Michelson's Interferometer — Question 1

Question 1

A moving Michelson mirror changes the optical path twice as much

A Michelson interferometer operates in air at wavelength $\lambda=632.8\,\mathrm{nm}$. One end mirror is translated away from the beam splitter while the other arm is fixed. A detector counts 250 complete fringe cycles. Neglect air-index changes.

1. Find the mirror displacement and the unwrapped phase change.

2. If the endpoint phase can be resolved to 0.02 of a fringe cycle, find the corresponding displacement resolution.

Question 1 - Solution

A moving Michelson mirror changes the optical path twice as much

(a) Method and calculation

$$\Delta\mathcal L=2\Delta x=N\lambda\Longrightarrow\boxed{\Delta x=250(632.8\,\mathrm{nm})/2=79.10\,\mu\mathrm m}.$$$$\boxed{\Delta\phi=2\pi N=500\pi\ \mathrm{rad}}.$$

(b) Interpretation and further calculation

$$\delta x=\frac\lambda2\delta N.$$$$\boxed{\delta x=(632.8\,\mathrm{nm})(0.02)/2=6.328\,\mathrm{nm}}.$$

Physical check

Each complete fringe cycle corresponds to a mirror displacement of half a wavelength because the light travels to the mirror and back.