1. Introduction

Due to the advantages of the unlicensed frequency spectrum, ease of deployment, small size, low cost, and high bandwidth (10 Mbps-10 Gbps), free-space optical (FSO) communication is widely deployed as a primary and secondary link for civilian, commercial, and military applications [1–3]. However, since the FSO link transmission medium is the atmosphere, it is by nature an open and unprotected link. In addition, the main lobe of the transmitter beam footprint is much wider than the dimensions of the receiver (Assuming a divergence angle of 1 mrad, the beam diameter for a 10 km link distance is 10 m), resulting in an FSO system that still risks having part of its radiated power captured by an eavesdropper. Scenarios in which FSO systems may be attacked have been reported in many previous articles [4–8]. With the development of high-powered Unmanned Aerial Vehicles (UAVs) and the miniaturization and lightweight of optical devices used for detection, it is entirely possible for UAVs equipped with special equipment (relay or Intelligent Reflecting Surfaces) to eavesdrop in FSO links [9–12]. Therefore, secure communication of the FSO systems remains a challenging task.

Recently, physical layer security (PLS) have been considered as a potential solution for countering adversary eavesdropping by exploiting the time-varying random nature of the wireless fading channel [4,6,13]. Compared to cryptography-based upper layer data encryption, PLS does not require additional bandwidth consumption. This feature results in more acceptance of using PLS for secure communication [14,15]. In this paper, we assume the three worst eavesdropping scenarios as shown in Fig. 1. Eve is between Alice and Bob, denoted as Scenario 1, Eve and Bob are in the same receiving plane, denoted as Scenario 2 and Eve is behind Bob, denoted as Scenario 3. The consideration of realistic eavesdropping scenarios has been a key issue in the research community on PLS security for FSO-based systems [4–7,16–18]. In PLS studies based on FSO systems and hybrid RF (radio frequency)/FSO systems, when FSO channels are considered in the system, authors typically assume that Scenario 2 is the more accepted [16–20]. However, it is a challenge to numerically investigate the PLS issues of the FSO system in Scenarios 1 and 3. In [4], Lopez-Martinez et al. investigate the positive non-zero secrecy capability (PNZSC) performance of eavesdroppers located close to legitimate transmitters and legitimate receivers respectively. Scenario 1 is beyond the scope of the discussion and is not mentioned in the article. In [6, Fig. 1], Lei et al. mentioned three possible eavesdropping scenarios. Among them, Scenario 1 and Scenario 3 are not mathematically modeled to discuss their security performance. In [7], Yun Ai et al. analyzed the PLS performance of the FSO system considering the impact of correlation under three different eavesdropping scenarios based on [4]. Due to the different distances of the legitimate receiver and the eavesdropper with respect to the transmitter, the path loss, pointing error, and atmospheric turbulence are in fact different on the main channel and the eavesdropping channel, respectively. Therefore, it is unreasonable to equate Scenario 1 and Scenario 3. Compared to [4,6,7], in this paper, we numerically study the PLS performance of the FSO system for possible eavesdropping scenarios. As a reaction to this challenge, we developed a general mathematical model to investigate the PLS issues, considering different eavesdropping scenarios.

 

Fig. 1. Diagram of Eve’s eavesdropping scenarios.

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There is no doubt that the reliability of FSO systems is susceptible to path loss, pointing errors, and atmospheric turbulence. Moreover, the PLS performance of FSO systems can also be affected by these factors. And, these randomness-induced fading are not foes but friends [21]. As far as path loss is concerned, it is considered in Scenario 2 [17,18,22]. Technically speaking, the differential effect of path loss on the secrecy capacity in this eavesdropping scenario can cancel each other out, i.e., the link distance is independent of the secrecy capacity in Scenario 2. But in Scenario 1 and Scenario 3, path loss is related to the link distance that has to be considered. Regarding the impact of pointing errors, in addition to the random angular jitter caused by building sway and building vibration, the non-zero boresight due to the thermal expansion of the building should also be considered [23]. In Scenario 2, the PLS performance study based on zero boresight pointing error is discussed in [16,19], and the non-zero boresight pointing error model that is more in line with the real scenario is not considered. Nevertheless, the most crucial factor affecting the PLS performance is the random variation of the refractive index of the atmospheric medium along the propagation path, i.e., atmospheric turbulence. In order to model this random fluctuation characteristic of the atmospheric channel under variable turbulence conditions, different kinds of mathematical models of the received irradiance probability density function (PDF) have been proposed so far [24–26]. Among them, the Log-Normal (LN) distribution for weak turbulent fluctuations and the Gamma-Gamma (G-G) distribution for weak to strong turbulent fluctuations are the most widely used. In [27], the authors describe the outage capacity performance of the FSO system considering zero-boresight pointing error and link blockage under the G-G distribution channel model. The authors consider that under the G-G distribution channel model the link must be interrupted whenever there is an obstacle in the link for blocking. As the fact stated in [28], the receiver can also receive the optical irradiance through the scattering component when atmospheric turbulence is modeled with the Málaga ($\mathcal {M}$)-distribution channel model in the case of blocking occurs. Therefore, considering link blockage in different eavesdropping scenarios, it is reasonable to model the attenuation of the optical signal caused by atmospheric turbulence with the $\mathcal {M}$-distribution channel model in this paper.

In our previous research article, we discussed the PLS performance of partially coherent beam (PCB)-based FSO systems through anisotropic non-Kolmogorov turbulent atmosphere in Scenario 2 [20]. The PLS performance of the FSO system can be improved by reducing the coherence of the light source, and with the development of measurement techniques, a growing body of experimental evidence and theoretical results demonstrate that both non-Kolmogorov properties and anisotropy are exhibited in atmospheric turbulence both in the stratosphere and within a few meters from the ground [29–31]. However, in Scenarios 1 and 3, compared to the published articles, the effect of anisotropic non-Kolmogorov atmospheric turbulence on the PLS performance of PCB-based FSO systems in the presence of partial link blockage is not considered.

To the best of our knowledge, there is no published literature discussing the PLS performance of PCB-based FSO communication system considering different eavesdropping scenarios and non-zero boresight pointing errors as well as path loss. An comprehensive study of different eavesdropping scenarios such as this work is important for the understanding and design guidelines of these systems. Motivated by the above, in this paper, we numerically explore the PLS performance of PCB-based FSO communication system over $\mathcal {M}$-distribution fading channel through anisotropic non-Kolmogorov turbulence considering three different eavesdropping scenarios over the whole FSO link. In this regard, the main contributions of this paper are the following:

2. System and channel models

2.1 System

In this paper, as shown in Fig. 1, we assume that in the considered FSO system the optical transmitter (Alice) sends a confidential message to the legitimate receiver (Bob), and there exists a potential eavesdropper (Eve) who can intercept a small portion of the power from Alice to decode the confidential information without being known by Alice at any location over the whole FSO link. Just as the legitimate transmitters and receivers in the FSO system model in common scenarios are static devices placed on top of buildings (except for pointing error effects), we also consider physically realizable devices suitable for eavesdropping to be static. This is the worst-case scenario we can consider when studying the eavesdropping scenario problem. In this paper, for the possible eavesdropping locations of the eavesdropper over the whole link, we define three different eavesdropping scenarios as follows

We consider the single-input single-output (SISO) FSO link of both Alice-Bob (the main channel) and Alice-Eve (the eavesdropping channel) using on-off keying (OOK) modulation with intensity-modulation and direct-detection (IM/DD) technique due to their simplicity and low cost. At the receiver, the optical-to-electrical conversion takes place, and the input-output (I/O) expression of the signal is as follows

2.2 Atmospheric turbulence model

The $\mathcal {M}$-distribution is suitable for all the atmospheric turbulence conditions of laser beam propagation in the atmosphere, as detailed in [36]. The main feature of this model is the definition of three components of laser beam propagation in the atmosphere as shown in Fig. 1: a line-of-sight (LOS) component (${U}_{L}$) and two scattering components, the first one of these two scattering components is the quasi-forward optical signal scattered by the eddies on the propagation axis ($U_{S}^{C}$), which is assumed to be coupled to the LOS component, while the second one is the classical independent off-axis non-LOS (NLOS) scattering component ($U_{S}^{G}$). In this paper, under Scenario 1, the LOS component and the scattered component coupled to the LOS are blocked by the eavesdropper and the legitimate receiver has to maintain link communication by receiving the optical irradiance through the off-axis NLOS scattering component. Scenario 3 is analyzed in the same way as Scenario 1, in contrast to Scenario 1 where Eve is blocked by Bob. Considering partial blockage, in [28], the normalized received irradiance ${{I}_{a,m}}$ of optical transmission through $\mathcal {M}$-distribution atmospheric turbulence is considered as the product of two independent random variables ${X}_{m}$ and ${Y}_{m}$, which represent irradiance fluctuations caused by large-scale and small-scale fluctuations, respectively, as follows

The PDF of the associated optical irradiance, $x$, can be expressed as

2.3 Path loss model

Path loss is caused by molecular absorption and aerosol scattering suspended in the air. The total path loss is given as [33]

2.4 Pointing error model

Following the similar analysis in [34], the fraction of the collected power at receiver can be approximated as in

In the most general case, we assume that both displacement are modeled as independent Gaussian random variables, i.e., ${{x}_{m}}\sim \mathcal {N}\left ( {{\mu }_{x,m}},\sigma _{x,m}^{2} \right )$ and ${{y}_{m}}\sim \mathcal {N}\left ( {{\mu }_{y,m}},\sigma _{y,m}^{2} \right )$. As described in [38], the Beckmann distribution can be approximated by a modified Rayleigh distribution as follows

2.5 Statistics of the combined effect

The joint PDF of channel gain ${{I}_{m}}={{I}_{a,m}}{{I}_{l,m}}{{I}_{p,m}}$ can be calculated as

Using (7), (16), and (18) can be rewritten as

Scenario 1 can be seen as a case of LOS blockage (Alice sends a confidential message to Bob in the presence of Eve eavesdropping in the middle of the FSO link). This means that Bob’s coherent component is blocked by Eve, i.e., in (3) when $b=0$, then $\Omega '=0$, and the signal can only be received by the incoherent component (scattered component) with the power of the independent scattered term ${\xi }_{g}$. As described in [28], the main channel parameters ${{p}_{B}}$ and ${{\tilde {m}}_{{{k}_{B}}}}$ mentioned above are forced to be 0 and 1, respectively. This shows that, regardless of the value of ${{\beta }_{M,B}}$ in the main channel, the only active Generalized-K sub-channel is the first-order channel in which the transmitted optical power is ${\xi }_{g}$, i.e., in Scenario 1, when $b=0$, then ${{\tilde {k}}_{B}}=1$, ${{p}_{B}}=0$, and ${{\tilde {m}}_{{{k}_{B}}}}=1$. Further, substituting the values of the above mentioned parameters ${{\tilde {k}}_{B}}$, ${{p}_{B}}$, and ${{\tilde {m}}_{{{k}_{B}}}}$ into (20), the novel CDF expression for the main channel under Scenario 1 is expressed as

Meanwhile, under Scenario 1, the eavesdropping channel is not blocked, and the PDF expression for the eavesdropping channel is obtained by substituting $m = E$ in (19) as follows

As shown in Fig. 1, in scenario 2, Bob and Eve are in the same receiving plane and they are not blocking each other. Similarly, the CDF expression for the main channel is obtained by substituting $m = B$ into (20) as follows

In scenario 2, since there is no blockage, the PDF expression for the eavesdropping channel is (22).

Similar to Scenario 1, as shown in Fig. 1, in Scenario 3, the main channel is blocked by no object, and the CDF expression of the main channel in Scenario 3 is (23). In Scenario 3, Eve receives the optical signal from Alice behind Bob, but the coherent component is blocked by Bob ($\Omega '=0$), i.e., when $b=0$, the parameters ${{\tilde {k}}_{E}}=1$, ${{p}_{E}}=0$, and ${{\tilde {m}}_{{{k}_{E}}}}=1$ of the eavesdropping channel mentioned in Scenario 1 are substituted into (19) to obtain the novel PDF expression of the eavesdropping channel as

The CDFs and PDFs of the main channel and the eavesdropping channel obtained in the above three eavesdropping scenarios will be used in the next section.

3. Performance analysis

3.1 Lower bound SOP

The secrecy rate denotes the maximum secrecy rate that can be achieved by the main channel in case of eavesdropping. The instantaneous secrecy rate ${C}_{s}$ of the considered eavesdropping model is defined as

Since the integral in Eq. (26) contains complex exponential function calculations, it is difficult to obtain an exact closed-form solution. According to [40], the lower bound of SOP ($SOP_{LB}$) can be obtained from (17) when $\gamma _{E}$ tends to $\infty$. Consequently, the $SOP_{LB}$ can be computed by

Substituting (21) and (22) into (27) and making use of the classical Meijer’s integral from two G-functions expressed in [39, Eq. (07.34.21.0011.01)], we obtain the $SOP_{LB}$ of Scenario 1 as

Substituting (22) and (23) into (27) and making use of the classical Meijer’s integral from two G-functions expressed in [39, Eq. (07.34.21.0011.01)], we obtain the $SOP_{LB}$ of Scenario 2 as

Substituting (23) and (24) into (27) and making use of the classical Meijer’s integral from two G-functions expressed in [39, Eq. (07.34.21.0011.01)], we obtain the $SOP_{LB}$ of Scenario 3 as

3.2 Asymptotic SOP analysis

FSO systems usually operate at a high SNR regime. Therefore, the approximation of the SOP lower-bound at a high SNR regime obtains a simpler asymptotic expression with an exponential form. At a high SNR regime, the SOP can be expressed as

Lema 1: The $SOP_{Asy}$ of the three Scenarios, $SOP_{Asy}^{S1}$, $SOP_{Asy}^{S2}$ and $SOP_{Asy}^{S3}$ are obtained as

Proof: See Appendix A.

4. Results and discussion

In this section, we show the simulation and analytic results of our system. If not specified, all the key system parameters are listed in Table 1, and the other parameters for the $\mathcal {M}$-distribution fading channel are set to $\Omega =0.5$, ${{\xi }_{gc}} =0.5$, and ${{\phi }_{L}}-{{\phi }_{C}}=\pi /2$. In this paper, we consider the value of ${\beta }_{M,m}$ as a natural number, so we round up the value obtained from (9). We use Monte-Carlo (MC) simulation method to verify the correctness of our derived analytical and asymptotic expressions. The number of simulated channel realizations is set to $10^8$.

Table 1. System parameters

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Remark 1: Generation of random variable (RV)

From Eq. (1), the combined effect gain between the transceivers is expressed as ${{I}_{m}}={{I}_{a,m}} {{I}_{p,m}} {{I}_{l,m}}={{X}_{m}}{{Y}_{m}}{{I}_{p,m}} {{I}_{l,m}}$. First, we generate Large-scale fluctuation ${X}_{m}$ follows Gamma distribution as ${X}_{m}$=gamrnd$({\alpha }_{M,m}, 1/{\alpha }_{M,m},10^8)$. Second, to generate the Small-scale fluctuation ${Y}_{m}$ RV, we first generate circular Gaussian complex RV, i.e., $U_{S}^{'}=\left ( {{U}_{S,comp}}+{{U}_{S,real}} \right )\sqrt {\frac {1-\Omega }{2}}$ to follow a Normal distribution as ${{U}_{S,real}}$= randn$(1,10^8)$ and ${{U}_{S,comp}}$= randn$(1,10^8)$, then, we generate $G$ following the Gamma distribution as $G$=gamrnd$({\beta }_{M,m}, 1/{\beta }_{M,m},10^8)$. The above RV are brought into ${Y}_{m}={{\left | b\left ( {{U}_{L}}+U_{S}^{C} \right )+U_{S}^{G} \right |}^{2}}$ to obtain the small-scale fluctuation. Third, the pointing error displacements are both independent Gaussian RVs for ${{x}_{m}}\sim \mathcal {N}\left ( {{\mu }_{x,m}},\sigma _{x,m}^{2} \right )$ and ${{y}_{m}}\sim \mathcal {N}\left ( {{\mu }_{y,m}},\sigma _{y,m}^{2} \right )$ brought into Eq. (13) to obtain ${I}_{p,m}$. ${I}_{l,m}$ is a constant value.

Since the optical transmission through the main channel and the eavesdropping channel is the same path, only their respective distances to the transmitter are different, so the atmospheric turbulence parameters (power-law $\alpha$, anisotropy coefficient $\xi$ and coupling parameter $\rho$) are assumed to be the same in this paper for all the scenarios considered. Furthermore, the main factors affecting the atmospheric turbulence parameters are wind speed and temperature as well as atmospheric pressure, therefore, it is reasonable to assume that the atmospheric conditions at two points separated by a very short distance are the same [24].

In all the figures, the MC simulation results fit well with the analytical and asymptotic results, and the asymptotic results match perfectly with the analytical results at a high SNR regime, which proves the correctness of our theoretical derivation. Furthermore, we can observe that the SOP performance $SOP_{Scenario3}>SO{{P}_{Scenario2}}>SO{{P}_{Scenario1}}$ for a given condition. This can be explained by the fact that in Scenario 1, the coherent component of Bob is blocked by Eve resulting in a less received power component of Bob compared to Scenario 2 in which Bob and Eve are not blocked by each other; in Scenario 3, the coherent component of Eve is blocked by Bob resulting in less received power component of Eve in Scenario 3, which results in the optimal SOP performance for Scenario 3. Finally, in the high SNR regime, the slope is the same in all the curves of the SOP performance with $\gamma _{B}$ (the slopes of the curves seem to be different under different conditions in Fig. 5 and Fig. 6; in fact, the slope is same as the value of $\gamma _{B}$ continues to increase, and since such a high SNR is meaningless in practical, it is not shown in this paper). To further verify the slope, we consider the horizontal boresight displacement $\mu _{x,B}$ = 0.5 m under Scenario 3 in Fig. 4(d), with the SOP values of $7.8203\times {{10}^{-5}}$ at $\gamma _{B}$ = 50 dB and $2.4378\times {{10}^{-5}}$ at $\gamma _{B}$ = 60 dB. The value of the slope is calculated as ${{\log }_{10}}7.8203\times {{10}^{-5}}-{{\log }_{10}}2.4378\times {{10}^{-5}}=0.5062$. This is consistent with the conclusion in [13], which verifies the theoretical asymptotic analysis.

The analytical, simulated, and asymptotic (at a high SNR regime) SOP versus Bob’s average transmitted electrical SNR ($\gamma _{B}$) of the FSO link are depicted in Fig. 2, which considers the variation of Eve’s eavesdropping location $L_{E}$ for different eavesdropping scenarios. First, the increase of $\gamma _{B}$ makes the SOP performance better in all scenarios. Second, in Fig. 2(a), the SOP performance gets better with the increasing $L_{E}$ in Scenario 3, and more interestingly we find that the SOP performance deteriorates and then gets better with the increasing $L_{E}$ in scenario 1; in Fig. 2(b), we set the $\gamma _{B}$ = 20 dB and other parameters remain unchanged to depict the SOP variation curve with $L_{E}$ for different source coherence lengths $l_{c}$ in scenario 1. This is because, in the PCB-based FSO systems, the scintillation index of the eavesdropping channel decreases and then increases as Eve’s eavesdropping location $L_{E}$ increases [41,42]. Also, in Fig. 2(b), we can observe that increasing the coherence length of the light source can also improve the SOP performance. Compared to [4] and [7], in Scenario 1, firstly, the SOP performance of the PCB-based FSO system can be derived numerically using our model. Second, the property that the scintillation index factor of the PCB-based FSO system becomes smaller and then larger as the communication distance becomes larger makes it necessary to focus on the worst position of the link where the eavesdropper is located when designing the PLS of the PCB-based FSO. This significant conclusion is not found in previous published literature.

 

Fig. 2. Impact of Eve’s eavesdropping location $L_{E}$ and source coherence length $l_{c}$ on the SOP performance. (a) SOP versus $\gamma _{B}$ for different eavesdropping scenarios and different Eve’s eavesdropping location $L_{E}$; (b) the SOP variation curve with $L_{E}$ for different source coherence lengths $l_{c}$ in scenario 1.

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In Fig. 3, we depict the variation of SOP versus $\gamma _{B}$ for different eavesdropping scenarios and different source coherence lengths $l_{c}$. In Scenario 1, we observe that increasing the source coherence length $l_{c}$ has a weak improvement on SOP performance. However, in Scenarios 2 and 3, the SOP performance can be significantly improved by increasing $l_{c}$. This is because in Scenario 1, the coherent component of the small-scale fluctuations of the main channel is blocked by Eve, and the optical irradiance can only be received through the non-coherent component. As described in [32], the contribution to the small-scale fluctuation scintillation index is obtained from the sum of all the $k$ sub-channels (from 1 to $\beta _{M,B}$), when link blockage is present, regardless of the $\beta _{M,B}$ value of the main channel, the only active Generalized-K sub-channel is the first-order one, in which the transmitted optical power is ${{\mu }_{1,B}}={{\xi }_{g}}$. Therefore, the reason for the small effect of $l_{c}$ on the SOP performance improvement under Scenario 1 is since in this case $l_{c}$ only affects the SOP performance of one sub-channel, i.e., the first-order one.

 

Fig. 3. SOP versus $\gamma _{B}$ for different eavesdropping scenarios and different source coherence length $l_{c}$.

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Figures 4(a)–4(d) depict the change in SOP as a function of Bob’s electrical SNR ($\gamma _B$) in the absence of fading for different eavesdropping scenarios and different values of power-law $\alpha$ in Fig. 4(a), anisotropy coefficient $\xi$ in Fig. 4(b), refractive index structure parameter $C_{n}^{2}$ in Fig. 4(c), and horizontal boresight displacement $\mu _{x,B}$ in Fig. 4(d). The same conclusions as in Fig. 3 can be drawn in all the plots of Fig. 4. In Fig. 4(a) and Fig. 4(b), we can see that an increase in the parameters $\alpha$ and $\xi$ leads to a better SOP performance, and conversely, in Fig. 4(c) and Fig. 4(d), an increase in parameters $C_{n}^{2}$ and $\mu _{x,B}$ deteriorates the SOP performance. This is because the increasing values of parameters $\alpha$ and $\xi$ imply better weather conditions, while the increasing values of $C_{n}^{2}$ and $\mu _{x,B}$ account for the deterioration of weather and the increasing parameters related to pointing errors, respectively. More importantly, in Scenario 1, changing the weather conditions and boresight displacement is not significant for improving the SOP performance, and only changing these parameters mentioned above is meaningful in Scenario 2 and Scenario 3. Compared to [16] and [43], in Scenario 2 although the PLS performance of $\mathcal {M}$-distribution is discussed, the effect of weather-related parameters and boresight displacement parameters related to pointing errors on PLS performance in the presence of link blockage in Scenarios 1 and 3 is not considered. Importantly, in Scenarios 1 and 3, it is concluded that these parameters have different effects on the PLS performance of the PCB-based FSO system. Compared to [16] and [43], our model is a generalized model used to evaluate the PLS of the FSO system.

The variation curves of SOP with Bob’s electrical SNR ($\gamma _B$) in the absence of fading for different eavesdropping scenarios and different values of jitter variance $\sigma _s$ are illustrated in Fig. 5. From Fig. 5, we can conclude that the increase of the value of $\sigma _s$ deteriorates the SOP performance in different scenarios. Compared to other parameters related to the light source and atmospheric turbulence, Scenario 1 is more sensitive to the change of parameter $\sigma _s$ with $\Delta SO{{P}_{Scenario3}}>\Delta SO{{P}_{Scenario2}}>\Delta SO{{P}_{Scenario1}}$. In principle, this is due to the fact that, unlike the parameters mentioned above, when analyzing the presence of blockage, the blockage does not cause a change of $\sigma _s$ on the received optical power of the FSO system. And, as described in (14), $\sigma _s$ is gradually increasing as the distance of the eavesdropper Eve from Alice increases ($L_{E}$), which corroborates our results.

Figure 6 presents the SOP performance as a function of Bob’s electrical SNR ($\gamma _B$) for different eavesdropping scenarios and coupling parameter $\rho$. The value of $\rho$ is taken from 0 to 1, as the fact described in [28], when $\rho$ = 0, the PDF model used to simulate atmospheric turbulence is the K distribution, and when $\rho$ = 1, the PDF model is the G-G distribution. When $\rho$ = 1 means that the link is completely blocked, communication is interrupted, so we use $\rho$ = 0.99 to approximate the G-G distribution. In Fig. 6(a), under Scenario 1, we find that the SOP performance deteriorates as the value of $\rho$ increases, in contrast, under Scenario 2 and Scenario 3 in Fig. 6(b) and Fig. 6(c), we find that SOP performance becomes better as the value of $\rho$ increases. This is because, in Scenario 1, the main channel is blocked by Eve, and the increase in the value of $\rho$ means that the proportion of optical power coupled into the blocked coherent component increases making the optical power received by Bob lower leading to the deterioration of SOP performance of the FSO system, while in Scenario 2 and Scenario 3, the main channel is not blocked by Eve and the increase in the value of $\rho$ means that the proportion of optical power coupled into the coherent component increases making the optical power received by Bob higher leading to the improvement of the SOP performance of the FSO system.

 

Fig. 4. SOP versus $\gamma _{B}$ for different eavesdropping scenarios and different values of (a) power law $\alpha$; (b) anisotropy coefficient $\xi$; (c) refractive index structure parameter $C_{n}^{2}$ and (d) horizontal boresight displacement $\mu _{x,B}$.

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Fig. 5. SOP versus $\gamma _{B}$ for different eavesdropping scenarios and different values of jitter variance $\sigma _s$; (a) Scenario 1, (b) Scenario 2, and (c) Scenario 3.

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Fig. 6. SOP versus $\gamma _{B}$ for different eavesdropping scenarios and different values of coupling parameter $\rho$; (a) Scenario 1, (b) Scenario 2, and (c) Scenario 3.

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5. Conclusion

In this paper, we have numerically further investigated the SOP performance of the FSO system based on our previous work, considering different eavesdropping scenarios and using a more suitable $\mathcal {M}$-distribution channel model for realistic scenarios for weak to strong atmospheric turbulence. First, the analytical expressions have been proved to be correct by the MC and the derived asymptotic expressions. Second, the slope of the curve describing the variation of the SOP performance with Bob’s electrical SNR ($\gamma _B$) is the same for all scenarios at a high SNR regime. Third, under Scenario 1, there are two important conclusions: (1) The SOP performance changes are not sensitive to source coherence length $l_{c}$, power-law $\alpha$, anisotropy coefficient $\xi$, refractive index structure parameter $C_{n}^{2}$ and horizontal boresight displacement $\mu _{x,B}$, but the SOP performance is sensitive to jitter variance $\sigma _s$ and coupling parameter $\rho$, as the values of $\sigma _s$ and $\rho$ increase the SOP performance deteriorates; (2) The SOP performance deteriorates and then gets better with the increasing $L_{E}$ of Eve’s eavesdropping location. Finally, under Scenario 2 and Scenario 3, the SOP performance is sensitive to all the mentioned parameters, and increasing the values of Eve’s eavesdropping location $L_{E}$, source coherence length $l_{c}$, power-law $\alpha$, anisotropy coefficient $\xi$ and coupling parameter $\rho$ can improve the SOP performance, while on the contrary, increasing the values of refractive index structure parameter $C_{n}^{2}$, horizontal boresight displacement $\mu _{x,B}$ and jitter variance $\sigma _s$ deteriorate the SOP performance. Of course, Eve and Bob can also exist in Scenario 1 and Scenario 3 where the coherent components are not blocked by each other, and the SOP of such a scenario can be derived by our previous work in [20], just by bringing in the distance of Eve’s eavesdropping location. The important findings of this paper can help telecom engineers in designing PLS schemes for FSO systems considering different eavesdropping scenarios.