By finding the trace of the appropriate coherency matrix, show that the average intensity transmitted by a quarter-wave plate followed by a polarization analyzer set at \(+45^{\circ}\) to the \(X\)-axis can be expressed as

\[ \bar{I}=\frac{1}{2}\left[\mathbf{J}_{x x}+\mathbf{J}_{y y}\right]+\operatorname{Im}\left\{\mathbf{J}_{x y}\right\} \]

where \(\mathbf{J}_{x x}, \mathbf{J}_{y y}\), and \(\mathbf{J}_{x y}\) are again elements of the coherency matrix of the incident light, and it has been assumed that the quarter-wave plate delays \(\mathbf{u}_{Y}\) with respect to \(\mathbf{u}_{X}\) by \(90^{\circ}\).