## 3 Noise Analysis and Optimizations of Photoelectric Detection Circuit
### Optimized Design of Photoelectric Detection Circuit  
According to the analysis in the previous section, a higher sensitivity of photodiode can result in a better SNR of the system. The wavelength in the Cs-OPM is about 894 nm. Therefore, the silicon photodiode Hamamatsu S2387-66R is used. Its sensitivity is about 0.55 A/W for the 894 nm wavelength source. Its main parameters are listed in Table 1.  
The cesium lamp and the atomic gas-cell determine the DC power of the incident light and the signal power at Larmor frequency. In general, the DC power of incident light (\(P_{DC}\)) is several hundred microwatts, and the signal power at Larmor frequency (\(P_{R}\)) is about several microwatts. In the model for our case, \(P_{DC}\) is 185 \(\mu\)W, and \(P_{R}\) is 2.4 \(\mu\)W. The feedback resistance with an optimized value is selected according to the sensitivity of the photodiode, the value of the photocurrent and the supply voltage of the circuit. From the parameters in Equation (9) and in Table 1, one can calculate the value of the DC photocurrent. It yields 0.1 mA. Thus, it should select a feedback resistance \(R_{f}\) of 100 k\(\Omega\) when the supply voltage is 12 V. According to the feedback resistance, the parameters of the amplifier and the input capacitance, the feedback capacitance \(C_{f}\) of 100 pF is chosen. When the light power and photodiode are fixed, it is very important to choose a proper amplifier. Table 2 shows several typical low-noise amplifiers with different equivalent voltage and current noise spectral density values. In the following section, the influence of amplifier noise on the total output noise will be calculated. By substituting the circuit parameters in Table 1 and the amplifier noise parameters in Table 2 into Equations (22) and (23), the total output noise and SNR of the circuit with different amplifiers can be obtained. The corresponding model results are shown in Figure 5. It can be seen that in the low-frequency domain, the circuit output noise with different amplifiers shows similar noise levels. However, by increasing the frequency, the difference gradually increases, and the circuit with a lower equivalent noise voltage has better performance. The circuit with LT1028 as the amplifier has the best noise performance and SNR in the full working frequency band. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Sensitivity (@ \(\lambda\)=894 nm), \(\eta\) & 0.55 A/W \\ Terminal capacitance, \(C_{i}\) & 4300 pF \\ Shunt resistance, \(R_{s}\) & 10 G\(\Omega\) \\ Dark current, \(l_{d}\) & 50 pA \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of S2387-66R. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **AD549** & **AD822** & **AD745** & **LT1028** \\ \hline equivalent input noise voltage (nV/\(\sqrt{\text{Hz}}\)) & 35 & 13 & 2.9 & 0.85 \\ equivalent input noise current (fA/\(\sqrt{\text{Hz}}\)) & 0.11 & 0.8 & 6.9 & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Typical low-noise amplifiers. The noise characteristic is further studied. Taking LT1028 as an example, the noise components of the circuit is modelled and analyzed, and the model results are shown in Figure 6. Due to the large light power, the photocurrent shot noise has a major contribution to the total noise outputs. At low frequencies, the circuit noise is mainly determined by the photocurrent shot noise. As the frequency increases, the amplifier equivalent noise voltage gradually dominates. From Figure 6, we can find that in the low frequency band, the effect of equivalent noise current component is greater than noise voltage source of the amplifier. In the high frequency band, it is the opposite. If the effect of photocurrent shot noise is not considered, for low magnetic field measurement, the corresponding Larmor signal is located in the lower frequency band, so the amplifier with low equivalent noise current should be selected.