RIS-Aided Mixed RF-FSO Wireless Networks: Secrecy Performance Analysis With Simultaneous Eavesdropping

In order to meet the demands of diverse services within sixth-generation networks across a range of industries, numerous approaches are being employed to address notable signal degradation resulting from channel obstruction, particularly in the realms of millimeter wave and sub-THz frequencies. One of these solutions is the utilization of reconfigurable intelligent surfaces (RISs), which can reflect or refract signals in the desired direction. This integration offers significant potential to improve the coverage area from a transmitter to a receiver. In this paper, we present a comprehensive framework for analyzing the secrecy performance of an RIS-aided mixed radio frequency (RF)-free space optics (FSO) system employing in terms of physical layer security (PLS). It is assumed that a secure message is transmitted from a RF transmitter to a FSO receiver through an intermediate relay. The RF link experiences Rician fading while the FSO link experiences Málaga distributed turbulence with pointing errors. Three different eavesdropping scenarios are examined: 1) RF-link eavesdropping, 2) FSO-link eavesdropping, and 3) simultaneous eavesdropping attack on both RF and FSO links. We evaluate the secrecy performance using analytical expressions to compute secrecy metrics such as the average secrecy capacity, secrecy outage probability, probability of strictly positive secrecy capacity, effective secrecy throughput, and intercept probability. Our results are confirmed via Monte-Carlo simulations and demonstrate that fading parameters, atmospheric turbulence conditions, pointing errors, and detection techniques play a crucial role in enhancing secrecy performance.

SECTION I. Introduction

A. Background and Literature Study

As the beyond fifth generation (B5G) and sixth generation (6G) of wireless communication have emerged, reconfigurable intelligent surfaces (RISs) to address the negative impacts of wireless channels have been explored as one of the most crucial technologies [1], [2], [3], [4], [5]. To create a truly intelligent environment, there is a significant strategy to control the wireless medium, i.e., signal reflection direction, in an intentional way [6]. To address this need, an RIS has been developed using passive components that can be programmed and managed through an RIS controller allowing it to reflect signals toward specific directions as required [7]. Furthermore, the mixed radio frequency (RF)-free space optical (FSO) systems are considered potential structures for the next-generation wireless networks [8]. The use of RISs in both RF and FSO transmissions can help solve signal blockage issues that arise in wireless communication channels.

FSO communications are seen as a promising option that can provide fast data transmission speeds and be applied in a range of scenarios such as serving as a backup to fiber, supporting wireless networks for backhauling, and aiding in disaster recovery efforts [9]. However, they are vulnerable to pointing errors and atmospheric conditions and are not suitable for transmitting information over long distances. Through the implementation of relaying strategy, the dual-hop RF-FSO mixed models merge the strengths of RF and FSO communication technologies [10], [11]. In [12], the authors demonstrated how pointing errors, atmospheric turbulence, and path loss affect a mixed FSO-RF system and provided insights for improving the design and operation of such systems. The authors of [13] derived analytical expressions for the outage probability (OP), average data rate, and ergodic capacity (EC) of the RF-FSO systems, and assessed the performance in the presence of multiple users with various data rate requirements. Recently, the authors of [14], [15], [16], and [17] enhanced the dual-hop performance by optimizing the system parameters and made it suitable for space-air-ground integrated networks.

There has been a lot of research in the literature where single RIS-aided systems have been investigated [18], [19], [20], [21], [22], [23], [24]. In [18], the accuracy and effectiveness of RIS-assisted systems in modifying wireless signals were evaluated by assuming practical factors such as phase shift and amplitude response that can affect their performance. In [19], it was demonstrated that RISs could improve system performance over a Nakagami-$m$ fading channel by examining signal-to-noise ratio (SNR) and channel capacity. The findings of the study also provide insights into how to optimize RIS-empowered communications in practical scenarios. The system performance of an RIS-aided network is assessed by the authors of [20] wherein they suggested that the number of reflecting elements used in the network does not affect the diversity gain. On the other hand, the system performance of an RIS-aided dual-hop network was analyzed in [25], [26], [27], [28], and [29]. For example, the authors of [25] conducted a study comparing RIS-equipped RF sources and RIS-aided RF sources, and suggested that mixed RF-FSO relay networks utilizing these two types of sources offer great potential for enhancing the performance of wireless communication networks in various environments, both indoors and outdoors. In [26], it was concluded that incorporating RIS in mixed FSO-RF systems can greatly enhance the coverage area. This is achieved by improving the signal quality and reducing the signal attenuation that may occur during transmission. However, it was observed that RISs can help mitigate the impact of interference from nearby channels in dual-hop communication systems with co-channel interference [27]. The authors of [28] proposed a study on the effect of different system parameters, including the number and placement of RIS elements, on the performance of an RIS-assisted communication system. Here, the authors concluded that the most effective RIS configuration for optimal performance depends on the specific communication scenario and network requirements. In [30], a RIS-assisted FSO-RF mixed model with hybrid automatic repeat request techniques was proposed where the OP and packet error rate was derived in closed form to evaluate the system performance.

Wireless communications face a significant challenge in terms of protecting the privacy of information because their inherent characteristics make them vulnerable to security threats [31]. Till date, the security of wireless communication has relied on different encryption and decryption techniques that take place in the higher levels of the protocol stack [32]. Newly suggested physical layer security (PLS) methods are now seen as a practical solution to stop unauthorized eavesdropping in wireless networks by utilizing the unpredictable nature of time-varying wireless channels [33], [34], [35], [36], [37], [38]. Recently, extensive research has been conducted to explore the secrecy performance of mixed RF-FSO systems. The authors of [39] concluded that using a mixed model offers better security compared to using RF or FSO technology alone, and they emphasized the importance of implementing appropriate security measures and techniques. Another study in [40] examined the secrecy performance of a mixed RF-FSO relay channel with variable gain while [41] provided insights into the secrecy performance of a cooperative relaying system and emphasized the significance of selecting suitable statistical models and security techniques. In [42], the authors examined the trade-off between security and reliability in a DF-based FSO-RF system, demonstrating that employing receiver diversity leads to improved secrecy performance. The reference [43] presents an analysis of the secrecy performance in a scenario where an RF backhaul system is augmented with a parallel FSO communication link to enhance data transmission security. In [44], the authors demonstrated a mixed FSO-RF cooperative system that takes simultaneous eavesdropping into account. Their findings revealed that a low pointing error and heterodyne detection (HD) at the receiver results in improved performance. Additionally, the authors in [45] evaluated the performance of the mixed RF-FSO system with a wireless-powered friendly jammer and analyzed the impact of different system parameters on secrecy performance. On the other hand, some challenges and limitations associated with the dual-hop model were identified in [46] including the impact of atmospheric turbulence on the FSO link’s performance and the importance of accurate channel estimation. Finally, a new model for the mixed RF-FSO channel was presented in [47] that takes into account arbitrary correlation, and the results showed that both correlation and pointing error could significantly affect the secure outage performance of the model. Although a lot of research has looked into the investigation of PLS analysis due to the RF-FSO mixed systems [39], [40], [41], [42], [43], [44], [45], [46], [47], the potential of RIS to improve confidentiality in wireless networks has not been extensively studied in the context of RIS-assisted RF-FSO systems. It is worth noting that recent research has explored the RIS-aided model, even in the context of the single hop only, which significantly differs from our proposed system. Many of these studies have focused on utilizing Rayleigh distributions [6], [48] and Nakagami-$m$ distributions [49] to assess secrecy performance. However, there are also studies such as [50] that have investigated the PLS of a non-orthogonal multiple access (NOMA)-based visible light communication-RF mixed system where the authors looked into the influence of RIS on secrecy performance. Furthermore, a different perspective was taken in [51], where the authors analyzed a high-altitude platform-based RF-FSO model. In this case, they considered Nakagami-$m$ and Rayleigh distributions for the RF link, and they assumed a Gamma-Gamma distribution for the FSO link.

B. Motivation and Contributions

Although RIS-aided mixed RF-FSO systems are strong contenders for upcoming B5G and 6G wireless networks and their diverse applications, there has been limited investigation into their capacity to maintain secrecy in the available literature. The current literature mainly focuses on mixed RF-FSO systems and does not fully investigate the security performance of RIS-assisted RF-FSO systems from a perspective of physical layer security (PLS) particularly when RIS is used in both links [52]. In this paper, we conduct a PLS analysis of the RIS-aided RF-FSO system configuration and evaluate its secrecy performance under the simultaneous influence of RF and FSO eavesdropping attacks, which, to the best of the authors’ knowledge, has not been inspected before for this type of configuration. In addition, since wireless channels experience frequent variation over time, assuming a Rician channel in the RF links would provide a more realistic environment to model the wireless propagation perfectly [53]. Meanwhile, the Málaga fading distribution applied to the FSO link in the system being examined produces reliable results, particularly in challenging atmospheric turbulence and pointing error scenarios [47]. Motivated by these advantages, we introduce a secure wireless network over the Rician-Málaga mixed RF-FSO fading channel model. The key contributions of this research are as follows:

• In the past few decades, numerous studies have investigated the secrecy performance of mixed RF-FSO systems, such as [39], [40], [41], [45], [46], [47], [54], and [55]. However, the secrecy performance of dual-hop systems that incorporate both RF and FSO links aided by RISs remains an open concept, with no research conducted on this specific configuration up to date. In this paper, we propose an RIS-aided mixed RF-FSO network in the presence of two different eavesdroppers accounting for their ability to intercept information transmitted through both RF and FSO links. It is important to highlight that while there has been a recent investigation into the secrecy performance of the RIS-assisted model [6], [48], [49], [50], [51], their studies are significantly different from our proposed structure in terms of both the system model and the statistical characteristics being considered.

• Firstly, we obtain the cumulative distribution function (CDF) of the dual-hop RF-FSO system under decode-and-forward (DF) relaying protocols by utilizing the CDF of each link. Furthermore, we develop the new analytical expressions of average secrecy capacity (ASC), the lower bound of secrecy outage probability (SOP), the probability of strictly positive secrecy capacity (SPSC), effective secrecy throughput (EST), and intercept probability (IP). These expressions are novel compared to the previous works because the proposed model is completely different from the existing RF-FSO literature.

• The derived expressions are utilized to obtain numerical results with specific configurations. Furthermore, we confirm the precision of the analytical results through Monte-Carlo (MC) simulations. This validation through simulation strengthens the reliability of our analysis.

• In an effort to increase the practicality of our analysis, we provide insightful remarks that shed light on the design of secure RIS-aided mixed RF-FSO relay networks. To ensure a more realistic analysis, we take into account the major impairments and features of both RF and FSO links. For instance, we incorporate the impacts of fading parameters and the number of reflecting elements for RF links, as well as atmospheric turbulence, detection techniques, and pointing error conditions for FSO links.

C. Organization

The paper is organized into several sections. Section II provides an introduction to the models of the system and channel that are utilized in the study. In Section III, the paper presents analytical expressions for five significant performance metrics including ASC, SOP, and the probabilities of SPSC, EST, and IP. Section IV provides enlightening discussions and numerous numerical examples. Finally, Section V serves as the conclusion to the paper.

SECTION II. System Model and Problem Formulation

As depicted in Fig. 1, we present the system model of a RIS-aided combined RF-FSO DF-based relaying system where a RIS-RF system forms the first hop and the second hop is composed of a RIS-FSO system. Since a source, denoted by $\mathcal {S}$ (ground control station, smartphone, etc), and a relay, denoted by $\mathcal {R}$ (tall building), cannot communicate directly due to obstructions, $\mathcal {S}$ link to $\mathcal {R}$ through a RIS, denoted $\mathcal {I}_{P}$ , mounted on a structure. Similarly, communication between $\mathcal {R}$ and a destination, denoted by $\mathcal {D}$ (satellite ground station, smartphones, drone, etc), is established through another RIS, i.e., $\mathcal {I}_{Q}$ ). A RIS works as an intermediate medium between $\mathcal {S}$ and $\mathcal {R}$ with a view to ensuring a line-of-sight path between two nodes. In practical applications, RF-FSO mixed systems offer versatile solutions to address specific communication challenges and enhance performance. The mixed models find use in industrial sensor networks, aerial surveillance drones, secure military communication, hybrid data centers, and internet-of-things (IoT)-enabled smart buildings, showcasing the adaptability and benefits of combining RF and FSO technologies to meet diverse communication needs. The unauthorized users, which are known as eavesdroppers, denoted by $\mathcal {E}$ (smartphone, drone, etc), attempt to intercept the confidential information that is transmitted from $\mathcal {S}$ to $\mathcal {D}$ . Based on the position of eavesdroppers, three different scenarios are considered where the eavesdropper attempts to overhear the communication between $\mathcal {S}$ and $\mathcal {D}$ via $\mathcal {R}$ .

• In Scenario-I (Fig. 1a), the eavesdropper, $\mathcal {E}_{P}$ , utilizes the RF link for wiretapping and both $\mathcal {R}$ and $\mathcal {E}_{P}$ attain analogous signal propagated from $\mathcal {I}_{P}$ .

• The Scenario-II (Fig. 1b) infers the eavesdropper, $\mathcal {E}_{Q}$ , at the FSO link and both $\mathcal {D}$ and $\mathcal {E}_{Q}$ obtain the resembling propagated signal from $\mathcal {I}_{Q}$ .

• In Scenario-III (Fig. 1c), the eavesdroppers $\mathcal {E}_{P}$ and $\mathcal {E}_{Q}$ both attempt concurrently to overhear the confidential information from both the RF and FSO links.

This model describes a passive eavesdropping scenario assuming that the RIS is oblivious to the CSI of eavesdroppers. Herein, $\mathcal {S}$ and $\mathcal {E}$ are equipped with a singular antenna, while $\mathcal {R}$ acts as a transceiver. $\mathcal {D}$ comprises a single photo-detector for optical wave reception, while $\mathcal {I}_{P}$ and $\mathcal {I}_{Q}$ have $\mathcal {N}_{1}$ and $\mathcal {N}_{2}$ reflecting elements, respectively. The surface RF networks using $\mathcal {S}-\mathcal {I}_{P}-\mathcal {R}$ and $\mathcal {S}-\mathcal {I}_{P}-\mathcal {E}_{P}$ links pursue the Rician fading distribution. $\mathcal {R}$ serves to convert the obtained RF signal and redirect it as optical signal to $\mathcal {D}$ in the presence of $\mathcal {E}_{Q}$ . Both the FSO links, $\mathcal {R}-\mathcal {I}_{Q}-\mathcal {D}$ and $\mathcal {R}-\mathcal {I}_{Q}-\mathcal {E}_{Q}$ experience Málaga turbulence with pointing error aided by an RIS, $\mathcal {I}_{Q}$ .

FIGURE 1.

System model of a combined RIS-aided dual-hop RF-FSO system with source ($\mathcal {S}$ ), relay ($\mathcal {R}$ ), the destination user ($\mathcal {D}$ ), and eavesdroppers ($\mathcal {E}_{P}$ and $\mathcal {E}_{Q}$ ).

A. SNRs of Individual Links

For Scenario-I, $h_{{s_{p}}}$ ($s_{p}=1, 2, \ldots, {\mathcal {N}_{1}}$ ) indicates the first hop channel gain between $\mathcal {S}$ and $\mathcal {I}_{P}$ in both $\mathcal {S}-\mathcal {I}_{P}-\mathcal {R}$ and $\mathcal {S}-\mathcal {I}_{P}-\mathcal {E}_{P}$ links. Similarly, $g_{{s_{p}}}$ and $n_{{s_{p}}}$ indicate the channel gains of the second hop from $\mathcal {I}_{P}$ to $\mathcal {R}$ and $\mathcal {E}_{P}$ , respectively. Hence, the signals at $\mathcal {R}$ and $\mathcal {E}_{P}$ are expressed by $$\begin{align*} y_{s,r}&=\left [{\sum _{s_{p}=1}^{\mathcal {N}_{1}} h_{{s_{p}}}e^{j\Phi _{s_{p}}}g_{{s_{p}}}}\right]x+w_{1}, \tag{1}\\ y_{s,e}&=\left [{\sum _{s_{p}=1}^{\mathcal {N}_{1}} h_{{s_{p}}}e^{j\Psi _{s_{p}}}n_{{s_{p}}}}\right]x+w_{2}, \tag{2}\end{align*}$$ respectively. For the channels of each particular link, we have $$\begin{align*} h_{{s_{p}}} & =\alpha _{{s_{p}}}e^{j\varrho _{s_{p}}} \\ g_{s_{p}} & =\beta _{s_{p}}e^{j\vartheta _{s_{p}}} \\ n_{{s_{p}}} & =\eta _{s_{p}}e^{j\delta _{s_{p}}}\end{align*}$$ where $\alpha _{s_{p}}$ , $\beta _{s_{p}}$ , and $\eta _{s_{p}}$ are the Rician distributed random variables (RVs), $\varrho _{{s_{p}}}$ , $\vartheta _{s_{p}}$ , and $\delta _{s_{p}}$ are the resembling phases of received signal gains. Moveover, $\Phi _{s_{p}}$ and $\Psi _{s_{p}}$ identifies the phases emanated by the $s_{p}$ -th reflecting element of the RIS. In this work, we consider $\Phi _{s_{p}}\in [0,2\pi$ ), $\Psi _{s_{p}}\in [0,2\pi$ ), and the range of reflection on the assembled fortuitous signal present at the $s_{p}$ -th element is deliberated as 1. The conveyed data from $\mathcal {S}$ is represented in this scenario by $x$ with the power $S_{s}$ and $w_{1}\sim \mathcal {\widetilde {M}}(0,M_{r})$ , $w_{2}\sim \mathcal {\widetilde {M}}(0,M_{e})$ are the additive white Gaussian noise (AWGN) samples with $M_{r}$ , $M_{e}$ indicating the power of noise for the relevant networks. Mathematically, (1) and (2) are expressed as $$\begin{align*} y_{s,r}& =\textbf {g}^{T}\Phi \,\textbf {h}\,x+w_{1}, \tag{3}\\ y_{s,e}& =\textbf {n}^{T}\Psi \,\textbf {h}\,x+w_{2}, \tag{4}\end{align*}$$ where the channel coefficient vectors are denoted by $$\begin{align*} \textbf {h} & =[h_{1}\,h_{2}\,\ldots \,h_{\mathcal {N}_{1}}]^{T}, \\ \textbf {g} & =[g_{1}\,g_{2}\,\ldots \,g_{\mathcal {N}_{1}}]^{T}, \\ \textbf {n} & =[n_{1}\,n_{2}\,\ldots \,n_{\mathcal {N}_{1}}]^{T}, \\ \Phi & =\text {diag}([e^{j\Phi _{1}}\,e^{j\Phi _{2}}\,\ldots \, e^{j\Phi _{\mathcal {N}_{1}}}]), \\ \Psi & =\text {diag}([e^{j\Psi _{1}}\,e^{j\Psi _{2}}\,\ldots \, e^{j\Psi _{\mathcal {N}_{1}}}]).\end{align*}$$ $\Phi$ and $\Psi$ are the diagonal matrices containing the transitions of phase employed by RIS components. The SNRs at $\mathcal {R}$ and $\mathcal {E}_{P}$ are expressed as $$\begin{align*} \gamma _{r}&=\left [{\sum _{s_{p}=0}^{\mathcal {N}_{1}}\alpha _{{s_{p}}} \beta _{s_{p}}e^{j\left ({\Phi _{s_{p}}-\varrho _{s_{p}}-\vartheta _{s_{p}}}\right)}}\right]^{2}\bar {\gamma }_{r}, \tag{5}\\ \gamma _{e_{p}}&=\left [{\sum _{s_{p}=0}^{\mathcal {N}_{1}}\alpha _{s_{p}}\eta _{s_{p}}e^{j\left ({\Psi _{s_{p}}-\varrho _{s_{p}}-{\delta _{s_{p}}}}\right)}}\right]^{2}\bar {\gamma }_{e_{p}}, \tag{6}\end{align*}$$ where the average SNR of the $\mathcal {S}-\mathcal {I}_{P}-\mathcal {R}$ link is denoted by $\bar {\gamma }_{r}=\frac {S_{s}}{M_{r}}$ and the average SNR of $\mathcal {S}-\mathcal {I}_{P}-\mathcal {E}_{P}$ link is represented by $\bar {\gamma }_{e_{p}}=\frac {S_{s}}{M_{e}}$ . Therefore, the ideal selection of $\Phi _{s_{p}}$ and $\Psi _{s_{p}}$ are $\Phi _{s_{p}}=\varrho _{s_{p}}+\vartheta _{s_{p}}$ and $\Psi _{s_{p}}=\varrho _{s_{p}}+\delta _{s_{p}}$ for obtaining maximized instantaneous SNRs. It is worth noting that similar to the research conducted by [56] and [57], we make the assumption of the most adverse eavesdropping scenario, where the eavesdropper possesses strong detection capabilities. Hence, the maximum possible SNRs at $\mathcal {R}$ and $\mathcal {E}_{P}$ are given, correspondingly, as $$\begin{align*} \gamma _{r}&=\left ({\sum _{s_{p}=0}^{\mathcal {N}_{1}}\alpha _{{s_{p}}} \beta _{s_{p}}}\right)^{2} \bar {\gamma }_{r}, \tag{7}\\ \gamma _{e_{p}}&=\left ({\sum _{s_{p}=0}^{\mathcal {N}_{1}}\alpha _{s_{p}}\eta _{s_{p}}}\right)^{2} \bar {\gamma }_{e_{p}}. \tag{8}\end{align*}$$

For Scenario-II, the received signals at $\mathcal {D}$ and $\mathcal {E}_{Q}$ are represented in a form similar to the expressions in (1)-​(4) and utilizing the same procedures for optimization, the received SNRs are given as $$\begin{align*} \gamma _{d}&=\left ({\sum _{r_{q}=0}^{\mathcal {N}_{2}}\xi _{{r_{q}}}\beta _{r_{q}}}\right)^{2}\bar {\gamma }_{d}, \tag{9}\\ \gamma _{e_{q}}&=\left ({\sum _{r_{q}=0}^{\mathcal {N}_{2}}\xi _{{r_{q}}}\eta _{r_{q}}}\right)^{2}\bar {\gamma }_{e_{q}}, \tag{10}\end{align*}$$ where $\xi _{{r_{q}}}$ , $\beta _{r_{q}}$ , and $\eta _{r_{q}}$ are Málaga distributed RVs, $\bar {\gamma }_{d}$ and $\bar {\gamma }_{e_{q}}$ are the average SNRs of the $\mathcal {R}-\mathcal {I}_{Q}-\mathcal {D}$ and $\mathcal {R}-\mathcal {I}_{Q}-\mathcal {E}_{Q}$ links, respectively. The received SNR at the destination utilizing a DF relay is given by $$\begin{equation*} \gamma _{eq} \cong \min \left \{{\gamma _{r}, \gamma _{d}}\right \}. \tag{11}\end{equation*}$$

B. PDF and CDF of RIS-Aided RF Links

The probability density function (PDF) and CDF of $\gamma _{j}$ , where $j\in (r,e_{p})$ are respectively expressed as [53] $$\begin{align*} f_{\gamma _{j}}(\gamma) &\simeq \frac {\gamma ^{\frac {a_{j}-1}{2}} \exp \left ({-\frac {\sqrt {\gamma }}{b_{j} \sqrt {{\bar {\gamma }_{j}}}}}\right)}{2 b_{j}^{a_{j}+1} \Gamma (a_{j}+1) {\bar {\gamma }_{j}}^{\frac {a_{j}+1}{2}}}, \tag{12}\\ F_{\gamma _{j}}(\gamma)& \simeq \frac {\gamma \left ({a_{j}+1, \frac {\sqrt {\gamma }}{b_{j} \sqrt {\bar {\gamma }_{j}}}}\right)}{\Gamma (a_{j}+1)}, \tag{13}\end{align*}$$ where $\gamma (\cdot, \cdot)$ is the lower incomplete Gamma function [58, Eq. (8.350.1)], $\Gamma (\cdot)$ is the Gamma operator, $a_{j}$ and $b_{j}$ are constants related to the mean and variance of the cascaded Rician random variable $\xi _{j}$ computed as $$\begin{align*} a_{j} & =\frac {\mathbb {E}^{2}\left [{\xi _{j}}\right]}{\mathrm {Var}(\delta _{j})}-1, \tag{14}\\ b_{j} & =\frac {\mathrm {Var}(\delta _{j})}{\mathbb {E}\left [{\xi _{j}}\right]}, \tag{15}\end{align*}$$ where $\xi _{m}$ is the sum of i.i.d. non-negative random variables, $$\begin{equation*} \mathbb {E}\left [{\xi _{j}}\right]=\mathbb {E}\left [{{\alpha _{s}}_{p}}\right] \mathbb {E}\left [{{\lambda _{s}}_{p}}\right], \tag{16}\end{equation*}$$ $\lambda \in (\beta,\eta)$ and $\mathbb {E}\left [{{\alpha _{s}}_{p}}\right]=\frac {1}{2} \sqrt {\frac {\pi \Omega _{1}}{K_{1}+1}} L_{1 / 2}\left ({- K_{1}}\right)$ , $\mathbb {E}\left [{{\lambda _{s}}_{p}}\right]=\frac {1}{2} \sqrt {\frac {\pi \Omega _{j}}{K_{j}+1}} L_{1 / 2}\left ({- K_{j}}\right)$ , $\text {Var}(\delta _{j})=\mathcal {N}_{1} \text { Var}\left ({\xi _{j}}\right)$ , $K_{1}$ and $\Omega _{1}$ denote the shape parameter and scale parameter, respectively, for the first hop of $\mathcal {S}-\mathcal {I}_{P}-\mathcal {R}$ link, for the other hop those are expressed by $K_{j}$ and $\Omega _{j}$ , correspondingly, $L_{1 / 2}(\cdot)$ denotes the Laguerre polynomial, i.e., $L_{1 / 2}(x)= e^{x / 2}\left [{(1-x) I_{0}\left ({\frac {-x}{2}}\right)-x I_{1}\left ({\frac {-x}{2}}\right)}\right]$ , and $I_{v}(\cdot)$ is the modified Bessel function of the first kind and order $v$ [58, Eq. (8.431)]. Further simplification of (16) gives $$\begin{align*} \mathbb {E}\left [{\xi _{j}}\right]&=\frac {\pi e^{-\frac {\left ({K_{1}+K_{j}}\right)}{2}}}{4} \sqrt {\frac {\Omega _{1} \Omega _{j}}{\left ({K_{1}+1}\right)\left ({K_{j}+1}\right)}} \\ &\quad \times \left [{\left ({K_{1}+1}\right) I_{0}\left ({\frac {K_{1}}{2}}\right)+K_{1} I_{1}\left ({\frac {K_{1}}{2}}\right)}\right] \\ &\quad \times \left [{\left ({K_{j}+1}\right) I_{0}\left ({\frac {K_{j}}{2}}\right)+K_{j} I_{1}\left ({\frac {K_{j}}{2}}\right)}\right]. \tag{17}\end{align*}$$ Notice that $\mathbb {E}\left [{\xi _{j}^{2}}\right]=\mathbb {E}\left [{{\alpha _{s}}_{p}^{2}}\right] \mathbb {E}\left [{{\lambda _{s}}_{p}^{2}}\right]=\Omega _{1} \Omega _{j}$ . The variance is computed as $$\begin{equation*} \mathrm {Var}\left ({\delta _{j}}\right)=\mathbb {E}\left [{\xi _{j}^{2}}\right]-\mathbb {E}^{2}\left [{\xi _{j}}\right]=\Omega _{1} \Omega _{j}-\mathbb {E}^{2}\left [{\xi _{j}}\right]. \tag{18}\end{equation*}$$

C. PDF and CDF of RIS-Aided FSO Links

The PDF of $\gamma _{p}$ , where $p\in (d,e_{q})$ , can be calculated as [59, Eq. (17)] $$\begin{equation*} f_{\gamma _{p}}(\gamma)=\int _{0}^{\infty } f_{\gamma _{h}}(t) f_{\gamma _{g_{p}}}\left ({\frac {\gamma }{t}}\right) \frac {1}{t} dt. \tag{19}\end{equation*}$$ The PDFs of $\mathcal {R}-\mathcal {I}_{Q}$ , $\mathcal {I}_{Q}-\mathcal {D}$ and $\mathcal {I}_{Q}-\mathcal {E}_{Q}$ links can be written as [60, Eq. (10)] $$\begin{align*} f_{\gamma _{i}}\left ({\gamma _{i}}\right)=\frac {\xi _{i}^{2} {A_{i}}}{2^{r} \gamma _{i}} \sum _{m_{i}=1}^{\beta _{i}} b_{m_{i}} \mathrm {G}_{1,3}^{3,0}\left [{{B_{i}}\left ({\frac {\gamma _{i}}{{\bar {\gamma _{i}}}}}\right)^{\frac {1}{r}} \biggl | \begin{array}{c} {\xi _{i}}^{2}+1 \\ {\xi _{i}}^{2}, {\alpha _{i}}, m_{i} \end{array}}\right], \tag{20}\end{align*}$$ where $i \in \{h, {g_{p}}\}$ , $$\begin{align*} A_{i} & \triangleq \frac {2 {\alpha _{i}}^{\alpha _{i} / 2}}{c^{1+\alpha _{i} / 2} \Gamma (\alpha _{i})}\left ({\frac {c\beta _{i}}{c\beta _{i}+\Omega ^{\prime }}}\right)^{\beta _{i}+\alpha _{i} / 2}, \\ a_{m_{i}} & \triangleq \left ({\!\!\!\begin{array}{c} \beta _{i}-1 \\ {m_{i}}-1 \end{array}\!\!\!}\right) \frac {\left ({c \beta _{i}+\Omega ^{\prime }}\right)^{1-{m_{i}} / 2}}{({m_{i}}-1) !}\!\!\left ({\frac {\Omega ^{\prime }}{c}}\right)\!\!^{{m_{i}}-1}\!\!\left ({\!\frac {\alpha _{i}}{\beta _{i}}\!}\right)^{\frac {m_{i}}{2}}, \\ {b_{m}}_{i} & ={a_{m}}_{i}[\alpha _{i} \beta _{i} /(c \beta _{i}+\Omega ^{\prime })]^{-(\alpha _{i}+m_{i}) / 2}, \\ B_{i} & =\xi _{i}^{2} \alpha _{i} \beta _{i}(c+\Omega ^{\prime }t) /[(\xi _{i}^{2}+1)(c \beta _{i}+\Omega ^{\prime })], \\ \Omega ^{\prime } & =\Omega +2 b_{0} \rho + 2 \sqrt {2 b_{0} \rho \Omega } \cos (\phi _{A}-\phi _{B}),\end{align*}$$ and $c=2b_{0}\left ({1-\rho }\right)$ indicates the average amount of power received by off-axis eddies from the dispersive element, $\alpha _{i}$ and $\beta _{i}$ are the turbulence parameters, $\bar {\gamma }_{i}$ is the average SNR, $\xi _{i}$ represents pointing error, $\Omega$ is the average power of LOS component, $b_{0}$ is the average power of the total scatter components, $\rho$ represents the quantity of scattering power coupled to the LOS component, $\phi _{A}$ and $\phi _{B}$ are the deterministic phases of the LOS and the coupled-to-LOS scatter terms, respectively, $r \in \{1,2\}$ determines if the transmission makes use of the heterodyne detection (HD) approach $(r=1)$ or the intensity modulation/direct detection (IM/DD) techniques ($r=2$ ) [60], and $\begin{aligned} \mathbf {G}_{p, q}^{m, n}\left [{z \mid \begin{array}{l}a_{p} \\ b_{q}\end{array}}\right] \end{aligned}$ is the Meijer’s G function [58, Eq. (9.301)]. From (20), we get $$\begin{align*} f_{\gamma _{h}}\left ({t}\right)\!\!&=\!\!\frac {\xi _{h}^{2} {A_{h}}}{2^{r} t} \sum _{m_{h}=1}^{\beta _{h}} b_{m_{h}}\,G_{1,3}^{3,0}\left [{{B_{h}}\left ({\frac {t}{\bar {\gamma }_{h}}}\right) ^{\frac {1}{r}} \!\! \biggl |\!\! \begin{array}{c} {\xi _{h}}^{2}+1 \\ {\xi _{h}}^{2}, {\alpha _{h}}, m_{h} \end{array}\!\!}\right], \tag{21}\\ f_{\gamma _{g_{p}}}\left ({{\frac {\gamma }{t}}}\right)\!\!&=\!\!\frac {\xi _{g_{p}}^{2} {A_{g_{p}}}t}{2^{r} \gamma }\, \!\!\sum _{m_{g_{p}}=1}^{\beta _{g_{p}}} \!\! b_{m_{g_{p}}} \\ & \quad \times G_{1,3}^{3,0}\left [{{B_{g_{p}}}\left ({\frac {\gamma }{t{\bar {\gamma }_{g_{p}}}}}\right)^{\frac {1}{r}} \!\! \biggl | \! \! \begin{array}{c} {\xi _{g_{p}}}^{2}+1 \\ {\xi _{g_{p}}}^{2}, {\alpha _{g_{p}}}, m_{g_{p}} \end{array}\!\!}\right], \tag{22}\end{align*}$$ where ${\bar {\gamma }_{h}}$ and ${\bar {\gamma }_{g_{p}}}$ are the average SNRs. In (22), the variable $t$ appears in the denominator. Utilizing the Meijer’s G function’s reflection characteristic [61] in (22), we get $$\begin{align*} &\hspace {-.5pc}f_{\gamma _{g_{p}}}\left ({{\frac {\gamma }{t}}}\right) \\ &=\frac {\xi _{g_{p}}^{2} {A_{g_{p}}}t}{2^{r} \gamma } \sum _{m_{g_{p}}=1}^{\beta _{g}} b_{m_{g_{p}}} \\ &\quad \times G_{3,1}^{0,3}\left [{\frac {1}{B_{g_{p}}}\left ({\frac {{t\bar {\gamma }_{g_{p}}}}{\gamma }}\right)^{\frac {1}{r}} \!\! \biggl | \!\! \begin{array}{c} 1-{\xi _{g_{p}}}^{2},1- {\alpha _{g_{p}}},1-m_{g_{p}}\\ {-\xi _{g_{p}}}^{2} \end{array}\!\!\!}\right]. \tag{23}\end{align*}$$ Substituting (21) and (23) into (19) then applying the change of variable $X=t^{\frac {1}{a}} \Rightarrow t=X^{a}$ and $d t= a X^{a-1} d X$ via utilizing [62, Eq. (2.24.1.1)], we obtain the exact unified PDF of end-to-end SNR as $$\begin{align*} &\hspace {-.5pc}f_{\gamma _{p}}(\gamma) \\ &=\frac {\xi _{h}^{2} {A_{h}}\xi _{g_{p}}^{2} {A_{g_{p}}}r}{2^{2r} \gamma } \sum _{m_{h}=1}^{\beta _{h}}\sum _{m_{g_{p}}=1}^{\beta _{g_{p}}} b_{m_{h}} b_{m_{g_{p}}} \\ &\quad \times G_{2,6}^{6,0}\left [{{B_{h}}{B_{g_{p}}}\left ({\frac {\gamma }{{\bar {\gamma }_{p}}}}\right)^{\frac {1}{r}} \!\!\biggl |\!\! \begin{array}{c} {1+\xi _{g_{p}}}^{2}, 1+\xi _{h}^{2} \\ {\xi _{h}}^{2}, {\alpha _{h}}, m_{h},\xi _{g_{p}}^{2}, \alpha _{g_{p}}, m_{g_{p}} \end{array}\!\!}\right], \tag{24}\end{align*}$$ where ${\bar {\gamma }_{p}}$ = $\bar {\gamma }_{h}\bar {\gamma }_{g_{p}}$ . The CDF of the end-to-end SNR can be written as $$\begin{equation*} F_{\gamma _{p}}(\gamma)=\int _{0}^{\gamma } f_{\gamma _{p}}(\gamma) d\gamma. \tag{25}\end{equation*}$$ By substituting (24) into (25), we obtain the CDF of $\gamma _{p}$ via utilizing [63, Eq. (07.34.21.0084.01)] as $$\begin{align*} F_{\gamma _{p}}(\gamma)&\!\!=\!\! \frac {\xi _{h}^{2} {A_{h}}\xi _{g_{p}}^{2} {A_{g_{p}}}}{2^{2r} }\!\! \sum _{m_{h}=1}^{\beta _{h}} \!\! \sum _{m_{g_{p}}=1}^{\beta _{g_{p}}}b_{m_{h}} b_{m_{g_{p}}} \frac {r^{\alpha _{h}+\alpha _{g_{p}}+m_{h}+m_{g_{p}}-2}}{2\pi ^{2(r-1)}} \\ &\quad \times G_{2r+1,6r+1}^{6r,1}\left [{\left ({\frac {B_{h{g_{p}}}}{\bar {\gamma }_{p}}}\right)\gamma \biggl | \begin{array}{c} {1,l_{g_{p}{}_{1}}, l_{h_{1}}} \\ l_{h_{2}}, l_{g_{p}{}_{2}}, 0 \end{array}}\right], \tag{26}\end{align*}$$ where $$\begin{align*} {B}_{h{g_{p}}} & =\frac {\left ({{B_{h}}{B_{g_{p}}}}\right)^{r}}{r^{4r}}, \\ l_{g_{p}{}_{1}} & =\left ({\frac {1+{\xi _{g_{p}}}^{2}}{r},\ldots, \frac {1+{\xi _{g_{p}}}^{2}+r-1}{r}}\right), \\ l_{h_{1}} & =\left ({\frac {1+{\xi _{h}}^{2}}{r}, \ldots, \frac {1+{\xi _{h}}^{2}+r-1}{r}}\right), \\ l_{g_{p}{}_{2}} & =\biggl (\frac {{\xi _{g_{p}}}^{2}}{r}, \ldots, \frac {{\xi _{g_{p}}}^{2}+r-1}{r}, \frac {\alpha _{g_{p}}}{r},\ldots, \frac {\alpha _{g_{p}}+r-1}{r}, \\ &\qquad \frac {m_{g_{p}}}{r}, \ldots, \frac {m_{g_{p}}+r-1}{r}\biggl), \\ l_{h_{2}} & =\biggl (\frac {\xi _{h}^{2}}{r}, \ldots, \frac {\xi _{h}^{2}+r-1}{r}, \frac {\alpha _{h}}{r}, \ldots, \frac {\alpha _{h}+r-1}{r}, \\ &\qquad \frac {m_{h}}{r}, \ldots, \frac {m_{h}+r-1}{r}\biggl).\end{align*}$$

D. CDF of End-To-End SNR for RIS-Aided Dual-Hop RF-FSO Link

The CDF of $\gamma _{eq}$ is expressed as $$\begin{equation*} F_{\gamma _{eq}}(\gamma)=F_{\gamma _{r}}(\gamma)+F_{\gamma _{d}}(\gamma)-F_{\gamma _{r}}(\gamma)F_{\gamma _{d}}(\gamma). \tag{27}\end{equation*}$$ Substituting (13) and (26) in (27) and carrying out algebraic calculations, the simplification of CDF of $\gamma _{eq}$ can be attained as $$\begin{align*} F_{\gamma _{eq}}(\gamma)&=\sum _{n=0}^{\infty } \frac {(-1)^{n}\left ({\frac {1}{b_{r} \sqrt {{\bar {\gamma }_{r}}}}}\right)^{a_{r}+1+n}}{n !\left ({a_{r}+1+n}\right)\Gamma \left ({a_{r}+1}\right)} + \frac {\xi _{h}^{2} {A_{h}}\xi _{g_{d}}^{2} {A_{g_{d}}}}{2^{2r} } \\ &\quad \times \sum _{m_{h}=1}^{\beta _{h}} \sum _{m_{g_{d}}=1}^{\beta _{g_{d}}}b_{m_{h}} b_{m_{g_{d}}} \frac {r^{\alpha _{h}+\alpha _{g_{d}}+m_{h}+m_{g_{d}}-2}}{2\pi ^{2(r-1)}} \\ &\quad \times G_{2r+1,6r+1}^{6r,1}\left [{\left ({\frac {B_{h{g_{p}}}}{\gamma _{2}}}\right)\gamma \biggl | \begin{array}{c} {1,l_{g_{d}{}_{1}}, l_{h_{1}}} \\ l_{h_{2}}, l_{g_{d}{}_{2}}, 0 \end{array}}\right] \\ &\quad \times \left ({1-\sum _{n=0}^{\infty } \frac {(-1)^{n}\left ({\frac {1}{b_{r} \sqrt {{\bar {\gamma }_{r}}}}}\right)^{a_{r}+1+n}}{n !\left ({a_{r}+1+n}\right)\Gamma \left ({a_{r}+1}\right)} }\right). \tag{28}\end{align*}$$ As per the comprehension discussed in the literature review section, the combination of RIS-assisted RF-FSO framework taking Rician and Málaga distributions into account has not yet been described in any current study within the literature. As a result, the expression found in (28) can be demonstrated to be unique. Also, the generalized depiction of both Rician and Málaga distribution drives this endeavor towards the goal of unifying the many existing models by treating them as prominent occurrences.

SECTION III. Performance Analysis

In this portion of the work, we attain the expressions for the suggested RIS-assisted RF-FSO network’s metrics of performance, namely ASC, the lower bound of SOP, probability of SPSC, IP, and EST.

A. Average Secrecy Capacity Analysis

ASC is the mean value of the instantaneous secrecy capacity, which can be stated analytically as [35], [40], and [64, Eq. (15)] $$\begin{equation*} ASC^{I}=\int _{0}^{\infty }\frac {1}{1+\gamma }\,F_{\gamma _{e_{p}}}(\gamma)\,\left [{1-F_{\gamma _{eq}}(\gamma)}\right]\,d\gamma. \tag{29}\end{equation*}$$ On substituting (26) and (28) into (29), ASC is derived as $$\begin{align*} ASC^{I}& = {\mathcal {X}_{1}}\biggl (\sum _{n=0}^{\infty }{\mathcal {U}_{1}} -\sum _{n=0}^{\infty }{\mathcal {X}_{2}}{\mathcal {U}_{2}}-\sum _{m_{h}=1}^{\beta _{h}} \sum _{m_{g}=1}^{\beta _{g_{d}}} {\mathcal {X}_{3}}{\mathcal {U}_{3}} \\ &\quad +\sum _{n=0}^{\infty } \sum _{m_{h}=1}^{\beta _{h}} \sum _{m_{g_{d}}=1}^{\beta _{g_{d}}} {\mathcal {X}_{2}}{\mathcal {X}_{3}} {\mathcal {U}_{4}}\biggl), \tag{30}\end{align*}$$ where $$\begin{align*} {\mathcal {X}_{1}} & = \frac {(-1)^{n}\left ({\frac {1}{b_{e_{p}}\sqrt {{\bar {\gamma }_{r}}}}}\right)^{a_{e_{p}}+n+1}}{n !\left ({a_{e_{p}}+n+1}\right)\Gamma \left ({a_{e_{p}}+1}\right)}, \\ {\mathcal {X}_{2}} & = \frac {(-1)^{n}\left ({\frac {1}{b_{e_{p}}\sqrt {{\bar {\gamma }_{r}}}}}\right)^{a_{r}+n+1}}{n !\left ({a_{r}+n+1}\right)\Gamma \left ({a_{r}+1}\right)}, \\ {\mathcal {X}_{3}} & =b_{m_{h}} b_{m_{g_{d}}} \frac {r^{\alpha _{h}+\alpha _{g_{d}}+m_{h}+m_{g_{d}}-2}}{2\pi ^{2(r-1)}},\end{align*}$$ and four integral expressions ${\mathcal {U}_{1}}$ , ${\mathcal {U}_{2}}$ , ${\mathcal {U}_{3}}$ and ${\mathcal {U}_{4}}$ are expressed as follows.

1) Derivation of ${\mathcal{U}_{1}}$

${\mathcal {U}_{1}}$ is expressed as $$\begin{equation*} {\mathcal {U}_{1}}=\int _{0}^{\infty }\frac {\gamma ^{\frac {a_{e_{p}}+n+1}{2}}}{1+\gamma } d\gamma. \tag{31}\end{equation*}$$ With the fulfillment of identity [58, Eq. (3.194.3)], ${\mathcal {U}_{1}}$ is attained as $$\begin{equation*} {\mathcal {U}_{1}}=\mathcal {B}\left ({\frac {a_{e_{p}}+n+3}{2},1-\frac {a_{e_{p}}+n+3}{2}}\right), \tag{32}\end{equation*}$$ where $\mathcal {B}\left ({.,.}\right)$ is the Beta function [58, Eq. (8.39)].

2) Derivation of ${\mathcal{U}_{2}}$

${\mathcal {U}_{2}}$ is expressed as $$\begin{equation*} {\mathcal {U}_{2}}=\int _{0}^{\infty }\frac {\gamma ^{\frac {a_{r}+a_{e_{p}}+2n+2}{2}}}{1+\gamma } d\gamma. \tag{33}\end{equation*}$$ Using an analogous method to the one used to derive ${\mathcal {U}_{1}}$ , ${\mathcal {U}_{2}}$ is closed in as $$\begin{align*} {\mathcal {U}_{2}}=\mathcal {B}\left ({\frac {a_{r}+a_{e_{p}}+2n+4}{2},1-\frac {a_{r}+a_{e_{p}}+2n+4}{2}}\right). \tag{34}\end{align*}$$

3) Derivation of ${\mathcal{U}_{3}}$

${\mathcal {U}_{3}}$ is expressed as $$\begin{align*} {\mathcal {U}_{3}}=\int _{0}^{\infty }\frac {\gamma ^{\frac {a_{e_{p}}+n+1}{2}}}{1+\gamma } G_{2r+1,6r+1}^{6r,1}\left [{\left ({\frac {B_{h{g_{p}}}}{\gamma _{2}}}\right)\gamma \biggl | \begin{array}{c} {1,l_{d_{1}}, l_{h_{1}}} \\ l_{h_{2}}, l_{d_{2}}, 0 \end{array}}\right] d\gamma. \tag{35}\end{align*}$$ The transformation of $\frac {1}{1+\gamma }$ into Meijer’s G function is done by utilizing the identity [62, Eq. (8.4.2.5)] and solving the integral upon utilization of [62, Eq. (2.24.1.1)], ${\mathcal {U}_{3}}$ is obtained as $$\begin{align*} {\mathcal {U}_{3}}&=\int _{0}^{\infty }{\gamma }^{\frac {a_{e_{p}}+n+1}{2}} G_{2r+1,6r+1}^{6r,1}\left [{\left ({\frac {B_{h{g_{p}}}}{\gamma _{2}}}\right)\gamma \biggl | \begin{array}{c} {1,l_{d_{1}}, l_{h_{1}}} \\ l_{h_{2}}, l_{d_{2}}, 0 \end{array}}\right] \\ &\quad \times G_{1,1}^{1,1}\!\!\left [{\!\!\gamma \biggl | \begin{array}{c} {0} \\ 0 \end{array}}\right]d\gamma =\left ({\frac {B_{h{g_{p}}}}{\gamma _{2}}}\right)^{-\alpha _{1}} \\ &\quad \times G_{6r+2,2r+2}^{2,6r+1}\!\left [{\!\frac {\gamma _{2}}{B_{h{g_{p}}}} \!\biggl | \!\!\begin{array}{c} {0,-\alpha _{1}-l_{h_{2}},-\alpha _{1}-l_{d_{2}},-\alpha _{1}} \\ 0, -\alpha _{1}-1,-\alpha _{1}-l_{d_{1}},-\alpha _{1}-l_{h_{1}} \end{array}\!\!\!}\right]\!, \tag{36}\end{align*}$$ where $\alpha _{1}=\frac {a_{e_{p}}+n+3}{2}$ .

4) Derivation of ${\mathcal{U}_{4}}$

${\mathcal {U}_{4}}$ is expressed as $$\begin{align*} {\mathcal {U}_{4}}&\!\!=\!\! \int _{0}^{\infty } \!\! \frac {1}{1+\gamma }{\gamma }^{\frac {a_{r}+a_{e_{p}}+2n+2}{2}} \\ &\quad \times G_{2r+1,6r+1}^{6r,1}\left [{\left ({\frac {B_{h{g_{p}}}}{{\bar {\gamma }_{d}}}}\right)\gamma \biggl | \begin{array}{c} {1,l_{d_{1}}, l_{h_{1}}} \\ l_{h_{2}}, l_{d_{2}}, 0 \end{array}}\right] d\gamma. \tag{37}\end{align*}$$ Using an analogous method to the one used to derive ${\mathcal {U}_{3}}$ , ${\mathcal {U}_{4}}$ is derived as $$\begin{align*} {\mathcal {U}_{4}}&=\int _{0}^{\infty } \!\!\!{\gamma }^{\frac {a_{e_{p}}+n+1}{2}} G_{2r+1,6r+1}^{6r,1}\left [{\left ({\frac {B_{h{g_{p}}}}{\gamma _{2}}}\right)\gamma \biggl | \begin{array}{c} {1,l_{d_{1}}, l_{h_{1}}} \\ l_{h_{2}}, l_{d_{2}}, 0 \end{array}}\right] \\ &\quad \times G_{1,1}^{1,1}\left [{\gamma \biggl | \begin{array}{c} {0} \\ 0 \end{array}}\right]d\gamma =\left ({\frac {B_{h{g_{p}}}}{\gamma _{2}}}\right)^{-\alpha _{2}-1} \\ &\quad \times G_{6r+2,2r+2}^{2,6r+1}\!\left [{\!\frac {\gamma _{2}}{B_{h{g_{p}}}} \!\biggl |\! \begin{array}{c} {0,-\alpha _{2}-l_{h_{2}},-\alpha _{2}-l_{d_{2}},-\alpha _{2}} \\ 0, -\alpha _{2}-1,-\alpha _{2}-l_{d_{1}},-\alpha _{2}-l_{h_{1}} \end{array}\!\!\!}\right]\!\!, \tag{38}\end{align*}$$ where $\alpha _{2}=\frac {a_{r}+a_{e_{p}}+2n+2}{2}$ .

B. Secrecy Outage Probability Analysis

1) Scenario-I (Lower Bound SOP)

In accordance with [65, Eq. (21)], the lower bound of SOP can be stated as $$\begin{align*} SOP^{I}=\text {Pr}\left \{{\gamma _{eq}\leq \phi \,\gamma _{\mathcal {E}}}\right \} =\int _{0}^{\infty }F_{\gamma _{eq}}(\phi \,\gamma)\,f_{\gamma _{e_{p}}}(\gamma)\,d\gamma, \tag{39}\end{align*}$$ where $\phi =2^{R_{s}}$ and $R_{s}$ denotes the target secrecy rate. Now, substituting (12) and (28) into (39), SOP is expressed finally as $$\begin{equation*} SOP^{I} =\frac {{{\mathcal {M}_{1}} {\mathcal {R}_{1}}+{\mathcal {M}_{2}}{\mathcal {R}_{2}}-{\mathcal {M}_{3}}{\mathcal {R}_{3}}}}{2 b_{e_{p}}^{a_{e_{p}}+1} \Gamma (a_{e_{p}}+1) {\bar {\gamma }_{e_{p}}}^{\frac {a_{e_{p}}+1}{2}}}, \tag{40}\end{equation*}$$ where $$\begin{align*} {\mathcal {M}_{1}} & =\sum _{n=0}^{\infty } \frac {(-1)^{n}}{n !\left ({a_{r}+1+n}\right)\Gamma (a_{r}+1)}\left ({\frac {\phi ^{\frac {1}{2}}}{b_{r} \sqrt {{\bar {\gamma }_{r}}}}}\right)^{a_{r}+1+n}, \\ {\mathcal {M}_{2}} & =\frac {\xi _{h}^{2} {A_{h}}\xi _{g_{d}}^{2} {A_{g_{d}}}}{2^{2r} }\sum _{m_{h}=1}^{\beta _{h}}\sum _{m_{g_{d}}=1}^{\beta _{g_{d}}}{b_{m_{h}}} {b_{m_{g_{d}}}} \\ &\quad \times \frac {r^{\alpha _{h}+\alpha _{g_{d}}+m_{h}+m_{g_{d}}-2}}{2\pi ^{2(r-1)}}, \\ {\mathcal {M}_{3}} & = \frac {{\mathcal {M}_{1}} \xi _{h}^{2} \xi _{g_{d}}^{2}} {2^{2r} }{A_{h}}{A_{g_{d}}}\!\!\! \sum _{m_{h}=1}^{\beta _{h}} \!\! \sum _{m_{g_{d}}=1}^{\beta _{g_{d}}} \!\!{b_{m_{h}}} {b_{m_{g_{d}}}} \\ &\quad \times \frac {r^{\alpha _{h}+\alpha _{g_{d}}+m_{h}+m_{g_{d}}-2}}{2\pi ^{2(r-1)}},\end{align*}$$ and derivations of the three integral terms ${\mathcal {R}_{1}}$ , ${\mathcal {R}_{2}}$ and ${\mathcal {R}_{3}}$ are expressed as follows.

2) Derivation of ${\mathcal{R}_{1}}$

${\mathcal {R}_{1}}$ is expressed as $$\begin{equation*} {\mathcal {R}_{1}}=\int _{0}^{\infty }{\gamma }^{\frac {a_{r}+a_{e_{p}}+n}{2}}e^{-}{\frac {\sqrt {\gamma }}{b_{e_{p}}\sqrt {{\bar {\gamma }_{e_{p}}}}} } d\gamma.\end{equation*}$$ ${\mathcal {R}_{1}}$ is derived by utilizing [58, Eq. (3.326.2)] as $$\begin{equation*} {\mathcal {R}_{1}}=2(a_{r}+a_{e_{p}}+n+1)!\left ({\frac {1}{b_{e_{p}}\sqrt {{\bar {\gamma }_{e_{p}}}}}}\right)^{-a_{r}-a_{e_{p}}-n-2}. \tag{41}\end{equation*}$$

3) Derivation of ${\mathcal{R}_{2}}$

${\mathcal {R}_{2}}$ is expressed as $$\begin{align*} {\mathcal {R}_{2}}&=\int _{0}^{\infty }{\gamma }^{\frac {a_{e_{p}}+1}{2}-1} G_{2r+1,6r+1}^{6r,1}\left [{\left ({\frac {B_{h{g_{p}}}\phi }{\gamma _{2}}}\right)\gamma \biggl | \begin{array}{c} {1,l_{d_{1}}, l_{h_{1}}} \\ l_{h_{2}}, l_{d_{2}}, 0 \end{array}}\right] \\ &\quad \times e^{\frac {-\sqrt {\gamma }}{b_{e_{p}}\sqrt {{\bar {\gamma }_{e_{p}}}}} } d{\gamma }\end{align*}$$ Now, through the use of a number of mathematical operations utilizing [62, Eq. (8.4.3.1) and (2.24.1.1)], ${\mathcal {R}_{2}}$ is derived as $$\begin{align*} {\mathcal {R}_{2}}&=\int _{0}^{\infty }{\gamma }^{\frac {a_{e_{p}}+1}{2}-1} G_{2r+1,6r+1}^{6r,1}\left [{\left ({\frac {B_{h{g_{p}}}\phi }{\gamma _{2}}}\right)\gamma \biggl | \begin{array}{c} {1,l_{d_{1}}, l_{h_{1}}} \\ l_{h_{2}}, l_{d_{2}}, 0 \end{array}}\right] \\ &\quad \times G_{0,1}^{1,0}\left [{\left ({\frac {\sqrt {\gamma }}{b_{e_{p}}\sqrt {{\bar {\gamma }_{e_{p}}}}}}\right) \biggl | \begin{array}{c} {-} \\ 0 \end{array}}\right]d{\gamma } \\ &={\mathcal {Z}_{1}}G_{6r+1,2r+3}^{3,6r}\hspace {-1mm}\left [{\hspace {-1mm}\frac {\gamma _{2}\left ({\frac {1}{b_{e_{p}}\sqrt {{\bar {\gamma }_{e_{p}}}}}}\right)^{2}}{4{B}_{h{g_{p}}}\phi }\biggl | \begin{array}{c} {l_{h_{3}},l_{d_{3}}, 1-\frac {a_{e_{p}}+1}{2}} \\ 0,-\frac {a_{e_{p}}+1}{2},l_{d_{4}}, l_{h_{4}} \end{array}\hspace {-2mm}}\right], \tag{42}\end{align*}$$ where ${\mathcal {Z}_{1}}={\pi }^{-\frac {1}{2}} \left ({\frac {\gamma _{2}}{B_{h{g_{p}}}\phi }}\right)^{\frac {a_{e_{p}}+1}{2}}, l_{h_{3}}=1-\frac {a_{e_{p}}+1}{2}-l_{h_{2}}, l_{d_{3}}=1-\frac {a_{e_{p}}+1}{2}-l_{d_{2}}$ , $l_{h_{4}}=1-\frac {a_{e_{p}}+1}{2}-l_{h_{1}}$ , and $l_{d_{4}}=1-\frac {a_{e_{p}}+1}{2}-l_{d_{1}}$ .

4) Derivation of ${\mathcal{R}_{3}}$

${\mathcal {R}_{3}}$ is expressed as $$\begin{align*} {\mathcal {R}_{3}}&=\int _{0}^{\infty }{\gamma }^{\frac {a_{r}+a_{e_{p}}+n}{2}}\mathrm {G}_{2r+1,6r+1}^{6r,1}\left [{\left ({\frac {B_{h{g_{p}}}\phi }{\gamma _{2}}}\right)\gamma \biggl | \begin{array}{c} {1,l_{d_{1}}, l_{h_{1}}} \\ l_{h_{2}}, l_{d_{2}}, 0 \end{array}}\right] \\ &\quad \times e^{\frac {-\sqrt {\gamma }}{b_{e_{p}}\sqrt {{\bar {\gamma }_{e_{p}}}}} }d{\gamma }.\end{align*}$$ Using an analogous method to the one used to derive ${\mathcal {R}_{2}}$ , ${\mathcal {R}_{3}}$ is derived as $$\begin{align*} {\mathcal {R}_{3}}&=\int _{0}^{\infty }{\gamma }^{\frac {a_{r}+a_{e_{p}}+n}{2}}\mathrm {G}_{2r+1,6r+1}^{6r,1}\left [{\left ({\frac {B_{h{g_{p}}}\phi }{\gamma _{2}}}\right)\gamma \biggl | \begin{array}{c} {1,l_{d_{1}}, l_{h_{1}}} \\ l_{h_{2}}, l_{d_{2}}, 0 \end{array}}\right] \\ &\quad \times \mathrm {G}_{0,1}^{1,0}\left [{\left ({\frac {\sqrt {\gamma }}{b_{e_{p}}\sqrt {{\bar {\gamma }_{e_{p}}}}}}\right) \biggl | \begin{array}{c} {-} \\ 0 \end{array}}\right]d{\gamma } \\ &={\mathcal {Z}_{2}} \mathrm {G}_{6r+1,2r+3}^{3,6r}\left [{\frac {\left ({\frac {1}{b_{e_{p}}\sqrt {{\bar {\gamma }_{e_{p}}}}}}\right)^{2}}{4{B}_{h{g_{p}}}\phi }\gamma _{2} \biggl | \begin{array}{c} {l_{h_{5}},l_{d_{5}}, 1-{\mathcal {Z}_{3}}} \\ 0,- {\mathcal {Z}_{3}},l_{d_{6}}, l_{h_{6}} \end{array}}\right], \tag{43}\end{align*}$$ where ${\mathcal {Z}_{2}}={\pi }^{-\frac {1}{2}} \left ({\frac {\gamma _{2}}{B_{h{g_{p}}}\phi }}\right)^{\mathcal {Z}_{3}}$ , ${\mathcal {Z}_{3}}=\frac {a_{r}+a_{e_{p}}+n+2}{2}$ , $l_{h_{5}}=1-{\mathcal {Z}_{3}}-l_{h_{2}}, l_{d_{5}}=1-{\mathcal {Z}_{3}}-l_{d_{2}}, l_{h_{6}}=1-{\mathcal {Z}_{3}}-l_{h_{1}}$ , and $l_{d_{6}}=1-{\mathcal {Z}_{3}}-l_{d_{1}}$ .

5) Scenario-I (Asymptotic SOP)

For a better understanding of our analytical method in high SNR region, we derive asymptotic expressions of our lower bound SOP by assuming the condition $\gamma _{2}\rightarrow \infty$ . In general, the asymptotic expression of (40) is presented as $$\begin{equation*} SOP^{I,\infty } =\frac {{{\mathcal {M}_{1}} {\mathcal {R}_{1}}+{\mathcal {M}_{2}}{\mathcal {R}_{4}}-{\mathcal {M}_{3}}{\mathcal {R}_{5}}}}{2 b_{e_{p}}^{a_{e_{p}}+1} \Gamma (a_{e_{p}}+1) {\bar {\gamma }_{e_{p}}}^{\frac {a_{e_{p}}+1}{2}}}, \tag{44}\end{equation*}$$ Here, $\mathcal {R}_{4}$ is formed by transforming the Meijer’s $G$ term in (42) via utilizing [60, Eq. (19)] as $$\begin{align*} {\mathcal {R}_{4}}&={\mathcal {Z}_{1}}\sum _{y=1}^{6r}\Gamma (l_{d_{4},y})\Gamma (l_{h_{4},y})\left[{\frac {\gamma _{2}\left ({\frac {1}{b_{e_{p}}\sqrt {{\bar {\gamma }_{e_{p}}}}}}\right)^{2}}{4{B}_{h{g_{p}}}\phi }}\right]^{-l_{d_{4},y}-l_{h_{4},y}} \\ &\quad \times \frac {\prod _{\Theta =1,\Theta \neq y}^{6r}\Gamma (l_{d_{4},\Theta }-l_{d_{4},y})\Gamma (l_{h_{4},\Theta }-l_{h_{4},y})} {\prod _{\Theta =3}^{2r+3}\Gamma (l_{d_{3},\Theta }-l_{d_{4},y})\Gamma (l_{h_{3},\Theta }-l_{h_{4},y})}. \tag{45}\end{align*}$$ Similarly, $\mathcal {R}_{5}$ is found from (43) as $$\begin{align*} {\mathcal {R}_{5}}&={\mathcal {Z}_{2}}\sum _{y=1}^{6r}\Gamma (l_{d_{6},y})\Gamma (l_{h_{6},y})\left[{\frac {\gamma _{2}\left ({\frac {1}{b_{e_{p}}\sqrt {{\bar {\gamma }_{e_{p}}}}}}\right)^{2}}{4{B}_{h{g_{p}}}\phi }}\right]^{-l_{d_{6},y}-l_{h_{6},y}} \\ &\quad \times \frac {\prod _{\Theta =1,\Theta \neq y}^{6r}\Gamma (l_{d_{6},\Theta }-l_{d_{6},y})\Gamma (l_{h_{6},\Theta }-l_{h_{6},y})} {\prod _{\Theta =3}^{2r+3}\Gamma (l_{d_{5},\Theta }-l_{d_{6},y})\Gamma (l_{h_{5},\Theta }-l_{h_{6},y})}. \tag{46}\end{align*}$$

6) Scenario-II (Lower Bound SOP)

The expression of SOP of RIS-aided combined RF-FSO channel when the eavesdropper is at the FSO link can be described as [11, Eq. (13)] $$\begin{align*} {SOP}^{II} &=\mathrm {Pr}\left \{{C_{s c} \leq R_{s }}\right \} =\mathrm {Pr}\left \{{\gamma _{e q} \leq \phi \gamma _{\mathcal {E}_{Q}}+\phi -1}\right \} \\ &=\int _{0}^{\infty } \int _{\phi \gamma +\phi -1}^{\infty } f_{\gamma _{e q}}\left ({\gamma }\right) f_{\gamma _{e_{q}}}\left ({\gamma }\right) d \gamma _{e q} d \gamma \\ &=\int _{0}^{\infty } F_{\gamma _{d}}\left ({\phi \gamma +\phi -1}\right) f_{\gamma _{e_{q}}}\left ({\gamma }\right) d \gamma \\ &\quad \times (1-F_{\gamma _{d}}(\phi -1))+F_{\gamma _{d}}(\phi -1). \tag{47}\end{align*}$$ where $C_{sc}$ denotes the channel capacity for the $\mathcal {S}-\mathcal {R}-\mathcal {D}$ link. Due to mathematical process of obtaining the precise SOP closed-form formula being challenging, we deduce the mathematical expression of SOP at the lower-bound considering the variable gain relaying scheme as [11, Eq. (14)] $$\begin{align*} {SOP}^{II}\geq {SOP}^{L} &=\mathrm {Pr}\left \{{\gamma _{e q} \leq \phi \gamma _{e_{p}}}\right \} \\ &=\int _{0}^{\infty } F_{\gamma _{d}}\left ({\phi \gamma }\right) f_{\gamma _{e_{q}}}\left ({\gamma }\right) d \gamma \\ &\quad \times (1-F_{\gamma _{d}}(\phi -1))+F_{\gamma _{d}}(\phi -1). \tag{48}\end{align*}$$ Substituting Eqs. (24) and (26) into (48) and utilizing [62, Eq. (2.24.1.1)] yields $$\begin{align*} {SOP}^{II} &={\mathcal {S}_{1}} \sum _{m_{h}=1}^{\beta _{h}}\sum _{m_{g_{d}}=1}^{\beta _{g_{d}}}\sum _{m_{g}{}_{e_{q}}=1}^{\beta _{g}{}_{e_{q}}}{\mathcal {S}_{2}} {b_{m_{h}}^{2}} b_{m_{g_{d}}} b_{m_{g}{}_{e_{q}}} \\ &\quad \times \mathrm {G}_{8r+1,8r+1}^{6r+1,6r}\!\! \left [{\frac {{{\mathcal {B}_{1}\gamma _{2}}}}{\bar {\gamma }_{e_{q}}\phi } \biggl | \!\! \begin{array}{c} {1-l_{h_{2}},1-l_{d_{2}}, 1,l_{e_{q_{1}}},l_{h_{1}}} \\ l_{h_{2}}, l_{e_{q_{2}}}, 0,1-l_{h_{1}},1-l_{d_{1}} \end{array}\!\!}\right] \\ &\quad \times (1-F_{\gamma _{d}}(\phi -1))+F_{\gamma _{d}}(\phi -1), \tag{49}\end{align*}$$ where $$\begin{align*} \mathcal{B}_1 & =\left(\frac{B_{g_{e_q}}}{B_{g_d}}\right)^r, \\ \mathcal{S}_1 & =\frac{\xi_h^4 \xi_{g_d}^2 \xi_{g_{e_q}}{ }^2 A_h{ }^2 A_{g_d} A_{g_{e_q}}}{2^{2 r}}, \\ \mathcal{S}_2 & =\frac{r^{2 \alpha_h+\alpha_{g_d}+\alpha_{g_{e_q}}+2 m_h+m_{g_d}+m_{g_{e_q}}-5}}{4 \pi^{4(r-1)}}, \\ l_{e_{q_1}} & =\left(\frac{1+\xi_{g_{e_q}}{ }^4}{r}, \ldots, \frac{1+\xi_{g_{e_q}}{ }^2+r-1}{r}\right), \\ l_{e_2}= & \left(\frac{\xi_{g_{e_q}}{ }^2}{r}, \ldots, \frac{\xi_{g_{e_q}}{ }^2+r-1}{r}, \frac{\alpha_{g_{e_q}}}{r}, \ldots,\right. \\ & \left.\frac{\alpha_{g_{e q}}+r-1}{r}, \frac{m_{g_{e q}}}{r}, \ldots, \frac{m_{g_{e q}}+r-1}{r}\right), \\ F_{\gamma_d}(\phi-1)= & \sum_{n=0}^{\infty} \frac{(-1)^n\left(\frac{1}{b_r \sqrt{\gamma_r}}\right)^{a_r+n+1}(\phi-1)^{\frac{a_r+n+1}{2}}}{n !\left(a_r+n+1\right)} . \end{align*}$$

7) Scenario-II (Asymptotic SOP)

Applying some mathematical operations on the Meijer’s G function in (49) by using [44, Eq. (29)], the asymptotic expression of lower bound SOP for Scenario-II is obtained as $$\begin{align*} {SOP}^{II,\infty } &={\mathcal {S}_{1}} \sum _{m_{h}=1}^{\beta _{h}}\sum _{m_{g_{d}}=1}^{\beta _{g_{d}}}\sum _{m_{g}{}_{e_{q}}=1}^{\beta _{g}{}_{e_{q}}}{\mathcal {S}_{2}} {b_{m_{h}}^{2}} b_{m_{g_{d}}} b_{m_{g}{}_{e_{q}}} \\ &\quad \times \sum _{v=1}^{6r}\frac {\prod _{\iota =1,\iota \neq v}^{6r}\Gamma (\mathcal {V}_{1,v}-\mathcal {V}_{1,\iota })}{\prod _{\iota =6r+1}^{8r+1}\Gamma (1+\mathcal {V}_{1,\iota }-\mathcal {V}_{1,v})} \\ &\quad \times \frac {\prod _{\iota =1}^{6r+1}\Gamma (1+\mathcal {V}_{2,\iota }-\mathcal {V}_{1,v})}{\prod _{\iota =6r+2}^{8r+1}\Gamma (\mathcal {V}_{1,v}-\mathcal {V}_{2,\iota })} \left [{\frac {{{\mathcal {B}_{1}\gamma _{2}}}}{\bar {\gamma }_{e_{q}}\phi }}\right]^{\mathcal {V}_{1,v}-1}, \tag{50}\end{align*}$$ where $\mathcal {V}_{1}=(1-l_{h_{2}},1-l_{d_{2}}, 1,l_{e_{q_{1}}},l_{h_{1}})$ and $\mathcal {V}_{2}=(l_{h_{2}}, l_{e_{q_{2}}}, 0,1-l_{h_{1}},1-l_{d_{1}})$ .

8) Scenario-III

The lower bound of SOP for RIS-aided mixed RF-FSO framework with simultaneous eavesdropping attack via the RF and FSO links can be defined as $$\begin{equation*} {} {SOP}^{III}= 1-(SOP_{1}\times SOP_{2}), \tag{51}\end{equation*}$$ where $$\begin{align*} SOP_{1}= 1-\int _{0}^{\infty } F_{\gamma _{r}}\left ({\phi \gamma }\right) f_{\gamma _{e_{p}}}\left ({\gamma }\right) d \gamma, \tag{52}\\ SOP_{2}= 1-\int _{0}^{\infty } F_{\gamma _{d}}\left ({\phi \gamma }\right) f_{\gamma _{e_{q}}}\left ({\gamma }\right) d \gamma. \tag{53}\end{align*}$$ By placing (12) and (13) into (52) and utilizing [58, Eq. (3.326.2)], $SOP_{1}$ is expressed as $$\begin{align*} SOP_{1}&=1-\frac {1}{b_{e_{p}}^{a_{e_{p}}+1}{\bar {\gamma }_{e_{p}}}^{\frac {a_{e_{p}}+1}{2}}\Gamma (a_{e_{p}}+1)} \\ &\quad \times \sum _{n=0}^{\infty } \frac {(-1)^{n} \left ({\frac {\phi ^{\frac {1}{2}}}{b_{r} \sqrt {{\bar {\gamma }_{r}}}}}\right)^{a_{r}+1+n}\left ({{b_{e_{p}}\sqrt {{\bar {\gamma }_{e_{p}}}}}}\right)^{T_{1}}\Gamma (T_{1})}{n !\left ({a_{r}+n+1}\right)\Gamma (a_{r}+1)}, \tag{54}\end{align*}$$ where $T_{1}=(a_{r}+a_{e_{p}}+n+2)$ . Plugging (24) and (26) into (53), utilizing [62, Eq. (2.24.1.1)] to conduct integration and facilitating the expression, $SOP_{2}$ is obtained as $$\begin{align*} SOP_{2}&=1-{\mathcal {S}_{1}} \sum _{m_{h}=1}^{\beta _{h}}\sum _{m_{g_{d}}=1}^{\beta _{g_{d}}}\sum _{m_{g}{}_{e_{q}}=1}^{\beta _{g}{}_{e_{q}}}{\mathcal {S}_{2}} {b_{m_{h}}^{2}} b_{m_{g_{d}}} b_{m_{g}{}_{e_{q}}} \\ &\quad \times \!\mathrm {G}_{8r+1,8r+1}^{6r+1,6r}\!\left [{\!\!{\mathcal {B}_{1}}\frac {\gamma _{2}}{\bar {\gamma }_{e_{q}}\phi }\biggl | \begin{array}{c} {1-l_{h_{2}},1-l_{d_{2}}, 1,l_{e_{q_{1}}},l_{h_{1}}} \\ l_{h_{2}}, l_{e_{q_{2}}}, 0,1-l_{h_{1}},1-l_{d_{1}} \end{array}\!\!}\right]. \tag{55}\end{align*}$$

C. Strictly Positive Secrecy Capacity Analysis

The probability of SPSC serves to be one of the critical performance metrics that assures a continuous data stream only when the secrecy capacity remains a positive value in order to maintain secrecy in optical wireless communication. Mathematically, probability of SPSC is characterized as [40, Eq. (25)] $$\begin{equation*} SPSC=\Pr \left \{{C_{sc}>0}\right \}=1-SOP|_{R_{s}=0}. \tag{56}\end{equation*}$$ Formulation of probability of SPSC is readily obtained through substitution of the SOP formula from (40), (49), and (51) into (56). Hence, $$\begin{align*} {SPSC}^{I}& =1-{SOP}^{I}|_{R_{s}=0}, \text {(Scenario-I)} \tag{57}\\ {SPSC}^{II}& =1-{SOP}^{II}|_{R_{s}=0}, \text {(Scenario-II)} \tag{58}\\ {SPSC}^{III}& =1-{SOP}^{III}|_{R_{s}=0}. \text {(Scenario-III)} \tag{59}\end{align*}$$

D. Effective Secrecy Throughput (EST)

The EST is a gauge of performance indicator that clearly incorporates both tapping channel dependability and indemnity constraints. It essentially measures the mean rate at which secure data gets transmitted from the source to the destination with no interception. EST can be expressed numerically as [10, Eq. (5)] $$\begin{equation*} EST=R_{S}(1-SOP). \tag{60}\end{equation*}$$ Formulation of EST is readily obtained through substitution of the SOP formula in (60). Hence, $$\begin{align*} {EST}^{I}& =R_{S}(1-{SOP}^{I}), \text {(Scenario-I)} \tag{61}\\ {EST}^{II}& =R_{S}(1-{SOP}^{II}), \text {(Scenario-II)} \tag{62}\\ {EST}^{III}& =R_{S}(1-{SOP}^{III}). \text {(Scenario-III)} \tag{63}\end{align*}$$

E. Intercept Probability (IP)

IP is another significant performance metric that offers additional insights into the secrecy performance of a communication system. It refers to the probability that an eavesdropper successfully intercepts the data maintained at the actual receiving device. In this situation, the eavesdropper’s chances of successfully intercepting the legitimate message are notably high. Mathematically, it can be written as [66, Eq. (33)] $$\begin{align*} IP&=\text {Pr}\left \{{C_{sc} < C_{e}}\right \} \\ &=\text {Pr}\left \{{\gamma _{eq} < \gamma _{\mathcal {E}}}\right \} =SOP|_{R_{s}=0}, \tag{64}\end{align*}$$ where $C_{e}$ defines the channel capacity for the eavesdropper links. Similar to the probability of SPSC, IP can be demonstrated by substituting the analytical expressions of SOP from (40), (49), and (51) into (64). Hence, $$\begin{align*} {IP}^{I}& ={SOP}^{I}|_{R_{s}=0}, \text {(Scenario-I)} \tag{65}\\ {IP}^{II}& ={SOP}^{II}|_{R_{s}=0}, \text {(Scenario-II)} \tag{66}\\ {IP}^{III}& ={SOP}^{III}|_{R_{s}=0}. \text {(Scenario-III)} \tag{67}\end{align*}$$

F. Significance of the Derived Expressions

In this study, the derived expressions for metrics such as ASC, SOP, SPSC, EST, and IP serve as quantitative indicators of the system’s secrecy performance. These expressions not only validate the proposed theoretical framework’s accuracy but also offer a deeper understanding of the relationships between secrecy metrics and system parameters. Through these expressions, it becomes evident how factors like fading characteristics, turbulence conditions, pointing errors, and attack scenarios impact the system’s secrecy performance. This analysis facilitates insights into the system’s ability to maintain secure communication, aiding its practical applicability. Additionally, the expressions provide valuable guidance for network designers, offering clear directions for design choices and optimization strategies.

SECTION IV. Numerical Results

In this section, we present some numerical results based on the deduced analytical expressions of secrecy performance indicators i.e. SOP, SPSC, IP, ASC, and EST. Our analytical results are demonstrated for all three eavesdropping scenarios. To corroborate those analytical outcomes, we also exhibit MC simulations via generating Rician and Málaga random variables in MATLAB. This simulation is performed by averaging 100,000 channel realizations to get each secrecy indicator value. Unless stated otherwise, all the analytical results are obtained by considering $K_{1}=K_{r}=K_{e_{p}}=2$ , $\Omega _{1}=\Omega _{r}=\Omega _{e_{p}}=3$ , $\alpha _{h}=\alpha _{g_{d}}=\alpha _{g_{e_{q}}}=2.296$ , $\beta _{h}=\beta _{g_{d}}=\beta _{g_{e_{q}}}=2$ , $\xi _{h}=\xi _{g_{d}}=\xi _{g_{e_{q}}}=6.7$ , $\mathcal {N}_{1}=2$ , $\bar {\gamma }_{r}=\bar {\gamma }_{d}=10$ dB, $\bar {\gamma }_{e_{p}}=\bar {\gamma }_{e_{q}}=-5$ dB, $R_{s}=0.1$ , $(\alpha,\beta)$ = (2.296,2), (4.2,3), (8,4) for strong, moderate, and weak turbulence, and $r=1$ . Due to the rapid convergence of the infinite series, we opt to utilize only the initial 25 terms to compute the values of the secrecy metrics.

The impact of average SNR on EST performance for a selected range of target secrecy rate ($R_{s}$ ) is investigated in Figs. 2 and 3.

FIGURE 2.

The ${EST}^{I}$ versus $R_{s}$ for selected values of $\bar {\gamma }_{r}$ .

FIGURE 3.

The ${EST}^{III}$ versus $R_{s}$ for selected values of $\bar {\gamma }_{r}$ .

Both outcomes are analyzed through a range of average SNR values for the RF main channel, denoted as $\bar {\gamma }_{r}$ , ranging from −10 dB to 10 dB. In this context, Fig. 2 depicts Scenario-I (${EST}^{I}$ ), while Fig. 3 corresponds to Scenario-III (${EST}^{III}$ ). It is observed that both figures devise concave down-shaped curves where EST rises to a certain value of $R_{s}$ then declines afterward. Several reasons are responsible for this event. A lower value of $R_{s}$ requires fewer security maintenance resources, leading to increased EST. Furthermore, larger $R_{s}$ exacerbates the channel condition by introducing additional noise, interference, and fading in the system that has a notable impact on EST. Supposedly, the EST vs $R_{s}$ relation in Figs. 2 and 3 demonstrate the optimization between the secured transmission rate of the system and the required resources for maintaining security.

The impact of the shape parameter ($K$ ) and scale parameter ($\Omega$ ) in Rician fading distribution on secrecy performance is investigated in Figs. 4 and 5.

FIGURE 4.

The ${IP}^{I}$ versus $\bar {\gamma }_{r}$ for selected values of $\Omega _{1}, \Omega _{r}, \Omega _{e_{p}}$ , and $\bar {\gamma }_{e_{p}}$ .

FIGURE 5.

The ${IP}^{III}$ versus $\bar {\gamma }_{r}$ for selected values of $K_{1}, K_{r}$ , and $\bar {\gamma }_{e_{p}}$ .

In Fig. 4, ${IP}^{I}$ in Scenario-I decreases when $\Omega _{1}$ and $\Omega _{r}$ from $\mathcal {S}-\mathcal {L_{P}}-\mathcal {R}$ link are increased from 2 to 4, therefore, the secrecy performance improves. This behavior occurs because a higher scale parameter value improves signal quality by reducing signal attenuation, making the communication channel more reliable. For the same reason, secrecy performance downturns when $\Omega _{e_{p}}$ rises from 2 to 4 as it declines security of the main channel by strengthening $\mathcal {S}-\mathcal {L_{P}}-\mathcal {E_{P}}$ link. Additionally, the channel with a larger shape parameter experiences a lesser fading effect. This is because the line-of-sight signal is much stronger than the scattered signal due to the higher value of $K$ . This phenomenon is also justified in Fig. 5 for Scenario-III where system performance improves due to the higher values of $K_{1}$ and $K_{r}$ . Furthermore, it can be concluded from both graphs that improved channel quality is attainable through larger Rician fading parameters and lower $\bar {\gamma }_{e_{p}}$ .

Fig. 6 demonstrates a comparison between two detection techniques (i.e. HD and IM/DD) at the receiver for Scenario-I.

FIGURE 6.

The ${SOP}^{I}$ versus $\bar {\gamma }_{r}$ for selected values of $K_{e_{p}}$ and $r$ .

Conventionally, the HD technique has the ability of frequency shifting in a high-frequency range. Thus, HD is less susceptible to wiretapping and contains more secrecy advantages than IM/DD for the secured wireless channel. Fig. 6 upholds this agreement and our analysis supports the results as testified in [40]. It can also be observed from Fig. 6 that a higher value of $K_{e_{p}}$ lessens the fading effect of $\mathcal {S}-\mathcal {L_{P}}-\mathcal {E_{P}}$ link, thereby reinforcing wiretapping capability while diminishing overall system performance. The saturation of SOP at high SNR levels is readily apparent in the figure which is to be expected. The observed saturation phenomenon is primarily a result of the dominance of the weakest RF hop in this proposed RF-FSO configuration. Similar outcomes also observed in [41].

Fig. 7 shows the effect on SOP (Scenario-I) for different values of the SNR of $\mathcal {S}-\mathcal {L_{P}}-\mathcal {R}$ link and shape parameter of $\mathcal {S}-\mathcal {L_{P}}-\mathcal {E_{P}}$ link, i.e., $\bar {\gamma }_{r}$ and $K_{e_{p}}$ . Outcome shows that $\bar {\gamma }_{r} = 25$ dB exhibits better secrecy performance than $\bar {\gamma }_{r} = 18$ dB. However, system performance downturns while $K_{e_{p}}$ is increasingly tuned from 3 to 5 in both SNR cases. Evidently, both higher $\bar {\gamma }_{r}$ and lower $K_{e_{p}}$ not only increases the strength of $\mathcal {S}-\mathcal {L_{P}}-\mathcal {R}$ link but also lessens the impact of $\mathcal {S}-\mathcal {L_{P}}-\mathcal {E_{P}}$ link, hence lower SOP results. Moreover, the asymptotic expression explained in (44) is also implied in this figure, and result shows that asymptotic values match perfectly with analytical and simulation values in the higher SNR region.

FIGURE 7.

The ${SOP}^{I}$ versus $\bar {\gamma }_{d}$ for selected values of $K_{e_{p}}$ and $\bar {\gamma }_{r}$ .

The impact of FSO eavesdropper for Scenario-II is represented in both Figs. 8 and 9. It is observed that $EST^{II}$ in Fig. 8 is higher for $\bar {\gamma }_{e_{q}}=0$ dB compared to $\bar {\gamma }_{e_{q}}=5$ dB. Moreover, $SOP^{II}$ in Fig. 9 decreases when $\bar {\gamma }_{e_{q}}$ is tuned from 5 dB to −5 dB. These results are obvious as higher SNR of $\mathcal {R}-\mathcal {I}_{Q}-\mathcal {E_{Q}}$ link constantly increases the wiretapping capability of $\mathcal {E_{Q}}$ that ultimately reduces the secrecy performance. It is also seen that the asymptotic results in Fig. 9 reveal exact match with the analytical results in higher $\bar {\gamma }_{r}$ . This outcome is evident as both analytical and asymptotic results are theoretically bound to be equal while investigating the channel at high SNR region.

FIGURE 8.

The ${EST}^{II}$ versus $\bar {\gamma }_{d}$ for selected values of $\alpha _{h}, \alpha _{g_{d}}, \beta _{h}, \beta _{g_{d}}$ , and $\bar {\gamma }_{e_{q}}$ .

FIGURE 9.

The ${SOP}^{II}$ versus $\bar {\gamma }_{d}$ for selected values of $\alpha _{h}, \alpha _{g_{d}}, \beta _{h}, \beta _{g_{d}}$ , and $\bar {\gamma }_{e_{q}}$ .

The influence of the pointing error in the Málaga distribution on secrecy performance is studied in Figs. 10–​12 due to all three scenarios. The findings clearly demonstrate that secrecy performance for three eavesdropping scenarios upturns when the pointing error index ($\xi$ ) increases from 1.1 (severe pointing error state) to 6.7 (negligible pointing error state). This outcome is consistent while varying both the average SNRs of the RF link (Figs. 10 and 12) and FSO link (Fig. 11). It is a notable fact that ${SOP}^{III}$ in Fig. 12 downturns when $\mathcal {N}_{1}$ increases from 1 to 4. This occurrence serves as evidence that having a larger number of reflecting elements for the RIS-assisted fading model progressively enhances the system performance.

FIGURE 10.

The ${ASC}^{I}$ versus $\bar {\gamma }_{r}$ for selected values of $\alpha _{h}, \alpha _{g_{d}},\beta _{h}, \beta _{g_{d}}, \xi _{h}$ , and $\xi _{g_{d}}$ .

FIGURE 11.

The ${IP}^{II}$ versus $\bar {\gamma }_{d}$ for selected values of $\alpha _{h}, \alpha _{g_{d}}, \beta _{h}, \beta _{g_{d}}, \xi _{h}$ , and $\xi _{g_{d}}$ .

FIGURE 12.

The ${SOP}^{III}$ versus $\bar {\gamma }_{r}$ for selected values of $\mathcal {N}_{1}, \xi _{h}$ and $\xi _{g_{d}}$ .

The effect of atmospheric turbulence conditions on EST, SOP, IP (Scenario-II), and ASC (Scenario-I) is investigated in Figs. 8–​11. It can be observed that regardless of the different wiretapping scenarios, the secrecy performance of the proposed model improves when the turbulence condition of the FSO link shifts from strong turbulence to weak turbulence as demonstrated in [65]. This is due to the fact that the weak turbulence conditions allow for better control and compensation of atmospheric effects, leading to improved system performance in the FSO link compared to the more challenging conditions of severe turbulence.

A comparative analysis of Scenario-I, Scenario-II and Scenario-III for the proposed system model is presented in Fig. 13-​14. In Fig. 13, the probability of SPSC is plotted against the average SNR of $\mathcal {R}-\mathcal {L_{Q}}-\mathcal {D}$ link, i.e., $\gamma _{2}$ utilizing the derived expressions of (57)-(59). Conventionally, an FSO link is more secure and has lower susceptibility to eavesdropping compared to the RF link [41]. This can be proved also from Fig. 14. From the figure, it is seen that SOP for Scenario-I is higher than Scenario-II. Our analysis confirms the validity of this statement. Likewise, Scenario-I is worse than Scenario-II since the RF link is usually more vulnerable to wiretapping than the FSO link. However, from the Fig. 13, it is seen that the worst case is displayed by ${SPSC}^{III}$ (Scenario-III). The reason behind this case is both $\mathcal {E_{P}}$ and $\mathcal {E_{Q}}$ remain active simultaneously in Scenario-III that eventually creates the strongest form of eavesdropping for the channel.

FIGURE 13.

The ${SPSC}$ versus $\bar {\gamma }_{d}$ for different eavesdropping scenarios.

FIGURE 14.

The ${SOP}$ versus $\bar { \gamma _{r}}$ for different eavesdropping scenarios.

A. Design Guidelines

In this section, some important guidelines that can be utilized in the design of practical RF-FSO mixed systems are provided.

• Fig. 12 demonstrates the notable performance enhancement achieved by incorporating RIS in both RF and FSO links, affirming their valuable contribution to the practical design of a RF-FSO mixed model.

• Given the modeling of the FSO link with a Málaga-distributed turbulent channel, employing the HD technique at the receiver side offers a superior estimation approach for RF-FSO systems, effectively mitigating turbulence-induced fading while maintaining a high SNR.

• The impact of pointing error is pivotal in RF-FSO mixed systems, as demonstrated in Fig. 11. To mitigate this effect, it is advisable to employ larger diameter transmitters and receivers [67].

SECTION V. Conclusion

This study aimed at analyzing the security performance of a DF-based RIS-aided RF-FSO communication system in the presence of wiretapping attacks in both RF and FSO networks. We derived closed-form expressions for various performance metrics, such as ASC, EST, IP, probability of SPSC, and lower-bound SOP, to efficiently evaluate the impact of each parameter on the secrecy performance. The study validated its analytical outcomes using MC simulations. Numerical results reveal that fading severity, pointing errors, and natural turbulence parameters have a significant impact on secrecy performance. The study also explores the trade-off between the target secrecy rate and the resources required to maintain security via EST. Moreover, the result highlights the superiority of HD over IM/DD for optical signal detection. We also conducted a comparative analysis of three proposed wiretapping scenarios and concluded that simultaneous wiretapping has a more detrimental effect on the secrecy performance than individual wiretapping. Finally, it is claimed that the FSO link is less susceptible to wiretapping than the RF link.