Coherence — Question 1

Question 1

A stated coherence-time convention sets a path-delay budget

For a source of spectral width $\Delta\nu=2.0\,\mathrm{GHz}$, define the estimated coherence time by $\tau_c=1/\Delta\nu$ and the vacuum coherence length by $\ell_c=c\tau_c$. Use $c=3.00\times10^8\,\mathrm{m/s}$. These are scale estimates, not a specified visibility contour.

1. Calculate both coherence scales and compare a $6.0\,\mathrm{cm}$ optical path difference with them.

2. Find the largest bandwidth satisfying a design requirement that this path delay be no greater than $\tau_c/4$.

Question 1 - Solution

A stated coherence-time convention sets a path-delay budget

(a) Method and calculation

$$\boxed{\tau_c=0.500\,\mathrm{ns},\quad\ell_c=0.150\,\mathrm m}.$$$$\tau=D/c=\boxed{0.200\,\mathrm{ns}=0.40\tau_c}.$$

(b) Interpretation and further calculation

$$\tau\leq\tau_c/4=1/(4\Delta\nu)\Longrightarrow\Delta\nu\leq1/(4\tau).$$$$\boxed{\Delta\nu_{\max}=1.25\,\mathrm{GHz}}.$$

Physical check

A bandwidth estimate alone does not fix an exact visibility: the spectral shape and the definition of coherence time also matter.