Michelson's Interferometer — Question 4

Question 4

A slight mirror tilt creates straight fringes across the output beam

A Michelson mirror tilts by a small angle $\alpha$, so the two recombined plane waves differ in direction by approximately $2\alpha$. At transverse coordinate y use $\delta(y)=\delta_0+2k\alpha y$, where $k=2\pi/\lambda$. At $\lambda=600\,\mathrm{nm}$ the measured fringe spacing is $p=3.00\,\mathrm{mm}$.

1. Infer the mirror tilt in radians and degrees.

2. Calculate the number of full fringe intervals across a $12.0\,\mathrm{mm}$ beam and the spacing if the mirror tilt doubles.

Question 4 - Solution

A slight mirror tilt creates straight fringes across the output beam

(a) Method and calculation

$$\delta(y+p)-\delta(y)=2k\alpha p=2\pi\Longrightarrow p=\lambda/(2\alpha).$$$$\boxed{\alpha=1.00\times10^{-4}\,\mathrm{rad}=0.0057296^\circ}.$$

(b) Interpretation and further calculation

$$\boxed{N=W/p=12.0/3.00=4\ \text{intervals}}.$$$$\alpha\to2\alpha\Longrightarrow\boxed{p\to p/2=1.50\,\mathrm{mm}}.$$

Physical check

The reflected ray turns through twice the mirror angle. Omitting that factor would double the inferred tilt.