1. Introduction

Recently, extensive research activities have focused on the investigation of underwater wireless optical imaging and optical communication, due to their advantages of wide band, large information capacity, fast data transmission rate, low delay, high confidentiality and low energy consumption [1–6]. However, the communication and imaging performance of underwater optical systems can be seriously affected by the oceanic turbulence. Moreover, the oceanic power spectrum of refractive-index fluctuation can basically describe the optical properties of oceanic turbulence. Therefore, the construction of accurate and practical spatial oceanic power spectrum of refractive-index fluctuation is of great significance for the theoretical study of underwater wireless optical communication and imaging.

The spatial oceanic power spectrum of optical turbulent refractive-index fluctuation in horizontal path has been studied by many researchers [7–9]. Under the assumption of the stable stratification, Nikishov [7] firstly established the spatial oceanic power spectrum of turbulent refractive-index fluctuation by the linear combination of three scalar spectra of the temperature, salinity, and coupling fluctuations. To eliminate the error of stable stratification assumption, Elamassie et al. [8] modified the Nikishov’s spatial power spectrum and built the spatial power spectrum of oceanic refractive-index fluctuation with the eddy diffusivity ratio. Considering the complexity of the power spectrum to the study of optical properties in oceanic turbulence, Yao et al. [9] derived an approximation spatial power spectrum of oceanic refractive-index fluctuation with reference to Nikishov’s spectrum. To improve the distortion part of Nikishov’s spatial power spectrum, Yi et al. [10] and Yao et al. [11] derived the modified spatial oceanic power spectrum based on Hill’s model [12]. And in previous work, we also developed the practical oceanic power spectrum of refractive-index fluctuation with the outer scale in unstable layer [13].

The displacement and wander of spatially limited light beam in oceanic turbulence have great significance for underwater communication, underwater plankton observation and underwater optical radar. Yue et al. [14] studied the beam wander variance of the Gaussian beam propagating in horizontal path of oceanic turbulence, and revealed that the beam wander in unstable stratification is more overestimated or underestimated than that in stable stratification. Cui et al. [15] analyzed the beam wander and the misaligned displacements in oceanic turbulence. Wu et al. [16] investigated the beam wander of Gaussian-Schell model beams propagation in the isotropic oceanic turbulence, and found that the effect of salinity fluctuations on beam displacement is more serious than that of temperature fluctuations. However, these investigations of beam wander were all based on the spatial oceanic power spectrum model of refractive-index fluctuation of optical turbulence along the horizontal path [14–16]. As is known to all, there is no research on the spatial oceanic power spectrum of optical turbulent refractive-index fluctuation in slant path. Hence, the establishment of the spatial oceanic power spectrum of refractive-index fluctuation in slant path to study the displacements of spatially limited light beam in slant path of oceanic turbulence is crucial.

In this paper, we developed the new dissipation rate of oceanic turbulent kinetic energies with depth induced by the wind-generated and tide-generated kinetic energies. Firstly, the oceanic temperature structure parameter with depth was established through the relationship among the temperature structure parameters, the temperature dissipation rate and the dissipation rate of turbulent kinetic energies. Then, the spatial oceanic power spectrum of optical turbulent refractive-index fluctuation in slant path is constructed by the new oceanic temperature structure parameter with depth. Based on the new oceanic power spectrum, we established the displacement variance model of Gaussian-beam in slant path of oceanic turbulence. Finally, the influence of light source parameters and oceanic turbulent parameters on the beam displacement of Gaussian-beam in slant path was investigated.

2. Oceanic power spectrum of refractive-index fluctuations in slant path

As for the investigation of beams propagation in slant oceanic path, the refractive-index structure parameter of oceanic turbulence in slant path $C_m^2(h )$, which includes temperature structure parameter, salinity structure parameter, together with coupling structure parameter of temperature and salinity, is very important to properly describe the fluctuations strength of optical turbulence for the optical wave. First of all, through analogy with the temperature structure parameter of atmospheric turbulence [17], the temperature structure parameter of oceanic turbulence in slant path can be defined as

Taking the two major factors of wind speed and tide velocity into consideration, we can structure the dissipation rate of oceanic turbulent kinetic energies with depth as [18,19]

Wind stress is one of the key factors for oceanic turbulence production in upper seawater of slant path. Strong wind and weak wind have different interference law on oceanic temperature dissipation rate. It is defined that wind speed less than or equal to $7\textrm{m/s}$ is weak wind while speed above $7\textrm{m/s}$ is strong wind [22]. The piecewise functions of oceanic temperature dissipation rate models in Ref. [22] are not conducive to study the beam propagation in oceanic turbulence. Thus, we construct the continuous dissipation rate of mean-squared temperature models in weak wind and strong wind near the upper seawater surface, respectively

Substituting Eqs. (3) and (2) into Eq. (1), the temperature structure parameter of oceanic turbulence in slant path can be rewritten as

Based on the oceanic power spectrum in the horizontal path [13], the spatial oceanic power spectrum of optical turbulent refractive-index fluctuations in slant path can be expressed as

For a fixed value of the depth h, the dissipation rate of mean-squared temperature in slant path ${\chi _T}(h )$ degrades to the dissipation rate of mean-squared temperature in horizontal path ${\chi _T}$, and the dissipation rate of mean-squared salinity in slant path ${\chi _S}(h )$ degrades to the dissipation rate of mean-squared salinity in horizontal path ${\chi _S}$. The concentration of salt hardly changes with depth in upper seawater [23]. Therefore, in the case of upper seawater, the dissipation rate of mean-squared salinity in slant path ${\chi _S}(h )$ is approximated to the dissipation rate of mean-squared salinity in horizontal path ${\chi _S}$. Utilizing the ratio of temperature and salinity to the distribution of refractive-index in the horizontal oceanic path $\varpi = A{B^{ - 1}}\sqrt {{\chi _T}/{\chi _S}}$ [7], we can derive the expression in slant oceanic channel of upper seawater $\varpi (h )= A{B^{ - 1}}\sqrt {{\chi _T}(h )/{\chi _S}}$. Consequently, the ratio of the temperature and salinity to the distribution of the refractive-index in upper seawater can be obtained as

In upper seawater of slant path, the variation of the dissipation rate of mean-squared temperature ${\chi _T}(h )$ and mean-squared salinity ${\chi _S}(h )$ with depth are mainly caused by temperature fluctuation and salinity fluctuation with depth respectively. We can assume that the eddy diffusivity of temperature ${K_T}$ and the eddy diffusivity of salinity ${K_S}$ hardly change with depth. Thus, in unstable stratification, the eddy diffusivity ratio of temperature to salinity with depth can be approximated by $\theta = {K_T}/{K_S}$. According to the relational expression in unstable stratification case, the eddy diffusivity ratio of temperature to salinity is expressed as [8,13]

The salinity structure parameter and the coupling structure parameter of temperature and salinity in slant path can be obtained as

Furthermore, the practical spatial oceanic power spectrum of optical turbulent fluctuations of refractive-index in slant path can be rewritten by substituting Eqs. (4), (7), (8) and (9) into Eq. (5) as

3. Displacement variance of Gaussian beam in slant path of oceanic turbulence

In order to explore the influence of oceanic turbulence in slant path on beam displacement in Fig. 1, it is very important to study the light beams propagation in slant path of oceanic turbulence by the scalar parabolic equation [24–26], which is defined as

 

Fig. 1. Schematic diagram of beam displacement in uplink transmission channel of oceanic turbulence.

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The position vector of the beam center of gravity in slant path of oceanic turbulence is expressed as [26]

The mutual coherence function is defined by [17]

To derive the displacements of spatially limited light beam in slant path of oceanic turbulence, the scalar parabolic equation of Eq. (12) is reconstructed as [17]

Replacing the old coordinates $({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2}} )$ with the new coordinates $({{\boldsymbol R} + {\boldsymbol \rho }/2,{\boldsymbol R} - {\boldsymbol \rho }/2} )$, the Eq. (15) will be modified as follows

According to the definition of Eq. (13) and Eq. (16) assuming ${\boldsymbol \rho } = 0$, the beam displacement vector of the center of gravity ${\rho _c}(z )$ can be expressed by the mutual coherence function as

In order to construct the form of the right-hand side of Eq. (18) by the mutual coherence function, we operate upon Eq. (16) by the operator $[{ - i/({k{P_0}} )} ]\nabla {\boldsymbol \rho }$ and then integrate over ${\boldsymbol R}$. After that, Eq. (16) becomes second-order ordinary differential equation of beam displacement ${\rho _c}(z )$ in the propagation direction z. The displacement of symmetrical beams [${\rho _c}(0 )= 0$, $d{\rho _c}(z )/{ {dz} |_{z = 0}} = 0$] propagation at plane $z = L$ in slant path of oceanic turbulence can be expressed as

In consideration of the random fluctuation of light intensity caused by oceanic turbulent in slant path, the mean square of the displacement vector of the beam center of gravity involving the Eq. (19) under the Markovian-random-process approximation has the solution by the ensemble average

In the Markovian process approximation, the correlation function of the random refractive index ${\varepsilon _1}({z,{\boldsymbol \rho }} )$ in slant path of oceanic turbulence is approximated as [25]

Assuming that the refractive index is delta correlated in the propagation direction and the field ${\varepsilon _1}({z,{\boldsymbol R}} )$ is statistically homogeneous and Gaussian distributed, and we substitute Eq. (21) into Eq. (20), the mean square of beam displacement will be represented as follows by making use of the Furutsu-Novikov formula

Using the Fourier transform definition relation, the Fourier representation of Gaussian beam is defined as follows [17,26]

Then, we substitute Eq. (23) into Eq. (22), the beam displacement can be rewritten as

For the sake of obtaining the analytical solution of beam displacement conveniently, it is necessary to use approximate expressions for $\left\langle {{{|{F({\kappa ,z} )} |}^2}} \right\rangle$. Consequently, we consider a focused Gaussian beam at distances z, the quantity $\left\langle {{{|{F({\kappa ,z} )} |}^2}} \right\rangle$ under the weak irradiance fluctuations based on Rytov theory can be expressed as [17,26]

To research the uplink propagation characteristics of Gaussian beam in oceanic turbulence, the normalized distance variable $\xi$ has the definition for uplink propagation as [17]

The quantity ${T_{oc}}$ of Eq. (25) in slant path of oceanic turbulence can be expressed as [17]

Using the geometrical optics approximation [17]

Furthermore, with the help of the integral expression [27]

Adapted to an uplink path of oceanic turbulence, the beam displacement of Eq. (24) has the solution as follows

Integrating over R by the following formula

Moreover, in order to integrate over $\kappa$ in Eq. (35), with the help of the integral expression in Eq. (30), the final simplified expression of the beam displacement in slant path is given as

4. Numerical analysis and discussion

In this section, we investigate the displacement of Gaussian beam center of gravity in slant path of oceanic turbulence by means of the numerical simulation. For the convenience and accuracy of numerical simulation, the parameters in the simulation experiment are set as follows except for certain parameters: $\beta = 0.72$, ${u_w} = 0.0012\textrm{m/s}$, ${C_D} = 0.0015$, ${\rho _a} = 1.2\textrm{kg/}{\textrm{m}^3}$, $u = 0.2\textrm{m/s}$, $\eta = 1\textrm{mm}$, $L = 1\textrm{50m}$, $J = 1$, $W = 1\textrm{m/s}$, $\vartheta = 6 \times {10^{ - 3}}$, ${T^{\prime}_0} = 0.003{}^\textrm{o}\textrm{C}$, $J_b^0 = 80\textrm{W/}{\textrm{m}^2}$, ${\rho _w} = 1025\textrm{kg/}{\textrm{m}^3}$, ${h_0} = 0.1\textrm{m}$, $\zeta = \pi /12$, $\lambda = 0.45\mu \textrm{m}$, $D = 1\textrm{0cm}$, ${c_\rho } = 3932\textrm{J} \cdot \textrm{k}{\textrm{g}^{ - 1}}\textrm{/}{}^\textrm{o}\textrm{C}$, $v = {10^{ - 4}}{\textrm{m}^\textrm{2}}\textrm{/s}$, $\alpha = 0.4$, ${L_0} = 10m$, $\varpi ={-} 0.2$, and ${w_0} = 5\textrm{cm}$.

Figure 2 shows the influence of wavelength $\lambda$ and inner scale $\eta$ on the mean-square beam displacement of Gaussian beam in slant path of oceanic turbulence. It can be seen from Fig. 2 that the beam displacement of Gaussian beam through oceanic turbulence in slant path increases with increasing inner scale $\eta$. The physical reason is that, the large inner scale of oceanic turbulence in slant channel can cause a large refractive-index fluctuations when the long-term spot size is much larger than the inner scale. Additionally, the range of wavelengths that covers the oceanic turbulence are selected. As the wavelength increases, the beam displacement of Gaussian beam in slant path of oceanic turbulence hardly changes. Therefore, we can conclude that the influence of wavelength on the beam displacement in slant path of upper seawater can be ignored.

 

Fig. 2. Mean-square beam displacement of Gaussian beam through oceanic turbulence in slant path versus the wavelength $\lambda$ for the values of different inner scale $\eta$.

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Since the oceanic power spectrum of turbulent refractive-index fluctuation in slant path is different under weak wind and strong wind conditions, we study the beam displacement with the change of propagation distance under six different wind speeds by using the weak wind model and strong wind model in Fig. 3, respectively. It’s found that the beam displacement decreases as the wind speed increases while increases as the propagation distance increases. Moreover, in the case of strong wind, the weak wind model is easier to overestimate the influence of wind speed than the strong wind model. In the case of weak wind, the strong wind model shows a grater tendency to underestimate the influence of wind speed than the weak wind model. Therefore, an appropriate wind speed model can more accurately reveal the effect of wind speed on beam displacement in slant channel of oceanic turbulence.

 

Fig. 3. Mean-square beam displacement of Gaussian beam through oceanic turbulence in slant path versus the propagation distance L for the values of different wind speed W (a) weak wind model, (b) strong wind model.

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Figure 4 illustrates the beam displacement of Gaussian beam in slant path of oceanic turbulence as a function of the tidal velocity of depth-averaged u and the ratio of temperature to salinity contributions $\varpi$. As shown in Fig. 4, the beam displacement slowly increases with the increase of the ratio of temperature to salinity contributions $\varpi$ when $- 3 < \varpi \le - 1$. However, the beam displacement rapidly increases as the ratio of temperature to salinity contributions $\varpi$ increases when $- 1 \le \varpi < 0$. This implies that the oceanic turbulence dominated by salinity fluctuation has more effect on the beam displacement in slant path than that dominated by temperature fluctuation. That is to say, the beam propagation in shallow-sea regions induces a smaller beam displacement than that in deep-sea regions. It also can be observed in Fig. 4 that the beam displacement decreases with the increase of the tidal velocity of depth-averaged u, which indicates that a large tidal velocity of depth-averaged u reduces the beam displacement. The reason for this phenomenon is that a large tidal velocity of depth-averaged corresponds to a large dissipation rate of oceanic turbulent kinetic energy, resulting in a weak oceanic turbulent fluctuation.

 

Fig. 4. Mean-square beam displacement of Gaussian beam through oceanic turbulence in slant path versus the tidal velocity of depth-averaged u for the values of different the ratio of temperature to salinity contributions $\varpi$.

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Figure 5 plots the beam displacement of Gaussian beam as a function of initial beam radius ${w_0}$ and receiver radius D. It’s seen that that a large receiver radius D causes a small beam displacement. Moreover, as the initial beam radius ${w_0}$ increases, the beam displacement of Gaussian beam in slant path of oceanic turbulence decreases when the beam radius ${w_0}$ is smaller than receiver radius D. But the beam displacement increases with increasing initial beam radius ${w_0}$ when the beam radius ${w_0}$ is larger than receiver radius D. Because oceanic turbulence in slant path and beam diffraction effect results in a broadening of the beam waist radius ${w_0}$, the beam displacement reaches the minimum value when the initial beam radius ${w_0}$ is slightly lower than the receiver radius. The reason for this phenomenon is that the aperture averaging effect of the beam displacement of the center of gravity.

 

Fig. 5. Mean-square beam displacement of Gaussian beam through oceanic turbulence in slant path versus the initial beam radius ${w_0}$ for the values of different receiver radius $D$.

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Figure 6 shows the beam displacement of Gaussian beam as a function of the zenith angle $\zeta$ and the depth below sea level of the uplink receiver ${h_0}$. It’s shown in Fig. 6 that the beam displacement of Gaussian beam in slant path of oceanic turbulence decreases as the zenith angle $\zeta$ increases. This is because when the receiver position and the transmission distance are fixed, the larger zenith angle is, the smaller seawater depth where the laser transmitter is located is. With the increase of seawater depth, the salinity fluctuation increases and the temperature fluctuation decreases, which leads to a large beam displacement. In addition, the beam displacement of Gaussian beam in slant path of oceanic turbulence increases with increasing depth below sea level of the uplink receiver. The reason for this phenomenon is that oceanic turbulence in deep-sea regions causes much greater refractive-index fluctuation than that in shallow-sea regions.

 

Fig. 6. Mean-square beam displacement of Gaussian beam through oceanic turbulence in slant path versus the depth below sea level of the uplink receiver ${h_0}$ for the values of different zenith angle $\zeta$.

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To explore the influence of oceanic turbulent parameters in slant path on the beam displacement, we investigate the beam displacement for different dissipation rate of mean-squared temperature ${\chi _T} = {({J_b^0} )^2}/({c_\rho^2\rho_w^2{u_w}} )$ and dissipation rate of mean-squared salinity ${\chi _S} = {A^2}{\chi _T}/{B^2}{\varpi ^2}$ in Fig. 7. It can be found from Fig. 7 that the beam displacement increases with the increase of dissipation rate of mean-squared temperature ${\chi _T}$ and dissipation rate of mean-squared salinity ${\chi _S}$. The reason for this phenomenon is that the small dissipation rate of mean-squared temperature and dissipation rate of mean-squared salinity correspond to weak fluctuation of oceanic turbulence in slant path, which results in the small beam displacement.

 

Fig. 7. Mean-square beam displacement of Gaussian beam through oceanic turbulence in slant path versus the dissipation rate of mean-squared temperature ${\chi _T}$ and the dissipation rate of mean-squared salinity ${\chi _S}$.

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5. Conclusion

In summary, we have proposed new temperature structure parameters of oceanic turbulence with depth in upper seawater under weak wind or strong wind. Then, the spatial oceanic power spectrum of optical turbulent refractive-index fluctuations in slant path was derived. Based on this new oceanic power spectrum, the displacement variance model of the beam center of gravity in slant path of oceanic turbulence was developed. Our results show that small depth below sea level of the uplink receiver and large zenith angle of the transmitter lead to small beam displacement. For a large enough receiver radius, the beam displacement variance can be effectively reduced by choosing the optimized initial beam radius. Oceanic turbulence in slant path with a small dissipation rate of mean-squared temperature and mean-squared salinity causes a small beam displacement. In the case of a large sea-surface wind speed and tidal velocity of depth-averaged, oceanic turbulence dominated by temperature fluctuations in shallow-sea regions induces a smaller beam displacement than that dominated by salinity fluctuations in deep-sea regions. This work provides a theoretical basis for underwater wireless optical communication and imaging system in slant path of oceanic turbulence.