## 2 Derivation of the Coherence Relationship
\right] \tag{9}\]  
It shows that the spectrum is bounded in the k-space due to the presence of \(w\left(\overrightarrow{\mathbf{r}}_{xy}^{*}\right)\). With the purpose of simplifying the imaging scenario under consideration, the ROI is set to a square with the size of \(L\) at the imaging range of \(z\) (Figure 1). Thus combining Equation (6) with (9) gives an approximation of the cut-off spatial frequency of the spectrum:  
\[\left|k_{x,y}\right|\leq k_{0}L/2z \tag{10}\]  
Equation (10) shows that the scenario geometry functions equivalently as a low-pass filter in the k-space of which the cut-off frequency varies with the AOV of the ROI. The filter shrinks with the increasing of the imaging range. Therefore, the spectrum of the fields in the ROI are bounded in a low-pass area (denoted as LPF for convenience) in the k-space due to not only the propagational components but also the finite AOV of the imaging scenario. In the next, the coherence relationship will be discussed. Before the discussion, we briefly introduce the electromagnetic model for MCI. In this model, the echo signals are formulated by the measurement mode \(h\) sampling the contrast function \(x\) of the object:  
\[ \begin{split} g&=\int_{S}hx\mathrm{d}^{2} \overrightarrow{\mathbf{r}}\\ h&=-\frac{\mathrm{i}\mathbf{k}_{0}}{Z_{0}}\overrightarrow{ \mathbf{E}}\cdot\overrightarrow{\mathbf{E}}_{\mathrm{Rx}}\\ x&=\widetilde{\epsilon}_{\mathrm{r}}-1=\epsilon_{ \mathrm{r}}-\mathrm{j}\sigma/\omega\epsilon_{0}-1\end{split}  \tag{11}\]  
where \(g\) is the echo, \(\epsilon_{\mathrm{r}}\) and \(\sigma\) are the relative permittivity and the conductivity of the object respectively, \(\overrightarrow{\mathbf{E}}\) is the field generated by the incident field \(\overrightarrow{\mathbf{E}}_{\mathrm{Tx}}\) illuminating the object and \(\overrightarrow{E}_{\mathrm{Rx}}\) is the field radiated by the receiving antennas with normalized excitation [24]. Comparing with the model used for electromagnetic inverse scattering imaging [23]:  
\[ \begin{split}\overrightarrow{\mathbf{E}}_{\mathrm{sca}}& =\int_{S}\overleftrightarrow{\mathbf{G}}\cdot\mathrm{O}\overrightarrow{ \mathbf{E}}\mathrm{d}^{2}\overrightarrow{\mathbf{r}}\\ \mathrm{O}&=k_{0}^{2}(\widetilde{\epsilon}_{\mathrm{ r}}-1)\end{split}  \tag{12}\]  
where \(\mathrm{O}\) is the same quantity as \(x\) omitting the constant \(k_{0}^{2}\), \(\overleftrightarrow{\mathbf{G}}\) is dyadic Green's function in free space and \(\overrightarrow{\mathbf{E}}_{\mathrm{sca}}\) is the scattering field, it shows that Equation (12) does not consider the generation of the echo. When the generation is considered, the measurement mode \(h\), which is the dot production \(\overrightarrow{\mathbf{E}}\cdot\overrightarrow{\mathbf{E}}_{\mathrm{Rx}}\) appears in the integral as a result of electromagnetic reciprocity:  
\[g\propto\int_{S^{\prime}}\overrightarrow{\mathbf{E}}_{\mathrm{inc}}\cdot \overrightarrow{\mathbf{J}}_{\mathrm{Rx}}\mathrm{d}^{2}\overrightarrow{\mathbf{r}}^ {\prime}=\int_{S^{\prime}}\int_{S}\overleftrightarrow{\mathbf{G}}\cdot\mathrm{O} \overrightarrow{\mathbf{E}}\mathrm{d}^{2}\overrightarrow{\mathbf{r}}^{\prime}\cdot \overrightarrow{\mathbf{J}}_{\mathrm{Rx}}\mathrm{d}^{2}\overrightarrow{\mathbf{r}}^{ \prime}\propto\int_{S}x\left(\overrightarrow{\mathbf{E}}\cdot\overrightarrow{ \mathbf{E}}_{\mathrm{Rx}}\right)\mathrm{d}^{2}\overrightarrow{\mathbf{r}} \tag{13}\]  
where \(\overrightarrow{\mathbf{E}}_{\mathrm{inc}}\) is \(\overrightarrow{\mathbf{E}}_{\mathrm{sca}}\) at the receiving antenna region \(S^{\prime}\) and \(\overrightarrow{\mathbf{J}}_{\mathrm{Rx}}\) is the virtual electric current density that generates \(\overrightarrow{\mathbf{E}}_{\mathrm{Rx}}\). Essentially, Equation (11) is obtained by the combination of Equation (12) and the reciprocity of the receiving antennas, while the scattering of the receiving antennas is omitted. A detailed derivation of Equation (11) can be found in Reference [5; 24].