## 2 Structure Topology
### Feed Network Topology  
A square topology is used for the EM power distribution network based on four SIW cavities. SIW cavities are formed by inductive windows. This feeding network is designed by considering several primary dimensions as follows. To control the leakage of the EM wave, the diameters of cylindrical metalized holes (vias) are selected by these conditions: \(\mathrm{S/d}\leq 2\), and \(\mathrm{d}/\lambda_{0}\leq 0.1\), where \(\lambda_{0}\) is the free-space wavelength at the center frequency. The resonance frequency is determined by the cavities' dimensions.  
The resonance frequency is determined by the cavity dimensions. The following relationships could be used to calculate the SIW cavity's initial dimensions: [39]  
\[f_{mnp}\frac{C}{2\sqrt{\varepsilon_{r}}}\sqrt{{(\frac{m}{Wcavity_{eff}})}^{2} +{(\frac{n}{Lcavity_{eff}})}^{2}+{(\frac{p}{h})}^{2}} \tag{1}\]  
\[Wcavity_{eff}=W_{cavity}-1.08\frac{{d_{1}}^{2}}{P_{1}}+0.1\frac{{d_{1}}^{2}}{ W_{Cavity}} \tag{2}\]  
\[Lcavity_{eff}=L_{cavity}-1.08\frac{{d_{1}}^{2}}{P_{1}}+0.1\frac{{d_{1}}^{2}}{ L_{cavity}} \tag{3}\]  
The resonant frequency of the proposed cavities is determined by \(f_{mnp}\). The designed cavities operate when \(m=n=1\) and \(p=0\).  
Cavity dimensions are represented by the parameters \(W_{cavity}\), \(L_{cavity}\) and h in the mentioned formula. Moreover, in free space, C is the speed of light, and \(\varepsilon_{r}\) (=3.55) is the dielectric relative permittivity. The resonant mode for proposed cavities is \(\mathrm{TE}_{110}\).  
To minimize the leakage in the SIW layers, vias are used with diameter \(\mathrm{d}=0.8\) mm, and the center-to-center spacing between adjacent vias is \(\mathrm{S}=1.6\) mm.  
After primary calculations, CST Microwave Studio was used to optimize the geometries of the proposed cavities in Figure 1 to obtain the desired values and best performance.  
Figure 1: Configuration of the proposed \(2\times 2\) patch array. (**a**) Exploded view, (**b**) top view, (**c**), back view, and top views of the (**d**) top layer, (**e**) middle layer, and (**f**) bottom layer.  
As shown in Figure 2, cavities are named 1, 2, 3, and 4. The specified area A contains cavities 1 and 2, while area B contains cavities 3 and 4. The power distribution in the feeding network is as follows. First, EM power flows to the SIW channel through an SMA connector 50 \(\Omega\). Then, it enters the cavities through an embedded slot. Cavities 1 and 2 are excited in phase (also cavities 3 and 4 are excited in phase), but there is a 180-degree phase difference between cavities in specified areas A and B.  
Figure 2: Simulated electric field distributions of the proposed array. (**a**) Top layer, (**b**) middle layer, and (**c**) bottom layer.  
Figure 2 shows the distribution of EM fields in the proposed structure considering the phase situation. The specified areas A and B have co-polarization radiation without phase difference, while the cross-polarization radiations have a 180-degree phase difference. Arrangement of the patches and the way the distribution of the EM fields in the feeding network contributes to phase cancelation leads to suppressing cross-polarization fields.