## 2 Principle
 \tag{6}\]  
In the CO, the optical signal is split into two channels (I and Q channels) by an optical splitter. In each channel, the orthogonally combined, polarization-multiplexed signals are aligned into one polarization direction after going through a PC and a Pol. The optical fields of channel 1 and channel 2 can be expressed as follows:  
\[[E_{1}\propto[\text{E}^{\prime}_{\text{X}\text{--QPSK}}\cos(\alpha)+\text{E} ^{\prime}_{\text{Y}\text{--QPSK}}\text{sin}(\alpha)\text{exp}(\text{j}\varphi _{1})] \tag{7}\]  
\[[E_{2}\propto[\text{E}^{\prime}_{\text{X}\text{--QPSK}}\cos(\gamma)+\text{E} ^{\prime}_{\text{Y}\text{--QPSK}}\text{sin}(\gamma)\text{exp}(\text{j}\varphi _{2})] \tag{8}\]  
where \(\alpha\) and \(\gamma\) are the polarization azimuths. \(\varphi_{1}\) and \(\varphi_{2}\) denote the phase differences between the two orthogonal polarization states introduced by the PCs of channel 1 and channel 2, respectively. The phase differences can be tuned from \(0^{\circ}\) to \(360^{\circ}\). In order to maximize the power of the combined signals, \(\alpha\) and \(\gamma\) are both set to \(45^{\circ}\). Then, the two linearly polarized signals are sent to two low-speed PDs for photoelectric conversion, respectively. Since the PDs have lower responsiveness at high frequencies, the RF and LO sidebands at the output of the PD can be neglected. The photocurrents at the IF frequency bands of the two paths are given by the following:  
\[ \begin{split}& I_{1}=\Re E_{1}\cdot E_{1}^{*}\\ &\propto 23\eta^{2}E_{0}^{2}\left\{\begin{array}{c}-J_{0}(m_{RF})J_{1 }(m_{IM})J_{2}(m_{0})cos[(\omega_{IM}-2\omega_{LO})t+\varphi_{1}+\theta_{3}- \theta_{1}]\\ -J_{1}(m_{RF})J_{0}(m_{IM})J_{2}(m_{0})cos[(\omega_{RF}-2\omega_{LO})t+\varphi _{1}+\theta_{3}-\theta_{2}]\end{array}\right\}\end{split}  \tag{9}\]\[I_{2}=\Re E_{2}\cdot E_{2}^{*} \tag{10}\] \[\propto 2\Re\eta^{2}E_{0}^{2}\left\{ \begin{array}{l}-J_{0}(m_{RF})I_{1 }(m_{IM})I_{2}(m_{0})\cos[(\omega_{IM}-2\omega_{LO})t+\varphi_{2}+\theta_{3}- \theta_{1}]\\ -I_{1}(m_{RF})J_{0}(m_{IM})I_{2}(m_{0})\cos[(\omega_{RF}-2\omega_{LO})t+ \varphi_{2}+\theta_{3}-\theta_{2}]\end{array} \right\}\]  
where \(\Re\) denotes the responsivity of the PDs. From Formulas (9) and (10), we can see that a two-channel photonic microwave phase shifter is realized. Moreover, the phase shift angles induced by dispersion do not affect the amplitude of the beating electrical signal (only the phase). Therefore, our proposed scheme can resist the dispersion-introduced power fading effect. In order to build a pair of quadrature downconverted channels, the PCs in the two channels should be independently and continuously adjusted. In channel 1, the PC does not introduce phase difference, namely \(\varphi_{1}=0\). In channel 2, the PC should be adjusted to introduce a \(90^{\circ}\) phase difference between the two polarization-multiplexed optical signals. Thus, the in-phase and quadrature IF signals can be obtained at the outputs of channel 1 and channel 2, respectively, which means that I/Q mixing is successfully realized. The photocurrents of the I and Q paths can be written as follows:  
\[I_{I}\propto 2\Re\eta^{2}E_{0}^{2}\left\{ \begin{array}{l}-J_{0}(m_{ RF})I_{1}(m_{IM})I_{2}(m_{0})\cos[(2\omega_{LO}-\omega_{IM})t+\theta_{3}- \theta_{1}]\\ -I_{1}(m_{RF})J_{0}(m_{IM})I_{2}(m_{0})\cos[(\omega_{RF}-2\omega_{LO})t+\theta _{3}-\theta_{2}]\end{array} \right\} \tag{11}\] \[I_{Q}\propto 2\Re\eta^{2}E_{0}^{2}\left\{ \begin{array}{l}-J_{0}(m_{ RF})I_{1}(m_{IM})I_{2}(m_{0})\sin[(2\omega_{LO}-\omega_{IM})t+\theta_{3}- \theta_{1}]\\ +I_{1}(m_{RF})J_{0}(m_{IM})I_{2}(m_{0})\sin[(\omega_{RF}-2\omega_{LO})t+ \theta_{3}-\theta_{2}]\end{array} \right\} \tag{12}\]  
In order to achieve image rejection, a low-frequency \(90^{\circ}\) EHC is connected to the two photodiode outputs. After combining the two photocurrents at a low-frequency \(90^{\circ}\) EHC, the target RF-induced IF signals remain in phase while the two image-induced IF signals are out of phase.