Intensity in Double-Slit Interference — Question 2

Question 2

Measured bright and dark levels determine the two beam intensities

At one screen region, fully coherent parallel-polarized beams produce $I_{\max}=25\,\mathrm{W/m^2}$ and $I_{\min}=1\,\mathrm{W/m^2}$. Assume no detector background and label the stronger beam 1.

1. Infer the individual intensities from the measured extrema.

2. Find the relative phases in $[0,2\pi)$ at which the total intensity is $19\,\mathrm{W/m^2}$.

Question 2 - Solution

Measured bright and dark levels determine the two beam intensities

(a) Method and calculation

$$\sqrt{I_1}+\sqrt{I_2}=5,\quad\sqrt{I_1}-\sqrt{I_2}=1.$$$$\boxed{I_1=9\,\mathrm{W/m^2},\quad I_2=4\,\mathrm{W/m^2}}.$$

(b) Interpretation and further calculation

$$I=I_1+I_2+2\sqrt{I_1I_2}\cos\delta=13+12\cos\delta.$$$$I=19\Longrightarrow\cos\delta=1/2\Longrightarrow\boxed{\delta=\pi/3,\ 5\pi/3}.$$

Physical check

An intensity alone usually leaves a twofold phase ambiguity because cosine is unchanged when the phase sign reverses.