## 2 Metamaterial Absorber Design
### Evaluation of MMA and Metamaterial Property Analysis  
The evaluation of the proposed MMA towards SSRR for achieving quad-band absorption is shown in Figure 3. In order to understand the absorption mechanism of the MMA impedance, analysis is vital. The reflection coefficient (\(S_{11}\)) depends considerably on the effective impedance (\(Z_{e}\)), as shown in Equation (4).  
\[S_{11}=\frac{Z_{e}-Z_{0}}{Z_{e}+Z_{0}} \tag{4}\]where \(Z_{e}=\sqrt{\mu_{0}\mu_{r}(\omega)/\varepsilon_{0}\varepsilon_{r}(\omega)}\ Z_{0}=377 \Omega=\sqrt{\mu_{0}/\varepsilon_{0}}\) is the free space impedance; and \(\mu_{0}\), \(\varepsilon_{0}\), \(\mu_{r}(\omega)\), and \(\varepsilon_{r}(\omega)\) are the free space permeability, free space permittivity, frequency-dependent permeability, and permittivity, respectively. The normalized impedance can be calculated by \(Z=Z_{eff}/Z_{o}=\sqrt{\mu_{r}/\varepsilon_{r}}\). Near-unity absorption is achieved by impedance matching with free space. The near-unity value of the real part and the near-zero value of the imaginary part represent the normalized impedance matched with free space [52; 53]. The relation between the absorption and metamaterial properties can be understood by calculating Equations (3) and (4) [54].  
\[A(\omega)=1-S_{11}^{2}=1-\left|\frac{Z_{e}-Z_{0}}{Z_{e}+Z_{0}}\right|^{2}=1- \left|\frac{\sqrt{\mu_{r}}-\sqrt{\varepsilon_{r}}}{\sqrt{\mu_{r}}+\sqrt{ \varepsilon_{r}}}\right| \tag{5}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline Parameter & Value (mm) \\ \hline
1 & 10 \\ a1 & 4 \\ a2 & 7.07 \\ a3 & 4.95 \\ b1 & 2.0 \\ b2 & 1.23 \\ g1 & 0.35 \\ g2 & 0.35 \\ w & 0.35 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters list.  
Figure 1: Patch design of the proposed MMA. (**a**) Perspective view, (**b**) simulation setup and (**c**) front view.  
The metamaterial attributes of the absorber are determined by the Nicolson-Ross-Weir (NRW) formula [55], where \(\omega=2\pi f\), and \(c\) is the velocity of light.  
\[\varepsilon_{r}=\frac{2}{\sqrt{\frac{\omega}{c}\times d}}\times\frac{1-(S_{21}+ S_{11})}{1+(S_{21}+S_{11})} \tag{6}\]  
\[\mu_{r}=\frac{2}{\sqrt{\frac{\omega}{c}\times d}}\times\frac{1-(S_{21}-S_{11}) }{1+(S_{21}-S_{11})} \tag{7}\]  
Initially (design 1), a square ring resonator (SRR) is placed on top of the substrate materials, achieving only 14% absorption at a 12.25 GHz resonance frequency with (single negative) SNG metamaterial properties. The absorption percentage slightly increases to 30% at a 12.30 GHz resonance frequency using two square ring resonators because of coupling capacitance between the two SRRs, where SNG metamaterial properties are achieved. In design 3, four splits are made in the middle of each SRR, which significantly increases the capacitance in the splits, resulting in 96.37% and 98.64% absorption peaks appearing at 13.01 GHz and 17.46 GHz resonance frequency, respectively, with DNG metamaterial properties. Three absorption peaks are achieved by inserting four microstrip lines in the  
Figure 3: Evaluation of proposed MMA unit cell.  
Figure 2: Absorption and s-parameter plot.  
outer side of the SSRR. This microstrip line generates another coupling capacitance, and therefore, another absorption peak is generated. However, at 12.62 GHz, 17.42 GHz, and 19.85 GHz, SNG metamaterial properties are achieved with 97.97%, 98.23%, and 97.90% absorption peaks, respectively. Finally, a Jerusalem cross was designed at the absorber's center to increase absorption. The absorption peaks are 97%, 99.51%, 99%, and 99.5% at 12.62 GHz, 14.12 GHz, 17.53 GHz, and 19.91 GHz, respectively. The DNG metamaterial properties appears at 12.62 GHz, 17.53 GHz, and 19.91 GHz. However, the metamaterial property is SNG at 14.12 GHz frequency. The simulated absorption plot for different designs is shown in Figure 4. Table 2 lists the metamaterial properties for various design evaluations. Metamaterial properties such as permittivity, permeability, refractive index, and normalized impedance of different designs are presented in Figure 5.