## 2 Numerical Analysis  
This section shows the design and development of the solar absorber. The solar absorber development is done using three layers in this design. The two substrate layers, namely SiO\({}_{2}\) and GaAs, are used. The selection of these substrates is done because of their wide availability for the solar absorber design and their absorbing capacity. The resonating layer is made up of titanium material. Titanium is a low-cost material compared to gold and silver. This low-cost behavior of this metal will improve its affordability for different classes of people. Figure 1 depicts the three-layer design, including the graphene layer as a sandwiching layer between the resonator and SiO\({}_{2}\) substrate. The optimization of the structural parameters has been investigated in this design, and the optimized parameters for different parameters are 1000 nm for the I-shaped resonator thickness, 1000 nm for the GaAs substrate, and 2500 nm for the SiO\({}_{2}\) substrate. There is no ground plane used in this design. Optimization of I-shape design parameters such as length, width, and edges is also performed. The two different metasurface designs, such as the I-shape design and the thin wire design, are also carried out, and their results are presented in the next section. The length and width of the substrate are 7000 nm and 7000 nm, respectively. I-shape length and width are 3000 nm and 3000 nm, respectively. The thin wire's length and width are 3000 nm and 500 nm, respectively.  
The graphene material is very important in the design of this absorber because it enhances the absorptance of the solar design by incorporating a monolayer of it between the resonator and substrate. The optical and electric properties are helping to improve the overall efficiency of the design. The change in graphene's chemical potential changes the conductivity of the monolayer, which will improve the absorption of the structure. The equation for these is presented in Equations (1)-(4) [27]. The absorption also depends on the angle of incidence, which is presented in Equations (5)-(11) [28]  
\[\varepsilon(\omega)=1+\frac{\sigma_{s}}{\varepsilon_{0}\omega\Delta} \tag{1}\]  
\[\sigma_{intra}=\frac{-je^{2}k_{B}T}{\pi\hbar^{2}(\omega-j2\Gamma)}\left(\frac {\mu_{c}}{k_{B}T}+2 ln\left(e^{-\frac{\mu_{c}}{k_{B}T}}+1\right)\right) \tag{2}\]  
\[\sigma_{inter}=\frac{-je^{2}}{4\pi\hbar}ln\left(\frac{2|\mu_{c}|-(\omega-j2 \Gamma)\hbar}{2|\mu_{c}|+(\omega-j2\Gamma)\hbar}\right) \tag{3}\]  
\[\sigma_{s}=\sigma_{inter}+\sigma_{intra} \tag{4}\]  
The angle of incidence and absorption relation  
\[r(\omega,\theta_{i})=\frac{\omega\cos\theta_{i}\prod_{00}(\omega,\theta_{i})} {2i\hbar ck^{2}+\omega\cos\theta_{i}\prod_{00}(\omega,\theta_{i})} \tag{5}\]  
\[\sigma_{||}(\omega,k)=-i\,\frac{\omega}{4\pi\hbar k^{2}}\prod_{00}(\omega,k) \tag{6}\]  
\[r(\omega,\theta_{i})=\frac{2\pi\cos\theta_{i}\sigma_{||}(\omega,k)}{c+2\pi \cos\theta_{i}\sigma_{||}(\omega,k)} \tag{7}\]  
\[\mathcal{R}(\omega,\theta_{i})=|r(\omega,\theta_{i})|^{2} \tag{8}\]\[\mathcal{R}(\omega,\theta_{i})=\frac{4\pi^{2}\cos^{2}\theta_{i}\left[\text{Re}^{2} \sigma_{||}(\omega,k)+\text{Im}^{2}\sigma_{||}(\omega,k)\right]}{\left[c+2\pi \cos\theta_{i}\text{Re}\sigma_{||}(\omega,k)\right]^{2}+4\pi^{2}\cos^{2}\theta_ {i}\text{Im}^{2}\sigma_{||}(\omega,k)} \tag{9}\]  
\[\mathcal{R}(\omega)=\mathcal{R}(\omega,0)=\frac{4\pi^{2}\left[\text{Re}^{2} \sigma(\omega)+\text{Im}^{2}\sigma(\omega)\right]}{\left[c+2\pi\text{Re} \sigma(\omega)\right]^{2}+4\pi^{2}\text{Im}^{2}\sigma(\omega)} \tag{10}\]  
\[A(\omega)=1-\mathcal{R}(\omega)-T(\omega) \tag{11}\]  
The results of all these analyses related to optimization are discussed in the following sections.