## 2 Operation and Principle
\begin{array}{ll}E_{\rm PBS_{2}}(t)&=\left[\begin{array}{c}E_{\rm PBS_{2}- X}(t)\\ E_{\rm PBS_{2}-Y}(t)\\ \end{array}\right]=\left[\begin{array}{c}\cos\left(\frac{\pi}{4}\right)\\ \sin\left(\frac{\pi}{4}\right)\\ \end{array}\right]E_{\rm WDM-channel1}(t)\\ &=\frac{E_{0}}{4}\left[\begin{array}{c}J_{-1}(m_{LO2})exp\left[j\left( \omega_{\rm c}-\omega_{f_{S}}-\omega_{LO2}\right)t\right]\\ J_{-1}(m_{LO2})exp\left[j\left(\omega_{\rm c}-\omega_{f_{S}}-\omega_{LO2} \right)t\right]\\ \end{array}\right]\end{array}  \tag{6}\]  
The four optical signals from PBS\({}_{1}\) and PBS\({}_{2}\) enter a Dpol-QOH, and the output signals can be represented as follows:  
\[E_{\rm QOH-X}(t)=\frac{E_{0}}{8}\left[\begin{array}{l}2J_{-1}(m_{RF})exp[-j( \omega_{\rm c}-\omega_{RF})t]+J_{-1}(m_{LO2})exp\left[j\left(\omega_{\rm c}- \omega_{f_{S}}-\omega_{LO2}\right)t\right]\\ 2J_{-1}(m_{RF})exp[-j(\omega_{\rm c}-\omega_{RF})t]-J_{-1}(m_{LO2})exp\left[j \left(\omega_{\rm c}-\omega_{f_{S}}-\omega_{LO2}\right)t\right]\\ 2J_{-1}(m_{RF})exp[-j(\omega_{\rm c}-\omega_{RF})t]+jJ_{-1}(m_{LO2})exp\left[j \left(\omega_{\rm c}-\omega_{f_{S}}-\omega_{LO2}\right)t\right]\\ 2J_{-1}(m_{RF})exp[-j(\omega_{\rm c}-\omega_{RF})t]-jJ_{-1}(m_\[E_{QHC-Y}(t)=\frac{E_{0}}{8}\left[ \begin{array}{c}2J_{-1}(m_{RF})exp[j(\omega_{c} -\omega_{RF})t]+J_{-1}(m_{LO2})exp\left[j\left(\omega_{c}-\omega_{fs}-\omega_{ LO2}\right)t\right]\\ 2J_{-1}(m_{RF})exp[j(\omega_{c}-\omega_{RF})t]-J_{-1}(m_{LO2})exp\left[j\left( \omega_{c}-\omega_{fs}-\omega_{LO2}\right)t\right]\\ 2J_{-1}(m_{RF})exp[j(\omega_{c}-\omega_{RF})t]+jJ_{-1}(m_{LO2})exp\left[j\left( \omega_{c}-\omega_{fs}-\omega_{LO2}\right)t\right]\\ 2J_{-1}(m_{RF})exp[j(\omega_{c}-\omega_{RF})t]-jJ_{-1}(m_{LO2})exp\left[j\left( \omega_{c}-\omega_{fs}-\omega_{LO2}\right)t\right]\end{array} \right] \tag{8}\]  
The four I/Q signals output after BPD can be represented as follows:  
\[ \begin{array}{l}i_{I(t)-X}\propto\frac{1}{4}\eta E_{0}^{2}J_{-1}(m_{RF})J_{- 1}(m_{LO1})\Big{\{}\cos[\left(\omega_{RF-U}-\omega_{fs}+\omega_{LO1}\right)t ]+\cos[\left(\omega_{fs}-\omega_{LO1}-\omega_{RF-L}\right)t]\Big{\}}\\ i_{Q(t)-X}\propto\frac{1}{4}\eta E_{0}^{2}J_{-1}(m_{RF})J_{-1}(m_{LO1})\{\sin[ \left(\omega_{RF-U}-\omega_{FS}+\omega_{LO1}\right)t]-\sin[\left(\omega_{FS}- \omega_{LO1}-\omega_{RF-L}\right)t]\}\\ i_{I(t)-Y}\propto\frac{1}{4}\eta E_{0}^{2}J_{-1}(m_{RF})J_{-1}(m_{LO1})\{\cos[ \left(\omega_{RF-U}-\omega_{FS}+\omega_{LO1}\right)t]+\cos[\left(\omega_{FS}- \omega_{LO1}-\omega_{RF-L}\right)t]\}\\ i_{Q(t)-Y}\propto\frac{1}{4}\eta E_{0}^{2}J_{-1}(m_{RF})J_{-1}(m_{LO1})\{\sin[ \left(\omega_{RF-U}-\omega_{FS}+\omega_{LO1}\right)t]-\sin[\left(\omega_{FS}- \omega_{LO1}-\omega_{RF-L}\right)t]\}\end{array}  \tag{9}\]  
Here, \(\eta\) is the photodetector's responsivity, \(\omega_{RF-U}\) represents the part of the RF signal spectrum with frequencies higher than the LO signal frequency, and \(\omega_{RF-L}\) represents the part of the RF signal spectrum with frequencies lower than the LO signal frequency. At this stage, the four I/Q signals still have spectral aliasing. In the X-polarized state, the LO\({}_{2}\) signal with a frequency of \(\omega_{c}-\omega_{fs}-\omega_{LO2}\), after down-conversion, outputs a mixed signal of the RF signal spectral components 8 and 6, and in the Y-polarized state, it outputs a mixed signal of the RF signal spectral components 9 and 11. The image rejection receiver with a Hartley structure can be utilized to achieve a double output of the image-rejected mixed signals [16]; the principle of image rejection is not elaborated in this paper. Therefore, the four frequency-different LO signals can ultimately demodulate 16 sub-channels, thus achieving the complete reception of the RF signal.