Michelson's Interferometer — Question 10

Question 10

Quadrature detection resolves a signed sub-fringe mirror displacement

A calibrated Michelson has two normalized output channels $C=\cos\phi$ and $S=\sin\phi$. Initially $(C,S)=(1,0)$. After a continuous motion known to produce $|\Delta\phi|<\pi$, the readings are $C=\sqrt3/2$ and $S=-1/2$. Positive mirror motion increases phase according to $\Delta\phi=4\pi\Delta x/\lambda$. Let $\lambda=600\,\mathrm{nm}$.

1. Infer the signed phase and mirror displacement.

2. If the phase estimate has standard uncertainty $\sigma_\phi=0.010\,\mathrm{rad}$, find the displacement uncertainty and the ambiguity if the motion bound were absent.

Question 10 - Solution

Quadrature detection resolves a signed sub-fringe mirror displacement

(a) Method and calculation

$$\Delta\phi=\operatorname{atan2}(S,C)=\boxed{-\pi/6}.$$$$\boxed{\Delta x=\frac{\lambda\Delta\phi}{4\pi}=-25.0\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$\boxed{\sigma_x=\frac{\lambda}{4\pi}\sigma_\phi=0.477465\,\mathrm{nm}}.$$$$\Delta\phi=-\pi/6+2\pi m\Longrightarrow\boxed{\Delta x=-25.0\,\mathrm{nm}+m(300\,\mathrm{nm}),\quad m\in\mathbb Z}.$$

Physical check

Quadrature identifies the phase quadrant, while continuous tracking or a prior motion bound supplies the integer fringe order.