Michelson's Interferometer — Question 3

Question 3

A fringe rate measures mirror speed but not its sign

A Michelson mirror moves at constant speed v in vacuum. The source wavelength is $600\,\mathrm{nm}$, and a stationary detector records 1200 full intensity cycles per second. Other phases are constant.

1. Derive the relation between fringe frequency and mirror velocity and infer its speed.

2. Calculate the displacement magnitude during a $0.25\,\mathrm s$ observation and explain why one intensity channel cannot determine the direction of motion.

Question 3 - Solution

A fringe rate measures mirror speed but not its sign

(a) Method and calculation

$$\phi(t)=\phi_0+\frac{4\pi vt}{\lambda},\quad f_{\mathrm{fringe}}=\frac{|\dot\phi|}{2\pi}=\frac{2|v|}{\lambda}.$$$$\boxed{|v|=f_{\mathrm{fringe}}\lambda/2=0.360\,\mathrm{mm/s}}.$$

(b) Interpretation and further calculation

$$\boxed{|\Delta x|=|v|T=90.0\,\mu\mathrm m,\quad N=fT=300}.$$$$\cos(\phi_0+\omega t)=\cos(-\phi_0-\omega t)\Longrightarrow\boxed{\text{direction needs additional phase information}}.$$

Physical check

A fringe counter measures the magnitude of phase winding; a second quadrature channel can establish its sign.