Coherence 2 - Time coherence

Time coherence is directly linked to the spectral width. Indeed, a perfectly monochromatic signal is coherent at all times: a maximum now ensures a maximum \(N\) periods later. Not so for a polychromatic source: two oscillations of slightly different frequency are in phase at \(t=0\) and become out of phase \(N\) periods later, as shown below.

Image from [BW].

This defines the coherence time (and the longitudinal coherence length \(\xi _l\)), via:
\begin{equation}
\label{shift}
\xi _l = N \lambda = (N - \frac{1}{2})(\lambda + \Delta \lambda) \Rightarrow N \simeq \frac{\lambda}{2 \Delta \lambda}\Rightarrow \xi _l \simeq \frac{\lambda^2}{2 \Delta \lambda}
\end{equation}

In conclusion, the larger the spectral width \(\Delta \lambda\), the shorter the longitudinal correlation length \(\xi _l\). Once again, the descriptions in the time and frequency domains are complementary.

In the next installment, we will look at time coherence in more detail, using the Michelson interferometer.