## 4 Theoretical Analysis of the Influence of AL on Coherent/Non-Coherent OOK Reception  
When the influence of AL on the OOK coherent receiver was considered, the total electric field of the OOK signal light and AL entering the coherent receiving system is expressed as  
\[E(t)=s(t)\sqrt{2P_{S}}\cos(\omega_{S}t+\varphi_{S}(t))+\sum\limits_{i=-(\bar{B}_{ o}/2\delta p)}^{(B_{o}/2\delta p)}\sqrt{2S_{aln}\delta v}\cos[(\omega_{S}+2\pi i \delta v)t+\varphi_{i}], \tag{17}\]  
where \(S_{aln}\)(W/Hz) is the PSD of the AL noise, which is a function of frequency and is considered to be the same in the optical bandwidth; \(B_{o}\) (Hz) is the FWHM of the optical filter, the FWHM of the optical filter is the same as the optical bandwidth in this study; \(\delta v\)(Hz) is the subdivided frequency interval; \(i\) represents the serial number of each frequency interval; \(\varphi_{i}\) represents the random phase at each frequency. The first term in Equation (17) is the signal light and the second term is the AL. In this study, it is considered that the PSD of the AL is flat within the optical bandwidth of the system. The LO light can be expressed as  
\[E_{LO}(t)=\sqrt{2P_{LO}}\cos(\omega_{LO}t+\varphi_{LO}(t)), \tag{18}\]  
At this time, the four light outputs of the optical 90\({}^{\circ}\) hybrid enter the balance detectors; all photogenerated currents can be expressed as follows after removing the DC term and sum frequency term. Compared with Equations (1) to (4), the change in the following expression is only because the AL is added to the signal light. \[i_{0}(t)=s^{2}(t)R\frac{P_{S}}{4}+i_{aln-aln}(t)+s(t)R\frac{\sqrt{P_{S}P_{LO}} }{2}\cos(\Delta\omega t+\Delta\varphi)+i_{aln-al}(t)+i_{aln-LO}(t)+n_{\text{ dc}}(t), \tag{19}\]  
\[i_{0}(t)=s^{2}(t)R\frac{P_{S}}{4}+i_{aln-aln}(t)+s(t)R\frac{\sqrt{P_{S}P_{LO}} }{2}\sin(\Delta\omega t+\Delta\varphi)+i_{S-aln}(t)+i_{aln-LO}(t)+n_{\text{ dc}}(t), \tag{20}\]  
\[i_{100}(t)=s^{2}(t)R\frac{P_{S}}{4}+i_{aln-aln}(t)-s(t)R\frac{\sqrt{P_{S}P_{LO} }}{2}\cos(\Delta\omega t+\Delta\varphi)+i_{S-aln}(t)-i_{aln-LO}(t)+n_{\text{ dc}}(t), \tag{21}\]  
\[i_{20}(t)=s^{2}(t)R\frac{P_{S}}{4}+i_{aln-aln}(t)-s(t)R\frac{\sqrt{P_{S}P_{LO} }}{2}\sin(\Delta\omega t+\Delta\varphi)+i_{S-aln}(t)-i_{aln-LO}(t)+n_{\text{ dc}}(t), \tag{22}\]  
The specific expressions in Equations (19)-(22) are expressed as  
\[i_{aln-aln}(t)=R\frac{S_{aln}\delta v}{4}\sum\limits_{i=-M}^{M}\sum\limits_{j= -M}^{M}\cos\left(2\pi(i-j)\delta vt+\varphi_{i}-\varphi_{j}\right), \tag{23}\]  
\[i_{S-aln}(t)=s(t)\frac{R\sqrt{P_{S}S_{aln}\delta v}}{2}\sum\limits_{i=-M}^{M} \cos[2\pi i\delta vt+\Delta\varphi_{iS}], \tag{24}\]  
\[i_{aln-LO}(t)=R\frac{\sqrt{P_{LO}S_{aln}\delta v}}{2}\sum\limits_{i=-M}^{M} \cos[(\Delta\omega+2\pi i\delta v)t+\Delta\varphi_{iLO}], \tag{25}\]  
\[i_{Qaln-LO}(t)=R\frac{\sqrt{P_{LO}S_{aln}\delta v}}{2}\sum\limits_{i=-M}^{M} \sin[(\Delta\omega+2\pi i\delta v)t+\Delta\varphi_{iLO}], \tag{26}\]  
where \(n_{0c}(t)\), \(n_{90c}(t)\), \(n_{180c}(t)\), and \(n_{270c}(t)\) are additive noise of the four current signals, respectively; \(i\) and \(j\) represent the serial number of each frequency interval, respectively; \(M=B_{o}/2\delta v\); \(\Delta\omega=\omega_{S}-\omega_{LO}\); \(\Delta\varphi=\varphi_{S}(t)-\varphi_{LO}(t)\); \(\Delta\varphi_{iS}=\varphi_{i}-\varphi_{S}(t)\); \(\Delta\varphi_{iLO}=\varphi_{i}-\varphi_{LO}(t)\). From Equations (19)-(22), it is observed that the AL-AL beat noise and the signal light-AL beat noise are common-mode noises. They can be effectively eliminated by subtraction. However, the AL-LO beat noise is a differential-mode noise, that cannot be reduced by subtraction. The current signals output by the I-channel BD and the Q-channel BD can be expressed as follows  
\[ \begin{split}& i_{l}(t)=i_{0}(t)-i_{180}(t)=s(t)R\sqrt{P_{S}P_{LO}} \cos(\Delta\omega t+\Delta\varphi)\\ &+R\sqrt{P_{LO}S_{aln}\delta v}\sum\limits_{i=-M}^{M}\cos[( \Delta\omega+2\pi i\delta v)t+\Delta\varphi_{iLO}]+n_{lc}(t)\end{split}