Coherence — Question 2

Question 2

Random phase noise reduces contrast without changing the average intensity

Two equal-intensity beams each contribute $I_0$. Their instantaneous intensity is $I=2I_0[1+\cos(\delta+\varphi)]$. The fluctuating phase has zero-mean Gaussian statistics with standard deviation $\sigma$, and $\langle e^{i\varphi}\rangle=e^{-\sigma^2/2}$. The exposure samples the distribution fully.

1. Derive the time-averaged intensity and fringe visibility $V=(I_{\max}-I_{\min})/(I_{\max}+I_{\min})$.

2. Find the largest phase standard deviation compatible with $V\geq0.90$.

Question 2 - Solution

Random phase noise reduces contrast without changing the average intensity

(a) Method and calculation

$$\langle\cos(\delta+\varphi)\rangle=\Re[e^{i\delta}\langle e^{i\varphi}\rangle]=e^{-\sigma^2/2}\cos\delta.$$$$\boxed{\langle I\rangle=2I_0[1+e^{-\sigma^2/2}\cos\delta],\quad V=e^{-\sigma^2/2}}.$$

(b) Interpretation and further calculation

$$e^{-\sigma^2/2}\geq0.90\Longrightarrow\sigma^2\leq-2\ln0.90.$$$$\boxed{\sigma_{\max}=0.45904\,\mathrm{rad}=26.30^\circ}.$$

Physical check

A uniform scan of the controlled phase still has mean intensity $2I_0$; noise reduces modulation rather than removing beam power.