## 2 Detecting Principle
### Theoretical Model  
Because the phase modulation of the reflected light is converted to amplitude modulation through interference, a detector was used to detect the optical power of the reflected light in the experiment [16,19]. The optical power of the incident light is known in the experiment, so the EOFM signal of the DUT can also be expressed as  
\[R\big{(}V_{p}\big{)}=\frac{P_{ref}}{P_{inc}}=R_{0}+\Delta R\big{(}V_{p}\big{)} \tag{3}\]  
where \(R\big{(}V_{p}\big{)}\) is the reflectivity of the DUT, \(P_{inc}\) and \(P_{out}\) are, respectively, the incident and reflected laser powers, \(R_{0}\) is the static reflectivity of the DUT without power, and \(\Delta R(V_{p})\) is the change in reflectivity due to the external voltage on the DUT, namely the EOFM signal of the DUT. Equation (3) shows that the EOFM signal of the DUT is essentially caused by the variation of the reflectivity of the DUT with the applied voltage. Therefore, we only need to build a physical model to calculate the reflectivity of the DUT, and then the corresponding EOFM signal values of the DUT under different voltages can be obtained using the proposed model. The detailed model-building process is given below. Semiconductor devices usually contain several differently doped regions. In order to provide even greater simplicity for the model, the device is simplified to a multilayer structure consisting of multiple parallel interfaces as depicted in Figure 2[15,20]. The incident light, \(E_{0}^{+}\), is reflected and transmitted at each interface and partially absorbed as it propagates through each layer of the medium. Finally, all reflected light interferes with each other to form the total reflected light, \(E_{0}^{-}\). The propagation process of the laser in the device is similar to the propagation process of the light in the multilayer film system. Therefore, the transfer matrix method commonly used to study the reflection properties of thin film systems is applied to the model. With this method, it is possible to calculate the total reflectivity \(R\) in multilayer structures [15,18]. Figure 1: Cross-sectional diagram of MOSFET. The optical properties of each layer in Figure 2 are characterized by the thickness \(d\) and the complex refractive index \(\widetilde{n}\). \(\widetilde{n}\) can be expressed as follows :  
\[\widetilde{n}=n-ik \tag{4}\]  
where \(n\) is the absolute refraction index of silicon, which consists of the refraction index of intrinsic silicon \(n_{0}\) and the free carrier term \(n_{fc}\) given by Equation (1), and \(k\) is the extinction coefficient. \(n\) and \(k\) are calculated as shown:  
\[n=n_{0}+n_{fc} \tag{5}\]  
\[k=\frac{\alpha\lambda}{4\pi} \tag{6}\]  
where \(\alpha\) is the absorption coefficient of silicon, which can be calculated as shown in the following formula:  
\[\alpha=\alpha_{0}+\alpha_{fc} \tag{7}\]  
For the absorption coefficient of intrinsic silicon \(\alpha_{0}\), we use experimental data on intrinsic silicon from the literature [21]. The refraction index of intrinsic silicon \(n_{0}\) in Equation (5) is a function of laser wavelength \(\lambda\) and temperature \(T\)[22]:  
\[n_{0}=\left[\varepsilon(T)+\frac{e^{-3\Delta L(T)}}{\lambda^{2}}\Big{(}0.894 8+4.3977\times 10^{-4}T+7.3835\times 10^{-8}T^{2}\Big{)}\right]^{0.5} \tag{8}\]  
with  
\[\varepsilon(T)=11.445+2.7739\times 10^{-4}T+1.705\times 10^{-6}T^{2}-8.1347 \times 10^{-10}T^{3} \tag{9}\]  
Furthermore, for \(293K<T<1600K\)  
\[\Delta L(T)=-0.071+1.887\times 10^{-6}T+1.934\times 10^{-9}T^{2}-4.554\times 10^{- 13}T^{3} \tag{10}\]  
In this paper, the temperature change caused by the laser thermal effect can be ignored owing to the small laser power. Therefore, we consider the temperature \(T=300K\). Although the free carrier term, \(\alpha_{fc}\), in Equation (7) can be calculated by Equation (2), in some cases the results calculated by Equation (2) deviate from the experimental data. Therefore, a fitting parameter is introduced in Equation (11) to better agree with the experimental values [20].