## 2 Antenna Design
The metamaterial unit cell was placed in a rectangular waveguide environment, with Port 1 positioned at a \(\lambda\)/4 distance from the structure facing the metallic elements, and Port 2 positioned at a \(\lambda\)/4 distance on the opposite side, facing the substrate, as shown in the simulation setup of Figure 4a. The port separation distance was calculated based on the center frequency of 3.65 GHz, resulting in \(\lambda\)/4 \(\approx\) 20.5 mm. This \(\lambda\)/4 distance ensures optimal impedance matching and minimizes reflections at the port interfaces, providing accurate S-parameter measurements. The simulation employs periodic boundary conditions to replicate the behavior of an infinite array. Electric boundary conditions (Et = 0) were applied along the \(x\)-axis, creating perfect electric conductor (PEC) walls, while magnetic boundary conditions (Ht = 0) were applied along the \(y\)-axis, creating perfect magnetic conductor (PMC) walls. Open boundary conditions with additional space were implemented at \(Z_{\text{max}}\) and \(Z_{\text{min}}\) to allow electromagnetic wave propagation in the z-direction. These boundary conditions simulate the periodic nature of the metamaterial structure while enabling proper wave propagation analysis. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Value (mm)** & **Parameter** & **Value (mm)** & **Parameter** & **Value (mm)** \\ \hline W\({}_{\text{u}}\) & 22 & a & 3.5 & d\({}_{\text{y}}\) & 9 \\ L\({}_{\text{u}}\) & 22 & b & 2.25 & W & 75 \\ W\({}_{\text{MTM}}\) & 13 & c & 4.95 & L & 75 \\ L\({}_{\text{MTM}}\) & 13 & d\({}_{\text{x}}\) & 9 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter values of the metamaterial unit cell and 3 \(\times\) 3 superstrate. Figure 3: Design of metamaterial (**a**) Unit Cell and (**b**) 3 \(\times\) 3 superstrate. The effective constitutive parameters (permittivity \(\varepsilon\), permeability \(\upmu\), and refractive index n) were extracted from the S-parameters (S\({}_{11}\) and S\({}_{21}\)) using the robust method proposed by [28]. This method improves upon the Nicolson-Ross-Weir approach by optimizing the effective slab boundaries. The effective slab thickness d was derived as 1.6 mm, corresponding to the FR-4 substrate thickness. The Nicolson-Ross-Weir relative permittivity and permeability are calculated using Equations (6) and (7). \[\varepsilon_{r}=\frac{\lambda_{0}^{2}}{\mu_{r}}\bigg{(}\frac{1-\Gamma}{1+ \Gamma}\bigg{)}\bigg{(}\frac{1}{\lambda_{c}^{2}}-\bigg{(}\frac{ln(T)}{2\pi d} \bigg{)}^{2}\bigg{)} \tag{6}\]  
\[\mu_{r}=\frac{1+\Gamma}{1-\Gamma}\sqrt{\frac{\lambda_{0}^{2}}{\varepsilon_{r}} \bigg{(}\frac{1}{\lambda_{c}^{2}}-\bigg{(}\frac{ln(T)}{2\pi d}\bigg{)}^{2} \bigg{)}} \tag{7}\]  
where:  
\(\Gamma=\frac{S_{11}+S_{21}-S_{12}-S_{21}}{2}\) is the reflection coefficient;  
\(T=\frac{S_{12}+S_{21}}{2}\) is the transmission coefficient;  
\(\lambda_{0}=\frac{\zeta}{f}\) free space wavelength;  
\(\lambda_{c}=\frac{2}{\sqrt{\big{(}\frac{m}{2}\big{)}^{2}+\big{(}\frac{\zeta}{ f}\big{)}^{2}}}\) is the cut-off wavelength. Where a and b are the width and height of the waveguide. For TE10 dominant mode \(\lambda_{c}=2a\). These were refined using [28] impedance z and refractive index n calculations given by Equations (8) and (9). \[z=\pm\sqrt{\frac{\left(1+S_{11}\right)^{2}-S_{21}^{2}}{\left(1-S_{11}\right)^ {2}-S_{21}^{2}}} \tag{8}\]  
Figure 4: (**a**) Two-port waveguide simulation setup. (**b**) S-parameter linear scale magnitude (left \(y\)-axis) and phase (right \(y\)-axis). (**c**) Extracted permittivity, permeability, and refractive index. \[n=\frac{1}{k_{0}d}\Big{\{}\Big{[}Im\Big{(}ln\Big{(}e^{ink_{0}d}\Big{)}\Big{)}+2m \pi\Big{]}-iRe\Big{(}ln\Big{(}e^{ink_{0}d}\Big{)}\Big{)}\Big{\}} \tag{9}\]  
where:  
\(e^{ink_{0}d}=X\pm i\sqrt{1-X^{2}}\) and \(X=\frac{1}{2S_{21}^{2}}\big{(}1-S_{11}^{2}+S_{21}^{2}\big{)}\) with the sign of z selected to ensure \(z^{\prime}\geq 0\) and adjusted via \(e^{ink_{0}d}=\frac{S_{21}}{1-S_{11}\frac{z}{z+1}}\) for \(n^{\prime\prime}\geq 0\).