Capacity of the Weakly Absorbent Turbulent Ocean Channel with the Coaxial Double-Position Power Gaussian Vortex

: Turbulence and absorption of seawater are two important factors affecting the signal transmission quality of underwater optical communication link. Here, we study the channel capacity and bit error rate of an underwater extinction communication link with a coaxial double-position power Gaussian vortex carrier based on Rytov approximation theory. The study ﬁnds that channel capacity and bit error rate are the nonlinear functions of the dimensionless structural parameter and reach maximum and minimum values at | α | = 1, respectively. The seawater absorption has a great inﬂuence on the channel capacity but not bit error rate. The communication link with large receiving aperture, small transmitting beam diameter, long wavelength of light source in a window, and more OAM channels has high channel capacity. number of OAM channels to higher channel capacity. The channel capacity decreases with the increase in initial beam width and the decrease in D. Our results also show that with the increase in transmission distance z , the topological charge of vortex, and ε , the BER of this link increases; however, the BET of the link increases with the increase in χ T and is minimum at α = 1 or − 1. Our paper also points out that we need to further study the inﬂuence of the helical direction of non-coaxial multi-helical combination vortices on the channel capacity of the communication link.


Introduction
Orbital angular momentum (OAM) is a new dimension that can be encoded, which can be adopted by optical [1][2][3][4] and acoustic [5][6][7] communication systems to improve the information capacity of links. Although acoustic OAM has the advantages of low loss of underwater communication and perfect theoretical [5][6][7] and experimental model compared with underwater optical OAM transmission, its fatal weakness is the low channel capacity and signal transmission rate of underwater acoustic communication link. Therefore, as underwater communication and detection technology requires higher communication link channel capacity and signal transmission rate, OAM optical communication has become one of the new focuses in the field of underwater communication.
However, due to the presence of various constituents in the ocean and oceanic turbulence, seawater scattering and absorption inevitably limit the transmission of information carried by OAM modes over optical communication links [8][9][10][11][12][13][14][15][16][17][18]. Therefore, in order to reduce the above negative effects of seawater and consider the relationship between signal transmission probability and channel capacity, a lot of studies have been conducted from the perspectives of OAM detection probability and link channel capacity. The disturbance degree of ocean turbulence to the detection probability of OAM mode of partially coherent Laguerre-Gaussian beams increases with the increase in beam radial mode order, OAM mode topological charge and ocean turbulence strength, and fully coherent vortex beams provided better performance than partially coherent ones [8]. The partially coherent elegant Laguerre-Gaussian beams are also more affected by turbulence as compared to the fully coherent elegant Laguerre-Gaussian beams, and one can choose optimum beam source to mitigate the influence of oceanic turbulence [9]. Although the detection probability of fractional OAM mode is lower than that of adjacent integer OAM mode, the channel capacity of the link can be increased by using fractional OAM mode when the available OAM topological charges are limited [10]. Compared with Laguerre-Gaussian beams, the Bessel Gaussian beam transmitted in turbulent seawater has stronger anti-turbulence and better OAM transmission performance [11]. For partially coherent modified Bessel the effects of the parameters of turbulent absorption seawater and beam structure on the channel capacity as well as BET. Section 5 provides the main conclusions of the paper.

Kolmogorov Ocean Spectrum and Random Coaxial Double-Position Power Gaussian Vortex
This section mainly discusses two parts of your content. First, we briefly discuss the seawater spectrum. Secondly, the random coaxial double-position power Gaussian vortex is established by Rytov theory.

Kolmogorov Ocean Spectrum
For a liner, isotropic, homogeneous, and clean absorbing seawater, the refractive index of seawater is [19][20][21] n oc = n r + in i (1) where n r is the mean value of real part of refractive index of seawater, n i is the mean value of imaginary part of refractive index of seawater that depends on the type of seawater, and i = √ −1 is the imaginary symbol. The turbulent characteristics of seawater are described by the following turbulent spectrum equation [24] Φ n (κ) = 0.267 where κ is the spatial frequency of turbulent fluctuations, ε is the rate of dissipation of kinetic energy per unit mass of fluid ranging from 10 −10 m 2 /s 3 to 10 −1 m 2 /s 3 , χ T is the dissipation rate of the mean-squared temperature and has the range from 10 −10 K 2 /s to 10 −2 K 2 /s, l 0 is the inner scale, κ 0 = 2π/L 0 , L 0 is the outer scale of turbulence, κ T = R T /l, κ S = R S /l 0 , and κ TS = R TS /l 0 . R j = , Q = 2.35, Pr T and Pr S , respectively, represent the Prandtl number of the temperature and salinity, Pr TS = 2Pr T Pr S /(Pr T + Pr S ), defines the ratio of temperature and salinity contributions to the refractive index spectrum, which can vary in the interval [−5,0] in the seawaters, = −5 temperature is dominant in the induction of optical turbulence, while = 0 salinity is dominant in the induction of optical turbulence.
Consider that the physical mechanism behind the oceanic turbulence and atmospheric turbulence are same and in conjunction with the above discussion, using Equation (5), we can just write Rytov variance for a plane wave in a homogeneous isotropic weakly turbulent ocean as Formally, Equation (6) is the same as the Rytov variance of plane waves in the turbulent atmosphere, except that C 2 εχ T is replaced by the structure constant of atmospheric turbulence C 2 n . Of course, from the turbulence effect theory of light transmission in the atmosphere [25], we can directly obtain the conditions that seawater is in a weak fluctuation state, that is, σ 2 1εχ T < 1.

Random Coaxial Double-Position Power Gaussian Vortex
In Figure 1, we give a schematic diagram of the rectangular coordinates (x , y , z ) and the communication link for signal transmission from the light source plane (x , y , z ) to the receiving plane (x, y, z).
Here, ( ) Consider that the physical mechanism behind the oceanic turbulence and atmospheric turbulence are same and in conjunction with the above discussion, using Equation (5), we can just write Rytov variance for a plane wave in a homogeneous isotropic weakly turbulent ocean as Formally, Equation (6) is the same as the Rytov variance of plane waves in the turbulent atmosphere, except that 2 T C εχ is replaced by the structure constant of atmospheric turbulence 2 n C . Of course, from the turbulence effect theory of light transmission in the atmosphere [25], we can directly obtain the conditions that seawater is in a weak fluctuation state, that is,

Random Coaxial Double-Position Power Gaussian Vortex
In Figure 1, we give a schematic diagram of the rectangular coordinates (x', y', z') and the communication link for signal transmission from the light source plane (x', y', z') to the receiving plane (x, y, z). In rectangular coordinates (x՛, y՛, z՛) and at light source plane, we consider two position power Gaussian vortexes that have the same amplitude at their waist [26] ( ) ( , e x p , , 0 l x y E x y x iy x y w where "+" represents right-spin vortex, "−" represents the left-spin vortex, 0 l is the topological charge of vortex, and 0 w is the waist of these vortexes. Essentially, ( ) 0 l x iy ′ ′ ± is a transmission factor and is a position power vortex. The position power vortex can be produced from "fork" hologram [27].
In the cylindrical coordinate system ( ) , , r z θ ′ ′ ′ these two position power Gaussian vortexes can be rewritten as follows  In rectangular coordinates (x , y , z ) and at light source plane, we consider two position power Gaussian vortexes that have the same amplitude at their waist [26] where "+" represents right-spin vortex, "−" represents the left-spin vortex, l 0 is the topological charge of vortex, and w 0 is the waist of these vortexes. Essentially, (x ± iy ) l 0 is a transmission factor and is a position power vortex. The position power vortex can be produced from "fork" hologram [27].
In the cylindrical coordinate system (r , θ , z ) these two position power Gaussian vortexes can be rewritten as follows where r = x 2 + y 2 , and θ = arctan(y /x ). Now, according to Equation (8), we form a coaxial double-position power Gaussian vortex with a normalized field and opposite helices in the source plane z = 0, as shown in the following equation (see Appendix A) J. Mar. Sci. Eng. 2021, 9, 1117 5 of 16 where is the incomplete gamma function, and α denotes a dimensionless structural parameter. When |α| = 1, the beam degenerates into left spin or right spin single-position power Gaussian vortex, respectively; when |α| = 1, the beam is a coaxial double-position power Gaussian vortex with reverse helix and different amplitude, and when |α| = 0, the beam is a coaxial double-position power Gaussian vortex with inverse spiral and equal amplitude. The field of the coaxial double-position power Gaussian vortex in free space at any receiving lane x = x, y = y, z = z is given by (see Appendix B) where Γ ν+µ+1 2 is gamma function, and 1 F 1 is confluent hypergeometric function. k is the wave number, and λ is the wavelength of light in the vacuum.
In weak turbulence and absorbent seawater and by Rytov approximation [25], the field of the RCDPPGV is expressed as (see Appendix C) where ψ (r, θ, z) represents the complex phase disturbance caused by seawater turbulence.

Bit Error Rate of the OAM Link and Channel Capacity
In this section, the series expansion method of RCDPPGV random spiral mode is used to obtain the mathematical model of bit error rate and channel capacity.

Bit Error Rate of OAM Links
To understand the OAM mode redistribution generated by turbulent seawater disturbance, we expanded the disturbed field into a series of the spiral harmonic exp (ilθ) [18,19] with Note that l = l 0 is the topological charge of an OAM crosstalk mode. The ensemble average of |a l | 2 represents the condition probability distribution of OAM modes in turbulent as well as absorbent seawater and at receiving plane, which has the form of where * denotes the complex conjugate, and ρ oc is the spatial coherence radius of waves propagating in isotropic and turbulent as well as absorbent seawater and is given by [13] Substituting Equation (9) into Equation (12), through integration operations, we obtain the probability distribution of OAM modes (see Appendix D) The receiving probability of OAM mode carried by RCDPPGV and measured by the detector with detection diameter D can be expressed as In addition, for sufficiently large transmit power, the signal-to-noise ratio (SNR) of OAM channel is approximately [28] SNR(l, l 0 ) = P l/l 0 ,l=l 0 For signal modulation of on-off keying (OOK), based on (18), the bit error rate (BER) of OAM channels is derived as [29] BER(l, where erfc(·) denotes the complementary error function.

Channel Capacity
Consider a line-of-sight communication link using OAM eigenstates in the range of topological charge number l 0 = 0, 1, . . . , M − 1 that means M dimension Hilbert space for different signal values. Further, by classical Shannon information theory of a stationary discrete memory-less system [3] and Equation (17), the channel capacity of a communication link with coaxial double-position power Gaussian vortex in weakly absorbent seawater can be given by [3] with the entropy of the signal vortex mode l 0 max and conditional entropy of the l mode for given signal mode l 0 max

Numeric Analysis
To explore the influence of beam structure parameters and seawater parameters on BER and the channel capacity of OAM links in isotropic turbulent as well as absorbent seawater, in this section, we numerically study the BER and channel capacity of the link with a RCDPPGV carrier. In the following analysis, we set the calculation parameters as: 34, and n i = 0.6741 × 10 −10 , unless other variable parameters are specified in calculation. Moreover, in the following numerical analysis, we first analyze the influence of system parameters and turbulent seawater on the BER of OAM channels and, then, analyze the influence of system parameters and turbulent seawater on channel capacity. Figure 2 shows variation curve of the BER of the random double-Gaussian optical vortex carrier communication link with the dissipation rate of the mean-squared temperature χ T and topological charge of the RCDPPGV vortex. The variation curve shows that with the increase in transmission distance z, the BER of this link increases. When the transmission distance is less than 50 m, BER increases rapidly and, then, tends to be flat. Additionally, a larger l 0 means a higher BER. The physical reason for this result is that when the topological charge (orbital angular momentum quantum number) of OAM signal mode is larger, the OAM energy level of the signal is also higher. Therefore, the probability of transition form to another OAM energy level is also higher when it is disturbed.   Under certain topological charge of vortex laser, we use Figure 3 to show the variation curve of BER of random double−Gaussian optical vortex carrier communication link with dimensionless parameter α. From Figure 2, we can see that BER has two minimum values at |α| = 1. This result indicates that the link with a single-position power Gaussian vortex has minimum BER. In Figure 4, we investigate the effects of the ε and 0 l on the BER of RCDPPGV carrier link for given topological charge of vortex. From Figure 3, we can see that the value of the BER decreases with an increase in the ε . The reason for this phenomenon is the increase in ε , the turbulence strength of seawater decreases, and the influence of turbulence on the transmission vortex decreases. In Figure 4, we investigate the effects of the ε and l 0 on the BER of RCDPPGV carrier link for given topological charge of vortex. From Figure 3, we can see that the value of the BER decreases with an increase in the ε. The reason for this phenomenon is the increase in ε, the turbulence strength of seawater decreases, and the influence of turbulence on the transmission vortex decreases.  In Figure 5, we investigate the effects of dissipation rate of the mean-squared temperature T χ on the channel capacity of the RCDPPGV carrier link for the given topological charge of vortex. It is clear from Figure 5 that the BER of the channel capacity increases with increasing T χ . The reason for this result comes from that with the increase in T χ , the turbulence strength of seawater increases, and the influence of turbulence on the transmission vortex increases. In Figure 5, we investigate the effects of dissipation rate of the mean-squared temperature χ T on the channel capacity of the RCDPPGV carrier link for the given topological charge of vortex. It is clear from Figure 5 that the BER of the channel capacity increases with increasing χ T . The reason for this result comes from that with the increase in χ T , the turbulence strength of seawater increases, and the influence of turbulence on the transmission vortex increases.

Effects of Seawater Turbulence and Beam Parameters on the BER
In Figure 5, we investigate the effects of dissipation rate of the mean-squared temperature T χ on the channel capacity of the RCDPPGV carrier link for the given topological charge of vortex. It is clear from Figure 5 that the BER of the channel capacity increases with increasing T χ . The reason for this result comes from that with the increase in T χ , the turbulence strength of seawater increases, and the influence of turbulence on the transmission vortex increases.    Figure 6 illustrates the impact of the dimensionless structural parameters α on the channel capacity of the RCDPPGV carrier with different values of the channel numbers M. Figure 6 shows that the channel capacity in the [−10, 10] interval of α is a nonlinear function of α and reaches two maximum values at |α| = 1. The difference value between the maximum channel capacity of each link increases with the number of OAM channels M. However, with the decrease in |α| value, channel capacity tends to be constant, this constant is channel capacity at |α| = 0. This result indicates that the link with singleposition power Gaussian vortex has maximum channel capacity. Similar to any other type of beam [9][10][11][12][13][14][15][16][17][18][19][20][21][22], Figure 6 shows that the channel capacity also increases with the increase in channel number. constant is channel capacity at α = 0. This result indicates that the link with single-position power Gaussian vortex has maximum channel capacity. Similar to any other type of beam [9][10][11][12][13][14][15][16][17][18][19][20][21][22], Figure 6 shows that the channel capacity also increases with the increase in channel number.  Figure 7 shows that channel capacity decays rapidly as absorption increases. Seawater absorption has a great influence on the channel capacity, so in the detector design of the underwater communication system, it should take into account the inconsistency between the absorption of the offshore area and the open sea area.   Figure 7 shows that channel capacity decays rapidly as absorption increases. Seawater absorption has a great influence on the channel capacity, so in the detector design of the underwater communication system, it should take into account the inconsistency between the absorption of the offshore area and the open sea area.   Figure 8 shows the channel capacity of the RCDPPGV carrier versus the initial beam width w 0 for different values of received diameter D. We can find that the channel capacity of the link with RCDGV carrier decreases with increase in the w 0 . This phenomenon can be explained as follows: when the initial beam size w 0 is larger, the difference of turbulent seawater within the beam diameter is also larger. Of course, the wave front distortion of the transmitting beam is also large, so that the difference between signal probability and crosstalk probability tends to zero [30]. Figure 8 also shows that the channel capacity of the link with RCDPPGV carrier increases with an increase in D. This result can be explained by the following reasons: because the channel of OAM mode with higher topological charge is farther away from the axis of communication link [1,26], the receiver with the larger receiving aperture can receive the information of the OAM mode channel with higher topological charge.
In Figure 9, we investigate the effects of the outer scale of turbulence L 0 and the inner scale of turbulence η on the channel capacity of RCDPPGV carrier. From Figure 5, we can see that the value of the channel capacity increases with an increase in η and a decrease in L 0 . This shows that the channel capacity increases with the increase in the inner scale of turbulence. According to the turbulence theory, with the increase in the inner scale of turbulence, the uniform area in the seawater increases when the beam passes through the channel, that is, the influence of the channel on the transmitted beam decreases. We can also find that the channel capacity decreases with the increase in the outer scale, but the range of change is small. The smaller the outer scale of turbulence is, the less wave front distortion will be generated, so the crosstalk probability will be smaller. The reasons for this phenomenon are as follows: When the turbulent outer scale increases, the deflection of light ray in the beam is growing, and the frequency of light rays going through different paths also increases. As a result, the optical path difference between the light rays increases, and the wave front distortion of the beam increases accordingly. However, because the refractive index difference is small in the outer scale of turbulence, the wave front distortion of the beam increases but not much.
with higher topological charge.
In Figure 8, we investigate the effects of the temperature-salinity contribution ratios and different wavelength λ on the channel capacity of the RCDPPGV carrier in the "seawater light transmission window". Similar to any other type of beam [9][10][11][12][13][14][15][16][17][18][19][20][21][22], Figure 8 shows that the channel capacity of the RCDPPGV carrier increases with increasing λ . This is because the long-wavelength beam has lower scintillations [25]. As ϖ increases, channel capacity decreases. This study is similar to studies for the single vortex information carrier [9][10][11][12][13][14][15][16][17][18][19][20][21][22], and salinity fluctuations have a greater impact on the channel capacity with a larger number of channels. In Figure 9, we investigate the effects of the outer scale of turbulence 0 L and the inner scale of turbulence η on the channel capacity of RCDPPGV carrier. From Figure 5, we can see that the value of the channel capacity increases with an increase in η and a decrease in 0 L . This shows that the channel capacity increases with the increase in the inner scale of turbulence. According to the turbulence theory, with the increase in the inner scale of turbulence, the uniform area in the seawater increases when the beam passes through the channel, that is, the influence of the channel on the transmitted beam decreases. We can also find that the channel capacity decreases with the increase in the outer scale, but the range of change is small. The smaller the outer scale of turbulence is, the less wave front distortion will be generated, so the crosstalk probability will be smaller. The reasons for this phenomenon are as follows: When the turbulent outer scale increases, the deflection of light ray in the beam is growing, and the frequency of light rays going through different paths also increases. As a result, the optical path difference between the light rays increases, and the wave front distortion of the beam increases accordingly. In Figure 10, we investigate the effects of dissipation rate of kinetic energy per unit mass of fluid and dissipation rate of the mean-squared temperature χ t on the channel capacity of the RCDPPGV carrier. It is clear from Figure 6 that the magnitude of the channel capacity decreases with increasing χ t and decreasing ε. The reason for this result comes from that with the increase in χ t and the decrease in ε, the turbulence of seawater increases, and the influence of turbulence on the beam increases [31]. However, because the refractive index difference is small in the outer scale of turbulence, the wave front distortion of the beam increases but not much.
In Figure 10, we investigate the effects of dissipation rate of kinetic energy per unit mass of fluid and dissipation rate of the mean-squared temperature t χ on the channel capacity of the RCDPPGV carrier. It is clear from Figure 6 that the magnitude of the channel capacity decreases with increasing t χ and decreasing ε . The reason for this result comes from that with the increase in t χ and the decrease in ε , the turbulence of seawater increases, and the influence of turbulence on the beam increases [31].

Conclusions
We focused on the propagation properties of OAM modes carried by the RCDPPGV carrier in absorptive, isotropic, and weak turbulent ocean link and derived the channel capacity of this link. The results show that seawater absorption has a great influence on the channel capacity of the optics communication link, and the larger the seawater absorption, the faster the absorption of channel capacity with the transmission distance. With the optics communication link with single-position power Gaussian vortex carrier, the channel capacity of the link reached an extreme value. The channel capacity increases with the increase in the ε and η, and the change in L 0 has little effect on the channel capacity. The channel capacity decreases with increasing and χ t . The large number of OAM channels M leads to higher channel capacity. The channel capacity decreases with the increase in initial beam width and the decrease in D. Our results also show that with the increase in transmission distance z, the topological charge of vortex, and ε, the BER of this link increases; however, the BET of the link increases with the increase in χ T and is minimum at α = 1 or −1. Our paper also points out that we need to further study the influence of the helical direction of non-coaxial multi-helical combination vortices on the channel capacity of the communication link.  ; ν + 1; − b 2 4a is confluent hypergeometric function. k is the wave number, and λ is the wavelength of light in the vacuum.

Appendix C
In weak turbulence and extinction seawater, by using the Rytov approximation [25] and Equation (10) where k = 2π λ (n r + in i ), and ψ(r, θ, z) represents the complex phase disturbance caused by seawater turbulence.