## 3 Noise Analysis and Optimizations of Photoelectric Detection Circuit  
### Noise Analysis of Photoelectric Detection Circuit  
The equivalent output noise spectral density of the photoelectric detection circuit in Figure 3 can be modeled as shown in Figure 4, where \(R_{i}\) is the equivalent parallel resistance of the photodiode, \(C_{i}\) is the parallel value of the junction capacitance of the photodiode. Besides, the circuit noise sources in Figure 4 consist of the photocurrent shot noise density \(I_{ns}\), the dark current shot noise density \(I_{nd}\), the thermal noise density of feedback resistor \(I_{nr}\), the equivalent input noise voltage density of operational amplifier \(E_{hop}\) and the equivalent input noise current density of operational amplifier \(I_{hop}\). Due to the large parallel resistance of photodiodes, the thermal noise generated by the resistance can be ignored compared to the photocurrent and dark current shot noise level, so it is not considered in this paper. The operating range of typical magnetometers for geomagnetic applications is 10,000 nT to 100,000 nT [20; 21; 22]. The frequency range of the Larmor signal for cesium magnetometer is 35 kHz to 350 kHz. The 1/f noise of the amplifier is not considered either.  
The noise voltage density produced by \(I_{ns}\) at the output of the amplifier is  
\[E_{ns}(f)=I_{ns}Z_{f}(f)=\sqrt{2 el_{\mathrm{DC}}}Z_{f}(f), \tag{16}\]  
where, \(e\) is the electron charge.  
The noise voltage density produced by \(I_{nd}\) at the output of the amplifier is  
\[E_{nd}(f)=I_{nd}Z_{f}(f)=\sqrt{2 el_{d}}Z_{f}(f), \tag{17}\]  
Figure 4: Equivalent noise model of photoelectric detection circuit.  
\(I_{d}\) is the dark current of the photodiode.  
The noise voltage density produced by the feedback resistor at the output is  
\[E_{nr}(f)=I_{nr}Z_{f}(f)=\sqrt{\frac{4 kT}{R_{f}}}Z_{f}(f), \tag{18}\]  
where \(k\) is the Boltzmann constant and \(T\) is the temperature in Kelvin.  
The noise voltage density produced by the equivalent input noise current of the operational amplifier at the output is  
\[E_{niop}(f)=I_{nop}Z_{f}(f), \tag{19}\]  
The noise voltage density produced by the equivalent noise voltage of the operational amplifier at the output is  
\[E_{nop}(f)=E_{nop}\bigg{(}1+\frac{Z_{f}(f)}{Z_{i}(f)}\bigg{)}, \tag{20}\]  
where  
\[Z_{i}(f)=\frac{R_{i}}{1+j2\pi fR_{i}C_{i}}, \tag{21}\]  
As the above five noises Equations (16)-(20) are uncorrelated, the total output noise voltage density is  
\[E_{n0}(f)=\sqrt{E_{ns}^{2}+E_{nd}^{2}+E_{nr}^{2}+E_{nop}^{2}+E_{nop}^{2}}. \tag{22}\]  
From Equations (15) to (22), the output SNR \(R_{sn}\) with the noise expressed per unit bandwidth of the photoelectric detection circuit can be obtained by  
\[R_{sn}(f)=\frac{\left|V_{R}(f)\right|}{E_{n0}(f)}=\frac{1}{\sqrt{R_{s}^{2}(f) +R_{d}^{2}(f)+R_{r}^{2}(f)+R_{iop}^{2}(f)+R_{cop}^{2}(f)}}, \tag{23}\]  
where  
\[R_{s}(f)=\sqrt{\frac{2 eP_{DC}}{P_{R}^{2}\eta}}, \tag{24}\]  
\[R_{d}(f)=\sqrt{\frac{2 el_{d}}{P_{R}^{2}\eta^{2}}} \tag{25}\]  
\[R_{r}(f)=\sqrt{\frac{4 kT}{R_{f}P_{R}^{2}\eta^{2}}} \tag{26}\]  
\[R_{iop}(f)=\frac{I_{nop}}{P_{R}\eta} \tag{27}\]  
\[R_{cop}(f)=\frac{E_{nop}}{P_{R}\eta}\bigg{|}\frac{1}{Z_{f}(f)}+\frac{1}{Z_{i} (f)}\bigg{|}=\frac{E_{nop}}{P_{R}\eta}\frac{\left|R_{i}+R_{f}+j2\pi fR_{i}R_{f }\big{(}C_{i}+C_{f}\big{)}\right|}{R_{i}R_{f}}. \tag{28}\]  
It can be seen from above that the output SNR of the photoelectric detection circuit is related to the sensitivity of photodiode, dark current, feedback resistance, amplifier equivalent noise voltage, amplifier equivalent noise current, input resistance, input capacitance, etc. According to Equations (23) and (24), it can be found that an enhanced photo sensitivity \(\eta\) can improve the SNR of the system. From Equations (27) and (28), reducing the equivalent noise voltage and noise current of the amplifier can improve the SNR. However, noise voltage and noise current are inversely related to each other, so it is very important to choose a proper amplifier.