## 4 FSO System Design  
As shown in Figure 3, the output of MZM passes through a pre-amplifier with gain \(G_{1}\) and the received signal passes through a post-amplifier with gain \(G_{2}\). Let \(P_{in}\) and \(P_{out}\) be the outputs of the MZM and post-amplifier, respectively. They are related by  
\[P_{out}=G_{1}AG_{2}P_{in}, \tag{12}\]  
where \(A\) is the path loss due to FSO link. As shown in Figure 1, \(A\) is different for three different scenarios: (i) satellite-to-satellite (sat-to-sat), (ii) satellite-to-ground (downlink), and (iii) ground-to-satellite (uplink) communications. In this paper, we assumed that a Gaussian laser beam is used at the transmitter. The path loss for the sat-to-sat communications is [23; 24; 25; 26]:  
\[A_{Sat.-Sat.}=\frac{D_{T}^{2}D_{R}^{2}}{L^{2}\lambda^{2}}\eta_{T}T_{T}(1-L_{P} )T_{R}\eta_{R}, \tag{13}\]  
where \(L\) is the distance between transmitter and receiver, \(\lambda\) is the wavelength, and \(D_{T}\) and \(D_{R}\) are the lens diameters of transmitter and receiver telescopes, respectively. \(T_{T}\) and \(T_{R}\) are the transmission factors (\(\leq 1\)) of the telescopes. \(\eta_{T}\) and \(\eta_{R}\) are the transmitter and receiver efficiency, respectively. \(L_{P}\) is the pointing loss due to misalignment of transmitter and receiver. The path loss of downlink is [23; 24; 25; 26]  
\[A_{downlink}=A_{Sat.-Sat.}\times 10^{-\frac{A_{plan}}{10}}, \tag{14}\]  
where \(A_{atm}\) is the attenuation of the atmosphere in [dB]. The beam divergence due to the atmosphere for the downlink is ignored as it is negligible [21]. For the uplink, the attenuation due to turbulence and beam divergence is [20; 21; 22; 23],  
\[A_{uplink}=\frac{D_{R}^{2}}{L^{2}\big{(}\theta_{T}^{2}+\theta_{atm}^{2}\big{)} }\eta_{T}T_{T}(1-L_{P})T_{R}\eta_{R}\times 10^{-\frac{A_{atm}}{10}}, \tag{15}\]  
where the divergence angle due to telescope at the transmitter is [23; 25]  
\[\theta_{T}=\frac{\lambda}{D_{T}}, \tag{16}\]  
and additional divergence due to turbulence is  
\[\theta_{atm}=\frac{\lambda}{r_{0}}, \tag{17}\]  
where \(r_{0}\) is the _Fried parameter_[23]. Let \(NF_{1}\) and \(NF_{2}\) be the noise figures of the pre- and post-amplifiers, respectively. The effective noise figure is [6]  
\[NF_{eq}=NF_{1}+\frac{NF_{2}}{G_{1}A}-\frac{1}{G_{1}}. \tag{18}\]  
Typically, \(G_{1}>>1\) and the last term in Equation (18) can be ignored. From Equation (18), we see that if \(G_{1}A<<1\) (which is typically the case), then the effective the noise figure is mainly determined by \(NF_{2}\) (i.e., first term on the right side of Equation (18) is also negligible compared to the second term) and, hence, an amplifier with relatively low noise figure should be used as the post-amplifier. Assuming that the noise of the optical amplifier and background noise to be white, then the noise power spectral density per polarization is  
\[\rho=\rho_{ASE,eq}+\rho_{bg}, \tag{19}\]  
where \(\rho_{bg}\) is the power spectral density (PSD) of background radiation noise and \(\rho_{ASE,eq}\) is the equivalent PSD of the amplified spontaneous emission (ASE) noise given by [6]  
\[\rho_{ASE,eq}=\big{(}G_{eq}NF_{eq}-1\big{)}\frac{hf}{2}, \tag{20}\]  
\[G_{eq}=G_{1}AG_{2}, \tag{21}\]  
where \(h\) is [PERSON]'s constant, and \(f=c/\lambda\) where \(c\) is the speed of light. The main source of background radiation is the sun, and its PSD [3,4] is,  
\[A_{Sat.-Sat.}=\frac{D_{T}^{2}D_{R}^{2}}{L^{2}\lambda^{2}}\eta_{T}T_{T}(1-L_{P} )T_{R}\eta_{R}, \tag{13}\]0  
with  
\[A_{Sat.-Sat.}=\frac{D_{T}^{2}D_{R}^{2}}{L^{2}\lambda^{2}}\eta_{T}T_{T}(1-L_{P} )T_{R}\eta_{R}, \tag{13}\]1  
and  
\[A_{Sat.-Sat.}=\frac{D_{T}^{2}D_{R}^{2}}{L^{2}\lambda^{2}}\eta_{T}T_{T}(1-L_{P} )T_{R}\eta_{R}, \tag{13}\]2  
where \(A_{eff}\) is the area of source that can be seen by lens of receiver telescope, \(k\) is [PERSON]'s constant, \(\lambda_{s}\) is the background radiation source wavelength at temperature \(T_{se}\), and \(L_{s}\) is distance between receiver and background radiation source.