## 3 Analytical Calculation of the EOFM Signal on the Diode  
Using all previous equations, we can now calculate the EOFM signal of a \(P^{+}N\) junction at \(300K\). Considering the wide application of the 1319 nm laser wavelength in the industry [11], we also used a 1319 nm laser wavelength in our calculations. The bandwidth \(\Delta\lambda=2\) nm of the laser source used in the experiment was also taken into account in our calculations. The calculated structure was a diode between the base and collector regions of the transistor \(Q_{1}\) in the CA3083 chip from Intersil. The relative parameters of the diode can be found in the literature [26]. For the reverse biased junction, the device was modeled with five layers as depicted in Figure 4. In Figure 4, the reverse biased diode is simplified to a structure with six interfaces, which are the air/substrate, substrate/SCR, SCR/collector, collector/SCR, SCR/base, and base/contact.  
During the modeling process, the \(P^{+}N\) junction was treated as an abrupt junction, where the SCR was completely depleted (depletion approximation). The SCR thickness between base and collector depends on the applied voltage; consequently, the different reflectivity values of the DUT can be obtained if the different voltage levels are applied. Additionally, the average doping distribution substituted for the actual doping distribution in this model. The average of the base region (\(P^{+}\)-type) doping concentration was set to \(5.279\times 10^{17}\) cm\({}^{-3}\), the collector region (\(N\)-type) doping concentration was set to \(3.141\times 10^{14}\) cm\({}^{-3}\), and the substrate (\(P\)-type) doping concentration was set to \(6.054\times 10^{12}\) cm\({}^{-3}\). At the last interface, the reflectivity \(R_{c}\) was calculated using Equation (15).  
Figure 5 gives the reflectivity variation \(\Delta R\) of the device as a function of the laser wavelength when the applied voltage \(V_{p}=4V\). The signal in Figure 5 is defined as the relative reflectivity variation, \(|\Delta R/R_{0}|\), expressed in parts per million (ppm), because the reflectivity variation value is very weak. This is also the reason why the EOFM signal of the device is very difficult to detect and requires special techniques in signal processing [27]. From Figure 5, the reflectivity variation, i.e., the EOFM signal, increases with the increase in laser wavelength. This is consistent with the phenomenon found in the experiment [14]. The calculation results in Figure 5 show why the 1319 nm laser wavelength is widely utilized in the industry [11].  
The EOFM signal \(V_{out}\) of the diode calculated for different reverse bias voltages applied is shown in Figure 6. When calculating \(V_{out}\) using Equation (24), the incident light power \(P\) was considered to be 11.48 mW, and the parameters of the detector used in the experiment were also employed. The responsivity \(\Re\) of the photodiode was set to \(0.85A/W\), and the transimpedance gain \(G\) of the photodiode was set to \(2.38\times 10^{3}V/A\). We can see from Figure 6 that the EOFM signal of the diode increases with the value of the reverse bias voltage. This is caused by the fact that the SCR thickness increases with the applied reverse bias voltage.  
Figure 4: Simplified reverse biased diode structure for the calculation.