Intensity in Double-Slit Interference — Question 5

Question 5

Four phase-stepped measurements reconstruct the interference phase

A detector measures $I(\alpha)=A+B\cos(\delta+\alpha)$ for imposed phase steps $\alpha=0,\pi/2,\pi,3\pi/2$. The four readings in that order are 14, 7, 6, and 13 in common intensity units. Assume A and B are constant during the sequence, with $B>0$.

1. Derive expressions for A, $B\cos\delta$, and $B\sin\delta$ from the four readings.

2. Calculate B, the phase in $[0,2\pi)$, and visibility.

Question 5 - Solution

Four phase-stepped measurements reconstruct the interference phase

(a) Method and calculation

$$\boxed{A=(I_0+I_{\pi})/2=10};\quad\boxed{B\cos\delta=(I_0-I_{\pi})/2=4}.$$$$\boxed{B\sin\delta=(I_{3\pi/2}-I_{\pi/2})/2=3}.$$

(b) Interpretation and further calculation

$$\boxed{B=\sqrt{4^2+3^2}=5,\quad\delta=\operatorname{atan2}(3,4)=0.64350\,\mathrm{rad}}.$$$$\boxed{V=B/A=0.50,\quad I_{\max}=15,\quad I_{\min}=5}.$$

Physical check

Both sine and cosine are recovered, resolving the sign ambiguity of a single intensity measurement.