## 4 Results and Discussion  
Figure 9 shows the phase delay in each measurement direction (\(\theta\): \(0^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\)) according to the relative position. The black lines represent the values derived through theoretical analysis (temperature field derivation through numerical analysis and phase delay calculation using Equations (12) and (13)) using the properties in Table 6, and the square points represent the values derived through the experiment. The experimental results are shown along with error bars. An error \begin{table}
\begin{tabular}{c c c} \hline \hline
**Density (\(\rho\), \(kg/m^{3}\))** & 3980 \\ \hline
**Specific heat capacity (\(c_{p}\), \(I/(kg\cdot K)\))** & 761 \\ \hline Thermal conductivity (\(k\), \(W/(mK)\)) &
\begin{tabular}{c} \(x\)-dir (\(k_{1,11}\)) \\ \end{tabular} & 32.5 \\ \hline \(y\)-dir (\(k_{1,22}\)) & 23.1 \\ \hline \(z\)-dir (\(k_{1,33}\)) & 32.5 \\ \hline \(\Delta\) absorption coefficient (\(\lambda\), \(m^{-1}\)) & 653.05 \\ \hline Thermo-optical coefficient (\(dn/dT\), \(k^{-1}\)) & \(1.30\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Properties of A-Plane Sapphire. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Frequency (\(Hz\))** & 20 \\ \hline
**Radius (\(mm\))** & 82.5 \\ \hline
**Relative position (Numerical) (\(mm\))** & 0.15–0.5 (interval: 0.02) \\ \hline
**Relative position (Experimental) (\(mm\))** & 0.15–0.5 (interval: 0.02) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Experimental setup value. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Thermal Conductivity (\(W\)/(\(mK\)))** & \
\begin{tabular}{c} **Measurement Temperature** \\ (\(K\)) \\ \end{tabular} & **Literature Author** &
\begin{tabular}{c} **Average Value** \\ (\(W\)/(\(mK\))) \\ \end{tabular} \\ \hline \multirow{4}{*}{\(x\)-_dir_ (\(k_{1,11}\)) and \(z\)-_dir_ (\(k_{1,33}\))} & 32.5 & 298 & [PERSON] et al [43] \\ \cline{2-4}  & 25.2 & 300 & [PERSON] et al [44] \\ \cline{2-4}  & 47 & 298 & [PERSON] et al [45] & 36.4 \\ \cline{2-4}  & 42.3 & 303 & [PERSON] [46] \\ \cline{2-4}  & 35 & 300 & [PERSON] et al [47] \\ \hline \multirow{4}{*}{\(y\)-_dir_ (\(k_{1,22}\))} & 30.3 & 298 & [PERSON] et al [43] \\ \cline{2-4}  & 23.1 & 300 & [PERSON] et al [44] & 27.1 \\ \cline{1-1} \cline{2-4}  & 31 & 298 & [PERSON] et al [45] & (\(\pm\) 4.1) \\ \cline{1-1} \cline{2-4}  & 24 & 300 & [PERSON] et al [48] \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Literature values of thermal conductivity for sapphire glass. bar represents the standard deviation of the 95% interval data among the obtained 2400 phase delay values at one relative position. It was found that the experimental and numerical analysis results had similar tendencies and were in good agreement. The experimental results showed that the error was relatively higher due to  
Figure 9: Comparison of experimental and theoretical results for various measurement directions. (a) \(\theta=0^{\circ}\). (b) \(\theta=30^{\circ}\). (c) \(\theta=45^{\circ}\). (d) \(\theta=60^{\circ}\). (e) \(\theta=90^{\circ}\). the influence of the pump beam in the area where the distance between the pump beam and the probe beam was small (near 0.15 mm), as explained above. On the other hand, in the area where the distance between the pump beam and the probe beam was large (near 0.50 mm), the error was relatively higher because the accuracy of the phase difference acquisition was reduced due to the low temperature gradient in the material. Using the experimental results, it is possible to determine the thermal conductivity through the phase curve method algorithm explained above through Figure 5. From the theoretical phase delay results derived while the thermal conductivity (\(k_{rand}\)) was varied and the phase delay results derived from the experiment, the RMSE was derived using Equation (14).