## 2 Derivation of the Coherence Relationship
With the aim of relating the mutual coherence \([\hat{\mathbf{H}}\hat{\mathbf{H}}^{\ast}]_{ij}\) to the equivalent radiation sources, we substitute Equation (14) into (15), which gives:  
\[[\hat{\mathbf{H}}\hat{\mathbf{H}}^{\ast}]_{ij}=\frac{1}{Z_{0}^{2}}\langle k_{i} \overrightarrow{\mathbf{E}}_{i}^{\text{Tx}}\left(\overrightarrow{\mathbf{r}} \right)\cdot\overrightarrow{\mathbf{E}}_{i}^{\text{Rx}}\left(\overrightarrow{\mathbf{r }}  
where'm' in the superscript denotes magnetic currents.  
In most microwave imaging scenarios, the AOV is a small angle, which leads to \(\overset{\leftrightarrow}{\mathbf{I}}-\hat{\boldsymbol{k}}\hat{\boldsymbol{k}} \approx\overset{\leftrightarrow}{\mathbf{I}}\) and \(r\approx z\). Applying these approximations to Equation (17), it is convenient to transform the independent variable of the right-hand-side (RHS) from \(\overset{\rightarrow}{\boldsymbol{r}}\) to \(\overset{\rightarrow}{\boldsymbol{k}}_{xy}^{*}\). Omitting constant coefficients, the transformed equation with the variable \(\overset{\rightarrow}{\boldsymbol{k}}_{xy}^{*}\) can therefore be formulated within the low-pass area (LPF):  
\[\left[\hat{\boldsymbol{\Pi}}\hat{\boldsymbol{\Pi}}^{*}\right]_{ij}\propto \langle k_{i}^{3}\mathrm{e}^{\mathrm{i}\phi\overset{\leftrightarrow}{ \boldsymbol{\Pi}}_{j}}\overset{\rightarrow}{\boldsymbol{\Pi}}_{i}\cdot \overset{\rightarrow}{\boldsymbol{\Pi}}_{i}^{\mathrm{Tx}},k_{j}^{3}\mathrm{e}^ {\mathrm{i}\phi\overset{\leftrightarrow}{\boldsymbol{\Pi}}_{j}}\overset{ \rightarrow}{\boldsymbol{\Pi}}_{j}\cdot\overset{\rightarrow}{\boldsymbol{ \Pi}}_{j}\rangle_{\mathrm{LPF}} \tag{18}\] \[\left[\hat{\boldsymbol{\Pi}}\hat{\boldsymbol{\Pi}}^{*}\right]_{ij} \propto\langle k_{i}^{3}\mathrm{e}^{\mathrm{i}\phi\overset{\leftrightarrow }{\boldsymbol{\Pi}}_{j}}\overset{\rightarrow}{\boldsymbol{\Pi}}_{i}^{\mathrm{ mx}}\cdot\overset{\rightarrow}{\boldsymbol{\Pi}}_{i}^{\mathrm{ mx},\mathrm{Rx}},k_{j}^{3}\mathrm{e}^{\mathrm{i}\phi\overset{\leftrightarrow}{ \boldsymbol{\Pi}}_{j}}\overset{\rightarrow}{\boldsymbol{\Pi}}_{j}\cdot \overset{\rightarrow}{\boldsymbol{\Pi}}_{j}\rangle_{\mathrm{LPF}}\]  
where the phase term in Equation (17) is denoted by \(\phi\) for convenience in expression:  
\[\phi=2k_{0}r=2 zk_{0}^{2}/k_{z}^{*} \tag{19}\]  
Furthermore, if the frequencies of the measurements are bounded in a narrow band, which makes \(\mathrm{e}^{\mathrm{i}\phi}\) negligible, an approximate form of Equation (18) can be yielded:  
\[\left[\hat{\boldsymbol{\Pi}}\hat{\boldsymbol{\Pi}}^{*}\right]_{ij}\propto \langle\overset{\rightarrow}{\boldsymbol{\Pi}}_{i}^{\mathrm{Tx}}\cdot \overset{\rightarrow}{\boldsymbol{\Pi}}_{i}^{\mathrm{Rx}}\cdot\overset{ \rightarrow}{\boldsymbol{\Pi}}_{j}^{\mathrm{Rx}}\cdot\overset{\rightarrow}{ \boldsymbol{\Pi}}_{j}^{\mathrm{Rx}}\rangle_{\mathrm{LPF}} \tag{20}\] \[\left[\hat{\boldsymbol{\Pi}}\hat{\boldsymbol{\Pi}}^{*}\right]_{ij} \propto\langle\overset{\rightarrow}{\boldsymbol{\Pi}}_{i}^{\mathrm{ mx},\mathrm{Tx}}\cdot\overset{\rightarrow}{\boldsymbol{\Pi}}_{i}^{\mathrm{ mx},\mathrm{Rx}},\overset{\rightarrow}{\boldsymbol{\Pi}}_{j}^{\mathrm{m}, \mathrm{Rx}}\cdot\overset{\rightarrow}{\boldsymbol{\Pi}}_{j}^{\mathrm{m}, \mathrm{Rx}}\rangle_{\mathrm{LPF}}\]  
For convenience in expression, we discretize the RHS of Equation (20) and rewrite it as:  
\[\left[\hat{\boldsymbol{\Pi}}\hat{\boldsymbol{\Pi}}^{*}\right]_{ij}\propto \left[\hat{\boldsymbol{\mathrm{S}}}\hat{\boldsymbol{\mathrm{S}}}^{*}\right]_{ij} \tag{21}\]  
where the source matrix \(\boldsymbol{\mathrm{S}}\) is composed of \(\langle\overset{\rightarrow}{\boldsymbol{\Pi}}^{\mathrm{Tx}}\cdot \overset{\rightarrow}{\boldsymbol{\Pi}}_{j}^{\mathrm{Rx}}\rangle_{\mathrm{LPF}}\) or \(\langle\overset{\rightarrow}{\boldsymbol{\Pi}}^{\mathrm{m},\mathrm{Tx}}\cdot \overset{\rightarrow}{\boldsymbol{\Pi}}^{\mathrm{m},\mathrm{Rx}}\rangle_{ \mathrm{LPF}}\) in discrete form:  
\[\left[\boldsymbol{\mathrm{S}}\right]_{ij}=\overset{\rightarrow}{ \boldsymbol{\Pi}}_{i}^{\mathrm{Tx}}\left(\overset{\rightarrow}{\boldsymbol{ \Pi}}_{j}\cdot\overset{\rightarrow}{\boldsymbol{\Pi}}_{i}^{\mathrm{Rx}} \right.\overset{\rightarrow}{\boldsymbol{\Pi}}_{j}^{\mathrm{Rx}}\left( \overset{\rightarrow}{\boldsymbol{\Pi}}_{j}\right) \tag{22}\] \[\left[\boldsymbol{\mathrm{S}}\right]_{ij}=\overset{\rightarrow}{ \boldsymbol{\Pi}}_{i}^{\mathrm{m},\mathrm{Tx}}\left(\overset{\rightarrow}{ \boldsymbol{\Pi}}_{j}\cdot\overset{\rightarrow}{\boldsymbol{\Pi}}_{i}^{ \mathrm{m},\mathrm{Rx}}\right.\overset{\rightarrow}{\boldsymbol{\Pi}}_{j}^{ \mathrm{m},\mathrm{Rx}}\left(\overset{\rightarrow}{\boldsymbol{\Pi}}_{j}\right)\] \[\overset{\rightarrow}{\boldsymbol{\overset{\rightarrow}{ \boldsymbol{\Pi}}}_{j}}\in\mathrm{LPF}\]  
Thus far, the mutual coherences in the measurement matrix are explicitly related to the spectral mutual coherences of the equivalent radiation sources:  
\[\hat{\boldsymbol{\Pi}}\hat{\boldsymbol{\Pi}}^{*}\propto\hat{\boldsymbol{ \mathrm{S}}}\hat{\boldsymbol{\mathrm{S}}}^{*} \tag{23}\]  
while in this relationship the imaging scenario functions equivalently as a low-pass filter in the k-space.