## 2 Background Theory and Method
### Temperature Distribution Analysis  
To derive the thermal conductivity by analyzing the phase delay, it is essential to compare the results of the probe beam deflection analysis based on the theoretical temperature analysis with the experiment results. For the temperature distribution analysis, a 3D single layer solid model with infinite lengths in the \(x\) and \(y\) directions and finite length in the \(z\) direction was selected, as shown in Figure 3. Regions 0 and 2 are air layers surrounding the measurement material (specimen) and Region 1 is the measurement specimen. As the temperature increase by the pump beam is very small and occurs in local areas, only conduction heat transfer was considered in this study. Therefore, the governing equation can be expressed by Equation (2) below. \[\
abla\cdot\left(\overset{\rightarrow}{k}_{k}\times\
abla T_{k}\right)-\rho_{ k}\epsilon_{p,k}\frac{\partial T_{k}}{\partial t}=Q_{i}\hskip 28.452756 pt(k=0,1,2) \tag{2}\]  
Figure 3: Schematic diagram of collinear deflection analysis. Figure 2: Schematic of phase delay. \[\vec{k}_{1}=\left( \begin{array}{ccc}k_{1,11}&k_{1,12}&k_{1,13}\\ k_{1,21}&k_{1,22}&k_{1,23}\\ k_{1,31}&k_{1,32}&k_{1,33}\end{array} \right)\qquad k_{1,ij}>0\ \,\ k_{1,ij}=0\ (i\
eq j), \qquad(i,j=1,2,3) \tag{3}\]  
where T is the temperature, \(k\) is the thermal conductivity, \(\rho\) is the density, \(C_{p}\) is the specific heat, and \(Q\) is the heat source by the pump beam. The values \(k\), \(\rho\), and \(C_{p}\) are thermal properties of the object of the measurement and thermal properties of the air layer. As the target material was thermally anisotropic, the thermal conductivity could be expressed in orthorhombic form, as shown in Equation (3), and an orthorhombic system composed of \(k_{1,11}\), \(k_{1,22}\), and \(k_{1,33}\) components without \(k_{1,ij}\big{|}_{i\
eq j}\) components was considered. As mentioned above, only conduction was considered in this study. Therefore, the boundary conditions can be expressed by Equation (4) below. \[ \begin{array}{c}T_{0}(x,y,0)=T_{1}(x,y,0)\qquad\qquad q_{0}(x,y,0)=q_{1}(x,y,0)\\ T_{1}(x,y,L)=T_{2}(x,y,L)\qquad\qquad q_{1}(x,y,L)=q_{2}(x,y,L)\\ q_{1}=-\Big{(}k_{1,11}\frac{\partial T}{\partial x}+k_{1,2}\frac{\partial T}{ \partial y}k_{1,33}\frac{\partial T}{\partial z}\Big{)}\\ q_{k}=-k_{k}\frac{\partial T}{\partial x}\qquad\qquad(k=0,2)\end{array}  \tag{4}\]  
The pump beam is a laser with Gaussian distribution and can be expressed as shown in Equation (5) according to the modulation frequency (\(f\)), radius (\(a\), \(1/e^{2}\) of maximum intensity), and light absorption coefficient of the material (\(\lambda\)). The pump beam reaches the material through the air layer. As the light absorption coefficient of the air layer is very small and the pump beam is absorbed only by the material, the heat source of the air layer can be expressed as shown in Equation (6). \(P_{1}\) is the intensity of the laser that finally reaches the material and can be expressed by Equation (7) using the initial intensity of the laser (\(P_{0}\)) and the reflectivity of the specimen (\(\varepsilon_{1}\)). \[Q_{1}(x,y,z,t)=-\lambda_{1}\frac{P_{1}}{4\pi a^{2}}e^{-\lambda_{1}z}e^{-(x^{2} +y^{2})/a^{2}}[1+\cos(\omega t)] \tag{5}\]  
\[Q_{k}(x,y,z,0)=0\qquad\qquad(k=0,2) \tag{6}\]  
\[P_{1}=P_{0}\times(1-\varepsilon_{1}) \tag{7}\]  
The heat source \(Q_{1}\) consists of a time-independent term and a time-dependent term with a constant period. Only the time-dependent term was considered because the time-independent term does not affect the phase delay. Therefore, the governing equation is a 3D unsteady-state heat conduction equation in which a periodic heat source exists over time. It can be converted into a form in which steady-state analysis is possible by applying a complex transformation method, as shown in Equation (8). \[