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Abstract
In this paper, the physical layer security (PLS) of fractional orbital angular momentum (OAM) multiplexing under atmospheric turbulence channels is studied. Based on the PLS theory, the secrecy capacities and the probabilities of positive secrecy capacities of fractional OAM (FrOAM) multiplexing systems with different topological charge intervals are analyzed. The influence of the eavesdropping ratio and the power allocation on secrecy capacities are compared. The simulation results show that, under the finite aperture limitation, the FrOAM multiplexing technique provides higher security over the integer OAM multiplexing in terms of the total secrecy capacities under weak and medium turbulence.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, with the increasing scarcity of radio spectrum resources, free space optical (FSO) communication has become an alternative for data centers and cellular networks with its advantages of broad bandwidth, high capacity and easy deployment [1,2]. While the FSO link has dramatically improved the communication capacity, the security issue of FSO systems has also received more attentions.
Since Wyner proposed the wiretap channel, the physical layer security (PLS) of communication has become a serious problem [3]. Despite the high directivity of laser beams in FSO links, eavesdroppers can still perform eavesdropping by intercepting a portion of the transmitted beam energy on the transmission path [4–6]. With the large-scale deployment of FSO links, PLS of optical wireless communications has also received much attention [7–9]. Lopez et al. studied the PLS of Gaussian optical communication based on On-Off Keying (OOK) modulation and the probability of positive secrecy capacity [10]. Besides, different cases of PLS in visible light communication (VLC) systems and FSO networks have been extensively presented [11–13].
The PLS issue of orbital angular momentum (OAM) multiplexing has received much attentions recently. Sun and Djordjevic studied the effects of power cost schemes and OAM transmit mode sets on the PLS of integer-order OAM multiplexing [14]. Since then, the PLS of various types of OAM multiplexing systems has been studied in depth, including terahertz (THz) FSO multiplexing systems [15–17], Bessel-Gauss multiplexing systems [18,19], and so on.
In FSO systems, OAM beams have been highly concerned with their stable propagation characteristics and additional angular momentum degrees [20–22]. OAM beams have not only received extensive attention in FSO communication systems, but also been applied in fibre-based optical quantum communication systems [23–25]. In recent years, with the further demand for communication capacity and security, researchers found that fractional orbital angular momentum (FrOAM) multiplexing can reach higher spectral efficiency than integer OAM multiplexing under finite aperture limitations [26,27]. FrOAM multiplexing can bring extra capacity gain, but it is more susceptible to atmospheric turbulence [28,29]. Therefore, the influence of atmospheric turbulence must be considered when analyzing the performance of FSO links. At present, many researches on OAM multiplexing under atmospheric turbulence channels adopt Kolmogorov turbulence. The influence of atmospheric turbulence on the communication capacity of OAM multiplexing has been quantified [30–32]. However, the PLS issue of FrOAM multiplexing under atmospheric turbulence channels still remains much to explore.
In this paper, the PLS of FrOAM multiplexing systems with different topological charge intervals under atmospheric turbulence channels are studied. The structure of the rest of this paper is as follows. The mode transmitting model of FrOAM beams is analyzed in Section 2. In Section 3, the system model and the atmospheric turbulence model are described. Simulation results are presented and analyzed in Section 4. Finally, the conclusions are drawn in Section 5.
2. Basic theory
Light beams carrying OAM include Laguerre-Gaussian (LG) beams, Bessel-Gaussian (BG) beams, Hypergeometric Gaussian (HyGG) beams, and so on. Among these OAM beams, LG beams are considered by many researchers as the best candidates for FSO link because of its natural generation and stable propagation characteristics. The optical field distribution of an LG beam in a cylindrical coordinate is [29]
In an OAM multiplexing system, the inner product of two designated OAM beams, which can be obtained through an optical correlator, is usually used to measure the interference between these two beams. The inner products of all the beams used in the system form the mode transmission matrix $\mathbf {H}$, which will be discussed in Section 3. When ignoring the atmospheric turbulence, the inner product of two LG beams in a finite circular aperture of radius $r_a$ is described as [33]
When analyzing the PLS of FrOAM multiplexing systems, the results of Eq. (2) are used to describe the mode transmission matrix of the eavesdropper who does not suffer from atmospheric turbulence when being located near the legitimate transmitter.
3. System model
3.1 Fundamental structure
The fundamental structure of an OAM multiplexing system for PLS analyzing is shown in Fig. 1. Multiple OAM beams are combined and transmitted coaxially through the transmitting aperture to the atmospheric channel. The transmitted coaxial beams are received at the receiver through the receiving aperture and splitted by beam splitters. The splitted beams are processed by optical correlators to obtain the inner product of the corresponding beams. Then the PLS characteristics can be analyzed.
Fig. 1. Schematic diagram of OAM multiplexing system under atmospheric turbulence channel. BS: beam splitter, M: mirror.
In this paper, we focus on the influence of the topological charges on the PLS of FrOAM systems. Hence, the OAM beams are set to be with the same wavelength. In order to receive most energy of the OAM beam at the receiving end, the size of the receiving aperture has to increase as the transmission distance increases. However, due to the hardware foundation, the lens size of some systems is limited to a certain range. The apertures are all circular.
3.2 Atmospheric turbulence model
The most influential factor in the FSO channel is atmospheric turbulence. The random variation of atmospheric turbulence distorts the distributions of OAM beams, which in turn affects the beam correlations and degrades the system performance. In order to simulate the effect of atmospheric turbulence, an appropriate number of atmospheric turbulence phase screens should be placed in the transmission path [11]. The beam transmission distance is set as 1km, and 11 turbulent phase screens are placed at equal intervals in the transmission range. The beam follows the free-space diffraction theorem between two screens. Then sequentially performing the simulated transmission of the atmospheric turbulence channel through all the atmospheric turbulence phase screens and free space. This process is also called split-step propagation method [34].
In this paper, the method in [35] is used to generate atmospheric turbulence phase screens. The Kolmogorov refractive index power spectral density is
where $\kappa = 2\pi ( f_x\mathbf {\hat {i}} +f_y\mathbf {\hat {j}})$ is the angular spatial frequency. $C_n^{2}$ is the refractive structure constant which characterizes the turbulence strength and coherence distance. The coherence distance can be expressed as $r_0=0.185\left (\lambda ^{2}/\int _z^{z+\Delta z}C_n^{2}(\xi )\textrm {d}\xi \right )^{3/5}$. The theoretical phase structure function corresponding to Eq. (6) is The simulation results of the phase distribution and the phase structure function are shown in Fig. 2. The coherence length $r_0$ of the turbulence is 0.1m and $z=1000$m. Figure 2(a) shows an example of the phase distribution of the phase screen. Figure 2(b) is a comparison of the simulated average phase structure function and the theoretical phase structure function of Eq. (7).Fig. 2. Simulation results of the atmospheric turbulence phase distribution with $r_0=0.1$m based on FFT compensation. (a) An example of the atmospheric turbulence phase distribution. (b) Comparison of the simulated and the theoretical phase structure functions.
It can be seen from Fig. 2(b) that the simulated phase screen structure function is very close to the theoretical value and meet the simulation conditions.
3.3 Capacity analytic model
An OAM multiplexing system with $N_\textrm {T}$ transmitted modes and $N_\textrm {R}$ detecting modes is equivalent to a multiple-input-multiple-output (MIMO) system with $N_\textrm {T}$ transmitting antennas and $N_\textrm {R}$ receiving antennas. In an integer OAM multiplexing system, $N_\textrm {T}$ must equals to $N_\textrm {R}$, because of the orthogonalities of the integer OAM beams. In an FrOAM multiplexing system $N_\textrm {T}$ and $N_\textrm {R}$ can be different. In this paper, only the case of $N_\textrm {T}=N_\textrm {R}=M$ is considered.
For an OAM multiplexing system with $N_\textrm {T}$ transmitted modes and $N_\textrm {R}$ detecting modes, the channel matrix $\mathbf {H}$ has the following form as
At different times, the phase distribution of the atmospheric turbulence will result in different channel matrices, which means that the capacity of the system varies randomly. Therefore, measuring the channel capacity under the atmospheric turbulence channel requires time averaging, that is
where $E\{\cdot \}$ is the mean function through time.4. Physical-layer security analysis
4.1 Secrecy capacity model
In this section we consider the scenario in which there are two legitimate peers, say Alice (transmitter) and Bob (receiver), and an eavesdropper, say Eve. It is assumed that Eve can get the mode set information of Alice and Bob. Moreover, considering the worst conditions for secure communication, Eve locates close to Alice so that the information received by Eve has access to perfect OAM channels without suffering from the turbulence. Eve gets the transmission information by intercepting a part of the transmission power. Assume that the power ratio intercepted by Eve is $r_e$, and the power ratio of Bob is $(1-r_e)$. The capacities of Bob and Eve are
The secrecy capacity is defined as the portion of the sent message that Eve cannot extract any useful information [14], which is calculated from Eq. (12) and Eq. (13) according to
4.2 Simulation results and analysis
In this section, the PLS characteristics of FrOAM multiplexing systems are simulated and analyzed. One integer OAM multiplexing system $S$ is simulated as a standard. Three FrOAM multiplexing systems, $S1$, $S2$ and $S3$ are analyzed in detail. The topological charge sets and the corresponding topological charge intervals are listed in Table 1. Since the topological charge interval is the key parameter to determine the interference between OAM beams and the number of OAM beams in a limited aperture of an FrOAM multiplexing system, the topology charges for each system are set to not greater than 2 for simplicity.
Table 1. Topological charge sets and key parameters of the simulated systems
The refractive structure constant set of the atmospheric turbulence is $C_n^{2}=\{1\times 10^{-16}, 5\times 10^{-16}, 1\times 10^{-15}, 5\times 10^{-15}, 1\times 10^{-14}, 5\times 10^{-14}, 1\times 10^{-13}, 5\times 10^{-13}\}$. The wavelengths of the beams are set to be the same as 1550nm. We performed 6,000 transmission experiments under each turbulence intensity. The channel capacities of different systems under different atmospheric turbulence conditions are averaged.
4.2.1 System capacity
Here we consider the performance of the systems using two different power cost schemes as those in [14]. The main difference between this paper and [14] is that the systems studied in this paper utilize FrOAM beams with topological charge intervals less than one. Ignore the energy loss during transmission and set the received signal-to-noise ratio of each mode as 20dB. The first scenario is the equal channel power (ECP) scheme. In this power allocation scenario, the received power of each mode is $P_\textrm {R}$. The total power of the system increases linearly with the number of transmission modes M and the total power of the system is $M\times P_\textrm {R}$. The second scenario is the fixed system power (FSP) scheme. In this power allocation scenario, the total power of the system is $P_\textrm {R}$. As the number of modes $M$ increases, the transmit power received by each mode decreases, and the received power received by each mode is $P_\textrm {R}/M$. The simulation results of the average system capacities of different systems with different power allocation schemes are shown in Fig. 3.
Fig. 3. Comparison of channel capacity between integer OAM multiplexing systems and FrOAM multiplexing systems with different power allocation schemes. ECP: equal channel power, FSP: fixed system power.
As can be seen from Fig. 3, in the ECP scenario, System $S3$ with 11 FrOAM beams and System $S2$ with 7 FrOAM beams perform better than the integer OAM multiplexing system $S$ in system capacity. The system capacities increase with the number of the multiplexed FrOAM beams under finite aperture limitation. Inversely, in the FSP scenario, an increase of the number of the FrOAM beams brings no benefit compared with the integer OAM multiplexing system $S$. Note that the capacities of the integer OAM multiplexing system $S$ are always higher than those of the FrOAM system $S1$ of the same number of OAM modes.
4.2.2 Secrecy capacity analysis
In this section, the aggregate secrecy capacities of each system under seven turbulence intensities are investigated. The power intercepted by Eve in each system is $1\%$. Aggregate secrecy capacity is the summation of the secrecy capacity for all multiplexed channels, and the results are shown in Fig. 4.
Fig. 4. Aggregate secrecy capacities of different transmitted modes sets varies with turbulence intensity. ECP: equal channel power scheme. FSP:fixed system power.
As can be seen from Fig. 4, under the finite aperture limitation, whether in the ECP scenario or the FSP scenario, the total secrecy capacity increases with the number of the FrOAM transmitted modes. It is worth noting that the secrecy capacity of the integer OAM multiplexing system $S$ is larger than the FrOAM multiplexing system $S1$ due to orthogonality. When $C_n^{2}\geq 5\times 10^{-13}\textrm {m}^{-2/3}$, there is a significant change. As the turbulence intensity increases, the total secrecy capacity of the system $S3$ with the most modes, i.e., the smallest topological charge interval, decreases faster than the others. The results mean that FrOAM multiplexing systems with small topological charge intervals are affected more under strong turbulence.
Since FrOAM multiplexing can provide more modes than integer OAM multiplexing under finite aperture limitations, it is foreseeable that multiplexing more FrOAM modes can bring secrecy capacity growth. However, in actual deployment, as the number of multiplexing modes increases, the complexity of the system increases, so it is necessary to consider the compromise between the secrecy capacity requirement and the system complexity.
Multiplexing more FrOAM modes can bring about an increase in aggregate secrecy capacity of the system under finite aperture limitations. However, as the number of multiplexing modes increases, the interference between FrOAM modes increases. To evaluate this kind of influence, the average channel capacities of different systems in the same channel power scenarios are calculated. The results are shown in Fig. 5.
Fig. 5. Comparison of the aggregate secrecy capacities and the average secrecy capacities per channel of different systems under ECP scheme. The solid lines are the aggregate secrecy capacities and the dotted lines are the average secrecy capacities per channel.
It can be evaluated from Fig. 5 that as the topological charge interval decreases, the average secrecy capacity that can be transmitted by each channel decreases. However, the transmission set with smaller interval has more transmission modes, and the aggregate capacity is larger than the systems with large intervals in weak turbulence and medium-intensity turbulence. In the strong turbulent region, the average secrecy capacity of each channel has a sharp drop in the system with small mode interval. Meanwhile, the aggregate secrecy capacity of the system with a smaller mode interval decreases steeply. Besides, it can be seen that integer OAM multiplexing always has an advantage in single channel capacity than FrOAM multiplexing.
In the RF domain, Eve and Bob use different channels, while in FSO systems, Eve shares the same information transmission path with Bob. For Eve, the closer to the emitter, the more possible it is to get the message. However, Bob’s received power will decrease as Eve intercepts more light power. When Bob has a significant continuous reduction in the receiving power, the existence of eavesdropping will be detected. Therefore, we set the maximum eavesdropping ratio of Eve to 0.1. We studied the secrecy capacities of the four systems $S$, $S1$, $S2$ and $S3$. The results are shown in Fig. 6.
Fig. 6. Changes in secrecy capacities of the systems with different topological charge sets and different interception ratios. (a) System $S$. (b) System $S1$. (c) System $S2$. (d) System $S3$. ECP: equal channel power. FSP: fixed system power.
It can be seen from Fig. 6 that the secrecy capacities decrease with Eve’s interception ratio increases. The secrecy capacities approach to zero when Eve’s interception ratio reaches a certain level during strong turbulence. At this point, the information acquired by Eve is more than that of Bob. Then the communication security of the system is no longer guaranteed. Communication between Alice and Bob will be severely damaged and they do not know the existence of Eve. The results also show that the ECP scheme has more advantages in secrecy capacity under weak and medium turbulence. In the strong turbulence region, when the power interception ratio of Eve is greater than $1\%$, FSP scheme performs better than ECP scheme in secrecy capacity.
4.2.3 Statistical feature of secrecy capacity
Turbulence disturbances cause the secrecy capacities of FrOAM multiplexing systems to vary randomly. It can be seen from Fig. 6 that when the turbulence intensity reaches a certain level, the information that Eve can receive is more than the information received by Bob, so the mean value of the secrecy capacity does not represent the security of the communication. Here, the probability of positive secrecy capacity is used to describe the reliability of secure communication, which is defined as [10]:
The results of the probabilities of positive secrecy capacities of the systems $S$, $S1$, $S2$ and $S3$ under different turbulence intensities are shown in Fig. 7.As can be seen from Fig. 7, as Eve’s interception ratio increases, the probability of positive secrecy capacity decreases. When the turbulence intensity is weak, the probability of positive secrecy capacity of each system is above $75\%$. As the turbulence intensity increases, the probability of positive secrecy capacity $P_\textrm {S}^{+}$ of each system decreases rapidly. When $C_n^{2}\geq 5\times 10^{-13}\textrm {m}^{-2/3}$, $P_\textrm {S}^{+}$ reaches zero at a certain interception ratio. In addition, it can be seen from Fig. 7 that under the same turbulence intensity, the probabilities of positive secrecy capacity of the systems with FSP are higher than those of the systems with ECP.
Fig. 7. Relation between Eve’s interception ratio and the probabilities of positive secrecy capacity of OAM multiplexing systems with different topological charge sets under different turbulence intensities. (a) $C_n^{2}=1\times 10^{-14}\textrm {m}^{-2/3}$. (b) $C_n^{2}=5\times 10^{-14}\textrm {m}^{-2/3}$. (c) $C_n^{2}=1\times 10^{-13}\textrm {m}^{-2/3}$. (d) $C_n^{2}=5\times 10^{-13}\textrm {m}^{-2/3}$. ECP: equal channel power. FSP: fixed system power.
5. Conclusion
In this paper, the physical layer security of FrOAM multiplexing communication systems under atmospheric turbulence channel is investigated. Firstly, the simulation results show that, under finite aperture limitations, FrOAM multiplexing can increase aggregate secrecy capacity in weak and medium turbulence regimes. Secondly, when the proportion of eavesdropping reaches a certain high level, legitimate communication link is not safe. Finally, we evaluated the probability of positive secrecy capacity. The results show that there is no obvious difference between integer OAM multiplexing and FrOAM multiplexing in the probability of positive secrecy capacity. Besides, the FSP power allocation scheme always reaches higher probability of positive secrecy capacity than the ECP power allocation scheme under the same turbulence intensity and eavesdropping intercept ratio. The work in this paper can be extended to OAM-MIMO mixed systems and wavelength division multiplexing systems.
Funding
National Natural Science Foundation of China (61671185).
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