Figure 1. The light path through a Michelson interferometer.

Interferometry refers to a family of techniques in which waves, usually electromagnetic, are superimposed in order to extract information about the waves. Interferometry is an important investigative technique in the fields of astronomy, fiber optics, engineering metrology, optical metrology, oceanography, seismology, chemistry, quantum mechanics, nuclear and particle physics, plasma physics, remote sensing, biomolecular interactions, surface profiling, microfluidics, mechanical stress/strain measurement, and velocimetry .^[1][2]

Interferometers are widely used in science and industry for the measurement of small displacements, refractive index changes, surface irregularities and the like. An astronomical interferometer consists of two or more separate telescopes that combine their signals, offering a resolution equivalent to that of a telescope of diameter equal to the largest separation between its individual elements.

Basic principles[link]

Figure 2. An idealized interferometric determination of wavelength obtained by looking at interference fringes between two coherent beams recombined after traveling different distances. (The square red emitter is a laser.)

Interferometry makes use of the principle of superposition to combine waves in a way that will cause the result of their combination to have some meaningful property that is diagnostic of the original state of the waves. This works because when two waves with the same frequency combine, the resulting pattern is determined by the phase difference between the two waves—waves that are in phase will undergo constructive interference while waves that are out of phase will undergo destructive interference. Most interferometers use light or some other form of electromagnetic wave.^[3]

Typically (see Fig. 1, the well-known Michelson configuration) a single incoming beam of coherent light will be split into two identical beams by a beam splitter (a partially reflecting mirror). Each of these beams travels a different route, called a path, and are recombined before arriving at a detector. The path difference, the difference in the distance traveled by each beam, creates a phase difference between them. It is this introduced phase difference that creates the interference pattern between the initially identical waves. If a single beam has been split along two paths, then the phase difference is diagnostic of anything that changes the phase along the paths. This could be a physical change in the path length itself or a change in the refractive index along the path.

Categories[link]

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

Heterodyne versus homodyne detection[link]

• In heterodyne detection, one modulates, usually by a frequency shift, one of two beams prior to detection. A special case of heterodyne detection is optical heterodyne detection, which detects the interference at the beat frequency. Since the modulation is known, the relative phase of the measured beat frequency can be measured very precisely even if the intensity levels of the beams are slowly drifting.^[4]
• In standard interferometry (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 light beams is measured.^[4]

Double path versus common path[link]

• 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.
• A common path interferometer is a class of interferometer in which the reference beam and sample beam travel along the same path. Examples include the Sagnac interferometer, the point diffraction interferometer, the lateral shearing interferometer, and the Zernike phase contrast interferometer.^[5]

Fig. 3 illustrates four common path interferometers: Sagnac, fibre optic gyroscope, point diffraction, and lateral shearing interferometers.

• Sagnac interferometers are sensitive to rotation. The first accounts of the effects of rotation on this form of interferometer were published in 1913 by Georges Sagnac, who mistakenly believed that his ability to detect a "whirling of the ether" disproved relativity theory.^[6]

The sensitivity of present-day Sagnac interferometers far exceeds that of Sagnac's original arrangement. The sensitivity to rotation is proportional to the area circumscribed by the counter-rotating beams, and fiber optic gyroscopes, the present-day descendants of the Sagnac interferometer, use thousands of loops of optical fibre rather than mirrors, such that even small to medium sized units easily detect the rotation of the Earth. Ring gyroscopes have important applications in inertial guidance systems.^[6][7]

• Another common path interferometer useful in lens testing and fluid flow diagnostics is the point diffraction interferometer (PDI), invented by Linnik in 1933.^[8][9] The reference beam is generated by diffraction from a small pinhole, about half the diameter of the Airy disk, in a semitransparent plate. Fig. 3 illustrates an aberrated wavefront focused onto the pinhole. The diffracted reference beam and the transmitted test wave interfere to form fringes. The common path design of the PDI brings to it a number of important advantages. (1) Only a single laser path is required rather than the two paths required by the Mach-Zehnder or Michelson designs. This advantage can be very important in large interferometric setups such as in wind tunnels that have long optical paths through turbulent media. (2) The common path design uses fewer optical components than double path designs, making alignment much easier, as well as reducing cost, size, and weight, especially for large setups.^[10] (3) While the accuracy of a double path interferometer is dependent on the precision with which the reference element is figured, careful design enables the generated reference beam of the PDI to be of guaranteed precision.^[11] A disadvantage is that the amount of light getting through the pinhole depends on how well the light can be focused onto the pinhole. If the incident wavefront is severely aberrated, very little light may get through.^[5] The PDI has seen use in various adaptive optics applications.^[12][13]
• Lateral shearing interferometry is a self-referencing method of wavefront sensing. Instead of comparing a wavefront with a reference wavefront, lateral shearing interferometry interferes a wavefront with a shifted version of itself. As a result, it is sensitive to the slope of a wavefront, not the wavefront shape per se. Applications of lateral shearing interferometry have included thin film analysis, collimation testing, and adaptive optics.^[14][15] Shearing interferometers, a general framework which includes the lateral shearing, Hartmann, Shack-Hartmann, rotational shearing, folding shearing, and aperture masking interferometers, are used in most of the wavefront sensors industrially developed.^[16]

File:Four common path interferometers.png

Figure 3. Four examples of common path interferometers.

Wavefront splitting versus amplitude splitting[link]

• 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.^[17]

Fig. 4 illustrates the operation of two wavefront splitting interferometers. Other examples of wavefront splitting interferometers include the Fresnel biprism, Lloyd's mirror and the Billet Bi-Lens.^[18]

• 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,^[19] with electrons,^[20][21] and with buckyball molecules large enough to be seen under an electron microscope ^[22]
• 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. Interestingly, the band of equal path length, nearest the mirror, is dark rather than bright. In 1834, Humphry Lloyd interpreted this effect as proof that the phase of a front-surface reflected beam is inverted.^[23][24]

Figure 4. Young's two-slit experiment and Lloyd's mirror

• An amplitude splitting interferometer uses a partial reflector to divide the amplitude of the incident wave into separate beams which are separated and recombined. Examples of amplitude splitting interferometers include the Fizeau, Michelson, Twyman–Green, Mach–Zehnder and Fabry–Pérot interferometers.^[25]

Fig. 5 illustrates the operation of three amplitude splitting interferometers.

• 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. A collimated beam of monochromatic light illuminates the two flats, and a beam splitter allows the fringes to be viewed on-axis.^[26][27]
• 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 in focus in any desired plane. For a wind tunnel study, as illustrated in Fig. 5, the fringes would customarily 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 microns, great care must be taken to equalize the optical paths or no fringes will be visible. 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.^[28][29]
• 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.) 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. 5, 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.^[30] In Fig. 5, 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.

File:Three amplitude-splitting interferometers.png

Figure 5. Three amplitude-splitting interferometers: Fizeau, Mach–Zehnder, and Fabry Perot

It is interesting to note that Michelson and Morley (1887)^[31] 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 environmental temperature control, the fringes showed continual 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 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. This was an early example of the use of white light to resolve the "2 pi ambiguity".

Applications[link]

Physics and astronomy[link]

• In physics, one of the most important experiments of the late 19^th century was the famous "failed experiment" of Michelson and Morley that served as an inspiration for special relativity. Michelson interferometers are used in tunable narrow band optical filters^[32] and as the core hardware component of Fourier transform spectrometers.^[33]

When used as a tunable narrow band filter, Michelson interferometers exhibit a number of advantages and disadvantages when compared with competing technologies such as Fabry–Pérot interferometers or Lyot filters. Michelson interferometers have the largest field of view for a specified wavelength, and are relatively simple in operation, since tuning is via mechanical rotation of waveplates rather than via high voltage control of piezoelectric crystals or lithium niobate optical modulators as used in a Fabry–Pérot system. Compared with Lyot filters, which use birefringent elements, Michelson interferometers have a relatively low temperature sensitivity. On the negative side, Michelson interferometers have a relatively restricted wavelength range and require use of prefilters which restrict transmittance.^[34]

Fig. 6 illustrates the operation of a Fourier transform spectrometer, which is essentially a Michelson interferometer with one mirror movable. (A practical Fourier transform spectrometer would substitute corner cube reflectors for the flat mirrors of the conventional Michelson interferometer, but for simplicity, the illustration does not show this.) An interferogram is generated by making measurements of the signal at many discrete positions of the moving mirror. A Fourier transform converts the interferogram into an actual spectrum.^[35]

Figure 7. A picture of the solar corona taken with the LASCO C1 coronagraph.

• Fabry-Pérot thin-film etalons are used in narrow bandpass filters capable of selecting a single spectral line for imaging; for example, the H-alpha line or the Ca-K line of the Sun or stars.

Fig. 7 shows a doppler image of the solar corona made using a tunable Fabry-Pérot interferometer to recover scans of the solar corona at a number of wavelengths near the FeXIV green line. The picture is a color coded image of the doppler shift of the line, which may be associated with the coronal plasma velocity towards or away from the satellite camera.

The Laser Interferometer Gravitational-Wave Observatory (LIGO) uses two 4-km Michelson-Fabry-Pérot interferometers for the detection of gravitational waves.^[36] In this application, the Fabry–Pérot cavity is used to store photons for almost a millisecond while they bounce up and down between the mirrors. This increases the time a gravitational wave can interact with the light, which results in a better sensitivity at low frequencies. Smaller cavities, usually called mode cleaners, are used for spatial filtering and frequency stabilization of the main laser.

• The Mach-Zehnder interferometer's relatively large and freely accessible working space, and its flexibility in locating the fringes has made it the interferometer of choice for visualizing flow in wind tunnels,^[37][38] and for flow visualization studies in general. It is frequently used in the fields of aerodynamics, plasma physics and heat transfer to measure pressure, density, and temperature changes in gases.

Mach-Zehnder interferometers are also used to study one of the most counterintuitive predictions of quantum mechanics, the phenomenon known as quantum entanglement.^[39][40]

Figure 8. The VLA interferometer

• An astronomical interferometer achieves high-resolution observations using the technique of aperture synthesis, mixing signals from a cluster of comparatively small telescopes rather than a single very expensive monolithic telescope.^[41]

Early radio telescope interferometers used a single baseline for measurement. Later astronomical interferometers, such as the Very Large Array illustrated in Fig 8, used arrays of telescopes arranged in a pattern on the ground. A limited number of baselines will result in insufficient coverage. This was alleviated by using the rotation of the Earth to rotate the array relative to the sky. Thus, a single baseline could measure information in multiple orientations by taking repeated measurements, a technique called Earth-rotation synthesis. Baselines thousands of kilometers long were achieved using very long baseline interferometry.^[41]

Astronomical optical interferometry has had to overcome a number of technical issues not shared by radio telescope interferometry. The short wavelengths of light necessitate extreme precision and stability of construction. For example, spatial resolution of 1 milliarcsecond requires 0.5 micron stability in a 100 m baseline. Optical interferometric measurements require high sensitivity, low noise detectors that did not become available until the late 1990's. Astronomical "seeing", the turbulence that causes stars to twinkle, introduces rapid, random phase changes in the incoming light, requiring kilohertz data collection rates to be faster than the rate of turbulence.^[42][43] Despite these technical difficulties, dozens of astronomical optical interferometers are now in operation offering resolutions down to the fractional milliarcsecond range.

• The wave character of matter can be exploited to build interferometers. The first examples of matter interferometers were electron interferometers, later followed by neutron interferometers. Around 1990 the first atom interferometers were demonstrated, later followed by interferometers employing molecules.

Electron holography is an imaging technique that photographically records the electron interference pattern of an object, which is then reconstructed to yield a greatly magnified image of the original object.^[44] This technique was developed to enable greater resolution in electron microscopy than is possible using conventional imaging techniques. The resolution of conventional electron microscopy is not limited by electron wavelength, but by the large aberrations of electron lenses.^[45]

Neutron interferometry has been used to investigate the Aharonov–Bohm effect, to examine the effects of gravity acting on an elementary particle, and to demonstrate a strange behavior of fermions that is at the basis of the Pauli exclusion principle: Unlike macroscopic objects, when fermions are rotated by 360° about any axis, they do not return to their original state, but develop a minus sign in their wave function. In other words, a fermion needs to be rotated 720° before returning to its original state.^[46]

Atom interferometry techniques are reaching sufficient precision to allow laboratory-scale tests of general relativity.^[47]

Engineering and applied science[link]

Figure 9. Optical flat interference fringes

• 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. 9 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. Since the fringes are being viewed at an angle, they are slightly distorted. When the flats are ready for sale, they will be mounted in a Fizeau interferometer for formal testing and certification.
• Fabry-Perot etalons are widely used in telecommunications, lasers and spectroscopy to control and measure the wavelengths of light. Dichroic filters are multiple layer 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.

• The Twyman–Green interferometer, invented by Twyman and Green in 1916, is a variant of the Michelson interferometer widely used to test optical components.^[48] The basic characteristics distinguishing it from the Michelson configuration are the use of a monochromatic point light source and a collimator. It is interesting to note that 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.^[49] 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. 10 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.^[50]
• Mach-Zehnder interferometers are being used in integrated optical circuits, in which light interferes between two branches of a waveguide that are externally 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 modulators to sensors^[51][52] to optical switches.^[53]
• The latest proposed extremely large astronomical telescopes, such as the Thirty Meter Telescope and the European 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 holograms (CGHs) have begun to supplement null correctors in test setups for complex aspheric surfaces. Fig. 11 illustrates how this is done. Unlike the figure, actual CGHs have line spacing on the order of 1 to 10 microns. 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.^[54]

File:Fizeau optical testing with computer generated hologram.png

Figure 11. Optical testing with a Fizeau interferometer and a computer generated hologram

• Ring laser gyroscopes (RLGs) and fibre optic gyroscopes (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.
• Doppler radar detectors are basically heterodyne detection devices that compare transmitted and reflected beams.
• 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),^[55] Vertical Scanning Interferometry(VSI),^[56] and Enhanced VSI (EVSI).^[57] 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. VSI uses white light and, rather than looking at the shape of the fringes, as does PSI, looks for best focus position; it provides less precise measurements than PSI but can be used on rough surfaces. EVSI represents a hybrid technique, using white light and measuring phase shifting.^[58]

File:Phase Shifting and Vertical Scanning Interferometry Animation.gif

Figure 12. Phase shifting and vertical scanning interferometers.

• Phase Shifting Interferometry addresses several issues associated with the classical analysis of interferograms. Classically, one measures the positions of the fringe centers. As seen in Fig. 9, fringe deviations from straightness and equal spacing provide a measure of the aberration. Determining the location of the fringe centers is difficult to automate, and any intensity variations across the interferogram will introduce error. 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. Finally, there is a phase problem: In Fig. 9, one can see that the tested surface deviates from flatness, but one cannot tell from this single image whether this deviation from flatness is concave or convex.^[59][60]

Phase shifting interferometry overcomes these limitations by not relying on finding fringe centers, but rather by collecting intensity data from every point of the CCD image sensor. As seen in Fig. 12, 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 piezoelectric transducer (PZT). Alternatively, precise phase shifts can be introduced by modulating the laser frequency.^[59][60] The captured images are processed by a computer to calculate the optical wavefront errors.

Figure 13. Lunate cells of Nepenthes khasiana visualized by Scanning White Light Interferometry (SWLI), a form of Vertical Scanning Interferometry.

• Vertical Scanning Interferometry is an example of low-coherence interferometry, which exploits the low coherence of white light. Interference will only be achieved when the path length delays of the interferometer are matched within the coherence time of the light source. VSI monitors the fringe contrast rather than the shape of the fringes. Fig. 12 illustrates a VSI microscope. The objective assembly contains a Mirau interferometer. 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.^[57][61] The chief benefit of low-coherence interferometry is that systems can be designed that do not suffer from the 2 pi ambiguity of coherent interferometry,^[62][63][64] and as seen in Fig. 13, which scans a 180μm x 140μm x 10μm volume, it is well suited to profiling steps and rough surfaces. The axial resolution of the system is determined by the coherence length of the light source and is typically in the micrometer range.^[65][66][67] 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.).^[68]

• 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.

Holographic interometry 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.^[69]

Eventually, several independent groups of experimenters in the mid-60's realized that the fringes encoded important information about dimensional changes occuring 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.

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.^[70]

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.^[70]

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.^[70]

Figure 14. ESPI fringes showing a vibration mode of a clamped square plate.

• 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. 14) The fringes are similar to those obtained in holographic interferometry.

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,^[71] 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.^[72][73]

• A method of establishing precise 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 metrologically traceable scale for geodetic networks measured by these instruments.^[74] (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.^[58]

Biology and medicine[link]

Figure 15. Typical optical setup of single point OCT

• Optical coherence tomography (OCT) is a medical imaging technique using low-coherence interferometry to provide tomographic visualization of internal tissue microstructures. As seen in Fig. 15, the core of a typical OCT system is a Michelson interferometer. One interferometer arm is focused onto the tissue sample and scans the sample in an X-Y longitudinal raster pattern. The other interferometer arm is bounced off a reference mirror. Reflected light from the tissue sample is combined with reflected light from the reference. Because of the low coherence of the light source, interferometric signal is observed only over a limited depth of sample. X-Y scanning therefore records one thin optical slice of the sample at a time. By performing multiple scans, moving the reference mirror between each scan, an entire three-dimensional image of the tissue can be reconstructed.^[75][76] Recent advances have striven to combine the nanometer phase retrieval of coherent interferometry with the ranging capability of low-coherence interferometry.^[58]

Figure 16. Toxoplasma gondii unsporulated oocyst, differential interference contrast (DIC), 100×.

• Phase contrast and differential interference contrast (DIC) microscopy are important tools in biology and medicine. Most animal cells and single-celled organisms have very little color, and their intracellular organelles are almost totally invisible under simple bright field illumination. These structures can be made visible by staining the specimens, but staining procedures are time-consuming and kill the cells. As seen in Fig. 16, phase contrast and DIC microscopes allow unstained, living cells to be studied.^[77] DIC also has non-biological applications, for example in the analysis of planar silicon semiconductor processing.
• Angle-resolved low-coherence interferometry (a/LCI) uses scattered light to measure the sizes of subcellular objects, including cell nuclei. This allows interferometry depth measurements to be combined with density measurements. Various correlations have been found between the state of tissue health and the measurements of subcellular objects. For example, it has been found that as tissue changes from normal to cancerous, the average cell nuclei size increases.^[78][79]

See also[link]

• List of types of interferometers
  • Fizeau interferometer
  • Michelson interferometer
  • Fabry–Pérot interferometer
• Aperture synthesis
• Astronomical interferometer
• Coherence
• Interference
• Optical coherence tomography
• Optical heterodyne detection
• Very Long Baseline Interferometry

References[link]