Intelligent Reflecting Surfaces for Underwater Visible Light Communications

Intelligent reflecting surfaces (IRSs) offer paradigm shift towards enhancing the capabilities of wireless communications. The use of this emerging technology in the realm of underwater wireless systems is a promising solution to overcome the limitations pertinent to such challenging environments. In this paper, we quantify the performance enhancement offered by the integration of IRS technology in the context of underwater optical wireless communication (OWC). Specifically, we derive a closed-form expression for the outage probability over log-normal channels, taking into consideration the underwater attenuation, pointing error, and turbulence effects. The underwater turbulent medium is characterized by the recently introduced Oceanic Turbulence Optical Power Spectrum (OTOPS) model that uses the practical values of average temperature and salinity concentration in earth basins. The presented numerical results take into account the effects of the turbulent medium as well as the communication system parameters (i.e., communication range, receiver aperture diameter, number of IRS). Our results show that IRSs can offer significant enhancement in the reliability of underwater OWC systems under attenuation, beam displacement, and turbulence effects. Moreover, the combined effect of using a large number of reflecting surfaces and a larger aperture diameter yields a more noticeable improvement.

underwater vehicles (AUVs) through reliable wireless links in order to facilitate seamless underwater operations such as undersea monitoring, marine life protection, oil and gas exploration, and navigation support, to name a few [1], [2]. Communication in IoUT can be established based on three different types of propagation media, namely: acoustic, radio frequency (RF), and optical. While acoustic signals can propagate for long distances, they typically only support low data rates. RF signals, on the other hand, support higher data rates but at the expense of decreased communication range. Visible light communication (VLC), as a subset of underwater optical wireless communication (UOWC), is particularly well-suited to underwater applications as it allows sufficiently high data rates and low latency at medium transmission ranges [3].
The continuing advancements in solid-state lighting and optical detectors are paving the way for wide adoption of UOWC systems. In UOWC, the information is encoded on the intensity of the light beams emitted by light-emitting diodes (LEDs) or laser diodes (LDs) and the receiver side employs photo-detectors (PDs) to detect the fluctuations in the received light intensity and translate it into a decodable signal [4]. UOWC is a viable solution for providing low-power, low-cost, high-speed underwater communications. For example, an UOWC link utilising LDs was demonstrated in [5] offering a data rate of 1.5 Gbps over a 20 m distance. In [6], adaptive bit-power loading discrete multi-tone modulation combined with nonlinear equalisation was shown to enhance the capacity of UOWC, achieving a 7.33 Gbps over a 15 m distance. A transmission distance of 56 m was realised in [7] based on frequency domain equalisation combined with a time-domain decision feedback noise predictor at the receiver. More recently, a record transmission range of 150 m offering a data rate of 500 Mbps was demonstrated in [8] based on combination of partial response shaping, interleaving, precoding, and Trellis coded modulation technology. It is noted that one of the obstacles of achieving ubiquitous UOWC connectivity in IoUT is the limitation on the transmission range as well as the susceptibility to turbulence caused by the random variations of the refractive index of water, which leads to both intensity and phase fluctuation of the optical beams. A possible solution is to use high-sensitivity detectors which relax the alignment requirement such as photon-counting receivers [9] and avalanche diodes [10]. Nonetheless, these types of detectors are typically characterised with low modulation bandwidth compared to traditional photo-detectors, which reduces the spectral efficiency and achievable capacity. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ The emerging concept of intelligent reflecting surfaces (IRSs) opens the door for the possibility of controlling and optimising the wireless medium in underwater communications. IRS technology offers a change of paradigm by introducing metasurface structures that can be programmed and reconfigured to achieve a specific response to the incident signals [11]. Utilising these structures, the optical radiation can be manipulated by introducing engineered responses that affect one or more of the light wave characteristics, i.e., amplitude, phase, polarisation, spatial power distribution, and wavefront shape. Based on this, the propagation of the wireless signal can be controlled in order to achieve specific quality-of-service (QoS) requirements in terms of throughput, coverage, reliability, security, etc [12]. Moreover, the use of multiple IRSs within the UOWC network allows the possibility of creating multi-hop transmission by directing and steering the light beams, establishing non-line-of-sight (NLoS) connections between the IoUT entities [13]. However, the feasibility of this approach is mainly dependent on the characteristics and capabilities of the employed reflecting surfaces. Recently, mitigating the turbulence effect by using IRSs was investigated for Gamma distributed channel and Nikishov's power spectrum model and the associated bit-error-rate (BER) performance of the system was presented [14]. However, there is a need to evaluate the IRS effect under various underwater turbulence conditions to get a better idea of the feasibility and effectiveness of this technology.
In this paper, we investigate the outage probability performance of IRS-assisted VLC links taking into account the effects of attenuation, pointing error, and turbulence. We model the underwater turbulent medium by the recently introduced Oceanic Turbulence Optical Power Spectrum (OTOPS) model that uses practical values for the average temperature and average salinity concentration in earth basins. Our results indicate that the integration of IRSs in the underwater medium can offer significant enhancement in the link reliability under attenuation, beam displacement, and turbulence effects.
The rest of the paper is organised as follows; Section II describes the system and channel model of IRS-assisted UOWC. The outage probability derivations are shown in Section III while Section IV presents and discusses the obtained numerical results. Finally, the conclusions are drawn in Section V.

II. SYSTEM AND CHANNEL MODEL
The block diagram of the investigated system model is given in Fig. 1. An UOWC link is configured between two platforms operating in the underwater medium and an IRS is used to create an alternative path between the transmitting and receiving entities. The communication between the underwater platforms is provided by means of a VLC link. The attenuation due to absorption and scattering phenomena is taken into account through Beer-Lambert law. The horizontal and vertical beam displacements are assumed to be independently Gaussian distributed and the pointing error is modeled by Rayleigh distribution. The Lognormal distributed channel model is chosen as the probability density function (PDF) of the underwater turbulence and the recently introduced OTOPS is used to characterise the turbulence power spectrum. The signal-to-noise-ratio (SNR)  dependent channel PDF and cumulative distribution function (CDF) are obtained by combining the effects of attenuation, pointing error and underwater turbulence. The performance of the UOWC system is analysed in terms of outage probability.

A. Attenuation
According to the Beer-Lambert's law, the attenuation due to absorption and scattering can be expressed as where c(λ) is the attenuation coefficient, λ is the wavelength and L is the link length. The attenuation coefficient c(λ) can be written as the sum of absorption and scattering coefficients, c(λ) = a(λ) + b(λ). Absorption remains the most dominant factor on optical beams in underwater medium and is mainly dependent on the chlorophyll concentration. The classification of waters is generally based on chlorophyll concentration, as given in Table I [15]. The absorption coefficient given in (1) is decomposed as [16], [17] where a w (λ) is the absorption coefficient of pure water (1/m) and is given for optically and chemically pure water depending on the wavelength [18], a cl (λ) = a 0 c (λ) × (C c /C 0 c ) 0.0602 is the absorption coefficient of chlorophyll, a 0 c (λ is the specific absorption coefficient of chlorophyll, C 0 c = 1 mg/m 3 is the chlorophyll concentration, C c is the total concentration of the chlorophyll in mg/m 3 , a f (λ) = a 0 f C f exp(−k f λ) is the absorption coefficient of fulvic acid, a 0 f = 35.959 m 2 /mg is the specific absorption of fulvic acid, (1) is given by [16], [17] where b w (λ) = 0.005826(400/λ) 4.322 is the scattering coefficient of pure water [18], b 0 s (λ) = 1.151302(400/λ) 1.7 is the scattering due to small particles, b 0 l (λ) = 0.3411(400/λ) 0.3 is the scattering due to large particles,

B. Pointing Error
The PDF of the pointing error is [19] , r a = D G /2 is the receiver circular aperture radius, D G is the receiver aperture diameter, σ s is the standart deviation of pointing error, and

C. Underwater Turbulence
The PDF of underwater turbulent channel modeled by lognormal distribution is found to be [20] is the log-amplitude variance, σ 2 I is the scintillation index and is given for propagating Gaussian beam and apertured receiver by [20] where σ 2 lnX (D G ) and σ 2 lnY (D G ) denote the log variances of large-scale and small-scale, respectively and they are expressed by [20] In the previous equations, σ 2 R is the Rytov variance of the plane wave, σ 2 B is the Rytov variance of the Gaussian beam wave, Ω G = 2L/kW 2 G is the parameter characterising the spot radius of the collecting lens, W G is the radius of the Gaussian lens and is the beam curvature parameter at receiver. The Rytov variances for plane and Gaussian beam waves are analytically obtained for underwater medium using OTOPS model as [21], [22] where k = 2π/λ is the wave number, L is the link length, β 0 = 0.72, ε is the energy dissipation rate, χ T is the temperature dissipation rate, χ S = d r χ T /H 2 and χ T S = 0.5(1 + d r )χ T /H are the ensemble-averaged variance for salinity and co-spectrum dissipation rates, d r is the eddy diffusivity ratio, H is the temperature-salinity gradient ratio, 2 F 1 (.) is the hypergeometric function, A and B are the linear coefficients depending on the average temperature T , average salinity concentration S . The eddy diffusivity ratio that is used for χ S and χ T S calculation is [23] where R p = |H|α T /β S is the density ratio, α T is the thermal expansion coefficient and β S is the saline contraction coefficient. The linear coefficients A and B are expressed by [24], [25] where n(T, S, λ) is the refractive index of seawater and empirically obtained as [26] n(T, where a 0 = 1.31405, a 1 = 1.779 × 10 −4 , a 2 = −1.05 × 10 −6 , a 3 = 1.6 × 10 −8 ), a 4 = −2.02 × 10 −6 , a 5 = 15.868, a 6 = 0.01155, a 7 = −0.00423, a 8 = −4382, a 9 = 1.1455 × 10 6 , T is the temperature and S is the salinity concentration. The OTOPS power spectrum is modeled by [24] Φ n (κ, T , where each spectrum is expressed by where η is the Kolmogorov microscale length, the nondimensional coefficients c i are given as where P r = μc p /σ T is Prandtl number, μ is the dynamic viscosity, c p is the specific heat, σ T is the thermal conductivity, 15)ρ] is Schmidt number and ρ is the water density. The details of the parameters and their derivations are given in [24], [25].

D. Light Propagation Model
Considering that a UOWC system using intensity modulation and direct detection scheme (IM/DD) operates in underwater medium and additive white Gaussian noise (AWGN) n 0 with zero mean and variance σ 2 n is incorporated to the transmitted signal at the receiver.
The received signal can be written as where h sr n = v n e −jφ n and g ru n = u n e −jϕ n represent the path gain of first (source to IRS) and second (IRS to user) links, v n = h a1n h p1n h l1n for the first link with length of L 1 and u n = h a2n h p2n h l2n for the second link with length of L 2 including turbulence, pointing error and attenuation effects, x is transmitted signal, β n ∈ [0, 1] is the reflection amplitude of the n th reflecting element, θ n ∈ [0, 2π] is the phase shift induced by the n th reflecting element, and N is the total number of reflecting elements in the IRS array. We can write (17) in matrix form as where h sr = [h sr 1 h sr 2 h sr 3 . . . h sr N ] T and g ru = [g ru 1 g ru 2 g ru 3 . . . g ru N ] T . Moreover, the phase shift and reflection applied by the IRS can be expressed as Consequently, (18) can be arranged as v n e −jφ n β n e jθ n u n e −jϕ n x + n 0 We assume that the IRS reflecting elements are set such that θ n = φ n + ϕ n in order to provide the maximum SNR. Based on that, (19) can be writen as We assume that all the IRS reflecting elements are identical having the same reflection coefficient, i.e., β n = β for n = 1, 2, . . ., N, then the instantaneous electrical SNR can be defined as where h n = v n u n since the phase has been compensated. The average SNR can be written as γ = P t /σ 2 n . The link length between transmitter and receiver can be approximated as L ≈ L 1 + L 2 . Then, assuming that random variables h 1 , h 2 , ..., h N are independent and identically distributed (i.i.d.), SNR expression becomes where h is the total channel state.

III. OUTAGE PROBABILITY ANALYSIS
The channel state including the underwater turbulence, beam displacement and attenuation effects can be written as h = h a h p h l . Then, the PDF of the combined channel state becomes [27] where f h|h a (h|h a ) is the conditional probability and is shown by

Then, inserting (5) and (24) into (23)
Then, (25) becomes Changing variable as δ = ln(h a ) √ 2σ 2 l , (27) takes the form To solve integral given in (28), we can use Eq. (3.322-1) of [28] that is given by . (29) Applying (29)to (28), we arrive In (30), obtained PDF depends on the combined channel state h including the attenuation, turbulence and pointing effects. We also note that the channel state h is only for one source to user link.Since we are assuming that the random variables h 1 , h 2 , . . ., h N are independent and identically distributed (h 1 = h 2 = . . . = h N = h), the PDF of the instantaneous SNR including all links through each IRS surface can be expressed depending on the average SNR and number of used IRS surfaces. Similar approximation is used in [14]. Using the transformation given in [14], [29], the PDF of SNR for lognormal distribution can be written as Using the relationships given in (22) and (31), the SNR dependent PDF can be obtained as The CDF of SNR γ is found by Inserting (32) into (33), we have There are two integral parts in (34). The first part can be taken as The second integration is challenging and changing parameters as

result in a new equation as
where Δ 1 = . Dividing integration in (35) and using some mathematical manipulations and the odd function property of erf, (35) turns into two-parts integral as It is seen that the u dependent integral in (36) The second part integration can be solved by using following equation that is given by [31] Applying (37) and (38) to (36), the CDF of underwater turbulent medium including the effects of attenuation, pointing error and turbulence can be found as levels as T = 15 • C and S = 20 ppt. The optical wave is the collimated Gaussian beam with F 0 = ∞ phase front radius of curvature and W 0 = 2 cm radius. The link length in underwater medium remains in the range of several tens of meters here, it is set to L = 20 m that is challenging but realistic for UOWC systems due to combined effects of different phenomena such as absorption, scattering, turbulence and pointing error. In underwater medium, the temperature dissipation rate χ T and the energy dissipation rate ε change in the range of χ T = 10 −4 − 10 −10 K 2 s −1 and ε = 10 −2 − 10 −10 m 2 s −3 , respectively. The values of χ T and ε are selected for their moderate levels as χ T = 10 −7 K 2 s −1 and ε = 10 −5 m 2 s −3 to see the improvement better with the IRS application.
Results are obtained by using MATLAB simulation environment. To verify the accuracy of our derivations, we compared our derivations with their initial counterparts for all steps. We first validated our derivations for PDF f h (h) by comparing initial equation (25) with the PDF derivation given in (30). We performed an additional validation for our derivations by comparing the initial equation of CDF F γ (γ) given in (34) with the analytical derivation of CDF given in (39). In both cases, it was observed that initial and derived equations match perfectly which indicates the accuracy of our results. Fig. 2 depicts the average outage probability variation depending on both the link length and the receiver aperture diameter. It can be seen that the outage probability increases with the increase in link length. Using an aperture with D G = 2 cm and N = 50 IRS reflecting elements, the outage probability takes the values of ∼ 2.4 × 10 −5 , ∼ 1.6 × 10 −4 , ∼ 7.1 × 10 −4 and ∼ 2.6 × 10 −3 for the link length values of L = 10 m, L = 20 m, L = 30 m and, L = 40 m, respectively. Since we focus on the combined effect of absorption, scattering, pointing error and turbulence in this study, results show that the practical effective distance of an UOWC system operating in an underwater medium remains in a few ten meters. The increasing trend of outage probability with the link length is also seen from Fig. 3.  Another conclusion from Fig. 2 is the significant reduction in outage probability with the receiver aperture averaging. For example, keeping the link length as L = 10 m and N = 50 IRS surfaces, outage probability falls from ∼ 1.6 × 10 −4 to ∼ 2.4 × 10 −5 changing the point receiver (D G = 0) with the D G = 2 cm apertured receiver. In Fig. 3, the benefit of using IRS for UOWC in underwater turbulent medium is observed. When link length is fixed as L = 20 m, outage probability takes the value of ∼ 2.4 × 10 −2 when no IRS element is used. However, outage probability maintains its reduction with the values of ∼ 6.5 × 10 −5 , ∼ 8.3 × 10 −6 , and ∼ 3.4 × 10 −6 with an increase in the number of IRS sequentially as N = 100, N = 500, and N = 1000. Changing the number of IRS from N = 0 up to N = 1000 reduces outage probability from order of 10 −2 to the order of (10 −7 − 10 −3 ) depending on the link distance. These results indicate that IRSs can provide a substantial improvement in the performance of the UOWC system. Another conclusion from Fig. 3 that using IRS may not yield benefit after a certain distance because the combined effect of attenuation, pointing error and turbulence becomes severe and outage probability values of IRS cases gradually merge to no IRS case depending on the number of IRS elements (This can be seen for N = 100 IRS in Fig. 3).
In Fig. 4, the outage performance of an UOWC system versus the number of IRS is plotted for various values of receiver aperture diameter. A monotonic decrease in outage probability with the increase in the number of IRS is seen. Using a receiver with D G = 1 cm aperture diameter, the average BER takes the values of ∼ 7 × 10 −3 for no IRS is used. However, outage probability decreases to ∼ 1.5 × 10 −4 , ∼ 1.9 × 10 −5 , ∼ 2.4 × 10 −6 , and   is seen then, outage probability decreases slower with the increase of a number of IRS surfaces. The larger the receiver aperture diameter the smaller the outage probability trend is also seen from Fig. 4. We note that the combined effect of using large number IRS and a larger aperture diameter yields more performance improvement. For example, the outage probability reaches the value of ∼ 6.7 × 10 −5 from ∼ 2.4 × 10 −2 by using N = 100 IRS elements for an UOWC system using a point receiver (D G = 0). However, outage probability decreases to ∼ 9.8 × 10 −6 when receiver aperture diameter is increased to D G = 2 cm and the number of IRS is still N = 100. Fig. 5 presents the outage probability variation with the temperature dissipation rate for various numbers of IRS. One can see from Fig. 5 that the outage probability stands smaller with the smaller values of temperature dissipation rate showing the weaker turbulent power spectrum strength and hence, less turbulence effect. Keeping the number of IRS as N = 100, the outage probability jumps from ∼ 2.1 × 10 −6 to ∼ 8 × 10 −4 when temperature dissipation rate raises from χ T = 10 −10 K 2 s −1 to χ T = 10 −4 K 2 s −1 . By observing the outage probability reduction from ∼ 7.6 × 10 −3 to ∼ 1.1 × 10 −6 with the increase of the number of IRS from N = 0 to N = 1000 for the fixed value of χ T = 1 × 10 −8 K 2 s −1 , the advantage of using IRS as a mitigation technique is obvious from Fig. 5. The effect of another dissipation rate, kinetic energy dissipation rate ε, is illustrated in Fig. 6. We observe that an increase in the kinetic  energy dissipation rate causes a fall in the outage probability showing that the performance of UOWC system improves when the underwater turbulent medium is more energetic. This can be verified by varying of outage probability from ∼ 4 × 10 −4 to ∼ 2.1 × 10 −5 with the change of kinetic energy dissipation rate from ε = 10 −10 m 2 s −3 to ε = 10 −2 m 2 s −3 keeping the number of IRS fixed as N = 100. Similar to the previous figures as a function of the number of IRS, the considerable reduction in the outage probability is also seen with the increase in the number of IRS in Fig. 6.
The log(P out ) variation of an UOWC depending on the average temperature T and average salinity concentration S as density plots in Fig. 7. From Fig. 7, it is seen that increase in both the average temperature (horizontal axes) and the average salinity concentration (vertical axes) cause a performance degradation up to a certain level. The outage probability value has an order of magnitude around ∼ 10 −8 for the number of IRS N = 50 when average temperature and average salinity concentration take their highest values then, outage probability decreases around ∼ 10 −10 when both parameters take their lowest values.
The reflection amplitude of IRS is generally assumed to be β = 1 (perfect reflector) for the sake of simplicity in almost all studies. However, the reflection amplitude is one of the most important factors that define the efficiency of the IRS implementation. For this purpose, Fig. 8 represents the outage  probability variation versus the number of IRS for various values of reflection amplitude of IRS. It is observed from Fig. 8 that the reflection amplitude becomes a performance improving factor and outage probability remains at lower levels when reflection coefficient aprroximates the value of β = 1.
In Fig. 9, the outage performance of an UOWC system is illustrated as function of average SNR for various values of beam waist. The significant performance improvement effect with the average SNR increase is observed. Keeping beam waist as ω b = 2 × r a , the outage probability drops from ∼ 1.5 × 10 −2 to ∼ 1.4 × 10 −3 when average SNR increases from γ = 20 dB to γ = 100 dB. It is also seen from Fig. 9 that an UOWC can benefit from the higher beam waist due to increased probability of collecting more of optical beam at the receiver aperture. While average SNR is γ = 40 dB, outage probability varies positively from ∼ 1.3 × 10 −2 to ∼ 7.4 × 10 −4 with the increase of beam waist from ω b = 1 × r a to ω b = 4 × r a .
Finally, the outage performance of an UOWC system is shown for different types of waters in Fig. 10. Since waters are classified based on chlorophyll concentration, the drastic effect of chlorophyll concentration is seen. Although N = 50 number of IRS elements are used, the outage probability can drop below ∼ 10 −6 level for average SNR values of γ = 60 dB, γ = 69 dB, and γ = 84 dB in pure, clear ocean and coastal waters, respectively. However, it is not possible to catch the outage probability below ∼ 10 −6 level in harbor water. These results show that underwater medium is still challenging for UOWC system and optimum distance remains as few ten meters even IRS is implemented.

V. CONCLUSION
The outage performance of a VLC-based UOWC system operating in an underwater turbulent medium and the effect of IRS implementation are theoretically analysed. The closed-form expression of outage probability including the IRS effect is obtained. Results show that the reliability enhancement offered by IRSs is undeniable. Using a sufficiently large number of IRS can cause a significant enhancement in the system performance. The reflection amplitude of the used IRS also remains an important factor in the outage probability performance of UOWC system. The outage probability tends to increase with the increase of link length, temperature dissipation rate, average temperature, and average salinity concentration. However, the outage probability starts to decrease with the increase of receiver aperture diameter, the number of IRS, the reflection amplitude of IRS, the kinetic energy dissipation rate and the average SNR. All these parameters need to be taken into consideration when designing and optimising IRS-assisted UOWC systems. Based on that, an optimal configuration of the number and reflection coefficients can be achieved.