# Measurement of Contrast and Spatial Resolution for the Photothermal Imaging Method
## 3 Numerical and Experimental Analysis
### Numerical Analysis  
In order to analyze the image quality of the photothermal imaging method, numerical analysis was conducted on specimens with the simple subsurface structure shown in Figure 4. The numerical analysis model had the boundary conditions presented in Figure 7, and it was conducted on the basis of the model established by [PERSON] et al. [3]. Here, the infinite condition means that the physical properties  
\begin{table}
\begin{tabular}{c c c c c} \hline  & 5 Hz & 20 Hz & 80 Hz & Ratio with \(L_{d}\) \\ \hline
**Thermal diffusion length**, \(L_{d}\) **(mm)** & 2.76 & 1.38 & 0.69 & - \\
**Pump-beam radius**, \(R_{b}\) **(mm)** & 0.40 & 0.20 & 0.10 & 0.145 \\
**First-layer thickness**, \(L_{f}\) **(mm)** & 2.00 & 1.00 & 0.50 & 0.725 \\
**Relative position**, \(X_{0}\) **(mm)** & 1.20 & 0.60 & 0.30 & 0.435 \\
**z-position of probe beam**, \(L_{h}\) **(mm)** & 0.60 & 0.30 & 0.15 & 0.218 \\ \hline \end{tabular}
\end{table}
Table 2: System conditions in Figure 6.  
Figure 6: Nondimensionalization of the scanning results using different modulation frequencies.  
of the adjacent material are infinite [25]. The size and shape of the numerical model specimen are the same as those in Figure 4, and the thickness of the top and bottom air layers is set to 15 mm. Equation (4) is the governing equation for the three areas that consist of three layers: the top air layer, the specimens, and the bottom air layer:  
\[\
abla^{2}T_{i}+\frac{1}{k_{i}}Q_{i}=\frac{1}{\alpha_{i}}\frac{\partial T_{i}}{ \partial t}\left(i=\text{top air}\left[0\right],\text{ sample}\left[1\right],\text{ bottom air}\left[2\right]\right) \tag{4}\]  
where the subscript \(i\) indicates each domain to be interpreted (0: top air layer, 1: sample, 2: bottom air layer), \(T_{i}\) is the temperature of the layer, \(k_{i}\) is the thermal conductivity of the layer, \(t\) is the time, and \(Q_{i}\) indicates the modulated heat source. Equation (5) is the heat source equation for each layer. Here, the light absorption coefficient (\(\lambda,\text{ cm}^{-1}\)) of copper is 623,100 cm\({}^{-1}\), \(a\)(m\({}^{2}\)) is the area of the pump beam, and \(x_{c}\) and \(y_{c}\) are the center coordinates of the pump beam, respectively, which are changed to implement laser scanning. \(y_{c}\) is fixed at 0 mm, and \(x_{c}\) is changed from 0 to 10 mm (0.1 mm intervals), as shown in Figure 4.  
\[\begin{array}{l}Q_{1}(x,y,z,t)=-\frac{P_{0}}{4\pi a^{2}}e^{-\lambda z}e^{- \left((x-x_{c})^{2}+(y-y_{c})^{2}\right)/a^{2}}e^{iwt}\\ Q_{0}(x,y,z,t)=0\\ Q_{2}(x,y,z,t)=0\\ \end{array} \tag{5}\]  
The specific parameters used in the numerical analysis are presented in Table 3. With the other parameters fixed, the numerical analysis was conducted with five different values of the first (copper) layer thickness (\(L_{f}\)) and four different values of the pump-beam radius (\(R_{b}\)).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Pump beam frequency, \(f\) (Hz)** & 80 & 1 step \\
**Thermal diffusion length, \(L_{d}\) (mm)** & 0.69 & 1 step \\
**First layer thickness, \(L_{f}\) (mm)** & 0.125–1.00 & 5 steps (equal intervals) \\
**Pump-beam radius, \(R_{b}\) (\(\mu\)m)** & 50–400 & 4 steps (equal intervals) \\
**Relative position, \(X_{0}\) (mm)** & 0.3 & 1 step \\ \(z\)-position of probe beam, \(L_{h}\) (mm) & 0.15 & 1 step \\ \hline \hline \end{tabular}
\end{table}
Table 3: Study parameters.  
Figure 7: Boundary conditions of the numerical analysis model.