Modeling and performance analysis of oblique underwater optical communication links considering turbulence effects based on seawater depth layering

Xiuyang Ji; Hongxi Yin; Lianyou Jing; Yanjun Liang; Jianying Wang

doi:10.1364/OE.453918

1. Introduction

Underwater wireless optical communication systems have become a research hotspot in recent years owing to their high speed, large bandwidth, and low latency [1,2]. The current commercial equipment is capable of providing underwater wireless optical communication services at rates of several Mbps over 150m [3], while a recent laboratory result reported a 50-m, 5Gbps UWOC system [4]. It is foreseeable that in the near future, UWOC systems will be available to provide real-time image and video transmission services over short and medium distances.

The underwater optical link is not only affected by absorption and scattering of seawater but also by the oceanic turbulence. The oceanic turbulence generates vortices of different scales with disordered and random motion velocities, sizes, and directions. These random vortices originate from random changes in the local temperature and salinity of seawater, causing random fluctuations in the refractive index of ocean water. Therefore, when a light is transmitted in turbulent oceans, its phase, light intensity and transmission direction will change randomly, resulting in undesirable effects such as beam wandering and light intensity scintillation [5], which significantly affect the reliability and effectiveness of underwater wireless optical communication.

Thus far, there have been several works focused on effect of turbulence towards underwater wireless optical channels. Some of these studies concentrate on solving closed-form solutions of the scintillation index by using Rytov theory [6–12]. This type of theoretical analysis usually only yields the variations of the intensity not the phase of the optical signal at the receiver end. The multi-phase screen method is an important measure for simulating the transmission of optical waves in a medium and has been employed to investigate the ocean turbulence optical channel [13–16], which can acquire the intensity as well as the phase of the optical signal reaching the receiver, and has been used in the study of optical orbital angular momentum communication systems [15,17–19].

Practically, variations of temperature and salinity of seawater will affect the intensity of turbulence. However, the research on oceanic optical channels has mostly employed their fixed parameters such as temperature, salinity etc., so that the intensity of turbulence is fixed over the transmission range, which are justified only for exploring horizontal links. The temperature and salinity distributions of seawater vary significantly at different depths. For instance, data from the Global Ocean Observing Buoy, Argo, in the Pacific basin (136.3^° W, 47.7^° N) on 21 February 1999, revealed that the average temperature and salinity of seawater were almost constant in the depth range of 0∼100 m. Nevertheless, in the depth range of 150∼300 m, the average salinity of seawater increased with depth while the temperature increased rapidly before decreasing [20]. However, in different ocean areas, the temperature variations with the depth of seawater approximately follow the same law, i.e., the temperature decreases with the increase in ocean water depth. According to this law, seawater can be divided into a mixed layer, thermocline, and deep layer in terms of ocean water depth [21].

The mixed layer is located at a depth of 0 to approximately 150 m. The temperature and salinity of seawater in the mixed layer hardly change with ocean water depth because the seawater in the mixed layer is strongly mixed by wind, waves, currents, and tides. The thermocline is located at a depth of approximately 200∼800 m, where the temperature and salinity will decrease rapidly with increasing depth. In the deep water layer at depths greater than 1000 m, the ocean water flows slowly and rarely circulate with the surface and upper layers, and thus, the temperature and salinity are relatively stable.

Such distributions of temperature and salinity inevitably lead to oceanic turbulent channels at different depths, demonstrating different optical properties, and hence, the performances of optical communications in the vertical and oblique links differ significantly from that in the horizontal link. Thus far, the research on oceanic optical channels has mostly focused on horizontal channels, and the study on vertical optical channels is relatively rare [22–26]. Although [27,28] consider vertical optical links, they do not provide sufficient analysis of oblique link channels in a more general sense. Since there are some practical applications for requiring to transmit data to the sea surface gateway on-board or communication buoy, from the deep ocean, such as a network of sensors fixed at the seafloor to monitor earthquakes and tsunamis and warn early or military surveillance networks deployed in some critical straits, oceanic wireless optical communication channels are mostly vertical and oblique links. The ocean is constantly in motion and governed by the earth's gravity, and even an already set horizontal optical communication link will be difficult to maintain to be horizontal. Consequently, the impact of depth-based seawater layering on the performance of optical communication systems must be taken into account.

In this study, seawater is first stratified based on the distribution of the average temperature and salinity of seawater with depth. Then, a model of an oceanic oblique optical channel in terms of multi-phase screens is proposed on account of seawater layering. Finally, the BERs (bit error rates) of the optical communication systems under different conditions, such as the depth of the optical transmitter, tilt angle to the vertical direction of the optical link, and transmission distance in different ocean areas, are analyzed. To ensure the accuracy of the data, we have employed the underwater temperature and salinity data provided by Argo and obtained the necessary ocean thermodynamic data through the TEOs-10 toolbox [29,30]. This research is the first to model vertical and oblique underwater optical channels with multi-phase screens, and the data are derived from Argo measurements of real oceans, which have important scientific value and practical significance.

2. Modeling of underwater turbulent oblique optical channel

2.1 Multi-phase screen method to simulate optical signal propagation in seawater oblique channel

The existing multi-phase screen method modelling a horizontal optical link is not appropriate for modelling the vertical and oblique links in the deep ocean for UWOC. If the optical receiver is aligned with the optical transmitter without considering scattering, the propagation of the optical signal from the transmitter to the receiver can be equivalent to the process of the optical signal passing through multiple phase screens, as shown in Fig. 1. Owing to the variation in seawater temperature and salinity with seawater depth, all phase screens are set to be perpendicular to the direction of gravity for the vertical optical link. The optical transmitter is located at a depth h_D below sea level, and it sends an optical signal to the receiver located at a link depth of h. The link distance of the optical signal is L, and L = h/cosβ. The optical signal passes through the N-phase screens and finally reaches the optical receiver. Assuming that the optical transmitter initially sends a Gaussian beam, the optical field at the optical transmitter can be expressed as follows:

(1)$${U_\textrm{0}}({x,y,z = 0} )= {A_0}\exp ( - \frac{{{x^2} + {y^2}}}{{{w^2}}}),$$

where w denotes the radius of the mode field and A₀ is the amplitude of the optical field. The propagation process of the optical signal in the adjacent phase screens is shown in Fig. 2, where k₀ denotes the wave number in free space in the coordinate system of XoZ, $k_0^2 = k_x^2 + k_y^2 + k_z^2$, and ${\tilde{k}_0}$ is the wave number of the optical signal in the propagation direction, corresponding to the coordinate system of $\tilde{X}o\tilde{Z}$. The term ${R_i}$ denotes the link length of the light beam transmitted from the i^th phase screen to the (i + 1)^th phase screen, ${R_i} = {{\Delta {h_i}} / {\cos \beta }}$, and the distance between the adjacent two screens is $\Delta {h_i}$. Assuming that the input optical signal of the i^th phase screen is ${U_{i,in}}({x,y,0} )$, the output optical signal after passing through the i^th phase screen becomes:

(2)$${U_{i,out}}({x,y,0} )= {U_{i,in}}({x,y,0} )\exp ({j{\phi_i}({x,y} )} ),$$

where $\phi {}_i({x,y} )$ denotes the phase function of the i^th phase screen.

Fig. 1. Schematic of multi-phase screen method to simulate the transmission process of optical signals in seawater.

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Fig. 2. Transmission process of optical signal between phase screens

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When the optical signal is transmitted between the i^th and (i + 1)^th phase screens, the position of reaching the (i + 1)^th phase screen is shifted because of the oblique projection of light in the seawater. Therefore, the coordinate system of the (i + 1)^th screens should be transformed, that is, $XoZ:({x,y,z} )\to {X^{\prime}}o{Z^{\prime}}:({{x^{\prime}},{y^{\prime}},{z^{\prime}}} )$, and the transformed coordinate system ${X^{\prime}}o{Z^{\prime}}:({{x^{\prime}},{y^{\prime}},{z^{\prime}}} )$ will be treated as the original coordinate system $XoZ$ at the (i + 1)^th propagation, and then, the relationship between the original and new coordinates can be expressed as follows:

(3)$$\begin{array}{l} x^{\prime} = x + \Delta {h_i}\tan \beta \\ z^{\prime} = z + \Delta {h_i}\textrm{ }\textrm{.}\\ y^{\prime} = y \end{array}$$

Meanwhile, because there is a tilt angle β between the direction of light propagation and direction normal to the phase screen, the spatial wave number of the light signal of the i^th phase screen, ${k_0}$, also needs to be transformed with the spatial wave number in the direction of light signal propagation, ${\tilde{k}_0}$, and the relationship between them can be expressed as follows:

(4)$$\left[ {\begin{array}{c} {{{\tilde{k}}_x}}\\ {{{\tilde{k}}_y}}\\ {{{\tilde{k}}_z}} \end{array}} \right] = \left[ {\begin{array}{ccc} {\cos \beta }&0&{\sin \beta }\\ 0&1&0\\ { - \sin \beta }&0&{\cos \beta } \end{array}} \right]\left[ {\begin{array}{c} {{k_x}}\\ {{k_y}}\\ {{k_z}} \end{array}} \right],{\tilde{k}_x} + {\tilde{k}_y} + {\tilde{k}_z} = {\tilde{k}_0}.$$

The input optical signal of the (i + 1)^th phase screen can be expressed as follows: [31,32]

(5)$${U_{i + 1,in}}({{x^{\prime}},{y^{\prime}}} )= {\mathrm{\mathbb{F}}^{ - 1}}\{{\mathrm{\mathbb{F}}[{{U_{i,out}}({x,y,0} )} ]\exp ({ - c{R_i} + j{{\tilde{k}}_z}{R_i}} )} \}[{{k_x},{k_y}} ],$$

where ${\tilde{k}_z} = {k_z}\cos \beta - {k_x}\sin \beta$, $\mathrm{\mathbb{F}}({\cdot} )$ denotes the Fourier transform for (x, y), and ${\mathrm{\mathbb{F}}}^{ - 1}({\cdot} )$ is the inverse Fourier transform for (k_x, k_y). The term c indicates the attenuation coefficient of seawater [m⁻¹], which is related to both the quality of seawater, wavelength of light λ, and depth z of seawater, which can be expressed as follows [33]

(6)$$c({\lambda ,z} )= {a_w}(\lambda )+ {a_f}\exp ({ - {k_f}\lambda } ){C_f}(z )+ {a_h}\exp ({ - {k_h}\lambda } ){C_h}(z )+ {a_c}({\lambda ,h} ){[{{C_c}(z )} ]^{0.602}},$$

where ${a_w}$ denotes the pure water absorption coefficient [m⁻¹], ${a_f}$ is the specific absorption coefficient of fulvic acid (35.959 m²/mg), ${a_h}$ is the specific absorption coefficient of humic acid (18.828 m²/mg), and ${a_c}$ is the specific absorption coefficient of chlorophyll [34]; ${k_f}$ = 0.0189 nm⁻¹ and ${k_h}$ = 0.01105 nm⁻¹. The term ${C_c}(z )$ indicates the concentration of chlorophyll-a, and

(6a)$${C_h}(z )= 0.19334{C_c}(z )\exp ({0.12343{C_c}(z )} ),$$

(6b)$${C_f}(z )= 1.70498{C_c}(z )\exp ({0.12327{C_c}(z )} ).$$

Eq. (6) shows that the higher the chlorophyll concentration, the higher the attenuation coefficient of seawater. In fact, the chlorophyll concentration reaches a maximum value at a water position in the range of 10∼120 m water depth in the mixed layer below the sea surface. When the chlorophyll concentration of the sea surface is in the range of 0.4∼0.8 mg/m³, the attenuation coefficient of seawater can reach 0.2 m⁻¹. At approximately 200 m deep and deeper depths, the chlorophyll concentration was almost 0 because of the absence of sunlight. Thus, at deeper depths of seawater, the attenuation coefficient can be as low as 0.0092 m⁻¹. To simplify the analysis and highlight the effects of turbulence, the attenuation coefficients of seawater in the thermocline and mixed layer are set as constants in this study.

2.2 Calculating the phase function $\phi {}_i({x,y} )$

The phase function is obtained using the spectral inversion method, and the phase of the i^th phase screen can be expressed as follows: [35]

(7)$${\phi _i}({x,y} )= \sum\nolimits_{{\kappa _x}} {\sum\nolimits_{{\kappa _y}} {\Re ({{\kappa_x},{\kappa_y}} )} } \sqrt {{F_{\phi ,i}}({{\kappa_x},{\kappa_y}} )} \exp [{j({{\kappa_x}x + {\kappa_y}y} )} ]\Delta {\kappa _x}\Delta {\kappa _y},$$

where κ_x and κ_y indicate the spatial wave numbers in the x and y directions, respectively, and $\Re ({{\kappa_x},{\kappa_y}} )$ denotes a Gaussian random complex function with zero mean and variance of 1. Considering the small difference of wave numbers in the horizontal direction, the spatial wave numbers can be regarded as isotropic for the sake of simplification, and then, there is ${F_{\phi ,i}}({{\kappa_x},{\kappa_y}} )\approx {F_{\phi ,i}}(\kappa )$, and ${F_{\phi ,i}}(\kappa )$ can be expressed as follows:

(8)$${F_{\phi ,i}}(\kappa )= 2{\pi ^2}k_0^2\Delta {h_i}{\Phi _n}(\kappa ),$$

where ${\Phi _n}(\kappa )$ denotes the power spectral function of the ocean turbulence channel, which can be expressed as follows: [36]

(9)$${\Phi _n}(\kappa )= {A^2}{\Phi _T}(\kappa )+ {B^2}{\Phi _S}(\kappa )\textrm{ + }2AB{\Phi _{TS}}(\kappa ),$$

where

(10)$$\left\{ \begin{array}{l} A\left( {\left\langle T \right\rangle ,\left\langle S \right\rangle ,\lambda } \right) = {a_2}\left\langle S \right\rangle + 2{a_3}\left\langle T \right\rangle \left\langle S \right\rangle + 2{a_4}\left\langle T \right\rangle + \frac{{{a_6}}}{\lambda }\\ B\left( {\left\langle T \right\rangle ,\left\langle S \right\rangle ,\lambda } \right) = {a_1} + {a_2}\left\langle T \right\rangle + {a_4}{\left\langle T \right\rangle^2} + \frac{{{a_5}}}{\lambda } \end{array} \right..$$

Here, ${a_1} = 1.779 \times {10^{ - 4}}$, ${a_2} ={-} 1.05 \times {10^{ - 6}}$, ${a_3} = 1.6 \times {10^{ - 8}}$, ${a_4} ={-} 2.02 \times {10^{ - 6}}$, ${a_5} = 0.01155$, and ${a_6} ={-} 0.00423$. The terms $\left\langle T \right\rangle$ and $\left\langle S \right\rangle$ denote the local average temperature and salinity of the seawater, respectively. The terms ${\Phi _T}(\kappa )$, ${\Phi _S}(\kappa )$, and ${\Phi _{TS}}(\kappa )$ signify the power spectrum of temperature, salinity, and co-effect of temperature and salinity, and they can be expressed uniformly as follows:

(11)$$\begin{aligned} {\Phi _m}(\kappa) &= \frac{{{\beta _\textrm{0}}{\chi _m}{\varepsilon ^{ - \frac{1}{3}}}}}{{4\pi }}{\kappa ^{ - \frac{{11}}{3}}}[{1 + 21.61{{({\kappa \eta } )}^{0.61}}c_m^{0.02} - 18.18{{({\kappa \eta } )}^{0.55}}c_m^{0.04}} ]\\ &\times \exp [{ - 174.90{{({\kappa \eta } )}^2}c_m^{0.96}} ]\textrm{ , } \end{aligned}$$

where $m \in [{T,S,TS} ]$, ${\beta _\textrm{0}}\textrm{ = 0}\textrm{.72}$, ${\chi _{TS}} = {{A{\chi _T}({1 + {d_r}} )} / {({2\omega B} )}}$, and ${\chi _S} = {{{A^2}{\chi _T}{d_r}} / {({{\omega^2}{B^2}} )}}$. The term η denotes the inner scale and ${\chi _T}$ is the dissipation rate of the mean-squared temperature in units of [K²·s⁻¹], which is relevant to the temperature gradient of seawater. The term ${\chi _T}$ can be expressed as follows: [37]

(12)$${\chi _T} = 6{K_T}\left\langle {{{\left( {\frac{{\partial T(z )}}{{\partial z}}} \right)}^2}} \right\rangle ,$$

where T(z) denotes the temperature as a function of seawater depth, and K_T is the dispersion coefficient of turbulence in the vertical direction, which is related to the flow velocity of seawater in the vertical direction. The term $\varepsilon $ indicates the turbulent kinetic energy dissipation rate in units of [m²·s⁻³]. Depending on the subscripts of $c_m^{}$, $c_m^{}$ can be obtained as follows:

(13)$$\left\{ \begin{array}{l} {c_T} = {0.072^{4/3}}{\beta_\textrm{0}}\textrm{P}{\textrm{r}^{ - 1}}\left( {\left\langle T \right\rangle ,\left\langle S \right\rangle } \right)\\ {c_S} = {0.072^{4/3}}{\beta_\textrm{0}}\textrm{S}{\textrm{c}^{ - 1}}\left( {\left\langle T \right\rangle ,\left\langle S \right\rangle } \right)\\ {c_{TS}} = {0.072^{4/3}}{\beta_\textrm{0}}{\textstyle{{\textrm{Pr}\left( {\left\langle T \right\rangle ,\left\langle S \right\rangle } \right) + {\textrm{S}_\textrm{C}}\left( {\left\langle T \right\rangle ,\left\langle S \right\rangle } \right)} \over {2\textrm{Pr}\left( {\left\langle T \right\rangle ,\left\langle S \right\rangle } \right){\textrm{S}_\textrm{C}}\left( {\left\langle T \right\rangle ,\left\langle S \right\rangle } \right)}}} \end{array} \right.,$$

where S(z) denotes the salinity as a function of the seawater depth. The term $\textrm{Pr}$ indicates the Planck number of temperature and $\textrm{Sc}$ is the Schmidt number of salinity, and their expressions are as follows:

(14)$$\left\{ \begin{array}{l} \textrm{Pr} = {{\mu \left( {\left\langle T \right\rangle ,\left\langle S \right\rangle } \right){c_p}} / {{\sigma_T}}}\\ \textrm{Sc} = \frac{{\mu {{\left( {\left\langle T \right\rangle ,\left\langle S \right\rangle } \right)}^2}}}{{5.954 \times {{10}^{ - 15}}\left( {\left\langle T \right\rangle + 273.15} \right)\rho \left( {\left\langle T \right\rangle ,\left\langle S \right\rangle } \right)}} \end{array} \right.,$$

where ${c_p}$ symbolizes the characteristic temperature in units of [J·kg⁻¹·K⁻¹], ${\sigma _T}$ is the thermal conductivity with the dimension of [W·m⁻¹·K⁻¹], and $\mu \left( {\left\langle T \right\rangle ,\left\langle S \right\rangle } \right)$ denotes the dynamic viscosity in units of [N·s·m⁻²], which can be calculated by using the TEOs-10 toolbox. The term ${d_r}$ indicates the eddy diffusion ratio, and $\omega $ is the correlation strength of the temperature–salinity perturbation. The relationship between $\omega $ and ${d_r}$ is as follows:

(15)$${d_r} = \left\{ \begin{array}{ll} |\omega |+ {|\omega |^{0.5}}{({|\omega |- 1} )^{0.5}}, &|\omega |\ge 1 \\ 1.85|\omega |- 0.85, &0.5 \le |\omega |< 1 \\ 0.15|\omega |, &|\omega |< 0.5 \end{array} \right.,$$

where $\omega $ can be expressed as follows:

(16)$$\omega = \frac{{{\alpha _c}H(z)}}{{{\beta _c}}}.$$

Here, ${\alpha _c}$ denotes the coefficient of the thermal expansion coefficient and ${\beta _c}$ is the saline contraction coefficient, which can also be calculated using the TEOs-10 data toolbox. The term $H(z)$ indicates the ratio of the variation of temperature and salinity with the depth of light transmission in seawater, which can be expressed as follows:

(17)$$H(z )= \frac{{{{dT(z)} / {dz}}}}{{{{dS(z)} / {dz}}}}.$$

2.3 Initial phase screen setup

The positions of the depths of the phase screens are determined according to the variations in temperature and salinity with depths. Here, the maximum distance between the phase screens ${h_C}$ is set, and the interval between the phase screens is adjusted in terms of the degree of variation of the temperature and salinity of the seawater with depth. First, the temperature and salinity are recorded at the location of the transmitter, h₀, and thereafter, the temperature and salinity at a deeper location are determined. When the temperature or salinity of the seawater has an absolute difference of ΔT or ΔS compared to the previous phase screen, the phase screen is set at this depth position. When the depth interval, h_i - h_i₋₁, reaches the maximum distance ${h_C}$, we set the depth position h_i as the i^th phase screen. Figure 3 shows the flowchart for setting each phase screen.

Fig. 3. Initial setup of the phase screens

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The simulation process for modelling the vertical and oblique optical channels using the multi-phase screen method is shown in Fig. 4. The individual processes, except for the initial setup, have been described in detail in Sections 2.1 and 2.2 respectively. The initial setup requires obtaining the distribution of seawater temperature and salinity at each depth, T(z), and S(z). Then, set the number of phase screens, N, and the interval Δh_i in accordance with the link depths, and determine the number of Fourier transform grids, M.

Fig. 4. Process of the oblique optical channel simulation in the ocean turbulence.

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2.4 Optical communication system performance with underwater oblique optical channel

Because the transmitted optical signal is influenced by ocean turbulence, we assume that the signal received by the optical receiver is ${U_R}({x,y} )$, and its signal strength can be expressed as follows: [35]

(18)$$I = U_R^ \ast ({x,y} )U_R^{}({x,y} ).$$

The superscript “*” indicates taking the complex conjugate. According to the definition of scintillation index,

(19)$$\sigma _I^2 = \frac{{\left\langle {{I^2}} \right\rangle }}{{{{\left\langle I \right\rangle }^2}}} - 1 = \frac{{\left\langle {\sum\limits_{m = 1}^M {\sum\limits_{n = 1}^M {{I^2}({m\Delta x,n\Delta y} )\Delta x\Delta y} } } \right\rangle }}{{{{\left\langle {\sum\limits_{m = 1}^M {\sum\limits_{n = 1}^M {I({m\Delta x,n\Delta y} )\Delta x\Delta y} } } \right\rangle }^2}}} - 1.$$

The probability density function of the light intensity reaching the optical receiver can be expressed as follows:

(20)$${p_I}(I )= \frac{1}{{I\sqrt {2\pi \sigma _I^2} }}\exp \left\{ { - \frac{{{{[{\ln (I )+ \sigma_I^2} ]}^2}}}{{2\sigma_I^2}}} \right\}.$$

For the on-off keying modulation, the average bit error rate (BER) of the system can be expressed as follows:

(21)$$\left\langle {\textrm{BER}} \right\rangle = \frac{1}{2}\int_0^\infty {{p_I}(I )\textrm{erfc}\left( {\frac{{\left\langle {\textrm{SNR}} \right\rangle \cdot I}}{{2\sqrt 2 }}} \right)} dI,$$

where SNR denotes the average signal-to-noise ratio of the receiver, and $\textrm{erfc(} \cdot \textrm{)}$ is the complementary error function.

3. Stratification conditions of seawater temperature and salinity with depth

The simulations employ the measured data of Argo buoy nodes with depths from 0 to 600 m, and the locations of these nodes are listed in Table 1 [29]. We plot the variation curves of temperature and salinity with depth, as shown in Fig. 5. The blue and red curves represent the variation in temperature and salinity of seawater with the depth of the ocean, respectively. It is observed from Fig. 5 that there are mixed layers and thermocline in seawater regardless of the ocean area, and however, the depths of the mixed layers and thermocline are different in each ocean area. For instance, the measured data at node 5902507 in Fig. 5(a) show that the depth of the mixed layer in the Tasman Sea can reach nearly 200 m, whereas that of the mixed layer at node 3901299 in Fig. 5(c) is only a few tens of meters, excluding the part of the sudden temperature drop at the seawater surface.

Fig. 5. Temperature and salinity versus depth for each node from Argo buoy measurements. (a) ID: 5902507, (b) ID: 4902912, (c) ID: 3901299, and (d) ID: 4901700.

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Table 1. Geographic location and date of Argo data acquisition employed in the simulation

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4. Performance analysis of optical communication system for oceanic oblique channel

Based on the model of the oblique channel for optical communication established in Section 2, the system performance simulation is performed by using the seawater data in Fig. 5, measured by Argo in several ocean areas. And the main parameters used in the simulations are listed in Table 2. When the optical transmitter is located at different depths of seawater, the BERs of the optical receivers with a vertical optical link length h in different locations in ocean water are as shown in Fig. 6. The simulation results of each data set are repeatedly obtained 10 times and averaged, and each data set is obtained under the same optical transmitting power (40 dBm).

Fig. 6. Average BERs for transmitters at different depths with distances in different oceans. (a) ID: 5902507, (b) ID: 4902912, (c) ID: 3901299, and (d) ID: 4901700.

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Table 2. Channel parameters in the simulation

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Figure 6 shows the performance of the optical communication system at different initial depths of the optical transmitter in different ocean locations. The solid curves indicate the simulation results obtained when the absorption coefficients of both the mixed layer and thermocline are all 0.053 m⁻¹, and the dashed curves indicate the simulation results obtained when the absorption coefficient of the mixed layer is 0.08 m⁻¹. To justified the results from our method, we have also obtained some of the numerical results by Monte Carlo simulation, they are labelled as ‘MC’ in the Fig. 6(a) and are shown as dotted lines in this diagram. Moreover, the Monte Carlo curves have been omitted in Fig. 6(b)–(d) in case there are too many, too dense and too messy curves. The term “FEC” in Fig. 6 denotes the threshold of BER for forward error correction, which is 3.8 × 10⁻³. It is observed from Fig. 6(a)–6(d) that the BERs of optical receivers all increase with an increase of the communication distance, and the initial depth of the optical transmitter can significantly affect the communication quality. For instance, the system BERs in Fig. 6(a) and 6(b) are the highest when the transmitters are located at a depth of 50 m, which are in the mixed layer. In fact, the surface seawater moves violently because of the effect of tides, waves, and currents such that the temperature and salinity of the seawater rarely change in the mixed layer. This implies that in the mixed layer, the turbulence strength is strong, and its effect on the communication system is more serious.

Comparing the system BER curves of Fig. 6(a)–6(d) under different depths of the transmitters, it is observed that the system BERs under different depths of transmitters in the thermocline (200∼500m) are generally lower than those under the different depths of transmitters in the mixed layer. However, in the thermocline, there may be fluctuations in temperature or salinity at some depths, which are influenced by factors such as dark currents, and probably cause locally constant temperature or constant salinity layers. The occurrence of constant temperature and constant salinity layers in the thermocline signifies that there is strong mixing of seawater in these layers, and the stronger turbulence exists in these layers than in other layers. Consequently, the BERs are higher at some locations of the thermocline, where both the temperature and salinity change slowly with depths. For instance, Fig. 6(c) shows a higher BER at a depth of 500 m than those at 200 m and 400 m. From Fig. 5(c), it is found that the temperature and salinity remain stable near a depth of 500 m, indicating that strong mixing of seawater occurs. The results in Fig. 6 also demonstrate that the mixed layer and thermocline are two different turbulent channels, and therefore, the difference between these two channels must be clarified for the design of vertical communication links and study of turbulence suppression schemes.

Figure 7 simulates the BERs of the horizontal and vertical links at different depths of the optical transmitters, and the data of node 5902507 on 9 August 2021 are selected. The parameters ε and H(z) in the horizontal link take their individual mean values at corresponding depths nearby. Comparing the BER curves of the horizontal link and that of the vertical link at the same transmitter depth, it is seen that the communication quality of the horizontal link is better than that of the vertical link when the optical transmitters are located in the mixed layer (50 m, 150 m). For example, when the optical transmitters are located at 50 m depth and the communication distance is 80 m, the BER of the vertical channel is nearly 10⁻⁴, whereas the BER of the horizontal channel under the same condition is only 10⁻⁵. This indicates that the communication performance of the vertical link in the mixed layer is worse than that in the horizontal link. When the transmitter is located in the thermocline (300 m), it is observed that the BER curves of the horizontal and vertical links almost overlap, implying that the difference in communication performance between the horizontal and vertical links in the thermocline is small.

Fig. 7. Comparison curves between BERs of horizontal links and those of vertical links. (Node: 5902507, date: 2021/08/09)

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Figure 8 shows the system BERs with different tilt angles of the optical links. Figure 9 gives the BER curves versus transmitter tilt angles for transmitters located at depths of 50m, 150m and 300m respectively when the link distances are 80m and 100m severally. Here, the measurement data of node 5902507 on 9 August 2021 are selected. When the tilt angle is 0°, the link is a vertical link, and link distance L equals link depth h. It is seen from Fig. 8 that the BER of the system has an overall decreasing trend with increasing tilt angle, regardless of the depth of the transmitter. Furthermore, it can be found from Figs. 8 and 9 that the BERs of the system do not vary significantly when the tilt angle is within 10°, and decrease significantly when the tilt angle changes from 10° to 20°. With the variation in tilt angle from 20° to 25°, the BER decreases by approximately two orders of magnitude. This result implies that when the tilt angle is less than 10°, the BER of an oblique link with link distance L will be almost identical to the BER of a vertical link with the same link depth (h = L). In other words, when the tilt angle is less than 10°, an oblique link can be treated as a vertical link.

Fig. 8. Variation curves of BERs for different tilt angles of the optical transmitter. (a) Depth of the optical transmitter = 50, (b) 150, and (c) 300 m. (Node: 5902507, date: 2021/08/09)

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Fig. 9. BERs versus transmitter tilt angles for different link distances and depths. (Node: 5902507, Date: 2021/08/09)

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5. Conclusion

In this study, a modelling method for optical communication systems using phase screens for vertical and oblique optical links is proposed based on seawater depth layering, and through Argo's measurement data of temperature and salinity in different ocean areas, the performance of the optical communication system with oceanic turbulence for different depths of optical transmitters and different tilt angles of optical links has been analyzed. Simulation results show that for the same link depth h, the vertical link in the mixed layer has poorer communication quality compared to the vertical link in the thermocline. Therefore, for the design and deployment of an underwater wireless optical communication system, the optical transmitter’s depth below the sea level needs to be considered to guarantee the communication quality of the optical link.

When the transmitter is at the same depth below sea level, the communication quality of the vertical link is worse than that of the horizontal link at the same link length. When the tilt angle is less than 10°, an oblique link with a link distance L can be treated as a vertical link with the same link depth (h = L). Additionally, when the tilt angle of the oblique link is greater than 20°, the system BER is significantly reduced with an increase in the tilt angle. Hence, the difference of impacts an oblique link and a vertical link of the optical channel performances is obvious.

In the thermocline, the variation of temperature and salinity with depth below the sea level reflects the degree of seawater mixing and the turbulence intensity in the seawater, and the BER is generally higher when the transmitter is at a location where temperature and salinity change more slowly with depth. Consequently, the local turbulence strength of the ocean at the thermocline can be estimated from the distribution of seawater temperature and salinity at different depths. This is important for the analysis of channel performance, design of turbulence suppression methods, and improvement of the quality of underwater optical communication.

Funding

National Natural Science Foundation of China (61871418, 61801079); Science and Technology on Underwater Information and Control Laboratory (6142218200408).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are available in Ref. [29,30].

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ID of nodes	Latitude	Longitude	Ocean areas	Climate Zone	Date
5902507	−41.63	161.52	Tasman Sea	Temperate zone	2021/08/09
4902912	31.78	−75.67	Hatteras Plain	Subtropical zone	2021/01/05
3901299	0.67	−120.67	Central Pacific Basin	Tropical zone	2020/11/19
4901700	24.46	−68.35	North America Basin	Tropical-temperate border	2021/10/30

Parameters	Value
Light Wavelength λ	532 nm
Communication link depth h	75–115 m
Rate of loss of temperature variance $χ_{T}$	10⁻⁴∼10⁻⁹ K²s⁻¹
Turbulent energy dissipation rate ε	10⁻⁵ m²s⁻³
Number of grids M	256
Attenuation coefficient c₁ for mixed layer (*)	0.08 m⁻¹
Attenuation coefficient c₂ for thermocline	0.053 m⁻¹
Maximum depth interval of phase screen $h_{C}$	10 m
Temperature interval of phase screen ΔT	2 °C
Salinity interval of phase screen ΔS	0.1 ppt
Number of simulations per data set	10

ID of nodes	Latitude	Longitude	Ocean areas	Climate Zone	Date
5902507	−41.63	161.52	Tasman Sea	Temperate zone	2021/08/09
4902912	31.78	−75.67	Hatteras Plain	Subtropical zone	2021/01/05
3901299	0.67	−120.67	Central Pacific Basin	Tropical zone	2020/11/19
4901700	24.46	−68.35	North America Basin	Tropical-temperate border	2021/10/30

Parameters	Value
Light Wavelength λ	532 nm
Communication link depth h	75–115 m
Rate of loss of temperature variance $χ_{T}$	10⁻⁴∼10⁻⁹ K²s⁻¹
Turbulent energy dissipation rate ε	10⁻⁵ m²s⁻³
Number of grids M	256
Attenuation coefficient c₁ for mixed layer (*)	0.08 m⁻¹
Attenuation coefficient c₂ for thermocline	0.053 m⁻¹
Maximum depth interval of phase screen $h_{C}$	10 m
Temperature interval of phase screen ΔT	2 °C
Salinity interval of phase screen ΔS	0.1 ppt
Number of simulations per data set	10

Modeling and performance analysis of oblique underwater optical communication links considering turbulence effects based on seawater depth layering

Abstract

1. Introduction

2. Modeling of underwater turbulent oblique optical channel

2.1 Multi-phase screen method to simulate optical signal propagation in seawater oblique channel

2.2 Calculating the phase function $\phi {}_i({x,y} )$

2.3 Initial phase screen setup

2.4 Optical communication system performance with underwater oblique optical channel

3. Stratification conditions of seawater temperature and salinity with depth

4. Performance analysis of optical communication system for oceanic oblique channel

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

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Figures (9)

Tables (2)

Equations (23)

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