## 2 Antenna Design
The current flows perpendicular to the width of slits and length of the slot, therefore the effective length increase is approximately the cumulative current detour. The effective contribution of the slits and slots to the electrical length is a fraction of their physical length, typically adjusted empirically to match observed frequency shifts and bandwidth broadening [25]. For slits and slots, the additional length is approximated as  
\[\Delta L_{s}\approx 0.15(S_{1x}+S_{2x}+S_{3x}+S_{4x}+S_{5}) \tag{2}\]  
where:  
\(S_{1x}=S_{2x}=S_{3x}=S_{4x}=\text{Slit width}\);  
\(S_{5}=\text{Slot length}\). Therefore, \(L_{\textit{eff}}\) for Equation (1) is given by Equation (3). \[L_{\textit{eff}}=L_{p}+2\Delta L+\Delta L_{s} \tag{3}\]  
The bandwidth is inversely proportional to the quality factor [26]:  
\[B.W=\frac{\Delta f}{f_{r}}\ \approx\ \frac{1}{Q} \tag{4}\]  
Slits and slots influence \(Q\) by affecting the individual quality factors [27]. The addition of slits and slots alters the radiation pattern and increases the effective radiating area and introduces higher-order modes which reduce the radiation quality factor by enhancing radiation losses. Slits and slots modify the current distribution, potentially increasing dielectric losses in localized areas, which lowers the dielectric quality factor. Introduction of slits and slots excites surface waves, increasing surface wave losses and lowering the surface wave quality factor. The total \(Q\) factor is:  
\[\frac{1}{Q}=\frac{1}{Q_{r}}+\frac{1}{Q_{d}}+\frac{1}{Q_{c}}+\frac{1}{Q_{\textit{ sw}}} \tag{5}\]  
where \(Q_{r},\ Q_{d},Q_{c},\ \textit{and}\ Q_{\textit{sw}}\) are radiation, dielectric, conductor, and surface wave quality factors, respectively. Overall dimensions of the unit element are \(37.5\times 37.5\times 1.6\) mm\({}^{3}\). Figure 2 clarifies the design stages, final design, and the corresponding S-parameters of the unit element. Table 2 provides detailed design parameters and their corresponding values in millimeters for the proposed antenna unit element. The proposed design achieves the required impedance bandwidth of more than 300 MHz (3.5-3.8 GHz). B. Metamaterial Unit Cell and Superstrate Design \begin{table}
\begin{tabular}{c c c c c c} \hline
**Parameter** & **Value (mm)** & **Parameter** & **Value (mm)** & **Parameter** & **Value (mm)** \\ \hline W\({}_{\text{g}}\) & 37.5 & F\({}_{\text{w}}\) & 2.98 & S\({}_{\text{4}}\) & 5 \\ L\({}_{\text{g}}\) & 37.5 & I\({}_{\text{L}}\) & 3 & S\({}_{\text{5}}\) & 9.6 \\ W\({}_{\text{P}}\) & 23.6 & S\({}_{\text{1}}\) & 11.95 & X & 1 \\ L\({}_{\text{P}}\) & 17.95 & S\({}_{\text{2}}\) & 7.5 & Y & 1.5 \\ F\({}_{\text{L}}\) & 11.5 & S\({}_{\text{3}}\) & 2.5 & & \\ \hline \end{tabular}
\end{table} 
Table 2: Parameter values of the single-element antenna. Figure 2: Unit Element (**a**) Proposed Design and (**b**) Design Stages with corresponding S-parameters. To enhance MIMO antenna performance parameters a superstrate consisting of a 3 \(\times\) 3 grid of identical structures is designed on a 1.6 mm thick FR-4 substrate as shown in Figure 3b. A single element of the grid is depicted in Figure 3a, which is based on a metallic rectangular patch of dimensions 13 \(\times\) 13 mm\({}^{2}\) with an octagram star-shaped slot at its center. The octagram star-shaped slot was selected after iterative simulations comparing rectangular, circular, and star-shaped slots. The design parameters of the MTM unit cell and 3 \(\times\) 3 MTM are listed in Table 3. The octagram design maximized resonance paths, resulting in a 3.5 dB gain enhancement. Starting from a conventional rectangular slot in a rectangular patch, the design was enhanced by superimposing a 45\({}^{\circ}\) rotated version of the slot, creating the octagram configuration. The star-shaped perforations were chosen over a simple slot loop as they create multiple resonant paths due to their multiple angular branches. These paths redirect the electromagnetic fields from the underlying MIMO radiators, resulting in enhanced gain and improved radiation characteristics.