## 2 Background Theory and Method
### Deflection Analysis  
The thermal conductivity of the transparent anisotropic material, which is the target material of this study, varies depending on the crystal coordinates \((x,\;y,\;z)\) of the material. As the accurate direction of the principal thermal conductivity of the specimen cannot be found by measurement through experiments, it is necessary to compare and analyze the experimental and analytical results. To this end, the crystal coordinates \((x,\;y,\;z)\) used for the analysis must be matched to the coordinates of the experimental setup \((X,\;Y,\;Z)\) using the coordinate transformation equation below [37].  
\[\begin{array}{l}x=X\cos\theta+Y\sin\theta\\ y=Y\cos\theta-X\sin\theta\\ z=Z\end{array} \tag{9}\]  
Through this process, the effective thermal conductivity can be derived according to the measurement direction (\(\theta\)). The theoretical thermal conductivity (\(k_{\theta,eff}\)) according to the measurement direction (\(\theta\)) can be expressed by the principal thermal conductivity and direction cosine, as shown in Equation (9) [39; 40; 41; 42]. \(k_{1,11}\), \(k_{1,22}\), and \(k_{1,33}\) are the principal thermal conductivities for each axis, and \(l,\;m,\;\) and \(n\) are the direction cosines for the \(x,\;y,\;and\;z\) axes, which are the crystal coordinates of the material, respectively.  
\[\begin{array}{l}k_{1,11}x^{2}+k_{1,22}y^{2}+k_{1,33}z^{2}=1\\ k_{0,eff}=k_{1,11}l^{2}+k_{1,22}m^{2}+k_{1,33}n^{2}\end{array} \tag{10}\]  
As explained above, the deflection angle of the probe beam (\(\psi\)) can be derived through Equation (1). The probe beam in collinear deflection passes through the upper air layer (region 0), material layer (region 1), and lower air layer (region 2) in sequence, and can be derived as the sum of the deflection angles (\(\psi\)) in each region as shown in Equation (11).  
\[\psi=\psi_{0}+\psi_{1}+\psi_{2} \tag{11}\]  
Figure 4: Comparison of analytical and numerical solutions. (a) Temperature change in x-direction (\(y=0,\;z=0\)). (b) Temperature change in z-direction (\(x=0,\;y=0\)).  
It is reasonable to assume that the refractivity (\(n\)) of Equation (1) and the temperature coefficient of the refractivity (\(\partial n/\partial T\)) are constants when the ambient temperature is constant. The deflection of the probe beam can be divided into two components perpendicular to the path (\(X\)-\(dir\), \(Y\)-\(dir\)), as shown in Figure 3, and can be finally expressed by Equation (12). In this study, only the X-direction deflection component (\(\psi_{axial}\)) of the probe beam was used.  
\[\begin{array}{l}\psi_{axial}=\frac{1}{n_{0}}\Big{(}\frac{\partial n}{ \partial T}\Big{)}_{0}\int_{-\infty}^{0}\Big{(}\frac{\partial T_{0}}{ \partial X}\Big{)}dZ+\frac{1}{n_{1}}\Big{(}\frac{\partial n}{\partial T}\Big{)} _{1}\int_{0}^{L_{1}}\Big{(}\frac{\partial T_{1}}{\partial X}\Big{)}dZ+\frac{1 }{n_{2}}\Big{(}\frac{\partial n}{\partial T}\Big{)}_{2}\int_{L_{1}}^{\infty} \Big{(}\frac{\partial T_{2}}{\partial X}\Big{)}dZ\\ \psi_{normal}=\frac{1}{n_{0}}\Big{(}\frac{\partial n}{\partial T}\Big{)}_{0} \int_{-\infty}^{0}\Big{(}\frac{\partial T_{0}}{\partial X}\Big{)}dZ+\frac{1 }{n_{1}}\Big{(}\frac{\partial n}{\partial T}\Big{)}_{1}\int_{0}^{L_{1}}\Big{(} \frac{\partial T_{1}}{\partial X}\Big{)}dZ+\frac{1}{n_{2}}\Big{(}\frac{ \partial n}{\partial T}\Big{)}_{2}\int_{L_{1}}^{\infty}\Big{(}\frac{\partial T _{2}}{\partial Y}\Big{)}dZ\end{array} \tag{12}\]  
As it is very difficult to accurately measure the deflection angle (\(\psi\)) through experiments, the phase delay (\(\chi\)) between the pump beam and probe beam is measured. The deflection angle (\(\psi\)) derived from the analysis is a complex number. Therefore, the phase delay (\(\chi\)) can be expressed as shown in Equation (13).  
\[\chi(X_{0},z,a,l,k)=\tan^{-1}\!\left(\frac{\text{Imag}(\psi)}{\text{Real}( \psi)}\right) \tag{13}\]