Intensity in Double-Slit Interference — Question 9

Question 9

Complementary output ports make interference energy accounting explicit

An ideal lossless balanced interferometer has output powers $P_+=P_{\mathrm{in}}(1+\cos\delta)/2$ and $P_-=P_{\mathrm{in}}(1-\cos\delta)/2$. Its input power is $12\,\mathrm{mW}$. Assume no losses and identical detector calibration.

1. Calculate both powers at $\delta=\pi/3$ and verify total-power conservation.

2. Derive the normalized difference signal and find both phase possibilities in $[0,2\pi)$ if the measured normalized difference is $-1/2$.

Question 9 - Solution

Complementary output ports make interference energy accounting explicit

(a) Method and calculation

$$\boxed{P_+=9.0\,\mathrm{mW},\quad P_-=3.0\,\mathrm{mW}}.$$$$P_++P_-=P_{\mathrm{in}}\Longrightarrow\boxed{P_{\mathrm{total}}=12\,\mathrm{mW}}.$$

(b) Interpretation and further calculation

$$\boxed{D=\frac{P_+-P_-}{P_++P_-}=\cos\delta}.$$$$D=-1/2\Longrightarrow\boxed{\delta=2\pi/3,\ 4\pi/3}.$$

Physical check

Destructive interference at one port is accompanied by increased power at the other port; it does not destroy electromagnetic energy.