derive the working of Mickelson interferometer
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The Michelson interferometer is a common configuration for optical interferometry and was invented by Albert Abraham Michelson. 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 (among other interferometer configurations) is employed in many scientific experiments and became well known for its use by Albert Michelson and Edward Morley in the famous Michelson–Morley experiment (1887)[1] 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.[2] 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).
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Figure 2. Path of light in Michelson interferometer.
A Michelson interferometer consists minimally of mirrors M1 & M2 and a beam splitter M. 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.
Figure 3. Formation of fringes in a Michelson interferometer
This photo shows the fringe pattern formed by the Michelson interferometer,using monochromatic light (sodium D lines).
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.[3]:17