Information Capacity of Turbulent and Absorptive Underwater Wireless Link with Perfect Laguerre–Gaussian Beam and Pointing Errors

Abstract

The model of information capacity for underwater wireless optical communication (UWOC) links with pointing errors and the carrier of perfect Laguerre–Gaussian (PLG) beam in absorbed and weakly turbulent seawater is modeled. Using this model, the influence of channel parameters on the propagation of PLG beams is numerically analyzed. We show that the pointing error significantly reduces the performance of the optical transmission system. Large topological charge and radial orders are not conducive to high information capacity transmission. Additionally, the information capacity decreases with the increase in the spectral absorption coefficient of seawater. With higher transmission power, the transmission of an optical transmission system with high average capacity can be realized. The information capacity is affected by both signal wavelength and seawater absorption coefficient. That is, in the case of small seawater absorption and short message channel, the channel capacity loss of the link with long signal wavelength is smaller. In the case of long channels and strong seawater absorption, the influence of signal wavelength on the average capacity can be ignored. The information capacity of the communication link is the highest, which is composed of the carrier with the OAM topological charge, and the radial orders are both 1, and the wavelength is 410 nm.

1. Introduction

The underwater wireless optical communication (UWOC) system has the advantages of high anti-interference, high security, high information capacity (large bandwidth) and low cost, which has attracted increasing attention [1,2]. However, the performance of the UWOC system, especially the capacity of communication information, is severely restricted by the combined effects of seawater absorption, water flow [3], turbulence [4,5] and pointing error [6]. So far, there have been a lot of reports on the information capacity of UOWC links affected by turbulent seawater, such as as Gaussian beam (GB) with large wavelength and waist radius propagation in weak and unstable stratification ocean turbulence with large edge scale turbulence, including small pointing error and large information capacity [4]. The channel capacity of a UOWC link with GB in anisotropic strong oceanic turbulence is always greater than that of a UOWC link in isotropic strong oceanic turbulence for the same system parameters [7]. The average capacity of a UOWC link with partially coherent GB is slightly larger in clear ocean than in coastal ocean [8], and the converging Gaussian beam array can effectively reduce the scintillation caused by ocean turbulence in a UWOC link between misaligned sensors [9]. Cheng et al. [10] found that quasi-nondiffracting Bessel–Gauss (BG) beams have obvious advantages over Laguerre–Gaussian (LG) beams in reducing turbulence and improving the performance of OAM optical Communication Links in the weak turbulence region. In a weak UOWC link, large-capacity ocean optical communication can be realized by using Bessel–Gaussian local wave (BGLV) [11] and Lommel–Gaussian local wave (LGLV) [12], and under certain parameters of a UOWC link, there exists the maximum effective waist. The channel capacity of the link can be increased by adopting a wider initial half-pulse width, smaller Bessel cone angle of BGLW and larger receiver aperture of the link [13]. Yang et al. [14] show that the transmission distance of OAM mode carried by POV [15,16] in four kinds of seawater is better than that of a Bessel–Gaussian beam [10]. For the link with coaxial double-position power Gaussian vortex carrier in weak turbulence, the link with high information capacity can be achieved by increasing the receiving aperture, reducing the diameter of the transmitting beam, adopting the long wavelength of the light source in the seawater window and multiple OAM channels [17]. In the link with strong oceanic turbulence and pointing error, the performance of the optics communication system can be improved significantly by a smaller rectangular QAM dimension in short distance, or a larger aperture diameter and the ratio of beam width to aperture radius [18]. There exists an optimizing half-beam width of perfect optical vortex (POV) link signal source of the UWOC link that is the function of the topological charge and transmission distance [19]. The UWOC link with perfect Laguerre–Gaussian (PLG) beam [20] can obtain high channel capacity by adopting low OAM topological charge and smaller aperture of transceiver system [21]. The channel capacity of free-space optical UWOC links is significantly enhanced with an aggregate transmission rate of 100 Gb/s by combining dual-wavelength wavelength division multiplexing and four-level pulse amplitude modulation [22]. Models for the outage probability and average channel capacity of the UWOC systems are proposed by the generalized Málaga fading model of weak-to-strong turbulent links [23,24]. The average symbol error probability and average channel capacity of the UWOC link with absorption, scattering and turbulence (modeled by Málaga distribution), and M-ary pulse position modulation (PPM), is under the constraints of the limited average power and peak power [25]. Compared with the LG beam and the BG beam, anti-diffraction and anti-attenuation of the random beam have a stronger ability to resist seawater turbulence and seawater absorption [26]. The existence of input-dependent noise severely degrades the channel capacity of the link with absorption and scattering losses [27]. Spatial diversity MIMO schemes can improve the UOWC link performance [28]. The POV beam could improve the capacity of the spatial-mode modulating, multiplexing and OAM-based UWOC link [29]. Choosing optimum beam source parameters is favorable to mitigate the influence of oceanic turbulence, and the partially coherent elegant LG beams are more affected by turbulence as compared with the fully coherent elegant LG beams [30].

As far as we know, the research on channel capacity affected by link pointing error in OAM underwater communication has not been reported in detail.

In this paper, the influence of absorption and weak turbulent seawater on the OAM mode of PLG beam propagation is studied from the point of view of the information capacity of the link when the link pointing error exists. In Section 2, the information capacity of the OAM communication link that has pointing errors is presented. The numerical simulations for demonstrating the relationships between the information capacity of the communication link and the parameters of the beam and turbulence are in Section 3. Finally, conclusions are given in Section 4.

2. Information Capacity of the Link with Perfect Laguerre–Gaussian Beam Carrier and Pointing Errors in Turbulent Seawater

2.1. Perfect Laguerre–Gauss Beam in Absorbing Turbulent Seawater

In the UWOC system model (Figure 1), the transmitter is composed of M-QAM modulator, light source and optical antenna. The receiver is made of an optical antenna, detector and demodulator.

In cylindrical polar coordinates $(r,\phi ,z),$ the normalized complex amplitude of the PLG beam at z plane from the source in absorbing seawater is represented by [19]

$$E_{m_0}(r,\phi ,z)=\frac{R_{m_0,p}(r,z)}{\sqrt{2\pi }}exp\left(\mathrm{i}{m}_0\phi +\mathrm{i}{k}_{0}nz\right)exp\left[-\mathrm{i}\left(2p+\left|m_0\right|+1\right){\delta }_{\mathrm{G}}-\frac{\alpha \left(\lambda \right)}{2}z\right],(m_0\ne 0)$$

(1)

with the complex amplitude of the PLG beam $R_{m_{,},p}(r,z)$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill R_{m_0,n}(r,z)& =\frac{1}{w\left(z\right)}\sqrt{\frac{4p!}{\left(p+\left|m_0\right|\right)!}}{\left(\frac{\sqrt{2}r}{w\left(z\right)}\right)}^{|m_0|}{\mathrm{L}}_p^{m_0}\left(\frac{2r^2}{w^2\left(z\right)}\right)\hfill \\ & \times exp\left[-\left(\frac{1}{w^2\left(z\right)}-\frac{k_{0}n}{2R\left(z\right)}\right)r^2\right]exp\left(-\frac{\mathrm{i}{k}_{0}n{r}^2}{2R\left(z\right)}\right),\hfill \end{array}\end{array}$$

(2)

where r is the radial cylindrical coordinate, $\phi$ is the azimuthally angle, z is the propagation distance, ${\mathrm{L}}_p^{m_0}$ represents the associated Laguerre polynomial, p is the radial orders and $m_0$ is the azimuthal orders, $m_0$ also notes the OAM topological charge of initial OAM modes, n is the refractive index of seawater and $\alpha \left(\lambda \right)$ represents the spectral absorption coefficient of seawater [20,22]; $w\left(z\right)=\frac{w_0}{2\sqrt{2p+\left(\left|m_0\right|+1\right)}}\sqrt{1+{\left(z/z_R\right)}^2}$ is the spot size of the PLG beam, $w_0$ is the waist of the Gaussian beam, $R\left(z\right)=z+z_{oR}^2/z$ is the radius of wave front curvature and $z_R=\frac{1}{2}{k}_0\sqrt{n^2+{\left(\frac{\lambda \alpha \left(\lambda \right)}{4\pi }\right)}^2}{w}_{m_{0}p}^2$ is the Raleigh range of PLG beam in absorbing seawater and ${\delta }_{\mathrm{G}}$ is the Gouy phase.

In Rytov approximation, we have the normalized complex amplitude of PLG vortex in turbulence absorptive seawater

$$\begin{array}{c}\hfill \begin{array}{c}\hfill E_m(r,\phi ,z)=E_{m_0}(r,\phi ,z)exp\left[\psi (r,\phi ,z)\right],\end{array}\end{array}$$

(3)

where $\psi (r,\phi ,z)$ is the random complex phase factor caused by seawater turbulence.

2.2. Information Capacity of the OAM Channel without Pointing Errors

As the PLG beam propagates through the turbulent ocean, the beam is disturbed by refractive index fluctuations that alter the OAM mode of this beam. The fluctuation of the spectral absorption coefficient perturbs the complex amplitude of the wave so that it is no longer guaranteed to be in the original eigenstate of the OAM. The resulting wave is now written as a superposition of eigenstates [11]

$$\begin{array}{c}\hfill \begin{array}{c}\hfill E_m(r,\phi ,z)=\frac{1}{\sqrt{2\pi }}\sum _m{b}_p^m(r,z)exp\left(\mathrm{i}m\phi \right)\end{array}\end{array}$$

(4)

where m is the topological charge of the signal ($m=m_0$) and crosstalk ($m\ne m_0)$ OAM modes, and $b_p^m(r,z)$ is the Fourier expansion coefficient, which is given by

$$\begin{array}{c}\hfill b_p^m\left(z\right)=\frac{1}{\sqrt{2\pi }}\sum _{m-\infty }^{\infty }{\int }_0^{2\pi }{E}_{m_0}(r,\phi ,z)exp\left[\psi (r,\phi ,z)\right]exp(-\mathrm{i}m\phi )\mathrm{d}\phi .\end{array}$$

(5)

Then, the probability distribution of the new OAM $l_z=m\hslash$ is obtained by summing the probabilities associated with that eigenvalue [20],

$$\begin{array}{c}\hfill p_{m/m_0}=\sum _p〈{\left|b_p^m(r,z)\right|}^2〉,\end{array}$$

(6)

where $〈·〉$ is the average of the seawater turbulence ensemble.

By Equations (5) and (6) and ref. [31], Equation (5) is expressed as [20]

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p_{m/m_0}& =\frac{1}{2\pi }\int \int E_{m_0}^{*}\left(r,{\phi }^{\prime },z\right)E_{m_0}(r,\phi ,z)\hfill \\ & \times exp\left\{-\left[1-cos\left(\phi -{\phi }^{\prime }\right)\right]\frac{2r^2}{{\rho }_0^2}-\mathrm{i}m\left(\phi -{\phi }^{\prime }\right)\right\}\mathrm{d}{\phi }^{\prime }\mathrm{d}\phi ,\hfill \end{array}\end{array}$$

(7)

where

$${\rho }_o={\left\{\frac{1}{3}{\pi }^2{k}_0^2\left[n^2+{\left(\frac{\lambda \alpha \left(\lambda \right)}{4\pi }\right)}^2\right]z{\int }_0^{\infty }{\kappa }^3{\mathsf{\Phi }}_n\left(\kappa \right)\mathrm{d}\kappa \right\}}^{-1/2},$$

(8)

In Equation (8), ${\mathsf{\Phi }}_n\left(\kappa \right)$ is the power spectrum of the fluctuations of the refractive index of seawater in stable stratification and anisotropic turbulence that is given by [4,11]

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_n(\kappa ,\zeta )& =\frac{0.033C_{ϵ{\chi }_T}^2{\zeta }^2{\kappa }^{-11/3}\left[1+4.6{\left(\kappa \eta \right)}^{2/3}\right]}{{(1-\varpi )}^2{\left({\kappa }^2+{\kappa }_0^2\right)}^{11/6}}\hfill \\ \hfill & \times \left[{\varpi }^{2}exp\left(-{\kappa }^2/{\kappa }_T^2\right)\right.\left.+exp\left(-{\kappa }^2/{\kappa }_S^2\right)-2\varpi exp\left(-{\kappa }^2/{\kappa }_{TS}^2\right)\right],0<\kappa <\infty ,\hfill \end{array}$$

(9)

where $\kappa$ represents the scalar spatial wave number of refractive index of seawater turbulence, $\zeta$ is the turbulence anisotropy index, $C_{\epsilon {\chi }_T}^2=0.809\times {10}^{-7}{\epsilon }^{-1/3}{\chi }_T{\overline{\omega }}^{-2}{(1-\overline{\omega })}^2\left({\mathrm{K}}^2/{\mathrm{m}}^{2/3}\right)$ is the structural constant of oceanic turbulence, $ϵ$ is the rate of dissipation of kinetic energy per unit mass of fluid, $ϵ$ is in the range $\left[{10}^{-10},{10}^{-1}\right]{\mathrm{m}}^2/{\mathrm{s}}^3$, ${\chi }_T$ is the dissipation rate of the mean-squared temperature ranging from ${10}^{-10}{\mathrm{K}}^2/\mathrm{s}$ to ${10}^{-2}{\mathrm{K}}^2/\mathrm{s}$ and $\overline{\omega }$ represents the ratio of temperature and salinity contributions to the refractive index fluctuations and varies in the interval $(-5,0)$ in seawater. ${\kappa }_0=2\pi /L_0$, ${\kappa }_T=R_T/\eta$, ${\kappa }_S=R_S/\eta$, ${\kappa }_{TS}=R_{TS}/\eta$, $\eta$ is turbulent inner scale, $L_0$ is the outer scale of turbulence, $R_j=\sqrt{3}{\left[W_j-1/3+1/\left(9W_j\right)\right]}^{3/2}/Q^{-3/2}(j=T,S,TS)$, $W_j={\left\{{\left[\frac{{Pr}_j^2{Q}^4}{{\left(6\beta \right)}^2}-\frac{{Pr}_j{Q}^2}{81\beta }\right]}^{1/2}-\left[\frac{1}{27}-\frac{{Pr}_j{Q}^2}{6\beta }\right]\right\}}^{1/3}$ and Q is the nondimensional constant; ${Pr}_T$ and ${Pr}_S$, respectively, represent the Prandtl number of the temperature and salinity, ${Pr}_{TS}=2{Pr}_T{Pr}_S/({Pr}_T+{Pr}_S)$.

The transverse spatial coherence radius of the spherical wave represented by Equations (8) and (9) is given by the following formula [20]

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\rho }_o& =\left\{\frac{2.144[n^2+{\left(\frac{\lambda \alpha \left(\lambda \right)}{4\pi }\right)}^2]C_{\epsilon {\chi }_T}^{2}z}{{\lambda }^2{\zeta }^2{(1-\varpi )}^2}{\kappa }_0^{1/3}\left[{\varpi }^2\mathrm{U}\left(2;\frac{7}{6};\frac{{\kappa }_0^2}{{\kappa }_T^2}\right)+\mathrm{U}\left(2;\frac{7}{6};\frac{{\kappa }_0^2}{{\kappa }_S^2}\right)\right.\right.\hfill \\ & \left.-2\varpi \mathrm{U}\left(2;\frac{7}{6};\frac{{\kappa }_0^2}{{\kappa }_{TS}^2}\right)\right]+9.862l_0^{2/3}{\kappa }_0\Gamma \left(\frac{7}{3}\right)\frac{\left[n^2+{\left(\frac{\lambda \alpha \left(\lambda \right)}{4\pi }\right)}^2\right]C_{\epsilon {\chi }_T}^{2}z}{{\lambda }^2{(1-\varpi )}^2}\hfill \\ & {\left.\left.\times \left[{\varpi }^2\mathrm{U}\left(\frac{7}{3};\frac{3}{2};\frac{{\kappa }_0^2}{{\kappa }_T^2}\right)+\mathrm{U}\left(\frac{7}{3};\frac{3}{2};\frac{{\kappa }_0^2}{{\kappa }_S^2}\right)-2\varpi \mathrm{U}\left(\frac{7}{3};\frac{3}{2};\frac{{\kappa }_0^2}{{\kappa }_{TS}^2}\right)\right]\right\}\right\}}^{-1/2},\hfill \end{array}\end{array}$$

(10)

where $\Gamma (·)$ is gamma function, and $\mathrm{U}(·)$ is the confluent hypergeometric function of the second kind.

Substitute Equations (1) and (2) into Equation (7), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p_{m/m_0}& =\frac{4p!exp[-\alpha (\lambda \left)z\right]}{{\left(2\pi \right)}^{2}w{\left(z\right)}^2\left(p+|m_0|\right)!}{\left(\frac{\sqrt{2}r}{w\left(z\right)}\right)}^{\left|2m_0\right|}exp\left[-2\left(\frac{1}{w^2\left(z\right)}-\frac{k_{0}n}{2R\left(z\right)}+\frac{1}{{\rho }_0^2}\right)r^2\right]\hfill \\ & \times {\left|{\mathrm{L}}_p^{m_0}\left(\frac{2r^2}{w^2\left(z\right)}\right)\right|}^2\left\{{\int }_0^{2\pi }{\int }_0^{2\pi }exp\left[2cos\left(\phi -{\phi }^{\prime }\right)\frac{r^2}{{\rho }_0^2}-\mathrm{i}\left(m-m_0\right)\left(\phi -{\phi }^{\prime }\right)\right]\mathrm{d}{\phi }^{\prime }\mathrm{d}\phi \right\}.\hfill \end{array}\end{array}$$

(11)

By the integral relationship of Equation (BI 277) in ref. [32], Equation (11) can be represented as

$$\begin{array}{c}\hfill p_{m/m_0}=\frac{4\pi p!exp[-\alpha (\lambda \left)z\right]}{w^2\left(z\right)\left(p+\left|m_0\right|\right)!}{\left(\frac{2r^2}{w^2\left(z\right)}\right)}^{\mid m_{0\mid }}exp\left(-\frac{r^2}{w_{eff}^2}\right){\mathrm{I}}_{m-m_0}\left(\frac{2r^2}{{\rho }_o^2}\right){\left|{\mathrm{L}}_p^{m_0}\left(\frac{2r^2}{w^2\left(z\right)}\right)\right|}^2,\end{array}$$

(12)

where $w_{eff}={\left(\frac{4\sqrt{2p+\left(\left|m_0\right|+1\right)}}{{\left(w_0\sqrt{1+{\left(z/z_R\right)}^2}\right)}^2}+\frac{2}{{\rho }_o^2}-\frac{\alpha \left(\lambda \right)z}{2R\left(z\right)}\right)}^{-1/2}$, where ${\mathrm{I}}_m(·)$ is the Bessel function of second kind with m order.

The probability distribution for $P_{m/m_0}$ of the OAM mode can be expressed as

$$\begin{array}{c}\hfill p\left(P_{m/m_0}\right)=\frac{P_{m/m_0}}{{\sum }_{q=-\infty }^{\infty }{P}_{q/q_0}},\end{array}$$

(13)

and

$$\begin{array}{c}\hfill P_{m/m_0}=2\pi {\int }_0^{D/2}{p}_{m/m_0}r\mathrm{d}r,\end{array}$$

(14)

where D is diametet of receiver.

2.3. Information Capacity and OAM Signal Detection Probability for Link with Pointing Errors

In order to obtain a theoretical model of the information capacity of underwater optical communication links, we first derive the theoretical relation of bit error rate of communication link with pointing error. When a Gaussian beam of beam width $w_0$ propagates from the transmitter to the detector plane z, the fraction of power collected by the receiver can be approximated as [33]

$$\begin{array}{c}\hfill f\left({\sigma }_s\right)\approx {|erf\left(v\right)|}^{2}exp\left(-\frac{2{\sigma }_s^2}{w_{eq}^2\left(z\right)}\right),w_{eff}\left(z\right)>3\frac{w_0}{\sqrt{2p+\left(\left|m_0\right|+1\right)}},\end{array}$$

(15)

where $w_{eff}\left(z\right)$ is the beam width of Gaussian beam at detector plane z, ${\sigma }_s$ is the radial displacement between the centers of the beam and detector, $v=\sqrt{\frac{\pi w_0^2}{4w_{eff\left(z\right)}^2\left(z\right)\left(2p+\left(\mid m_0+1\right)\right)}}$ is the ratio between the aperture radius $w_0$ and the beam width $w_{eff}\left(z\right)$, $erf\left(x\right)=\frac{2}{\sqrt{\pi }}{\int }_0^{x}exp\left(-t^2\right)\mathrm{d}t$ is the error function and $w_{eff}\left(z\right)$ is the equivalent beam width defined by $w_{eq}^2\left(z\right)=w_{eff}^2\left(z\right)\frac{\sqrt{{\pi |erf\left(v\right)|}^2}}{2vexp\left(-v^2\right)}$.

The probability distribution for $f\left[f\left({\sigma }_s\right)\right]$ can be expressed as [33]

$$\begin{array}{c}\hfill f\left[f\left({\sigma }_s\right)\right]=\frac{{\gamma }^2}{{|erf\left(v\right)|}^{2{\gamma }^2}}f{\left({\sigma }_s\right)}^{{\gamma }^2-1}.\end{array}$$

(16)

In Equation (16), $\gamma =w_{eq}\left(z\right)/2{\sigma }_s$ and ${\sigma }_s$ is the standard deviation of transverse displacement of link pointing (or pointing error) at the receiver.

The intensity of $P_{m,m_0,p}=P_{m/m_0}f\left({\sigma }_s\right)$ can be expressed as

$$\begin{array}{c}\hfill p_{m,m_0,p}={\int }_{P_{m/m_0}f\left({\sigma }_s\right)/{|erf\left(v\right)|}^2}^{\infty }f\left(P/P_{m/m_0}\right)p\left(P_{m/m_0}\right)\mathrm{d}{P}_{m/m_0},\end{array}$$

(17)

where

$$\begin{array}{c}\hfill f\left(P/P_{m/m_0}\right)=\frac{{\gamma }^2}{P_{m/m_0}{|erf\left(v\right)|}^{2{\gamma }^2}}{\left(\frac{P_{m,m_0,p}}{P_{m/m_0}}\right)}^{{\gamma }^2-1}.\end{array}$$

(18)

Substitute Equation (18) into Equation (17), and rewrite Equation (18) as

$$\begin{array}{cc}\hfill p_{m,m_0,p}& =\frac{{\gamma }^2}{{|erf\left(v\right)|}^{2{\gamma }^2}{\sum }_{q=-\infty }^{\infty }{P}_{q/q_0}}{\int }_{P_{m/m_0}f\left({\sigma }_s\right)/|erf{\left(v\right)}^2|}^{\infty }{P}_{m/m_0}{}^{1-{\gamma }^2}\mathrm{d}{P}_{m/m_0}\hfill \\ \hfill & =\frac{{\gamma }^{2}f\left({\sigma }_s\right)}{\left({\gamma }^2-2\right){|erf\left(v\right)|}^4}p\left(P_{m/m_0}\right).\hfill \end{array}$$

(19)

For $m\ne m_0$, $p_{m{m}_0,p}=p_{m\ne m_0,p}$ is the crosstalk probability of the OAM mode, which describes the probability of the photon migrating from the OAM signal mode to the new OAM mode, and $p_{m{m}_0,p}=p_{m=m_0,p}$ is the signal probability of the OAM mode.

When the transmit power is large sufficient, the signal-to- noise- crosstalk ratio (SNCR) of the OAM channel is approximately [34]

$$SNC{R}_{m,m_0}=\frac{p_{m=m_0,p}}{{\sum }_{m=-\infty }^{\infty }{p}_{m\ne m_0,p}+\frac{N_0}{P_{TX}}},$$

(20)

where ${\sum }_{m=-\infty }^{\infty }{p}_{m\ne m_0,p}$ is OAM crosstalk noise. $N_0/P_{TX}$ represents the ratio of channel noise and transmit power.

For signal modulation of M-QAM, based on SNCR, the symbol error probabilities of the M-QAM OAM channel are, respectively, expressed as [1,34]

$$\begin{array}{cc}\hfill BE{R}_{M-QAM}& =\frac{2}{{log}_{2}M}\left(1-\frac{1}{\sqrt{M}}\right)erfc\sqrt{\frac{3{log}_{2}M}{2(M-1)}SNC{R}_{m,m_0}}\hfill \\ \hfill & \times \left[1-\frac{1}{2}\left(1-\frac{1}{\sqrt{M}}\right)effc\sqrt{\frac{3{log}_{2}M}{2(M-1)}SNC{R}_{m,m_0}}\right].\hfill \end{array}$$

(21)

The information capacity of the OAM channel with signal modulation of M-QAM can be described by the following formula

$$\begin{array}{cc}\hfill C_{M-QAM}& ={log}_{2}N+\left(1-BE{R}_{M-QAM}\right){log}_2\left(1-BE{R}_{M-QAM}\right)\hfill \\ \hfill & +BE{R}_{M-QAM}{log}_2\frac{BE{R}_{M-QAM}}{N-1},\hfill \end{array}$$

(22)

where $N=-m,-m+1,\cdots ,0,\cdots m-1,m$.

3. Numeric Analysis

In this section, we analyze the information capacity of the UWOC link with PLG beam in turbulence and absorbable seawater through numerical simulation. Among other things, in the numerical analysis below, we select the parameters listed in

Table 1. Major system parameters.

Symbol	Parameters	Value
${Pr}_T$	Prandtl number of the temperature	0.72
${Pr}_S$	Prandtl number of the salinity	700
Q	Nondimensional constant	2.5
$\epsilon$	Rate of dissipation of kinetic energy per unit mass of fluid	${10}^{-3}{\mathrm{m}}^2/{\mathrm{s}}^3$
${\chi }_T$	Dissipation rate of the mean-squared temperature	${10}^{-7}{\mathrm{K}}^2/\mathrm{s}$
$\overline{\omega }$	Ratio of temperature and salinity	−4.5
$k_0$	Wave number	$2\pi /\lambda$
$\eta$	Inner scale	0.001 m
$L_0$	Outer scale	10 m
$m_0$	OAM topological charge	1
p	Radial order	0
D	Receiver diameter	0.05 m
$w_0$	Half-beam width	0.012 m
$\lambda$	Wavelength	410 nm
z	Propagation distance	100 m
M	Modulation order	256
${\sigma }_s$	Pointing errors	0.03 m
$\zeta$	Anisotropy index	2

from the available experiments [12,20,35].

First of all, to compare whether the scheme using PLG vortex as carrier is more conducive to reducing the link information capacity loss caused by absorbing turbulent seawater than the scheme using POV as carrier in UWOC, the numerical curves of the channel capacity of the PLG carrier and POV carrier links with the evolution of turbulence strength and transmission distance are presented in Figure 2 and Figure 3, respectively.

The relation diagram with the dissipation rate of the mean-squared temperature ${\chi }_T$ and the dissipation rate of kinetic energy per unit mass $ϵ$ of fluid as independent variables and information capacity as function is drawn in Figure 2. Figure 2 reveals that the information capacity increases with the increase in $ϵ$ or the decrease in ${\chi }_T$, which can be explained since the larger $ϵ$ or the smaller ${\chi }_T$ mean a weaker seawater turbulence, and PLG forms a smaller wavefront distortion. Comparing Figure 2a,b, we can see that the loss of the channel capacity of the PLG carrier link is less than that of the PLG carrier link under the same turbulence conditions and the same other link parameters (see

Table 2. Performance comparison between PLG system and POV system.

Parameters	PLG Vortex Information Capacity (B/s)	POV Vortex Information Capacity (B/s)
$\epsilon ={10}^{-1}{\mathrm{m}}^2/{\mathrm{s}}^3,{\chi }_T={10}^{-9}{\mathrm{K}}^2/\mathrm{s}$	3.907	$3.907$
$\epsilon ={10}^{-1}{\mathrm{m}}^2/{\mathrm{s}}^3,{\chi }_T={10}^{-8}{\mathrm{K}}^2/\mathrm{s}$	3.854	$3.809$
$\epsilon ={10}^{-1}{\mathrm{m}}^2/{\mathrm{s}}^3,{\chi }_T={10}^{-7}{\mathrm{K}}^2/\mathrm{s}$	3.253	$3.200$
$\epsilon ={10}^{-3}{\mathrm{m}}^2/{\mathrm{s}}^3,{\chi }_T={10}^{-9}{\mathrm{K}}^2/\mathrm{s}$	3.899	$3.881$
$\epsilon ={10}^{-3}{\mathrm{m}}^2/{\mathrm{s}}^3,{\chi }_T={10}^{-8}{\mathrm{K}}^2/\mathrm{s}$	3.479	$3.403$
$\epsilon ={10}^{-3}{\mathrm{m}}^2/{\mathrm{s}}^3,{\chi }_T={10}^{-7}{\mathrm{K}}^2/\mathrm{s}$	3.006	$2.98$
$z=50\mathrm{m},\alpha =0.0027$	3.386	$3.078$
$z=50\mathrm{m},\alpha =0.0440$	3.192	$3.009$

).

Figure 3 shows the influence of the signal wavelength, propagation distance and seawater absorption on the propagation performance of the OAM mode. In the case of small seawater absorption and short message channel, the information capacity loss of the link with long signal wavelength is smaller. This result comes from the fact that the longer the signal wavelength is, the smaller the signal scintillation generated by turbulence is [30]. However, in the case of a long channel and strong seawater absorption, the influence of signal wavelength on the information capacity can be ignored. Additionally, the greater the seawater absorption, the more serious the loss of link channel capacity. Comparing Figure 3a,b, we can see that similar to Figure 2, the loss of channel capacity of the PLG carrier link is less than that of the PLG carrier link at the same transmission distance and the same other link parameters. The specific comparison values are shown in Table 2.

Figure 4 shows the relationship between the information capacity and transverse pointing errors for different signal wavelength. The results of Figure 4 indicate that with the increase in pointing errors ${\sigma }_s$, the information capacity of links decreases. The reason is that when the pointing errors increases, part of the signal deviates from the receiver and cannot be received, which leads to a decrease in information capacity. In addition, as the pointing error increases, the average capacity of long-wavelength signals decays faster than that of short-wavelength signals, indicating that long-wavelength signals are more susceptible to pointing error. This is because seawater absorbs the signal corresponding to long wavelengths strongly, making the signal reaching the receiver weaker, resulting in long-wavelength signals being more susceptible to pointing errors.

Figure 5 indicates the information capacity of OAM link as a function of the inner and outer scales of seawater turbulence. Figure 5 shows that the information capacity increases as the inner scale of the turbulence increases and decreases as the outer scale of the turbulence decreases. This is due to the fact that as the inner scale of the turbulence increases, the uniform area in the seawater increases, and the wavefront distortion of the OAM signal mode through the channel decreases, that is, the transmission of the OAM signal mode increases. On the other hand, the transmission channel with large outer turbulence scale will cause large random deflections of the light rays, which results in large random optical path differences between the sub-beams of the OAM mode and, eventually, will lead to large wavefront distortion of the OAM mode and produce large OAM crosstalk.

Figure 6 indicates the relationship between the information capacity and pointing errors for the different receiver diameter D. Figure 6 shows that as the pointing errors is small, the link with smaller receiver aperture has a larger information capacity. The reasons for the results are: (1) The average radius of the OAM modes of each order that make up the PLG beam increases with the increase in the absolute value of the topological charge of the OAM mode. (2) The topological load of the signal OAM mode in Figure 6 is $\left|1\right|$, and the topological charge of the crosstalk mode is $>\left|1\right|$. (3) The average channel capacity is calculated as the information capacity per unit area in the receiving plane. However, as the pointing error is large, the link with larger receiver aperture has a larger average capacity. Furthermore, when the pointing error is further increased to a sufficient size, the link channel capacity decreases slowly and tends to be independent of the receive aperture size. The physical basis of this result is as follows: when the receiving aperture of the communication system is larger, the background noise outside the signal it collects is larger.

In Figure 7, we show the information capacity of the OAM link as the function of half-beam width for different pointing errors under the condition that the receiver aperture is given. It can be seen from Figure 7 that there exists optimal ${\tilde{w}}_{\mathrm{opt}}\left(z\right)$ with maximum information capacity as the function of the channel length. In addition, the information capacity and maximum information capacity decrease with the increase in ${\sigma }_s$. Additionally, the optimal ${\tilde{w}}_{\mathrm{opt}}\left(z\right)$ increases as with an increase in ${\sigma }_s$.

In Figure 8, the relationship between the ratio of temperature and salinity $\varpi$ and the information capacity of the link under different turbulence anisotropy index is studied. From Figure 8, it can be clearly found that, similar to communication links with other light beams as carriers [10,12,13,14], with the increase in $\varpi$, the information capacity decreases because of the increase in $\varpi$. In addition, with the increase in turbulence anisotropy index, the information capacity increases. This shows the larger anisotropic index $\zeta$ is conducive to the transiting of signal vortex modes.

Figure 9 shows the relationship between information capacity and topological charge number $m_0$ with the different radial order of PLGs p. As can be seen from Figure 9, with the increase in the radial order of PLGs p, the information capacity decreases. This is because the increase in p leads to more rings at the receiver, which disperses the signal strength and leads to the decrease in the received information. In addition, with the increase in $m_0$, the information capacity also decreases. This is because when the OAM topological charge $m_0$ increases, the dark spots of light intensity in the receiving plane are large, leading to the decrease in the amount of information detected by the receiver at the receiver.

Figure 10 shows the influence of the ratio of noise to system power $P_{TX}/N_0$ on the information capacity of the link. In Figure 10, with the increase in the ratio of noise to system power $P_{TX}/N_0$, the information capacity increases. Because $P_{TX}/N_0$ increases, which means $N_0$ decreases, or $P_{TX}$ increases, when $N_0$ decreases, and system noise decreases, it leads to a decrease in interference with the signal. Similarly, when $P_{TX}$ increases, the power of the system increases, and the signal is not susceptible to noise. In addition, with the increase kin channel number N, the average capacity also increases.

4. Conclusions

This paper focuses on the influence of the anisotropy of ocean turbulence, and the pointing error of the system on the information capacity of an optical communication link with PLG vortex beam as the carrier in the channel of turbulence absorbing seawater does not take into account the scattering of seawater under 256-QAM modulation. The information capacity of the optics communication link is derived based on the Rytov approximation for weak turbulence. By comparing the loss of channel capacity of PLG and POV carrier links under the same conditions, which carrier is more suitable to be used as the optical communication carrier in the absorbing turbulent sea channel is studied. We came to the important conclusion that the link composed of the PLG carrier is better than that composed of the POV carrier. In low-absorption turbulent seawater with a short communication link, the transmission signal with longer wavelength is beneficial to reduce the loss of information capacity of communication links. The information capacity decreases with the increase in transmission distance, pointing errors, dissipation rate of the mean-squared temperature, topological charges $m_0$ of OAM mode, PLG’s radial order and the outer scale of seawater turbulence, The information capacity of the OAM channel increases with the increase in the rate of dissipation of kinetic energy per unit mass of fluid, channel numbers, anisotropic index of turbulence, SNR of transceiver system and inner scale of seawater turbulence. Moreover, the long wavelength (or high absorption coefficient) signal and large receiver aperture are more susceptible to the pointing errors.