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Abstract
The influence of the ocean depth and anisotropic tilt angle on vertical underwater wireless optical communication (UWOC) systems is considered in this study. We propose a power spectrum model of oceanic turbulence with an anisotropic tilt angle for the first time. Thereafter, the expression of the scintillation index is derived for a spherical wave propagating over anisotropic oceanic turbulence in the vertical link. In addition, considering the temperature and salinity, relevant data of the Atlantic and Pacific oceans at different depths are selected to study further the effect of ocean depth on the scintillation index. The results indicate that the scintillation index strongly depends on the ocean depth and anisotropic tilt angle. Moreover, the scintillation index is also related to other parameters, such as temperature and salinity, kinematic viscosity, the anisotropic factor, optical wavelength, and propagation distance. The presented results can be beneficial in designing optical wireless communication systems in the ocean environment.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
To meet the increasing demands of underwater communication, such as port security, environmental monitoring, data collection, etc., a reliable underwater wireless communication scheme is greatly needed [1–3]. Traditionally, underwater acoustic and radio frequency (RF) communications have been widely used in underwater environments, but these encounter bottlenecks because of the disadvantages of insecurity, constrained bandwidth, and low transmission speed. Recently, underwater wireless optical communication (UWOC) has been standing out for its high transmission rate and high security, and is envisioned as having tremendous potential in underwater communication. In 2016, a low-power UWOC system was established by Shen et al. [4], offering data rates of 2 Gbps at 12 m and 1.5 Gbps at 20 m, with the bit error rate (BER) well beyond forward error correction standards. After that, a maximum transmission capacity of 10 Gbps is achieved by using 16-quadrature amplitude modulation-orthogonal frequency division modulation [5,6].
Although the UWOC system is established in many underwater scenarios, its development is restricted due to the complexity and strong interference of the underwater environment. Therefore, the performance of the UWOC system is severely impaired, and the communication link can even be interrupted. Generally speaking, the optical wave in the oceanic environment is mainly affected by the following impacts: absorption, scattering, pointing errors, and oceanic turbulence. Physically, the photons interact with water molecules and other particulate matter in the water, leading to absorption and scattering [7]. The misalignment effect caused by random water movement is called pointing errors. The average channel capacity under a synthetic channel fading model, including absorption, scattering, and oceanic turbulence effects, was considered by Zou et al. [8]. In [9], the authors considered the effects of oceanic turbulence and pointing errors on optical links, and the results demonstrated that pointing errors can significantly degrade the performance of the UWOC system.
In addition, the oceanic turbulence caused by fluctuations in temperature and salinity severely degrades the performance of the UWOC link. Many undesired effects, such as beam expansion, jitter, and intensity fluctuations, occur when the optical beam propagates in oceanic turbulence, resulting in a further drop in beam quality. The scintillation index used to characterize the irradiance fluctuations of the optical beam has been extensively studied. Ata et al. proposed the scintillation index models for plane waves and spherical waves in oceanic turbulence, and proved that the scintillation index of spherical waves is smaller than that of plane waves [10]. The scintillation index of the plane and spherical waves was further investigated by Elamassie et al. when the eddy diffusivity of temperature and salinity are not equal [11]. Based on a linear combination of temperature, salinity and coupling spectra, a new oceanic spectrum of refractive index fluctuations in unstable stratified oceans was presented to study the scintillation index of Gaussian beams [12]. The results revealed that the contributing factors of temperature and salinity significantly affect the scintillation index, and that weak oceanic turbulence with greater kinematic viscosity can obtain a smaller scintillation index.
Since non-Gaussian statistics govern turbulent fluctuations, the fourth-order statistics contain additional information about the structure of turbulence and must be estimated in several applications. The most important fourth-order statistical quantity of a wave is the scintillation index, the normalized variance of the fluctuating intensity. The scintillation index, also called the Rytov variance, represents the change in refractive index caused by turbulence, which leads to the channel fading and further affects the UWOC system performance.
The pioneering studies on the scintillation index of the UWOC system mainly consider the horizontal link under the assumption that the turbulence intensity is fixed at a certain depth. However, this simplification may not suitable for the vertical link since the temperature and salinity vary with the depth of the ocean [13,14]. Normally, there is a distinct ocean stratification in the vertical direction. More specifically, the water with different salinity and temperature forms an unmixed layer. Therefore, the turbulence intensity for the optical wave in the ocean environment varies over the vertical link. To analyze the BER of the UWOC system in the vertical link, Elamassie et al. further modeled the vertical underwater link as a multi-level cascade [15], and the results demonstrated that the temperature variation has a more significant impact on the intensity of oceanic turbulence than the pressure. Ylimaz et al. considered the propagation range of 120 m and established a four-layer vertical link model [16].
Noting that the above studies mainly focused on the isotropic ocean turbulence in the vertical link. For a real scenario, the oceanic turbulence is anisotropic due to the Earth’s rotation [17]. Previously, the effect of the fluctuations intensity in anisotropic oceanic turbulence has been discussed by Baykal et al. [18,19]. After that, Ata et al. further analyzed the system performance of an asymmetric Gaussian beam in anisotropic oceanic turbulence [20]. The results show that the performance of the UWOC system in anisotropic oceanic turbulence is better than that in isotropic turbulence. For the anisotropic turbulence model, the long axis of the turbulence cell is normally assumed parallel to the ground. This is a special case for the real condition, and an anisotropic tilt angle exists; that is, the long axis of the turbulence cell is not parallel to the ground [21,22]. In [23], the analytical expression for the scintillation index of an optical wave in the horizontal link with the anisotropic tilt angle under weak turbulence was derived.
According to the above descriptions, pioneering studies have provided fundamental investigations on the optical wave propagating in anisotropic oceanic turbulence. However, these studies on the vertical UWOC system in weak turbulence are still in their infancy. Specifically, the scintillation index model for the optical wave in oceanic turbulence, considering the effect of the anisotropic tilt angle, is missing in many studies. Moreover, the inherent relationship between the scintillation index of the UWOC system and the variation of the temperature and salinity caused by ocean depth should be further scrutinized.
The major contributions of this work can be summarized as follows:
- • A power spectrum model of oceanic turbulence with an anisotropic tilt angle is proposed for the first time.
- • The expression for the scintillation index of the UWOC system in the vertical link is proposed. Thereafter, we reveal the influence of oceanic turbulence with the anisotropic tilt angle on the UWOC system.
- • The influence of temperature and salinity at different depths in the Atlantic and Pacific oceans is revealed.
The structure of this paper is organized as follows. The scintillation index model for the UWOC system is described in Section 2. Section 3. presents the simulation results and detailed discussion, followed by concluding remarks in Section 4.
2. Scintillation index model for the UWOC system
2.1 System model
This paper depicts a point-to-point vertical link for the UWOC in Fig. 1. The transmitter is located at a depth of $d_0$ from the ocean surface, the receiver is placed under the ocean at a depth of $h$, and the propagation distance of the UWOC link is $L$. For the sake of simplicity, we assume that the considered optical link is immersed in the oceanic turbulence, which can be modeled as a continuous and unmixed layer with multiple uniform cells (turbulence cells) [24]. According to the Taylor frozen turbulence hypothesis [25], oceanic turbulence is considered a heterogeneous medium composed of multiple homogeneous cells. This assumption is reasonable since different salinity and temperature form an unmixed layer in the real ocean environment. In addition, the oceanic turbulence is considered anisotropic, which is suitable for the real ocean environment due to the rotation of the Earth [17]. As discussed in the last section, the performance of the UWOC system is susceptible to anisotropic oceanic turbulence. This phenomenon can be interpreted as the random inhomogeneity of the refractive index in the anisotropic turbulence, resulting in beam expansion, jitter, and even intensity fluctuations when the optical wave propagates through anisotropic oceanic turbulence. Note that the normalized variance of the intensity fluctuations, referred to as the scintillation index, is a significant metric for evaluating the performance of the UWOC system and is derived in the following.
2.2 Anisotropic oceanic turbulence spectrum model
An anisotropic oceanic turbulence spectrum model is greatly needed for deriving the scintillation index model for the UWOC system in the real oceanic environment. Here, we propose an anisotropic oceanic turbulence spectrum model based on the well-known isotropic spectrum model, which has been extensively employed in pioneering studies [26,27]. The conventional isotropic power spectrum of the fluctuations in the refractive index is given by [28]
Table 1. The definition of some parameters in Eq. (1).
The temperature–salt index, $\omega$, in Eq. (1) represents the contribution of temperature and salinity fluctuations of the refractive index distribution, which is defined as $\omega =\frac {\alpha \left (d T_{0} / d z\right )}{\beta \left (d S_{0} / d z\right )}$ , where $\beta$ is the saline contraction coefficient, and $dT_0/dz$ and $dS_0/dz$ denote the temperature and salinity differences of the vertical UWOC link at positions of $l_1$ and $l_2$, respectively, as shown in Fig. 1. Note that the value of $\omega$ ranges from $-5$ to 0 [30]. When $\omega$ tends to $-5$, it refers to temperature-induced refraction, and it refers to salinity-induced refraction when $\omega$ tends to 0. The eddy diffusivity ratio $d_r$ in Eq. (1), as the function of $\omega$, is given by
In Eq. (1), $\delta =\dfrac {3}{2} C_{1}^{2}(\kappa \eta )^{\frac {4}{3}}+C_{1}^{3}(\kappa \eta )^{2}$ , and $\eta$ is the Kolmogorov microscale length, which is given by $\eta =\left (v^{3} / \epsilon \right )^{1 / 4}$. Here, $v$ denotes the kinematic viscosity, which is defined as the ratio of the dynamic viscosity of the ocean to the density at the same temperature, i.e.,
where $\mu _{o}$ is the dynamic viscosity of the ocean and is a function of the temperature in some specific regions [31,32], which is represented byNote that the power spectrum of the oceanic turbulence is also affected by the density of the ocean, $\rho _{o}$, as shown in Eq. (3). A widely used expression for the density can be found in many pioneering studies as [33]
Some representative values for the parameters in Eq. (5) can be found in [31] and are tabulated in Table 2 as follows.
Table 2. Representative values for the parameters in Eq. (5)
In addition, the Prandtl numbers for the temperature and salinity are expressed as $P_T$ and $P_S$, respectively, in Eq. (1), and $P_{TS}$ is one-half of the harmonic mean of $P_T$ and $P_S$. $P_T=vD_{T}^{-1}$, $P_S=vD_{S}^{-1}$, where $D_T$ and $D_S$ denote the molecular diffusivity of the temperature and salinity [34], and can be represented as
where $k$ denotes the thermal conductivity, which is given by [31]Employing the concept of anisotropy in atmospheric turbulence together with the spectral density of the index of refraction fluctuations of isotropic oceanic turbulence [10,35], the spectral density of the index of refraction fluctuation of anisotropic oceanic turbulence can be expressed as [36]
We have $K=\sqrt {K_{x}^{2}+K_{y}^{2}+K_{z}^{2}}=\sqrt {\mu _{x}^{2}\kappa _{x}^{2}+\mu _{y}^{2}\kappa _{y}^{2}+\kappa _{z}^{2}}$, where $\mu _{x}$ and $\mu _{y}$ are the anisotropic factors of oceanic turbulence in the $x$ and $y$ directions, and $\kappa _{x}$, $\kappa _{y}$, and $\kappa _{z}$ are the $x$, $y$, and $z$ components of the spatial frequency, respectively. Note that we use the Markov approximation here for simplification [37], which is usually used for the wave propagation in random media, assuming that the refractive index is delta-correlated at any pair of points located along the direction of propagation. By invoking the Markov approximation, the value of the spatial wavenumber component along the direction of propagation $\kappa _{z}$ is assumed as $0$. Therefore, Eq. (10) can be rewritten as
As a function of $\kappa$, the parameter $\delta$ in Eq. (1) can be further transformed into $\delta _{a}$ in Eq. (11) with $\delta _{a}=\dfrac {3}{2} C_{1}^{2} \eta ^{\frac {4}{3}}\left [\left (\mu _{x} \kappa _{x}\right )^{2}+\left (\mu _{y} \kappa _{y}\right )^{2}\right ]^{2 / 3}+C_{1}^{3} \eta ^{2}\left [\left (\mu _{x} \kappa _{x}\right )^{2}+\left (\mu _{y} \kappa _{y}\right )^{2}\right ]$.
As mentioned in the introduction, the anisotropic tilt angle $\gamma$, which is defined as the angle between the plane where the long axes of ellipsoid are located and the horizontal plane. Counterclockwise is positive, and the value range of $\gamma$ is $0 \mathrm {^{\circ }}$ to $180 \mathrm {^{\circ }}$ degrees, as shown in Fig. 2. It is easily seen from Fig. 2 that due to an anisotropic tilt angle, the turbulent cell rotates counterclockwise around the $x$-axis by an angle $\gamma$. In Fig. 2, $xOy$-plane is the horizontal plane, and $Oz$ represents the beam’s propagation direction in the vertical link. In addition, since the horizontal axis length of the turbulence cell is usually one to several times the vertical axis length, we assume $OB=\mu s$, $OA=s$, where the anisotropy factor $\mu$ represents the ratio of the horizontal axis and vertical axis of the turbulence. Therefore, the anisotropic turbulence cell can be assumed to be an ellipsoid, consistent with many studies on anisotropic turbulence [21,22,35]. Note that the turbulence cells in this paper are simplified to statistically uniform, which should be further considered with the unstable situations in future studies.
Fig. 2. The anisotropic turbulence cell for the vertical link of the UWOC system with the anisotropic tilt angle.
In the following study, the spectrum of the oceanic turbulence with the consideration of the anisotropic tilt angle will be proposed. To reveal the turbulence effect in the presence of an anisotropic tilt angle, we devote ourselves to reshaping the turbulence power spectrum in Eq. (11) with the consideration of the anisotropic tilt angle. The anisotropic factors $\mu _{x}$ and $\mu _{y}$ are defined as [23]
Then, we transform the coordinate system for the spectrum $\Phi _{n}\left (\kappa _{x}, \kappa _{y}\right )$ into an isotropic coordinate system by using the following substitutions:
Therefore, we further have
Substituting Eq. (13) into Eq. (11), the oceanic anisotropic turbulent spectrum with an anisotropic tilt angle can therefore be written as
An anisotropic oceanic turbulence spectrum model with anisotropic tilt angles is derived in Eq. (16), which is suitable for the real oceanic environment.
2.3 Scintillation index model
The scintillation index of a spherical wave in anisotropic oceanic turbulence can be calculated using the extended Rytov theory as [39]
By inserting Eqs. (15) and (16) into (19), the scintillation index with an anisotropic tilt angle can be expressed as
Inserting Eqs. (17) into (18), the scintillation index can be rewritten as
Let $Q_1=\dfrac {C_0}{C_{1}^{2} P_{T}}$, $Q_2=\dfrac {C_0}{C_{1}^{2} P_{S}}$, $Q_3=\dfrac {C_0}{C_{1}^{2} P_{TS}}$, and assume $a_n=\dfrac {3}{2} Q_n C_{1}^{2} \eta ^{4/3}$, $b_n=Q_n C_1^{3} \eta ^{2}$, a unified expression for $I_1$, $I_2$, $I_3$ in Eqs. (20)–22 with $n=1, 2, 3$ can be recast as
Using Euler’s formula with $\cos (x)=\operatorname {Re}[\exp (-j x)]$, Eq. (23) becomes
With the assistance of the Maclaurin series [40], the exponential function in Eq. (24) can be further expanded as $\exp \left (-a_{n} q^{\frac {4}{3}}\right )=\sum _{m=0}^{\infty } \dfrac {\left (-a_{n}\right )^{m}}{m !} q^{\frac {4 m}{3}}$. Therefore, we have
For ease of calculation, Eq. (25) can be further reshaped as
By employing the Gamma function as $\Gamma (n)=\int _{0}^{\infty } x^{n-1} \exp (-x) d x$, the expressions in Eq. (27) can be recast as
Therefore, $I_n$ can be recast by inserting Eq. (28) into Eq. (26) as
Finally, substituting Eqs. (29) into (19), the scintillation index for a spherical wave propagating in anisotropic oceanic turbulence in the presence of an anisotropic tilt angle can be obtained.
According to the derived scintillation index model for the vertical optical link propagating in anisotropic oceanic turbulence with an anisotropic tilt angle, we find that the scintillation index is directly related to the kinematic viscosity, $v$, the contribution ratio of temperature and salinity, $\omega$, the ocean depth, $h$, the anisotropic factor, $\mu$, optical wavelength, $\lambda$, and propagation distance, $L$. On the one hand, the kinematic viscosity, the contribution ratio of temperature and salinity, and the anisotropic factor can be classified as ocean parameters. Note that the kinematic viscosity related to temperature and salinity is determined by different ocean depths in different oceans. On the other hand, the system parameters, including the optical wavelength and propagation distance, have a negligible impact on the scintillation index. Detailed simulations are conducted in the next section to evaluate the composite effects of ocean and system parameters on the scintillation index.
3. Numerical results and discussion
This section presents the numerical results for the scintillation index of an underwater optical communication vertical link with spherical waves in weak turbulence versus different parameters. The critical parameters in the simulations are listed in Table 3.
Table 3. Key parameters used in the calculations.
The influence of the wavelength, $\lambda$, and anisotropic factor, $\mu$, on the scintillation index versus the anisotropic tilt angle is illustrated in Figs. 3 and 4. In these two figures, a salinity of 35 PPT and a temperature of $20 \mathrm {^{\circ }C}$ are selected. It is observed that an increase in the wavelength will result in a decrease in the scintillation index. This tendency means that the beam with a large wavelength is more suitable for the UWOC system. Figure 4 also demonstrates that the scintillation index decreases as the anisotropic factor in the oceanic turbulence become larger. When the anisotropic tilt angle, $\gamma =90^{\circ }$, for instance, $\mu =4$ offers a scintillation index value of 0.15 compared to a value of 0.17 at $\mu =2$. This phenomenon can be interpreted as the turbulence cells with higher anisotropy acting as lenses with a higher radius of curvature, leading to a reduction of amplitude fluctuations, as a consequence of which the scintillation index decreases [41]. Therefore, we conclude that anisotropic turbulence is helpful for spherical waves propagating in the ocean.
Fig. 3. Scintillation index of an optical beam in oceanic turbulence versus the anisotropic tilt angle for different wavelengths.
Fig. 4. Scintillation index of an optical beam in oceanic turbulence versus the anisotropic tilt angle for different anisotropic factors.
The scintillation index versus the dissipation rate of the mean-squared temperature, $\chi _{T}$, and the salinity–temperature contribution factor, $\omega$, under three anisotropic tilt angles, is shown in Fig. 5. Note that the color bar on the left side of each subfigure denotes the value of the scintillation index. Obviously, the scintillation index increases as the mean-squared temperature dissipation rate increases. A more considerable value of $\chi _{T}$, which generally exists in regions of high-energy turbulence, leads to larger scintillation. However, the salinity–temperature contribution factor induces a complicated variation for the scintillation index. In addition, the scintillation index in Fig. 5 increases when $\omega$ is small. Note that a noticeable color change towards warm colors, denoted by the chartreuse, happens in these three subfigures. These phenomena mean that the scintillation index increases slowly when the turbulence is dominated by temperature. However, with the increase of $\omega$, the scintillation index increases rapidly when salinity dominates. The definition of $d_r$ can explain such a tendency for the scintillation index versus the salinity–temperature contribution factor in Eq. (2). Comparing these subfigures, the UWOC system with a larger anisotropic tilt angle has severe beam intensity fluctuations for a fixed dissipation rate of the mean–squared temperature and a fixed salinity–temperature contribution factor. This conclusion has also been proven in Figs. 3 and 4.
Fig. 5. Scintillation index versus the dissipation rate of the mean-squared temperature and the salinity–temperature contribution factor. (a) $\gamma =30^{\circ }$, (b) $\gamma =60^{\circ }$, (c) $\gamma =90^{\circ }$.
The variations of the scintillation index of optical spherical waves versus kinematic viscosity are presented for different propagation distances in Fig. 6. It is evident that the increased viscosity reduces the intensity fluctuations of a spherical beam in anisotropic turbulence. The increase in kinematic viscosity reduces the power spectrum in Eq. (16), affecting the scintillation index. Besides, a longer propagation distance yields a more extensive scintillation index at a constant viscosity. More importantly, as the propagation distance increases, the variation of the scintillation index increases. For instance, when $L=6$ m, the increase in kinematic viscosity from $1\times 10^{-6}$ to $1\times 10^{-5}$ results in a 0.014 decrease in the scintillation index. Note that the kinematic viscosity reduces the impact on the scintillation index less when $L=2$ m than with $L = 4$ and 6 m. This phenomenon can be attributed to the optical signal encountering less underwater turbulence when the link distance is small. Therefore, the kinematic viscosity directly influences the scintillation index but is strongly coupled with the propagation distance.
Fig. 6. Scintillation index of an optical beam in oceanic turbulence versus the kinematic viscosity for different propagation distances.
Figure 7(a) shows the relationship between the propagation distance and the scintillation index under different salinity. Note that the salinity changes depending on the geographical region. Therefore, we set the salinity ranges from 30-40 PPT, with a temperature of $20 ^{\circ }\mathrm {C}$, according to [42]. Figure 7(a) indicates that the increase in the propagation distance causes the scintillation index to increase, as verified in Fig. 6. In addition, the scintillation index decreases as the oceanic turbulence has lower salinity. Moreover, it is observed that the salinity gradient also affects the scintillation index. The relationship between the salinity gradient and scintillation index is further analyzed in Fig. 7(b). It is evident that the spherical beam’s intensity fluctuates more as the salinity gradient increases. The differences in the scintillation index between the propagation distances of 2, 4, and 6 m also increase as the salinity gradient increases.
Fig. 7. Scintillation index of an optical beam in oceanic turbulence versus (a) propagation distance for different salinity values and (b) salinity gradient for various propagation distances.
Figure 8(a) shows the influence of the propagation distance for different temperatures on the scintillation index. The salinity is fixed as 35 PPT in this part. It is obvious that the scintillation index increases as the propagation distance increases. Under the same propagation distance, the higher the temperature, the larger the scintillation index is. This phenomenon implies that the beam intensity has fewer fluctuations in the seawater with a lower temperature. In addition, Fig. 8(b) presents the scintillation index versus the temperature gradient. An increase in the temperature gradient results in an increased scintillation index.
Fig. 8. Scintillation index of an optical beam in oceanic turbulence versus (a) propagation distance for different temperatures and (b) temperature gradient for different propagation distances.
In the above studies, we have revealed the influence of temperature and salinity on the scintillation index. To further provide some engineering insight into a real UWOC system, we investigated the scintillation index variation for the optical wave propagation in a specific ocean under different depths and latitudes. The thermohaline characteristics of varying depths of the Atlantic Ocean and the Pacific Ocean at high latitude ($60^{\circ }$–$90^{\circ }$) are shown in Fig. 9, where all the data is collected from pioneering studies [13,16]. It is observed from Fig. 9 that both the temperature and salinity show depth-dependency, particularly up to 2000 m, with the majority of changes in the first 500 m. These profiles show the relationship between the scintillation index and ocean depth in Fig. 10.
Fig. 9. Temperature and salinity profiles of the Atlantic Ocean and the Pacific Ocean at high latitudes.
Fig. 10. Scintillation index of an optical beam in oceanic turbulence versus the depth for different latitudes: (a) the Atlantic Ocean, (b) the Pacific Ocean, and (c) both the Atlantic Ocean and the Pacific Ocean at high latitude.
Figure 10 illustrates the scintillation index of optical waves in the Atlantic Ocean and the Pacific Ocean at different latitudes versus the ocean depth. It is observed that the variation of the scintillation index of the UWOC system can be categorized into two parts. On the one hand, in the offshore area, i.e., where the ocean depth is less than 500 m, the UWOC system has the most considerable scintillation index in the low-latitude ocean and the smallest in the high-latitude ocean. On the other hand, when the ocean depth is greater than 500 m, the scintillation index in mid-latitude ocean regions is larger than those in the low-latitude and high-latitude oceans. Note that these variations in the scintillation index result from the interpretation of the temperature and salinity. Significantly, temperature plays a dominant role. When the ocean depth is less than 500 m, the ocean surface temperature is higher because solar radiation first reaches the ocean surface. Due to the low thermal conductivity of the ocean, the temperature decreases with depth. Therefore, the low-latitude areas have the highest temperature, followed by mid-latitude regions and high-latitude regions with the lowest temperature. However, these temperature differences at various depths no longer change when the depth is greater than 500 m. More specifically, the ocean temperature in the mid-latitude ocean areas is the highest because of the interaction of the ocean currents, while the temperature in the high-latitude regions is still the lowest.
In addition, Fig. 10(c) shows the variation of the scintillation index with ocean depth in both the Atlantic and Pacific Oceans in high-latitude ocean regions. It can be observed that the scintillation index of the UWOC link in high-latitude ocean areas exhibits complex changes. First, when the depth is less than 250 m, the scintillation index decreases rapidly with the increase of depth. Then, the scintillation index increases briefly when the depth increases from 250 to 500 m. Finally, when the depth is greater than 500 m, the scintillation index decreases slowly with increasing depth. These tendencies are strongly related to temperature and depth, as described in the last paragraph. The transient increase in the scintillation index is due to the temperature inversion phenomenon, mainly caused by geothermal action [13]. Moreover, regardless of the depth, the scintillation index of the UWOC link in the Atlantic Ocean is more prominent than that in the Pacific Ocean. This tendency is reasonable since the salinity of the Atlantic Ocean is higher than that of the Pacific Ocean [43–45].
Figure 11 presents the scintillation index as a function of anisotropic tilt angle $\gamma$ under different depths. The temperature and salinity of the high-latitude ocean at depths of 1000, 2000, and 3000 m are selected from Fig. 9. As demonstrated in Fig. 11(a) and (b), the scintillation index first increases and then decreases with the increasing anisotropic tilt angle. These curves are symmetric about the benchmark line $\gamma =90^{\circ }$. Note that these scintillation index variations can be attributed to the changes in the radius of curvature of the interface between the propagating beam and the turbulence cell since the turbulence cell is usually ellipsoid [23]. With the increase of the anisotropic tilt angle, the focusing properties of the turbulence are changed. The decrease in the radius of curvature of the interface leads to an increased scintillation index. In addition, the scintillation index decreases with increasing depth in the deep-sea region, which is consistent with the conclusion in Fig. 10. Moreover, from Fig. 11(c), it can be observed that the scintillation index of the UWOC system in the Atlantic Ocean is more significant than that in the Pacific Ocean.
Fig. 11. Scintillation index of an optical beam in oceanic turbulence versus anisotropic tilt angle for different depths: (a) the Atlantic Ocean, (b) the Pacific Ocean, and (c) the comparison of the Atlantic Ocean and the Pacific Ocean at a depth of 1000 m.
4. Conclusion
In this paper, a new oceanic turbulence power spectrum model with an anisotropic tilt angle is established. The expression for the scintillation index of the UWOC system in the vertical link is proposed. According to the deduced expression, the influence of the channel parameters and system parameters on the UWOC system is investigated. It is found that the anisotropic tilt angle has a significant effect on the UWOC system in the vertical link. As the angle increases, the scintillation index first increases and then decreases. In addition, the values of the scintillation index are symmetrical about the anisotropic tilt angle of 90$\circ$. Moreover, the effect of the ocean depth cannot be ignored. When the ocean depth is less than 500 m, the UWOC system has the most considerable scintillation index in the low-latitude and the smallest in the high-latitude ocean. When the ocean depth is greater than 500 m, the scintillation index in mid-latitude ocean regions is the largest. All these results are helpful for the design of the vertical UWOC system.
Funding
National Natural Science Foundation of China (61831008, 62027802, 62271202); The Open Foundation of State Key Laboratory of Integrated Services Networks (Xidian University) (ISN23-01); The Major Key Project of PCL (PCL2021A03-1); Shanghai Space Innovation Fund (SAST2020-054); Young Elite Scientist Sponsorship Program by CAST.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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