Editing Interferometry (section)

==Categories==
Interferometers and interferometric techniques may be categorized by a variety of criteria:

===Homodyne versus heterodyne detection <span class="anchor" id="Heterodyne detection"></span>===
In [[homodyne detection]], the interference occurs between two beams at the same wavelength (or [[carrier frequency]]). The phase difference between the two beams results in a change in the intensity of the light on the detector. The resulting intensity of the light after mixing of these two beams is measured, or the pattern of interference fringes is viewed or recorded.<ref name=Rudiger/> Most of the interferometers discussed in this article fall into this category.

The [[heterodyne]] technique is used for (1) shifting an input signal into a new frequency range as well as (2) amplifying a weak input signal (assuming use of an active [[Electronic mixer|mixer]]). A weak input signal of frequency f<sub>1</sub> is [[Frequency mixer|mixed]] with a strong reference frequency f<sub>2</sub> from a [[local oscillator]] (LO). The nonlinear combination of the input signals creates two new signals, one at the sum f<sub>1</sub>&nbsp;+&nbsp;f<sub>2</sub> of the two frequencies, and the other at the difference f<sub>1</sub>&nbsp;−&nbsp;f<sub>2</sub>. These new frequencies are called '''heterodynes'''. Typically only one of the new frequencies is desired, and the other signal is filtered out of the output of the mixer. The output signal will have an intensity proportional to the product of the amplitudes of the input signals.<ref name=Rudiger/>

The most important and widely used application of the heterodyne technique is in the [[superheterodyne receiver]] (superhet), invented in 1917-18 by U.S. engineer [[Edwin Howard Armstrong]] and French engineer [[Lucien Lévy]]. In this circuit, the incoming [[radio frequency]] signal from the antenna is mixed with a signal from a local oscillator (LO) and converted by the heterodyne technique to a lower fixed frequency signal called the [[intermediate frequency]] (IF). This IF is amplified and filtered, before being applied to a [[detector (radio)|detector]] which extracts the audio signal, which is sent to the loudspeaker.<ref name=superhet>{{cite web |last=Poole |first=Ian |title=The superhet or superheterodyne radio receiver |url=http://www.radio-electronics.com/info/rf-technology-design/superheterodyne-radio-receiver/basics-tutorial.php|publisher=Radio-Electronics.com|access-date=22 June 2012}}</ref>

[[Optical heterodyne detection]] is an extension of the heterodyne technique to higher (visible) frequencies.<ref name=Rudiger>{{cite web |last=Paschotta |first=Rüdiger |title=Optical Heterodyne Detection|url=http://www.rp-photonics.com/optical_heterodyne_detection.html |publisher=RP Photonics Consulting GmbH|access-date=1 April 2012}}</ref>  While optical heterodyne interferometry is usually done at a single point it is also possible to perform this widefield.<ref>{{cite journal|last=Patel|first=R.|author2=Achamfuo-Yeboah, S. |author3=Light R.|author4=Clark M.|title=Widefield heterodyne interferometry using a custom CMOS modulated light camera|journal=Optics Express|date=2011|volume=19|issue=24|pages=24546–24556|url=https://www.osapublishing.org/oe/abstract.cfm?uri=oe-19-24-24546|bibcode=2011OExpr..1924546P|doi=10.1364/OE.19.024546|pmid=22109482|doi-access=free}}</ref>

===Double path versus common path{{anchor|Double path}}===
{{See also|Common-path interferometer}}
[[File:Four common path interferometers.png|750px|thumb|Figure 4. Four examples of common-path interferometers]]

A '''double-path interferometer''' is one in which the reference beam and sample beam travel along divergent paths.  Examples include the [[Michelson interferometer]], the [[Twyman–Green interferometer]], and the [[Mach–Zehnder interferometer]]. After being perturbed by interaction with the sample under test, the sample beam is recombined with the reference beam to create an interference pattern which can then be interpreted.<ref name=HariharanBasics2007/>{{rp|13–22}}

A ''[[common-path interferometer]]'' is a class of interferometer in which the reference beam and sample beam travel along the same path. Fig.&nbsp;4 illustrates the [[Sagnac interferometer]], the [[fibre optic gyroscope]], the [[point diffraction interferometer]], and the [[Shearing interferometer|lateral shearing interferometer]]. Other examples of common path interferometer include the [[Zernike phase-contrast microscope]], [[Common-path interferometer#Selected examples|Fresnel's biprism]], the [[zero-area Sagnac interferometer|zero-area Sagnac]], and the [[scatterplate interferometer]].<ref name=Malacara2006>{{Cite book | last1 = Mallick | first1 = S. | last2 = Malacara | first2 = D. | doi = 10.1002/9780470135976.ch3 | chapter = Common-Path Interferometers | title = Optical Shop Testing | pages = 97 | year = 2007 |isbn=978-0-470-13597-6}}</ref>

===Wavefront splitting versus amplitude splitting===
====Wavefront splitting inferometers====
A wavefront splitting interferometer divides a light wavefront emerging from a point or a narrow slit (''i.e.'' spatially coherent light) and, after allowing the two parts of the wavefront to travel through different paths, allows them to recombine.<ref name=Verma>{{cite book|last=Verma|first=R.K.|title=Wave Optics|date=2008|publisher=Discovery Publishing House|isbn=978-81-8356-114-3|pages=97–110|url=https://books.google.com/books?id=uuzPcVS_dc0C&pg=PA97}}</ref> Fig.&nbsp;5 illustrates [[Young's interference experiment]] and [[Lloyd's mirror]]. Other examples of wavefront splitting interferometer include the Fresnel biprism, the Billet Bi-Lens, diffraction-grating Michelson interferometer,<ref>{{cite journal | last1=Kolesnichenko |first1=Pavel| last2=Wittenbecher |first2=Lukas| last3=Zigmantas |first3=Donatas |date=2020|title= Fully symmetric dispersionless stable transmission-grating Michelson interferometer |journal = Optics Express| volume= 28|issue=25 |pages=37752–37757| doi = 10.1364/OE.409185 |doi-access=free |pmid=33379604 |bibcode=2020OExpr..2837752K }}</ref> and the [[Rayleigh interferometer]].<ref name=OPI>{{cite web |title=Interferential Devices – Introduction|url=http://www.optique-ingenieur.org/en/courses/OPI_ang_M02_C05/co/Cours_M02C05_1.html |publisher=OPI – Optique pour l'Ingénieur |access-date=1 April 2012}}</ref>

[[File:Young's two-slit experiment and Lloyd's mirror.png|thumb|750px|Figure 5. Two wavefront splitting interferometers]]

In 1803, [[Young's interference experiment]] played a major role in the general acceptance of the wave theory of light. If white light is used in Young's experiment, the result is a white central band of [[constructive interference]] corresponding to equal path length from the two slits, surrounded by a symmetrical pattern of colored fringes of diminishing intensity. In addition to continuous electromagnetic radiation, Young's experiment has been performed with individual photons,<ref>{{cite journal | author-link=Geoffrey Ingram Taylor|first=Sir Geoffrey |last=Ingram Taylor |title=Interference Fringes with Feeble Light| journal = [[Mathematical Proceedings of the Cambridge Philosophical Society]] | volume=15|page=114 |date=1909| url=https://archive.org/details/proceedingsofcam15190810camb/page/114/mode/2up | access-date=7 December 2024}}</ref> with electrons,<ref>{{cite journal | last1 = Jönsson | first1 = C  | date = 1961 | title = Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten| journal = Zeitschrift für Physik | volume = 161 | issue = 4 | pages = 454–474 | doi = 10.1007/BF01342460 |bibcode = 1961ZPhy..161..454J | s2cid = 121659705 }}</ref><ref>{{cite journal | last1=Jönsson |first1=C |date=1974|title= Electron diffraction at multiple slits |journal = American Journal of Physics| volume= 4|issue=1 |pages=4–11| bibcode = 1974AmJPh..42....4J |doi = 10.1119/1.1987592 }}</ref> and with [[buckyball]] molecules large enough to be seen under an [[electron microscope]].<ref name=Buschhorn2004>{{cite book |title=Fundamental Physics – Heisenberg and Beyond: Werner Heisenberg Centennial Symposium "Developments in Modern Physics" |date=2004 |publisher=Springer |isbn=978-3-540-20201-1 |chapter-url=https://books.google.com/books?id=oLMCFnkFIdoC&pg=PA35|author1=Arndt, M. |author2=Zeilinger, A. |chapter=Heisenberg's Uncertainty and Matter Wave Interferometry with Large Molecules |editor1=Buschhorn, G. W. |editor2=Wess, J.|pages=35–52}}</ref>

[[Lloyd's mirror]] generates interference fringes by combining direct light from a source (blue lines) and light from the source's reflected image (red lines) from a mirror held at grazing incidence. The result is an asymmetrical pattern of fringes. The band of equal path length, nearest the mirror, is dark rather than bright. In 1834, Humphrey Lloyd interpreted this effect as proof that the phase of a front-surface reflected beam is inverted.<ref name=Carroll2010>{{cite web |last=Carroll |first=Brett|title=Simple Lloyd's Mirror |url=http://www.aapt.org/Programs/contests/upload/carroll.pdf|publisher=American Association of Physics Teachers |access-date=5 April 2012}}</ref><ref name=Serway2010>{{cite book|author=Serway, R.A.|author2=Jewett, J.W.|title=Principles of physics: a calculus-based text, Volume 1|date=2010|publisher=Brooks Cole |isbn=978-0-534-49143-7 |pages=905 |url=https://books.google.com/books?id=1DZz341Pp50C&pg=PA904}}</ref>

====Amplitude-splitting inferometers====
[[File:Three amplitude-splitting interferometers.svg|thumb|750px |Figure 6. Three amplitude-splitting interferometers: [[Fizeau interferometer|Fizeau]], [[Mach-Zehnder interferometer|Mach–Zehnder]], and [[Fabry Pérot interferometer|Fabry Pérot]].]]
An amplitude splitting interferometer uses a partial reflector to divide the amplitude of the incident wave into separate beams which are separated and recombined. 

The [[Fizeau interferometer]] is shown as it might be set up to test an [[optical flat]]. A precisely figured reference flat is placed on top of the flat being tested, separated by narrow spacers. The reference flat is slightly beveled (only a fraction of a degree of beveling is necessary) to prevent the rear surface of the flat from producing interference fringes. Separating the test and reference flats allows the two flats to be tilted with respect to each other. By adjusting the tilt, which adds a controlled phase gradient to the fringe pattern, one can control the spacing and direction of the fringes, so that one may obtain an easily interpreted series of nearly parallel fringes rather than a complex swirl of contour lines. Separating the plates, however, necessitates that the illuminating light be collimated. Fig&nbsp;6 shows a collimated beam of monochromatic light illuminating the two flats and a beam splitter allowing the fringes to be viewed on-axis.<ref name=Guideline>{{cite web |title=Guideline for Use of Fizeau Interferometer in Optical Testing |url=http://engineer.jpl.nasa.gov/practices/2404.pdf |publisher=NASA |access-date=8 April 2012 |archive-url=https://web.archive.org/web/20180925131401/https://engineer.jpl.nasa.gov/practices/2404.pdf |archive-date=25 September 2018 |url-status=dead }}</ref><ref name=OPIFizeau>{{cite web |title=Interferential devices – Fizeau Interferometer |url=http://www.optique-ingenieur.org/en/courses/OPI_ang_M02_C05/co/Contenu_25.html |publisher=Optique pour l'Ingénieur.|access-date=8 April 2012}}</ref>

The [[Mach–Zehnder interferometer]] is a more versatile instrument than the Michelson interferometer. Each of the well separated light paths is traversed only once, and the fringes can be adjusted so that they are localized in any desired plane.<ref name=HariharanBasics2007/>{{rp|18}} Typically, the fringes would be adjusted to lie in the same plane as the test object, so that fringes and test object can be photographed together. If it is decided to produce fringes in white light, then, since white light has a limited [[coherence length]], on the order of [[micrometre|micrometers]], great care must be taken to equalize the optical paths or no fringes will be visible.  As illustrated in Fig.&nbsp;6, a compensating cell would be placed in the path of the reference beam to match the test cell. Note also the precise orientation of the beam splitters. The reflecting surfaces of the beam splitters would be oriented so that the test and reference beams pass through an equal amount of glass. In this orientation, the test and reference beams each experience two front-surface reflections, resulting in the same number of phase inversions. The result is that light traveling an equal optical path length in the test and reference beams produces a white light fringe of constructive interference.<ref name=Zetie>{{cite web |author=Zetie, K.P. |author2=Adams, S.F. |author3=Tocknell, R.M. |title=How does a Mach–Zehnder interferometer work?|url=http://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/zetie_et_al_mach_zehnder00.pdf |publisher=Physics Department, Westminster School, London |access-date=8 April 2012}}</ref><ref name=Ashkenas1950>{{cite thesis |last=Ashkenas |first=Harry I. |title=The design and construction of a Mach–Zehnder interferometer for use with the GALCIT Transonic Wind Tunnel. Engineer's thesis |date=1950 |publisher=California Institute of Technology |doi=10.7907/D0V1-MJ80 |url=https://thesis.library.caltech.edu/1483/|type=engd }}</ref>

The heart of the [[Fabry–Pérot interferometer]] is a pair of partially silvered glass optical flats spaced several millimeters to centimeters apart with the silvered surfaces facing each other. (Alternatively, a Fabry–Pérot ''etalon'' uses a transparent plate with two parallel reflecting surfaces.)<ref name=HariharanBasics2007/>{{rp|35–36}} As with the Fizeau interferometer, the flats are slightly beveled. In a typical system, illumination is provided by a diffuse source set at the [[focal plane]] of a collimating lens. A focusing lens produces what would be an inverted image of the source if the paired flats were not present, i.e., in the absence of the paired flats, all light emitted from point A passing through the optical system would be focused at point A'. In Fig.&nbsp;6, only one ray emitted from point A on the source is traced. As the ray passes through the paired flats, it is multiply reflected to produce multiple transmitted rays which are collected by the focusing lens and brought to point A' on the screen. The complete interference pattern takes the appearance of a set of concentric rings. The sharpness of the rings depends on the reflectivity of the flats. If the reflectivity is high, resulting in a high [[Q factor]] (i.e., high finesse), monochromatic light produces a set of narrow bright rings against a dark background.<ref name=Betzler>{{cite web|last=Betzler|first=Klaus|title=Fabry–Perot Interferometer|url=http://www.fen.bilkent.edu.tr/~aykutlu/msn513/fibersensors/fabryperot.pdf|publisher=Fachbereich Physik, Universität Osnabrück|access-date=8 April 2012}}</ref> In Fig.&nbsp;6, the low-finesse image corresponds to a reflectivity of 0.04 (i.e., unsilvered surfaces) ''versus'' a reflectivity of 0.95 for the high-finesse image.

Fig.&nbsp;6 illustrates the Fizeau, Mach–Zehnder, and Fabry–Pérot interferometers. Other examples of amplitude splitting interferometer include the [[Michelson interferometer|Michelson]], [[Twyman–Green interferometer|Twyman–Green]], Laser Unequal Path, and [[Linnik interferometer]].<ref name=Nolte>{{cite book|last=Nolte|first=David D.|title=Optical Interferometry for Biology and Medicine |date=2012 |publisher=Springer |isbn=978-1-4614-0889-5 |pages=17–26 |url=https://books.google.com/books?id=emFmnDhq9EwC&pg=PA17|bibcode=2012oibm.book.....N}}</ref>

====Michelson-Morley====
Michelson and Morley (1887)<ref name=Michelson1887>{{cite journal|last=Michelson|first=A.A.|author2=Morley, E.W.|title=On the Relative Motion of the Earth and the Luminiferous Ether|journal=American Journal of Science|date=1887|volume=34|issue=203|pages=333–345|url=http://www.aip.org/history/exhibits/gap/PDF/michelson.pdf|doi=10.2475/ajs.s3-34.203.333|bibcode=1887AmJS...34..333M|s2cid=124333204|access-date=2012-04-09|archive-date=2016-03-07|archive-url=https://web.archive.org/web/20160307150033/https://www.aip.org/history/exhibits/gap/PDF/michelson.pdf|url-status=dead}}</ref> and other early experimentalists using interferometric techniques in an attempt to measure the properties of the [[luminiferous aether]], used monochromatic light only for initially setting up their equipment, always switching to white light for the actual measurements. The reason is that measurements were recorded visually. Monochromatic light would result in a uniform fringe pattern. Lacking modern means of [[air conditioning|environmental temperature control]], experimentalists struggled with continual fringe drift even though the interferometer might be set up in a basement. Since the fringes would occasionally disappear due to vibrations by passing horse traffic, distant thunderstorms and the like, it would be easy for an observer to "get lost" when the fringes returned to visibility. The advantages of white light, which produced a distinctive colored fringe pattern, far outweighed the difficulties of aligning the apparatus due to its low [[coherence length]].<ref name=Miller1933>{{cite journal|author=Miller, Dayton C.|title=The Ether-Drift Experiment and the Determination of the Absolute Motion of the Earth|journal=Reviews of Modern Physics|volume=5|issue=3|date=1933|pages=203–242|doi=10.1103/RevModPhys.5.203|bibcode=1933RvMP....5..203M |s2cid=4119615 |quote=White light fringes were chosen for the observations because they consist of a small group of fringes having a central, sharply defined black fringe which forms a permanent zero reference mark for all readings.}}</ref> This was an early example of the use of white light to resolve the "2&nbsp;pi ambiguity".