## 2 System and Channel Models  
In this paper, we consider a DH FSO communication system with a relay that utilizes a DF-based relaying scheme. We also consider that the relay is harvesting energy from the received optical signal through SLIPT with a time-splitting technique. Both source-to-relay and relay-to-destination nodes use FSO communication links that undergo weather conditions (atmospheric attenuation), atmospheric turbulence, and pointing error effects. To explore the physical layer security of the proposed communication system, we also assume an eavesdropper that tries to intercept confidential information at the relay. We consider that the eavesdropper is located close to the legitimate relay in the divergence region of the laser beam, and that the eavesdropper can intercept enough optical power that can threaten the security of the transmitted information.  
We assume that, as illustrated in Figure 1, the source node converts digitally modulated information signal, \(\mathcal{X}_{m}(t)\), to an optical signal. A DC bias \(\mathcal{B}\in(\mathcal{B}_{\min},\mathcal{B}_{\max})\) is added to \(\mathcal{X}_{m}(t)\) to ensure a non-negative optical signal, where \(\mathcal{B}_{\min}\) and \(\mathcal{B}_{\max}\) are the minimum and maximum DC bias values, respectively. Let \(\mathcal{P}_{S}\) be the electrical power of the source node to transmit the optical signal \(\mathcal{S}(t)\). Then, \(\mathcal{S}(t)\) can be described as  
\[\mathcal{S}(t)=\sqrt{\mathcal{P}_{S}}\{\zeta\mathcal{X}_{m}(t)+ \mathcal{B}\}, \tag{1}\]  
where \(\zeta\) is the electrical-to-optical conversion coefficient at the source node. To avoid clipping due to the nonlinear behavior of the laser, \(\zeta\) is given as [34]  
\[\zeta\leq\min\left(\frac{\mathcal{B}-\mathcal{B}_{\min}}{v}, \frac{\mathcal{B}_{\max}-\mathcal{B}}{v}\right), \tag{2}\]  
where \(v\) is the power of \(\mathcal{X}_{m}(t)\). The electrical signal available at the output of the photo-detector (P-D) at the relay, \(y_{R}(t)\), and at the eavesdropper, \(y_{E}(t)\), can be described as  
\[y_{i}(t)=\eta_{i}\mathcal{R}_{i}A_{i}h_{i}\mathcal{S}(t)+n_{i}( t),\quad i\in\{R,E\}, \tag{3}\]  
where \(\eta_{i}\) is the optical-to-electrical conversion coefficient. \(\mathcal{R}_{i}\) and \(A_{i}\) are the responsivity and the detection area of the P-D, respectively. \(h_{i}\) is the channel gain of the FSO link, and \(n_{i}\) stands for a zero-mean additive white Gaussian noise (AWGN) with power \(\sigma_{i}^{2}\). Noises at the P-D are the summation of many transmitted-signal-independent random variables. In the case of FSO systems with wide-field-of-view P-Ds, noises are adopted as AWGN with zero mean and variance \(\sigma^{2}\). Please refer to [35] (Chapter 2) on more about the statistical modeling of the noises. Furthermore, the received signal at the destination node can be written as  
\[y_{D}(t)=\sqrt{\mathcal{P}_{R}}\eta_{D}\mathcal{R}_{D}A_{D}h_{ D}\zeta_{D}\mathcal{X}_{m}(t)+n_{D}(t), \tag{4}\]  
where \(\mathcal{P}_{R}\) is the transmit power at the relay node, \(\zeta_{D}\) is the electrical-to-optical conversion coefficient at the relay, \(\mathcal{X}_{m}\) is the information signal transmitted by the relay to the destination, and \(n_{D}\) is a zero-mean AWGN with power \(\sigma_{D}^{2}\).