## 2 Detecting Principle
### Theoretical Model
\[\alpha_{fc}=2.2\frac{q^{3}\lambda^{2}}{4\pi^{2}c^{3}\epsilon_{0}n_{0}}\left( \frac{\Delta N_{e}}{m_{ce}^{+}{}^{2}\mu_{e}}+\frac{\Delta N_{h}}{m_{ch}^{+}{ }^{2}\mu_{h}}\right) \tag{11}\]  
where \(\mu_{e}\) and \(\mu_{h}\) are the electron and hole mobilities, respectively. The electron and hole mobilities in silicon are a function of the doping concentration and temperature. Many analytical expressions for calculating mobility can be found in the literature. For the calculations in this work, Equations (12) and (13) are used. Some  
Figure 1:0 Schematic of the multilayer system. \(E_{0}^{+}\) is the electricity vector of the incident light, and \(E_{0}^{-}\) is the electricity vector of the reflected light. parameters in Equation (12) are modified to make the calculations consistent with the experimental data [20,23]:  
Figure 1:1  
Figure 1:2  
where \(T_{n}=T/300\) is the normalized temperature, and \(N\) is the free carrier's density. By using Equations (4)-(13), we can calculate the complex refractive index of each layer in Figure 1:3 Then \(E_{0}^{+}\) and \(E_{0}^{-}\) are obtained by the Fresnel coefficient matrix method [24]:  
\[\left[ \begin{array}{c}E_{0}^{+}\\ E_{0}^{-}\end{array} \right]=\frac{1}{t_{1}t_{2}\cdots t_{l+1}}\left[ \begin{array} []{cc}1&\rho_{1}\\ \rho_{1}&1\end{array} \right]\prod_{j=1}^{l}\left[ \begin{array}{cc}e^{i\delta _{j}}&\rho_{j+1}e^{i\delta_{j}}\\ \rho_{j+1}e^{-i\delta_{j}}&e^{-i\delta_{j}}\end{array} \right]\left[ \begin{array} []{c}E_{l+1}^{+}\\ 0\end{array} \right] \tag{14}\]  
where \(\rho_{1}=\frac{\widetilde{n_{0}}-\widetilde{n_{1}}}{\widetilde{n_{0}}+ \widetilde{n_{1}}}\) and \(t_{1}=\frac{2\widetilde{n_{0}}}{\widetilde{n_{0}}+\widetilde{n_{1}}}\) are the Fresnel coefficients of interface 1, and \(\delta_{1}=\frac{2\pi}{\lambda}\widetilde{n_{1}}\,d_{1}\) is the phase factor. In Figure 2, the reflection coefficients \(\rho\) of all interfaces except the last one can be calculated by the complex refractive index \(\widetilde{n}\). The reflectivity of the last interface in Figure 2 is contributed by each elementary part of the laser spot coverage (see Figure 3), and we can not directly calculate the reflection coefficient \(\rho_{l+1}\) using the complex refractive index. To solve this problem, we first calculate the reflectivity \(R_{c}\) of the last interface, and we use the obtained reflectivity to calculate the reflection coefficient \(\rho_{l+1}\). Taking into account the Gaussian distribution of the laser intensity, the reflectivity of the last interface is expressed as the sum of the reflectivity of each part of the laser coverage multiplied by the Gaussian distribution profile of the laser intensity and the ratio of each part area to the laser spot area:  
\[R_{c}=\sum_{j=1}^{H}R_{j}\frac{2e^{2}S_{j}}{(e^{2}-1)\pi^{2}\omega_{0}^{4}} \int\int_{x_{1j},y_{1j}}^{x_{2j},y_{2j}}e^{-\frac{2x_{1}^{2}+y_{1}^{2}}{\omega _{0}^{2}}}dxdy \tag{15}\]  
where \(H\) is the total number of structures covered by the laser beam, \(S_{j}\) is the area of each part, \(\left(x_{1j},y_{1j}\right)\) and \(\left(x_{2j},y_{2j}\right)\) are the spatial coordinates of each part, and \(\omega_{0}\) is the spot radius of the focused laser. Figure 3: Illustration of the laser spot covering many elementary parts. After \(R_{c}\) is obtained by Equation (15), the reflection coefficient \(\rho_{l+1}\) at the last interface in Figure 2 can be calculated as shown in the following formula:  
\[\rho_{l+1}=\sqrt{R_{c}} \tag{16}\]  
All the parameters in Equation (14) are now given in detail, so the total reflection coefficient \(\rho\) of the multilayer system in Figure 2 can be obtained through Equation (14):  
\[\rho=\frac{E_{0}^{+}}{E_{0}^{-}}=\frac{C}{A} \tag{17}\]  
Therefore, the total reflectivity \(R\) of the multilayer system is  
\[R=|\rho|^{2} \tag{18}\]  
The above analysis process neglects the spectral line width of the laser, which should be taken into account when using the transfer matrix [25].