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.