Metre (unit)

This article is about Metre (unit). For other uses of the term Metre, please see Metre (disambiguation).

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 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. According to this definition, lengths may be 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. As a consequence of this relation, measurement errors in frequency are mirrored in the wavelengths, and for any specific choice of source they share the same relative standard uncertainty. A list of standard sources and measurement uncertainties is maintained by the BIPM.

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. As a result of the present definition if a transit time approach is used, length measurements are not subject to knowledge of the source frequency, but are subject to the error in measuring transit times. In the case of a wavelength approach using an interferometer, length measurements are subject to the uncertainty in determining source frequency and various interferometer errors. Dominant among these are light beam alignment, collimation and fractional fringe determination. An additional uncertainty in both methods is relating the medium used in a particular measurement to a reference vacuum, taken in SI units to be the classical vacuum, sometimes referred to as the refractive index correction.

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 metre is 3.281 feet.

History

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

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.

After 1960 and continuing to the present, lengths are measured in terms of the number of wavelengths of light that can be fitted into them. The figure shows schematically how length is determined: 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.

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.

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. This interferometer methodology has an experimental uncertainty related to determining the precise frequency of the source. The measurement also requires careful specification of the medium in which the light propagates, which must be related to a reference medium of ideal vacuum.

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. 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.

Precision measurement of lengths still are made in units of wavelengths of light using Michelson interferometry, with today's difference being that the wavelength unit is converted to metres using λ = c₀ / f, rather than by the older more error-prone method involving a comparison with a selected standard source with a wavelength defined in metres. If a comparison of sources is to be made, it is more accurately done by comparing frequencies than by comparing wavelengths. An added advance, unrelated to the change in definition, is that today's sources are much more stable and their frequencies are better known.

Practical measurement

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.

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

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. Measurements are made in units of wavelengths λ at a known frequency f, and related to the metre using λ = c₀ / f. By using sources of several wavelengths to generate sum and difference beat frequencies, absolute distance measurements become possible. Multiple frequencies are used also to make corrections for the non-ideal properties of the medium (departures from classical vacuum), for example, by adding frequencies sensitive to the presence of water vapor.

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. The present best value for the lattice parameter of silicon, denoted a, is:

a = 543.102 064(14) × 10⁻¹² m.

Similar techniques can provide the dimensions of small structures repeated in large periodic arrays like a diffraction grating.

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

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

with V the electrical voltage drop traversed by the electron, m_e 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.

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. 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.

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.

Traditional units

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).