The Michelson interferometer is a common configuration for optical interferometry and was invented by the American physicist Albert Abraham Michelson in 1887. Using a beam splitter, a light source is split into two arms. Each of those light beams is reflected back toward the beamsplitter which then combines their amplitudes using the superposition principle. The resulting interference pattern that is not directed back toward the source is typically directed to some type of photoelectric detector or camera. For different applications of the interferometer, the two light paths can be with different lengths or incorporate optical elements or even materials under test.
The Michelson interferometer is employed in many scientific experiments and became well known for its use by Michelson and Edward Morley in the famous Michelson–Morley experiment (1887) in a configuration which would have detected the Earth's motion through the supposed luminiferous aether that most physicists at the time believed was the medium in which light waves propagated. The null result of that experiment essentially disproved the existence of such an aether, leading eventually to the special theory of relativity and the revolution in physics at the beginning of the twentieth century. In 2015, another application of the Michelson interferometer, LIGO, made the first direct observation of gravitational waves. That observation confirmed an important prediction of general relativity, validating the theory's prediction of space-time distortion in the context of large scale cosmic events (known as strong field tests).
Configuration
A Michelson interferometer consists minimally of mirrors M1 & M2 and a beam splitter M (although a diffraction grating is also used). In Fig 2, a source S emits light that hits the beam splitter (in this case, a plate beamsplitter) surface M at point C. M is partially reflective, so part of the light is transmitted through to point B while some is reflected in the direction of A. Both beams recombine at point C' to produce an interference pattern incident on the detector at point E (or on the retina of a person's eye). If there is a slight angle between the two returning beams, for instance, then an imaging detector will record a sinusoidal fringe pattern as shown in Fig. 3b. If there is perfect spatial alignment between the returning beams, then there will not be any such pattern but rather a constant intensity over the beam dependent on the differential pathlength; this is difficult, requiring very precise control of the beam paths.
Fig. 2 shows use of a coherent (laser) source. Narrowband spectral light from a discharge or even white light can also be used, however to obtain significant interference contrast it is required that the differential pathlength is reduced below the coherence length of the light source. That can be only micrometers for white light, as discussed below.
If a lossless beamsplitter is employed, then one can show that optical energy is conserved. At every point on the interference pattern, the power that is not directed to the detector at E is rather present in a beam (not shown) returning in the direction of the source.
As shown 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.
Source bandwidth
White light has a tiny coherence length and is difficult to use in a Michelson (or Mach–Zehnder) interferometer. Even a narrowband (or "quasi-monochromatic") spectral source requires careful attention to issues of chromatic dispersion when used to illuminate an interferometer. The two optical paths must be practically equal for all wavelengths present in the source. This requirement can be met if both light paths cross an equal thickness of glass of the same dispersion. In Fig. 4a, the horizontal beam crosses the beam splitter three times, while the vertical beam crosses the beam splitter once. To equalize the dispersion, a so-called compensating plate identical to the substrate of the beam splitter may be inserted into the path of the vertical beam. In Fig. 4b, we see using a cube beam splitter already equalizes the pathlengths in glass. The requirement for dispersion equalization is eliminated by using extremely narrowband light from a laser.
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. Single longitudinal mode lasers are highly coherent and can produce high contrast interference with differential pathlengths of millions or even billions of wavelengths. On the other hand, using white (broadband) light, the central fringe is sharp, but away from the central fringe the fringes are colored and rapidly become indistinct to the eye.
Early experimentalists attempting to detect the Earth's velocity relative to the supposed luminiferous aether, such as Michelson and Morley (1887) and Miller (1933), used quasi-monochromatic light only for initial alignment and coarse path equalization of the interferometer. Thereafter they switched to white (broadband) light, since using white light interferometry they could measure the point of absolute phase equalization (rather than phase modulo 2π), thus setting the two arms' pathlengths equal.Michelson (1881) wrote, "... when they [the fringes using sodium light] were of convenient width and of maximum sharpness, the sodium flame was removed and the lamp again substituted. The screw m was then slowly turned till the bands reappeared. They were then of course colored, except the central band, which was nearly black."Shankland (1964) wrote concerning the 1881 experiment, p. 20: "The interference fringes were found by first using a sodium light source and after adjustment for maximum visibility, the source was changed to white light and the colored fringes then located. White-light fringes were employed to facilitate observation of shifts in position of the interference pattern." And concerning the 1887 experiment, p. 31: "With this new interferometer, the magnitude of the expected shift of the white-light interference pattern was 0.4 of a fringe as the instrument was rotated through an angle of 90° in the horizontal plane. (The corresponding shift in the Potsdam interferometer had been 0.04 fringe.)" More importantly, in a white light interferometer, any subsequent "fringe jump" (differential pathlength shift of one wavelength) would always be detected.
Applications
The Michelson interferometer configuration is used in a number of different applications.
Fourier transform spectrometer
Fig. 5 illustrates the operation of a Fourier transform spectrometer, which is essentially a Michelson interferometer with one movable mirror. (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. Fourier transform spectrometers can offer significant advantages over dispersive (i.e., grating and prism) spectrometers under certain conditions. (1) The Michelson interferometer's detector in effect monitors all wavelengths simultaneously throughout the entire measurement. When using a noisy detector, such as at infrared wavelengths, this offers an increase in signal-to-noise ratio while using only a single detector element; (2) the interferometer does not require a limited aperture as do grating or prism spectrometers, which require the incoming light to pass through a narrow slit in order to achieve high spectral resolution. This is an advantage when the incoming light is not of a single spatial mode. For more information, see Fellgett's advantage.
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. 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. 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. Fig. 6 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.
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. A similar scheme has been used by Tajammal M in his PhD thesis (Manchester University UK, 1995) to balance two arms of an LDA system. This system used fibre optic direction coupler.
Gravitational wave detection