Consider a single-mode laser emitting light described by the analytic signal

\[ \mathbf{u}(t)=\exp (-j[2 \pi \bar{v} t-\theta(t)]) \]

(a) Assuming that \(\Delta \theta=\theta\left(t_{2}\right)-\theta\left(t_{1}\right)\) is an ergodic random process, show that the autocorrelation function of \(\mathbf{u}(t)\) is given by

\[ \boldsymbol{\Gamma}_{\mathbf{U}}\left(t_{2}, t_{1}\right)=e^{-j 2 \pi \bar{v} \tau} \mathbf{M}_{\Delta \theta}(1) \]

where \(\mathbf{M}_{\Delta \theta}(\omega)\) is the characteristic function of \(\Delta \theta\).

(b) Show that for a zero-mean Gaussian \(\theta(t)\) arising from a stationary instantaneous frequency process,

\[ \boldsymbol{\Gamma}_{\mathbf{U}}(\tau)=e^{-j 2 \pi \bar{v} \tau} e^{-(1 / 2) D_{\theta}(\tau)} \]

where \(\tau=t_{2}-t_{1}\) and \(D_{\theta}(\tau)\) is the structure function of the phase process \(\theta(t)\).