## 2 Background Theory and Methods
### Temperature Distribution and Phase Delay Analysis  
The phase delay between the pump beam and probe beam was measured experimentally by the rear-side photothermal deflection method. Temperature distribution information is required to obtain the phase delay through theoretical analysis. To acquire this information, we performed a numerical analysis of a three-dimensional unsteady heat conduction equation with periodic thermal sources shown in Equation (3). Since the increase in specimen temperature due to the pump beam was small, the effects of convection and radiation generated on the front and rear surfaces of the specimen were ignored [33]. Figure 3 shows a schematic diagram including the target materials with the thin light absorption film. The axis perpendicular to the surface of the specimen was set to be the z-axis. Regions 0 and 3 are the air layer around the specimen, which are regions with insufficient optical absorption. Region 1 represents the thin light absorption film and region 2 holds the target materials. \[\
abla^{2}T_{i}-\frac{1}{\alpha_{i}}\frac{\partial T_{i}}{\partial t}=-\frac{ Q_{i}}{k_{i}}\ (i=\text{front air}\ [0],\ \text{light absorption thin film}\ [1],\ \text{target material}\ [2],\ \text{rear air}\ [3]) \tag{3}\]  
\[Q_{1}(x,y,z,t)=\frac{\lambda_{1}P_{0}(1-R_{1})}{4\pi a^{2}}e^{-\frac{(x^{2}+y^{ 2})}{a^{2}}-\lambda_{1}z}[1+\cos(\omega t)] \tag{4}\]  
Figure 2: **(a)** Schematic diagram of a rear-side photothermal deflection method; **(b)** an example of phase difference derivation. \[Q_{2}(x,y,z,t)=\frac{\lambda_{2}P_{0}(1-R_{1})(1-R_{2})}{4\pi a^{2}}e^{-\frac{(x^{ 2}+y^{2})}{a^{2}}-\lambda_{2}(z-L_{1})-\lambda_{1}L_{1}}[1+\cos(\omega t)]=0 \tag{5}\]  
\[P_{1}=P_{0}(1-R_{1}) \tag{6}\]  
\[P_{2}=P_{0}(1-R_{1})(1-R_{2}) \tag{7}\]  
\(Q_{1}(x,y,z,t)\) and \(Q_{2}(x,y,z,t)\) represent the heat fluxes of regions 1 and 2, respectively, while \(\lambda_{1}\) and \(\lambda_{2}\) denote the optical absorption coefficients of the respective regions. \(R_{1}\) and \(R_{2}\) represent the reflexibility of the boundary between regions 0 and 1 and the boundary between regions 1 and 2, respectively. Since the heat flux of the laser is absorbed entirely by the light absorption film, the heat flux of region 2 can be regarded as 0. Equations (8)-(11) summarizes the equations by substituting the heat flux of the region where the optical absorption occurs. \[\
abla^{2}T_{0}-\frac{1}{\alpha_{0}}\frac{\partial T_{0}}{\partial t}=-\frac{ Q_{0}}{k_{0}}=0 \tag{8}\]  
\[\
abla^{2}T_{1}-\frac{1}{\alpha_{1}}\frac{\partial T_{1}}{\partial t}=-\frac{ Q_{1}}{k_{1}}=-\frac{\lambda_{1}P_{1}}{4\pi k_{1}a^{2}}e^{-\frac{t^{2}}{a^{2}}- \lambda_{1}z}[1+\cos(\omega t)] \tag{9}\]  
\[\
abla^{2}T_{2}-\frac{1}{\alpha_{2}}\frac{\partial T_{2}}{\partial t}=-\frac{ Q_{2}}{k_{2}}=0 \tag{10}\]  
\[\
abla^{2}T_{3}-\frac{1}{\alpha_{3}}\frac{\partial T_{3}}{\partial t}=-\frac{ Q_{3}}{k_{3}}=0 \tag{11}\]  
As shown in Equations (12)-(14), boundary conditions with the same temperature and heat flux are applied to the boundaries of each region. \[Q_{1}(x,y,z,t)=\frac{\lambda_{1}P_{0}(1-R_{1})}{4\pi a^{2}}e^{-\frac{(x^{2}+y^{ 2})}{a^{2}}-\lambda_{1}z}[1+\cos(\omega t)] \tag{4}\]0  
\[Q_{1}(x,y,z,t)=\frac{\lambda_{1}P_{0}(1-R_{1})}{4\pi a^{2}}e^{-\frac{(x^{2}+y^{ 2})}{a^{2}}-\lambda_{1}z}[1+\cos(\omega t)] \tag{4}\]1  
\[Q_{1}(x,y,z,t)=\frac{\lambda_{1}P_{0}(1-R_{1})}{4\pi a^{2}}e^{-\frac{(x^{2}+y^{ 2})}{a^{2}}-\lambda_{1}z}[1+\cos(\omega t)] \tag{4}\]2  
\[Q_{1}(x,y,z,t)=\frac{\lambda_{1}P_{0}(1-R_{1})}{4\pi a^{2}}e^{-\frac{(x^{2}+y^{ 2})}{a^{2}}-\lambda_{1}z}[1+\cos(\omega t)] \tag{4}\]3  
\[Q_{1}(x,y,z,t)=\frac{\lambda_{1}P_{0}(1-R_{1})}{4\pi a^{2}}e^{-\frac{(x^{2}+y^{ 2})}{a^{2}}-\lambda_{1}z}[1+\cos(\omega t)] \tag{4}\]4  
\[Q_{1}(x,y,z,t)=\frac{\lambda_{1}P_{0}(1-R_{1})}{4\pi a^{2}}e^{-\frac{(x^{2}+y^{ 2})}{a^{2}}-\lambda_{1}z}[1+\cos(\omega t)] \tag{4}\]5 Schematic of rear-side photothermal deflection method.