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:
\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.

No comments:

Post a Comment