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By the end of this section, you will be able to:
  • Explain changes in fringes observed with a Michelson interferometer caused by mirror movements
  • Explain changes in fringes observed with a Michelson interferometer caused by changes in medium

The Michelson interferometer    (invented by the American physicist Albert A. Michelson , 1852–1931) is a precision instrument that produces interference fringes by splitting a light beam into two parts and then recombining them after they have traveled different optical paths. [link] depicts the interferometer and the path of a light beam from a single point on the extended source S, which is a ground-glass plate that diffuses the light from a monochromatic lamp of wavelength λ 0 . The beam strikes the half-silvered mirror M, where half of it is reflected to the side and half passes through the mirror. The reflected light travels to the movable plane mirror M 1 , where it is reflected back through M to the observer. The transmitted half of the original beam is reflected back by the stationary mirror M 2 and then toward the observer by M.

Picture A shows a schematic drawing of the Michelson interferometer. Picture B is the planar view of the Michelson interferometer. A light beam from the laser passes through the screen S with the slit. It strikes the half-silvered mirror M, where half of it is reflected to the side and half passes through the mirror. The reflected light travels to the movable plane mirror M1, where it is reflected back through M to the observer. The transmitted half of the original beam is reflected back by the stationary mirror M2 and then toward the observer by M.
(a) The Michelson interferometer. The extended light source is a ground-glass plate that diffuses the light from a laser. (b) A planar view of the interferometer.

Because both beams originate from the same point on the source, they are coherent and therefore interfere. Notice from the figure that one beam passes through M three times and the other only once. To ensure that both beams traverse the same thickness of glass, a compensator plate C of transparent glass is placed in the arm containing M 2 . This plate is a duplicate of M (without the silvering) and is usually cut from the same piece of glass used to produce M. With the compensator in place, any phase difference between the two beams is due solely to the difference in the distances they travel.

The path difference of the two beams when they recombine is 2 d 1 2 d 2 , where d 1 is the distance between M and M 1 , and d 2 is the distance between M and M 2 . Suppose this path difference is an integer number of wavelengths m λ 0 . Then, constructive interference occurs and a bright image of the point on the source is seen at the observer. Now the light from any other point on the source whose two beams have this same path difference also undergoes constructive interference and produces a bright image. The collection of these point images is a bright fringe corresponding to a path difference of m λ 0 ( [link] ). When M 1 is moved a distance Δ d = λ 0 / 2 , this path difference changes by λ 0 , and each fringe moves to the position previously occupied by an adjacent fringe. Consequently, by counting the number of fringes m passing a given point as M 1 is moved, an observer can measure minute displacements that are accurate to a fraction of a wavelength, as shown by the relation

Δ d = m λ 0 2 .
Picture shows a photograph of the fringes produced with a Michelson interferometer. Fringes are visible as alternating dark and light circles.
Fringes produced with a Michelson interferometer. (credit: “SILLAGESvideos”/YouTube)

Precise distance measurements by michelson interferometer

A red laser light of wavelength 630 nm is used in a Michelson interferometer. While keeping the mirror M 1 fixed, mirror M 2 is moved. The fringes are found to move past a fixed cross-hair in the viewer. Find the distance the mirror M 2 is moved for a single fringe to move past the reference line.

Strategy

Refer to [link] for the geometry. We use the result of the Michelson interferometer interference condition to find the distance moved, Δ d .

Solution

For a 630-nm red laser light, and for each fringe crossing ( m = 1 ) , the distance traveled by M 2 if you keep M 1 fixed is

Δ d = m λ 0 2 = 1 × 630 nm 2 = 315 nm = 0.315 μ m .

Significance

An important application of this measurement is the definition of the standard meter. As mentioned in Units and Measurement , the length of the standard meter was once defined as the mirror displacement in a Michelson interferometer corresponding to 1,650,763.73 wavelengths of the particular fringe of krypton-86 in a gas discharge tube.

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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