## 1 Introduction
In Reference [16], the researchers proposed a spatial spectral analysis on the coverage of the Ewald sphere and obtained a convolutional relationship between the transmitting and receiving antennas on the sphere. The analysis in spectral domain (k-space) is very illuminating because it gives an intuitive and approximate approach of evaluating the imaging performance of MCI systems. However, the authors did not formulate the coherences in the frame of the k-space analysis. Recently, some researchers achieved average mutual coherence minimization of the measurement matrix in a two-dimensional (2D) planar imaging scenario [17]. In that work, the minimization is implemented using augmented Lagrangian method, while the positions of the antennas serve as the optimization variables. Despite the impressive results, the formation of the coherences is not investigated and the imaging scenario in which the antennas fully surround the region of interest (ROI) may not be applicable for many microwave imaging applications where the angle of view (AOV) is limited. Overall, the research gap between the mutual coherences in the measurement matrix and the antennas needs to be addressed by rigorous electromagnetic analysis. However, to the best of our knowledge, relevant research is still inadequate. Apart from microwave imaging, in the fields of optics and wireless communication, the researches on electromagnetic degree of freedom (EM DOF) may shed light on the investigation of the mutual coherences. In inverse problems, the EM DOF indicates the maximum amount of information that can be retrieved from echoes in the presence of noise [18]. For an imaging system, the number of EM DOF can be associated with the number of the system's effective eigenvalues that suffice for reconstruction and thus in negative relationship with the mutual coherences of the measurements. According to Reference [19], the number of EM DOF is strongly dependent on the geometry of imaging scenario and will saturate to a upper-bound due to the presence of noise when continuously expanding the transmit or receive volumes. In Reference [20], the researchers proposed wavevector-aperture-product (WAP) which characterizes the number of EM DOF in a scattering environment, giving a clearer view of the scenario geometry's influence on the EM DOF. These works on EM DOF imply that the mutual coherences in the measurement matrix may be related to the spatial spectrum of the radiation sources in MCI applications, while the geometry of the imaging scenario constitutes a constraint condition in this relationship. Inspired by the above works, we relate the mutual coherences in the measurement matrix to the antennas by formulating the electric fields of the antennas' equivalent sources via spectral Green's dyad. In our derivation, the antennas are replaced by the equivalent sources with the purpose of simplifying the problem under consideration. These sources can be real, induced or fictitious, that is, sources that appears as a result of electromagnetic equivalence theorem. Our work shows that the mutual coherences are dependent on the spectral coherences of the sources, while the imaging scenario acts equivalently as a low-pass filter in the k-space. Growing the imaging range narrows this spectral constraint and thus leads to surges in the mutual coherences, which is in accordance with previous work in which the number of EM DOF is bounded by the WAP. Full-wave electromagnetic simulations are performed for validation, where a set of spatially modulated magnetic currents and a set of typical cELC apertures are introduced as fictitious and real radiation sources respectively. In the validation, the coherence level of a matrix is evaluated by the average mutual coherence that is calculated between rows and the one that is calculated between columns, respectively. The rest of this paper is organized as follows. In Section 2, the derivation of the coherence relationship is presented. The full-wave validation is performed in Section 3.