## 2 Background Theory and Method
### Numerical Investigation  
Based on Equation (8), the temperature distribution was analyzed using COMSOL Multiphysics 5.2a, which is a commercial numerical analysis software program. The governing equation and boundary conditions were applied as explained in Equations (2)-(8). Geometry was constructed using the values shown in Table 2. To verify the numerical analysis model, node dependency was evaluated and compared with the analytical solution results of the mirage deflection method performed by [PERSON] et al. [29; 36; 37]. The material used for evaluating node dependency and verifying the numerical analysis model was copper. The thermal properties used and pump beam radius are summarized in Table 3.  
Table 4 shows the results derived through the node dependency evaluation. It was found that the temperature in the \(x\)-axis direction on the material surface (\(y=0,\ z=0\)) according to the relative position converged when the number of the nodes was equal to or higher than approximately 970,000. In this study, approximately 1,150,000 nodes were used for the analysis.  
Figure 4 compares the analytical solution results of the mirage deflection method for copper with the results of the numerical analysis model constructed in this study. The temperature change (at  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{5}{c}{**Temperature (\({}^{\circ}\)C)**} \\ \cline{3-6}
**Number of Nodes** & **Computing Time** & \multicolumn{5}{c}{**X-Position**} \\ \cline{3-6}  & & **0 mm** & **0.15 mm** & **0.3 mm** & **0.45 mm** \\ \hline
26,901 & 10s & 2.5412 & 2.3569 & 1.3154 & 0.8852 \\ \hline
87,451 & 22s & 2.6542 & 2.1516 & 1.2987 & 0.8751 \\ \hline
203,401 & 49s & 2.7567 & 2.1159 & 1.2256 & 0.9045 \\ \hline
348,145 & 105s & 2.8457 & 2.2548 & 1.4146 & 0.8912 \\ \hline
549,081 & 120s & 2.7542 & 2.1987 & 1.3554 & 0.8759 \\ \hline
815,425 & 223s & 3.0157 & 2.2247 & 1.4254 & 0.8945 \\ \hline
976,005 & 253s & 3.2187 & 2.4194 & 1.4610 & 0.9153 \\ \hline
1,156,393 & 379s & 3.2235 & 2.4243 & 1.4646 & 0.9192 \\ \hline
1,357,741 & 3,868s & 3.2234 & 2.4245 & 1.4648 & 0.9190 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Node dependency evaluation.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Length (\(mm\))** \\ \hline  & \(L_{x}\) & 20 \\ \hline  & \(L_{y}\) & 20 \\ \hline  & Region 0 & 6 \\ \cline{2-3} \(L_{z}\) & Region 1 & 3 \\ \cline{2-3}  & Region 2 & 6 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Geometry setup for numerical investigation and validation.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Copper [38]** & **Air [3]** \\ \hline Density (\(\rho,\ kg/m^{3}\)) & 8930 & 1.1614 \\ \hline Specific heat capacity (\(c_{P},J/(kg\cdot K)\)) & 385 & 1007 \\ \hline Thermal conductivity (\(k,\ W/(mK)\)) & 398 & 0.0263 \\ \hline Absorption coefficient (\(\lambda,\ m^{-1}\)) & \(7.01\times 10^{7}\) & - \\ \hline Thermo-optical coefficient (\(\partial n/\partial T,\ K^{-1}\)) & 2.22 & \(2.55\times 10^{-5}\) \\ \hline Pump beam radius (\(a,\ \mu m\)) & & 100 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Thermal properties for validation.  
\(y=0,\;z=0\)) in the \(x\)-axis direction on the material surface (Figure 4a) and the temperature changes (at \(x=0,\;y=0\)) in the air layer and inside the material (Figure 4b) were in good agreement with analytical solution (for temperature distribution) of mirage deflection.