Interferometry

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

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.)
See also: Interference (wave propagation)

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

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

Heterodyne versus homodyne detection

Double path versus common path

See also: Common path interferometer
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]
Figure 3. Four examples of common path interferometers.

Wavefront splitting versus amplitude splitting

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
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 shown as it might be set up for a wind tunnel study using white light. 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 is 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 are oriented so that the test and reference beams pass through an equal amount of glass. In addition, 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.
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

Figure 6. Fourier transform spectroscopy.

Physics and astronomy

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.[34]
Figure 7. A picture of the solar corona taken with the LASCO C1 coronagraph which employed a tunable Fabry-Perot 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.
Fig. 7 shows a doppler image of the solar corona made using a tunable Fabry-Pérot interferometer.
The Laser Interferometer Gravitational-Wave Observatory (LIGO) uses two 4-km Michelson-Fabry-Pérot interferometers for the detection of gravitational waves.[35]
Figure 8. The VLA interferometer
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.[38]
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.[39][40] Despite these technical difficulties, dozens of astronomical optical interferometers are now in operation offering resolutions down to the fractional milliarcsecond range.
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.[41] 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.[42]
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.[43]
Atom interferometry techniques are reaching sufficient precision to allow laboratory-scale tests of general relativity.[44]

Engineering and applied science

Figure 9. Optical flat interference fringes
Figure 10. Twyman-Green Interferometer.
Figure 11. Optical testing with a Fizeau interferometer and a computer generated hologram
Figure 12. Phase shifting and vertical scanning interferometers. The white light interferogram is vastly oversimplified in this drawing. It actually consists of a moving rainbow of fringes generated by multiple wavelengths, obtaining peak fringe contrast as a function of scan position, that is, the red portion of the object beam interferes with the red portion of the reference beam, the blue interferes with the blue, and so forth. In other words, a very large amount of data is available in white-light interferograms.
  • 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.[57][58]
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.[57][58] 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. As seen in Fig. 12, 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.[55][59] 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,[60][61][62] 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.[63][64][65] 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.).[66]
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.[67]
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.[68]
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.[68]
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.[68]
Figure 14. ESPI fringes showing a vibration mode of a clamped square plate.
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,[69] 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.[70][71]

Biology and medicine

Figure 15. Typical optical setup of single point OCT
Figure 16. Toxoplasma gondii unsporulated oocyst, differential interference contrast (DIC), 100×.

See also

References

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