The output of a single-mode, well-stabilized laser is passed through a spatially distributed phase modulator (or a phase-only spatial light modulator that is changing with time). The field observed at point \(P_{k}\) at the output of the spatial light modulator is of the form

\[ \mathbf{u}\left(P_{k}, t\right)=\sqrt{I_{k}} \exp \left[-j\left(2 \pi v_{0} t-\phi\left(P_{k}, t\right)\right)\right] \]

where \(v_{0}\) is the laser frequency, \(I_{k}\) is the intensity at \(P_{k}\), and \(\phi\left(P_{k}, t\right)\) is the phase modulation imparted to the wave at point \(P_{k}\). The phase modulation is chosen to be a zero-mean, stationary Gaussian random process. Noting that \(\Delta \phi=\phi\left(P_{1}, t\right)-\phi\left(P_{2}, t\right)\) is also a zero-mean, stationary Gaussian process, show that the second-order coherence function of the modulated wave is

\[ \boldsymbol{\Gamma}_{12}\left(t_{1}-t_{2}\right)=\sqrt{I_{1} I_{2}} e^{-j 2 \pi v_{0}\left(t_{1}-t_{2}\right)} e^{-\sigma_{\phi}^{2}\left[1-\gamma_{\phi}\left(P_{1}, P_{2} ; t_{1}-t_{2}\right)\right]} \]

where \(\sigma_{\phi}^{2}\) is the variance of \(\phi(P, t)\) (assumed independent of \(P\) ) and \(\gamma_{\phi}\) is the normalized cross-correlation function of \(\phi\left(P_{1}, t\right)\) and \(\phi\left(P_{2}, t\right)\).