## 3 Metamaterial Simulation and Methodology  
The various models were utilized to investigate different property parameters for the metamaterial. The CST Microwave Studio simulation software was used to realize the performance and compute the S-parameters of the proposed unit cell. The finite integration technique (FIT), including simulation and analysis, was used in this study following from Reference [34]. The numerical approach of the FIT is used to extract the transmission coefficient (S\({}_{21}\)) and reflection coefficient (S\({}_{11}\)), which indicate return loss and insertion loss,  
Figure 4: S21 response for different (**a**) polarization angles; (**b**) oblique incident angles. respectively. The two waveguide ports are utilized to generate electromagnetic waves and transmitted in the \(Z\)-axis direction. The boundary conditions are realized to induce the negative electric and negative magnetic responses to the \(X\) and \(Y\)-axis. The polarization happens while propagating the incoming electromagnetic wave through the E-field and H-field. The \(X\)-axis is approved as a perfect electric conductor and the \(Y\)-axis as a perfect magnetic conductor. For free space simulation, a frequency domain solver-based simulation was utilized. This metamaterial structure implies the equivalent LC circuit where the split gap acts as a capacitor and the resonator strip plays as an inductor. The inductance of the acting inductor and capacitance of the acting capacitor were modified with the width of the metallic part and the split gap. The simulation was performed from the 1 GHz to 14 GHz microwave range. The boundary conditions and the wave ports are shown in Figure 5.  
The property of effective permittivity (\(\epsilon_{\mathrm{r}}\)), permeability (\(\mu_{\mathrm{r}}\)), and refractive index (\(\eta\)) are extracted from the S-parameters using the Robust technique [35]. Throughout the whole procedure, Equations (1)-(7) were performed to calculate the parameters [36]. From Equations (1) and (2), the reflection coefficient (\(S_{11}\)) and transmission coefficient (\(S_{21}\)) can be formed. \[S_{11}=\frac{R_{01}\ (1-\mathbf{e}^{j2 nk_{0}d})}{1-R_{01}^{2}\ \mathbf{e}^{j2 nk_{0}d}} \tag{1}\]  
\[S_{21}=\frac{(1-R_{01}^{2})\mathbf{e}^{jnk_{0}d}}{1-{R_{01}}^{2}\ \mathbf{e}^{j2k_{0}d}} \tag{2}\]  
where \(R_{01}=\frac{z-1}{z+1}\). Figure 5: Simulation setup of metamaterial unit cell in computer simulation technology (CST). The metamaterial is identified as a passive medium. In such a situation, the real component and the imaginary portion of the refractive index and impedance are determined in the following manner. \[\text{z (real)}\geq 0 \tag{3}\]  
\[\eta\text{ (imaginary)}\geq 0 \tag{4}\]  
The refractive index (\(\eta\)) and the impedance (z) are derived from Equations (1) and (2),  
\[z=\pm\sqrt{\frac{\left(1+{S_{11}}\right)^{2}-{S_{21}}^{2}}{\left(1-{S_{11}} \right)^{2}-{S_{21}}^{2}}} \tag{5}\]  
\[\text{e}^{intb0d}=X\;\pm i\sqrt{1-X^{2}} \tag{6}\]  
where \(X=\frac{1}{2{S_{21}}(1-{S_{11}}^{2}+{S_{21}}^{2})}\). \(\eta\) denotes refractive index value and can be achieved from Equation (7) using Equation (6)  
\[\eta=\frac{1}{k_{0}d}\{\left[\left[\ln\left(\text{e}^{intb_{0}d}\right)\right] ^{\prime\prime}+2m\pi\right]-i\left[\ln\left(\text{e}^{intb_{0}d}\right)\right] ^{\prime}\} \tag{7}\]  
where \(m\) denotes an inextricable link to the index of \(\eta^{\prime}\) and \(k\) denotes the free space wave vector. The thickness of the proposed structure is denoted by \(d\). The imaginary component of \(\eta\) is known just once, while the real part is complicated by the subfunctions of the logarithm function. To calculate the true fraction of effective parameters, the usual extraction procedure is utilized. From the effective parameter analysis, the resonance frequencies are displayed. The real part of the effective parameters is determined using the conventional extraction procedure. The effective parameter analysis reveals the resonance frequencies.