3.5 The Michelson Interferometer

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. Figure 3.16 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 ${\lambda }_0$. 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 ${\text{M}}_1$, where it is reflected back through M to the observer. The transmitted half of the original beam is reflected back by the stationary mirror ${\text{M}}_2$ and then toward the observer by M.

Figure 3.16 (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 ${\text{M}}_2$. 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 $2d_1-2d_2$, where $d_1$ is the distance between M and ${\text{M}}_1$, and $d_2$ is the distance between M and ${\text{M}}_2$. Suppose this path difference is an integer number of wavelengths $m{\lambda }_0$. 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 $m{\lambda }_0$ (Figure 3.17). When ${\text{M}}_1$ is moved a distance $\text{Δ}d={\lambda }_0\text{/}2$, this path difference changes by ${\lambda }_0$, 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 ${\text{M}}_1$ is moved, an observer can measure minute displacements that are accurate to a fraction of a wavelength, as shown by the relation

Figure 3.17 Fringes produced with a Michelson interferometer. (credit: “SILLAGESvideos”/YouTube)

Example 3.6

Measuring the Refractive Index of a Gas

In one arm of a Michelson interferometer, a glass chamber is placed with attachments for evacuating the inside and putting gases in it. The space inside the container is 2 cm wide. Initially, the container is empty. As gas is slowly let into the chamber, you observe that dark fringes move past a reference line in the field of observation. By the time the chamber is filled to the desired pressure, you have counted 122 fringes move past the reference line. The wavelength of the light used is 632.8 nm. What is the refractive index of this gas?

Strategy

The $m=122$ fringes observed compose the difference between the number of wavelengths that fit within the empty chamber (vacuum) and the number of wavelengths that fit within the same chamber when it is gas-filled. The wavelength in the filled chamber is shorter by a factor of n, the index of refraction.

Solution

The ray travels a distance $t=2\text{cm}$ to the right through the glass chamber and another distance t to the left upon reflection. The total travel is $L=2t$. When empty, the number of wavelengths that fit in this chamber is

$$N_0=\frac{L}{{\lambda }_0}=\frac{2t}{{\lambda }_0}$$

where ${\lambda }_0=632.8\text{nm}$ is the wavelength in vacuum of the light used. In any other medium, the wavelength is $\lambda ={\lambda }_0\text{/}n$ and the number of wavelengths that fit in the gas-filled chamber is

$$N=\frac{L}{\lambda }=\frac{2t}{{\lambda }_0\text{/}n}.$$

The number of fringes observed in the transition is

$$\begin{array}{cc}m& =N-N_0,\hfill \\ & =\frac{2t}{{\lambda }_0\text{/}n}-\frac{2t}{{\lambda }_0},\hfill \\ & =\frac{2t}{{\lambda }_0}\left(n-1\right).\hfill \end{array}$$

Solving for $\left(n-1\right)$ gives

$$n-1=m\left(\frac{{\lambda }_0}{2t}\right)=122\left(\frac{632.8\times {10}^{\mathrm{-9}}\text{m}}{2(2\times {10}^{\mathrm{-2}}\text{m})}\right)=0.0019$$

and $n=1.0019$.

Significance

The indices of refraction for gases are so close to that of vacuum, that we normally consider them equal to 1. The difference between 1 and 1.0019 is so small that measuring it requires a correspondingly sensitive technique such as interferometry. We cannot, for example, hope to measure this value using techniques based simply on Snell’s law.