## 4 Calculation of Retrieved Effective Physical Parameters  
To explain the perfect absorption properties and tunable mechanism, the impedance theory is applied to determine the retrieved effective physical parameters from Equations (1)-(4) of the design structure. CST (MICROWAVE STUDIO) is used to extract the S-parameters. Compared to the Drude-Lorentz approach, S-parameters for complex structures yield more accurate permittivity and permeability values [49]. It is possible to write the system's S-parameters [50]. Figure 1 depicts the schematic metasurface with a typically incident plane wave. \[S_{11}=\frac{R_{01}\left(1-e^{i2 mk_{0}d}\right)}{1-R_{01}^{2}e^{i2 mk_{0}d}} \tag{1}\]  
\[S_{21}=\frac{\left(1-R_{01}^{2}\right)e^{i2 mk_{0}d}}{1-R_{01}^{2}e^{i2 mk_{0}d}} \tag{2}\]  
Figure 5: (**a**) The real part of the effective permittivity and (**b**) imaginary parts of effective permittivity for varying VO\({}_{2}\) conductivities. Figure 6: The absorption spectra of the design structure at \(1.5\times 10^{5}\) S/m VO\({}_{2}\) conductivities. Solving Equations (1) and (2) gives impedance (Z)  
\[=\pm\sqrt{\frac{\left(1+s_{11}\right)^{2}-s_{21}^{2}}{\left(1-s_{11}\right)^{2}-s _{21}^{2}}} \tag{3}\]  
\[e^{2 mk_{0}d}=\frac{S_{21}}{1-S_{11}\frac{z-1}{z+1}} \tag{4}\]  
\[n=\frac{1}{K_{0}d}[\{\left\{\left|ln(e^{ink_{0}d})\right|^{\prime\prime}+2m \pi\right\}-i[ln(e^{ink_{0}d})]^{\prime}\}] \tag{5}\]  
where (\(. )^{\prime\prime}\) represent the complex element and (\(o\))\({}^{\prime}\) represent real part of complex number. The refractive index is denoted by \(n\); \(z\) = impedance; \(k_{0}\) = wavenumber; \(d\) = maximum unit element length; \(k_{0}\) = wavenumber; \(m\) = branch due to sinusoidal periodicity, and electric and magnetic field components are denoted by \(E\) and \(H\), respectively. The following expressions link effective physical parameters with each other:  
\[\varepsilon_{effective}=\frac{n}{z} \tag{6}\]  
\[\mu_{\text{effective}}=nz \tag{7}\]  
Figure 7 shows the effective physical characteristics for perfect absorption when \(\sigma\) = 2 \(\times\) 10\({}^{5}\) S/m. The proposed absorber exhibits a negative refractive index and a positive impedance, indicating that the hypothesized metasurface behavior is in the band of interest. It can be seen from Figure 7a that in the 4 THz-5.7 THz frequency band, permittivity is negative. Figure 7 also presents the proposed THz absorber's refractive index and impedance. Complete absorption can occur when the absorber's effective impedance is equal to the free space impedance (377 Ohm). The fundamental electromagnetic theory demonstrates that the surface reflection is reduced to zero when a structure's effective permeability and permittivity are equal to those of free space. Figure 8 shows the relative impedance of the proposed absorber for different VO\({}_{2}\) conductivities. For VO\({}_{2}\) with a conductivity of 2 \(\times\) 10\({}^{5}\) S/m, the real part of the impedance is close to 1 (equivalent to free space) in the frequency range of 2.7-5.7 THz, while the imaginary parts are close to zero. Consequently, the parametrically optimized geometry of the metasurface's impedance is equal to that of free space. It satisfies the strategy requirements for a perfect absorber. To realize THz absorption, we start with a single VO\({}_{2}\) arc-shaped rectangle, which is placed on top of a quartz substrate to form a conventional metamaterial perfect absorber (MPAs) unit cell structure, as shown in Figure 9a, while in Figure 10a, the relative impedance is mismatched, and, due to this, there is no absorption. When the number of arc-shaped VO\({}_{2}\) patches increases on the top of the quartz substrate, the absorption bandwidth increases because the impedance becomes closer to that of the free space, as shown in Figure 10b,c. In the case of four identical arc-shaped VO\({}_{2}\) patches, the absorption efficiency and bandwidth increase, as seen in Figures 9d and 10d, respectively.