An idealized model of the normalized power spectral density of a gas laser oscillating in \(N\) equal-intensity axial modes is

\[ \widehat{\mathcal{G}}(v)=\frac{1}{N} \sum_{n=-(N-1) / 2}^{(N-1) / 2} \delta(v-\bar{v}+n \Delta v) \]

where \(\Delta v\) is the mode spacing (equal to the speed of light/( \(2 \times\) cavity length for axial modes)), \(\bar{u}\) is the frequency of the central mode, and \(N\) has been assumed odd for simplicity.

(a) Show that the envelope of the visibility function for such light is given by

\[ \mathcal{V}(\tau)=\left|\frac{\sin (N \pi \Delta v \tau)}{N \sin (\pi \Delta v \tau)}\right| \]

(b) Plot \(\mathcal{V}(\tau)\) vs. \(\Delta v \tau\) for \(N=3\) and \(0 \leq \tau \leq 1 / \Delta v\).