## 2 Derivation of the Coherence Relationship  
Consider a typical microwave imaging scenario as shown in Figure 1. As in most MCI applications, the AOV of the scenario is finite. In Figure 1, \(S^{\prime}\) and \(S\) denote the antenna volumes and the ROI respectively. According to electromagnetic equivalence theorem, the antennas are replaced by equivalent electric current density \(\overset{\rightarrow}{\boldsymbol{J}}\) or magnetic current density \(\overset{\rightarrow}{\boldsymbol{J}}_{\text{m}}\) (i.e., equivalent radiation sources). Utilizing 2D spectral [PERSON]'s dyad, we can formulate the electric fields in the corresponding 2D k-space [21]:  
\[ \begin{split}\overset{\rightarrow}{\boldsymbol{E}}(\overset{ \rightarrow}{\boldsymbol{k}}_{xy},z)=-\frac{k_{0}Z_{0}\phi^{kz}}{2 kz}( \overset{\leftrightarrow}{\boldsymbol{I}}-k\boldsymbol{k})\overset{ \rightarrow}{\boldsymbol{J}}\\ \overset{\rightarrow}{\boldsymbol{E}}(\overset{\rightarrow}{ \boldsymbol{k}}_{xy},z)=\frac{k_{0}\phi^{kz}}{2 kz}\boldsymbol{k}\times \overset{\rightarrow}{\boldsymbol{J}}_{\text{m}}\end{split}  \tag{1}\]  
where \(\overset{\leftrightarrow}{\boldsymbol{I}}\) is the unit dyad, \(Z_{0}\) and \(k_{0}\) are the wave impedance and the spatial frequency in free space, \(\boldsymbol{k}\) denotes the unit vector of the wave vector \(\overset{\rightarrow}{\boldsymbol{k}}\) and \(k_{z}=\sqrt{k_{0}^{2}-k_{xy}^{2}}\). Equation (1) shows that the transverse spatial frequency \(k_{xy}\) in the propagational components of the fields is bounded by \(k_{0}\) because of the presence of \(\phi^{kz}\). Figure 1: Imaging Scenario. Transforming the fields radiated by \(\widetilde{\widetilde{\widetilde{\boldsymbol{J}}}}\) in Equation (1) into spatial domain gives:  
\[\widetilde{\boldsymbol{E}}\Big{(}\overset{\rightarrow}{\boldsymbol{r}}_{xy},z \Big{)}=-\frac{k_{0}Z_{0}}{4\pi}\int_{-\infty}^{\infty}\frac{\big{(}\overset{ \leftrightarrow}{\boldsymbol{I}}-\hat{\boldsymbol{k}}\hat{\boldsymbol{k}} \big{)}\cdot\overset{\rightarrow}{\widetilde{\boldsymbol{J}}}}{k_{z}} \mathrm{e}^{\mathrm{i}\overset{\rightarrow}{\boldsymbol{k}}\cdot\overset{ \rightarrow}{\boldsymbol{r}}}\mathrm{d}^{2}\overset{\rightarrow}{\boldsymbol{k} }_{xy} \tag{2}\]  
According to Reference [22], the integral in Equation (2) can be solved by stationary phase approximation because \(\overset{\rightarrow}{\boldsymbol{k}}\cdot\overset{\rightarrow}{ \boldsymbol{r}}\) is stationary at the critical points distrubuting in the ROI. Define \(\varphi=\overset{\rightarrow}{\boldsymbol{k}}\cdot\overset{\rightarrow}{ \boldsymbol{r}}\), then it gives the gradient vector \(\
abla_{\overset{\rightarrow}{\boldsymbol{k}}_{xy}}\)\(\varphi\) and Hessian matrix \(\mathbf{Hess}\) as:  
\[ \begin{split}\
abla_{\overset{\rightarrow}{\boldsymbol{k}}_{xy}} \varphi=\left[\begin{array}{cc}x-k_{x}z/k_{z}&y-k_{y}z/k_{z}\end{array} \right]^{\mathrm{T}}\\ \mathbf{Hess}=-\frac{z}{k_{x}^{2}}\left[\begin{array}{cc}k_{x}^{2}+k_{z}^{2}& k_{x}k_{y}\\ k_{x}k_{y}&k_{y}^{2}+k_{z}^{2}\end{array}\right]\end{split}  \tag{3}\]  
Equation (3) shows that the critical points \(\overset{\rightarrow}{\boldsymbol{k}}_{xy}^{*}\) are give by \(\left[ \begin{array}{cc}k_{0}x/r&k_{0}y/r\end{array} \right]^{\mathrm{T}}\) where \(x\) and \(y\) vary in the ROI. These points correspond to the positions in the ROI where particular plane-wave components are illuminated [22]. According to Reference [23], at the critical points, Equation (2) integrates to:  
\[