= Optical coherence time
{title2=$\tau_c$}
= Coherence time
{synonym}
An <optical coherence time> summarizes the delay scale of <temporal optical coherence>, but requires a convention. Two useful choices are the first positive delay $\tau_{1/e}$ where $|g^{(1)}|=e^{-1}$, for a monotonically decaying envelope, and the integral definition
$$
\tau_{\mathrm{int}}=\int_{-\infty}^{\infty}|g^{(1)}(\tau)|^2\,d\tau.
$$
The latter counts correlation over both signs of delay and may diverge for ideal discrete lines. https://www.rp-photonics.com/coherence_time.html[Paschotta, Coherence Time: exponential time constant and integral definition]{ref}.
For the stipulated Lorentzian and Gaussian profiles in <temporal optical coherence>, direct substitution and a <Gaussian integral> give
$$
\begin{array}{c|cc}
\text{profile}&\tau_{1/e}&\tau_{\mathrm{int}}\\
\hline
\text{Lorentzian}&1/(\pi\Delta\nu)&1/(\pi\Delta\nu)\\
\text{Gaussian}&2\sqrt{\ln2}/(\pi\Delta\nu)&\sqrt{2\ln2/\pi}/\Delta\nu
\end{array}
$$
The respective dimensionless coefficients are $0.318310$ for both Lorentzian conventions, and $0.530021$ and $0.664282$ for the Gaussian conventions. The familiar scale $1/\Delta\nu$ is consequently an estimate, not a convention-independent equality. A source with narrow discrete lines can have repeated revivals instead of one monotonically decaying envelope, so recording the full $|g^{(1)}(\tau)|$ can be more informative than one <optical coherence time>.
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