Show that under quasimonochromatic conditions, mutual intensity (mathbf{J}left(P_{1}, P_{2} ight)) obeys a pair of Helmholtz equations [

Show that under quasimonochromatic conditions, mutual intensity \(\mathbf{J}\left(P_{1}, P_{2}\right)\) obeys a pair of Helmholtz equations

\[ \begin{aligned} & abla_{1}^{2} \mathbf{J}\left(P_{1}, P_{2}\right)+\bar{k}^{2} \mathbf{J}\left(P_{1}, P_{2}\right)=0 \\ & abla_{2}^{2} \mathbf{J}\left(P_{1}, P_{2}\right)+\bar{k}^{2} \mathbf{J}\left(P_{1}, P_{2}\right)=0 \end{aligned} \]

where \(\bar{k}=2 \pi / \bar{\lambda}\) and

\[ abla_{1}^{2}=\frac{\partial^{2}}{\partial x_{1}^{2}}+\frac{\partial^{2}}{\partial y_{1}^{2}}+\frac{\partial^{2}}{\partial z_{1}^{2}}, \quad abla_{2}^{2}=\frac{\partial^{2}}{\partial x_{2}^{2}}+\frac{\partial^{2}}{\partial y_{2}^{2}}+\frac{\partial^{2}}{\partial z_{2}^{2}} \]