# Metre (unit)

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The metre (American meter) is the basic unit of length measurement in the International System of Units (SI) commonly known as the metric system. The metre is abbreviated as m, appearing after the quantity. The metre has been adopted as the fundamental unit of length in almost all countries of the world, with the United States of America being a notable exception.

Currently, the SI units define the metre in terms of the speed of light and the second, such that the metre is the length traveled by light in the reference medium of classical vacuum in the time interval of 1/299 792 458 second.[1][2]

An easily realized and reproducible measurement is a major goal of metrology. The practicality of adoption of a time of flight as a measure of length depends critically upon the experimental fact that the speed of light in vacuum is a universal constant to within experimental error. Its practicality is further enhanced because corrections introduced by using real media, such as terrestrial vacuum, or even air, are very small.

The metre is also the basis of the SI units of area and volume, which are the square metre (m2) and cubic metre (m3), respectively.

The foot is defined in terms of the metre as exactly 0.3048 metres.[3]

## Realizations

An attempt to fashion a realization of the metre to serve as a reference length must use a real medium, and so the verification of any laboratory version of the "standard metre" is subject to uncertainties in characterizing the medium, in addition to various uncertainties of interferometry, and to uncertainties in measuring the frequency of the source.[4] A commonly used medium is air, and NIST has set up an on-line calculator to convert wavelengths in vacuum to wavelengths in air.[5] As described by NIST, in air the uncertainties in characterizing the medium are dominated by errors in finding temperature and pressure, and errors in the theoretical formulas used are secondary.[6] By implementing a refractive index correction such as this, an approximate realization of the standard metre can be implemented in air using the formulation of the metre as 1 579 800.762046±(3.3 ×10−5) wavelengths of helium-neon laser light in vacuum, for example, and converting the wavelengths in vacuum to wavelengths in air.[7] (The uncertainty in number of wavelengths corresponds to an uncertainty of Δf=10 kHz in the frequency of the laser.[8]) Of course, air is only one possible medium to use in setting up an approximate standard metre, and any partial vacuum can be used, or some inert atmosphere like helium gas, provided the appropriate corrections for refractive index are implemented.[9]

## History

The metre was initially adopted as a unit of measure in France in 1790, during the French Revolution.

The original definition of the metre was the length of a pendulum with a half-period of 1 second, but was changed in 1791 to be the length of a prototype bar which was supposed to be 1/10 000 000 of the length of the meridian of Paris from the north pole to the equator. Since then, the metre has been redefined a number of times.[10]

After 1960 and until 1983, lengths were measured in principle in terms of the number of wavelengths of light that could be fitted into them. The precise definition of the metre was established in the 11th CGPM of 1960 as 1 650 763.73 wavelengths of the 2p10−5d5 line of krypton-86.[11]

In 1972 KM Evenson and co-workers at the then National Bureau of Standards (NIST today) found the standard krypton line to be such an asymmetric function of frequency that several equally plausible choices for wavelengths could be determined, and they suggested measurements could be made 100 times more accurately using a methane stabilized He-Ne laser.[12] That assessment was confirmed by others.[13] These uncertainties in determining the frequency of the specific 1960 standard source, coupled with limitations in use of its particular wavelength, led in 1983 (along with other considerations spelled out in the final decision) to the switch to the frequency-independent standard based upon the transit time of light. This switch freed the definition of the metre from restriction to any specific source, allowing the source to be selected to suit the requirements of the measurement at hand. Perhaps the main benefit of the change in definition was to define the metre in a manner that did not require further revision as light sources and measurement methodology advanced.[14]

Although time-of-transit measurements are used today, for example in practical methods like GPS, precision measurements of intermediate lengths commonly are determined in wavelengths λ of a standard source and converted to metres using the relation λ= c0 / f , where c0 = 299 792 458m/s is the defined speed of light in classical vacuum and f is the frequency of the source. One consequence of the relation λ= c0 / f is that measurement errors in finding the source frequency are mirrored in the source wavelength, and for any specific choice of source they share the same relative standard uncertainty.[15] A list of standard sources and measurement uncertainties is maintained by the BIPM.[16]

The difference between today's use of the interferometer and that of the past is that the wavelength unit is converted directly to metres using λ = c0 / f, rather than by the older more error-prone intermediary of comparison with a selected standard source with a wavelength defined in metres. Comparison of sources using wavelengths introduces interferometer errors that are not introduced using a direct comparison of frequencies.[17] An added change in today's interferometry, unrelated to the change in definition, is that today's sources are much more stable and their frequencies are better known.

## Practical measurement

The next subject is how the definition of the metre is to be implemented in practice. The most commonly used approaches are discussed first: the transit-time methods, and the interferometer methods. For objects like crystals and diffraction gratings, diffraction patterns are used. Other methods for small-dimensional measurement are only mentioned.

### Transit-time measurement

The basic idea behind a transit-time measurement of length is to send a signal from one end of the length to be measured to the other, and back again. The time for the round trip is the transit time Δt, and the length ℓ is then 2ℓ = Δt/v, with v the speed of propagation of the signal, assumed the same in both directions. If light is used for the signal, its speed depends upon the medium in which it propagates; in SI units the speed is a defined value c0 in the reference medium of classical vacuum. Thus, when light is used in a transit-time approach, length measurements are not subject to knowledge of the source frequency (apart from possible frequency dependence of the correction to relate the medium to classical vacuum), but are subject to the error in measuring transit times, in particular, errors introduced by the response times of the pulse emission and detection instrumentation. An additional uncertainty is the refractive index correction relating the medium used to the reference vacuum, taken in SI units to be the classical vacuum. A refractive index of the medium larger than one slows the light.

Transit time measurement underlies most radio navigation systems for boats and aircraft, for example, radar and the nearly obsolete Long Range Aid to Navigation LORAN-C. For example, in one radar system, pulses of electromagnetic radiation are sent out by the vehicle (interrogating pulses) and trigger a response from a responder beacon. The time interval between the sending and the receiving of a pulse is monitored and used to determine a distance. In the global positioning system a code of ones and zeros is emitted at a known time from multiple satellites, and their times of arrival are noted at a receiver along with the time they were sent (encoded in the messages). Assuming the receiver clock can be related to the synchronized clocks on the satellites, the transit time can be found and used to provide the distance to each satellite. Receiver clock error is corrected by combining the data from four satellites.[18]

Such techniques vary in accuracy according to the distances over which they are intended for use. For example, LORAN-C is accurate to about 6km, GPS about 10m, enhanced GPS, in which a correction signal is transmitted from terrestrial stations (that is, differential GPS (DGPS)) or via satellites (that is, Wide Area Augmentation System (WAAS)) can bring accuracy to a few meters or < 1 meter, or, in specific applications, centimeters. Time-of-flight systems for robotics (for example, Light Detection and Ranging LIDAR and Laser Detection and Ranging LADAR) aim at lengths of 10-100m and have an accuracy of about 5-10 mm[19]

### Interferometer measurements

(PD) Image: John R. Brews
Measuring a length in wavelengths of light using an interferometer.

In many practical circumstances, and for precision work, measurement of dimension using transit time measurements is used only as an initial indicator of length and is refined using an interferometer.[20][21] The figure shows schematically how length is determined using a Michelson interferometer: the two panels show a laser source emitting a light beam split by a beam splitter (BS) to travel two paths. The light is recombined by bouncing the two components off a pair of corner cubes (CC) that return the two components to the beam splitter again to be reassembled. The distance between the left-hand corner cube and the beam splitter is compared to that separation on the fixed leg as the left-hand spacing is adjusted to match the length of the object to be measured.[22]

In the top panel the path is such that the two beams reinforce each other after reassembly, leading to a strong light pattern (sun). The bottom panel shows a path that is made a half wavelength longer by moving the left-hand mirror a quarter wavelength further away, increasing the path difference by a half wavelength. The result is the two beams are in opposition to each other at reassembly, and the recombined light intensity drops to zero (clouds). Thus, as the spacing between the mirrors is adjusted, the observed light intensity cycles between reinforcement and cancellation as the number of wavelengths of path difference changes, and the observed intensity alternately peaks (sun) and dims (clouds). This behavior is called interference and the machine is called an interferometer. By counting fringes it is found how many wavelengths long the measured path is compared to the fixed leg. In this way, measurements are made in units of wavelengths λ at a known frequency f, and related to the metre using λ = c0 / f. With c0 a defined value, the error in the measurement of length is determined by the error in measuring the frequency of the light source.

This methodology for length determination requires a careful specification of the wavelength of the light used, and is one reason for employing a laser source where the wavelength can be held stable. Regardless of stability, however, the precise frequency of any source has linewidth limitations.[23] Other significant errors are introduced by the interferometer itself; in particular: errors in light beam alignment, collimation and fractional fringe determination.[24] Resolution using wavelengths is in the range of ΔL/L ≈ 10−9 – 10−11 depending upon the length measured, the wavelength and the type of interferometer used.[24]

The measurement also requires careful specification of the medium in which the light propagates. A refractive index correction is made to relate the medium used to the reference vacuum, taken in SI units to be the classical vacuum. These refractive index corrections can be found more accurately by adding frequencies, for example, frequencies at which propagation is sensitive to the presence of water vapor. This way non-ideal contributions to the refractive index can be measured and corrected for at another frequency using established theoretical models.

It may be noted again, by way of contrast, that the transit-time measurement of length is independent of any knowledge of the source frequency, except for a possible dependence of the correction relating the measurement medium to the reference medium of classical vacuum, which may indeed depend on the frequency of the source. Where a pulse train or some other wave-shaping is used, a range of frequencies may be involved.

#### Electronic length measurements

(PD) Image: John R. Brews
A modulated wave resulting from adding two sine waves of nearly identical wavelength and frequency.

By using sources of several wavelengths to generate sum and difference beat frequencies, absolute distance measurements become possible.[25] The distance is measured in terms of the wavelength of the modulating wave rather than the carrier wave. A commonly used example in both space x and time t is that of the superposition of two waves of almost the same wavelength λ and frequency f:[26]

${\displaystyle F(x,\ t)=\sin \left[2\pi \left({\frac {x}{\lambda -\Delta \lambda }}-(f+\Delta f)t\right)\right]}$${\displaystyle +\sin \left[2\pi \left({\frac {x}{\lambda +\Delta \lambda }}-(f-\Delta f)t\right)\right]}$
${\displaystyle \approx 2\cos \left[2\pi \left({\frac {x}{\lambda _{mod}}}-\Delta f\ t\right)\right]\ \sin \left[2\pi \left({\frac {x}{\lambda }}-ft\right)\right]\ ,}$

which uses the trigonometric formula for the addition of two sine waves, and the approximation Δλ<<λ:

${\displaystyle {\frac {1}{\lambda \pm \Delta \lambda }}={\frac {1}{\lambda }}\ {\frac {1}{1\pm \Delta \lambda /\lambda }}\approx {\frac {1}{\lambda }}\mp {\frac {\Delta \lambda }{\lambda ^{2}}}.}$

Here the modulation wavelength λmod is given by:[26][27]

${\displaystyle \lambda _{mod}={\frac {\lambda ^{2}}{\Delta \lambda }}\ .}$

The modulation wavelength is double that of the envelope itself because each half-wavelength of the modulating cosine wave governs both positive and negative values of the modulated sine wave. By varying the wavelength difference Δλ the value of the measurement unit λmod can be changed. A range of wavelengths from about 40m to 490nm are in use.[28] Accuracy is somewhere between 20mm and 60mm depending upon the distances and the instrument.[29]

### Diffraction measurements

For small objects, different methods are used that also depend upon determining size in units of wavelengths. For instance, in the case of a crystal, atomic spacings can be determined using X-ray diffraction.[30] The present best value for the lattice parameter of silicon, denoted a, is:[31]

a = 543.102 0504(89) × 10−12 m,

corresponding to a resolution of ΔL/L ≈ 3 × 10−10. Similar techniques can provide the dimensions of small structures repeated in large periodic arrays like a diffraction grating.[32]

Such measurements allow the calibration of electron microscopes, extending measurement capabilities. For non-relativistic electrons in an electron microscope, the de Broglie wavelength is:[33]

${\displaystyle \lambda _{e}={\frac {h}{\sqrt {2m_{e}eV}}}\ ,}$

with V the electrical voltage drop traversed by the electron, me the electron mass, e the elementary charge, and h the Planck constant. This wavelength can be measured in terms of inter-atomic spacing using a crystal diffraction pattern, and related to the metre through an optical measurement of the lattice spacing on the same crystal. This process of extending calibration is called metrological traceability.[34]

### Other techniques

Measuring dimensions of localized structures (as opposed to large arrays of atoms like a crystal), as in modern integrated circuits, is done using the scanning electron microscope. This instrument bounces electrons off the object to be measured in a high vacuum enclosure, and the reflected electrons are collected as a photodetector image that is interpreted by a computer. These are not transit time measurements, but are based upon comparison of Fourier transforms of images with theoretical results from computer modeling. Such elaborate methods are required because the image depends on the three-dimensional geometry of the measured feature, for example, the contour of an edge, and not just upon one- or two-dimensional properties. The underlying limitations are the beam width and the wavelength of the electron beam (determining diffraction), determined, as already discussed, by the electron beam energy.[35] The calibration of these scanning electron microscope measurements is tricky, as results depend upon the material measured and its geometry. A typical wavelength is 0.5 Å, and a typical resolution is about 4 nm.

Other small dimension techniques are the atomic force microscope, the focused ion beam and the helium ion microscope. Calibration is attempted using standard samples measured by transmission electron microscopy.[36]

## Other systems of units

In some systems of units, unlike the current SI system, lengths are fundamental units (for example, wavelengths in the older SI units and bohrs in atomic units) and are not defined by times of transit. Even in such units, however, the comparison of two lengths can be made by comparing the two transit times of light along the lengths. Such time-of-flight methodology may or may not be more accurate than the determination of a length as a multiple of the fundamental length unit.

The foot has been defined in the United States to equal exactly 0.3048 m, though an earlier definition of the survey foot was 1200/3937 m, which is different than the current definition by about one part in 500,000. The inch, being 1/12 foot, is exactly 0.0254 m, or 2.54 cm.

The modern Chinese chǐ (市尺), or "Chinese foot", has been defined to equal exactly one-third of a meter. The Hong Kong chek (尺) is exactly 0.371475 m. The Japanese kanejaku (曲尺) was defined as 10/33 m in 1891.

The Spanish vara was fixed at about 835.9 mm in 1801; however, the vara was defined in California as 838.2 mm (33 inches), and in Texas as 846.666 mm (33 1/3 inches).

## References

1. Resolution 1 of the 17th meeting of the Conférence Générale des Poids et Mesures (CGPM) (1983). 1983 definition of the metre. International Bureau of Weights and Measures (BIPM). Retrieved on 2011-03-19.
2. The classical vacuum as a reference medium is described in Werner S. Weiglhofer and Akhlesh Lakhtakia (2003). “§ 4.1 The classical vacuum as reference medium”, Introduction to complex mediums for optics and electromagnetics. SPIE Press, 28, 34, 65. ISBN 9780819449474.  and Tom G. MacKay (2008). “Electromagnetic Fields in Linear Bianisotropic Mediums”, Emil Wolf: Progress in Optics, Volume 51. Elsevier. ISBN 9780444520388.
3. A conversion table for the SI units is maintained by NIST: Ambler Thompson and Barry N Taylor (March, 2008). Length. NIST guide for the use of the international system of units; Section B.9: Factors for units listed by kind of quantity or field of science. NIST. Retrieved on 2011-04-09. Advice on rounding conversions is provided here.
4. A more detailed listing of errors can be found in John S Beers, William B Penzes (December 1992). §4 Re-evaluation of measurement errors. NIST length scale interferometer measurement assurance; NIST document NISTIR 4998 9 ff. Retrieved on 2011-12-17.
5. The formulas used in the calculator and the documentation behind them are found at Engineering metrology toolbox: Refractive index of air calculator. NIST (September 23, 2010). Retrieved on 2011-12-16. The choice is offered to use either the modified Edlén equation or the Ciddor equation. The documentation provides a discussion of how to choose between the two possibilities.
6. §VI: Uncertainty and range of validity. Engineering metrology toolbox: Refractive index of air calculator. NIST (September 23, 2010). Retrieved on 2011-12-16.
7. The BIPM maintains a listing of standard frequencies for use in setting up the metre: Recommended values of standard frequencies. BIPM (9 September 2010). Retrieved on 2011-12-18. One of these is the 633nm line of iodine, taken to correspond to 632 991 212.58 fm with a relative uncertainty of 2.1 × 10−11. See Iodine (λ≈633nm). MEP (Mise en Pratique) 2003. BIPM (2003). Retrieved on 2011-12-18.
8. Maintaining the SI unit of length. National Research Council Canada (2010-02-05). Retrieved on 2011-12-18.
9. F. B. Dunning, Randall G. Hulet (1997). “Physical limits on accuracy and resolution: setting the scale”, Atomic, molecular, and optical physics: electromagnetic radiation, Volume 29, Part 3. Academic Press. ISBN 0124759777. “The error [introduced by using air] can be reduced tenfold if the chamber is filled with an atmosphere of helium rather than air.”
10. For a brief history, see Historical context of the SI. The NIST reference on constants, units and uncertainty. NIST. Retrieved on 2011-03-08.
11. Resolution 6 of the 11th meeting of the CGPM (1960). 1960 definition of the metre. International Bureau of Weights and Measures (BIPM). Retrieved on 2011-03-19.
12. K. M. Evenson, J. S. Wells, F. R. Petersen, B. L. Danielson, and G. W. Day (1972). "Speed of Light from Direct Frequency and Wavelength Measurements of the Methane-Stabilized Laser". Phys Rev Lett vol 29: 1346-1349. DOI:10.1103/PhysRevLett.29.1346. Research Blogging. Some historical remarks are found in: L Hollberg et al. (2005). "Optical frequency/ wavelength references". J. Phys B: At, Mol Opt Phys vol 38: pp. S469-S495.
13. A list of other confirmations is in Table 1 of the article K.M. Baird, D.S. Smith and BG Whitford (1979). "Confirmation of the currently accepted value 299 792 458 m/s for the speed of light". Optics Communications vol 31 (No 3): pp. 367-368. DOI:10.1016/0030-4018(79)90216-5. Research Blogging.
14. For example, see Optical frequency comb: The measurement of optical frequencies. National Research Council of Canada (2009). Retrieved on 2011-04-03. A more technical discussion is found in Th. Udem and F. Riehle (2007). “Frequency comb applications and optical frequency standards”, TW Hänsch, et al., eds.: Metrologia e costanti fondamentali; Volume 166 of Proceedings of the International School of Physics "Enrico Fermi". IOS Press, pp. 317 ff. ISBN 1586037846.
15. For example, see TJ Quinn (2003). "Practical realization of the definition of the metre, including recommended radiations of other optical frequency standards (2001)". Metrologia vol 40: pp. 103-133.
16. A web site listing current recommendations and uncertainties is maintained by the BIPM at Recommended values of standard frequencies. BIPM (2010-09-09). Retrieved on 2011-04-10. along with links to update documentation as updates occur.
17. Miao Zhu, John L Hall (1997). “Chapter 11: Precise wavelength measurements of tunable lasers”, Thomas Lucatorto et al. eds.: Experimental method in the physical sciences. Academic Press, pp. 311 ff. ISBN 0124759777. “While frequency-stabilized tunable lasers can give sub-Hertz line widths, their frequencies measured via interferometric methods are less accurate by many orders. Therefore, in ultrahigh-resolution measurements, say higher than 10−10, direct optical frequency measurement […] is probably a better choice.”
18. A brief rundown is found at Donald Clausing (2006). “Receiver clock correction”, The Aviator's Guide to Navigation, 4rth ed. McGraw-Hill Professional. ISBN 9780071477208.
19. Robert B Fisher and Kurt Konolige (2008). “§22.1.4: Time-of-flight range sensors”, Bruno Siciliano, Oussama Khatib, eds.: Springer handbook of robotics. Springer, pp. 528 ff. ISBN 354023957X.
20. For an overview, see for example, Walt Boyes (2008). “Interferometry and transit time methods”, Instrumentation reference book. Butterworth-Heinemann, p. 89. ISBN 0750683082.
21. An example of a system combining the pulse and interferometer methods is described by Jun Ye (2004). "Absolute measurement of a long, arbitrary distance to less than an optical fringe". Optics Letters vol. 29 (No. 10): p. 1153.
22. The corner cube reflects the incident light in a parallel path that is displaced from the beam incident upon the corner cube. That separation of incident and reflected beams reduces some technical difficulties introduced when the incident and reflected beams are on top of each other. For a discussion of this version of the Michelson interferometer and other types of interferometer, see Joseph Shamir (1999). “§8.7 Using corner cubes”, Optical systems and processes. SPIE Press, pp. 176 ff. ISBN 0819432261.
23. An atomic transition is affected by disturbances, such as collisions with other atoms and frequency shifts from atomic motion due to the Doppler effect, leading to a range of frequencies for the transition referred to as a linewidth. Corresponding to the uncertainty in frequency is an uncertainty in wavelength. In contrast, the speed of light in ideal vacuum is not dependent upon frequency at all.
24. A discussion of interferometer errors is found in the article cited above: Miao Zhu, John L Hall (1997). “Chapter 11: Precise wavelength measurements of tunable lasers”, Thomas Lucatorto et al. eds.: Experimental method in the physical sciences. Academic Press, pp. 311 ff. ISBN 0124759777.
25. S. K. Roy (2004). “§4.4 Basic principle of electronic distance measurement”, Fundamentals of Surveying. PHI Learning Pvt. Ltd, p. 68. ISBN 8120312600.
26. Blair Kinsman (2002). Wind Waves: Their Generation and Propagation on the Ocean Surface, Reprint of Prentice-Hall 1965. Courier Dover Publications, p. 186. ISBN 0486495116.
27. Mark W. Denny (1993). Air and Water: The Biology and Physics of Life's Media. Princeton University Press, p. 289. ISBN 0691025185.
28. C Venkatramaiah (1996). “§18.7.1 Principles of EDM”, Textbook of Surveying. Universities Press (India) Ltd., p. 622. ISBN 817371021X.
29. Satheesh Gopi, R. Sathikumar, N. Madhu (2007). “§8.13 EDM accuracies”, Advanced Surveying: Total Station, GIS And Remote Sensing. Pearson Education India, p. 172. ISBN 8131700674.
30. Peter J. Mohr, Barry N. Taylor, David B. Newell (2008). "CODATA recommended values of the fundamental physical constants: 2006". Rev Mod Phys vol. 80: pp. 633-730. See section 8: Measurements involving silicon crystals, p. 46.
31. Lattice parameter of silicon. The NIST reference on constants, units and uncertainty. National Institute of Standards and Technology. Retrieved on 2011-04-04.
32. A discussion of various types of gratings is found in Abdul Al-Azzawi (2006). “§3.2 Diffraction gratings”, Physical optics: principles and practices. CRC Press, pp. 46 ff. ISBN 0849382971.
33. (2009) “Electron wavelength and relativity”, High-resolution electron microscopy, 3rd ed. Oxford University Press, p. 16. ISBN 0199552754.
34. See Metrological traceability. BIPM. Retrieved on 2011-04-10.
35. Michael T. Postek (2005). “Photomask critical dimension metrology in the scanning electron microscope”, Syed Rizvi: Handbook of photomask manufacturing technology. CRC Press, pp. 457 ff. ISBN 0824753747.  and Harry J. Levinson (2005). “Chapter 9: Metrology”, Principles of lithography, 2nd ed. SPIE Press, pp. 313 ff. ISBN 0819456608.
36. NG Orji et al. (2007). "TEM calibration methods for critical dimension standards". Proc. of SPIE vol. 6518. DOI:10.1117/12.713368. Research Blogging.