Performance Analysis of Reconfigurable Intelligent Surface-Assisted Underwater Wireless Optical Communication Systems

With the in-depth exploration of the ocean, the Internet of Underwater Things (IoUT) technology has attracted increasing attention. Underwater wireless optical communication (UWOC) provides the possibility of massive data transmission for IoUT. The underwater optical link of UWOC can be interrupted by the obstruction of marine animals and plants, seamounts, and some underwater equipment. In this paper, we present a reconfigurable intelligent surface (RIS) assisted UWOC system to solve the problem of link occlusion. The cascaded turbulence channel fading coefficients from source to destination through RIS are modeled as Gamma-Gamma distribution, and the pointing errors caused by beam jitter and RIS jitter are considered. In the RIS-assisted UWOC systems with intensity modulation and direct detection (IM-DD), we apply the approximate distribution of the sum of Gamma-Gamma random variables to derive the probability density function (PDF) of the instantaneous received signal-to-noise ratio (SNR) for the first time. Based on the PDF, the novel closed-form expression of outage probability is given, and the outage probability analysis results show that when the number of reflecting elements is very large (i.e., tends to infinity), the outage probability asymptotically approaches zero. Subsequently, the closed-form expressions of the finite-SNR diversity order, the asymptotical diversity order, and the convergence speed of finite-SNR diversity order to asymptotical diversity order are proposed. Additionally, the ergodic channel capacity expression is investigated. Finally, we give simulation results to validate our derived results and analyze the impacts of the number of reflecting elements of RIS and the pointing errors on the above system performance indicators.


Performance Analysis of Reconfigurable Intelligent
Surface-Assisted Underwater Wireless Optical Communication Systems Qi Zhang , Dian-Wu Yue , Senior Member, IEEE, and Xian-Ying Xu Abstract-With the in-depth exploration of the ocean, the Internet of Underwater Things (IoUT) technology has attracted increasing attention.Underwater wireless optical communication (UWOC) provides the possibility of massive data transmission for IoUT.The underwater optical link of UWOC can be interrupted by the obstruction of marine animals and plants, seamounts, and some underwater equipment.In this paper, we present a reconfigurable intelligent surface (RIS) assisted UWOC system to solve the problem of link occlusion.The cascaded turbulence channel fading coefficients from source to destination through RIS are modeled as Gamma-Gamma distribution, and the pointing errors caused by beam jitter and RIS jitter are considered.In the RIS-assisted UWOC systems with intensity modulation and direct detection (IM-DD), we apply the approximate distribution of the sum of Gamma-Gamma random variables to derive the probability density function (PDF) of the instantaneous received signal-to-noise ratio (SNR) for the first time.Based on the PDF, the novel closed-form expression of outage probability is given, and the outage probability analysis results show that when the number of reflecting elements is very large (i.e., tends to infinity), the outage probability asymptotically approaches zero.Subsequently, the closed-form expressions of the finite-SNR diversity order, the asymptotical diversity order, and the convergence speed of finite-SNR diversity order to asymptotical diversity order are proposed.Additionally, the ergodic channel capacity expression is investigated.Finally, we give simulation results to validate our derived results and analyze the impacts of the number of reflecting elements of RIS and the pointing errors on the above system performance indicators.Index Terms-Underwater wireless optical communication, reconfigurable intelligent surface, outage probability, diversity order, ergodic channel capacity.

I. INTRODUCTION
A S A promising concept, the Internet of Underwater Things (IoUT) is a powerful technology for realizing the smart ocean.The goal of IoUT is to connect the underwater sensor networks and autonomous underwater vehicles (AUVs) through a Qi Zhang and Dian-Wu Yue are with the College of Information Science and Technology, Dalian Maritime University, Dalian 116026, China (e-mail: zhangqi0081@dlmu.edu.cn;dwyue@dlmu.edu.cn).
Xian-Ying Xu is with the School of Information Science and Engineering, Dalian Polytechnic University, Dalian 116034, China (e-mail: xuxiany@ dlpu.edu.cn).
Although acoustic signals can propagate over long distances with small signal attenuation, they usually have low bandwidth (typically on the order of Kbps), high delay, poor directivity and may affect marine organisms.On the other hand, RF signals support higher data rates (typically on the order of Mbps) and have low power consumption, but at the expense of decreased communication range and high signal attenuation.Compared with underwater acoustic and radio frequency communications, underwater wireless optical communication (UWOC) has the lowest link delay, the highest transmission rate (typically on the order of Gbps), and the lowest implementation cost.It is particularly well-suited to high-speed and large-capacity underwater data transmission [4], [5].Moreover, UWOC can be combined with the existing RF communications to establish mixed RF-UWOC systems for realizing the information transmission between the IoUT and other platforms.For example, [6] studied a dual-hop RF-UWOC transmission system and analyzed the outage probability and bit error rate (BER).The secrecy performance for a dual-hop mixed RF-UWOC system was discussed in [7] and [8].In conclusion, UWOC shows broad application prospects in IoUT systems.UWOC systems require that the transmitting light source and the receiving detector must be aligned.Considering the high transmission rate of UWOC, even short-term blocking or occlusion can lead to sudden communication interruption, resulting in the loss of large amounts of transmission information.This situation is fatal to a reliable IoUT system.In the underwater environment, marine animals and plants, seamounts, and offshore platform equipment can occlude or block the optical link between the light source and the detector, increasing the possibility of optical information loss.The emerging concept of reconfigurable intelligent surfaces (RISs) opens the door for reducing the adverse effects of obstacles and blockages on underwater wireless optical links.
RIS is a new and revolutionary technology that was initially used for RF communication links.Specifically, the RIS is a planar surface including multiple low-cost passive reflecting elements, each of which can independently control the amplitude and/or phase changes of the incident signals [9].It is used to reconfigure the wireless propagation environment through software-controlled reflection [9].This technology can be used in beyond 5 G and 6 G RF mobile communication systems [10], [11], [12], [13], [14].RIS for optical communications is different from that for RF communications.The hardware implementation of RIS for optical signals can be divided into three main categories, namely, metasurface-, mirror array-, and liquid crystal-based RIS [15].
At present, RISs for optical signals are mainly used in indoor visible light communications (VLC) and free space optical communications (FSOC).For instance, considering metasurfacebased and mirror array-based RIS models, [16] studied the temporal characteristics of the RIS-based indoor VLC channel using radiometric concepts.[17] devoted to investigating the effect of VLC RIS and putting forward a joint resource management method for a RIS-aided VLC system to maximize the overall spectral efficiency.[18] investigated physical layer security in a RIS-aided VLC system with multiple legitimate users and one eavesdropper.[19] designed a RIS-assisted system to improve the coverage of hybrid VLC/RF indoor wireless communication.Recently, the liquid crystal-based RIS has been studied in the VLC system.[20] proposed for the first time the concept of receivers using tunable liquid crystal-based RIS in VLC.In [21], a liquid crystal RIS-based VLC receiver design was proposed, and its operating principles and the channel model for a VLC system with such a receiver were provided.
For the RIS-assisted FSOC system, some research results are also presented.In [22], with the aim to increase the communication coverage and improve the system performance, RISs are considered in an FSOC setup, in which both atmospheric turbulence and pointing errors are considered.Considering link distances and jitter ratios at the RIS position, [23] carried out a performance analysis of RIS-aided FSOC links.In [24], a controllable multi-branch FSOC system based on optical RISs was proposed, and the asymptotic average BER and the outage probability were derived.[25] reported the performance analysis of multiple RISs-assisted FSOC and investigated the symbol error rate (SER) performance by utilizing the selection of the best RIS from the multiple available RISs.
Although some work has been done on RISs for optical signals in VLC and FSOC, research on RIS-assisted underwater communication systems still needs to be made available [26], [27].[28] proposed a RIS-assisted dual-hop mixed RF-UWOC system, but RIS is used in the RF link.[26] brought the concept of RIS to UWOC links, and the central limit theorem was applied for performance evaluation.[27] studied the outage probability over log-normal channels in a RIS-assisted UWOC system.In an underwater environment, ocean turbulence has a strong impact on the performance of UWOC systems.Turbulence fading is not considered in the indoor VLC [16], [17], [18], [19], [20], [21], so the results of VLC are not applicable to UWOC systems.Further, we have summarized the contributions of existing research related to the RIS-assisted FSOC and UWOC systems in Table I.From Table I, some works consider the single-reflecting element RIS [23], [24].Compared with single-reflecting element RIS, the modeling and analysis of multi-reflecting element RIS channel are more challenging.In existing works of [22] and [26], the central limit theorem is commonly used to analyze the channel of multi-reflecting element RIS.However, the results obtained by the central limit theorem are only suitable for the case with a large number of reflecting elements.For the case with a small number of RIS reflecting elements, the results revealed that the difference between the analytical and simulation results is very large, and it is difficult to evaluate the system performance accurately.
Against the above background, we establish the multireflecting element RIS-assisted UWOC system model in the environment with obstacles, and the fading coefficients of the cascaded turbulence channels from source to destination through RIS are modeled as Gamma-Gamma distribution.To the best of our knowledge, such a system setup in conjunction with the proposed analysis has not yet been investigated.The contributions made by this paper are enumerated below: r We present a general framework to analyze the perfor- mance of the multi-reflecting element RIS-assisted UWOC systems.In addition, the ocean turbulence fading and the pointing errors caused by beam jitter and RIS jitter are considered.
r For the RIS-assisted UWOC systems, the probability den- sity function (PDF) of instantaneous SNR has been derived by using a novel method, i.e., the one with the approximate distribution of the sum of Gamma-Gamma random variables.Unlike the results obtained by the central limit theorem, the derivation results based on the approximate distribution method can accurately evaluate the system performance and are not limited by the number of reflecting elements.
r Novel closed-form expressions quantifying the outage probability, the finite-SNR diversity order, the asymptotical diversity order, the convergence speed of finite-SNR diversity order to asymptotical diversity order, and the ergodic capacity of the RIS-assisted UWOC systems have been derived.Notably, this is the first time to evaluate the finite-SNR diversity order and its convergence speed in RIS-assisted wireless optical communication systems.r The accuracy of the expressions derived in the paper has been verified using Monte Carlo simulation.The effects of the number of reflecting elements and pointing errors on the considered system have been analyzed quantitatively.The rest of the paper is organized as follows: Section II describes the system and channel model of the RIS-assisted UWOC system.The system performance analysis is given in Section III.In Section IV, the simulation results are investigated.Finally, the conclusions are shown in Section V.

II. SYSTEM AND CHANNEL MODELS
As illustrated in Fig. 1, we propose a RIS-assisted UWOC communication system composed of an optical source, an optical RIS, and a destination.Due to the obstacles, such as the schools of fish, submarines, and seamounts, the line-of-sight (LOS) link (i.e., direct link) between the source and destination is blocked.The communication is only carried out through the RIS, which can be strategically placed under buoys or fixed on underwater reefs to provide rich non-line-of-sight (NLOS) connections between the source and destination.At present, there are three types of RIS for optical communication, i.e., metasurface-, mirror array-, and liquid crystal-based RISs.As the component of the optical receiver, the liquid crystal-based RISs cannot change the optical channel path.It is mainly used in a short-range indoor VLC system, which is not suitable for the scene we are considering.In this paper, considering optical wireless communication with intensity modulation and direct detection (IM-DD) and according to [24], for the metasurface-and mirror array-based RISs, we can regard these two types of RISs between the input beam and the output beam as a black box from the perspective of the impact on the performance of the optical communication system.
The RIS consists of N reflecting elements.Let x denote the transmission signal from the source with average electric transmission power of P te .The factors that affect the power loss of the received optical signal at the receiver include underwater path loss, turbulence channel fading, and the pointing error caused by the jitter of the transmitter and RIS.Therefore, the received signal at the destination can be written as where h ln , h tn and h pn are the path loss, underwater turbulence channel fading, and the channel fading caused by pointing error of the path from the source to the destination through the n th reflecting element, respectively.ρ n ∈ [0, 1] is the reflection coefficient of the n th reflecting element.h sr n l and h r n d l represent the path loss from the source to the n th reflecting element and from the n th reflecting element to the destination, respectively.Similarly, h sr n t and h r n d t represent the corresponding turbulence fading coefficients of the two paths respectively.n 0 ∼ N (0, σ 2 n ) is the additive white Gaussian noise (AWGN).

A. Path Loss
In the underwater environment, the path loss is the propagation attenuation of optical signals, which is mainly caused by the absorption and scattering effects of various components of seawater, including water molecules, phytoplankton, nonpigment suspended particles, and colored dissolved organic matter.The general theoretical models of path loss include Beer-Lambert's law and radiative transfer equation (RTE) [4], [5].RTE can describe the energy conservation of an optical wave passing through a steady medium.However, since the RTE is an integro-differential equation involving independent variables [29], finding an exact analytical solution is very difficult for many practical UWOC applications [4].Thus, few analytical RTE models have been given in recent years.Compared with RTE, Beer-Lambert's law provides the simplest and most widely used scenario to describe the propagation attenuation of optical signals in the underwater environment.In order to facilitate subsequent analysis, we approximate path loss by the exponential attenuation model of Beer-Lambert's law, which ignores the multiple scattering effect [30].
According to Beer-Lambert's law, we obtain h i l = exp(−cL i ), i ∈ {sr n , r n d}, where c is the extinction coefficient which is the sum of absorption and scattering coefficients.Under the fixed wavelength and a stable water quality environment, c is a constant.L sr n and L r n d are link distances from source to the n th reflecting element and from the n th reflecting element to destination, respectively.The separation between reflecting elements is much smaller than the link distance.Then, it can be considered that the link distances from the source to the N reflecting elements are the same, i.e., L sr n = L 1 (n = 1, 2, . . ., N).Similarly, L r n d = L 2 (n = 1, 2, . . ., N).Therefore, the path loss coefficient is considered identical for the N paths, and we can get

B. Underwater Turbulence
denotes the cascaded turbulence channel fading coefficient from the source to the destination through the n th reflecting element.h sr n t and h r n d t are characterized by Gamma distribution, and their PDF are given as [31] where Γ(•) is the Gamma function, α and β are the turbulence parameters from source to RIS and RIS to destination, respectively.By fixing h sr n t and using the change of variable, h r n d t = h tn /h sr n t , the conditional PDF can be obtained as Then, the PDF of h tn can be given as (6) where K p (•) is modified Bessel function of the second kind with order p.In (6), we obtain that the cascaded turbulence channel fading coefficient from the source to the destination through the n th reflecting element (i.e., h tn ) is modeled by a Gamma-Gamma distribution.h t1 , h t2 , . . ., h tn , . . ., h tN are independent and identically distributed (i.i.d).Under the assumption of plane wave propagation, the turbulence parameters, α and β, can be calculated as [exp(0.225σ 2 ) − 1] −1 [26], [31], where σ 2 B i (i = 1, 2) represents the Rytov variance of the two paths, which can be given in [32].

C. Pointing Error
In RIS-assisted wireless optical communication systems, h pn is the channel fading coefficient caused by the pointing error of the path from the source to the destination through the n th reflecting element, which is affected by beam jitter and RIS jitter [24].Beam jitter means that the beam vibrates due to jitter at the source, and RIS jitter refers to the normal vector deflection of the reflecting surface caused by the jitter of the RIS.Since the size of RIS is much smaller than the communication link distance and the reflecting elements are located on the same reflecting surface, they experience identical pointing error.Then, we can get h p can be approximated by [33] and according to the pointing error model in [24], the PDF of h p can be given as where and is the fraction of the collected power at R = 0, R is the pointing error radial displacement at the receiver, are variances of pointing error angle and deflection error angles, respectively.

A. PDF of Instantaneous SNR
We assume that all RIS reflecting elements have the same reflection coefficient, that is, Then, the instantaneous SNR can be given as Replacing ( 2) and ( 7) in (11), the instantaneous SNR can be expressed as where path loss h l is deterministic in contrast to h p and h t , which are random fading variables.Thus, we let γ = P te ρ 2 h 2 l /σ 2 n denote fading-free SNR, i.e., average SNR [34], [35].h t = N n=1 h tn represents the sum of i.i.d.Gamma-Gamma random variables, and its PDF can be calculated as (13), shown at the bottom of this page, but there is no closed-form solution for the multi-dimensional integration in (13).Several methods were used in the literature to obtain approximate solutions.Some of these literatures convert the integration into an infinite sum [36], and some use the central limit theorem to obtain approximate PDF [22], [26].However, these methods are complex and have limited applications.Here, we apply an alternative method to solve the PDF of h t .
According to [37], the sum of N identically distributed Gamma-Gamma random variables, i.e., Y = N n=1 X n with X n ∼ ΓΓ(α x , β x , Ω x ) (ΓΓ(•) denotes Gamma-Gamma distribution), can be accurately approximated by another Gamma-Gamma random variable with parameters (α y , β y , Ω y ) [37].The parameters are given by where and μ nm is the correlation coefficient between X n and X m .Since h t1 , h t2 , . . ., h tn , . . ., h tN are independent and identically distributed, the correlation coefficient between h tn and h tm is zero, and we have h tn ∼ ΓΓ(α, β, 1) from (6).Therefore, according to ( 14) and ( 15), we obtain where α h = Nα and β h = Nβ.Then, the PDF of h t can be expressed as For the convenience of subsequent calculation, we use Eq.(03.04.26.0009.01) in [38] to rewrite the Bessel function in (17) as where G m,n p,q (•|•) is Meijer-G function.Substituting ( 18) into (17), we obtain (19) shown at the bottom of the next page.Then, the property of Meijer-G function of Eq. (07.34.16.0001.01) in [38] is used within (19), it can be calculated as Let h = h p h t denote the joint channel fading coefficient and the PDF of h can be calculated as where f h|h p (h|h p ) is the conditional PDF.Substituting (9) and ( 20) into (21), we can obtain (22), shown at the bottom of this page.For the Meijer-G function in (22), the variable h p appears at the denominator.
Replacing ( 23) in (22), we can obtain (24) shown at the bottom of this page.Then, we solve the integration of ( 24) with the help of Eq. (07.34.21.0084.01) in [38] and apply the reflection property of the Meijer-G function to get a Meijer-G function with a numerator-based variable h.Consequently, the PDF of h can be expressed as Based on f h (h), the PDF of instantaneous SNR γ = γh 2 can be calculated as Substituting ( 25) into ( 26), the closed-form expression of f γ (γ) can be obtained as (27) shown at the bottom of this page.

B. Outage Probability
Since the coherence time of the underwater optical turbulence channel is long (10 −5 ∼ 10 −2 second [39]), it exhibits quasi-static characteristics [40].For a typical transmission rate of hundreds of megabits per second, fading remains constant over thousands of consecutive bits or even more.Therefore, the communication link may be temporarily interrupted due to deep fading.For the quasi-static channel, outage probability is a more appropriate performance metric than BER [40].
The outage probability is defined as the probability that the instantaneous SNR γ is lower than the threshold SNR γ th .It can be given by Substituting ( 27) into (28), we have the integral equation shown in (29) shown at the bottom of the next page.Using the integration of Meijer-G function of Eq. (07.34.21.0084.01) in [38], the closed-form expression of outage probability can be expressed as (30) shown at the bottom of the next page, where . Further, we investigate the outage probability when the number of reflecting elements is large, i.e., N → ∞.In this case, the inner argument of the Meijer-G function in (30) tends to infinity, i.e., Utilizing Eq. (07.34.06.0018.01) in [38], the Meijer-G function can be expanded, and P out (N ) N →∞ can be expressed as (31) shown at the bottom of the next page.When N → ∞, we have (Nα + 1)/2 ≈ Nα/2, and applying the recursive property of the Gamma function, i.e., Γ(1 + x) = xΓ(x), (31) can be simplified as According to the asymptotic series expansion of Gamma function, i.e., Γ(x) x→∞ ≈ √ 2πx x− 1 2 exp(−x), we have Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
In addition, we define the outage probability performance gain as By analyzing the relationship between G(N ) and N , the performance gain brought by increasing the number of reflecting elements can be obtained.

C. Diversity Order and Its Convergence Speed
In this subsection, we consider the finite-SNR diversity order, the asymptotical diversity order under high-SNR, and the convergence speed of finite-SNR diversity order to asymptotical diversity order.While the asymptotic analysis proposed in earlier studies is important for understanding the maximum diversity order, such diversity order may not be available in the actual range of SNR.Furthermore, different systems with the same asymptotical diversity order may have different diversity gains in the range of finite SNR values.These motivate the analysis of finite-SNR diversity gain.Therefore, in addition to the asymptotic diversity order, we derive the closed-form expressions of the finite-SNR diversity order and its convergence speed in the RIS-assisted wireless optical communication systems for the first time.
Diversity order is the negative slope of the outage probability curve on a log-log scale, which determines how fast the outage probability changes with SNR.The finite-SNR diversity order is defined as [40] DO Substituting ( 30) into (37), we obtain where In (38), we have With the help of the derivative of Meijer-G function of Eq. (07.34.20.0002.01) in [38] along with the chain rule of differentiation, we can get (41) Substituting ( 39) and ( 41) into (40) and replacing (40) in (38), we obtain the closed-form expression of the finite-SNR diversity order as In the following, we consider asymptotically high-SNR to derive the diversity order, i.e., asymptotical diversity order (ADO), which is defined as Under the assumption of high-SNR, i.e., γ → ∞, the inner argument of Meijer-G function in (30) goes to zero, i.e., α 2 h β 2 h γ th /(16N 2 A 2 0 γ) → 0.Then, applying Eq. (07.34.06.0006.01) in [38] and the recursive property of the Gamma function, the Meijer-G function can be expanded into Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
the form of a sum of multiple terms.Therefore, P out (γ) γ→∞ can be written as (44) shown at the bottom of this page.From (44), when γ → ∞, the highest order term of γ dominates the outage probability.Thus, the lower order terms can be ignored, then (44) can be simplified as (45) shown at the bottom of this page, where min(•) denotes the minimum value.Substituting (45) into ( 43), we get the expression of asymptotical diversity order as Let us make some comments on the diversity order of the proposed system.
1) The asymptotical diversity order under high-SNR depends not only upon the turbulence parameters but also on the pointing error and the number of reflecting elements.
3) If the number of reflecting elements increases, resulting in . Hence, the asymptotical diversity order is dominated only by the pointing error.If the number of reflecting elements continues to increase, the asymptotical diversity order will remain the same.
Here, it is important to note that the asymptotical diversity order of ( 46) is obtained if all reflecting channels experience identical pointing error, which is justified from a practical point of view.
In addition, the convergence speed of finite-SNR diversity order to asymptotical diversity order is an important indicator for evaluating the diversity order performance of the system [41], [42].The convergence speed (CS) can be obtained by normalizing the gradual change in the finite-SNR diversity order with respect to the asymptotical diversity order.Mathematically speaking, it can be defined as According to (42), we get where and we have Using the derivative of Meijer-G function of Eq. (07.34.20.0002.01) in [38] along with the chain rule of differentiation, we obtain (51) Then, Substituting ( 41) and ( 51) into (50) and sorting out (47), ( 48) and (50), we obtain the closed-form expression of the convergence speed as (52) shown at the bottom of the next page.

D. Ergodic Channel Capacity
Channel capacity reflects the maximum information transmission rate on the time-varying wireless channel.Ergodic channel capacity can be calculated as where τ = exp(1)/(2π) for IM-DD scheme.Using Eq. (01.04.26.0003.01) in [38], the logarithm in (53) can be written in Meijer-G function, and we get Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Substituting ( 27) and ( 54) into (53), we obtain the integral equation as (55) shown at the bottom of this page.By applying Eq. (07.34.21.0013.01) in [38] for the integration of the product of two Meijer-G functions in (55), we obtain the closed-form expression of the ergodic channel capacity as (56) shown at the bottom of this page.

IV. SIMULATION RESULTS
In this section, we verify the derived closed-form expressions through Monte Carlo simulation.Unless otherwise stated, we consider threshold SNR γ th = 15 dB, detector aperture diameter D R = 5 cm, noise variance σ 2 n = −100 dBm, the reflection coefficient of reflecting elements ρ = 1, the extinction coefficient c = 0.15 m −1 , wavelength λ= 530 nm, the link distance L 1 = L 2 = 50 m, the dissipation rate of mean-squared temperature [32] , the dissipation rate of turbulent kinetic energy per unit mass of fluid [32] Based on these parameters, the turbulence parameters are set to α = 4.37 and β = 1.24.
We first confirm the accuracy of the approximate distribution of the sum of Gamma-Gamma random variables, i.e., h t = N n=1 h tn ∼ ΓΓ(α h , β h , N).In Fig. 2, we give the exact cumulative distribution function (CDF) and the approximate CDF for different numbers of reflecting elements.The exact CDF is the empirical CDF based on the sum of the N i.i.d randomly generated Gamma-Gamma random variables from (6), while the approximate CDF is the empirical CDF based on the randomly generated Gamma-Gamma random variables from (17).From Fig. 2, it can be found that the approximate CDFs are well-matched with the exact CDFs.Further, Kolmogorov Smirnov statistic [43] is used as goodness of fit.This quantifies the vertical distance between the approximate CDF and the exact CDF, and the distance should not be greater than the so-called maximum difference D m .For 1 × 10 4 samples we used, D m = 0.0136 and confidence level of 0.05 are usually selected [43].We find that the distance is smaller than D m for different numbers of reflecting elements.Therefore, it can be concluded that the approximate distribution we applied is accurate.
In Fig. 3, we analyze the outage probability performance of the RIS-assisted UWOC system with different numbers of reflecting elements.We consider the number of reflecting elements of N = 2, 4, 8, 16, 24, 32, 40, 48, 56.It can be observed from Fig. 3 that the analysis results in our derived closed-form expression of (30) are consistent with the simulation results, which verifies the accuracy of the derivation.Even if the number of reflection elements is small, such as N = 2 and N = 4, our results still accurately evaluate the system performance.From Fig. 3, the outage probability decreases with the increase in the number of reflecting elements of RIS.For example, SNR of γ = 21 dB is required for N = 8 in order to achieve a target outage probability of P out = 10 −3 .The SNR decreases to 13.8 dB, 9.7 dB and 7 dB, and an extra SNR of 7.2 dB, 11.3 dB and 14 dB is required for N = 16, N = 24 and N = 32.It can also be shown from Fig. 3 that when the number of IRS reflecting elements reaches a certain number, the outage probability performance of the UWOC system is significantly improved.However, the outage probability performance will improve more slowly with the increase in the number of reflecting elements.This is also consistent with the results in Fig. 4. From Fig. 4, the outage probability decreases with the increase of N , but the decline speed becomes slower.The outage probability performance gain G(N ) also decreases with the increase in the number of reflecting elements.When the number of reflective elements is relatively large, G(N ) tends to be stable.Although G(N ) tends to be stable with the increase of N , the value of G(N ) is always greater than 1, i.e., G(N ) > 1.This means that the outage probability always decreases with the increase of N .Therefore, it can be concluded that when N → ∞, the outage probability approaches zero.This is consistent with our analysis of P out (N ) N →∞ .In Fig. 5, we consider the outage probability with different jitter values.Outage probability is obtained based on (30) and, as expected, it is well matched with the simulation results.Our results further quantify the outage probability performance degradation associated with the increase in pointing errors caused by the increase in jitter values.It is observed from Fig. 5 that the increase in σ θ and σ φ will all lead to the deterioration of outage probability performance.For instance, SNR of γ = 10 dB is required in order to achieve a target outage probability of P out = 10 −3 when σ θ = 1 × 10 −3 and σ φ = 1 × 10 −3 .The required SNR climbs to 12.5 dB and 17 dB, In Fig. 6, we present the diversity order with different numbers of reflecting elements, and verify the closed-form expressions for the finite-SNR diversity order in (42) and the asymptotical diversity order in (46), respectively.As a benchmark, we include the finite-SNR diversity order given by numerical evaluation in Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.(37).The plot shows that our derived closed-form expression for the finite-SNR diversity order in (42) provides an exact match to the diversity order in (37).This proves the correctness of our derivation.At a lower SNR level, the finite-SNR diversity order increases with the increase in SNR and the number of reflecting elements.For example, the finite-SNR diversity order at 10 dB is given as 0.43, 1.29, 3.22 and 4.29 for the numbers of reflecting elements of N = 2, 4, 8, 12.At a higher SNR level, the diversity order does not change significantly with the increase in SNR.It reaches the upper limit of the diversity order, i.e., the asymptotical diversity order obtained by (46).In this scenario, the asymptotical diversity order depends on the number of reflecting elements, the turbulence parameters, and the pointing errors.Here, we set σ θ = 1 × 10 −3 , σ φ = 1 × 10 −3 , then ξ 2 is calculated as 8.76.According to (46), when N = 2 and 4, we obtain Nβ < ξ 2 .In this case, the asymptotical diversity order depends on the number of reflecting elements and the turbulence parameter, and ADO = 1 2 min(ξ 2 , Nα, Nβ) = the number of reflecting elements of N reaches a specific value, if we continue to increase N , the diversity order performance will not be improved.In Fig. 7, we show the convergence speed of the finite-SNR diversity order to the asymptotical diversity order.Similar to Fig. 6, a perfect match can be observed between the convergence speed expression derived in (52) and its numerical calculation based on (47).With the increase in SNR, the convergence speed gradually tends to zero, which indicates that the finite-SNR diversity order has reached the asymptotical diversity order.It is further observed that the more reflecting elements, the faster convergence to the asymptotical diversity order.Although the systems with N = 8 and N = 12 have the same asymptotical diversity order in Fig. 6, the system with N = 12 converges faster, i.e., it can reach the asymptotical diversity order under lower SNR.
In Fig. 8, we investigate the diversity order with different jitter values.In Fig. 9, we show the corresponding convergence speed.With the increase in jitter values, the effect of the pointing error becomes larger, which leads to the reduction of finite-SNR diversity order.For example, when σ θ = 1 × 10 −3 and σ φ = 1 × 10 −3 , the finite-SNR diversity order is 2.0 at γ = 20 dB.If σ θ increases to 1.5 × 10 −3 and 2 × 10 −3 , the finite-SNR diversity order falls to 1.84 and 1.51.Additionally, if we set σ φ = 1 × 10 −3 , the same asymptotical diversity order is obtained when σ θ = 1 × 10 −3 and 1.5 × 10 −3 .In the two cases, ξ 2 can be calculated as 8.76 and 5.38, and Nβ < ξ 2 , the asymptotical diversity order depends on the value of Nβ, then we obtain ADO = 1 2 min(ξ 2 , Nα, Nβ) = 1 2 Nβ = 2.48.Although they have the same asymptotical diversity order, the convergence speed of the system with σ θ = 1.5 × 10 −3 is slower Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.  in Fig. 9.In Figs. 8 9, as the jitter value increases, the asymptotical diversity order decreases, and the convergence speed becomes faster.Because we have ξ 2 < Nβ, and the asymptotical diversity order depends on the value of ξ 2 .At this time, according to (46) and (52), it can also be obtained that the decrease in ξ 2 can lead to the decrease in the asymptotical diversity order and the increase in the convergence speed.To sum up, we can draw insightful conclusions.When the asymptotical diversity order is dominated by the number of reflecting elements and the minimum turbulence parameter, i.e., ξ 2 > N min(α, β), the increase in pointing error level will not change the asymptotical diversity order.However, it can reduce the finite-SNR diversity order and the convergence speed.When the asymptotical diversity order is dominated by the pointing error, i.e., ξ 2 < N min(α, β), the increase in the pointing error level will reduce the finite-SNR diversity order and the asymptotical diversity order, while the convergence speed will increase.
In Fig. 10, we investigate the ergodic channel capacity with different numbers of reflecting elements.The simulation results match well with the analysis results of (56), which verifies the accuracy of the derivation of ergodic capacity.We can find that increasing the number of reflecting elements can significantly improve the ergodic capacity.For example, the ergodic capacity is 5.74 bit • s −1 • Hz −1 at γ = 20 dB when N = 2.However, when N = 4, 8, 16, 24, 32, the ergodic capacity is increased to 7.99, 10.19, 12.30, 13.45 and 14.35 bit • s −1 • Hz −1 at γ = 20 dB, respectively.It can also be observed that for the cases of N = 8, 16, 24, 32, the increment of ergodic capacity decreases with the increase of N at the same SNR.In other words, the increase speed of ergodic capacity slows down.

V. CONCLUSION
In this paper, we investigated a RIS-assisted UWOC system with IM-DD in an underwater environment with obstacles.The cascaded turbulence channel fading coefficients from source to destination through RIS are modeled as Gamma-Gamma distribution, and the pointing error caused by beam jitter and RIS jitter has been considered.The probability density function of instantaneous received SNR is proposed by using the approximate distribution of the sum of Gamma-Gamma random variables, and the novel closed-form expression of outage probability is derived.Then, based on the outage probability result, the finite-SNR diversity order, the asymptotical diversity order, and the convergence speed of finite-SNR diversity order to asymptotical diversity order have been analyzed.Furthermore, the ergodic channel capacity is also investigated.Finally, the simulation results are presented to confirm the accuracy of our derivations.Analytical and simulation results show that increasing the number of reflecting elements improves the outage probability performance and the ergodic channel capacity, while the pointing error deteriorates the system performance.When the number of reflecting elements is very large (i.e., tends to infinity), the outage probability asymptotically approaches zero.Additionally, appropriately increasing the number of reflecting elements can improve the finite-SNR diversity order and its convergence speed, and the asymptotical diversity order depends on the number of reflecting elements, the pointing error, and the minimum turbulence parameter.

Manuscript received 26
March 2024; revised 16 May 2024; accepted 28 May 2024.Date of publication 31 May 2024; date of current version 10 June 2024.This work was supported in part by the National Natural Science Foundation of China under Grant 62301108 and in part by the Science and Technology Program of Liaoning Province under Grant 2023JH26/10300010.(Corresponding author: Dian-Wu Yue.)

Fig. 2 .
Fig. 2. Exact and approximate CDFs for different numbers of reflecting elements.

TABLE I EXISTING
RESEARCH ON RIS-ASSISTED FSOC AND UWOC SYSTEMS To obtain a Meijer-G function with a numerator-based variable h p , we utilize the reflection property of the Meijer-G function shown as Eq.(07.34.16.0002.01) in [38], and we have