What does the coherence time signifies?

Question

I was reading a book named "Laser Physics And Applications by LV Tarasov". In page 19, It tells that

The degree of non-monochromaticity and the time interval $\tau$ consistent with the length of the wave trains of this wave are related as $$\epsilon = 1/\tau v_{0} $$ and we know that $$ \epsilon = \Delta v / v_{0}.$$ So we have $$ \Delta v = 1/\tau. $$ The quantity $\tau$ is referred to as coherence time. The longer the coherence time the higher the coherence of beam of light.

Where $v_{0}$ is the central frequency of the light (laser), $\epsilon$ is the degree of non-monochromaticity and $\Delta v $ is the bandwidth.

But What does the coherence time signifies here?

Answer 1

Coherence time ($\tau$) is a measure of the correlation between phases of a wave at two different instances of time. It essentially tells you if a traveling beam remains 'in-phase' with itself after a certain time. Similarly, you can also define a coherence length ($L=c\tau$) which is the distance covered by the beam until it can be considered to be in-phase with itself. Lasers tend to have a high degree of coherence with single-mode He-Ne lasers having coherence lengths of about ~100 meters!

The origin of such coherence measures is simply due to uncertainty. A monochromatic laser is also not perfectly monochromatic and has a distribution of frequencies with some width $\Delta\nu$ which gives rise to a coherence time $\Delta\tau=1/\Delta\nu$. A more sophisticated evaluation of the coherence time can be carried out by calculating the temporal coherence function:

\begin{equation} g(\tau)=\frac{\langle E^{*}(t) E(t+\tau)\rangle}{\langle E^{*}(t) E(t)\rangle} \end{equation}

where E(t) is your oscillating wave function. This shall be a decaying function of the time delay $\tau$. Hence, $\tau$ for which $g(\tau)$ decays to $e^{-1}$ ($\approx37\%$) of its initial value gives you your coherence time.