Consider a light wave that has \(X\) - and \(Y\)-polarization components of its electric field at point \(P\) given by

\[ \begin{aligned} & \mathbf{u}_{X}(t)=\exp \left[-j 2 \pi\left(\bar{v}-\frac{\Delta v}{2}\right) t\right] \\ & \mathbf{u}_{Y}(t)=\exp \left[-j 2 \pi\left(\bar{v}+\frac{\Delta v}{2}\right) t\right] \end{aligned} \]

(a) Show that at time \(t\), the electric vector makes an angle

\[ \theta(t)=\tan ^{-1}\left\{\frac{\cos \left[2 \pi\left(\bar{v}+\frac{\Delta v}{2}\right) t\right]}{\cos \left[2 \pi\left(\bar{v}-\frac{\Delta v}{2}\right) t\right]}\right\} \]

with respect to the \(X\)-axis, and thus the polarization direction is entirely deterministic.

(b) Show that such light has a coherency matrix that is identical with that of natural light, for which the polarization direction is entirely random.