## 3 Unit Cell Analysis  
The MTM unit cell has been simulated using a computer simulation technology (CST) microwave studio based on the finite integration technique (FIT). The simulation setup of the proposed MTM unit cell is illustrated in Figure 1b. The unit cell is placed between two waveguide ports along the negative and positive \(z\)-axis. The perfect electric conductor boundary was placed perpendicular to the \(x\)-axis, whereas the ideal magnetic conductor is perpendicular to the \(y\)-axis. A tetrahedral mesh of frequency domain solver with an adaptive mesh scheme was used for both the array configuration and unit cell investigation. A total of 50 ohms normalized impedance was set with a 2.5-14 GHz operating frequency.  
The S-parameters (reflection co-efficient \(S_{11}\) and transmission co-efficient \(S_{21}\)) for both the elementary item and array structures were studied to understand the electromagnetic characteristics of the proposed MTM unit cell. The electromagnetic properties of MTM depend on the dimensions of the unit cell, and the working wavelength in the media should be larger than that of the unit cell wavelength [16]. The simulated S-parameters of the unit cell are exported to Math CAD. The \(S_{11}\) and \(S_{21}\) of the proposed unit cell is illustrated in Figure 2. The proposed unit cell is polarization independent and transparent to the transmission peak at 8.4 GHz with a half-power (3-dB) bandwidth of 1.3 GHz denoting a left-handed band. In the transmission co-efficient (\(S_{21}\)), the resonant is at 12.3 GHz. The major findings of the proposed unit cell is the enhancement of the metamaterial magnetic response, from the larger overall current, and self-resonance and overlap responses with respect to conventional SRRs designs. The reflection co-efficient (\(\Gamma\)) is calculated as:  
\[\Gamma=\frac{(z_{o}-1)}{(z_{o}+1)} \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Parameters** & **Dimension (mm)** & **Design Parameters** & **Dimension (mm)** \\ \hline
1 & 7 & t & 0.50 \\ w & 7 & w\({}_{1}\) & 6 \\ d & 0.25 & w\({}_{2}\) & 3 \\ g & 0.50 & w\({}_{3}\) & 2 \\ g\({}_{1}\) & 0.25 & w\({}_{4}\) & 1 \\ g\({}_{2}\) & 0.25 & t\({}_{1}\) & 0.25 \\ g\({}_{3}\) & 0.125 & p\({}_{1}\) & 4 \\ P & 5 & d\({}_{1}\) & 0.25 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Design specifications of the metamaterial unit cell.  
Figure 1: (**a**) Proposed metamaterials (MTM) unit cell; (**b**) Simulation setup of proposed unit cell.  
where \(z_{o}\) is the relative impedance determined as the square root of the ratio of effective permittivity and permeability:  
\[z_{o}=\sqrt{\frac{\mu_{r}}{\varepsilon_{r}}} \tag{2}\]  
The reflection co-efficient (\(S_{11}\)) and transmission co-efficient (\(S_{21}\)) can be determined as:  
\[S_{11}=\frac{(1-\Gamma^{2})z}{1-\Gamma^{2}z^{2}} \tag{3}\]  
\[S_{21}=\frac{(1-z^{2})\Gamma}{1-\Gamma^{2}z^{2}} \tag{4}\]  
By using Nicolson-Ross-Weir approach [17], the effective parameters such as the relative permittivity (\(\varepsilon_{r}\)), the permeability (\(\mu_{r}\)) and refractive index (\(\eta_{r}\)) can be retrieve using \(S_{11}\) and \(S_{21}\). The calculation is as follows:  
\[\text{Permittivity},\ \varepsilon_{r}=\frac{2}{jk_{o}d}*\frac{1-V_{1}}{1+V_{1}}= \frac{c}{j\pi fd}*\frac{1-V_{1}}{1+V_{1}} \tag{5}\]  
\[\text{Permeability},\ \mu_{r}=\frac{2}{jk_{o}d}*\frac{1-V_{2}}{1+V_{2}}=\frac{c}{j \pi fd}*\frac{1-V_{2}}{1+V_{2}} \tag{6}\]  
where, \(k_{o}=\frac{\omega}{c}=\frac{2\pi f}{c}\), d is slab thickness and \(c\) is the speed of light.  
\[V_{1}=S_{21}+S_{11} \tag{7}\]  
\[V_{2}=S_{21}-S_{11} \tag{8}\]  
The refractive index (\(\eta_{r}\)) can be calculated directly from the permittivity (\(\varepsilon_{r}\)), the permeability (\(\mu_{r}\)) using the transmission and reflection co-efficient as:  
\[\eta_{r}=\sqrt{\varepsilon_{r}\mu_{r}}=\frac{c}{j\pi fd}*\sqrt{\left\{\frac{ \left(S_{21}-1\right)^{2}-S_{11}^{2}}{\left(S_{21}+1\right)^{2}-S_{11}^{2}} \right\}} \tag{9}\]