## 2 Operation and Principle
The output signal from the AOM can be represented as follows:  
\[E_{AOM}(t)=\frac{E_{0}}{\sqrt{2}}\exp\left[j\Big{(}\omega_{c}-\omega_{f_{s}} \Big{)}t\right] \tag{3}\]  
After the frequency shift, the optical carrier enters a DPMZM and is modulated by two local oscillator (LO) signals, \(V_{LO1}(t)=V_{LO1}\sin(\omega_{LO1}t)\) and \(V_{LO2}(t)=V_{LO2}\sin(\omega_{LO2}t)\)  
Figure 2: Spectral diagram of channel reception; **(a)**: before channelization, **(b)** after channelization. Lets denote the amplitudes of LO\({}_{1}\) and LO\({}_{2}\) to be \(V_{LO1}\) and \(V_{LO2}\), respectively, and their angular frequencies as \(\omega_{LO1}\) and \(\omega_{LO2}\), respectively. Both sub-Mach-Zehnder modulators in the upper and lower arms of the DPMZM operate at the minimum transmission point to perform carrier-suppressed double-sideband (CS-DSB) modulation, while the main modulator operates at the maximum transmission point. Assuming that the higher-order optical sidebands with small amplitudes are negligible, the output optical signal from the DPMZM can be represented as follows:  
\[ \begin{array}{ll}E_{\rm DPMZM}(t)&=\frac{E_{0}(t)}{4}\exp\left[j\left(\omega _{\rm c}-\omega_{f_{S}}\right)t\right]\\ &\cdot\left[\begin{array}{l}I_{1}(m_{LO1})\exp(j\omega_{LO1}t)+J_{-1}(m_{LO 1})\exp(-j\omega_{LO1}t)\\ +J_{1}(m_{LO2})\exp(j\omega_{LO2}t)+J_{-1}(m_{LO2})\exp(-j\omega_{LO2}t)\\ \end{array}\right]\end{array}  \tag{4}\]  
From Equation (4), it is evident that the DPMZM outputs two sets of dual-line optical combs with different FSRs. Utilizing two wavelength division multiplexers (WDMs) with identical parameters, WDM\({}_{1}\) and WDM\({}_{2}\), the positive and negative first-order optical sidebands relative to the optical carrier are separated for both the upper and lower paths. The separated RF copy signals and LO signals then enter the same dual-polarization image rejection receiver (DP-IRR) for signal demodulation. The channel reception spectrum is illustrated in Figure 2. It should be noted that at this point, half of the channel requires a simple spectral inversion in the digital domain. Below, we will proceed with the derivation using the WDM's channel 1 as an example, with the understanding that the demodulation process for other channels is identical to that of channel 1. As can be seen from Figure 2a, channel 1 of the WDM can filter out the LO signal at frequency \(\omega_{\rm c}-\omega_{f_{S}}-\omega_{LO2}\), as well as several adjacent spectral components that are centered around the RF signal frequencies 7 and 10. For ease of analysis, in the following derivation, the complete RF signal will represent these spectral components of the RF signal. Adjusting PC\({}_{3}\) such that the optical signal output from WDM\({}_{1}\) forms a 45\({}^{\circ}\) angle with the principal axis of PBS\({}_{1}\) results in PBS\({}_{1}\) outputting two RF copy signals with different polarization states, namely the X-polarized and Y-polarized signals. The signals output from PBS\({}_{1}\) can be represented as follows:  
\[E_{\rm PBS_{1}}(t)=\left[ \begin{array}{c}E_{\rm PBS_{2}-X}(t)\\ E_{\rm PBS_{2}-Y}(t)\\ \end{array} \right]=\left[ \begin{array}{c}E_{0-POLM_{1}}(t)\cdot exp[-j( \omega_{\rm c}-\omega_{RF})t]\\ E_{-1-POLM_{1}}(t)\cdot exp[j(\omega_{\rm c}-\omega_{RF})t]\\ \end{array} \right] \tag{5}\]  
By the same principle, adjusting PC\({}_{4}\) ensures that the optical signal output from WDM\({}_{2}\) is aligned at a 45\({}^{\circ}\) angle to the principal axis of PBS\({}_{2}\). At this angle, PBS\({}_{2}\) outputs polarization-multiplexed signals that can be represented as follows:  
\[