Michelson interferometer

Figure 1. A Michelson interferometer for use on an optical table.

The Michelson interferometer is the most common configuration for optical interferometry and was invented by Albert Abraham Michelson. An interference pattern is produced by splitting a beam of light into two paths, bouncing the beams back and recombining them. The different paths may be of different lengths or be composed of different materials to create interference fringes on a back detector. Michelson, along with Edward Morley, used this interferometer in the famous Michelson-Morley experiment (1887)^[1] in a failed attempt to demonstrate the effect of the hypothetical "aether wind" on the speed of light. Their experiment left theories of light based on the existence of a luminiferous aether without experimental support, and served ultimately as evidence for special relativity.

Configuration

Figure 2. Path of light in Michelson interferometer.

A Michelson interferometer consists of two highly polished mirrors M₁ & M₂. In Fig 2, a source S emits light that hits a beam splitter (in this case, a half-silvered mirror), surface M, at point C. M is partially reflective, so one beam is transmitted through to point B while the other is reflected in the direction of A. Both beams recombine at point C' to produce an interference pattern (assuming proper alignment) visible to the observer at point E. To the observer at point E, the effects observed would be the same as those produced by placing surfaces A and B' (the image of B on the surface M) on top of each other. Fig. 2 shows use of a monochromatic source. White light can also be used, provided that the path lengths are carefully equalized, a requirement due to the short coherence length of white light (on the order of a micrometer).

Energy is conserved, because there is a redistribution of energy at the central beam-splitter in which the energy at the destructive sites is re-distributed to the constructive sites. The effect of the interference is to alter the share of the reflected light which heads for the detector, the part which misses the detector, and the part which reflected back to the source.

Figure 3. Formation of fringes in a Michelson interferometer

As seen in Fig. 3a and 3b, the observer has a direct view of mirror M₁ seen through the beam splitter, and sees a reflected image M'₂ of mirror M₂. The fringes can be interpreted as the result of interference between light coming from the two virtual images S'₁ and S'₂ of the original source S. The characteristics of the interference pattern depend on the nature of the light source and the precise orientation of the mirrors and beam splitter. In Fig. 3a, the optical elements are oriented so that S'₁ and S'₂ are in line with the observer, and the resulting interference pattern consists of circles centered on the normal to M₁ and M'₂ (fringes of equal inclination). If, as in Fig. 3b, M₁ and M'₂ are tilted with respect to each other, the interference fringes will generally take the shape of conic sections (hyperbolas), but if M₁ and M'₂ overlap, the fringes near the axis will be straight, parallel, and equally spaced (fringes of equal thickness). If S is an extended source rather than a point source as illustrated, the fringes of Fig. 3a must be observed with a telescope set at infinity, while the fringes of Fig. 3b will be localized on the mirrors.^[2]:17

White light has only a very limited coherence length. When using a white light source, the two optical paths must be equal for all wavelengths. To meet this requirement, both the longitudinal and transverse light paths must cross an equal thickness of glass of the same dispersion. In Fig. 4a, the longitudinal beam crosses the beam splitter three times, while the transverse beam crosses the beam splitter once. To equalize the path lengths, a compensating plate identical to the beam splitter, but without the semireflective coat, is inserted into the path of the transverse beam.^[2]:16 In Fig. 4b, we see that a cube beam splitter is self-compensating.

The extent of the fringes depends on the coherence length of the source. In Fig. 3b, the yellow sodium light used for the fringe illustration consists of a pair of closely spaced lines, D₁ and D₂, implying that the interference pattern will blur after several hundred fringes. Highly monochromatic sources, such as lasers, yield interference patterns that can remain distinct over many millions of fringes. In Fig. 4, the central fringes are sharp, but the fringe patterns rapidly become indistinct.

Figure 4. Michelson interferometers using a white light source

If one uses a half-silvered mirror as the beam splitter, as in Fig. 4a, the horizontally traveling beam will undergo a front-surface reflection at the mirror, and a rear-surface reflection at the beam splitter. The vertically traveling beam will undergo two front surface reflections at the beam splitter and the mirror. At each front-surface reflection, the light will undergo a phase inversion. Since light traveling the two paths will undergo a different number of phase inversions, when the two paths differ by a whole number (including 0) of wavelengths, there will be destructive interference and a weak signal at the detector.^[3] If they differ by a whole number and a half wavelengths (e.g., 0.5, 1.5, 2.5 ...) there will be constructive interference and a strong signal. Results will differ if a cube beam-splitter is employed, as in Fig. 4b, since a cube beam-splitter makes no distinction between a front- and rear-surface reflection. In Fig. 4a, the central fringe representing equal path length is dark, while in Fig. 4b, the central fringe is bright.

Applications

The best known application of the Michelson Interferometer is the Michelson-Morley experiment, the unexpected null result of which was an inspiration for special relativity.^[4] It is interesting to note that Michelson and Morley (1887)^[1] and other early experimentalists using interferometric techniques in an attempt to measure the properties of the luminiferous aether, used (partially) 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. Purely monochromatic light would result in a uniform fringe pattern. Lacking modern means of 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, (See Fig. 4) 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 what is known as the "2 pi ambiguity". Use of partially monochromatic light (yellow sodium light) during initial alignment enabled the researchers to locate the position of equal path length, more or less easily, before switching to white light.^{[note 1]}

Figure 5. Fourier transform spectroscopy.

Michelson's configuration can be used for an assortment of different applications. The Michelson Interferometer has been used for the detection of gravitational waves, as a tunable narrow band filter, and as the core of Fourier transform spectroscopy. There are also some interesting applications as a "nulling" instrument that is used for detecting planets around nearby stars. For most purposes, however, the geometry of the Mach–Zehnder interferometer is more useful.

Fig. 5 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.^[5] Fourier transform spectrometers offer significant advantages over dispersive (i.e. grating and prism) spectrometers. (1) The Michelson interferometer's detector in effect monitors all wavelengths simultaneously throughout the entire measurement, increasing the integration time and the total number of photons monitored; (2) the interferometer does not require as much limitation in aperture as do grating or prism spectrometers, which require the incoming light to pass through a narrow slit. The result is that Fourier transform spectrometers are much more sensitive and offer higher signal-to-noise ratios than dispersive spectrometers.^[6] For more information, see Fellgett's advantage.

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. The reliability of Michelson interferometers has tended to favor their use in space applications, while the broad wavelength range and overall simplicity of Fabry–Pérot interferometers has favored their use in ground-based systems.^[7]

A further application is to produce a delay line interferometer that converts phase modulation into amplitude modulation in DWDM networks.

Astronomical interferometry is principally conducted using Michelson (and sometimes other type) interferometers. Principle operational interferometric observatories which use this type of instrumentation include VLTI, NPOI, and CHARA.

Figure 6. Typical optical setup of single point OCT

Another application of the Michelson Interferometer is in optical coherence tomography (OCT), a medical imaging technique using low-coherence interferometry to provide tomographic visualization of internal tissue microstructures. As seen in Fig. 6, 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.^[8][9] Recent advances have striven to combine the nanometer phase retrieval of coherent interferometry with the ranging capability of low-coherence interferometry.^[10]

Atmospheric and Space Applications

The Michelson Interferometer has played an important role in studies of the upper atmosphere, revealing temperatures and winds, employing both space-borne, and ground-based instruments, by measuring the Doppler widths and shifts in the spectra of airglow and aurora. For example, the Wind Imaging Interferometer, WINDII,^[11] on the Upper Atmosphere Research Satellite, UARS, (launched on September 12, 1991) measured the global wind and temperature patterns from 80 to 300 km by using the visible airglow emission from these altitudes as a target and employing optical Doppler interferometry to measure the small wavelength shifts of the narrow atomic and molecular airglow emission lines induced by the bulk velocity of the atmosphere carrying the emitting species. The instrument was an all-glass field-widened achromatically and thermally compensated phase-stepping Michelson interferometer, along with a bare CCD detector that imaged the airglow limb through the interferometer. A sequence of phase-stepped images was processed to derive the wind velocity for two orthogonal view directions, yielding the horizontal wind vector.

The principle of using a polarizing Michelson Interferometer as a narrow band filter was first described by Evans ^[12] who developed a birefringent photometer where the incoming light is split into two orthogonally polarized components by a polarizing beam splitter, sandwiched between two halves of a Michelson cube. This led to the first polarizing wide-field Michelson interferometer described by Title and Ramsey ^[13] which was used for solar observations; and led to the development of a refined instrument applied to measurements of oscillations in the sun's atmosphere, employing a network of observatories around the Earth known as the Global Oscillations Network Group (GONG).^[14]

The Polarizing Atmospheric Michelson Interferometer, PAMI, developed by Bird et al.,^[15] and discussed in Spectral Imaging of the Atmosphere,^[16] combines the polarization tuning technique of Title and Ramsey ^[13] with the Shepherd et al. ^[17] technique of deriving winds and temperatures from emission rate measurements at sequential path differences, but the scanning system used by PAMI is much simpler than the moving mirror systems in that it has no internal moving parts, instead scanning with a polarizer external to the interferometer. The PAMI was demonstrated in an observation campaign ^[18] where its performance was compared to a Fabry-Perot spectrometer, and employed to measure E-region winds.

More recently, the Helioseismic and Magnetic Imager (HMI), on the Solar Dynamics Observatory, employs two Michelson Interferometers with a polarizer and other tunable elements, to study solar variability and to characterize the Sun's interior along with the various components of magnetic activity. HMI takes high-resolution measurements of the longitudinal and vector magnetic field over the entire visible disk thus extending the capabilities of its predecessor, the SOHO's MDI instrument (See Fig. 7).^[19] HMI produces data to determine the interior sources and mechanisms of solar variability and how the physical processes inside the Sun are related to surface magnetic field and activity. It also produces data to enable estimates of the coronal magnetic field for studies of variability in the extended solar atmosphere. HMI observations will help establish the relationships between the internal dynamics and magnetic activity in order to understand solar variability and its effects.^[20]

In one example of the use of the MDI, Stanford scientists reported the detection of several sunspot regions in the deep interior of the Sun, 1–2 days before they appeared on the solar disc.^[21] The detection of sunspots in the solar interior may thus provide valuable warnings about upcoming surface magnetic activity which could be used to improve and extend the predictions of space weather forecasts.

Variations

Figure 8. Twyman-Green Interferometer.

Nonlinear Michelson interferometer

Also known as a "Step-phase Michelson interferometer", this is a generalized Michelson interferometer in which one mirror in one arm is replaced with a Gires–Tournois interferometer^[22] (Interferometer useful for the compression of frequency-modulated light pulse) or Gires–Tournois etalon. The field coming from Gires–Tournois etalon interferes with the plane field reflected from the ordinary reflector. Because the phase change from the Gires–Tournois etalon depends on wavelength and shows step-like behavior, nonlinear Michelson interferometer has particular applications. One notable application in fiber-optic communications is an optical interleaver.

Both mirrors in a Michelson interferometer can be replaced with two Gires–Tournois etalons. Such a nonlinear Michelson interferometer exhibits stronger nonlinearity, which can be used to construct an asymmetric optical interleaver.

Twyman-Green interferometer

The Twyman-Green interferometer is a variation of the Michelson interferometer used to test small optical components, invented and patented by Twyman and Green in 1916. 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 available light sources had limited coherence length. Michelson pointed out that constraints on geometry forced by the 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.^[23] Decades later, the advent of laser light sources answered Michelson's objections.

The use of a figured reference mirror in one arm allows the Twyman-Green interferometer to be used for testing various forms of optical component, such as lenses or telescope mirrors.^[24] Fig. 8 illustrates a Twyman-Green interferometer set up to test a lens. A point source of monochromatic light 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.^[25]

Laser unequal path interferometer

The "LUPI" is a Twyman-Green interferometer that uses a coherent laser light source. The high coherence length of a laser allows unequal path lengths in the test and reference arms and permits economical use of the Twyman-Green configuration in testing large optical components.

See also

• List of types of interferometers
• LIGO Laser Interferometer Gravitational-Wave Observatory
• NPOI
• GEO600
• VIRGO

Notes

References

External links

• Diagrams of Michelson interferometers
• Application of a step-phase interferometer in optical communication
• European Gravitational Observatory
• A satellite view of the VIRGO interferometer
• A useful free software, to simulate and understand the Michelson interferometer principles, made by students of Faculty of Engineering of the University of Porto