## 2 Materials and Methods
There is a slight difference between the length of the two opposite arms to make their resonance frequencies different from each other. In addition, when the same dimensions of both antennas were selected, both antennas can radiate at 4.3 GHz. This situation was also simulated, and anticipated results were obtained. The reflective surface consisting of 24 unit elements was designed according to the requirements of the phase-frequency relationship between the surface and the antennas at the upper layer (the top antenna). Its substrate material is FR4 with a dielectric constant of \(\epsilon_{r}\) = 4.3 with a thickness of 1.6 mm. It is 109.74 mm wide and 164.62 mm long, as seen in Figure 4. The dimensions are given in Table 2. At the very beginning of the reflective surface design, the basic square-unit cell dimensions were obtained by using the EBG fundamentals, whose equivalent circuit diagram can be seen in Figure 5. \begin{table}
\begin{tabular}{c c c c c c} \hline
**a** & **b** & **c** & **d** & **e** & **f** \\ \hline
109.74 & 146.5 & 15.47 & 15.14 & 11.57 & 11.57 \\ \hline
**g** & **h** & **i** & **j** & **k** & **1** \\ \hline
1.13 & 0.8 & 18.03 & 2.37 & 29.94 & 1.69 \\ \hline
**m** & **n** & **o** & **p** & **q** & \\ \hline
2.56 & 27.1 & 3.16 & 1.72 & 8.49 & \\ \hline \end{tabular}
\end{table} 
Table 1: Dimensions of the antenna (mm). Figure 3: Dimensions of antenna front face with a close-up view. The proposed unit cell was designed with the changes made in the continuation of the study, resulting the response given in Figure 5b,c [35; 36]. \[f_{\mathit{ebg}}=\frac{1}{2\cdot\pi\sqrt{L_{\mathit{ebg}}\cdot C_{\mathit{ebg}}}} \tag{1}\]  
Here, \(C_{\mathit{ebg}}\) is the capacitance, \(L_{\mathit{ebg}}\) is the inductance, and \(f_{\mathit{ebg}}\) is the resonance frequency. The EBG width, dielectric height, and the distance between two unit cells are denoted by \(w_{\mathit{ebg}}\), \(h_{\mathit{ebg}}\), and \(g_{\mathit{ebg}}\), respectively. The approximate equivalent circuit of the proposed surface is given when looking at the cross-sectional area along with its tangential surface wave transmission behavior in Figure 5, which approves the simulation results of the surface. The capacitance \(C\) occurs in the gaps formed by the absence of vias in the unit cells. Capacitance (\(C_{\mathit{inp}}\)) is created by cuts on the surface that increase the total capacitance,  
Figure 4: Dimensions of reflective surface (**a**) and its unit element (**b**). Figure 5: Equivalent circuit diagram of EBG unit cell (**a**) [18], the approximate equivalent circuit diagram of the proposed surface (**b**), tangential surface wave transmission behavior derived from the equivalent circuit of the proposed surface (**c**). causing a drop in frequency. \(C_{s}\) symbolizes the capacity caused by the separation of unit cells. The engraved parts from the surface edges reduce the inductance and hence increase the resonance frequency. In the figure, \(L_{up}\) represents the inductance of the surface. While applying these instruments, it was considered that the total unit cell size remains almost the same. While trying to reach the desired resonance frequency, if the inductance was reduced by clipping the edges of the unit cell more and more without opening the gaps on the surface, there would be a dramatic increase in the resonance frequency. With the gaps opened on the surface, smoother transitions in frequency could be obtained. For a visual explanation, the frequency responses are shown in Figure 6 for a square unit cell of the same dimensions, a unit cell with no slots on it, and the proposed unit cell. The frequencies at which the zero-degree phase response of the unit cell occurs are shown in Figure 6. While designing the reflective surface, a unit cell simulation was run by locating the source \(\lambda/2\) far from the unit cell surface (Figure 6).