Michelson's Interferometer — Question 8

Question 8

Thermal expansion of one arm produces a measurable phase drift

One Michelson mirror is mounted at the end of a support of length $L=0.150\,\mathrm m$ and linear expansion coefficient $\alpha=23.0\times10^{-6}\,\mathrm{K^{-1}}$. The beam splitter end is fixed, the other arm is thermally stable, and light travels in vacuum at $\lambda=600\,\mathrm{nm}$. The support warms uniformly by $0.0200\,\mathrm K$.

1. Calculate the mirror motion, phase drift, and number of fringe cycles.

2. Find the temperature increase required for one complete fringe cycle.

Question 8 - Solution

Thermal expansion of one arm produces a measurable phase drift

(a) Method and calculation

$$\Delta x=\alpha L\Delta T\Longrightarrow\boxed{\Delta x=69.0\,\mathrm{nm}}.$$$$\boxed{\Delta N=2\Delta x/\lambda=0.230,\quad\Delta\phi=2\pi\Delta N=0.460\pi\ \mathrm{rad}}.$$

(b) Interpretation and further calculation

$$2\alpha L\Delta T_1=\lambda.$$$$\boxed{\Delta T_1=\lambda/(2\alpha L)=0.0869565\,\mathrm K}.$$

Physical check

The calculation isolates mechanical expansion. In an air-filled instrument, temperature-dependent refractive index could add another optical-path contribution.