Metre

==== Speed of light definition ====
To further reduce uncertainty, the 17th CGPM in 1983 replaced the definition of the metre with its current definition, thus fixing the length of the metre in terms of the [[second]] and the [[speed of light]]:<ref name="Res1" /><ref>{{Cite web|last=BIPM|date=20 May 2019|title=Mise en pratique for the definition of the meter in the SI|url=https://www.bipm.org/documents/20126/41489670/SI-App2-metre.pdf/0e011055-9736-d293-5e56-b8b1b267fd68?version=1.8&t=1637238031486&download=false|website=BIPM}}</ref>
:: The metre is the length of the path travelled by light in vacuum during a time interval of {{gaps|1|/|299|792|458}} of a second.

This definition fixed the speed of light in [[vacuum]] at exactly {{val|299792458}} metres per second<ref name="Res1">{{cite web |url=https://www.bipm.org/en/committees/cg/cgpm/17-1983/resolution-1 |title=17th General Conference on Weights and Measures (1983), Resolution 1. |access-date=2022-12-07}}</ref> (≈{{val|300000|u=km/s}} or ≈1.079 billion km/hour<ref>The exact value is {{val|299792458|u=m/s}} = {{val|1079252848.8|u=km/h}}.</ref>). An intended by-product of the 17th CGPM's definition was that it enabled scientists to compare lasers accurately using frequency, resulting in wavelengths with one-fifth the uncertainty involved in the direct comparison of wavelengths, because interferometer errors were eliminated. To further facilitate reproducibility from lab to lab, the 17th CGPM also made the iodine-stabilised [[helium–neon laser]] "a recommended radiation" for realising the metre.<ref name="recommendations-2" /> For the purpose of delineating the metre, the BIPM currently considers the HeNe laser wavelength, {{nowrap|''λ''{{sub|HeNe}}}}, to be {{val|632.99121258|u=nm}} with an estimated relative standard uncertainty (''U'') of {{val|2.1|e=-11}}.<ref name="recommendations-2" /><ref name="uncertainty">The term "relative standard uncertainty" is explained by NIST on their web site: {{cite web |title=Standard Uncertainty and Relative Standard Uncertainty |work=The NIST Reference on constants, units, and uncertainties: Fundamental physical constants |url=http://physics.nist.gov/cgi-bin/cuu/Info/Constants/definitions.html |publisher=NIST |access-date=2011-12-19}}</ref><ref>[[#NRC2010|National Research Council 2010]].</ref>

This uncertainty is currently one limiting factor in laboratory realisations of the metre, and it is several orders of magnitude poorer than that of the second, based upon the caesium fountain [[atomic clock]] ({{nowrap|1=''U'' = {{val|5|e=-16}}}}).<ref>[[#NIST2011|National Institute of Standards and Technology 2011]].</ref> Consequently, a  realisation of the metre is usually delineated (not defined) today in labs as {{val|1579800.762042|(33)}} wavelengths of helium–neon laser light in vacuum, the error stated being only that of frequency determination.<ref name="recommendations-2">{{cite web |title=Iodine (&lambda; ≈ 633 nm) |publisher=BIPM |url=http://www.bipm.org/utils/common/pdf/mep/M-e-P_I2_633.pdf |work=Mise en Pratique |year=2003 |access-date=2011-12-16}}</ref> This bracket notation expressing the error is explained in the article on [[Standard uncertainty#Measurements|measurement uncertainty]].

Practical realisation of the metre is subject to uncertainties in characterising the medium, to various uncertainties of interferometry, and to uncertainties in measuring the frequency of the source.<ref name="Beers2" /> A commonly used medium is air, and the [[National Institute of Standards and Technology]] (NIST) has set up an online calculator to convert wavelengths in vacuum to wavelengths in air.<ref name="NIST_calculator">The formulas used in the calculator and the documentation behind them are found at {{cite web |url=http://emtoolbox.nist.gov/Wavelength/Documentation.asp |title=Engineering metrology toolbox: Refractive index of air calculator |date=23 September 2010 |publisher=NIST |access-date=2011-12-16}} The choice is offered to use either the [http://emtoolbox.nist.gov/Wavelength/Edlen.asp modified Edlén equation] or the [http://emtoolbox.nist.gov/Wavelength/Ciddor.asp Ciddor equation]. The documentation provides [http://emtoolbox.nist.gov/Wavelength/Documentation.asp#EdlenorCiddor a discussion of how to choose] between the two possibilities.</ref> As described by NIST, in air, the uncertainties in characterising the medium are dominated by errors in measuring temperature and pressure. Errors in the theoretical formulas used are secondary.<ref name="errors">{{cite web |url=http://emtoolbox.nist.gov/Wavelength/Documentation.asp#UncertaintyandRangeofValidity |title=§VI: Uncertainty and range of validity |work=Engineering metrology toolbox: Refractive index of air calculator |date=23 September 2010 |publisher=NIST |access-date=2011-12-16}}</ref>

By implementing a refractive index correction such as this, an approximate realisation of the metre can be implemented in air, for example, using the formulation of the metre as {{val|1579800.762042|(33)}} wavelengths of helium–neon laser light in vacuum, and converting the wavelengths in vacuum to wavelengths in air. Air is only one possible medium to use in a realisation of the metre, and any [[partial vacuum]] can be used, or some inert atmosphere like helium gas, provided the appropriate corrections for refractive index are implemented.<ref name="Dunning">{{cite book |title=Atomic, molecular, and optical physics: electromagnetic radiation, Volume 29, Part 3 |chapter=Physical limits on accuracy and resolution: setting the scale |chapter-url=https://books.google.com/books?id=FV4Y39AGYuYC&pg=PA316 |page=316 |first1=F. B. |last1=Dunning |first2=Randall G. |last2=Hulet |isbn=978-0-12-475977-0 |publisher=Academic Press |year=1997 |quote=The error [introduced by using air] can be reduced tenfold if the chamber is filled with an atmosphere of helium rather than air.}}</ref>

The metre is ''defined'' as the path length travelled by light in a given time, and practical laboratory length measurements in metres are determined by counting the number of wavelengths of laser light of one of the standard types that fit into the length,{{#tag:ref|The BIPM maintains a list of recommended radiations on their web site.<ref name="recommendations-1">{{cite web |title=Recommended values of standard frequencies |url=http://www.bipm.org/en/publications/mep.html |publisher=BIPM |date=9 September 2010 |access-date=2012-01-22}}</ref><ref>[[#NPL2010|National Physical Laboratory 2010]].</ref>}} and converting the selected unit of wavelength to metres. Three major factors limit the accuracy attainable with laser [[Interferometry|interferometers]] for a length measurement:<ref name="Beers2">
A more detailed listing of errors can be found in {{cite web |work=NIST length scale interferometer measurement assurance; NIST document NISTIR 4998 |title=§4 Re-evaluation of measurement errors |first1=John S |last1=Beers |first2=William B |last2=Penzes |url=https://www.nist.gov/calibrations/upload/4998.pdf  |access-date=2011-12-17 |date=December 1992 |pages=9 ''ff'' }}
</ref><ref name="Webster2">[[#Zagar1999|Zagar, 1999, pp. 6–65''ff'']].</ref>
* uncertainty in vacuum wavelength of the source,
* uncertainty in the refractive index of the medium,
* [[least count]] resolution of the interferometer.

Of these, the last is peculiar to the interferometer itself. The conversion of a length in wavelengths to a length in metres is based upon the relation
: <math> \lambda = \frac{c}{n f} ,</math>
which converts the unit of wavelength ''&lambda;'' to metres using ''c'', the speed of light in vacuum in m/s. Here ''n'' is the [[refractive index]] of the medium in which the measurement is made, and ''f'' is the measured frequency of the source. Although conversion from wavelengths to metres introduces an additional error in the overall length due to measurement error in determining the refractive index and the frequency, the measurement of frequency is one of the most accurate measurements available.<ref name="Webster2" />

The CIPM issued a clarification in 2002:
{{Blockquote|text=Its definition, therefore, applies only within a spatial extent sufficiently small that the effects of the non-uniformity of the gravitational field can be ignored (note that, at the surface of the Earth, this effect in the vertical direction is about 1 part in {{val|e=16}} per metre). In this case, the effects to be taken into account are those of special relativity only.}}