## 15 October 2013

### 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:

\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}

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.