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Length measurement, distance measurement, or range measurement (ranging) refers to the many ways in which length, distance, or range can be measured. The most commonly used approaches are the rulers, followed by transit-time methods and the interferometer methods based upon the speed of light.
For objects such as crystals and diffraction gratings, diffraction is used with X-rays 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.
Standard rulers
The ruler the simplest kind of length measurement tool: lengths are defined by printed marks or engravings on a stick. The metre was initially defined using a ruler before more accurate methods became available.
Gauge blocks are a common method for precise measurement or calibration of measurement tools.
For small or microscopic objects, microphotography where the length is calibrated using a graticule can be used. A graticule is a piece that has lines for precise lengths etched into it. Graticules may be fitted into the eyepiece or they may be used on the measurement plane.
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, assuming that is 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.[1]
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 10 m, 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 metres or < 1 metre, or, in specific applications, tens of centimetres. Time-of-flight systems for robotics (for example, Laser Detection and Ranging LADAR and Light Detection and Ranging LIDAR) aim at lengths of 10–100 m and have an accuracy of about 5–10 mm.[2]
Interferometer measurements
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.[3][4] Generally, transit time measurements are preferred for longer lengths, and interferometers for shorter lengths.[5]
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.[6] 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 = c0 / f. With c0 a defined value of 299,792,458m/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.[7][8][9]
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.[10] Other significant errors are introduced by the interferometer itself; in particular: errors in light beam alignment, collimation and fractional fringe determination.[5][11] Corrections also are made to account for departures of the medium (for example, air)[12] from the reference medium of classical vacuum. 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.[11]
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.[13] The present best value for the lattice parameter of silicon, denoted a, is:[14]
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.[15]
Such measurements allow the calibration of electron microscopes, extending measurement capabilities. For non-relativistic electrons in an electron microscope, the de Broglie wavelength is:[16]
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.[17] 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.[18]
Far and moving targets
Ranging is technique that measures distance or slant range from the observer to a target, especially a far and moving target.
Other devices which measure distance using trigonometry are stadiametric, coincidence and stereoscopic rangefinders. Older methodologies that use a set of known information (usually distance or target sizes) to make the measurement, have been in regular use since the 18th century.
Special ranging makes use of actively synchronized transmission and travel time measurements. The time difference between several received signals is used to determine exact distances (upon multiplication by the speed of light). This principle is used in satellite navigation. In conjunction with a standardized model of the Earth's surface, a location on that surface may be determined with high accuracy. Ranging methods without accurate time synchronization of the receiver are called pseudorange, used, for example, in GPS positioning.
With other systems ranging is obtained from passive radiation measurements only: the noise or radiationsignature of the object generates the signal that is used to determine range. This asynchronous method requires multiple measurements to obtain a range by taking multiple bearings instead of appropriate scaling of active pings, otherwise the system is just capable of providing a simple bearing from any single measurement.
Combining several measurements in a time sequence leads to tracking and tracing. A commonly used term for residing terrestrial objects is surveying.
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.[19] 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.
Nuclear Overhauser effect spectroscopy (NOESY) is a specialized type of nuclear magnetic resonance spectroscopy where distances between atoms can be measured. It is based on the effect where nuclear spin cross-relaxation after excitation by a radio pulse depends on the distance between the nuclei. Unlike spin-spin coupling, NOE propagates through space and does not require that the atoms are connected by bonds, so it is a true distance measurement instead of a chemical measurement. Unlike diffraction measurements, NOESY does not require a crystalline sample, but is done in solution state and can be applied to substances that are difficult to crystallize.
Dashed black lines: Uncertain calibration ladder step.
The cosmic distance ladder (also known as the extragalactic distance scale) is the succession of methods by which astronomers determine the distances to celestial objects. A direct distance measurement of an astronomical object is possible only for those objects that are "close enough" (within about a thousand parsecs) to Earth. The techniques for determining distances to more distant objects are all based on various measured correlations between methods that work at close distances and methods that work at larger distances. Several methods rely on a standard candle, which is an astronomical object that has a known luminosity.
The ladder analogy arises because no single technique can measure distances at all ranges encountered in astronomy. Instead, one method can be used to measure nearby distances, a second can be used to measure nearby to intermediate distances, and so on. Each rung of the ladder provides information that can be used to determine the distances at the next higher rung.
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.
In physics, interference is a phenomenon in which two coherent waves are combined by adding their intensities or displacements with due consideration for their phase difference. The resultant wave may have greater intensity or lower amplitude if the two waves are in phase or out of phase, respectively. Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves, gravity waves, or matter waves as well as in loudspeakers as electrical waves.
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 terahertz, between the infrared and the ultraviolet.
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.
In optics, the refractive index of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings. Wavelength is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. Wavelength is commonly designated by the Greek letter lambda (λ). The term "wavelength" is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.
UV spectroscopy or UV–visible spectrophotometry refers to absorption spectroscopy or reflectance spectroscopy in part of the ultraviolet and the full, adjacent visible regions of the electromagnetic spectrum. Being relatively inexpensive and easily implemented, this methodology is widely used in diverse applied and fundamental applications. The only requirement is that the sample absorb in the UV-Vis region, i.e. be a chromophore. Absorption spectroscopy is complementary to fluorescence spectroscopy. Parameters of interest, besides the wavelength of measurement, are absorbance (A) or transmittance (%T) or reflectance (%R), and its change with time.
X-ray fluorescence (XRF) is the emission of characteristic "secondary" X-rays from a material that has been excited by being bombarded with high-energy X-rays or gamma rays. The phenomenon is widely used for elemental analysis and chemical analysis, particularly in the investigation of metals, glass, ceramics and building materials, and for research in geochemistry, forensic science, archaeology and art objects such as paintings.
In the physical sciences, the wavenumber, also known as repetency, is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. It is analogous to temporal frequency, which is defined as the number of wave cycles per unit time or radians per unit time.
Interferometry is a technique which uses the interference of superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important investigative technique in the fields of astronomy, fiber optics, engineering metrology, optical metrology, oceanography, seismology, spectroscopy, quantum mechanics, nuclear and particle physics, plasma physics, biomolecular interactions, surface profiling, microfluidics, mechanical stress/strain measurement, velocimetry, optometry, and making holograms.
Transmission electron microscopy (TEM) is a microscopy technique in which a beam of electrons is transmitted through a specimen to form an image. The specimen is most often an ultrathin section less than 100 nm thick or a suspension on a grid. An image is formed from the interaction of the electrons with the sample as the beam is transmitted through the specimen. The image is then magnified and focused onto an imaging device, such as a fluorescent screen, a layer of photographic film, or a sensor such as a scintillator attached to a charge-coupled device.
The resolution of an optical imaging system – a microscope, telescope, or camera – can be limited by factors such as imperfections in the lenses or misalignment. However, there is a principal limit to the resolution of any optical system, due to the physics of diffraction. An optical system with resolution performance at the instrument's theoretical limit is said to be diffraction-limited.
Time of flight (ToF) is the measurement of the time taken by an object, particle or wave to travel a distance through a medium. This information can then be used to measure velocity or path length, or as a way to learn about the particle or medium's properties. The traveling object may be detected directly or indirectly.
The Mach–Zehnder interferometer is a device used to determine the relative phase shift variations between two collimated beams derived by splitting light from a single source. The interferometer has been used, among other things, to measure phase shifts between the two beams caused by a sample or a change in length of one of the paths. The apparatus is named after the physicists Ludwig Mach and Ludwig Zehnder; Zehnder's proposal in an 1891 article was refined by Mach in an 1892 article. Demonstrations of Mach–Zehnder interferometry with particles other than photons had been demonstrated as well in multiple experiments.
Phase-contrast imaging is a method of imaging that has a range of different applications. It measures differences in the refractive index of different materials to differentiate between structures under analysis. In conventional light microscopy, phase contrast can be employed to distinguish between structures of similar transparency, and to examine crystals on the basis of their double refraction. This has uses in biological, medical and geological science. In X-ray tomography, the same physical principles can be used to increase image contrast by highlighting small details of differing refractive index within structures that are otherwise uniform. In transmission electron microscopy (TEM), phase contrast enables very high resolution (HR) imaging, making it possible to distinguish features a few Angstrom apart.
Nanometrology is a subfield of metrology, concerned with the science of measurement at the nanoscale level. Nanometrology has a crucial role in order to produce nanomaterials and devices with a high degree of accuracy and reliability in nanomanufacturing.
Fourier-transform infrared spectroscopy (FTIR) is a technique used to obtain an infrared spectrum of absorption or emission of a solid, liquid, or gas. An FTIR spectrometer simultaneously collects high-resolution spectral data over a wide spectral range. This confers a significant advantage over a dispersive spectrometer, which measures intensity over a narrow range of wavelengths at a time.
A measuring instrument is a device to measure a physical quantity. In the physical sciences, quality assurance, and engineering, measurement is the activity of obtaining and comparing physical quantities of real-world objects and events. Established standard objects and events are used as units, and the process of measurement gives a number relating the item under study and the referenced unit of measurement. Measuring instruments, and formal test methods which define the instrument's use, are the means by which these relations of numbers are obtained. All measuring instruments are subject to varying degrees of instrument error and measurement uncertainty. These instruments may range from simple objects such as rulers and stopwatches to electron microscopes and particle accelerators. Virtual instrumentation is widely used in the development of modern measuring instruments.
A common-path interferometer is a class of interferometers in which the reference beam and sample beams travel along the same path. Examples include the Sagnac interferometer, Zernike phase-contrast interferometer, and the point diffraction interferometer. A common-path interferometer is generally more robust to environmental vibrations than a "double-path interferometer" such as the Michelson interferometer or the Mach–Zehnder interferometer. Although travelling along the same path, the reference and sample beams may travel along opposite directions, or they may travel along the same direction but with the same or different polarization.
As described here, white light interferometry is a non-contact optical method for surface height measurement on 3D structures with surface profiles varying between tens of nanometers and a few centimeters. It is often used as an alternative name for coherence scanning interferometry in the context of areal surface topography instrumentation that relies on spectrally-broadband, visible-wavelength light.
The operation of a photon scanning tunneling microscope (PSTM) is analogous to the operation of an electron scanning tunneling microscope, with the primary distinction being that PSTM involves tunneling of photons instead of electrons from the sample surface to the probe tip. A beam of light is focused on a prism at an angle greater than the critical angle of the refractive medium in order to induce total internal reflection within the prism. Although the beam of light is not propagated through the surface of the refractive prism under total internal reflection, an evanescent field of light is still present at the surface.
References
↑ A brief rundown is found at Donald Clausing (2006). "Receiver clock correction". The Aviator's Guide to Navigation (4thed.). McGraw-Hill Professional. ISBN978-0-07-147720-8.
↑ 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. ISBN978-0-8194-3226-1.
↑ 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.
↑ For example, the index of refraction of air can be found based upon entering a wavelength in vacuum into the calculator provided by NIST: "Refractive index of air calculator". Engineering metrology toolbox. NIST. September 23, 2010. Retrieved 2011-12-08.
↑ Peter J. Mohr; Barry N. Taylor; David B. Newell (2008). "CODATA recommended values of the fundamental physical constants: 2006". Rev Mod Phys. 80 (2): 633–730. arXiv:0801.0028. Bibcode:2008RvMP...80..633M. doi:10.1103/revmodphys.80.633. See section 8: Measurements involving silicon crystals, p. 46.
↑ Mark H. Jones; Robert J. Lambourne; David John Adams (2004). An introduction to galaxies and cosmology. Cambridge University Press. pp.88 ff. ISBN978-0-521-54623-2. Relating one step on the distance ladder to another involves a process of calibration, that is, the use of an established method of measurement to give absolute meaning to the relative measurements provided by some other method.
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