## 2 Antenna Design
The real part \(n^{\prime}\) branch was determined iteratively using Equation (9) ensuring continuity and satisfying \(|n^{\prime}z^{\prime\prime}|\leq n^{\prime\prime}z^{\prime}\). The final mu and epsilon were obtained as \(\mu\)= \(nz\) and \(\epsilon\)= \(n\)/z, with results validated across the frequency range. Figure 4b illustrates the magnitude of the S-parameters on a linear scale, along the left \(y\)-axis, and the phase characteristics of the proposed unit cell as obtained through waveguide analysis along the right \(y\)-axis. The reflection coefficients \(|\)S\({}_{11}\) \(|\) and \(|\)S\({}_{22}\) \(|\) demonstrate values below 0.25 approaching zero across the operational bandwidth, while the transmission coefficients \(|\)S\({}_{12}\) \(|\) and \(|\)S\({}_{21}\) \(|\) exhibit near-unity magnitude throughout the frequency range. Figure 4c presents the extracted constitutive parameters, revealing negative values for both relative permittivity and permeability across the band of interest. The designed 3 \(\times\) 3 superstrate sheet will be positioned above the radiating antennas at a distance of 0.182\(\lambda_{0}\) (15 mm), where \(\lambda_{0}\) represents the wavelength calculated at 3.65 GHz. The interelement spacing of 0.11\(\lambda_{0}\) (9 mm) was selected to achieve optimal mutual coupling reduction while maintaining constructive interference between adjacent unit elements, resulting in enhanced gain and port isolation for the MIMO antenna system. C. Antenna Array with MTM Superstrate Design  
Figure 5a shows a four-port MIMO antenna with a 3 \(\times\) 3 metamaterial superstrate designed to enhance key performance indicators. The superstrate (Figure 3b, subsection B) is positioned 0.18\(\lambda_{0}\) above the array (\(\lambda_{0}\) calculated at 3.65 GHz). Initially, a \(\lambda_{0}\)/4 distance (approximately 20.5 mm at 3.65 GHz) was selected as the starting point, as this quarter-wavelength point is known to provide optimal impedance matching where reflections from the superstrate and antenna can constructively interfere, minimizing return loss. Following this initial choice, iterative parametric distances were investigated during the design phase to refine the height. The chosen distance of 0.18\(\lambda_{0}\) was found to provide a balance between enhanced gain and improved port isolation, while maintaining acceptable impedance matching (S\({}_{11}\)\(<\)\(-\)10 dB) across the operating band. The antenna is constructed on FR-4 with a thickness of 1.6 mm, a dielectric constant of 4.3, and a loss tangent of 0.02. The height of the antenna is 1.6 mm, but the overall antenna dimensions with the MTM superstrate are 0.913\(\lambda_{0}\)\(\times\) 0.913\(\lambda_{0}\)\(\times\) 0.22\(\lambda_{0}\). The rectangular radiating elements designed in the first subsection of this section, as shown in Figure 2, are placed orthogonal to each other. The distance between each element is set to 0.2\(\lambda_{0}\). The fabricated assembly consists of the MTM positioned above the MIMO antenna, separated by M3 nylon spacers at each corner (see Figure 5b. Table 4 provides optimized values for the positioning of each radiating element, the distance between MTM unit elements, and the gap between the MIMO antenna and the MTM superstrate. Alternative methods for improving isolation in MIMO antennas, such as the use of a circular resonator (a parasitic element) in the same plane as [29], have been explored. This approach, effective at higher frequencies, enhances isolation without increasing thickness but may \begin{table}
\begin{tabular}{c c c c c c} \hline
**Parameter** & **Value (mm)** & **Parameter** & **Value (mm)** & **Parameter** & **Value (mm)** \\ \hline H & 15 & D & 18 & d\({}_{\rm x}\) & 9 \\ W & 75 & h & 1.6 & d\({}_{\rm y}\) & 9 \\ L & 75 & & & & \\ \hline \end{tabular}
\end{table} 
Table 4: Parameter values of a four-port MIMO with a 3 \(\times\) 3 superstrate. compromise other key MIMO parameters, such as gain or impedance matching.