## 5 Simulation Results  
The numerical results from Monte Carlo simulations are presented in this section to demonstrate the correctness of the analytical derivations and provide insights into how various link parameters affect the performance. All parameters of FSO links utilized here are obtained from [7]; the FSO link distance is 3500 m, and the wavelength is \(\lambda=1550\) nm. The parameters \(a\) and \(b\) for various turbulence conditions can be obtained using (2-4), as in [7]. More specifically, we consider three sets \((a,b)\) of the FSO link, (4.59, 7.09), (2.33, 4.53), and (1.43, 3.49), which correspond to the weak, medium, and strong turbulence conditions, respectively. Additionally, unless otherwise specified, the average SNRs of the two hops are equal in all circumstances (i.e., \(\bar{\mu}_{1}=\bar{\mu}_{2}=\bar{\mu}\)). Figure 2 shows the OP versus \(\bar{\mu}\) for both \(r=1\) and \(r=2\) under weak, moderate, and strong turbulence conditions to illustrate the consistency between the simulation and analytical results. As can be observed, the OP of the proposed system is higher when the IM/DD is used instead of HD. Additionally, increasing the turbulence degree results in higher OP. Figure 3 shows the OP performance of the dual-hop and single-link scenarios where the single link is split into two equal-distance links. As can be seen, due to the minimized pointing error impairments, using dual-hop FSO transmission links can greatly enhance system performance compared to a single FSO link in the medium- to high-SNR range. Figure 4 illustrates the performance of the proposed system with HD under moderate turbulence conditions for the pointing error conditions of \(z_{1}\), \(z_{2}\), and \(z_{3}\), which are 1.1, 1.8, and 4.4. Additionally, the average SNR of the first hop is fixed at two different values \(\beta_{1}\) of 20 and 36 dB. As expected, the OP performance improves when \(\tilde{\mu}_{1}\) increases. Furthermore, the OP performance at low \(\tilde{\mu}_{1}=20\) dB can be observed differently in two SNR regimes. When \(\beta_{2}\) is lower than 30 dB, the OP decreases as the SNR increases. In contrast, an error floor occurs when \(\beta_{1}>30\) dB, since the first link becomes dominant in this SNR regime. Figure 4: OP versus \(\tilde{\mu}_{2}\) for fixed values of \(\tilde{\mu}_{1}\). Figure 3: OP versus \(\tilde{\mu}\) for single FSO and dual-hop FSO links. Figure 2: OP versus \(\tilde{\mu}\) with different turbulence conditions. Figure 5 shows the EC performance for both detection modes in the cases of low, moderate, and high turbulence. As can be seen, strong turbulence conditions result in a lower EC value than those obtained under moderate and and weak turbulence conditions. In addition, the HD outperforms the IM/DD under the same channel conditions, and the degree of performance degradation under different channel conditions with the IM/DD is more distinguishable than for the HD. The BER of the proposed system is given in Figure 6 with different turbulence conditions and two modulation schemes OOK and BPSK. Similarly to the OP and EC performances, various atmospheric turbulences affect the BER performance significantly. It is also clear that the system with BPSK modulation can maintain a better performance than the OOK modulation. Finally, Figure 7 shows a Monte Carlo simulation of the BER performance of the proposed system under both \(\mathcal{F}\) and GG turbulence distributions with the impact of pointing errors. More specifically, we set \(a=2.60\) and \(b=3.84\) for the case of \(\mathcal{F}\) turbulence, while \(\alpha=3.00\) and \(\beta=3.00\) for the case of GG turbulence for a fair comparison [6]. In addition, the varying pointing error level is set, with the values of \(z\) being 1.1, 1.8, and 4.4 when \(z_{1}\) and \(z_{2}\) are the same and the value is \(z\). As can be seen, the performance of the proposed system with both distributions are the same, which means that the system performance is independent of the turbulence channel models.