## 4 Theoretical Analysis of the Influence of AL on Coherent/Non-Coherent OOK Reception
, \tag{27}\]\[i_{Q}(t)=i_{0c}(t)-i_{20t}(t)=s(t)R\sqrt{P_{S}P_{LO}}\sin(\Delta\omega t+\Delta\varphi)\] \[+R\sqrt{P_{LO}S_{aln}\delta\
u}\sum\limits_{i=-M}^{M}\sin[(\Delta \omega+2\pi i\delta\
u)t+\Delta\varphi_{ILO}]+n_{QC}(t)\enspace, \tag{28}\]  
where \(n_{lc}(t)=n_{0c}(t)-n_{180c}(t)\); \(n_{QC}(t)=n_{90c}(t)-n_{270c}(t)\). From these two equations, when the signal light contains AL and the OOK coherent receiving system reaches the shot noise limit, the system noise is mainly AL-LO beat noise and shot noise. The corresponding double sideband PSD of cosine function \(A\cos(2\pi f_{0}t+\theta)\) is expressed as  
\[S_{\cos}(f)=\frac{A^{2}}{4}\delta(f-f_{0})+\frac{A^{2}}{4}\delta(f+f_{0}), \tag{29}\]  
As the optical bandwidth of the AL is much larger than the detector electrical bandwidth, and the frequency difference between the light signal and the LO is small, the frequency component in AL that can generate beat current with signal light is symmetrical about the central frequency of the signal light. Similarly, the frequency component in AL that can generate beat current with LO is also symmetrical about the central frequency of the LO. Thus, the current expression of the AL-LO beat term of channel I can also be approximated as  
\[i_{aln-LO}(t)=R\sqrt{P_{LO}S_{aln}\delta\
u}\sum\limits_{i=-M}^{M}\cos[2\pi i \delta\
u t+\Delta\varphi_{ILO}], \tag{30}\]  
In Equation (30), each frequency has both positive and negative terms. Their phases are random and cannot be combined directly. They can only be the sum of the PSD. Thus, the SSPSD of the AL-LO beat noise of Channel I is expressed as  
\[S_{aln-LO}(f)=\frac{1}{2}R^{2}P_{LO}S_{aln}\times 2=R^{2}P_{LO}S_{aln}, \tag{31}\]  
The current variance can be calculated according to the SSPSD of the AL-LO beat noise and the noise bandwidth of the detector using Equation (32). \[\sigma_{Laln-LO}^{2}=R^{2}P_{LO}S_{aln}B_{e}, \tag{32}\]  
where \(B_{e}\) denotes the noise bandwidth of the detector. The current variance corresponding to the channel I shot noise is expressed as  
\[\sigma_{Is}^{2}=eR(s(t)P_{s}+S_{aln}B_{o}+P_{LO})B_{e}, \tag{33}\]  
where \(e\) is the electronic charge and \(B_{o}\) is the receiver optical bandwidth. In Equation (33), \(P_{LO}\) is far greater than the other two terms. The ratio of AL-LO beat noise current variance and shot noise current variance can be expressed as  
\[ratio=R^{2}P_{LO}S_{aln}B_{e}/(eR(s(t)P_{S}+S_{aln}B_{o}+P_{LO})B_{e})\approx RS _{aln}/e, \tag{34}\]  
There is only a phase difference of \(\pi/2\) between channels Q and I. Thus, the AL-LO beat noise current variance and shot noise in channel Q can also be expressed as Equation (34). The SNR expressions for channels I and Q are the same. Thus, the coherent OOK sensitivity loss expression introduced by AL noise can be expressed as follows in conditions dominated by LO shot noise. \[SNRloss_{co}=10\times\log 10\left(\frac{RS_{aln}+e}{e}\right), \tag{35}\]  
Equation (35) represents the loss of SNR when the AL enters the coherent OOK receiving system. Due to the subtraction function of the BDs, the noise introduced is mostly AL-LO beat noise. The influence of AL on a non-coherent OOK receiver was considered. In this study, we chose the EDFA, PIN, and optical filter to receive OOK signals non-coherently. The OOK signal light containing AL after passing through the EDFA can be expressed as  
\[E_{EDFA}(t)=\sqrt{G}\left(s(t)\sqrt{2P_{S}^{*}}\cos(\omega_{S}t+ \varphi_{S}(t))+\sum\limits_{i=-M}^{M}\sqrt{2S_{aln}\delta\
u}\cos[(\omega_{S}+ 2\pi i\delta\
u)t+\varphi_{i}]\right)\enspace, \tag{36}\] \[+R\sqrt{P_{LO}S_{aln}\delta\
u}\sum\limits_{i=-M}^{M}\sin[(\Delta \omega+2\pi i\delta\
u)t+\Delta\varphi_{ILO}]+n_{QC}(t)\enspace, \tag{28}\]0where \(G\) is the gain of the EDFA; \(S_{\text{size}}\) is the PSD of the ASE of the EDFA, considered to be flat within the optical bandwidth of the system; \(I\) represents the serial number of each frequency interval of the ASE; and \(\varphi_{l}\) represents the random phase at each frequency interval of the ASE.