Editing Interferometry (section)

===Engineering and applied science===
[[File:Optical flat interference fringes.jpg|thumb|300px|Figure 13. Optical flat interference fringes. ''(left)'' flat surface, ''(right)'' curved surface.]]
[[Image:Optical flat interference.svg|thumb|upright=1.6|How interference fringes are formed by an optical flat resting on a reflective surface.  The gap between the surfaces and the [[wavelength]] of the light waves are greatly exaggerated.]]

Newton (test plate) interferometry is frequently used in the optical industry for testing the quality of surfaces as they are being shaped and figured. Fig.&nbsp;13 shows photos of reference flats being used to check two test flats at different stages of completion, showing the different patterns of interference fringes. The reference flats are resting with their bottom surfaces in contact with the test flats, and they are illuminated by a monochromatic light source. The light waves reflected from both surfaces interfere, resulting in a pattern of bright and dark bands. The surface in the left photo is nearly flat, indicated by a pattern of straight parallel interference fringes at equal intervals. The surface in the right photo is uneven, resulting in a pattern of curved fringes. Each pair of adjacent fringes represents a difference in surface elevation of half a wavelength of the light used, so differences in elevation can be measured by counting the fringes. The flatness of the surfaces can be measured to millionths of an inch by this method. To determine whether the surface being tested is concave or convex with respect to the reference optical flat, any of several procedures may be adopted. One can observe how the fringes are displaced when one presses gently on the top flat. If one observes the fringes in white light, the sequence of colors becomes familiar with experience and aids in interpretation. Finally one may compare the appearance of the fringes as one moves ones head from a normal to an oblique viewing position.<ref name=Mantravadi2006>{{Cite book | last1 = Mantravadi | first1 = M. V. | last2 = Malacara | first2 = D. | doi = 10.1002/9780470135976.ch1 | chapter = Newton, Fizeau, and Haidinger Interferometers | title = Optical Shop Testing | pages = 1 | year = 2007 |isbn=978-0-470-13597-6 }}</ref> These sorts of maneuvers, while common in the optical shop, are not suitable in a formal testing environment. When the flats are ready for sale, they will typically be mounted in a Fizeau interferometer for formal testing and certification.

Fabry-Pérot etalons are widely used in [[telecommunications]], [[lasers]] and [[spectroscopy]] to control and measure the wavelengths of light. [[Dichroic filters]] are multiple layer [[Thin-film interference|thin-film]] etalons. In telecommunications, [[wavelength-division multiplexing]], the technology that enables the use of multiple wavelengths of light through a single optical fiber, depends on filtering devices that are thin-film etalons. Single-mode lasers employ etalons to suppress all [[optical cavity]] modes except the single one of interest.<ref name=HariharanBasics2007/>{{rp|42}}

[[File:Twyman-Green interferometer.png|thumb|300px|right|Figure 14. Twyman–Green Interferometer]]

The Twyman–Green interferometer, invented by Twyman and Green in 1916, is a variant of the Michelson interferometer widely used to test optical components.<ref name=Malacara2006_2>{{Cite book | last1 = Malacara | first1 = D. | chapter = Twyman–Green Interferometer | doi = 10.1002/9780470135976.ch2 | title = Optical Shop Testing | pages = 46–96 | year = 2007 |isbn=978-0-470-13597-6}}</ref> The basic characteristics distinguishing it from the Michelson configuration are the use of a monochromatic point light source and a collimator. Michelson (1918) criticized the Twyman–Green configuration as being unsuitable for the testing of large optical components, since the light sources available at the time had limited [[coherence length]]. Michelson pointed out that constraints on geometry forced by limited coherence length required the use of a reference mirror of equal size to the test mirror, making the Twyman–Green impractical for many purposes.<ref name=Michelson1918>{{Cite journal | doi = 10.1073/pnas.4.7.210 | last1 = Michelson | first1 = A. A. | title = On the Correction of Optical Surfaces | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 4 | issue = 7 | pages = 210–212 | year = 1918 | pmid = 16576300 | pmc = 1091444 |bibcode = 1918PNAS....4..210M | doi-access = free }}</ref> Decades later, the advent of laser light sources answered Michelson's objections. (A Twyman–Green interferometer using a laser light source and unequal path length is known as a Laser Unequal Path Interferometer, or LUPI.) Fig.&nbsp;14 illustrates a Twyman–Green interferometer set up to test a lens. Light from a monochromatic point source is expanded by a diverging lens (not shown), then is collimated into a parallel beam. A convex spherical mirror is positioned so that its center of curvature coincides with the focus of the lens being tested. The emergent beam is recorded by an imaging system for analysis.<ref name=OPITwyman>{{cite web |title=Interferential Devices – Twyman–Green Interferometer |url=http://www.optique-ingenieur.org/en/courses/OPI_ang_M02_C05/co/Contenu_31.html |publisher=OPI – Optique pour l'Ingénieur |access-date=4 April 2012}}</ref>

Mach–Zehnder interferometers are being used in [[integrated optical circuit]]s, in which light interferes between two branches of a [[waveguide]] that are externally [[modulation|modulated]] to vary their relative phase. A slight tilt of one of the beam splitters will result in a path difference and a change in the interference pattern. Mach–Zehnder interferometers are the basis of a wide variety of devices, from [[RF modulator]]s to sensors<ref name=Heideman1992>{{Cite journal | last1 = Heideman | first1 = R. G. | last2 = Kooyman | first2 = R. P. H. | last3 = Greve | first3 = J. | doi = 10.1016/0925-4005(93)87008-D | title = Performance of a highly sensitive optical waveguide Mach–Zehnder interferometer immunosensor | journal = Sensors and Actuators B: Chemical | volume = 10 | issue = 3 | pages = 209–217 | year = 1993| bibcode = 1993SeAcB..10..209H | citeseerx = 10.1.1.556.5526 }}</ref><ref name=Oliver2005>{{Cite journal | last1 = Oliver | first1 = W. D. | last2 = Yu | first2 = Y. | last3 = Lee | first3 = J. C. | last4 = Berggren | first4 = K. K. | last5 = Levitov | first5 = L. S. | last6 = Orlando | first6 = T. P. | title = Mach–Zehnder Interferometry in a Strongly Driven Superconducting Qubit | doi = 10.1126/science.1119678 | journal = Science | volume = 310 | issue = 5754 | pages = 1653–1657 | year = 2005 | pmid =  16282527|arxiv = cond-mat/0512691 |bibcode = 2005Sci...310.1653O | s2cid = 46509116 }}</ref> to [[optical switch]]es.<ref name=Nieradko2006>{{Cite journal | last1 = Nieradko | first1 = Ł. | last2 = Gorecki | first2 = C. | last3 = JóZwik | first3 = M. | last4 = Sabac | first4 = A. | last5 = Hoffmann | first5 = R. | last6 = Bertz | first6 = A. | doi = 10.1117/1.2203366 | title = Fabrication and optical packaging of an integrated Mach–Zehnder interferometer on top of a movable micromirror | journal = Journal of Micro/Nanolithography, MEMS, and MOEMS | volume = 5 | issue = 2 | pages = 023009 | year = 2006 |bibcode = 2006JMM&M...5b3009N }}</ref>

The latest proposed [[extremely large telescope|extremely large astronomical telescopes]], such as the [[Thirty Meter Telescope]] and the [[Extremely Large Telescope]], will be of segmented design. Their primary mirrors will be built from hundreds of hexagonal mirror segments. Polishing and figuring these highly aspheric and non-rotationally symmetric mirror segments presents a major challenge. Traditional means of optical testing compares a surface against a spherical reference with the aid of a [[null corrector]]. In recent years, [[computer-generated hologram]]s (CGHs) have begun to supplement null correctors in test setups for complex aspheric surfaces. Fig.&nbsp;15 illustrates how this is done. Unlike the figure, actual CGHs have line spacing on the order of 1 to 10&nbsp;μm. When laser light is passed through the CGH, the zero-order diffracted beam experiences no wavefront modification. The wavefront of the first-order diffracted beam, however, is modified to match the desired shape of the test surface. In the illustrated Fizeau interferometer test setup, the zero-order diffracted beam is directed towards the spherical reference surface, and the first-order diffracted beam is directed towards the test surface in such a way that the two reflected beams combine to form interference fringes. The same test setup can be used for the innermost mirrors as for the outermost, with only the CGH needing to be exchanged.<ref name=Burge2010>{{cite book |last=Burge |first=J. H. |author2=Zhao, C. |author3=Dubin, M.  |chapter=Measurement of aspheric mirror segments using Fizeau interferometry with CGH correction |title=Modern Technologies in Space- and Ground-based Telescopes and Instrumentation |journal=Proceedings of SPIE |date=2010 |volume=7739 |pages=773902 |doi=10.1117/12.857816 |chapter-url=http://www.loft.optics.arizona.edu/documents/journal_articles/Jim_Burge_Measurement_of_aspheric_mirror_segments_using_Fizeau_interferometry_with_CGH_correction.pdf|bibcode=2010SPIE.7739E..02B |s2cid=49323922 }}</ref>

[[File:Fizeau optical testing with computer generated hologram-en.svg|thumb|550px|Figure 15. Optical testing with a Fizeau interferometer and a computer generated hologram]]

[[Ring laser gyroscope]]s (RLGs) and [[fibre optic gyroscope]]s (FOGs) are interferometers used in navigation systems.  They operate on the principle of the [[Sagnac effect]]. The distinction between RLGs and FOGs is that in a RLG, the entire ring is part of the laser while in a FOG, an external laser injects counter-propagating beams into an [[optical fiber]] ring, and rotation of the system then causes a relative phase shift between those beams. In a RLG, the observed phase shift is proportional to the accumulated rotation, while in a FOG, the observed phase shift is proportional to the angular velocity.<ref name=Anderson1994>{{cite journal |last=Anderson |first=R. |author2=Bilger, H.R. |author3=Stedman, G.E.  |title="Sagnac effect" A century of Earth-rotated interferometers |journal=Am. J. Phys. |date=1994 |volume=62 |issue=11 |pages=975–985 |url=http://signallake.com/innovation/andersonNov94.pdf|access-date=30 March 2012 |doi=10.1119/1.17656|bibcode = 1994AmJPh..62..975A }}</ref>

In telecommunication networks, heterodyning is used to move frequencies of individual signals to different channels which may share a single physical transmission line. This is called [[frequency division multiplexing]] (FDM). For example, a [[coaxial cable]] used by a [[cable television]] system can carry 500 television channels at the same time because each one is given a different frequency, so they don't interfere with one another. Continuous wave (CW) [[doppler radar]] detectors are basically heterodyne detection devices that compare transmitted and reflected beams.<ref name=Golio2007>{{cite book|last=Golio|first=Mike|title=RF and Microwave Applications and Systems|date=2007|publisher=CRC Press|isbn=978-0-8493-7219-3|pages=14.1–14.17 |url=https://books.google.com/books?id=fNJLcL1LBpEC&pg=SA14-PA1 |access-date=27 June 2012}}</ref>

Optical heterodyne detection is used for coherent [[Doppler lidar]] measurements capable of detecting very weak light scattered in the atmosphere and monitoring wind speeds with high accuracy. It has application in [[Fiber-optic communication|optical fiber communications]], in various high resolution spectroscopic techniques, and the self-heterodyne method can be used to measure the linewidth of a laser.<ref name=Rudiger/><ref name=Paschotta2>{{cite web|last=Paschotta|first=Rüdiger|title=Self-heterodyne Linewidth Measurement|url=http://www.rp-photonics.com/self_heterodyne_linewidth_measurement.html|publisher=RP Photonics|access-date=22 June 2012}}</ref>

[[File:FrequencyComb-measurement.svg|thumb|300px|Figure 16. Frequency comb of a mode-locked laser. The dashed lines represent an extrapolation of the mode frequencies towards the frequency of the carrier–envelope offset (CEO). The vertical grey line represents an unknown optical frequency. The horizontal black lines indicate the two lowest beat frequency measurements.]]

Optical heterodyne detection is an essential technique used in high-accuracy measurements of the frequencies of optical sources, as well as in the stabilization of their frequencies. Until a relatively few years ago, lengthy frequency chains were needed to connect the microwave frequency of a [[Atomic clock|cesium or other atomic time source]] to optical frequencies. At each step of the chain, a [[frequency multiplier]] would be used to produce a harmonic of the frequency of that step, which would be compared by heterodyne detection with the next step (the output of a microwave source, far infrared laser, infrared laser, or visible laser). Each measurement of a single spectral line required several years of effort in the construction of a custom frequency chain. Currently, optical [[frequency comb]]s have provided a much simpler method of measuring optical frequencies. If a mode-locked laser is modulated to form a train of pulses, its spectrum is seen to consist of the carrier frequency surrounded by a closely spaced comb of optical [[sideband]] frequencies with a spacing equal to the pulse repetition frequency (Fig.&nbsp;16). The pulse repetition frequency is locked to that of the [[frequency standard]], and the frequencies of the comb elements at the red end of the spectrum are doubled and heterodyned with the frequencies of the comb elements at the blue end of the spectrum, thus allowing the comb to serve as its own reference. In this manner, locking of the frequency comb output to an atomic standard can be performed in a single step. To measure an unknown frequency, the frequency comb output is dispersed into a spectrum. The unknown frequency is overlapped with the appropriate spectral segment of the comb and the frequency of the resultant heterodyne beats is measured.<ref name=NRCCanada>{{cite web |title=Optical Frequency Comb |url=http://www.nrc-cnrc.gc.ca/eng/projects/inms/optical-comb.html |publisher=National Research Council, Canada |access-date=23 June 2012 |url-status=dead |archive-url=https://web.archive.org/web/20120305160807/http://www.nrc-cnrc.gc.ca/eng/projects/inms/optical-comb.html |archive-date=5 March 2012 }}</ref><ref name=RPP_Combs>{{cite web |title=Frequency Combs |last=Paschotta |first=Rüdiger |url=http://www.rp-photonics.com/frequency_combs.html |publisher=RP Photonics|access-date=23 June 2012}}</ref>

One of the most common industrial applications of optical interferometry is as a versatile measurement tool for the high precision examination of surface topography. Popular interferometric measurement techniques include Phase Shifting Interferometry (PSI),<ref>{{Cite book | last1 = Schmit | first1 = J. | editor-first1 = Gordon M. | editor-first2 = Osuk Y. | editor-first3 = Malgorzata | editor-first4 = Graeme T. | editor-last1 = Brown | editor-last2 = Kwon | editor-last3 = Kujawinska | editor-last4 = Reid | chapter = Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension | doi = 10.1117/12.140770 | title = Proceedings of SPIE | volume = 1755 | pages = 202–201 | year = 1993| series = Interferometry: Techniques and Analysis | s2cid = 62679510 }}</ref> and Vertical Scanning Interferometry(VSI),<ref name=Larkin1996>{{cite journal |last=Larkin |first=K.G. |title=Efficient nonlinear algorithm for envelope detection in white light interferometry |journal=Journal of the Optical Society of America |date=1996 |volume=13 |pages=832–843 |issue=4 |doi=10.1364/JOSAA.13.000832 |url=http://www.nontrivialzeros.net/KGL_Papers/23_Neat_Algorithm_JOSAA96.pdf |access-date=1 April 2012 |bibcode=1996JOSAA..13..832L |citeseerx=10.1.1.190.4728 |archive-date=10 March 2020 |archive-url=https://web.archive.org/web/20200310165204/https://nontrivialzeros.net/KGL_Papers/23_Neat_Algorithm_JOSAA96.pdf |url-status=dead }}</ref> also known as scanning [[white light interferometry]] (SWLI) or by the ISO term [[coherence scanning interferometry]] (CSI),<ref>ISO. (2013). 25178-604:2013(E): Geometrical product specification (GPS) – Surface texture: Areal – Nominal characteristics of non-contact (coherence scanning interferometric microscopy) instruments (2013(E) ed.). Geneva: International Organization for Standardization.</ref> CSI exploits [[Coherence (physics)|coherence]] to extend the range of capabilities for interference microscopy.<ref name=Harasaki2000/><ref>{{cite journal | last1 = De Groot | first1 = P | year = 2015 | title = Principles of interference microscopy for the measurement of surface topography | doi = 10.1364/AOP.7.000001 | journal = Advances in Optics and Photonics | volume = 7 | issue = 1 | pages = 1–65 | bibcode = 2015AdOP....7....1D }}</ref> These techniques are widely used in micro-electronic and micro-optic fabrication. PSI uses monochromatic light and provides very precise measurements; however it is only usable for surfaces that are very smooth. CSI often uses white light and high numerical apertures, and rather than looking at the phase of the fringes, as does PSI, looks for best position of maximum fringe contrast or some other feature of the overall fringe pattern.  In its simplest form, CSI provides less precise measurements than PSI but can be used on rough surfaces. Some configurations of CSI, variously known as Enhanced VSI (EVSI), high-resolution SWLI or Frequency Domain Analysis (FDA), use coherence effects in combination with interference phase to enhance precision.<ref name=Olszak/><ref>{{cite journal|last1=de Groot|first1=Peter|last2=Deck|first2=Leslie|title=Surface Profiling by Analysis of White-light Interferograms in the Spatial Frequency Domain|journal=Journal of Modern Optics|date=1995|volume=42|issue=2|pages=389–401|doi=10.1080/09500349514550341|bibcode = 1995JMOp...42..389D }}</ref>

[[File:Phase Shifting and Vertical Scanning Interferometry Animation.gif|thumb|550px|Figure 17. Phase shifting and Coherence scanning interferometers]]

Phase Shifting Interferometry addresses several issues associated with the classical analysis of static interferograms. Classically, one measures the positions of the fringe centers. As seen in Fig.&nbsp;13, fringe deviations from straightness and equal spacing provide a measure of the aberration. Errors in determining the location of the fringe centers provide the inherent limit to precision of the classical analysis, and any intensity variations across the interferogram will also introduce error. There is a trade-off between precision and number of data points: closely spaced fringes provide many data points of low precision, while widely spaced fringes provide a low number of high precision data points. Since fringe center data is all that one uses in the classical analysis, all of the other information that might theoretically be obtained by detailed analysis of the intensity variations in an interferogram is thrown away.<ref name=Newport/><ref name=Graham/> Finally, with static interferograms, additional information is needed to determine the polarity of the wavefront: In Fig.&nbsp;13, one can see that the tested surface on the right deviates from flatness, but one cannot tell from this single image whether this deviation from flatness is concave or convex. Traditionally, this information would be obtained using non-automated means, such as by observing the direction that the fringes move when the reference surface is pushed.<ref name=Schreiber2006>{{Cite book | last1 = Schreiber | first1 = H. | last2 = Bruning | first2 = J. H. | doi = 10.1002/9780470135976.ch14 | chapter = Phase Shifting Interferometry | title = Optical Shop Testing | pages = 547 | year = 2007 |isbn=978-0-470-13597-6}}</ref>

Phase shifting interferometry overcomes these limitations by not relying on finding fringe centers, but rather by collecting intensity data from every point of the [[Charge-coupled device|CCD]] image sensor. As seen in Fig.&nbsp;17, multiple interferograms (at least three) are analyzed with the reference optical surface shifted by a precise fraction of a wavelength between each exposure using a [[Piezoelectricity|piezoelectric transducer]] (PZT). Alternatively, precise phase shifts can be introduced by modulating the laser frequency.<ref>Sommargren, G. E. (1986). US Patent 4,594,003.</ref> The captured images are processed by a computer to calculate the optical wavefront errors. The precision and reproducibility of PSI is far greater than possible in static interferogram analysis, with measurement repeatabilities of a hundredth of a wavelength being routine.<ref name=Newport>{{cite web |title=Phase-Shifting Interferometry for Determining Optical Surface Quality |url=http://www.newport.com/Phase-Shifting-Interferometry-for-Determining-Opt/979098/1033/content.aspx |publisher=Newport Corporation |access-date=12 May 2012 |archive-date=7 November 2012 |archive-url=https://web.archive.org/web/20121107000722/http://www.newport.com/Phase-Shifting-Interferometry-for-Determining-Opt/979098/1033/content.aspx |url-status=dead }}</ref><ref name=Graham>{{cite web|title=How Phase Interferometers work|url=http://www.grahamoptical.com/phase.html|publisher=Graham Optical Systems |date=2011 |access-date=12 May 2012}}</ref> Phase shifting technology has been adapted to a variety of interferometer types such as Twyman–Green, Mach–Zehnder, laser Fizeau, and even common path configurations such as point diffraction and lateral shearing interferometers.<ref name=Schreiber2006/><ref name=Ferraro2007>{{cite web|author=Ferraro, P.|author2=Paturzo, M.|author3=Grilli, S.|title=Optical wavefront measurement using a novel phase-shifting point-diffraction interferometer|url=http://spie.org/x8369.xml|publisher=SPIE|access-date=26 May 2012|date=2007}}</ref> More generally, phase shifting techniques can be adapted to almost any system that uses fringes for measurement, such as holographic and speckle interferometry.<ref name=Schreiber2006/>

[[File:Nepenthes khasiana lunate cells.jpg|thumb|300px|Figure 18. Lunate cells of ''[[Nepenthes khasiana]]'' visualized by Scanning White Light Interferometry (SWLI)]]
[[File:Twyman-Green interferometer set up as white light scanner.svg|thumb|300px|Figure&nbsp;19. Twyman–Green interferometer set up as a white light scanner]]

In [[coherence scanning interferometry]],<ref>P. de Groot, J., "Interference Microscopy for Surface Structure Analysis", in Handbook of Optical Metrology, edited by T. Yoshizawa, chapt.31, pp. 791-828, (CRC Press, 2015).</ref> interference is only achieved when the path length delays of the interferometer are matched within the coherence time of the light source. CSI monitors the fringe contrast rather than the phase of the fringes.<ref name=HariharanBasics2007/>{{rp|105}} Fig.&nbsp;17 illustrates a CSI microscope using a [[Mirau interferometer]] in the objective; other forms of interferometer used with white light include the Michelson interferometer (for low magnification objectives, where the reference mirror in a Mirau objective would interrupt too much of the aperture) and the [[Linnik interferometer]] (for high magnification objectives with limited working distance).<ref name=Schmit2006>{{Cite book | last1 = Schmit | first1 = J. | last2 = Creath | first2 = K. | last3 = Wyant | first3 = J. C. | doi = 10.1002/9780470135976.ch15 | chapter = Surface Profilers, Multiple Wavelength, and White Light Intereferometry | title = Optical Shop Testing | pages = 667 | year = 2007 |isbn=978-0-470-13597-6}}</ref> The sample (or alternatively, the objective) is moved vertically over the full height range of the sample, and the position of maximum fringe contrast is found for each pixel.<ref name=Harasaki2000>{{cite journal |last=Harasaki |first=A. |author2=Schmit, J. |author3=Wyant, J. C. |title=Improved vertical-scanning interferometry |journal=Applied Optics |date=2000 |volume=39 |issue=13 |pages=2107–2115 |url=http://www.optics.arizona.edu/jcwyant/pdf/Published_Papers/Optical_Profiler/Improved%20Vertical-Scanning%20Interferometry.pdf |access-date=21 May 2012 |bibcode=2000ApOpt..39.2107H |doi=10.1364/AO.39.002107 |pmid=18345114 |url-status=dead |archive-url=https://web.archive.org/web/20100725084235/http://www.optics.arizona.edu/jcwyant/pdf/Published_Papers/Optical_Profiler/Improved%20Vertical-Scanning%20Interferometry.pdf |archive-date=25 July 2010 |hdl=10150/289148 |hdl-access=free }}</ref><ref name=Veeco>{{cite web |title=HDVSI – Introducing High Definition Vertical Scanning Interferometry for Nanotechnology Research from Veeco Instruments |url=http://www.azom.com/article.aspx?ArticleID=4778 |publisher=Veeco |access-date=21 May 2012 |url-status=dead |archive-url=https://web.archive.org/web/20120409041953/http://www.azom.com/article.aspx?ArticleID=4778 |archive-date=9 April 2012 }}</ref> The chief benefit of coherence scanning interferometry is that systems can be designed that do not suffer from the 2&nbsp;pi ambiguity of coherent interferometry,<ref name="Pluciński2008">{{cite journal|last=Plucinski|first=J.|author2=Hypszer, R. |author3=Wierzba, P. |author4=Strakowski, M. |author5=Jedrzejewska-Szczerska, M. |author6=Maciejewski, M. |author7= Kosmowski, B.B. |title=Optical low-coherence interferometry for selected technical applications|journal=Bulletin of the Polish Academy of Sciences |date=2008 |volume=56 |issue=2 |pages=155–172 |url=http://bulletin.pan.pl/(56-2)155.pdf|access-date=8 April 2012}}</ref><ref name=Yang2002>{{cite journal |last=Yang |first=C.-H. |author2=Wax, A |author3=Dasari, R.R. |author4= Feld, M.S.  |title=2π ambiguity-free optical distance measurement with subnanometer precision with a novel phase-crossing low-coherence interferometer |journal=Optics Letters |date=2002 |volume=27 |issue=2 |pages=77–79 |url=http://authors.library.caltech.edu/3326/1/YANol02b.pdf|bibcode = 2002OptL...27...77Y |doi = 10.1364/OL.27.000077 |pmid=18007717 |s2cid=9524638 }}</ref><ref name=Hitzenberger2001>{{Cite journal | last1 = Hitzenberger | first1 = C. K. | last2 = Sticker | first2 = M. | last3 = Leitgeb | first3 = R. | last4 = Fercher | first4 = A. F. | title = Differential phase measurements in low-coherence interferometry without 2pi ambiguity | journal = Optics Letters | volume = 26 | issue = 23 | pages = 1864–1866 | year = 2001 | pmid = 18059719 | doi=10.1364/ol.26.001864 |bibcode = 2001OptL...26.1864H }}</ref> and as seen in Fig.&nbsp;18, which scans a 180μm&nbsp;x&nbsp;140μm&nbsp;x&nbsp;10μm volume, it is well suited to profiling steps and rough surfaces. The axial resolution of the system is determined in part by the coherence length of the light source.<ref>Wojtek J. Walecki, Kevin Lai, Vitalij Souchkov, Phuc Van, SH Lau, Ann Koo Physica Status Solidi C Volume 2, Issue 3, Pages 984–989</ref><ref>W. J. Walecki et al. "Non-contact fast wafer metrology for ultra-thin patterned wafers mounted on grinding and dicing tapes" Electronics Manufacturing Technology Symposium, 2004. IEEE/CPMT/SEMI 29th International Volume, Issue, July 14–16, 2004 Page(s): 323–325</ref> Industrial applications include in-process [[surface metrology]], roughness measurement, 3D surface metrology in hard-to-reach spaces and in hostile environments, profilometry of surfaces with high aspect ratio features (grooves, channels, holes), and film thickness measurement (semi-conductor and optical industries, etc.).<ref name=Lumetrics>{{cite web|title=Coating Thickness Measurement|url=http://www.lumetrics.com/coatingthicknessmeasurement.html|publisher=Lumetrics, Inc.|access-date=28 October 2013|archive-date=29 October 2013|archive-url=https://web.archive.org/web/20131029201451/http://www.lumetrics.com/coatingthicknessmeasurement.html|url-status=dead}}</ref><ref name=Novacam>{{cite web|title=Typical profilometry measurements|url=http://www.novacam.com/applications/|publisher=Novacam Technologies, Inc.|access-date=25 June 2012}}</ref>

Fig.&nbsp;19 illustrates a [[Twyman–Green interferometer]] set up for white light scanning of a macroscopic object.

[[Holographic interferometry]] is a technique which uses [[holography]] to monitor small deformations in single wavelength implementations. In multi-wavelength implementations, it is used to perform dimensional metrology of large parts and assemblies and to detect larger surface defects.<ref name=HariharanBasics2007/>{{rp|111–120}}

Holographic interferometry was discovered by accident as a result of mistakes committed during the making of holograms. Early lasers were relatively weak and photographic plates were insensitive, necessitating long exposures during which vibrations or minute shifts might occur in the optical system. The resultant holograms, which showed the holographic subject covered with fringes, were considered ruined.<ref name=HolographyRU>{{cite web|title=Holographic interferometry|url=http://www.holography.ru/files/holinte2.htm |date=2008 |publisher=Oquagen |access-date=22 May 2012}}</ref>

Eventually, several independent groups of experimenters in the mid-60s realized that the fringes encoded important information about dimensional changes occurring in the subject, and began intentionally producing holographic double exposures. The main [[Holographic interferometry]] article covers the disputes over priority of discovery that occurred during the issuance of the patent for this method.<ref name=Hecht1998>{{cite book|last=Hecht|first=Jeff|title=Laser, Light of a Million Uses|date=1998|publisher=Dover Publications, Inc.|isbn=978-0-486-40193-5|pages=229–230|url=https://books.google.com/books?id=Dcg8KG75WoIC&pg=PA229}}</ref>

Double- and multi- exposure holography is one of three methods used to create holographic interferograms. A first exposure records the object in an unstressed state. Subsequent exposures on the same photographic plate are made while the object is subjected to some stress. The composite image depicts the difference between the stressed and unstressed states.<ref name=Fein1997>{{cite journal|last=Fein|first=H|title=Holographic Interferometry: Nondestructive tool|journal=The Industrial Physicist|date=September 1997|pages=37–39|url=http://www.aip.org/tip/INPHFA/vol-3/iss-3/p37.pdf|url-status=dead|archive-url=https://web.archive.org/web/20121107000833/http://www.aip.org/tip/INPHFA/vol-3/iss-3/p37.pdf|archive-date=2012-11-07}}</ref>

Real-time holography is a second method of creating holographic interferograms. A holograph of the unstressed object is created. This holograph is illuminated with a reference beam to generate a hologram image of the object directly superimposed over the original object itself while the object is being subjected to some stress. The object waves from this hologram image will interfere with new waves coming from the object. This technique allows real time monitoring of shape changes.<ref name=Fein1997/>

The third method, time-average holography, involves creating a holograph while the object is subjected to a periodic stress or vibration. This yields a visual image of the vibration pattern.<ref name=Fein1997/>

<gallery mode="packed" heights="255">
File:SAR Kilauea topo interferogram.jpg|Figure 20. InSAR Image of Kilauea, Hawaii showing fringes caused by deformation of the terrain over a six-month period.
File:ESPIvibration.jpg|Figure 21. ESPI fringes showing a vibration mode of a clamped square plate
</gallery>

[[Interferometric synthetic aperture radar]] (InSAR) is a radar technique used in [[geodesy]] and [[remote sensing]]. Satellite [[synthetic aperture radar]] images of a geographic feature are taken on separate days, and changes that have taken place between radar images taken on the separate days are recorded as fringes similar to those obtained in holographic interferometry. The technique can monitor centimeter- to millimeter-scale deformation resulting from earthquakes, volcanoes and landslides, and also has uses in structural engineering, in particular for the monitoring of subsidence and structural stability. Fig&nbsp;20 shows Kilauea, an active volcano in Hawaii. Data acquired using the space shuttle Endeavour's X-band Synthetic Aperture Radar on April 13, 1994 and October 4, 1994 were used to generate interferometric fringes, which were overlaid on the X-SAR image of Kilauea.<ref name=NASA_JPL1999>{{cite web |title=PIA01762: Space Radar Image of Kilauea, Hawaii |url=http://photojournal.jpl.nasa.gov/catalog/PIA01762 |date=1999 |publisher=NASA/JPL |access-date=17 June 2012}}</ref>

[[Electronic speckle pattern interferometry]] (ESPI), also known as TV holography, uses video detection and recording to produce an image of the object upon which is superimposed a fringe pattern which represents the displacement of the object between recordings. (see Fig.&nbsp;21) The fringes are similar to those obtained in holographic interferometry.<ref name=HariharanBasics2007/>{{rp|111–120}}<ref>Jones R & Wykes C, Holographic and Speckle Interferometry, 1989, Cambridge University Press</ref>

When lasers were first invented, [[laser speckle]] was considered to be a severe drawback in using lasers to illuminate objects, particularly in holographic imaging because of the grainy image produced. It was later realized that speckle patterns could carry information about the object's surface deformations. Butters and Leendertz developed the technique of speckle pattern interferometry in 1970,<ref>{{Cite journal | last1 = Butters | first1 = J. N. | last2 = Leendertz | first2 = J. A. | doi = 10.1088/0022-3735/4/4/004 | title = A double exposure technique for speckle pattern interferometry | journal =  Journal of Physics E: Scientific Instruments | volume = 4 | issue = 4 | pages = 277–279 | year = 1971|bibcode = 1971JPhE....4..277B }}</ref> and since then, speckle has been exploited in a variety of other applications. A photograph is made of the speckle pattern before deformation, and a second photograph is made of the speckle pattern after deformation. Digital subtraction of the two images results in a correlation fringe pattern, where the fringes represent lines of equal deformation. Short laser pulses in the nanosecond range can be used to capture very fast transient events. A phase problem exists: In the absence of other information, one cannot tell the difference between contour lines indicating a peak ''versus'' contour lines indicating a trough. To resolve the issue of phase ambiguity, ESPI may be combined with phase shifting methods.<ref name=Dvorakova>{{cite journal|last=Dvořáková|first=P.|author2=Bajgar, V. |author3=Trnka, J. |title=Dynamic Electronic Speckle Pattern Interferometry in Application to Measure Out-Of-Plane Displacement|journal=Engineering Mechanics|date=2007|volume=14|issue=1/2|pages=37–44|url=http://www.im.fme.vutbr.cz/pdf/14_1_037.pdf}}</ref><ref name=Moustafa>{{cite journal|last=Moustafa|first=N. A.|author2=Hendawi, N.|title=Comparative Phase-Shifting Digital Speckle Pattern Interferometry Using Single Reference Beam Technique|journal=Egypt. J. Sol.|date=2003|volume=26|issue=2|pages=225–229|doi=10.21608/ejs.2003.150160|url=http://egmrs.powweb.com/EJS/PDF/vo262/225.pdf|access-date=22 May 2012|doi-access=free}}</ref>

A method of establishing precise [[geodesy|geodetic]] baselines, invented by [[Yrjö Väisälä]], exploited the low coherence length of white light. Initially, white light was split in two, with the reference beam "folded", bouncing back-and-forth six times between a mirror pair spaced precisely 1 m apart. Only if the test path was precisely 6 times the reference path would fringes be seen. Repeated applications of this procedure allowed precise measurement of distances up to 864 meters. Baselines thus established were used to calibrate geodetic distance measurement equipment, leading to a [[metrology|metrologically]] traceable scale for [[geodetic network]]s measured by these instruments.<ref>{{cite book |author=Buga, A. |author2=Jokela, J. |author3=Putrimas, R. |title=Environmental Engineering, The 7th International Conference |publisher=Vilnius Gediminas Technical University |pages=1274–1280 |chapter=Traceability, stability and use of the Kyviskes calibration baseline–the first 10 years |chapter-url=http://dspace.vgtu.lt/bitstream/1/638/1/buga_et_al_traceability.pdf |access-date=9 April 2012}}</ref> (This method has been superseded by GPS.)

Other uses of interferometers have been to study dispersion of materials, measurement of complex indices of refraction, and thermal properties. They are also used for three-dimensional motion mapping including mapping vibrational patterns of structures.<ref name=Olszak>{{cite web|author=Olszak, A.G.|author2=Schmit, J.|author3=Heaton, M.G.|title=Interferometry: Technology and Applications|url=http://www.bruker-axs.com/fileadmin/user_upload/PDF_2011/application_notes/Interferometry_Technology_and_Applications_SOM_AN47.pdf|publisher=Bruker|access-date=1 April 2012}}</ref>