Light from a partially polarized pseudothermal source is found to have a coherence matrix of the form

\[ \underline{\mathbf{J}}=\bar{I}\left[\begin{array}{ll} 1 / 2 & -1 / 6 \\ -1 / 6 & 1 / 2 \end{array}\right] \]

This light falls on a photoevent counting photodetector, for which the counting time is short compared with the coherence time ( \(T \ll \tau_{c}\) ).

(a) What is the probability density function of the total classical intensity \(I\) that falls on the photodetector?

(b) Using only Mandel's formula and the result of (a), find the probability distribution \(P(K)\) of the number of photoevents counted in time \(T\). Express the results in terms of \(K\) and \(\bar{K}\).