Let the field emitted by a multimode laser oscillating in \(N\) equals strength and independent modes be represented by

\[ \mathbf{u}(t)=\sum_{n=1}^{N} \exp \left[-j\left(2 \pi v_{n} t-\theta_{k}(t)\right)\right] \]

where the \(\theta_{k}(t)\) are statistically independent and uniformly distributed over \((-\pi, \pi)\). Show that the ratio of the standard deviation of total intensity to the mean intensity is given by

\[ \begin{equation*} \frac{\sigma_{I}}{\bar{I}}=\sqrt{1-\frac{1}{N}} \tag{p.4-1} \end{equation*} \]