# Design Procedure of a Frequency Reconfigurable Metasurface Antenna at mmWave Band
## 3 Design Procedure  
According to [50], ring resonators are described as two identical half guided wavelength \(\lambda_{g}/2\), which establish standing waves in parallel. Figure 4 depicts a schematic view of the dual band square SRR structure.  
Since \(L_{T1}\) and \(L_{T2}\) in Figure 4 represents the total path transversed by EM waves in the internal and external ring respectively, and the fundamental mode length \(\lambda_{g}/2\) is represented as,  
\[L_{T1}=\frac{\lambda_{g}}{2},\ \ \ L_{T2}=\frac{\lambda_{g}}{2}. \tag{1}\]  
Figure 4: Proposed metasurface unit cell. (**a**) Top View; (**b**) Perspective view.  
The guided wavelength at resonant frequency \(f_{o}\) is represented by the following equation [50]  
\[\lambda_{g}=\frac{\lambda_{o}}{\sqrt{\epsilon_{\text{eff}}}}=\frac{c}{f_{o}\sqrt{ \epsilon_{\text{eff}}}}, \tag{2}\]  
therefore, the internal and external ring guided wavelengths \(\lambda_{g1}\) & \(\lambda_{g2}\) respectively are given as  
\[\lambda_{g1}=\frac{c}{f_{1}\sqrt{\epsilon_{\text{eff}}}},\ \ \ \lambda_{g2}=\frac{c}{f_{2}\sqrt{\epsilon_{\text{eff}}}}, \tag{3}\]  
where resonance frequencies of the internal and external rings are defined by \(f_{1}\) and \(f_{2}\) respectively. The effective dielectric constant is calculated as [51]  
\[\epsilon_{\text{eff}}=\frac{\epsilon_{r}+1}{2}+\frac{\epsilon_{r}-1}{2}\Big{[} 1+\frac{12h}{W}\Big{]}^{-1/2}\qquad\text{for}\quad\frac{W}{h}>1, \tag{4}\]  
where \(\epsilon_{r}\) represents the dielectric constant. To attain the internal and external ring average length at any specified resonant frequency, Equation (2) is substituted into Equation (1) to give [50]  
\[L_{\text{T1}}=\frac{c}{2f_{1}\sqrt{\epsilon_{\text{eff}}}},\ \ \ L_{T2}=\frac{c}{2f_{2}\sqrt{\epsilon_{\text{eff}}}}. \tag{5}\]  
Furthermore, since the resonators exhibit the same standing waves at the fundamental modes, Equation (5) is used for both one-port and two-port resonators [50].  
From the specified substrate information, the dielectric constant \(\epsilon_{r}\) and height \(h\) are used to calculate the width \(W\) of the ring resonator. For a given characteristic impedance \(Z_{o}\), the width was represented as;  
\[W=\begin{cases}\frac{8\epsilon^{4}}{2^{2}A-2}h&\text{for}\quad\frac{W}{h}<2,\\ \frac{2}{\pi}\Bigg{[}B-1-\ln(2B-1)+\frac{\epsilon_{r}-1}{2\epsilon_{r}}\Bigg{(} \ln(B-1)+0.39-\frac{0.61}{\epsilon_{r}}\Bigg{)}\Bigg{]}h&\text{for}\quad\frac {W}{h}>2,\end{cases} \tag{6}\]  
where  
\[A=\frac{Z_{o}}{60}\sqrt{\frac{\epsilon_{r}+1}{2}}+\frac{\epsilon_{r}-1}{ \epsilon_{r}+1}\Bigg{(}0.23+\frac{0.11}{\epsilon_{r}}\Bigg{)}, \tag{7}\]  
\[B=\frac{377\pi}{2Z_{o}\sqrt{\epsilon_{r}}}. \tag{8}\]  
From Equation (5), the internal and external side lengths \(l_{1}\) and \(l_{2}\) of each ring of the SRR can be calculated by,  
\[l_{1}=L_{\text{T1}}+\frac{W}{2},\ \ \ l_{2}=L_{\text{T2}}+\frac{W}{2}. \tag{9}\]  
The permittivity and permeability are calculated from the refractive index (\(\eta\)) and impedance values \(z\) using Equations (10) and (11) in order to confirm if the designed SRR represents a double negative metamaterial.  
\[\epsilon =\frac{\eta}{z}, \tag{10}\] \[\mu =\eta z, \tag{11}\]  
where \(\eta\) and \(z\) are denoted as [49, 52]  
\[\eta=\frac{1}{kI}\Re\{\ln(e^{inkI_{\text{out}}})\}-\Im\{\ln(e^{inkI_{\text{out }}})\}, \tag{12}\]  
\[z=\frac{(1+S_{11}^{2})^{2}-S_{21}^{2}}{(1-S_{11}^{2})^{2}-S_{21}^{2}}. \tag{13}\]From Equations (12) and (13), the reflection and transmission coefficients are represented by \(S_{11}^{2}\) and \(S_{21}^{2}\), respectively, \(\Re\{\cdot\}\) and \(\Im\{\cdot\}\) denote the real and imaginary components, \(k\) represents the wave number, and \(e^{jnkl_{\text{out}}}\) is represented as [53]:  
\[e^{jnkl_{\text{out}}}=\frac{S_{21}^{2}}{1-S_{11}\left(\frac{z-1}{z+1}\right)}. \tag{14}\]