Length measurement
Encyclopedia
Length measurement is implemented in practice in an amazing variety of ways. This article is restricted to only a few methods, in particular, those used with SI units. The most commonly used approaches are the transit-time methods and the interferometer methods based upon the speed of light
. For objects such as crystals and diffraction grating
s, diffraction
is used with X-ray
s and electron beams. Measurement techniques for three-dimensional structures very small in every dimension use specialized instruments such as ion microscopy coupled with intensive computer modeling.
For a discussion of astronomical methods for determining cosmological distances, see the article Cosmic distance ladder
.
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.
Such techniques vary in accuracy according to the distances over which they are intended for use. For example, LORAN-C is accurate to about 6 km, 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, tens of centimeters. Time-of-flight systems for robotics (for example, Laser Detection and Ranging LADAR and Light Detection and Ranging LIDAR
) aim at lengths of 10-100m and have an accuracy of about 5–10 mm
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 corner cube serves to displace the incident from the reflected beam, which avoids some complications caused by superposing the two beams. 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 compare 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 (bright sun) and dims (dark 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 corresponding to a particular atomic transition
. The length in wavelengths can be converted to a length in units of metres if the selected transition has a known frequency f. The length as a certain number of wavelengths λ is related to the metre using λ = . With c0 a defined value of 299,792,458 m/s, the error in a measured length in wavelengths is increased by this conversion to metres by the error in measuring the frequency of the light source.
By using sources of several wavelengths to generate sum and difference beat frequencies
, absolute distance measurements become possible.
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. Other significant errors are introduced by the interferometer itself; in particular: errors in light beam alignment, collimation and fractional fringe determination. Corrections also are made to account for departures of the medium (for example, air) from the reference medium of classical vacuum. Resolution using wavelengths is in the range of ΔL/L ≈ depending upon the length measured, the wavelength and the type of interferometer used.
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.
corresponding to a resolution of ΔL/L ≈ 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 microscope
s, extending measurement capabilities. For non-relativistic electrons in an electron microscope, the de Broglie wavelength is:
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. The use of metrological traceability to connect different regimes of measurement is similar to the idea behind the cosmic distance ladder
for different ranges of astronomical length. Both calibrate different methods for length measurement using overlapping ranges of applicability.
s, 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 transform
s 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.
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
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
. For objects such as crystals and diffraction grating
Diffraction grating
In optics, a diffraction grating is an optical component with a periodic structure, which splits and diffracts light into several beams travelling in different directions. The directions of these beams depend on the spacing of the grating and the wavelength of the light so that the grating acts as...
s, diffraction
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...
is used with X-ray
X-ray
X-radiation is a form of electromagnetic radiation. X-rays have a wavelength in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz and energies in the range 120 eV to 120 keV. They are shorter in wavelength than UV rays and longer than gamma...
s and electron beams. Measurement techniques for three-dimensional structures very small in every dimension use specialized instruments such as ion microscopy coupled with intensive computer modeling.
For a discussion of astronomical methods for determining cosmological distances, see the article Cosmic distance ladder
Cosmic distance ladder
The cosmic distance ladder is the succession of methods by which astronomers determine the distances to celestial objects. A real direct distance measurement of an astronomical object is possible only for those objects that are "close enough" to Earth...
.
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. If light is used for the signal, its speedSpeed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
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
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
of the medium larger than one slows the light.
Transit-time measurement underlies most radio navigation systems for boats and aircraft, for example, radar
Radar
Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...
and the nearly obsolete Long Range Aid to Navigation LORAN-C
LORAN
LORAN is a terrestrial radio navigation system using low frequency radio transmitters in multiple deployment to determine the location and speed of the receiver....
. 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
Global Positioning System
The Global Positioning System is a space-based global navigation satellite system that provides location and time information in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more GPS satellites...
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 6 km, GPS about 10m, enhanced GPS, in which a correction signal is transmitted from terrestrial stations (that is, differential GPS
Differential GPS
Differential Global Positioning System is an enhancement to Global Positioning System that provides improved location accuracy, from the 15-meter nominal GPS accuracy to about 10 cm in case of the best implementations....
(DGPS)) or via satellites (that is, Wide Area Augmentation System
Wide Area Augmentation System
The Wide Area Augmentation System is an air navigation aid developed by the Federal Aviation Administration to augment the Global Positioning System , with the goal of improving its accuracy, integrity, and availability...
(WAAS)) can bring accuracy to a few meters or < 1 meter, or, in specific applications, tens of centimeters. Time-of-flight systems for robotics (for example, Laser Detection and Ranging LADAR and Light Detection and Ranging LIDAR
LIDAR
LIDAR is an optical remote sensing technology that can measure the distance to, or other properties of a target by illuminating the target with light, often using pulses from a laser...
) aim at lengths of 10-100m and have an accuracy of about 5–10 mm
Interferometer measurements
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. Generally, transit time measurements are preferred for longer lengths, and interferometers for shorter lengths.The figure shows schematically how length is determined using a Michelson interferometer
Michelson interferometer
The Michelson interferometer is the most common configuration for optical interferometry and was invented by Albert Abraham Michelson. An interference pattern is produced by splitting a beam of light into two paths, bouncing the beams back and recombining them...
: the two panels show a laser source emitting a light beam split by a beam splitter
Beam splitter
A beam splitter is an optical device that splits a beam of light in two. It is the crucial part of most interferometers.In its most common form, a rectangle, it is made from two triangular glass prisms which are glued together at their base using Canada balsam...
(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 corner cube serves to displace the incident from the reflected beam, which avoids some complications caused by superposing the two beams. 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 compare 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 (bright sun) and dims (dark 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 corresponding to a particular atomic transition
Atomic spectral line
In physics, atomic spectral lines are of two types:* An emission line is formed when an electron makes a transition from a particular discrete energy level of an atom, to a lower energy state, emitting a photon of a particular energy and wavelength...
. The length in wavelengths can be converted to a length in units of metres if the selected transition has a known frequency f. The length as a certain number of wavelengths λ is related to the metre using λ = . With c0 a defined value of 299,792,458 m/s, the error in a measured length in wavelengths is increased by this conversion to metres by the error in measuring the frequency of the light source.
By using sources of several wavelengths to generate sum and difference beat frequencies
Beat (acoustics)
In acoustics, a beat is an interference between two sounds of slightly different frequencies, perceived as periodic variations in volume whose rate is the difference between the two frequencies....
, absolute distance measurements become possible.
This methodology for length determination requires a careful specification of the wavelength of the light used, and is one reason for employing a laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
source where the wavelength can be held stable. Regardless of stability, however, the precise frequency of any source has linewidth limitations. Other significant errors are introduced by the interferometer itself; in particular: errors in light beam alignment, collimation and fractional fringe determination. Corrections also are made to account for departures of the medium (for example, air) from the reference medium of classical vacuum. Resolution using wavelengths is in the range of ΔL/L ≈ depending upon the length measured, the wavelength and the type of interferometer used.
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.
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. The present best value for the lattice parameter of silicon, denoted a, is:-
- a = 543.102 0504(89) × 10−12 m,
corresponding to a resolution of ΔL/L ≈ Similar techniques can provide the dimensions of small structures repeated in large periodic arrays like a diffraction grating
Diffraction grating
In optics, a diffraction grating is an optical component with a periodic structure, which splits and diffracts light into several beams travelling in different directions. The directions of these beams depend on the spacing of the grating and the wavelength of the light so that the grating acts as...
.
Such measurements allow the calibration of electron microscope
Electron microscope
An electron microscope is a type of microscope that uses a beam of electrons to illuminate the specimen and produce a magnified image. Electron microscopes have a greater resolving power than a light-powered optical microscope, because electrons have wavelengths about 100,000 times shorter than...
s, extending measurement capabilities. For non-relativistic electrons in an electron microscope, the de Broglie wavelength is:
with V the electrical voltage drop traversed by the electron, me the electron mass, e the elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
, and h the Planck constant
Planck constant
The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...
. 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. The use of metrological traceability to connect different regimes of measurement is similar to the idea behind the cosmic distance ladder
Cosmic distance ladder
The cosmic distance ladder is the succession of methods by which astronomers determine the distances to celestial objects. A real direct distance measurement of an astronomical object is possible only for those objects that are "close enough" to Earth...
for different ranges of astronomical length. Both calibrate different methods for length measurement using overlapping ranges of applicability.
Other techniques
Measuring dimensions of localized structures (as opposed to large arrays of atoms like a crystal), as in modern integrated circuitIntegrated circuit
An integrated circuit or monolithic integrated circuit is an electronic circuit manufactured by the patterned diffusion of trace elements into the surface of a thin substrate of semiconductor material...
s, is done using the scanning electron microscope
Scanning electron microscope
A scanning electron microscope is a type of electron microscope that images a sample by scanning it with a high-energy beam of electrons in a raster scan pattern...
. 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 transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
s 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
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...
), 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
Atomic force microscope
Atomic force microscopy or scanning force microscopy is a very high-resolution type of scanning probe microscopy, with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the optical diffraction limit...
, the focused ion beam
Focused ion beam
Focused ion beam, also known as FIB, is a technique used particularly in the semiconductor industry, materials science and increasingly in the biological field for site-specific analysis, deposition, and ablation of materials. An FIB setup is a scientific instrument that resembles a scanning...
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
Atomic units
Atomic units form a system of natural units which is especially convenient for atomic physics calculations. There are two different kinds of atomic units, which one might name Hartree atomic units and Rydberg atomic units, which differ in the choice of the unit of mass and charge. This article...
) 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
See also
- DistanceDistanceDistance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...
- Ellipsometry#Imaging ellipsometry
- Length scaleLength scaleIn physics, length scale is a particular length or distance determined with the precision of one order of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot affect each other and are said to decouple...
- Low-energy electron microscopyLow-energy electron microscopyLow-energy electron microscopy, or LEEM, is an analytical surface science technique invented by Ernst Bauer in 1962, however, not fully developed until 1985. LEEM is a technique used by surface scientists to image atomically clean surfaces, atom-surface interactions, and thin films...
- Orders of magnitude (length)
- Pulse-Doppler radarPulse-doppler radarPulse-Doppler is a 4D radar system capable of detecting both target 3D location as well as measuring radial velocity . It uses the Doppler effect to avoid overloading computers and operators as well as to reduce power consumption...
- Range ambiguity resolutionRange ambiguity resolutionRange ambiguity resolution is a technique used with medium Pulse repetition frequency radar to obtain range information for distances that exceed the distance between transmit pulses.This signal processing technique is required with pulse-Doppler radar....