## 2 Design Methodology  
The geometrical structure of the ultra-wideband THz absorber is shown in Figure 1a. It consists of three layers; a VO\({}_{2}\) arc-shaped patch on the top, a metal ground plane of gold (Au) placed at the bottom, and an insulating spacer (quartz) as the middle dielectric layer. The final optimized geometrical parameters are determined as a result of parametric scanning. \(p=75\)\(\upmu\)m, t = 0.2 \(\upmu\)m, h = 11 \(\upmu\)m, s1 = s2 = 17 \(\upmu\)m, c = 1 \(\upmu\)m, b = 5 \(\upmu\)m a = w1 = 4 \(\upmu\)m, w = 23 \(\upmu\)m, and L = 17 \(\upmu\)m. The thickness of the top layer VO\({}_{2}\) is 0.2 \(\upmu\)m. In our simulations, we used a quartz substrate with a permittivity of 2.25 [46]. The conductivity of gold (Au) is \(\sigma=5.8\times 10^{7}\) S/m. The optical properties of VO\({}_{2}\) in the THz range [34; 35; 36; 37] are described using the Drude model, which may be stated as:  
\[\varepsilon(\omega)=\varepsilon_{\infty}-\frac{\omega_{p}^{2}(\sigma)}{\omega ^{2}+i\,\tau\omega}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**References**} & **Absorption** & \multirow{2}{*}{**Absorptance**} & **Tunable Range** & \multirow{2}{*}{**Material**} \\  & **Bandwidth (THz)** & & & \\ \hline
[39] & 0.34 & \(>\)90\% & 30–100 & VO\({}_{2}\) \\ \hline
[40] & 0.66 & \(>\)90\% & 30–98 & VO\({}_{2}\) \\ \hline
[41] & 0.53 & \(>\)90\% & 5–100 & VO\({}_{2}\) \\ \hline
[42] & 1.24 & \(>\)90\% & 15–96 & VO\({}_{2}\) \\ \hline
[43] & 0.56 & \(>\)90\% & 9–99.3 & VO\({}_{2}\) and graphene \\ \hline
[44] & 1 The permittivity at infinite frequency is represented by \(\varepsilon_{\infty}=12\). The collision frequency (\(\gamma\)) is 5.75 \(\times\) 10\({}^{13}\) rad/s, while the plasma frequency, \(\omega_{\rho}(\sigma_{0})\), depends on conductivity. As a function of \(\sigma\), plasma frequency \(\omega_{\rho}(\sigma)\) can be represented as:  
\[\omega_{p^{2}}(\sigma)=\frac{\sigma}{\sigma_{0}}\,\omega_{p}^{2}\,\sigma_{0}\]  
with \(\sigma=3\,\times\,10^{5}\) S/m and \(\omega_{\rho}(\sigma_{0})=1.4\,\times\,10^{15}\) rad/s.  
In this work, thermally, VO\({}_{2}\) changes from an insulator to a metal state, with conductivity increasing from 200 S/m to 2 \(\times\,10^{5}\) S/m. At approximately 340 K, it is known that VO\({}_{2}\) can undergo a phase shift from insulator to metal [54]. CST Microwave Studio was used to simulate the electromagnetic response of the proposed absorber over the frequency range of 2-7 THz. The simulation employs a configuration for adaptive tetrahedron mesh refinement. To model an infinite array, periodic boundary conditions ire used with an electric field (E-field) along the \(x\)-axis and a magnetic field (H-field) along the \(y\)-axis. To incident a linearly polarized wave, floquet ports are used along the \(z\)-axis.  
The absorption is computed as follows as \(\mathrm{A}(\omega)=1-\mathrm{R}(\omega)-\mathrm{T}(\omega)\), where \(\mathrm{R}(\omega)=|\mathrm{S}_{11}|^{2}\) and \(\mathrm{T}(\omega)=|\mathrm{S}_{21}|^{2}\). \(\mathrm{S}_{11}\) and \(\mathrm{S}_{21}\) are numerically calculated reflection and transmission  
Figure 1: (**a**) \(3\,\times\,3\) array of the design metasurface; (**b**) the top view of the schematic unit cell; (**c**) three-dimensional view of the proposed planar ultra-wideband unit cell absorber.  
coefficients. Since the thickness of the back metallic layer (Au = 0.2 \(\upmu\)m) is much larger than the skin depth, the transmittance T(\(\omega\)) is close to 0, as shown in Figure 2 (T(\(\omega\)) = S21 = 0). As a result, the absorptivity is as follows:  
\[\text{A}(\omega)=1-\text{R}(\omega)\]  
Therefore, the concept of impedance-matching theory achieves a highly efficient broadband absorption from a range of 2.7 to 5.7 THz.