Coherence — Question 9

Question 9

A phase-diffusion model gives an exponential coherence envelope

Assume the phase difference accumulated over delay $\tau$ is Gaussian with zero mean and variance $2D|\tau|$, where $D=2.0\times10^8\,\mathrm{s^{-1}}$. Use $\langle e^{iX}\rangle=e^{-\operatorname{Var}(X)/2}$ for zero-mean Gaussian X. Equal-intensity beams have visibility equal to the modulus of this average.

1. Derive the visibility and find the delay at which it equals $e^{-1}$.

2. Calculate the maximum vacuum optical path difference allowed for $V\geq0.80$, using $c=3.00\times10^8\,\mathrm{m/s}$.

Question 9 - Solution

A phase-diffusion model gives an exponential coherence envelope

(a) Method and calculation

$$V(\tau)=\exp[-(2D|\tau|)/2]=\boxed{e^{-D|\tau|}}.$$$$D\tau_{1/e}=1\Longrightarrow\boxed{\tau_{1/e}=5.00\,\mathrm{ns}}.$$

(b) Interpretation and further calculation

$$e^{-D|\tau|}\geq0.80\Longrightarrow|\tau|\leq-\ln(0.80)/D.$$$$\boxed{|\tau|_{\max}=1.11572\,\mathrm{ns},\quad\Delta\mathcal L_{\max}=0.334715\,\mathrm m}.$$

Physical check

The visibility decay is exponential in delay even though each phase increment has a Gaussian distribution, because its variance grows linearly with delay.