Dutton |last3=Behroozi |first3=C. H. |last4=Hau |first4=L. V. |author-link4=Lene Hau |year=2001 |title=Observation of coherent optical information storage in an atomic medium using halted light pulses |journal=Nature |volume=409 |issue=6819 |pages=490–493 |doi=10.1038/35054017 |pmid=11206540 |bibcode = 2001Natur.409..490L |s2cid=1894748 |url=http://www.seas.harvard.edu/haulab/publications/pdf/Stopped_Light_2001.pdf }} * {{Cite journal |last1=Bajcsy |first1=M. |last2=Zibrov |first2=A. S. |last3=Lukin |first3=M. D. |year=2003 |title=Stationary pulses of light in an atomic medium |journal=Nature |volume=426 |issue=6967 |pages=638–641 |doi=10.1038/nature02176 |pmid=14668857 |arxiv = quant-ph/0311092 |bibcode = 2003Natur.426..638B |s2cid=4320280 }} * {{Cite web |last=Dumé |first=B. |year=2003 |title=Switching light on and off |url=http://physicsworld.com/cws/article/news/18724 |work=[[Physics World]] |publisher=Institute of Physics |access-date=8 December 2008 |archive-date=5 December 2008 |archive-url=https://web.archive.org/web/20081205051203/http://physicsworld.com/cws/article/news/18724 |url-status=dead }}</ref> The opposite, group velocities exceeding ''c'', was proposed theoretically in 1993 and achieved experimentally in 2000.<ref>See, for example: * {{Cite journal |first=R. Y. |last=Chiao |author-link=Raymond Chiao |title=Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations |journal=[[Physical Review A]] |volume=48 |year=1993 |issue=1 |pages=R34–R37 |doi=10.1103/PhysRevA.48.R34 |pmid=9909684 |bibcode=1993PhRvA..48...34C }} * {{Cite journal |first1=L. J. |last1=Wang |first2=A. |last2=Kuzmich |first3=A. |last3=Dogariu |title=Gain-assisted superluminal light propagation |journal=[[Nature (journal)|Nature]] |volume=406 |pages=277–279 |year=2000 |issue=6793 |url=https://www.nature.com/articles/35018520 |doi=10.1038/35018520 |pmid=10917523 |bibcode=2000Natur.406..277W |s2cid=4358601 |url-access=subscription }} * {{Cite news |last=Whitehouse |first=D. |date=19 July 2000 |title=Beam Smashes Light Barrier |url=http://news.bbc.co.uk/2/hi/science/nature/841690.stm |work=BBC News |access-date=9 February 2022 }} * {{Cite web |first=Greg |last=Gbur |author-link=Greg Gbur |title=Light breaking its own speed limit: how 'superluminal' shenanigans work |url=https://skullsinthestars.com/2008/02/26/light-breaking-its-own-speed-limit-how-superluminal-shenanigans-work/ |date=26 February 2008 |access-date=9 February 2022 }}</ref> It should even be possible for the group velocity to become infinite or negative, with pulses travelling instantaneously or backwards in time.<ref name="Milonni" /> None of these options allow information to be transmitted faster than ''c''. It is impossible to transmit information with a light pulse any faster than the speed of the earliest part of the pulse (the front velocity). It can be shown that this is (under certain assumptions) always equal to ''c''.<ref name="Milonni" /> {{Clear}} It is possible for a particle to travel through a medium faster than the phase velocity of light in that medium (but still slower than ''c''). When a [[charged particle]] does that in a [[dielectric]] material, the electromagnetic equivalent of a [[shock wave]], known as [[Cherenkov radiation]], is emitted.<ref>{{Cite journal |last=Cherenkov |first=Pavel A. |author-link=Pavel Alekseyevich Cherenkov |year=1934 |title=Видимое свечение чистых жидкостей под действием γ-радиации |trans-title=Visible emission of pure liquids by action of γ radiation |journal=[[Doklady Akademii Nauk SSSR]] |volume=2 |page=451}} Reprinted: {{Cite journal |last=Cherenkov |first=P. A. |date=1967 |title=Видимое свечение чистых жидкостей под действием γ-радиации |trans-title=Visible emission of pure liquids by action of γ radiation |journal=Usp. Fiz. Nauk |volume=93 |issue=10 |page=385 |doi=10.3367/ufnr.0093.196710n.0385}}, and in {{Cite book |title=Pavel Alekseyevich Čerenkov: Chelovek i Otkrytie |trans-title=Pavel Alekseyevich Čerenkov: Man and Discovery |editor1=A. N. Gorbunov |editor2=E. P. Čerenkova |location=Moscow |publisher=Nauka |date=1999 |pages=149–153}}</ref>
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