33.8.1 The Michelson Interferometer

Albert Michelson, the first American scientist to receive a Nobel Prize, invented an optical interferometer, shown in FIGURE 33.24, in which an incoming light wave is divided by a beam splitter, a partially silvered mirror that reflects half the light but transmits the other half. The two waves then travel toward mirrors $M_{1}$ and $M_{2}$ . Half of the wave reflected from $M_{1}$ is transmitted through the beam splitter, where it recombines with the reflected half of the wave returning from $M_{2}$ . The superimposed waves travel on to a light detector, originally a human observer but now more likely an electronic photodetector.

The figure illustrates a Michelson interferometer.

FIGURE 33.24 Full Alternative Text

After separating, the two waves travel distances $r_{1} = 2 L_{1}$ and $r_{2} = 2 L_{2}$ before recombining, with the factors of 2 appearing because the waves travel to the mirrors and back again. Thus the path-length difference between the two waves is

Δ r = 2 L_{2} - 2 L_{1}

(33.35)

The conditions for constructive and destructive interference between the two recombined beams are the same as for double-slit interference: $Δ r = m λ$ and $Δ r = (m + \frac{1}{2} λ)$ , respectively. Thus constructive and destructive interference occur when

\begin{matrix} \begin{matrix} Constructive : & L_{2} - L_{1} = m \frac{λ}{2} \\ Destructive : & L_{2} - L_{1} = ​ ​ ​ ​ (m + \frac{1}{2}) \frac{λ}{2} \end{matrix} & m = 0, 1, 2, \dots \end{matrix}

(33.36)

You might expect the interferometer output to be either “bright” or “dark.” Instead, a viewing screen shows the pattern of circular interference fringes seen in FIGURE 33.25. Our analysis was for light waves that impinge on the mirrors exactly perpendicular to the surface. In an actual experiment, some of the light waves enter the interferometer at slightly different angles and, as a result, the recombined waves have slightly altered path-length differences $Δ r$ . These waves cause the alternating bright and dark fringes as you move outward from the center of the pattern. Their analysis will be left to more advanced courses in optics. Equations 33.36 are valid at the center of the circular pattern; thus there is a bright or dark central spot when one of the conditions in Equations 33.36 is true.

A photograph of the interference fringes produced by a Michelson interferometer.

FIGURE 33.25 Full Alternative Text

Mirror $M_{2}$ can be moved forward or backward by turning a precision screw, causing the central spot to alternate in a bright-dark-bright-dark-bright cycle that is easily seen or monitored by a photodetector. Suppose the interferometer is adjusted to produce a bright central spot. The next bright spot will appear when $M_{2}$ has moved half a wavelength, increasing the path-length difference by one full wavelength. The number $Δ m$ of maxima appearing as $M_{2}$ moves through distance $Δ L_{2}$ is

Δ m = \frac{Δ L_{2}}{λ / 2}

(33.37)

Very precise wavelength measurements can be made by moving the mirror while counting the number of new bright spots appearing at the center of the pattern. The number $Δ m$ is counted and known exactly. The only limitation on how precisely $λ$ can be measured this way is the precision with which distance $Δ L_{2}$ can be measured. Unlike $λ$ , which is microscopic, $Δ L_{2}$ is typically a few millimeters, a macroscopic distance that can be measured very accurately using precision screws, micrometers, and other techniques. Michelson’s invention provided a way to transfer the precision of macroscopic distance measurements to an equal precision for the wavelength of light.

Interferometers are also playing a key role in the development of optical computers because the bright and dark outputs of constructive and destructive interference can be the 1’s and 0’s of binary code.

EXAMPLE 33.7 Measuring the wavelength of light

An experimenter uses a Michelson interferometer to measure one of the wavelengths of light emitted by neon atoms. She slowly moves mirror $M_{2}$ until 10,000 new bright central spots have appeared. (In a modern experiment, a photodetector and computer would eliminate the possibility of experimenter error while counting.) She then measures that the mirror has moved a distance of 3.164 mm. What is the wavelength of the light?

MODEL An interferometer produces a new maximum each time $L_{2}$ increases by $λ / 2$ .

SOLVE The mirror moves $Δ L_{2} = 3.164 mm = 3.164 \times 10^{- 3} m$ . We can use Equation 33.37 to find

λ = \frac{2 Δ L_{2}}{Δ m} = 6.328 \times 10^{- 7} m = 632.8 nm

REVIEW A measurement of $Δ L_{2}$ accurate to four significant figures allowed us to determine $λ$ to four significant figures. This happens to be the neon wavelength that is emitted as the laser beam in a helium-neon laser.

33.8.1 The Michelson Interferometer

FIGURE 33.24 A Michelson interferometer.

FIGURE 33.25 Photograph of the interference fringes produced by a Michelson interferometer.

EXAMPLE 33.7 Measuring the wavelength of light