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

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

where \(\mathbf{J}_{x x}, \mathbf{J}_{y y}\), and \(\mathbf{J}_{x y}\) are elements of the coherency matrix of the incident light.