1. Introduction

With the explosive increases of the Internet-based services, the sufficient transmission capacity is urgent request in various area [1,2]. Recently, there is keen interest in underwater optical communication (UWOC) systems because it can provide higher transmission bandwidth, higher data rate and low latency [3,4]. In real scenario, due to the underwater turbulence and the elevation angle of received light beam, the spot light diffuses as the propagation distance increases, and thus provide the opportunity for the eavesdroppers to wiretap the transmission link [5,6]. Therefore, in order to realize reliable communication, secrecy issues of UWOC links should be seriously considered.

Over the last several decades, orbital angular momentum (OAM) multiplexing is emerging as a promising candidate technique for the rapid development of high-capacity and high-security optical wireless communication (OWC) [7–10]. Given that the OAM beams with helical wave-front have an infinite-dimensional orthogonal basis theoretically, they can provide a new degree of freedom, hence could be utilized to dramatically enhancing transmission capacity and spectral efficiency of UWOC systems [11,12]. However, the wave-front of OAM is susceptible to the underwater turbulence-induced spatial aberrations, hence severely degrade the reliability of communication [13]. Moreover, the scintillation index originates from the temperature and the salinity fluctuations gives rise to intermodal crosstalk among different OAM modes, which could further deteriorate the security performance of communication system.

In order to avoid the intermodal crosstalk between different OAM modes, the longitudinal OAM multiplexing (LOAMM) system constructed by frozen waves (FW) is proposed for free space optical (FSO) communication due to its resistance to the disturbance from atmospheric turbulence [14]. FW is constituted by a class of non-diffracting Bessel beams with equal-frequency but different longitudinal wave numbers [15]. The OAM modes can be transmitted simultaneously and the topological charge can be controlled independently along the direction of propagation according to predesigned rules [16]. Nevertheless, the intensity attenuation of reference-channel induced by underwater turbulence cannot be tackled effectively without adaptive optics (AO) technique cooperation [8].

Due to the properties of helical phase fronts, the wave-front cannot be measured directly by the Shack-Hartmann wave-front sensor [17]. To circumvent this, the separate Gaussian probe beam aided AO technique is proposed [18]. The probe beam occupy one polarization state orthogonal to the other transmitted OAM beams. The distorted phase information of the probe beam can be retrieved from measured intensity pattern by means of phase retrieval algorithm and the corrected phase profiles can be employed for all the transmitted OAM modes [7]. The defocus-measurements-based phase retrieval algorithms with intensity-only data have been considered as an effective effort to reconstruct missing phase of object [19–21]. However, the multiple measurements, support constraint setting and time-consumption of algorithm convergence due to the initial processing will limit the deployment of these methods.

It has been widely known that the information-theoretic security for single input single output (SISO) OWC link can be characterized in terms of the probability density function (PDF) and cumulative density function (CDF) [22,25]. The statistical distribution model has been shown to efficiently characterize the reference-channel PDF of single OAM mode transmission link [26]. In 2019, Chen et al. proposed a dual Johnson $S_{B}$ (DJSB) statistical distribution model to characterize the PDF for reference-channel irradiance of the OAM mode in FSO propagation model [27]. In 2020, Amhoud et al. used the generalized Gamma distribution (GGD) model for modelling the reference-channel irradiance as well as the intermodal crosstalk irradiance of OAM modes [28].

Explicitly, we observe that in most cases of practical wiretap channels based on multiplexed Laguerre-Gaussian (LG) modes, the eavesdropper could simultaneous eavesdrop the information from different channel, therefore significantly deteriorate the security performance [8,10]. It is possible to overcome the deficiency by dint of the multiplexed FW (MFW) as a consequence of the characteristics of independent longitudinal pattern. Furthermore, although several of the aforementioned distribution models provide a good fit for reference-channel irradiance of OAM in most cases, the fitting goodness can be further improved by means of novel distribution model. In this work, we propose a defocus measurements aided AO (DMA-AO) technique for turbulence compensation in LOAMM underwater communication system operating in an intensity modulation and direct detection (IM/DD) setup to improve the performance of physical layer security. The primary contributions and novelty of this paper are summarized as follows:

In a nutshell, our contributions are boldly and explicitly contrasted to the published literature in Table 1. The remainder of this paper is organized as seen in Fig. 1. In Section II-A, the description of the LOAMM-UWOC system based on the DMA-AO technique is given. In Section II-B, a couple of phase retrieval algorithms for the DMA-AO technique are introduced to mitigate the channel impairment, and the performance for the phase estimation of the probe beam is analyzed in detail. In Section III, the GJSB distribution model is put forward for characterizing the PDF of reference-channel with/without compensation and the parameters of PDF are deduced using the maximum-likelihood estimation method. In Section IV, we derive the analytical expression of the probability of SPSC for a single OAM channel in terms of CDF. Furthermore, the ASC and the probability of SPSC for the LOAMM-UWOC system are evaluated attained by mode purity and intermodal crosstalk. Finally, in Section V, we offer our conclusion.

 

Fig. 1. The structure of this treatise

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2. Concept and principle

2.1 LOAMM-UWOC system establishment

The LOAMM communication system is constructed by superposition of FW carrying OAM modes [14], in which FW are composed of copropagating Bessel beams with equal frequency but slightly shifted longitudinal wavenumbers [15,16]. For FW, the spatial distribution of optical field in turbulence-free environment can be represented as:

Here, the $F_{\ell }(z)$ can be recognized as Fourier-like series and $A_{\ell m}$ is the Fourier coefficient. As a result, the FW can propagate along the axis as periodic waveforms with the predesigned period of $L$. By adjusting $F_{\ell }(z)$ and $L$, the longitudinal field distribution of LOAMM system can be determined. In this paper, the morphological function $F_{\ell }(z)$ is addressed within $L=1\mathrm {m}$ as:

The underwater turbulence effects, which are caused by random variations in temperature, salinity and inhomogeneity of the refractive index [30,31] constitute unavoidable deteriorations in LOAMM UWOC systems. The phase perturbations caused by turbulence will give rise to intermodal crosstalk, thus destroying the orthogonality of different OAM modes [32]. In order to characterize the phase fluctuation, the multiple-phase-screens-based method is proposed and introduced in the seminal contribution of [33]. The underwater refractive index power spectral density (PSD) proposed by Nikishov can be represented as [34]:

To characterize the strength of underwater turbulence, the Rytov variance is commonly adopted, which is defined as [39, eq. (26)]:

The turbulence strength is commonly classified as $\sigma _{R}^{2}<1$ for weak turbulence and $\sigma _{R}^{2}\gg 1$ for strong turbulence [40].

In order to simulate the underwater turbulence channel based on multiple phase screens model, the coherent diameter and scintillation index are necessary parameters to determine the phase PSD of each layer. For the plane wave, the spatial coherence can be formulated as [41, eq. (10)]:

On the basis of Eq. (8), the phase PSD in Eq. (6) can be rewritten as:

For further considering the distribution of each phase screen, the scintillation index of plane wave for underwater turbulence is introduced, which can be expressed as [42, eq. (6)]:

In order to determine the turbulence strength of each phase screen, the $r_{0}$ of Eq. (8) plays the role of link between the various phase screens, within which the effective coherence width of $i^{th}$ layer is defined as

With the aid of $r_{0i}$, the Eqs. (8) and (10) can be rewritten as follows:

Based on Eqs. (8)–(13), the spatial coherence of each screen can be calculated by using the function fmincon of MATLAB. A more detailed description for the establishment of multiple phase screens model has been invoke in [37]. Furthermore, in order to simulate FW propagating through underwater turbulence channel, angular transform relying on split step Fourier transform method is utilized and the procedures presented in [12] has been followed by us.

In Fig. 2, we conceive a LOAMM UWOC system equipped with DMA-AO technique to investigate the security features. At the transmitter (Alice), the laser relying on an amplitude modulator (AM) generates the modulated signal for FW carrying different OAM modes. The OAM modes are pre-set over a finite space interval with desired longitudinal intensity profile. The MFW propagate through underwater channel constructed by random multi-phase screens and the contaminated MFW are compensated by probe beam assisted DMA-AO technique. In our simulations, the probe beam experiences the same environment as the MFW and the defocus measurement procedure and phase estimation is described in detail in Fig. 3. At the receiver (Bob), the decontaminated MFW are detected relying on the conjugate holograms and then converted to the corresponding electrical signal by avalanche photodiodes (APD). We consider the worst scenario for secure communication, where the eavesdropper (Eve) locates close to the Alice to facilitate wiretapping perfect information without suffering from the turbulence [10].

 

Fig. 2. A LOAMM UWOC communication system with DMA-AO technique showing the transmitter (Alice), channel model and receiver (Bob) in the presence of an eavesdropper (Eve) locating close to Alice.

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Fig. 3. Schematic setup of the DMA-AO system for turbulence compensation.(LD: Laser Diode, Col: Collimator, PBS: Polarized Beam splitter, L: Convex Lens, PH: Pinhole, SLM: Spatial Light Modulator, Pol: Polaroid, M: Mirror, VOA: Variable Optical Attenuator, CCD: Charge-Coupled Devices, SMF: Single-Mode Fiber.)

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2.2 Probe beam assisted DMA-AO technique

In this section, we introduce a novel probe beam assisted DMA-AO method for turbulence compensation. The mathematical process of phase retrieval relying on defocus measurements is described. Moreover, the implementations of different non-convex algorithm including Gechberg-Saxton-Hybrid-Input-Output (GS-HIO) Algorithm, Wirtinger Flow (WF) algorithm and Reweighted Wirtinger Flow (RWF) algorithm are presented and the performance of different algorithm are evaluated critically.

2.2.1 Defocus measurement aided adaptive optics method

Figure 3 shows the schematic setup for DMA-AO technique. It can be seen from the figure that the FW is generated by using an amplitude spatial light modulator (SLM), in which a two-dimensional computer-generated hologram (CGH) is mapped onto it. The expanded Gaussian probe beam is added to the FW with different polarization and both the probe beam and FW propagating through the same turbulence channel. At the receiver, the distorted probe beam is captured and recorded by charge-coupled devices in different defocus plane. Our aim is to retrieve the phase of contaminated probe beam by means of iterative non-convex searching algorithm and mitigate the distorted FW. To address this issue, we first conceive the mathematical model of defocus measurements for distorted probe beam.

For defocus measurements scenario, let us consider that the contaminated probe Gaussian beam propagates along positive z-axis from the source plane at convex lens to the observation plane at defocus position. On the basis of the extended Huygens-Fresnel principle [40], the field of defocus plane can be expressed as:

The term $G(\mathbf {s}, \mathbf {r} ; L_{0})$ in Eq. (15) denotes the free-space Green function, $L_{0}$ represents propagation distance from the convex lens, the $U(\mathbf {s}, z_0)$ in Eq. (16) denotes the complex distribution of distorted probe beam and $\psi (\mathbf {s}, \mathbf {u})$ is complex phase perturbation of a plane wave. The Eq. (16) can be implemented by using multiple phase screen model, which has been given detailed description in section 2.1. The $\phi \left (\mathbf {s}\right )$ in Eq. (17) represents the phase shifter for the response of convex lens in spatial domain. The $\lambda$ is the wavelength of transmitted beam, the $r_{0}$ is the focal length of convex lens, the $\phi _{0}$ represents the reference phase and the $\theta$ denotes the incident angle of probe beam.

2.2.2 Non-convex phase retrieval algorithms

For the non-convex phase retrieval problem, our purpose is to find an optimal complex vector $\boldsymbol {x} \in \mathbb {C}^{n}$ from magnitudes of its linear measurements $\boldsymbol {b_{i}}=\left |\boldsymbol {a}_{i}^{*} \boldsymbol {x}\right |^2, i=1,2, \ldots, m$. In order to solve this problem, we formulate a least-squares optimization problem as:

From the alternating projection view of point, the relationship between $\boldsymbol {z}_{k+1}$ and $\boldsymbol {z}_{k}$ can be defined as [43]:

By iteratively utilizing the operator, the estimator of $z$ can be deduced. In our simulations, we have used lsqr function given in the MATLAB software. Furthermore, the GS-HIO algorithm [45,46] associated with zero pad preprocessing is performed to update the $z_{k}$ in each iteration. As in the GS-HIO processing, $K$ is the maximum number of iterations, $K_{p}$ is the iterations for each GS-HIO and $K_{1}$ is the iterations for GS algorithm, which can be expressed as:

As for the WF-based algorithm, the Eq. (18) can be solved by using a proper initialization. The objection function of the WF can be described as follows:

Refer to the literature [47], we update the $z_{k}$ by a initialization, which can be expressed as:

In order to solve the objection function, the gradient descent approach is established to iteratively estimate the $z_{k}$. Relying on the initialization $z_{0}$, the formulation can be defined as:

Empirically, in order to prevent the WF algorithm from being stagnated into the saddle points, Yuan et al. [48] put forward RWF algorithm and introduce a weight coefficient to search the global optimum. As expected, the objection function can be reformulated as:

2.2.3 Performance analysis for different phase retrieval algorithms

To evaluate the performance of different phase retrieval algorithm, we compare sum squared error (SSE) and probability of success of probe beam with different defocus distance and different number of defocus plane. The default parameters of numerical simulations for characterizing the LOAMM UWOC system with DMA-AO technique are set as Table 2. The SSE definition of recovered amplitude is given as below:

In addition, the probability of success for complex distribution of distorted probe beam can be defined as:

Table 1. Novelty in FW-based LOAMM-UWOC system with AO technique

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Table 2. Default parameters settings for LOAMM UWOC system based on DMA-AO technique

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In Fig. 4, the SSE performance of contaminated probe beam as a function of iterations is depicted. The defocus distance and number of defocus plane are considered to determine the quality of retrieval processing. The number of defocus plane with 5 and 7 are chosen as examples, respectively. Meanwhile, the defocus distance are set as 0.01m, 0.05m and 0.09m respectively to investigate the influence on SSE convergence status. Observe from Fig. 4 that the SSE performs better as the number of defocus plane increases. Moreover, as the inner defocus distance increases, the SSE converges to a steady state faster for the reason that the correlation decreases as the defocus distance gets further. Furthermore, the RWF and GS-HIO algorithms outperform the WF under the same simulation condition.

In Fig. 5, the probability of success of complex distribution for distorted probe beam is shown. The recovery success is plot as a function of defocus plane number and we observe that the successful reconstructions rate from amplitude-only data enhance as the defocus measurements increase. It can be observed from Fig. 5 that the RWF algorithm generally has quite an superior performance at defocus distances of 0.01m and 0.05m. However, the GS-HIO algorithm exceed the RWF and WF case when distance between inner defocus plane reaches 0.09m. It can be concluded from the results that the GS-HIO algorithm is more sensitive to the correlation among defocus measurements and three intensity-only measurements can achieve accurate phase reconstruction of probe beam when the defocus distance is suitable.

 

Fig. 4. The SSE metrics of probe beam obtained from Eq. (32) as a function of iterations with 5 and 7 measured defocus plane, respectively. The defocus distance is set as $(\mathrm {a})$ $\boldsymbol {0.01}\mathrm {m}$, $(\mathrm {b})$ $\boldsymbol {0.05}\mathrm {m}$, $(\mathrm {c})$ $\boldsymbol {0.09}\mathrm {m}$.

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Fig. 5. The probability of success of probe beam obtained from Eq. (33) as a function of number of defocus plane with different defocus distance, where $(\mathrm {a})$ $\boldsymbol {0.01}\mathrm {m}$, $(\mathrm {b})$ $\boldsymbol {0.05}\mathrm {m}$, $(\mathrm {c})$ $\boldsymbol {0.09}\mathrm {m}$.

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In Fig. 6, the reconstructed phase of probe beam relying on different phase retrieval algorithm is given. We consider the defocus distance set to 0.05m and 0.09m as two examples in our analysis. Observe that the phase pattern seen in the first column and second column in Fig. 6, upon increasing defocus distance, the phase can be recovered better with less defocus plane measurements. Similar to the results in Fig. 5, the best performance of reconstructed phase is attained by RWF when defocus distance set to 0.05m, whereas the GS-HIO algorithm performs best with less defocus measurements when defocus distance set to 0.09m.

 

Fig. 6. The reconstructed phase of probe beam under different phase retrieval algorithms based on 2 to 7 defocus planes, where $(\mathrm {a})$ WF with defocus distance $\boldsymbol {0.05}\mathrm {m}$; $(\mathrm {b})$ WF with defocus distance $\boldsymbol {0.09}\mathrm {m}$; $(\mathrm {c})$ RWF with defocus distance $\boldsymbol {0.05}\mathrm {m}$; $(\mathrm {d})$ RWF with defocus distance $\boldsymbol {0.09}\mathrm {m}$; $(\mathrm {e})$ GS-HIO with defocus distance $\boldsymbol {0.05}\mathrm {m}$; $(\mathrm {f})$ GS-HIO with defocus distance $\boldsymbol {0.09}\mathrm {m}$.

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For the sake of visual comparison, the normalized longitudinal amplitude patterns of LOAMM for $\left \{+1,+3,+5,+7\right \}$ are portrayed in Fig. 7 with the condition of (a) vacuum environment, (b) underwater turbulence environment without DMA-AO technique and (c) underwater turbulence environment with DMA-AO technique. The phase retrieval algorithm is set to GS-HIO with defocus distance for 0.09 under five defocus measurements. To elaborate on the Fig. 7(a), (b) and (c), the DMA-AO technique can dramatically mitigate the distortion of FW, hence improving the performance of LOAMM. To further visualize the full dynamics of this evolution, the Visualization 1, Visualization 2, and Visualization 3 exhibit the transverse profiles of LOAMM along the propagation direction within 1m for the setting condition (a), (b) and (c) in Fig. 7.

 

Fig. 7. Simulation propagation observation of LOAMM for $l=\left \{+1,+3,+5,+7\right \}$. The normalized side-view of the FW in the x-z plane for $(\mathrm {a})$ vacuum environment (see Visualization 1); $(\mathrm {b})$ underwater turbulence environment without DMA-AO compensation technique (see Visualization 2); $(\mathrm {c})$ underwater turbulence environment with DMA-AO compensation technique (see Visualization 3). The phase retrieval algorithm is set to GS-HIO with defocus distance for 0.09m under 5 defocus measurements.

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The Fig. 8 illustrates the PDF of reference-channel irradiance of LG beams and FW for $\left \{l=+1,+3,+5,+7\right \}$ with and without DMA-AO compensation technique and the irradiance measurement procedure is realizing by 50000 samples. The reference-channel irradiance can be defined as:

 

Fig. 8. The PDF of reference-channel irradiance of LG beams and FW for $\left \{l=+1,+3,+5,+7\right \}$ measured with and without DMA-AO compensation technique, where $(\mathrm {a})$ for LG beams without DMA-AO compensation technique; $(\mathrm {b})$ for LG beams with DMA-AO compensation technique; $(\mathrm {c})$ for FW without DMA-AO compensation technique; $(\mathrm {d})$ for FW with DMA-AO compensation technique. The phase retrieval algorithm is set to GS-HIO with defocus distance for 0.09m under 5 defocus measurements.

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3. Mixture GJSB distribution fitting compared with simulations

3.1 Mixture GJSB distribution

To characterize the PDF of reference-channel irradiance with closed-form and mathematically tractable formula, the GJSB distribution model is proposed and established, which is a weighted sum of generalized Gamma and Johnson $S_B$ distributions and the expression can be defined as:

The function $f(I ; [\vartheta,\tau ])$ and $g(I ; [a, b, c])$ represent the Johnson $S_B$ and generalized Gamma distributions, respectively where $\omega$ denotes the mixture coefficient satisfying $0<\omega <2$. The $\vartheta$ and $\tau >0$ in Johnson $S_B$ distribution joint control the shape of the distribution, where $\vartheta$ being associated with the symmetry. The $a$ and $b$ in generalized Gamma distributions denote the shape parameters, while $c$ represents the scale parameter.

3.2 Maximum likelihood estimates for GJSB distribution

In this section, our aim is to obtain optimal parameters for the Eq. (35) to realize the best fit with the Monte-Carlo simulations. In order to address this issue, the expectation maximization algorithm is adopted to find maximum likelihood estimates of GJSB distribution model, which has been detailed described in [50]. To construct and solve the associated equations, random irradiance realizations of reference-channel $I_{i}=\left |C_{l,i}\right |^{2}, i=1, \ldots, 50000$ associated with a hidden unobserved binary variable $s_{i}$ is given as follow:

We can obtain the results conveniently by maximizing the log-likelihood function, which are expressed as:

Upon substituting Eqs. (36) and (37) into Eq. (39) and by utilizing [51,52], the log-likelihood function can be rewritten as:

Based on [50,53], the estimated parameters $a$, $b$, $c$, $\vartheta$, $\tau$ and $\omega$ can be derived by taking the derivatives of functional $\ell$ with the following equations as:

 

Fig. 9. Distributions of the reference-channel for LG beams carrying OAM modes in the set $l=\left \{+1,+3,+5,+7\right \}$ with and without DMA-AO compensation technique.The measured data are depicted with fitted PDFs via the GJSB, GGD, DJSB, JSB and WB distributions under the conditions listed in Table 2.

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Fig. 10. Distributions of the reference-channel for FW carrying OAM modes in the set $l=\left \{+1,+3,+5,+7\right \}$ with and without DMA-AO compensation technique.The measured data are depicted with fitted PDFs via the GJSB, GGD, DJSB, JSB and WB distributions under the conditions listed in Table 2.

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Table 3. Estimated parameters of the GJSB for measured LG samples of $\mathbf {l={+1,+3,+5,+7}}$ and goodness of fits tests without and with AO compensation in UWOC

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Table 4. Estimated parameters of the GJSB for measured FW samples of $\mathbf {l={+1,+3,+5,+7}}$ and goodness of fits tests without and with AO compensation in UWOC

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4. Security performance analysis

In this section, the security performances are analyzed for both SISO and MIMO system operating in an IM/DD on-off keying (OOK) modulated setup with DMA-AO compensation technique. The underwater simulation setup is listed in Table 2 and the probe beam assisted GS-HIO phase retrieval algorithm with 5 defocus measurements and 0.09m defocus distance is adopted for DMA-AO mitigation. Furthermore, The security performance of LOAMM system is compared to the corresponding MLG system under the identical conditions. We consider scenarios that the effects of underwater turbulence and weak absorption degrade the performance of the system, while the beam scattering and inter symbol interference (ISI) have little impact on it. The probability of SPSC of SISO OAM UWOC system is investigated relying on the CDF. Moreover, the ASC and probability of SPSC of MIMO OAM UWOC system are explored on the basis of mode purity and intermodal crosstalk of transmitted OAM modes.

Assuming that the OAM beam propagating through the underwater channel under IM/DD setup. At the receiver, the signal is contaminated by additive white Gaussian noise (AWGN) due to the photon detectors, which can be defined as [57]:

Still referring to Eqs. (50) and (51), the $\gamma _{0, k}$ is the instantaneous electrical SNR free from underwater turbulence. $k=e$ (Eve) denotes eavesdropping receiver in the UWOC link and $k=b$ (Bob) represents information receiver. $r_{k}$ is the percentage of power extracted from Eve and Bob, which satisfying the desired relationship as $r_{e}+r_{b}\leqslant 1$. The term $e^{-\delta d_{k}}$ represents the intensity attenuation due to beam absorption induced by underwater environment, in which $\delta$ is the attenuation loss constant and $d_{k}, k=e,b$ is the propagation distance from transmitter to Eve and Bob, respectively.

In realistic scenario, the secrecy capacity is defined for the condition that the eavesdropper is incapable to obtain effective information. In accordance with [58], the total secrecy capacity of system is defined as:

4.1 Probability of SPSC for the SISO system

In this section, we analyze the probability of SPSC of SISO OAM UWOC system with and without DMA-AO compensation technique. The probability of SPSC regarded as a probabilistic indicator is represented as [22,59]:

Upon substituting Eqs. (50), (51) and (52) into Eq. (53), the probability of SPSC can be derived as:

Considering the scenario that the Eve is located close to the transmitter, the propagation distance between Alice and Eve will be satisfied as $d_{e}\approx 0$. For this reason, we assume that the Eve is free from underwater turbulence, which can be viewed as $\gamma _{e}=\gamma _{0, e}$. According to the PDF and CDF of reference-channel random fluctuations due to underwater turbulence, which have been analyzed in section 3, we can obtain the definition of probability of SPSC from Alice to Bob as follows [22]:

In Fig. 11, the probability of SPSC of SISO UWOC system for LG beams and FW with and without DMA-AO compensation technique are depicted. Observed from Fig. 11(a) that the probability of SPSC decreases as the $r_{e}$ increases and the secrecy curve become extremely abrupt when the $r_{e}$ exceed the threshold. This phenomenon can explained for the reason that the probability of SPSC has a binary behavior as in the Gaussian wiretap channel setup [22]. Explicitly, the FW outperform the LG beams except for the $l=\left \{+1\right \}$ due to the FW’s resistance to underwater turbulence. Furthermore, as shown in Fig. 11(b), the probability of SPSC is improved as a benefit of the corrected OAM beams and the FW perform much better than LG beams. In a nutshell, the non-diffractive characteristic of FW seems a non-negligible factor to establish a more secure communication link.

 

Fig. 11. The probability of SPSC vs fraction of Eve intercept $r_{e}$ for SISO OAM UWOC system. The transmitted OAM modes are set to $l=\left \{+1,+3,+5,+7\right \}$ and the comparisons between LG beams and FW are illustrated in $(\mathrm {a})$ without DMA-AO compensation technique; $(\mathrm {b})$ with DMA-AO compensation technique.

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4.2 ASC for the MIMO system

In this section, we consider an OOK IM/DD OAM-based UWOC system equipped with $l=\left \{+1,+3,+5,+7\right \}$ carrying different transmitted data. Assuming that the OAM beams are transmitted and received by $N_{T}$ and $N_{R}$ optical antennas, respectively. Due to the underwater turbulence, the reference-channel power of transmitted OAM modes will spread into other OAM modes, thus degrading the mode purity and causing crosstalk among different OAM modes, which deteriorates the performance of MIMO system.

The mode purity and crosstalk of MLG system and LOAMM system are illustrated in Fig. 12. It can be observed from the figure that the normalized power of the reference-channel leak into other transmitted channel after propagating through the underwater turbulence environment. Obviously, the leakage of power for adjacent OAM modes are higher than other OAM modes. It can be concluded from Fig. 12(b) and (d) that with the aid of DMA-AO compensation technique, the power leakage from transmitted OAM modes significantly improved. Moreover, the FW have an advantage over LG beams in crosstalk suppression between different OAM modes.

 

Fig. 12. Normalized mode purity and intermodal crosstalk of transmitted OAM modes for $l=\left \{+1,+3,+5,+7\right \}$ based on 50000 Monte-Carlo simulation results. $(\mathrm {a})$ MLG UWOC system without DMA-AO compensation technique; $(\mathrm {b})$ MLG UWOC system with DMA-AO compensation technique; $(\mathrm {c})$ LOAMM UWOC system without DMA-AO compensation technique; $(\mathrm {d})$ LOAMM UWOC system with DMA-AO compensation technique.

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In our simulations, we consider the scenario that eavesdropper Eve located close to the transmitter, which has been widely investigated in the wireless link [60] and FSO link [10], the LOAMM system constructed by FW has the advantages over MLG system described as follows:

By substituting Eqs. (57) and (58) into Eq. (52), ASC metric for MIMO system can be derived. It should be specified that one and only one channel of the LOAMM system can simultaneously eavesdrop and we set $l=+1$ as wiretap channel. Furthermore, the leakage of power from transmitted OAM modes will not cause interference to other channels.

In Fig. 13, the ASC performance of MIMO system based on MLG beams and MFW with and without DMA-AO compensation method are portrayed. It can be seen from the figure that the ASC decrease as the $r_{e}$ increases but the curves tend to zero when the SNR reaches a certain level. For the MIMO system, the ASC of LOAMM system outperform the one of MLG system with/without AO mitigation and the ASC increases as the SNR increases. Nonetheless, due to the proximity of the eavesdropping point to the transmitter and free from turbulence, the MLG scheme with higher SNR is less likely to ensure secure communication when the fraction Eve increases. Conversely, the LOAMM system performs better with higher SNR because of the avoidance of mutual crosstalk between different OAM modes and lower leakage of power for reference-channel.

 

Fig. 13. The ASC vs fraction of Eve intercept $r_{e}$ for MIMO OAM UWOC system. The transmitted OAM modes set for $l=\left \{+1,+3,+5,+7\right \}$ are carrying different information and the comparisons between MLG system and LOAMM system constructed by MFW are illustrated in $(\mathrm {a})$ without DMA-AO compensation technique; $(\mathrm {b})$ with DMA-AO compensation technique.

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4.3 Probability of SPSC for the MIMO system

In this section, we analyze the probability of SPSC of MIMO system with/without DMA-AO compensation method. In order to characterize the probability of SPSC for MIMO system, we adopt Monte-Carlo simulations to calculate Eq. (53) with the expression as:

The reliability of secure communication with and without DMA-AO compensation technique is described in Fig. 14. Similar to the results of Fig. 13 that the LOAMM system constituted by MFW have an advantage over MLG system. Still for MLG system, the performance of probability of SPSC deteriorate faster as the $r_{e}$ increases due to the simultaneous eavesdropping occurring close to the transmitter without being affected by underwater turbulence and beam absorption. Meanwhile, compared to the LOAMM system, the MLG system with higher SNR degrades the performance of probability of SPSC due to the eavesdropper close to the transmitter in the absence of turbulence. As expected, the performances for both the MLG system and LOAMM system achieve significant improvement with the DMA-AO compensation method.

 

Fig. 14. The probability of SPSC vs fraction of Eve intercept $r_{e}$ for MIMO OAM UWOC system. The transmitted OAM modes set for $l=\left \{+1,+3,+5,+7\right \}$ are carrying different information and the comparisons between MLG system and LOAMM system constructed by MFW are illustrated in $(\mathrm {a})$ without DMA-AO compensation technique; $(\mathrm {b})$ with DMA-AO compensation technique.

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5. Conclusion

In this work, the LOAMM system compensated by probe beam assisted DMA-AO technique is proposed and applied to UWOC based on multiple phase screens model. Compared to traditional MLG system, the security performance enhancement of the LOAMM system is analyzed. We first investigate the performance of different phase retrieval algorithms for DMA-AO technique. The retrieved phase of probe beam achieve higher accuracy under more defocus measurements and larger defocus distance. Furthermore, the mixture GJSB distribution model is proposed to characterize the reference-channel irradiance of transmitted OAM modes. The mathematical tractable closed-form expressions of PDF and CDF for GJSB distribution are derived. Compared to other classic distribution model, the GJSB distribution model yielded at least as good, or even better performance. Moreover, the probability of SPSC for SISO is deduced relying on the PDF of GJSB distribution. Based on the average reference-channel and crosstalk matrix, the ASC and probability of SPSC for MIMO system are explored. In a nutshell, the LOAMM system outperform the MLG system with and without DMA-AO compensation technique for providing reliable secure UWOC.