Coherence — Question 4

Question 4

A Gaussian spectrum predicts a Gaussian visibility envelope

For a stationary source, the normalized temporal coherence is $g(\tau)=\int S(\nu)e^{i2\pi\nu\tau}d\nu/\int S(\nu)d\nu$. Its spectrum is Gaussian about $\nu_0$ with standard deviation $\sigma_\nu$. Use the Gaussian-transform identity to obtain $g(\tau)=e^{i2\pi\nu_0\tau}e^{-2\pi^2\sigma_\nu^2\tau^2}$. Equal beams have $V=|g|$.

1. Derive the positive delay at which visibility falls to one half.

2. For $\sigma_\nu=1.0\,\mathrm{GHz}$, calculate that delay and the corresponding vacuum optical path difference with $c=3.00\times10^8\,\mathrm{m/s}$.

Question 4 - Solution

A Gaussian spectrum predicts a Gaussian visibility envelope

(a) Method and calculation

$$V(\tau)=e^{-2\pi^2\sigma_\nu^2\tau^2},\quad V(\tau_{1/2})=1/2.$$$$\boxed{\tau_{1/2}=\frac{\sqrt{2\ln2}}{2\pi\sigma_\nu}}.$$

(b) Interpretation and further calculation

$$\boxed{\tau_{1/2}=0.18739\,\mathrm{ns}}.$$$$\boxed{D_{1/2}=c\tau_{1/2}=5.6217\,\mathrm{cm}}.$$

Physical check

The quoted bandwidth is a standard deviation, not a full width at half maximum; mixing those conventions would change the numerical coherence scale.