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| '''Length measurement''' is implemented in practice in many ways. 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 beam]]s. Measurement techniques for three-dimensional structures very small in every dimension use specialized instruments such as [[Focused ion beam#FIB imaging|ion microscopy]] coupled with intensive computer modeling. | | Hello, I'm Lyda, a 25 year old from Gravatai, Brazil.<br>My hobbies include (but are not limited to) Squash, Computer programming and watching Supernatural.<br>xunjie 精神的な創造性とスタイルガイドへの概念の解釈に、 |
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| For a discussion of astronomical methods for determining cosmological distances, see the article [[Cosmic distance ladder]].
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| ==Transit-time measurement==
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| 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 of light|speed]] depends upon the medium in which it propagates; in [[SI units]] the speed is a defined value ''c''<sub>0</sub> in the reference medium of [[Vacuum#Electromagnetism|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 [[Vacuum#Electromagnetism|classical vacuum]]. A [[refractive index]] of the medium larger than one slows the light.
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| 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|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|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.<ref name= GPS/>
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| Such techniques vary in accuracy according to the distances over which they are intended for use. For example, LORAN-C is accurate to about {{nowrap|6 km,}} GPS about {{nowrap|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 meters or {{nowrap|< 1 meter,}} or, in specific applications, tens of centimeters. Time-of-flight systems for robotics (for example, Laser Detection and Ranging [[Laser rangefinder|LADAR]] and Light Detection and Ranging [[LIDAR]]) aim at lengths of {{nowrap|10 - 100 m}} and have an accuracy of about {{nowrap|5 – 10 mm}}<ref name=Siciliano/>
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| ==Interferometer measurements==
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| [[File:Michelson interferometer with corner cubes.png|thumb|Measuring a length in wavelengths of light using an [[interferometer]].]]
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| 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.<ref name=Boyes/><ref name=Ye/> Generally, transit time measurements are preferred for longer lengths, and interferometers for shorter lengths.<ref name=Yoshizawa/>
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| 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.<ref name=CC/> 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.
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| 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 (wave propagation)|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 spectral line|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'' = {{nowrap|''c<sub>0</sub> / f''}}. With ''c<sub>0</sub>'' 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.
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| By using sources of several wavelengths to generate sum and difference [[Envelope_(waves)#Example:_Beating_waves|beat frequencies]], absolute distance measurements become possible.<ref name=Zheng/><ref name=Roy/><ref name=Paul/>
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| 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.<ref name= frequency/> Other significant errors are introduced by the interferometer itself; in particular: errors in light beam alignment, collimation and fractional fringe determination.<ref name=errors/><ref name=Yoshizawa/> Corrections also are made to account for departures of the medium (for example, air<ref name=air/>) from the reference medium of [[Vacuum#Electromagnetism|classical vacuum]]. Resolution using wavelengths is in the range of ΔL/L ≈ {{nowrap|10<sup>−9</sup> – 10<sup>−11</sup>}} depending upon the length measured, the wavelength and the type of interferometer used.<ref name=errors/>
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| 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 [[Vacuum#Electromagnetism|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.
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| 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.
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| ==Diffraction measurements==
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| 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]].<ref name=Mohr/> The present best value for the lattice parameter of silicon, denoted ''a'', is:<ref name=silicon/>
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| ::a = 543.102 0504(89) × 10<sup>−12</sup> m,
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| corresponding to a resolution of ΔL/L ≈ {{nowrap|3 × 10<sup>−10</sup>.}} Similar techniques can provide the dimensions of small structures repeated in large periodic arrays like a [[diffraction grating]].<ref name=grating/>
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| 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:<ref name=Spence/>
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| :<math>\lambda_e = \frac{h}{\sqrt{2m_e e V}} \ , </math>
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| with ''V'' the electrical voltage drop traversed by the electron, ''m<sub>e</sub>'' 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]]''.<ref name=traceability>
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| See {{cite web |title=Metrological traceability |publisher=BIPM |url=http://www.bipm.org/en/bipm/calibrations/traceability.html |accessdate=2011-04-10}}
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| </ref> 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.<ref name=Adams/>
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| ==Other techniques==
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| Measuring dimensions of localized structures (as opposed to large arrays of atoms like a crystal), as in modern [[integrated circuit]]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.<ref name=IC_linewidth>
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| {{cite book |title=Handbook of photomask manufacturing technology |author=Michael T. Postek |editor=Syed Rizvi |url=http://books.google.com/books?id=0Smk9-VI1fcC&pg=PA485 |pages=457 ''ff'' |chapter=Photomask critical dimension metrology in the scanning electron microscope |url=http://books.google.com/books?id=0Smk9-VI1fcC&pg=PA457 |isbn=0-8247-5374-7 |year=2005 |publisher=CRC Press}} and {{cite book |title= Principles of lithography |chapter=Chapter 9: Metrology |author=Harry J. Levinson |url=http://books.google.com/books?id=EjMpqEy07bsC&pg=PA313 |pages=313 ''ff'' |isbn=0-8194-5660-8 |year=2005 |edition=2nd ed |publisher=SPIE Press}}
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| </ref>
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| The calibration of these scanning electron microscope measurements is tricky, as results depend upon the material measured and its geometry. A typical wavelength is {{nowrap|0.5 Å,}} and a typical resolution is about {{nowrap|4 nm.}}
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| 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 microscope]] (TEM).<ref name=Orji>
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| {{cite journal |title=TEM calibration methods for critical dimension standards |author=NG Orji ''et al.'' |url=ftp://129.6.13.25/pub/eeel/cresswell/orji_spie_2007.pdf |year=2007 |journal= Proc. of SPIE |volume= 6518 |doi=10.1117/12.713368}}
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| </ref>
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| ==Other systems of units==
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| 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
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| == References ==
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| {{Reflist|refs=
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| <ref name=Adams>
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| {{cite book |title=An introduction to galaxies and cosmology |author=Mark H. Jones, Robert J. Lambourne, David John Adams |url=http://books.google.com/books?id=36K1PfetZegC&pg=PA88 |pages=88 ''ff'' |quote=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.|isbn=0-521-54623-0 |publisher=Cambridge University Press |year=2004}}
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| </ref>
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| <ref name=air>
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| For example, the index of refraction of air can be found based upon entering a wavelength in [[classical vacuum|vacuum]] into the calculator provided by NIST: {{cite web |title=Refractive index of air calculator |work= Engineering metrology toolbox |url=http://emtoolbox.nist.gov/Wavelength/Documentation.asp |publisher=NIST |date=September 23, 2010|accessdate=2011-12-08}}
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| </ref>
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| <ref name=Boyes>
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| For an overview, see for example, {{cite book |title=Instrumentation reference book |author=Walt Boyes |url=http://books.google.com/books?id=ZvscLzOlkNgC&pg=PA89 |page= 89 |chapter=Interferometry and transit-time methods |isbn=0-7506-8308-2 |year=2008 |publisher=Butterworth-Heinemann}}
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| </ref>
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| <ref name=CC>
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| 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 {{cite book |title=Optical systems and processes |author=Joseph Shamir |url=http://books.google.com/books?id=C7MQjJPcfUQC&pg=PA176 |chapter=§8.7 Using corner cubes |isbn=0-8194-3226-1 |year=1999 |publisher=SPIE Press |pages=176 ''ff''}}
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| </ref>
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| <ref name=errors>
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| A discussion of interferometer errors is found in the article cited above: {{cite book |title=Experimental method in the physical sciences |chapter=Chapter 11: Precise wavelength measurements of tunable lasers |author=Miao Zhu, John L Hall |isbn=0-12-475977-7 |year=1997 |publisher=Academic Press |editor=Thomas Lucatorto ''et al.'' eds.|url=http://books.google.com/books?id=FV4Y39AGYuYC&pg=PA311 |pages=311 ''ff''}}
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| </ref>
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| <ref name= frequency>
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| 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.
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| </ref>
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| <ref name=grating>
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| A discussion of various types of gratings is found in {{cite book |title=Physical optics: principles and practices |author=Abdul Al-Azzawi |url=http://books.google.com/books?id=-qJCp026f-wC&pg=PA46 |pages=46 ''ff'' |chapter=§3.2 Diffraction gratings |isbn=0-8493-8297-1 |year=2006 |publisher=CRC Press}}
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| </ref>
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| <ref name= GPS>
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| A brief rundown is found at {{cite book |title=The Aviator's Guide to Navigation |chapter=Receiver clock correction |author=Donald Clausing |url=http://www.amazon.com/Aviators-Guide-Navigation-Donald-Clausing/dp/0071477209/ref=sr_1_1?s=books&ie=UTF8&qid=1302390225&sr=1-1#reader_0071477209 |isbn=978-0-07-147720-8 |publisher=McGraw-Hill Professional |year=2006 |edition=4th}}
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| </ref>
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| <ref name=Zheng>
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| {{cite book |title=Optical Frequency-Modulated Continuous-Wave (FMCW) Interferometry |author=Jesse Zheng |isbn=0-387-23009-2 |year=2005 |publisher=Springer |url=http://books.google.com/books?id=b9vz97IrRa8C&pg=PA1}}
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| </ref>
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| <ref name=Mohr>
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| {{cite journal |title=CODATA recommended values of the fundamental physical constants: 2006 |journal=Rev Mod Phys |volume= 80 |pages= 633–730 |year=2008 |url=http://arxiv.org/abs/0801.0028 |author= Peter J. Mohr, Barry N. Taylor, David B. Newell}} See section 8: Measurements involving silicon crystals, p. 46.
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| </ref>
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| <ref name=Paul>
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| {{cite book |title=Basic Surveying |author=W Whyte, R Paul |isbn=0-7506-1771-3 |publisher=Laxton's |chapter=§7.3 Electromagnetic distance measurement |pages=136 ''ff'' |url=http://www.amazon.com/gp/reader/0750617713/ref=sib_dp_pt#reader-link |isbn=0-7506-1771-3 |year=1997 |edition=4th}}
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| </ref>
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| <ref name=Roy>
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| {{cite book |title=Fundamentals of Surveying |author=SK Roy |isbn=0-7506-1771-3 |publisher=Laxton's |chapter=§7.3 Electromagnetic distance measurement |pages=136 ''ff'' |url=http://books.google.com/books?id=BxYeF-gQ1bMC&pg=PR62 |pages=62 ''ff'' |chapter=§4.4 Basic principles of electronic distance measurement |isbn=81-203-4198-8 |publisher=PHI Learning Pvt. Ltd. |edition=2nd |year=2010}}
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| </ref>
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| <ref name=Siciliano>
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| {{cite book |title=Springer handbook of robotics |author=Robert B Fisher and Kurt Konolige |editor=Bruno Siciliano, [[Oussama Khatib]], eds. |url=http://books.google.com/books?id=Xpgi5gSuBxsC&pg=PA528 |pages=528 ''ff'' |chapter=§22.1.4: Time-of-flight range sensors |isbn= 354023957X |=Springer |year=2008}}
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| </ref>
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| <ref name=silicon>
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| {{cite web |title=Lattice parameter of silicon |url=http://physics.nist.gov/cgi-bin/cuu/Value?asil|search_for=silicon |work=The NIST reference on constants, units and uncertainty |publisher=[[National Institute of Standards and Technology]] |accessdate=2011-04-04}}
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| </ref>
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| <ref name=Spence>
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| {{cite book |title=High-resolution electron microscopy |page= 16 |chapter=Electron wavelength and relativity |url=http://www.amazon.com/High-Resolution-Microscopy-Monographs-Chemistry-Materials/dp/0199552754/ref=sr_1_1?s=books&ie=UTF8&qid=1302394530&sr=1-1#reader_0199552754 |edition=3rd ed |year=2009 |isbn=0-19-955275-4 |publisher=Oxford University Press}}
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| </ref>
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| <ref name=Ye>
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| An example of a system combining the pulse and interferometer methods is described by {{cite journal |author=Jun Ye |title=Absolute measurement of a long, arbitrary distance to less than an optical fringe |url=http://jila.colorado.edu/yelabs/pubs/scienceArticles/2004/sArticle_2004_05_absolutedistance.pdf |journal=Optics Letters |year=2004 |volume= 29 |issue= 10 |page= 1153 }}
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| </ref>
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| <ref name=Yoshizawa>
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| {{cite book |title=Handbook of optical metrology: principles and applications |author=René Schödel|editor=Tōru Yoshizawa |url=http://books.google.com/books?id=DdzBQsqPbzcC&pg=PA366 |page=366 |chapter=Chapter 15: Length and size |volume=Volume 10 |isbn=0-8493-3760-7 |year=2009 |publisher=CRC Press}}
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| </ref>
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| }}
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| == See also ==
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| * [[Distance]]
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| * [[Ellipsometry#Imaging ellipsometry]]
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| * [[Frequency-modulated continuous-wave radar]] (FMCW)
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| * [[Length scale]]
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| * [[Low-energy electron microscopy]]
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| * [[Orders of magnitude (length)]]
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| * [[Pulse-Doppler radar]]
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| * [[Range ambiguity resolution]]
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| {{citizendium|title=Metre (unit)}}
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| [[Category:SI units]]
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| [[Category:Concepts in physics]]
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| [[Category:Interferometers]]
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| [[Category:Interference]]
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| [[Category:Metrology]]
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| [[Category:X-rays]]
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| [[Category:Scientific techniques]]
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| [[Category:Diffraction]]
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| [[Category:Scanning probe microscopy]]
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