1. Introduction

Underwater wireless information transfer is of great interest to the military, industry, and the scientific community, as it plays an important role in tactical surveillance, pollution monitoring, oil control and maintenance, offshore explorations, climate change monitoring, and oceanography research [1]. Recently, underwater wireless optical communication (UWOC) systems have been established between submarine devices, and these have proved to be promising alternative schemes for underwater environments due to their advantages of very high data rates and low power and mass requirements [2]. Despite advantages of UWOCs, one main disadvantage of UWOCs is the signal degradation because of the presence of scattering, absorption, and turbulence in water. Besides other effects, oceanic turbulence manifests itself in the form of intensity fluctuations measured by the scintillation index [3]. In other words, scintillation induced by oceanic turbulence is one of the main impairments of UWOCs and can result in performance degradation.

In terrestrial free-space optical (FSO) communication systems, it has been proved that scintillation induced by atmospheric turbulence can be reduced by using array laser beams [4–13]. In [4], by means of numerical simulations, the statistical properties of the optical power fluctuations induced by the incoherent superposition of multiple transmitted laser beams in a terrestrial free-space optical communication link was analyzed. The scintillation reduction achieved by using multiple beams was evaluated. In [5], using the Huygens-Fresnel principle, the on-axis scintillation index was formulated for partially coherent multiple Gaussian beams in weak atmospheric turbulence, and results were obtained for partially coherent annular and partially coherent flat-topped Gaussian beams in homogeneous horizontal paths. In [6], the scintillation properties of Airy beam arrays in atmospheric turbulence was investigated. It was shown that the scintillation of an Airy beam array is significantly reduced through numeric simulations. In [7], a method to reduce turbulence-induced scintillation by using an incoherent beam array composed of beams with non-uniform polarization was explored. In [8], scintillation index of partially coherent multi-Gaussian array beams was analytically derived and numerically evaluated in extremely strong atmospheric turbulence. Partially coherent annular array and flat-topped array beams were considered in the evaluations. It was shown that such array beams have smaller scintillation when compared to the non-array partially coherent multi-Gaussian beam. In [9], multiple-input single-output (MISO) systems were employed in free-space optical links to mitigate the degrading effects of atmospheric turbulence. The scintillation was formulated as a function of transmitter and receiver coordinates in the presence of weak atmospheric turbulence by using the extended Huygens-Fresnel principle. In [10], the scintillation index and the receiver aperture averaging factor in MISO FSO systems by using a partially coherent laser beam array and finite sized receiver aperture were evaluated. In [11], multiple-input multiple-output (MIMO) techniques were employed in FSO links to mitigate the turbulence-induced scintillation. a comprehensive performance analysis of MIMO FSO systems with Gaussian beams and finite-sized detectors was performed. In [12], the on-axis scintillation indices of the incoherent combination of a 2D Airy beam array (ICCAB) and the coherent combination of a 2D Airy beam array (CCAB) at the receiver plane in atmospheric turbulence were compared experimentally. It was shown that ICCAB has a smaller scintillation index than that of CCAB in the same turbulent condition. In [13], on the basis of the extended Huygens-Fresnel principle, a semi-analytical expression for describing the on-axis scintillation index of a partially coherent flat-topped (PCFT) array laser beams through non-Kolmogorov maritime and terrestrial atmospheric environments on a slant path was derived.

Similar to atmospheric turbulence, oceanic turbulence can also lead to scintillation and further result in performance degradation of UWOC systems [14–20]. Since optical signal transmitting by array laser beams is considered an effective scintillation reduction method in terrestrial FSO communications, it is reasonable to employ array laser beams in UWOCs [21–23]. In [21], MISO technique was employed in UWOC links to mitigate the scintillation induced by oceanic turbulence. Numerical results revealed that MISO UWOC systems show smaller scintillation indices than single-input single-output (SISO) UWOC systems. In [22], the performance of UWOC systems which were made up of the partially coherent flat-topped (PCFT) array laser beams were investigated in detail. It was shown that UWOC systems using the PCFT array laser beams have a considerable advantage compared to the UWOC systems with single PCFT or Gaussian laser beams and also Gaussian array beams in the sense of scintillation index reduction. In [23], a tilted MISO UWOC system consists of a beam array and an aperture averaged detector was proposed. The proposed titled MISO method is different from the traditional parallel MISO method. In the tilted MISO UWOC system, the beams are individually tilted to be directed at the center of receiver. While, in the parallel MISO UWOC system, the beam propagation directions are parallel to each other. It was found that titled MISO UWOC systems can provide smaller scintillation indices than SISO UWOC systems and parallel MISO UWOC systems.

All the above-mentioned research investigated scintillation of beam array UWOCs on the basis of perfect alignment between the transceivers in UWOCs. However, pointing errors may be existed between underwater transceivers because of the following two reasons. The first reason is that, in the presence of turbulence, the instantaneous center of the beam (point of maximum irradiance or "hot spot") is randomly displaced in the receiver plane, producing what is commonly called beam wander [24]. The second reason is that slight changes in the position and posture may easily happen to suspended underwater platforms, such as a submarine or an unmanned underwater vehicle (UUV), and further make the actual pointing directions of underwater platform-mounted transceivers deviate from expected pointing directions. In consideration of the existence of pointing errors in UWOCs, it is meaningful to study the effect of beam array method on reducing scintillation in UWOCs between misaligned transceivers. This motivates our work here. In this paper, we propose the convergent beam array to decrease scintillation in UWOCs between misaligned transceivers. In the proposed convergent beam array, the propagation directions of beams aim at different target points in the receiver transverse plane. We are able to control the optical field distributions of the convergent beam array in the receiver transverse plane by adjusting the distributions of the target points in the receiver transverse plane.

The remainder of this paper is organized as follows. In Section 2, we describe the parallel, tilted, and convergent beam array systems. We also analyze the spatial relationships in beam array systems. In Section 3, we review the power spectrum of refractive index fluctuations and the multistep wave optics simulation theory in oceanic turbulence. Further, we calculate the longitudinal and radial distance between the misaligned transceivers in UWOCs with pointing errors. In Section 4, we present and analyze the wave optics simulation results. Finally, in Section 5, we conclude the paper.

2. Beam array systems

In this section, we describe beam array systems. In particular, we propose the convergent beam array. We also analyze the spatial relationships between the transmitter and the individual beam in beam array systems.

The schematic diagrams of beam array systems are given in Fig. 1. The transmitter of a beam array consists of $N$ equal beams. In the transmitter transverse plane, the $N$ beams are located symmetrically on a ring with the radius ${{R}_{0}}$ and the center of the ring. The $N$ beams on the ring have equal angle separation $d\gamma$. The receiver is located on the propagation direction of the central beam. The angle separation $d\gamma$ is given by

 

Fig. 1. The schematic diagrams of (a) parallel, (b) tilted, and (c) convergent beam array systems.

Download Full Size | PDF

For the parallel beam array, as shown in Fig. 1(a), the propagation directions of $N$ beams are parallel to each other. For the tilted beam array, as shown in Fig. 1(b), the propagation directions of $N$ beams all point to a common target point in the receiver transverse plane. For the convergent beam array, as shown in Fig. 1(c), the propagation directions of $N$ beams are slanted inwards and slightly. The propagation directions of $N$ beams point to $N$ different target points in the receiver transverse plane.

As shown in Fig. 2, in order to describe the spatial relationship between the transmitter and the beams, we set the central beam coordinate system or the transmitter coordinate system ${{O}_{0}}-{{X}_{0}}{{Y}_{0}}{{Z}_{0}}$ and the $ith\left ( 0\le i\le N-1 \right )$ beam coordinate system ${{O}_{i}}-{{X}_{i}}{{Y}_{i}}{{Z}_{i}}$. The subscript $i$ denotes the central beam when $i=0$, while the subscript $i$ denotes the surrounding beam when $1\le i\le N-1$. For the transmitter coordinate system ${{O}_{0}}-{{X}_{0}}{{Y}_{0}}{{Z}_{0}}$, the origin ${{O}_{0}}$ is the center of the central beam and the transmitter. The plane ${{X}_{0}}{{O}_{0}}{{Y}_{0}}$ is the transmitter transverse plane. The axis ${{Z}_{0}}$ is perpendicular to the transmitter transverse plane ${{X}_{0}}{{O}_{0}}{{Y}_{0}}$, and the axis ${{Z}_{0}}$ is the propagation direction of the central beam. For the $ith$ beam coordinate system ${{O}_{i}}-{{X}_{i}}{{Y}_{i}}{{Z}_{i}}$, the origin ${{O}_{i}}$ is the center of the $ith$ beam. The axis ${{Z}_{i}}$ is perpendicular to the plane ${{X}_{i}}{{O}_{i}}{{Y}_{i}}$, and the axis ${{Z}_{i}}$ is the propagation direction of the $ith$ beam.

 

Fig. 2. The spatial relationships between the transmitter and the beams.

Download Full Size | PDF

The spatial relationship between the transmitter coordinate system ${{O}_{0}}-{{X}_{0}}{{Y}_{0}}{{Z}_{0}}$ and the $ith$ beam coordinate system ${{O}_{i}}-{{X}_{i}}{{Y}_{i}}{{Z}_{i}}$ can be described by the position coordinates $\left ( d{{x}_{i}},d{{y}_{i}},d{{z}_{i}} \right )$ and the angles ${{\alpha }_{i}},{{\beta }_{i}}$ and ${{\gamma }_{i}}$. In the transmitter coordinate system ${{O}_{0}}-{{X}_{0}}{{Y}_{0}}{{Z}_{0}}$, the position coordinates of the origin ${{O}_{i}}$ of the $ith$ beam coordinate system ${{O}_{i}}-{{X}_{i}}{{Y}_{i}}{{Z}_{i}}$ are $\left ( d{{x}_{i}},d{{y}_{i}},d{{z}_{i}} \right )$. The angles ${{\alpha }_{i}},{{\beta }_{i}}$ and ${{\gamma }_{i}}$ are the rotation angles of the $ith$ beam coordinate system ${{O}_{i}}-{{X}_{i}}{{Y}_{i}}{{Z}_{i}}$. The angle ${{\gamma }_{i}}$ is the angle between the axis ${{X}_{0}}$ and the projection of the axis ${{X}_{i}}$ in the plane ${{X}_{0}}{{O}_{0}}{{Y}_{0}}$. The angle ${{\gamma }_{i}}$, and the distance $d{{x}_{i}},d{{y}_{i}}$ and $d{{z}_{i}}$ are given by

It is assumed that the point $P$ is an arbitrary point. The position coordinates of the point $P$ in the transmitter coordinate system ${{O}_{0}}-{{X}_{0}}{{Y}_{0}}{{Z}_{0}}$ are$\left ( {{x}_{0}},{{y}_{0}},{{z}_{0}} \right )$. The position coordinates of the point $P$ in the $ith$ beam coordinate system ${{O}_{i}}-{{X}_{i}}{{Y}_{i}}{{Z}_{i}}$ are $\left ( {{x}_{i}},{{y}_{i}},{{z}_{i}} \right )$. The position coordinates $\left ( {{x}_{i}},{{y}_{i}},{{z}_{i}} \right )$ are

To aim at a target point in the receiver transverse plane, the $ith$ beam need to change the posture. When the point $P$ is the target point, the $ith$ beam changes the rotation angles according to Eq. (3) to make the position coordinates $\left ( {{x}_{i}},{{y}_{i}},{{z}_{i}} \right )$ of the point $P$ satisfy ${{x}_{i}}=0$ and ${{y}_{i}}=0$ in the $ith$ beam coordinate system ${{O}_{i}}-{{X}_{i}}{{Y}_{i}}{{Z}_{i}}$.

In the convergent beam array, the $N$ target points in the receiver transverse plane of the $N$ beams are different. The optical field distributions of the convergent beam array in the receiver transverse plane are able to be controlled by adjusting the distributions of the $N$ target points of the $N$ beams. For example, in a convergent beam array, the $N$ target points of the $N$ beams can be assumed to be located symmetrically on a ring with the radius ${{r}_{0}}$ and its center in the receiver transverse plane. The position coordinates of the central target point are $\left ( 0,0,L \right )$ in the transmitter coordinate system ${{O}_{0}}-{{X}_{0}}{{Y}_{0}}{{Z}_{0}}$, where $L$ is the distance between the transmitter transverse plane and the receiver transverse plane. Furthermore, the $N$ target points on the ring have equal angle separation $d\theta =d\gamma$, and the azimuth angle of the $jth\left ( 0\le j\le N-1 \right )$ target point with respect to the axis ${{X}_{0}}$ is ${{\theta }_{j}}$. The subscript $j$ denotes the central target points when $j=0$, while the subscript $j$ denotes the surrounding target point when $1\le j\le N-1$. Specifically, when it is assumed that ${{\theta }_{j}}={{\gamma }_{i}}$, the position coordinates of the $jth$ surrounding target point in the transmitter coordinate system ${{O}_{0}}-{{X}_{0}}{{Y}_{0}}{{Z}_{0}}$ can be expressed as $\left ( {{r}_{0}}\times \cos {{\gamma }_{i}},{{r}_{0}}\times \sin {{\gamma }_{i}},L \right )$, and the rotation angles ${{\alpha }_{i}}$ and ${{\beta }_{i}}$ of the $ith$ surrounding beam can be obtained by

3. Propagation in oceanic turbulence

In this section, we review the power spectrum of refractive index fluctuations induced by oceanic turbulence, and review the multistep wave optics simulation for beams propagating in turbulent ocean. Then, we analyze the longitudinal and radial distance between the misaligned transceivers in UWOCs with pointing errors.

3.1 Power spectrum of refractive index fluctuations

In turbulent ocean, the refractive index fluctuations that are referred to as optical turbulence, are able to introduce random perturbations in the wave front phase, amplitude and angle of arrival. The refractive index fluctuations in turbulent ocean are controlled by fluctuations of temperature and salinity. In this paper, the simulation for beams propagation in oceanic turbulence is based on the power spectrum of refractive index fluctuations developed by [25]. Moreover, in order to introduce the outer scale ${{L}_{0}}$ of the turbulence, we replace the factor ${{\kappa }^{{-11}/{3}\;}}$ with ${{\left ( {{\kappa }^{2}}+\kappa _{0}^{2} \right )}^{{-11}/{6}\;}}$ [26], where $\kappa$ is the magnitude of the spatial wavenumber vector $\boldsymbol {\kappa }$, i.e. $\kappa =\left | \boldsymbol {\kappa } \right |$, and ${{\kappa }_{0}}={2\pi }/{{{L}_{0}}}\;$. Therefore, the power spectrum ${{\Phi }_{n}}\left ( \kappa \right )$ of refractive index fluctuations is given by

We choose ${{\Pr }_{T}}=7,{{\Pr }_{S}}=700$, and then we obtain ${{\Pr }_{TS}}=13.86$. The parameter ${{d}_{r}}$ denotes the ratio of saline eddy diffusivity to thermal eddy diffusivity. The ratio ${{d}_{r}}$ is typically obtained through

3.2 Multistep wave optics simulation in oceanic turbulence

In this paper, the beams are all assumed to be Gaussian beams in the wave optics simulation. The $ith$ beam in the transmitter transverse plane can be expressed as

In order to simulate beams propagating through oceanic turbulence, the split-step approach is used in the wave optics simulation. The oceanic turbulence path is divided into ${{n}_{s}}$ equal segments by $\left ( {{n}_{s}}-1 \right )$ transmittance planes. The phase screen is located at the center of each segment to account for the effect of the oceanic turbulence in each segment. The optical field ${{U}_{{{k}_{p}}+1}}\left ( {{r}_{{{k}_{p}}}}+1 \right )$ in the $\left ( {{k}_{p}}+1 \right )th$ transmittance plane can be given by [28]

The transmission function $H\left ( {{\kappa }_{x}},{{\kappa }_{y}} \right )$ is given by

To obtain the function $\varphi \left ( x,y \right )$, we firstly generate the complex Gaussian random matrix $h\left ( {{\kappa }_{x}},{{\kappa }_{y}} \right )$. Then the result is filtered by the phase power spectrum density (PSD) ${{F}_{\phi }}\left ( {{\kappa }_{x}},{{\kappa }_{y}} \right )$ of the oceanic turbulence. Finally, the function $\varphi \left ( x,y \right )$ is obtained by inverse Fourier transform. The function $\varphi \left ( x,y \right )$ can be expressed as [29]

3.3 Spatial relationship between misaligned transceivers

To analyze the spatial relationship between the misaligned transceivers in view of the pointing errors ${{r}_{PE}}$ in UWOCs, we set the coordinate system $UOV$ in the receiver transverse plane. As shown in Fig. 3, the origin $O$ is the center of the receiver transverse plane. The axis $U$ is parallel to the axis ${{X}_{0}}$, and the axis $V$ is parallel to the axis ${{Y}_{0}}$. The origin ${{O}_{e}}$ is the center of beams in the aligned UWOC link, and is coincides with the center $O$ of the receiver transverse plane. The origin ${{O}_{a}}$ is the center of beams in the misaligned UWOC link. It is assumed that the azimuth angle of the origin ${{O}_{a}}$ is ${{\theta }_{PE}}\left ( 0\le {{\theta }_{PE}}\le 2\pi \right )$. Thus, in the coordinate system $UOV$, the position coordinates of the origin ${{O}_{a}}$ are $\left ( {{r}_{PE}}\times \cos {{\theta }_{PE}},{{r}_{PE}}\times \sin {{\theta }_{PE}} \right )$.

 

Fig. 3. Pointing errors in the receiver transverse plane.

Download Full Size | PDF

In the misaligned UWOC link, for the $ith$ beam, the longitudinal distance and the radial distance are ${{D}_{Li}}$ and ${{D}_{Ri}}$ respectively. In Eq. (3), by substituting ${{x}_{0}}=-{{r}_{PE}}\times \cos {{\theta }_{PE}},{{y}_{0}}=-{{r}_{PE}}\times \sin {{\theta }_{PE}}$ and ${{z}_{0}}=L$, the longitudinal distance ${{D}_{Li}}$ and the radial distance ${{D}_{Ri}}$ are given by

4. Wave optics simulation results and analyses

In this section, we provide the wave optics simulation results of scintillation indices of single beam UWOC systems and beam array UWOC systems in oceanic turbulence.

The scintillation index $\sigma _{I}^{2}$ in the receiver transverse plane is describes by

In the simulation, we set $N=1$ and for single beam UWOC systems and beam array UWOC systems respectively. For single beam UWOC systems, the $N=1$ beam is located at the center of the transmitter transverse plane. For beam array UWOC systems, the $N=5$ beams are located symmetrically on a ring with the radius ${{R}_{0}}=0.15m$ and the center of the ring in the transmitter transverse plane. Furthermore, for convergent beam array UWOC systems, the $N=5$ target points of the $N=5$ beams are set to be located symmetrically on a ring with the radius ${{r}_{0}}=0.05m$ and its center in the receiver transverse plane. The beam width parameters of the beams are set to be ${{W}_{0}}=0.05m$. The wavelength parameters of the beams are set to be $\lambda =0.53\mu m$. Furthermore, the grid size of the source plane and the observation plane are set to be $512\times 512$ points in the simulation. To computing each result data point of scintillation indices, $1000$ realizations are used in the simulation. In addition, $10$ phase screens are used in the simulation [30]. When the link length $L=100m$, the distance between the adjacent phase screens is set to be $10m$. To match with weak turbulence condition, the results for the values of the scintillation index in UWOCs without pointing errors are limited to less than one [14]. We also set parameters related to oceanic turbulence as follows, $\alpha =2.56\times {{10}^{-4}}{l}/{\deg }\;$, $\eta ={{10}^{-3}}m$, ${{\chi }_{T}}={{10}^{-5}}{{{K}^{2}}}/{{{s}^{3}}}\;$, $\varepsilon ={{10}^{-2}}{{{m}^{2}}}/{{{s}^{3}}}\;$, and $\omega =-1$.

Firstly, in Fig. 4 and Fig. 5, we demonstrate the normalized intensity distributions at different locations along the propagation direction of a single beam UWOC system and beam array UWOC systems. In addition, in Fig. 6, we illustrate the beam spreading of a single beam UWOC system and beam array UWOC systems in oceanic turbulence. The beam spreading ${{W}_{BS}}\left ( z \right )$ is defined as [30],

 

Fig. 4. Normalized intensity distributions at different locations along the propagation direction of (a) single beam, (b) parallel beam array, (c) tilted beam array, and (d) convergent beam array UWOC systems in environments without turbulence.

Download Full Size | PDF

 

Fig. 5. Normalized intensity distributions at different locations along the propagation direction of (a)single beam, (b)parallel beam array, (c)tilted beam array, and (d)convergent beam array UWOC systems in oceanic turbulence.

Download Full Size | PDF

 

Fig. 6. Beam spreading ${{W}_{BS}}\left ( z \right )$ of single beam and beam array UWOC systems versus link length $L$ in oceanic turbulence.

Download Full Size | PDF

In Fig. 4, assuming that the link length $L$ varies from $0m$ to $100m$, we present the normalized intensity distributions of single beam, parallel beam array, tilted beam array, and convergent beam array UWOC systems in environments without turbulence in Figs. 4(a), 4(b), 4(c) and 4(d). It is apparent that the propagation directions of $N=5$ beams in the parallel beam array are parallel to each other. The propagation directions of $N=5$ beams in the tilted beam array all point to one common target point in the receiver transverse plane. The propagation directions of $N=5$ beams in the convergent beam array are slanted inwards and point to $N=5$ different target points in the receiver transverse plane. Therefore, the spot in the receiver transverse plane of the convergent beam array UWOC system is bigger than that of tilted beam array UWOC system.

In Fig. 5, assuming that the link length $L$ varies from $0m$ to $100m$, we provide the normalized intensity distributions of single beam, parallel beam array, tilted beam array, and convergent beam array UWOC systems in oceanic turbulence in Figs. 5(a), 5(b), 5(c) and 5(d). It is shown that the normalized intensity distributions of tilted and convergent beam array UWOC systems tend to be concentrated in a central area of the screen as the link length $L$ increases. Moreover, the concentrated area of normalized intensity distributions in convergent beam array UWOC system is bigger than that in tilted beam array UWOC system.

In Fig. 6, assuming that the link length $L$ varies from $0m$ to $100m$, we provide the beam spreading ${{W}_{BS}}\left ( z \right )$ of single beam, parallel beam array, tilted beam array, and convergent beam array UWOC systems in oceanic turbulence. It can be seen that the beam spreading of single beam and parallel beam array UWOC systems rise gradually as the link length $L$ increases. Furthermore, the beam spreading of tilted and convergent beam array UWOC systems decrease with the link length $L$. It is clear that the beam spreading of convergent beam array UWOC system is always greater than that of the tilted beam array UWOC system, and is always smaller than that of parallel beam array UWOC system.

Then, in Fig. 7, we illustrate the normalized intensity distributions in the receiver transverse plane of single beam and beam array UWOC systems with different pointing errors. In Fig. 7, we obtain simulation results under the assumption that the link length $L=100m$ and the diameter of the receiver aperture $D=0.25m$. In Fig. 7(a), assuming the pointing errors ${{r}_{PE}}=0$ and the azimuth angle of pointing errors direction in the receiver transverse plane ${{\theta }_{PE}}=0$, we summarize the normalized intensity distributions in the receiver transverse plane of single beam UWOC system, and parallel, tilted and convergent beam array UWOC systems. Next, assuming ${{r}_{PE}}=0.1m$, and ${{\theta }_{PE}}=0,{\pi }/{12}\;,{\pi }/{6}\;$, and ${\pi }/{4}\;$ in Figs. 7(b), 7(c), 7(d) and 7(e), we depict the normalized intensity distributions of single beam UWOC system, and parallel, tilted and convergent beam array UWOC systems. Whatever the value of ${{\theta }_{PE}}$ is, the normalized intensity distributions of single beam and beam array UWOC systems will be similar to one of the cases in Figs. 7(b), 7(c), 7(d) and 7(e) due to the symmetry of the location of beams in the beam array. In Fig. 7, it is shown that pointing errors can lead to off-centered intensity distributions in the receiver transverse plane. Apparently, in UWOCs with pointing errors, the coverage in the receiver transverse plane of single beam, and tilted and convergent beam array UWOC systems are smaller than their counterpart in UWOCs without pointing errors. Both of tilted beam array and convergent beam array can provide more concentrated intensity contributions. Moreover, the coverage in the receiver transverse plane of the tilted beam array is equal to that of the single beam. The coverage in the receiver transverse plane of the convergent beam array is bigger than that of the single beam. For example, in the cases in Figs. 7(b), 7(c), 7(d) and 7(e), the coverage in the receiver transverse plane of the convergent beam array is about triple that of the single beam. Therefore, the coverage in the receiver transverse plane of the convergent beam array is bigger than that of the tilted beam array. We can find that the convergent beam array is able to show both concentrated intensity distributions and broad coverage taking into account pointing errors.

 

Fig. 7. Normalized intensity distributions in the receiver transverse plane of single beam UWOC systems and parallel, tilted, and convergent beam array UWOC systems with different pointing errors.

Download Full Size | PDF

Finally, in Fig. 8 and Fig. 9, taking into consideration of pointing errors, we study the effect of beam array on reducing scintillation induced by oceanic turbulence.

 

Fig. 8. Scintillation indices of single beam and beam array UWOC systems versus pointing errors ${{r}_{PE}}$ with (a) ${{\theta }_{PE}}=0$ and (b) ${{\theta }_{PE}}={\pi }/{4}\;$.

Download Full Size | PDF

 

Fig. 9. Scintillation indices of single beam and beam array UWOC systems versus link length in different UWOC links (a) ${{r}_{PE}}=0.08m$ and ${{\theta }_{PE}}=0$, and (b) ${{r}_{PE}}=0.08m$ and ${{\theta }_{PE}}={\pi }/{4}\;$.

Download Full Size | PDF

In Fig. 8, assuming the diameter of the receiver aperture $D=0.1m$ and the link length $L=100m$, we plot scintillation indices of single beam UWOC system, and parallel, tilted and convergent beam array UWOC systems as a function of pointing errors ${{r}_{PE}}\left ( 0m\le {{r}_{PE}}\le 0.15m \right )$. The azimuth angle of the pointing error direction in the receiver transverse plane are ${{\theta }_{PE}}=0$ in Fig. 8(a) and ${{\theta }_{PE}}={\pi }/{4}\;$ in Fig. 8(b). In addition, in Fig. 8, we set the radius of the ring which consists of target points in the receiver transverse plane ${{r}_{0}}={{r}_{PE}}$ for the convergent beam array. In Fig. 8(a), we can find that scintillation indices of single beam and tilted beam array UWOC systems increase as pointing errors increase. The reason of the phenomena is because that intensity distributions of single beam and tilted beam array UWOC systems are centralized and away from the center of the receiver transverse plane when pointing errors existed. As pointing errors increase, scintillation indices of parallel beam array UWOC system increase firstly and then decrease. This is because that when the center of the receiver transverse plane is far from the spot of the central beam, the center is close to a spot of a surrounding beam simultaneously. Obviously, tilted and convergent beam array UWOC systems have smaller scintillation indices when pointing errors are small. Meanwhile, parallel and convergent beam array UWOC systems have smaller scintillation indices when pointing errors are large. As expected, convergent beam array UWOC system can decrease scintillation indices effectively when pointing errors are taken into consideration. In Fig. 8(b), we can find that scintillation indices of single beam UWOC system and beam array UWOC systems increase with pointing errors. Similarly, convergent beam array is able to provide the smallest scintillation indices in UWOCs with pointing errors. Furthermore, scintillation indices of beam array UWOC systems with ${{\theta }_{PE}}={\pi }/{4}\;$ are greater than that of beam array UWOC systems with ${{\theta }_{PE}}=0$. This phenomenon can be understood by the reason that the distance between the center of the receiver transverse plane and the nearest spot of a surrounding beam in beam array UWOC systems with ${{\theta }_{PE}}={\pi }/{4}\;$ is larger than that in beam array UWOC systems with ${{\theta }_{PE}}=0$.

In Fig. 9, assuming $D=0.1m$ and ${{r}_{0}}=0.05m$, we show scintillation indices of single beam UWOC system, and parallel, tilted, and convergent beam array UWOC system with respect to the link length $L$ in UWOC links with different pointing errors. Figure 9(a) and Fig. 9(b) provide the results of UWOCs with the pointing errors ${{r}_{PE}}=0.08m$ and ${{\theta }_{PE}}=0$, and UWOCs with the pointing errors ${{r}_{PE}}=0.08m$ and ${{\theta }_{PE}}={\pi }/{4}\;$. We can see that scintillation indices of single beam and beam array UWOC systems increase with link length. Obviously, single beam and parallel beam array UWOC systems have greater scintillation indices. Meanwhile, convergent beam array UWOC system can provide the smallest scintillation indices. Since the distance between the center of the receiver transverse plane and the nearest spot of a surrounding beam in beam array UWOC systems with ${{\theta }_{PE}}={\pi }/{4}\;$ are bigger than that in beam array UWOC systems with ${{\theta }_{PE}}=0$, we can also find that scintillation indices of beam array UWOC systems with ${{\theta }_{PE}}={\pi }/{4}\;$ are bigger than that of beam array UWOC systems with ${{\theta }_{PE}}=0$. Obviously, convergent beam array can always degrade scintillation of UWOC system in view of pointing errors.

5. Conclusion

In order to mitigate the influence of turbulence induced scintillation in UWOCs between misaligned transceivers, we have proposed the convergent beam array. In the proposed convergent beam array, the propagation directions of beams aim at different target points in the receiver transverse plane. By adjusting the distributions of the target points, we can control the optical field distributions of the convergent beam array in the receiver transverse plane. In this paper, first, we have proposed the convergent beam array, and analyzed spatial relationships between the transmitter and the individual beam in the beam array systems. Then, in view of pointing errors, we have formulated the longitudinal and radial distance in misaligned UWOC links. Finally, we have provided the intensity distributions and scintillation of beam array systems in the receiver transverse plane by multistep wave optics simulation. Simulation results reveal the effectiveness of the convergent beam array on scintillation reduction. In UWOCs without pointing errors, tilted beam array $\left ( {{r}_{0}}=0 \right )$ and convergent beam array $\left ( 0<{{r}_{0}}<{{R}_{0}} \right )$ UWOC systems have smaller scintillation than parallel beam array $\left ( {{r}_{0}}={{R}_{0}} \right )$ and single beam UWOC systems. The parameters ${{R}_{0}}$ is the radius of the ring which is located in the transmitter transverse plane and consists of beams in the transmitter transverse plane. The parameter ${{r}_{0}}$ is the radius of the ring which is located in the receiver transverse plane and consists of target points of the beams in a beam array. In particular, in the UWOC with pointing errors ${{r}_{PE}}$, compared with tilted beam array, the convergent beam array $\left ( {{r}_{0}}\approx {{r}_{PE}} \right )$ is able to effectively reduce scintillation induced by oceanic turbulence. The results obtained in this paper may be helpful in the investigation of scintillation reduction and design of UWOC systems.