## 3 Full-Wave Validation and Discussion  
In this section, we validate the proposed coherence relationship by three numerical full-wave electromagnetic simulations and a commercial full-wave simulation (Ansoft HFSS). The imaging scenario in the simulations is set as shown in Figure 1 with \(D=L=300\) mm while the imaging range varies. During the comparisons in each simulation with varying ranges, other parameters in this simulation remain invariable. In the numerical simulations, a square which is filled by homogeneous, monochromatic (\(\lambda~{}=~{}10\) mm) and unitarily polarized magnetic currents is introduced as the equivalent radiation sources of the antenna in the region \(S^{\prime}\). For simplification of these simulations, this antenna is used for both transmitting and receiving. Because the sources in Equation (22) are formulated in the k-space, we apply spatial frequency modulations to the source in order to demonstrate the relationship or the low-pass filter or generate enough samples of the sources that are ergodic in the k-space. Different spatial frequency modulations are applied for specific purpose in respective simulation. The electric fields in the ROI are calculated via spatial Green's dyad:  
\[ \begin{split}\overset{\leftrightarrow}{\mathbf{G}}\Big{(}\overset{ \rightarrow}{\mathbf{r}}_{i},\overset{\rightarrow}{\mathbf{r}}_{j}\Big{)}=\frac{ \mathrm{g}^{\mathrm{i}_{0}R_{ij}}}{4\pi R_{ij}}\Bigg{[}&\left(1+ \frac{\mathrm{i}}{k_{0}R_{ij}}-\frac{1}{k_{0}^{2}R_{ij}^{2}}\right)\overset{ \leftrightarrow}{\mathbf{I}}+\left(-1-\frac{\mathrm{i}\mathrm{j}}{k_{0}R_{ij}} +\frac{3}{k_{0}^{2}R_{ij}^{2}}\right)\hat{\mathbf{R}}_{ij}\hat{\mathbf{R}}_{ij}\Bigg{]} \\ &\overset{\leftrightarrow}{\mathbf{G}}_{\mathbf{m}}\Big{(}\overset{ \rightarrow}{\mathbf{r}}_{i},\overset{\rightarrow}{\mathbf{r}}_{j}\Big{)}=\
abla \times\overset{\leftrightarrow}{\mathbf{G}}\Big{(}\overset{\rightarrow}{\mathbf{r}}_ {i},\overset{\rightarrow}{\mathbf{r}}_{j}^{\prime}\Big{)}\end{split}  \tag{24}\]  
The matrix \(\mathbf{H}\) is generated by the inner productions of the electric fields according to Equation (14). Firstly, we start with the validation of the low-pass filter in the k-space by the first numerical simulation. For explicit demonstration, the source is simultaneously modulated by a set of spatial frequencies. These spatial frequencies are uniformly distributed in the low-pass area in the k-space at the imaging range of 1 m (see Equation (10)). Then implementing Fourier transformation to the source and the measurement modes gives a demonstration of the low-pass filter:  
In Figure 2, the spatial spectrum of the modulated source is shown in (a), where each spectrum with respective modulation frequencies can be evidently distinguished. The spectra of the measurement modes at the imaging range of 1 m, 2 m and 4 m are shown in (b), (c) and (d) respectively. By comparison, it shows that the components with high spatial frequency are substantially attenuated with the increasing of the imaging range, which approximately conforms to Equation (10). This result validates that the imaging scenario acts equivalently as a spectral low-pass filter of which the cut-off frequency decreases with the increasing of the imaging range. In order to demonstrate the coherence properties of the measurement modes and the source, we perform the second numerical simulation in which the modulation of the source is changed. In this simulation, the source is modulated by a spatial frequency that is linearly increased from \(-30\pi\) to \(30\pi\) rad/m, while the directions of these frequencies are identical. In each measurement, only one modulation frequency is applied. According to Fourier analysis, the spectral coherence of the source can be reduced by applying the spatial modulation that shifts its spatial spectrum. If the proposed coherence relationship (i.e., Equation (20)) is valid, the mutual coherence between the measurement modes would decrease with the increasing of the distance between the spatial modulation frequencies applied to the source.