## 2 Modeling
### Model Calibration  
This section describes the calibration of the proposed model. The basic idea is to make the form of \(q_{\text{total}}^{{}^{\prime\prime}}(x)\) match the distribution of IR rays. In this case, an IR power meter can be used to capture the heat flux of an IR heater. The IR distribution data can be obtained by measuring the IR intensity from the center to the end of the entrance length at different positions. The used IR power meter and details of the heat flux measurement process are described in Appendix A. The measured data were normalized using the lamp power. For the calibration, it is convenient to define a normalized energy distribution probability \((q_{\text{norm}}^{{}^{\prime\prime}}(x))\) and normalizing coefficient \((K_{n})\).  
\[q_{\text{norm}}^{{}^{\prime\prime}}(x)=\frac{q_{\text{total}}^{{}^{\prime\prime }}(x)}{\langle\psi|\hat{H}|\psi\rangle}=\sum_{n=1}^{\infty}\frac{n^{2}\pi^{2} \hbar^{2}}{ml^{3}}K_{n}^{2}\sin^{2}\frac{n\pi x}{l}, \tag{13}\]where:  
\[\sum_{n=1}^{\infty}K_{n}^{2}=\frac{1}{\langle\psi|\hat{H}|\psi\rangle}\text{ and }c_{n}=\frac{K_{n}}{\sqrt{\sum_{n=1}^{\infty}K_{n}^{2}}}.\]  
Equation (13) sets the mass of the imaginary energy particle to be \(m=\frac{n^{2}\hbar^{2}}{\hbar^{3}}\), then \(q_{\text{norm}}^{{}^{\prime\prime}}(x)\) becomes:  
\[q_{\text{norm}}^{{}^{\prime\prime}}(x)=\sum_{n=1}^{\infty}n^{2}K_{n}^{2}\sin^{ 2}\frac{n\pi x}{l}. \tag{14}\]  
The normalizing coefficient, \(K_{n}\), is used to calibrate the model. Because \(q_{\text{norm}}^{{}^{\prime\prime}}(x)\) is normalized by the total energy flux of the lamp \(\langle\psi|\hat{H}|\psi\rangle\), the probability distribution condition is represented as below:  
\[\int_{0}^{l}q_{\text{norm}}^{{}^{\prime\prime}}(x)dx=1. \tag{15}\]  
\(q_{\text{norm}}^{{}^{\prime\prime}}(x)\) can be calibrated by the normalized measured data using the least square method [34], and the calibrated values of \(K_{n}\) and \(C_{n}\) for focal length 20 mm and 40 mm cases are summarized in Table 1. In this work, the 1 st, 2 nd, and 3 rd eigenmodes were used because it was confirmed that using the three modes is sufficient to capture the data well in the calibration. In Figure 4a,b, the normalized measurement data and calibrated models for the focal length of 20 mm and 40 mm are presented; the model follows the IR distribution well. The 20 mm focal length (Figure 4a) results in a narrower concentration of energy than the 40 mm focal length case (Figure 4b). Details are discussed in the discussion section. From the calibrated \(q_{\text{norm}}^{{}^{\prime\prime}}\) in Figure 4, the total heat flux distribution, \(q_{\text{total}}^{{}^{\prime\prime}}(x)\), can be obtained by considering \(I_{0}\) based on Equation (12) for each focal length. The total heat flux distributions, \(q_{\text{total}}^{{}^{\prime\prime}}(x)\), for the focal lengths of 20 mm and 40 mm are shown in Figure 5a,b, respectively. \(q_{\text{total}}(x)\) in Equations (10) and (11) can be implemented into a FEM code for electrical-thermal simulations of the focused IR heating conditions.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{6}{c}{**Normalizing Coefficient**} & \multicolumn{3}{c}{**Component Coefficient**} \\ \hline Focal Length \(f\) (mm) & \(K_{1}\) & \(K_{2}\) & \(K_{3}\) & \(\left|C_{1}\right|^{2}\) & \(\left|C_{2}\right|^{2}\) & \(\left|C_{3}\right|^{2}\) \\ \hline
20 & \(2.224\times 10^{-1}\) & \(5.041\times 10^{-3}\) & \(3.226\times 10^{-2}\) & 0.979 & 0 & 0.021 \\
40 & \(2.008\times 10^{-1}\) & \(6.794\times 10^{-5}\) & \(2.277\times 10^{-2}\) & 0.987 & 0 & 0.013 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Calibration of the model for the IR heater.  
Figure 4: Calibration of the new model: (**a**) Focal length 20 mm; (**b**) Focal length 40 mm.