## 2 Principle
Thus, the sub-harmonic single-sideband (SH-SSB) modulation of the LO signal is achieved. The optical field of the lower arm can be expressed as follows:  
\[E_{Y-QPSK}=\frac{\sqrt{2}\eta_{E_{\text{int}}}}{4}\left[ \begin{array}{l}\exp(jm_{0}(\sin\omega_{LO}t)+\exp(-jm_{0}(\sin\omega_{LO}t )\\ +j[\exp(jm_{0}(\sin\omega_{LO}t+\frac{\pi}{4})+\exp(-jm_{0}(\sin\omega_{LO}t+ \frac{\pi}{4}))\end{array} \right]\\ =\frac{\sqrt{2}\eta_{E_{\text{int}}}}{4}\left[ \begin{array}{l}2J_{2}(m_{0}) \exp(-j\omega_{LO}t)+2J_{2}(m_{0})\exp(j2\omega_{LO}t)+2J_{0}(m_{0})\\ +j[\begin{array}{l}2J_{2}(m_{0})\exp(-j(2\omega_{LO}t+\frac{\pi}{2}))\\ +2J_{2}(m_{0})\exp(j(2\omega_{LO}t+\frac{\pi}{2}))+2 jJ_{0}(m_{0})\end{array} \right]\\ = \frac{\sqrt{2}\eta_{E_{\text{int}}}}{2}[2J_{2}(m_{0})\exp(-j2\omega_{LO}t)+( 1+j)J_{0}(m_{0})]\end{array} \right] \tag{2}\]  
where \(m_{0}=\pi V_{LO}/2V_{\pi}\) represents the modulation index of the LO signal. \(V_{\pi}\) denotes the half-wave voltage of the modulator. The output signals from the upper and lower arms are orthogonally combined through PBC. The orthogonally combined, polarization-multiplexed optical signal can be expressed as follows:  
\[E_{DP-QPSK}=E_{X-QPSK}\vec{e}_{x}+E_{Y-QPSK}\vec{e}_{y} \tag{3}\]  
where \(e_{x}\) and \(e_{y}\) are mutual orthogonal basis vectors. Then, the polarization-multiplexed optical signal is amplified by an erbium-doped fiber amplifier (EDFA). To avoid power saturation of the photodiode (PD), the EDFA often works in an automatic power control (APC) mode. The amplified optical signals are then transmitted to the CO through a length of single-mode fiber (SMF). When a signal transmits through a dispersive device, the phase of the signal will be changed. Generally, the dispersive fiber-induced phase shift can be expressed as \(\theta=\beta L\), where \(\beta\) denotes the propagation constant and L is the length of the fiber. Expanding \(\beta\) in the Taylor series, we have the following:  
\[\theta=\beta_{0}(\omega_{c})L+L\beta_{1}(\omega_{c})(\omega-\omega_{c})+\tfrac {1}{2}L\beta_{2}(\omega_{c})(\omega-\omega_{c})^{2}+... \tag{4}\]  
where \(\beta_{m}(\omega_{c})\) represents the m-order derivatives of \(\beta\) with respect to the optical angular frequency. \(\beta_{2}(\omega_{c})=-2\pi cD(\omega_{c})/\omega_{c}^{2}\), where c is the speed of light, and D is the dispersion coefficient at \(\omega_{c}\). As the third and higher derivatives of \(\beta\) are small, they are ignored. Thus, when the nth-order optical sideband--whose angular frequency is \(\omega_{c}+n\Omega\)--transmits over a dispersive fiber, the dispersion-induced phase shift can be expressed as follows:  
\[\theta=\beta_{0}(\omega_{c})L+L\beta_{1}(\omega_{c})(n\Omega)+\tfrac{1}{2}L \beta_{2}(\omega_{c})(n\Omega)^{2} \tag{5}\]  
It can be perceived from the above equations that the dispersion-induced phase shift is only related to the frequency and the length of the fiber. Therefore, after transmitting over a dispersive link, the signal in Formula (3) can be rewritten as follows:  
\[ \begin{split}& E_{SMF}=\underbrace{\sqrt{2}\eta E_{\text{int}}(t) \left[\begin{array}{c}-J_{0}(m_{RF})J_{1}(m_{IM})\exp(-j\omega_{IM}t)\exp(j \theta_{1})-\\ J_{1}(m_{RF})J_{0}(m_{IM})\exp(-j\omega_{RF}t)\exp(j\theta_{2})\end{array} \right]}_{\text{E}_{\text{Y}}\text{--QPSK}}\vec{e}_{y}\\ &\underbrace{\frac{\sqrt{2}\eta E_{\text{int}}}{2}[2J_{2}(m_{0})\exp(-j \omega_{\text{LO}}t)\exp(j\theta_{3})+(1+j)J_{0}(m_{0})\theta_{0}]}_{\text{E} _{\text{Y}}\text{--QPSK}}\vec{e}_{y}\\ &\text{where}\left\{\begin{array}{c}\theta_{0}=\beta_{0}(\omega_{c})L\\ \theta_{1}=\beta_{0}(\omega_{c})L+\beta_{1}(\omega_{c})(\omega_{IM})L+\tfrac{1} {2}\beta_{2}(\omega_{c})(\omega_{IM})^{2}L\\ \theta_{2}=\beta_{0}(\omega_{c})L+\beta_{1}(\omega_{c})(\omega_{RF})L+\tfrac{1} {2}\beta_{2}(\omega_{c})(\omega_{RF})^{2}L\\ \theta_{3}=\beta_{0}(\omega_{c})L+\beta_{1}(\omega_{c})(2\omega_{LO})L+\tfrac{1 }{2}\beta_{2}(\omega_{c})(2\omega_{LO})^{2}L\end{array}\right.\end{split}