## 2 Derivation of the Coherence Relationship
\begin{split}\overset{\rightarrow}{\boldsymbol{E}}(\overset{ \rightarrow}{\boldsymbol{r}}_{xy},z)&\approx-\frac{k_{0}Z_{0}}{4\pi} \Bigg{\{}\mathrm{e}^{\mathrm{i}\varphi(\overset{\rightarrow}{\boldsymbol{k} }_{xy})}|\mathrm{det}[\mathbf{Hess}(\overset{\rightarrow}{\boldsymbol{k}}_{ xy}^{*})]|^{-\frac{1}{2}}\mathrm{e}^{\mathrm{i}\overset{\rightarrow}{\boldsymbol{ \pi}}\mathrm{sign}[\mathbf{Hess}(\overset{\rightarrow}{\boldsymbol{k}}_{xy}^{*} )]}2\pi\Bigg{[}\frac{(\overset{\leftrightarrow}{\boldsymbol{1}}-\boldsymbol {k}\boldsymbol{k})\cdot\overset{\rightarrow}{\boldsymbol{\widetilde{ \boldsymbol{J}}}}}{k_{z}}\Bigg{]}_{\overset{\rightarrow}{\boldsymbol{k}}_{xy}} +o\Big{(}\frac{1}{k_{0}}\Big{)}\Bigg{\}}\\ &\approx-\frac{k_{0}Z_{0}}{2r}\mathrm{e}^{\mathrm{i}(k_{0}r-\frac{\pi}{ 4})}\bigg{[}(\overset{\leftrightarrow}{\boldsymbol{1}}-\hat{\boldsymbol{k}} \hat{\boldsymbol{k}})\cdot\overset{\rightarrow}{\boldsymbol{\widetilde{ \boldsymbol{J}}}}\bigg{]}_{\overset{\rightarrow}{\boldsymbol{k}}_{xy}}\end{split}  \tag{4}\]  
Similarly, by the same procedures, the fields radiated by \(\overset{\rightarrow}{\boldsymbol{\widetilde{\boldsymbol{J}}}}_{\mathrm{m}}\) can be obtained:  
\[\overset{\rightarrow}{\boldsymbol{E}}\Big{(}\overset{\rightarrow}{ \boldsymbol{r}}_{xy},z\Big{)}\approx\frac{k_{0}}{2r}\mathrm{e}^{\mathrm{i}(k_{ 0}r-\frac{\pi}{4})}\bigg{(}\hat{\boldsymbol{k}}\times\overset{\rightarrow}{ \boldsymbol{\widetilde{\boldsymbol{J}}}}_{\mathrm{m}}\bigg{)}_{\overset{ \rightarrow}{\boldsymbol{k}}_{xy}} \tag{5}\]  
Equations (4) and (5) formulate \(\overset{\rightarrow}{\boldsymbol{E}}\) radiated by \(\overset{\rightarrow}{\boldsymbol{\widetilde{\boldsymbol{J}}}}\) and \(\overset{\rightarrow}{\boldsymbol{\widetilde{\boldsymbol{J}}}}_{\mathrm{m}}\) respectivley in the 2D k-space. Before we proceed, it is necessary to investigate the influence of the scenario geometry on the spatial spectrum of the fields in the ROI. In order to formulate the spectrum, we define a window function \(w\Big{(}\overset{\rightarrow}{\boldsymbol{r}}_{xy}\Big{)}\) in the ROI:  
\[w\Big{(}\overset{\rightarrow}{\boldsymbol{r}}_{xy}\Big{)}=\left\{ \begin{array}[ ]{cc}1,&\overset{\rightarrow}{\boldsymbol{r}}_{xy}\in S\\ 0,&\overset{\rightarrow}{\boldsymbol{r}}_{xy}\
otin S\end{array} \right. \tag{6}\]  
Implementing inverse Fourier transformation to Equation (5) with (6) yields:  
\[\overset{\rightarrow}{\boldsymbol{\widetilde{\boldsymbol{E}}}}_{w}\bigg{(} \overset{\rightarrow}{\boldsymbol{k}}_{xy},z\Big{)}=\int_{-\infty}^{\infty} \bigg{(}w\overset{\rightarrow}{\boldsymbol{E}}\bigg{)}\mathrm{e}^{- \mathrm{i}\overset{\rightarrow}{\boldsymbol{k}}_{xy}\cdot\overset{ \rightarrow}{\boldsymbol{r}}_{xy}}\mathrm{d}^{2}\overset{\rightarrow}{ \boldsymbol{r}}_{xy} \tag{7}\]  
By the similar procedures implemented previously, applying stationary phase approximation to Equation (7) gives the amplitude of the spectrum:  
\[ \begin{split}|\overset{\rightarrow}{\boldsymbol{\widetilde{ \boldsymbol{E}}}}_{w}(\overset{\rightarrow}{\boldsymbol{k}}_{xy},z)|\approx \pi\mathbb{Z}_{0}|(\overset{\leftrightarrow}{\boldsymbol{\widetilde{ \boldsymbol{I}}}}-\hat{\boldsymbol{k}}\hat{\boldsymbol{k}})\cdot\overset{ \rightarrow}{\boldsymbol{\widetilde{\boldsymbol{J}}}}|w(\overset{ \rightarrow}{\boldsymbol{r}}_{xy}^{*})\\ |\overset{\rightarrow}{\boldsymbol{\widetilde{\boldsymbol{E}}}}_{w}( \overset{\rightarrow}{\boldsymbol{k}}_{xy},z)|\approx\pi|\hat{\boldsymbol{k}} \times\overset{\rightarrow}{\boldsymbol{\widetilde{\boldsymbol{J}}}}_{ \mathrm{m}}|w(\overset{\rightarrow}{\boldsymbol{r}}_{xy}^{*})\end{split}  \tag{8}\]where the critical points \(\overrightarrow{\mathbf{r}}_{xy}^{*}\) distruibuting in the 2D k-space are given by:  
\[\overrightarrow{\mathbf{r}}_{xy}^{*}=\left[ \begin{array}{c}k_{x}\sqrt{\frac{k_{ x}^{2}x^{2}+k_{0}^{2}2^{2}}{\left(k_{0}^{2}-k_{x}^{2}\right)\left(k_{0}^{2}-k_{y}^{2} \right)}}\\ k_{y}\sqrt{\frac{k_{x}^{2}y^{2}+k_{0}^{2}z^{2}}{\left(k_{0}^{2}-k_{y}^{2} \right)\left(k_{0}^{2}-k_{y}^{2}\right)}}\end{array}