# Experimental Study on Partially Coherent Optical Coherent Detection
## Theoretical Analysis
### Polarization States and Mixing Efficiency  
To investigate the impact of the polarization state of an EGSM beam on mixing efficiency, the Jones matrix is utilized to simulate the 90\({}^{\circ}\) spatial optical mixer principle. It compares the optical mixer's mixing efficiency in various polarization states. Figure 2 displays the optical mixer's schematic diagram [18]. A balanced heterodyne detection schematic diagram is shown in Figure 2. The local oscillator emits a laser beam that first travels through the polarizer P1 with an \(x\)-axis polarization direction. After that, it travels through a quarter wave plate, where it forms an angle to the \(x\)-axis of 45 degrees. The signal light laser beam travels through polarizer P2, which is in the same polarization direction as P1, and after passing through a half wave plate, it forms an angle difference of 22.5\({}^{\circ}\) with respect to the \(x\)-axis. Subsequently, the two light beams are divided into four beams via PBS1 and PBS2, respectively, after being  
Figure 2: Optical path diagram of a 90\({}^{\circ}\) optical mixer. combined through BS. The I and Q electrical signals can then be output using a balanced detector after the four light signals have formed a 90\({}^{\circ}\) phase difference. \(E_{s}\) stands for signal light and \(E_{LO}\) for local oscillator light. \(k_{1}\) and \(k_{2}\) represent the polarization component of signal light, \(k_{3}\) and \(k_{4}\) represent the polarization component of local polarized light. \(\varphi(t)\) represents the signal light's phase. \(\psi\) is the local oscillator light's phase. \(\tau_{\parallel}\) and \(\rho_{\perp}\) represent the phase factor generated after the polarization splitting prism, representing the change in phase. After calculation, the four output results are \(I_{0}\), \(I_{90}\), \(I_{180}\), and \(I_{270}\), respectively. \[I_{0}=\frac{1}{2}\left\{k_{1}^{2}|E_{s}|^{2}+k_{4}^{2}|E_{LO}|^{2}+2k_{1}k_{4} |E_{s}E_{LO}|\exp\left\{i\left[\varphi(t)+\tau_{\parallel}-\psi-\rho_{\perp}- \frac{\pi}{4}\right]\right\}\right\} \tag{19}\]  
\[I_{180}=\frac{1}{2}\left\{k_{2}^{2}|E_{s}|^{2}+k_{3}^{2}|E_{LO}|^{2}-2k_{2}k_{3 }|E_{s}E_{LO}|\exp\left\{i\left[\varphi(t)+\rho_{\perp}-\psi-\tau_{\parallel} -\frac{\pi}{4}\right]\right\}\right\} \tag{21}\]  
\[I_{270}=\frac{1}{2}\left\{k_{1}^{2}|E_{s}|^{2}+k_{4}^{2}|E_{LO}|^{2}-2k_{1}k_{4 }|E_{s}E_{LO}|\exp\left\{i\left[\psi+\rho_{\perp}+\frac{\pi}{4}-\varphi(t)- \tau_{\parallel}\right]\right\}\right\} \tag{22}\]  
Based on the four output photocurrents obtained above, the DC signal \(I_{I}\) and AC signal \(I_{Q}\) can be calculated for subsequent information restoration and phase locking, respectively. It is evident that, following passage through a PBS, the two light paths' polarization states (p- and s-polarization) separate, allowing the p-polarized light to be fully transmitted and the s-polarized light to be reflected at a 45-degree angle. Both the local oscillator light and the signal light's vertical and parallel components enter two branches, respectively, creating two mixed beams with perpendicular polarization directions. When \(\tau_{\parallel}\) and \(\rho_{\perp}\) satisfy \(\tau_{\parallel}-\rho_{\perp}=\pm\frac{\pi}{4}\), a phase delay of 90\({}^{\circ}\) can be achieved. The signal and local oscillator light's incident on the mixer can be expressed as follows, when the signal light is linearly polarized. \[E_{s}=E_{s}(x,y,z)\left( \begin{array}{c}k_{1}\\ k_{2}\end{array} \right)\exp(i\phi(t)) \tag{23}\]  
\[E_{LO}=E_{LO}(x,y,z)\left( \begin{array}{c}k_{3}\\ k_{4}\end{array} \right)\exp(i\varphi(t)) \tag{24}\]  
The signal light's parallel and vertical polarization components are denoted by \(k_{1}\) and \(k_{2}\), while the local oscillator light's parallel and vertical polarization components are represented by \(k_{3}\) and \(k_{4}\), which, respectively, fulfill the equations \({k_{1}}^{2}+{k_{2}}^{2}=1\) and \({k_{3}}^{2}+{k_{4}}^{2}=1\).