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Transmission characteristics of Gaussian array beams in seawater-to-air propagating incorporating turbulence media and foam layer

Xing Du, Hao Du, Zhenyang Xiong, Guixuan Ding, Hui Feng, Sheng Wang, Qiang Liu, and Zhongzhen Jia

Open Access Open Access

Abstract

This paper investigates the propagation of Gaussian array beams (GABs) through seawater-to-air in the presence of oceanic turbulence, atmospheric turbulence, and wave foams. Specifically, we focus on the intensity distribution of diverse typical GAB structures (ring, multi-ring, and rectangle). Then, an innovative intensity analysis model to calculate the average intensity in each medium is proposed. Moreover, we experimentally verify the proposed method by examining the intensity fading characteristic of Gaussian beams in the seawater-to-air path. Our results show that the peak intensity is primarily affected by the refraction in the ocean and foam layer, rather than air layer. The difference of theoretical and experimental values are less than 0.13 for the peak intensity. Moreover, the intensity distributions are more significantly affected by ocean turbulence but less influenced by wind speed.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

In recent decades, laser communication has gained significant attention due to its potential values in ocean prospecting. However, such communication is limited by underwater environment, and overcoming these obstacles has become a widespread concern [1,2]. In practical maritime laser communication systems, as depicted in Fig. 1, the laser beams often transmit from unmanned underwater vehicle (UUVs) to unmanned aerial vehicles (UAUs) through the seawater-to-air link.

 figure: Fig. 1.

Fig. 1. GABs transmission from seawater-to-air.

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To overcome the disadvantages of individual beam, such as quality, distance, and coverage area, research on multi-beams has attracted widespread attention [3–11]. As a result, Gaussian array beams (GABs) have become one of the research highlights in underwater optical communication applications [4,5].

Recently, a massive number of studies on the GABs propagating in turbulent medium were carried out. In particular, Ji investigated the area of coherent beam propagation through non-Kolmogorov turbulence [12,13], and Tang gave the region of linear Gaussian array beams in ocean turbulence [14]. Moreover, Lu derived the expression for average intensity of M × N rectangular GABs in oceanic turbulence [15]. And, the exploration of the diverse beam structures (include T-type and Y-type structures) performance characteristics have also attracted attention [16–20]. Additionally, Yousefi analyzed the average intensity distribution of partially coherent flat-topped laser beam in oceanic turbulence [21]. Furthermore, Mao derived analytical formulas for the average intensity of the multi-radial Gaussian-Schell-model array beams in oceanic turbulence [22]. However, all the above studies focused on the average intensity analysis of a specific construction of the GABs, and a unified analysis method for the average intensity of the general GABs is still lacking.

To feed the needs of seawater-to-air laser communication, a valid framework for studying the beam propagation properties is of great significance. To do that, we display the GABs transmission from seawater-to-air in Fig. 1. In fact, the propagating characteristics of GABs are determined by beam layouts, oceanic turbulence, atmospheric turbulence and ocean waves [ 23]. Considering the afore-mentioned problems in the seawater-to-air propagation path, calculating the average intensity remains an open problem. And, to the best of our knowledge, the framework for intensity calculation that considers both turbulent mediums and oceanic waves has not been reported yet. To build the above-mentioned framework, there is a need to study the GABs propagation properties on the seawater-to-air path. However, to our knowledge, there have been few studies on the beams propagation properties that consider both turbulent mediums and foam layer in the seawater-to-air path.

In this research, aiming to the three representative beam arrays, an analytical model of GABs propagating through seawater-to-air is established. The properties of GABs in seawater-to-air propagation path are studied based on the Rytov approximation and extended Huygens-Fresnel principle. We believe that the results will be helpful in the design of the optical communication systems through seawater-to-air.

The rest of this study is organized as follows. In Section 2, the analytical expressions of the average intensity are given. The numerical results are analyzed in Section 3 and conclusions are drawn in Section 4.

2. Seawater-to-air analytical expressions

The GABs propagating through seawater-to-air are assumed to be represented as Fig. 2. As displayed in Fig. 2(A), the GABs propagate from seawater-to-air. And, Fig. 2(B) shows the GABs propagating in xoz plane; Fig. 2(C) presents the light field propagating in ocean-air turbulence. Here, the oceanic turbulence exists on the propagation path of EF and FG, the intensity attenuation at the seawater-air interface and foam layer is taken into consideration on the path GH; the atmospheric turbulence exists on the path HI and IJ.

 figure: Fig. 2.

Fig. 2. Diagram of GABs transmitted through seawater-to-air propagation path.

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Figure 2(D)(a) represents the random distribution of GABs, Fig. 2(D)(b)-2D(d) are three topical layouts of GABs. Figure 2(D)(e) and Fig. 2(D)(f) are dual-beam type and T-type of GABs. Figure 2(E) illustrates the experimental setup in our study. Here, RRing represents the radius of the Ring beams, R1, R2 is the radius of inner and outer circles in the Multi-ring layout, respectively. And RRectangleis the radius of envelope circle of the Rectangle beams. H1 is the height of oceanic turbulence layer, H2 is the height of seawater-to-air composite layer, h is the wave height, d0 is the thickness of foam layer. L is the distance between field source plane and seawater-air interface.

2.1 Light field propagating in ocean-air turbulence

It can be shown from Fig. 2(C) that the light field in receptive plane is affected by refractive index fluctuations in the turbulent medium (ocean and air). Then, the expression for the average intensity of GABs for the general case can be obtained based on the Huygens-Fresnel principle and Rytov approximation.

According to Rytov approximation [24], we rewrite the light field in turbulent mediums as

$${E^{(\wp )}} = \left\{ \begin{array}{l} {E^{(FS)}}{F_1},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{oceanic}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{turbulence}\\ {E^{(FS)}}{F_1}{G_1},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{atmospheric}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{turbulence} \end{array} \right.$$
where F1, G1 denote the first-order complex phase perturbations of oceanic and atmospheric turbulences, respectively. E(℘) is the light field in turbulent mediums and E(FS) is the light field in free space.

Using the Huygens-Fresnel principle, we can obtain the field intensity at the point (x, y, z) in free space.

$${E^{(FS)}}({{\boldsymbol \rho }_a},z) ={-} \frac{{ikexp (ikz)}}{{2\pi z}}\int\!\!\!\int {{d^2}{{\boldsymbol \rho }_0}{E_0}({{\boldsymbol \rho }_0},0)} \times \exp \left[ {\frac{{ik}}{{2z}}{{({{\boldsymbol \rho }_a} - {{\boldsymbol \rho }_0})}^2}} \right]$$
where k is the wave number, E0 (ρ0,0) denotes the field intensity of the point (ρ0,0) and (ρ0,0) = (x0, y0, 0) in the field source plane.

Due to the complex phase F1 and G1 are related to spatial coordinates, the complex phase perturbations can be written as

$${F_1} = \exp ({\psi ^{(o)}}({{\boldsymbol \rho }_a},{{\boldsymbol \rho }_1},z))$$
$${G_1} = \exp ({\psi ^{(a)}}({{\boldsymbol \rho }_a},{{\boldsymbol \rho }_1},z))$$
where, ρa = (x, y), ρ1 = (ρx1, ρy1), ρ2 = (ρx2, ρy2).

Here, the cross-spectral density function in the ocean-air turbulence can be represented as

$$W({{\boldsymbol \rho }_{a1}},{{\boldsymbol \rho }_{a2}},z) = {E^{(\wp )}}({{\boldsymbol \rho }_{a1}},z){E^{(\wp )}}^\ast ({{\boldsymbol \rho }_{a2}},z)$$
where, ρa1 = (x1, y1), ρa2 = (x2, y2).

Thus, we can obtain the average intensity by substituting ρa1 = ρa2 = ρ into Eq. (5).

$${I^{(o - a)}}({\boldsymbol \rho },z) = W({\boldsymbol \rho },{\boldsymbol \rho },z)$$

2.2 Gaussian array beams propagating through ocean-air turbulence

2.2.1 Gaussian array beams propagating through ocean-air turbulence

Moreover, GABs for random distribution case are shown in Fig. 2(D)(a), three topical structures of GABs are shown Fig. 2(D)(b)—2D(d). The cross-spectral density function of the GABs for general case in source plane (z = 0) can be written as [15]

$${W^{(0)}}({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},0} )= \mathop {\mathop \sum \limits^N }\limits_{i = 1} {\mkern 1mu} \mathop {\mathop \sum \limits^N }\limits_{j = 1} {\mkern 1mu} \exp \left[ { - \frac{{{{({{{\boldsymbol \rho }_1} - {{\boldsymbol r}_i}} )}^2} + {{({{{\boldsymbol \rho }_2} - {{\boldsymbol r}_j}} )}^2}}}{{w_0^2}}} \right], $$
where w0is the waist width of GABs when z = 0, ρ1 = (ρx1, ρy1), ρ2 = (ρx2, ρy2), ri = (xi, yi), rj = (xj, yj). Here, the cross-spectral density function (CSDF) of diverse array structures can be derived by specifying the coordinates of the central points of each beam. For instance, the CSDF of ring type can be obtained by assuming xi = Rcosθi, yi = Rsinθi, xj = Rcosθj, yj = Rsinθj. and the T-type, Y-type, and dual-beam type can also be obtained by the coordinates transform.

Based on the Eqs. (1)–(7), for the case of GABs propagating through oceanic and atmospheric turbulences, the average intensity of GABs is written as

$${I^{(o - a)}}({\boldsymbol \rho },z) = \left\{ \begin{array}{l} {(\frac{k}{{2\pi z}})^2}\;\int\!\!\!\int {{d^2}{{\boldsymbol \rho }_1}\int\!\!\!\int {{d^2}{{\boldsymbol \rho }_2}} } {W^{(0)}}({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},0} )\times \;exp \left\{ {\frac{{ik}}{{2z}}[{{{({{\boldsymbol \rho } - {{\boldsymbol \rho }_1}} )}^2} - {{({{\boldsymbol \rho } - {{\boldsymbol \rho }_2}} )}^2}} ]} \right\}\\ \times \;{\left\langle {\exp [{{\psi^{(ocean)}}^ \ast ({{\boldsymbol \rho },{{\boldsymbol \rho }_1},z} )+ {\psi^{(ocean)}}({{\boldsymbol \rho },{{\boldsymbol \rho }_2},z} )} ]} \right\rangle_m}{\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} z \le L\\ \\ {(\frac{k}{{2\pi z}})^2}\;\int\!\!\!\int {{d^2}{{\boldsymbol \rho }_1}\int\!\!\!\int {{d^2}{{\boldsymbol \rho }_2}} } {W^{(0)}}({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2},0} )\times \;exp \left\{ {\frac{{ik}}{{2z}}[{{{({{\boldsymbol \rho } - {{\boldsymbol \rho }_1}} )}^2} - {{({{\boldsymbol \rho } - {{\boldsymbol \rho }_2}} )}^2}} ]} \right\}\\ \times \;{\left\langle {\exp [{{\psi^{(ocean)}}^ \ast ({{\boldsymbol \rho },{{\boldsymbol \rho }_1},L} )+ {\psi^{(ocean)}}({{\boldsymbol \rho },{{\boldsymbol \rho }_2},L} )} ]} \right\rangle_m}\\ \times \;{\left\langle {\exp [{{\psi^{(air)}}^ \ast ({{\boldsymbol \rho },{{\boldsymbol \rho }_1},z - L} )+ {\psi^{(air)}}({{\boldsymbol \rho },{{\boldsymbol \rho }_2},z - L} )} ]} \right\rangle_m}{\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} z > L{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \end{array} \right.$$
where ρ = (x, y), <…>m represents the average over the ensemble of statistical realizations of the turbulent medium, which can be expressed as [25]
$$\begin{aligned}&\exp {\left\langle {\left[ {\psi ^{{(medium)}*} \left( {{\boldsymbol \rho },{\boldsymbol \rho }_1,u_\alpha } \right) + \psi ^{(medium)}\left( {{\boldsymbol \rho },{\boldsymbol \rho }_2,u_\alpha } \right)} \right]} \right\rangle }_m \\ &= \exp \left\{ {-4\pi ^2k^2u_\alpha \int_0^1 {\int_0^\infty \kappa} {\rm \Phi }_{_n }^{medium} (\kappa )\left[ {1-J_0\left( {\kappa \xi \left| {{\boldsymbol \rho }_1-{\boldsymbol \rho }_2} \right|} \right){\rm d}\kappa {\rm d}\xi } \right]} \right\} \end{aligned}$$
where J0 (•) is the Bessel function of the first kind and order zero, (uα = L, z–L), $\mathrm{\Phi }_{_n}^{medium}(\kappa )$ is the spatial power spectrum of the refractive-index fluctuations in the turbulent mediums (ocean and air), κ is the magnitude of spatial wave number [25]. When the turbulence medium is oceanic or atmospheric turbulence, $\mathrm{\Phi }_{_n}^{medium}(\kappa )$ can be represented as
$$\mathrm{\Phi }_n^{ocean}(\kappa ) = 0.388 \times {10^{ - 8}}{\varepsilon ^{ - 1/3}}{\kappa ^{ - 11/3}} \times [{1 + 2.35{{(\kappa \eta )}^{2/3}}} ]\frac{{{\chi _T}}}{{{w^2}}}({{w^2}{e^{ - {A_T}\delta }} + {e^{ - {A_S}\delta }} - 2w{e^{ - {A_{TS}}\delta }}} )$$
$$\mathrm{\Phi }_n^{air}(\kappa ) = 0.033C_n^2{\kappa ^{ - 11/3}}\exp [{ - ({{\kappa^2}/\kappa_m^2} )} ], $$
where ε is the rate of dissipation of turbulent kinetic energy per unit mass of fluid, η is the Kolmogorov microscale (inner scale), χT denotes the rate of dissipation of mean-square temperature, and w is the parameter related to temperature and salinity, AT = 1.863 × 10−2, AS = 1.9 × 10−4, ATS = 9.41 × 10−3, and δ = 8.284 (κη) 4/3 + 12.978 (κη)2 [26]. And $C_n^2$ is the refractive-index structure constant of atmospheric turbulence, take the value of 10−14 m-2/3, κm = 5.92/l0, with l0 the atmospheric turbulence inner scale, the typical value of l0 = 0.01 m is taken in the numerical calculations [27].

When κξ|ρ1 − ρ2| << 1, J0 (κξ|ρ1 − ρ2|) can be approximated to the first two terms in the power series expansion [15], i.e.,

$${J_0}({\kappa \xi |{{{\boldsymbol \rho }_1} - {{\boldsymbol \rho }_2}} |} )\approx 1 - \frac{1}{4}{(\kappa \xi |{{{\boldsymbol \rho }_1} - {{\boldsymbol \rho }_2}} |)^2}, $$

Upon substituting from Eq. (10)–(12) into Eq. (9), and assuming

$${\boldsymbol I}_{_\alpha }^{^{(o - a)}} = \left\{ \begin{array}{l} - \frac{{{\pi^2}{k^2}z}}{3}\int_0^\infty {{\kappa^3}\mathrm{\Phi }_n^{ocean}(\kappa )} \textrm{d}\kappa ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} z \le L\\ - \frac{{{\pi^2}{k^2}}}{3}\int_0^\infty {{\kappa^3}\textrm{[}L\mathrm{\Phi }_n^{ocean}(\kappa ) + (z - L)\mathrm{\Phi }_n^{air}(\kappa )]} \textrm{d}\kappa ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} z > L \end{array} \right.$$

Then, we rewrite Eq. (8)

$$\begin{array}{l} {I^{(o - a)}}({\boldsymbol \rho },z) = {(\frac{k}{{2\pi z}})^2}\;\int\!\!\!\int {{d^2}{{\boldsymbol \rho }_1}\int\!\!\!\int {{d^2}{{\boldsymbol \rho }_2}} } \mathop {\mathop \sum \limits^N }\limits_{m = 1} {\mkern 1mu} \mathop {\mathop \sum \limits^N }\limits_{n = 1} {\mkern 1mu} \exp \left[ { - \frac{{{{({{{\boldsymbol \rho }_1} - {{\boldsymbol r}_m}} )}^2} + {{({{{\boldsymbol \rho }_2} - {{\boldsymbol r}_n}} )}^2}}}{{w_0^2}}} \right]\\ \times \;exp \left\{ {\frac{{ik}}{{2z}}[{{{({{\boldsymbol \rho } - {{\boldsymbol \rho }_1}} )}^2} - {{({{\boldsymbol \rho } - {{\boldsymbol \rho }_2}} )}^2}} ]} \right\} \times \;exp \{{{\boldsymbol I}_{_\alpha }^{(o - a)}{{|{{{\boldsymbol \rho }_1} - {{\boldsymbol \rho }_2}} |}^2}} \}\end{array}. $$

Introducing two variables of integration ${\boldsymbol S} = ({{\boldsymbol \rho }_1} + {{\boldsymbol \rho }_2})/2,\;{\boldsymbol T} = {{\boldsymbol \rho }_2} - {{\boldsymbol \rho }_1}$, where S = (sx, sy), T = (tx, ty), the formula can be rewritten as

$$\begin{array}{l} {I^{(o - a)}}({\boldsymbol \rho },z) = {(\frac{k}{{2\pi z}})^2}\;\mathop {\mathop \sum \limits^N }\limits_{m = 1} {\mkern 1mu} \mathop {\mathop \sum \limits^N }\limits_{n = 1} {\mkern 1mu} \int\!\!\!\int {\int\!\!\!\int {d{s_x}d{t_x}d{s_y}d{t_y}} } \\ \times \exp \left[ { - \frac{{2s_x^2 + t_x^2/2 - 2({x_m} + {x_n}){s_x} + ({x_m} - {x_n}){t_x} + (x_m^2 + x_n^2)}}{{w_0^2}}} \right]\\ \times \exp \left[ { - \frac{{2s_y^2 + t_y^2/2 - 2({y_m} + {y_n}){s_y} + ({y_m} - {y_n}){t_y} + (y_m^2 + y_n^2)}}{{w_0^2}}} \right]\\ \times \exp \left\{ {\frac{{ik}}{z}(x{t_x} - {s_x}{t_x} + y{t_y} - {s_y}{t_y})} \right\} \times \;exp \{{{\boldsymbol I}_{_\alpha }^{(o - a)}(t_x^2 + t_y^2)} \}\end{array}. $$

According to the integral formula

$$\int {\exp ( - {p^2}{x^2} + qx)} dx = \frac{{\sqrt \pi }}{p}\exp (\frac{{{q^2}}}{{4{p^2}}}), $$
and after tedious integral calculations, the average intensity can be written as
$${I^{(o - a)}}({\boldsymbol \rho },z)\textrm{ = }\sum\limits_{m = 1}^N {\sum\limits_{n = 1}^N {\prod\limits_{\vartheta = x,y} {\frac{{\exp ({{A_\vartheta }{\vartheta^2} + {B_\vartheta }\vartheta + {C_\vartheta }} )}}{{{D_\vartheta }}}} } }, $$
$${A_\vartheta } ={-} \frac{{2\alpha _{\vartheta 3}^2}}{{w_0^2\alpha _{\vartheta 3}^2 + \frac{1}{{w_0^2}} + 2{\boldsymbol I}_{_\alpha }^{(o - a)}}}$$
$${B_\vartheta } ={-} \frac{{{\alpha _{\vartheta 0}}\alpha _{\vartheta 3}^2}}{{w_0^2\alpha _{\vartheta 3}^2 + \frac{1}{{w_0^2}} + 2{\boldsymbol I}_{_\alpha }^{(o - air)}}} - \frac{{i{\alpha _{\vartheta 1}}{\alpha _{\vartheta 3}}}}{{\frac{{w_0^4\alpha _{\vartheta 3}^2}}{2} + {\boldsymbol I}_{_\alpha }^{(o - a)}w_0^2 + \frac{1}{2}}}$$
$${C_\vartheta } ={-} \frac{{\alpha _{\vartheta 0}^2\alpha _{\vartheta 3}^2}}{{16\beta _\vartheta ^2}} + \frac{{\alpha _{\vartheta 0}^2}}{{8w_0^2}} - \frac{{i{\alpha _{\vartheta 0}}{\alpha _{\vartheta 1}}{\alpha _{\vartheta 3}}}}{{4w_0^2\beta _\vartheta ^2}} + \frac{{\alpha _{\vartheta 1}^2}}{{4w_0^4\beta _\vartheta ^2}} - \frac{{{\alpha _{\vartheta 2}}}}{{w_0^2}}$$
$${D_\vartheta } = {{\sqrt {\alpha _{\vartheta 3}^2 + \frac{1}{{w_0^4}} + \frac{2}{{w_0^2}}{\boldsymbol I}_{_\alpha }^{(o - a)}} } / {\left( {\frac{k}{{2z}}} \right)}}$$
$${\beta _\vartheta } = \sqrt {\frac{{w_0^2\alpha _{\vartheta 3}^2}}{2} + \frac{1}{{2w_0^2}} + {\boldsymbol I}_{_\alpha }^{(o - a)}}$$
$$\alpha _{\vartheta 0} = -2\left( {\vartheta _m + \vartheta _n} \right),\,\alpha _{\vartheta 1} = \vartheta _m-\vartheta _n,\,\alpha _{\vartheta 2} = m\vartheta _m + n\vartheta _n,\,\alpha _{\vartheta 3} = \displaystyle{k \over {2z}}$$

2.2.2 Probability model of GABs propagating through seawater-to-air

The intensity attenuation caused by the foam layer on sea surface and rough seawater-air interface is a very vital issue. We consider the transmittance of GABs propagating through the seawater-air interface and the foam layer, which is denoted as L(2)w and L(3)w, respectively [28,29]. And we assume that the temperature difference of air-seawater is moderate and the atmosphere is stable, the foam coverage is denoted as C.

The foam coverage C is defined as: On a defined area Ω, the ratio of wave superficial area with foam covered Ac to total wave superficial area AΩ, which is represented as C = Ac/AΩ.

We assume that the event F denotes the beams encounter foams when propagating through the foam layer, while the $\overline F$ is the inverse events of F. According to the definition of foam coverage, we can obtain the probability of event F and $\overline F$.

$$P_F = \displaystyle{{A_c} \over {A_\Omega }} = C,\,P_{\overline F } = 1-\displaystyle{{A_c} \over {A_\Omega }} = 1-C$$

By using the total probability theorem, we can obtain the modified transmittance L(3)w

$${L^{(3)}}_w = C{L^{(3)^{\prime}}}_w + (1 - C){L^{(3)^{\prime\prime}}}_w$$
where, ${L^{(3)^{\prime}}}_w$ is the transmittance with foam covering when propagating through the foam layer, ${L^{(3)^{\prime\prime}}}_w$ is the transmittance without foam covering.

Based on the above-mentioned derivation, we can obtain the modified expressions of average intensities,

$$\left\{ \begin{array}{l} {I^{(FG)}}({\boldsymbol \rho },z)\; = {I^{(o - a)}}({\boldsymbol \rho },z){L^{(1)}}_w\\ {I^{(GH)}}({\boldsymbol \rho },z)\; = {I^{(o - a)}}({\boldsymbol \rho },{H_1} + h){L^{(2)}}_w\\ {I^{(HI)}}({\boldsymbol \rho },z)\; = {I^{(o - a)}}({\boldsymbol \rho },z){L^{(2)}}_w{L^{(3)}}_w \end{array} \right., $$

Here, we assume L(1)w = 1, L(2)w = 0.83, ${L^{(3)^{\prime}}}_w = 0.53$, ${L^{(3)^{\prime\prime}}}_w = 1$. And, I (FG)(ρ, z), I (GH)(ρ, z), and I(HI)(ρ, z) denote the average intensities in the propagating path FG, GH and HI respectively. And the foam coverage C = 2.32 × 10−6 $U_{10}^{3.4988}$[30,31], where U10 denotes the wind speed above the sea surface at the distance of 10 m, which is in the unit of m/s. I (2)(ρ, z) is represented as a constant cause the foam layer is assumed thin enough.

On the other hand, the ocean waves are random under normal conditions [30,31]. Then, we assume that the distribution of the wave height is the normal distribution. And the distribution function of the wave height can be written as

$$F(h) = \int\limits_0^h {\frac{1}{{\sqrt {2\pi } \sigma }}{e^{ - \frac{{{{(h - \mu )}^2}}}{{2{\sigma ^2}}}}}dh}. $$
where µ and σ represent the average and variance of the normal distribution, respectively. In this paper, µ and σ take the value of 5 m and 1 m2. To fit the physical model, we assume that the lower limit of the integral is zero, here $h \in [{0,{H_2}} ]$.

Figure 2(A) presents the GABs propagate through the rough surface of ocean wave with foams covered. We assume that the beams are in the seawater, foam, and air sub-layers (corresponding to FG, GH and HI, respectively, in Fig. 2(B)) are events A, B, and C, respectively. And the probability of events A, B, and C can be written as:

$${P_A}\textrm{ = }P\{{{I^\mathrm{\ast }}({\boldsymbol \rho },z) = {I^{(FG)}}({\boldsymbol \rho },z)} \}\textrm{ = }P\{{{H_\textrm{1}} < z < {H_1} + h} \}\textrm{ = }F(z) - F(z - h), $$

Similarly, we can obtain

$${P_B}\textrm{ = }P\{{{I^\mathrm{\ast }}({\boldsymbol \rho },z) = {I^{(GH)}}({\boldsymbol \rho },z)} \}\;\textrm{ = }F(z - h) - F(z - h - {d_0}), $$
$${P_C}\textrm{ = }P\{{{I^\mathrm{\ast }}({\boldsymbol \rho },z) = {I^{(HI)}}({\boldsymbol \rho },z)} \}\;\textrm{ = }F(z - {H_2}) - F(z - h - {d_0}). $$

The average intensity ${I^\mathrm{\ast }}({\boldsymbol \rho },z)$ in the middle layer is a random variable due to the wave height is random. Then, we can obtain mean average intensity $\overline {{I^\mathrm{\ast }}({\boldsymbol \rho },z)}$, according to the total probability theorem,

$$\overline {{I^\mathrm{\ast }}({\boldsymbol \rho },z)} = {P_A}{I^{(FG)}}({\boldsymbol \rho },z) + {P_B}{I^{(GH)}}({\boldsymbol \rho },z) + {P_C}{I^{(HI)}}({\boldsymbol \rho },z). $$

According to the above derivation, we can obtain the average intensity I (ρ, z) that is suitable for all the propagation path (EF, FG, GH, HI, IJ).

$$I({\boldsymbol \rho },z) = \left\{ \begin{array}{l} {I^{(o - a)}}({\boldsymbol \rho },z),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \le z < {H_1}\\ \overline {{I^\mathrm{\ast }}({\boldsymbol \rho },z)} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {H_1} \le z < {H_1} + {H_2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \\ {I^{(o - a)}}({\boldsymbol \rho },z){L^{(2)}}_w{L^{(3)}}_w,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {H_1} + {H_2} \le z < {H_1} + {H_2} + {H_3} \end{array} \right.$$

The numerical simulations in Section 3 are all based on Eq. (26).

3. Numerical results and experiment investigation

In this section, the peak intensity and average intensity distribution of the three representative GABs are evaluated in numerical simulations and experiment investigations. And the propagation properties of the three topical GABs are analyzed comparatively.

3.1 Numerical simulations

The numerical simulations of the peak intensity and average intensity are illustrated with the following parameter values from [14]. Unless there is a specified explanation in captions of figures, the parameters of a coherent Gaussian beam and oceanic turbulence are set as λ = 1.06 µm, w0 = 5 mm, NRing = NMulti-ring = NRectangle = 16, RRing = RMulti-ring = RRectangle = 0.03 m, µ = 5 m, σ = 1 m2, η = 1 mm, ε = 10−6 m2/s3, w = −2.5, and χT = 10−7 K2/s. H1 = 50 m, H2 = 10 m, H3 = 140 m, d0 = 0.02 m.

3.1.1 Numerical simulations for peak intensity

To compare the performance of the three structures, the peak intensity versus the transmission distance z under different wind speed U10 is drawn in Fig. 3. Moreover, the peak intensity of each structure increases as the wind speed decreases. And it can be shown from the diagram that the rectangular array presents the best performance in the range 0–80 m, while the multi-annular array beams perform best within the range 100–200 m. We can indicate that rectangle array beams are more strongly superimposed at short distances and more uniform. Thus, the peak intensity of the rectangle structure is larger, relative to the other two structures (Ring and Multi-ring), in the near distance. Meanwhile, turbulence exerts a more pronounced effect on the rectangular configuration, dissipating a greater amount of energy compared to other two structures. Therefore, the faster decline in performance at longer distances is attributed to the more increased cripple of turbulence on the rectangle structure compared with other two structures (Ring and Multi-ring), leading to a more pronounced weakening of the peak intensity. The results may be used in the modulation processes for communication applications.

 figure: Fig. 3.

Fig. 3. Schematic of GABs versus distance z with different wind speed (U10 = 39, 35, 31, and 21 m/s).

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In the range 50–100 m, the performance advantage of the rectangular array reduces with the propagation distance increases. And the peak intensity rectangular array shows a larger decrease rate than that in the range 0–20 m. Therefore, the effective transmission distance is more sensitive to the refraction attenuation in the middle layer than the turbulence attenuation in the atmospheric layer. It can be concluded that the peak intensity is reduced by the refraction of the seawater-to-air interface and the foam layer, as well as the oceanic turbulence.

We present the peak intensity versus the propagation distance in Fig. 4. We can see from Figs. 4(a)–4(c) that under χT = 10−4 K2/s the three beam structures have similar peak intensity values. When the parameter χT reduces to 10−7 K2/s, the rectangular array presents best performance in the short-distance propagation (0–80 m), and the multi-ring array performs best in the range 80–150 m. When the parameter χT reduces to 10−10 K2/s, the rectangular array delivers better performance than the other structures. It can also be found from Figs. 4(d)–4(f) that under ε = 10−3 m2/s3 the rectangular array performs best in short distance (0–140 m); when the parameter ε decreases, the multi-ring array presents the best performance in middle distance (75–125 m). Figures 4(g)–4(i) show the peak intensity of three array beams versus transmission distance z. The rectangular structure performs best in the short distance (0–15 m) when w = −0.1. Within the scope of 50–200 m, the difference in the three array beams’ intensities is slight, and with decreasing salinity w the drop ratio of the peak intensity decreases with transmission distance drop; in other words, the curve is flatter. All in all, during GABs propagation in seawater-to-air path, the peak intensity declines due to the refraction at the seawater-air interface. In addition, the turbulent medium brings out the fluctuation of refractive index, which is also a primary reason for the peak intensity attenuation.

 figure: Fig. 4.

Fig. 4. Schematic of GABs versus distance z with different turbulence parameter χT, ε, and w.

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3.1.2 Numerical simulations for average intensity

We then study the distribution of average intensity of rectangular beam due to its best performance. Figure 5(A) shows the propagating behavior of rectangular GABs structure (3D distributions and corresponding contour plot) at different propagation distances. We can find from Fig. 5(A) that the rectangular GABs structures will gradually change their shapes to that of the quasi-Gaussian beam as the propagation distance z increases. However, with the propagation distance increasing, the GABs spots spread and overlap with interference and produce one new Gaussian beam spot. At a short distance (z = 50 m), the bright spots lie in the XY plane rectangularly. At a middle distance (z = 150 m), the individual bright spots disappear and a new rectangular bright spot appears. At a far enough distance (z = 200 m), the rectangular structure GABs and single Gaussian beam show similar distribution.

 figure: Fig. 5.

Fig. 5. 2D and 3D distributions of average intensity of rectangular, T-type, and 1 × 2-type GABs propagating through seawater-to-air path at different propagation path lengths.

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Considering the comprehensiveness to be studied, we present the distribution simulation analysis of the average intensity of the T-shaped structure (Fig. 5(B)) and dual-beam (Fig. 5(C)). Figure 5(B), (C) shows the propagation behavior of T and dual-beam (3D distribution and corresponding contour map) at different propagation distances.

In Fig. 5 (B), it can be conclude that the T-shaped structure undergoes gradual transformations as the propagation distance (z) increases. However, as the propagation distance increases, leading to interference and the emergence of novel evolution. Proximal to the source (z = 50 m), the bright spot is situated within the XY plane. At intermediate distances (z = 150-250 m), the original singular luminous focus diminishes, giving rise to irregular bright spots. At a considerable propagation distance (z = 300 m), the distribution of the GAB and single Gaussian beams within the T-shaped structure aligns, although this distance exceeds that observed for the rectangular structure. Compared with the rectangular GABs, the evolution distance of T-type is longer. However, when considering the pointing error or spot drift, the rectangular beam exhibits a more uniform distribution and demonstrates superior performance.

The transformations of the dual-beam configuration are evident in Fig. 5(C), exhibiting gradual modifications with the ascending propagation distance. Concurrently, an augmentation in the propagation distance results in the dispersion and overlap of dual-beam, engendering novel evolutionary patterns. Specifically, at relatively short distances (z = 50 m), the luminous is situated within the XY plane. Upon reaching intermediate distances (z = 150 m), the original luminous focus dissipates, giving way to the emergence of a distinct focal point. the dual-beam and single Gaussian beams show a similar distribution, but this distance is farther than the rectangular structure. However, in the contemplation of tracking error or spot drift, the performance of the dual-beam type falls short when compared with rectangular and T-type arrays. Nonetheless, its merits lie in a simplistic structure that facilitates easy implementation.

Figure 6 presents the propagating behavior of the rectangular GABs structure (3D distributions and corresponding contour plot) with different rates of dissipation of mean-square temperature χT. The average intensity increases as χT decreases. For the smaller value of χT [i.e., Fig. 6(d)], the bright spots lie in the XY plane rectangularly. For the middle value of χT [i.e., Fig. 6(c)], the bright spots disappear and a new bright spot emerges in the XY plane. For the larger value of χT [i.e., Fig. 6(b)], the average intensity profile is similar to the Gaussian distribution. It can be concluded that the average intensity distribution becomes a Gaussian-like distribution from the initial multiple independent Gaussian distributions as the value of χT goes from small to large.

 figure: Fig. 6.

Fig. 6. 2D and 3D intensity distributions of rectangular structure GABs at z = 150 m with different values of χT (χT = 10−4 K2/s, 10−6 K2/s, 10−7 K2/s, and 10−10 K2/s, from left to right).

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Figure 7 illustrates the propagating behavior of the rectangular GABs structure (3D distributions and corresponding contour plot) with different temperature and salinity related parameters w. When the smaller value of w is obtained [i.e., Figs. 7(c) and (d)], the bright spots lie in the XY plane rectangularly. When the middle value of w is obtained [i.e., Fig. 7(b)], the bright spots disappear and a new bright spot emerges in the XY plane. When the larger value of w is obtained [i.e., Fig. 7(a)], the average intensity profile is similar to the Gaussian distribution. In addition, the average intensity increases as the parameter w of the rectangular GABs structure decreases. Here, w denotes the ratio of temperature to salinity contribution to the refractive index spectrum, and thus it can be concluded that the average intensity distribution is more influenced by ocean turbulence with salinity-induced parameters than with temperature-induced ones.

 figure: Fig. 7.

Fig. 7. 2D and 3D intensity distributions of rectangular structure GABs at z = 150 m with different values of w (w = −0.1, −1.5, −2.5, and −4.9, from left to right).

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Figure 8 draws the propagating behavior of the rectangular GABs structure (3D distributions and corresponding contour plot) with different rates of dissipation of turbulent kinetic energy ε. As can be seen from the Fig. 8, the Gaussian beams propagating in oceanic turbulence can keep their initial shapes at the larger rate of dissipation of turbulent kinetic energy ε [Fig. 8(a)]. As the rate of dissipation of turbulent kinetic energy ε increases, the beams will lose its initial bright spots and evolve into a Gaussian-like beam.

 figure: Fig. 8.

Fig. 8. 2D and 3D intensity distributions of rectangular structure GABs at z = 150 m with different values of ε (ε = 10−3 m2/s3, 10−5 m2/s3, 10−7 m2/s3, and 10−9 m2/s3, from left to right).

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In general, the intensity distribution of GABs transmitting through seawater-to-air is more affected by the oceanic turbulence rather than atmospheric turbulence. Moreover, the atmospheric turbulence tends to be a less important factor of intensity reduction at a far distance. Additionally, the physics reason of the evolution behavior may be that the oceanic turbulence increases the variance of probability distribution of photons in the beam.

3.2 Experiment investigation

3.2.1 Experimental setup

Figure 9 and Fig. 2(E) show the experimental setup for detecting the light intensity in receive plane. As displayed in Fig. 9, the experimental setup is implemented in a transparent water tank with dimensions 30 × 80 × 150 cm3. In the this section, the selection of dual-beam (as one of the focal point of our investigation) serves as a means to effectively validate our proposed model. At the transmitting side, the 1 × 2 beam array is consisted of two 525 green laser diodes with maximum output power of 2 W to ensure a constant initial power. To simulate the underwater turbulence, two underwater motors is used. In this experiment, the rotational speed of the motor was set to 300 r/min. And, the seawater crystalline salts are added to the water tank to simulate the liquid environment in the ocean. Additionally, we keep the temperature in the water tank constant by the temperature controller. Meanwhile, the laser beam profiler and optical power meter is employed at the receiver side to detect the intensity distribution and peak intensity. In this experiment, we first measure the intensity distribution after the laser beam transmits through the water-to-air path. Then we measure the peak intensity when the laser beam transmits through the water-to-air path. Further, the effect of turbulence and water-air interface on the intensity distribution and peak intensity are investigate.

 figure: Fig. 9.

Fig. 9. Experimental setup for the beam intensity variation in water-to-air path.

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3.2.2 Experimental results for peak intensity

As illustrated in Fig. 10, the red line represents the experimental value of peak intensity of the double laser beam transmitting through the water-to-air path, the blue line represents that of the single laser beam propagates in water-to-air path. In the water path, the peak intensity decreases for 0.37 and 0.55 when use single and double laser beam, respectively. And, in the water-to-air interface, the peak intensity decreases for 0.08 and 0.17, respectively. In the air path, the peak intensity decreases for 0.08 and 0.07, respectively. Accordingly, the black dotted line represents the simulated value of the double laser beam, the blue dotted line represents that of the single laser beam in water-to-air path. We can find that, as we expect, the peak intensity in all case decreases with increasing distance z. The theoretical and experimental values differ by 0.07 and 0.13 for the single and double beam, respectively. The peak light intensity of double beams is stronger than that of single beam both in theory and experiment. Moreover, the light intensity decays induced by the water is more than that by the air. And this experimental phenomenon fits with the downward trend in numerical results. It experimentally verifies the proposed theoretical model of the GABs propagating through seawater to air. Meanwhile, the results here are consistent with the trend in Fig. 5, further confirming the validity of the theoretical model.

 figure: Fig. 10.

Fig. 10. Schematic of peak intensity versus distance z.

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3.2.3 Experimental results for average intensity

As displayed in Fig. 11, 2D and 3D diagram of the average intensity at various propagation path length is shown. Here, we obtain the values of the intensity distribution of the receptive plane experimentally. And one can find that the light energy is less concentrated and the light spot is smaller when the laser beam transmits in water-to-air path, as shown in Fig. 11(2-f)–(2-h). Meanwhile, the energy is more concentrated when the laser beam transmits in air path, as shown in Fig. 11(2-i) and Fig. 11(2-j). In the atmosphere path, the light intensity is less than 0.4. The experimental results in Fig. 11 support the conclusion from Fig. 7. It can be concluded that the turbulences in water and atmosphere affect the variance of light intensity distribution. Moreover, the light intensity is attenuated by the refraction effect at the water-air interface.

 figure: Fig. 11.

Fig. 11. Schematic of average intensity versus propagation distance z. (z = 0.4, 0.8, 1.2, 1.6, 2.0 m, from left to right)

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4. Conclusions

In this study, the transmission characteristics of Gaussian array beams propagating through seawater-to-air are examined. In the mathematical model, the foam layer, the random ocean wave and three oceanic turbulence parameters are taken into consideration. Meanwhile, the properties of the intensity versus distance, wind speed, and turbulence parameters are investigated. Moreover, we validate this method by examining the intensity distribution of GABs in seawater-to-air path with comparison to experimental data. It can be concluded that the intensity is more influenced by the relative strength of temperature and salinity fluctuation and the rate of dissipation of mean-square temperature rather than the rate of dissipation of turbulent kinetic energy per unit mass of fluid. Moreover, the intensity decreases with the increase of wind speed. The rectangle GABs are with the largest peak intensity in oceanic turbulence layer while the three types of GABs present the similar peak intensity in atmospheric turbulence layer and middle layer. And, the T-shape GABs also exhibits potential practicality. Furthermore, the effective communication distance is related to the three oceanic turbulence factors and the beam structures as well as wind speed. Through experiment investigation, the consistency between experimental findings and simulation results has been verified, through the utilization of dual beams, affirming the accuracy of the theoretical model. In the future study, more attention should be paid to the oceanic wave modeling and seawater-to-air laser communication channel analysis.

Future work

As we move forward, our research will encompass a comprehensive exploration of various shapes, with a specific focus on the T-type and Y-type beam. Recognizing the considerable potential inherent in the T shape and Y shape, our forthcoming efforts will delve deeper into related content.

Funding

Natural Science Foundation of Hainan Province (622MS105); National Natural Science Foundation of China-Shandong Joint Fund (U23A20336).

Acknowledgement

We extend heartfelt appreciation for the tremendous assistance provided by David Doxaran and reviewers in theoretical framework and experimental set. Especially in the introduction of T and Y shape, which improve the broader prospective on the paper.

Disclosures

The authors declare no conflicts of interest.

Data availability

All data generated or analyzed during this study are included in this article.

References

1. S. He, N. Wang, M. Ho, et al., “Design of a new stress wave communication method for underwater communication,” IEEE Trans. Ind. Electron. 68(8), 7370–7379 (2021). [CrossRef]  

2. J. Xu, A. Lin, X. Yu, et al., “Underwater laser communication using an OFDM-modulated 520-nm laser diode,” IEEE Photon. Technol. Lett. 28(20), 2133–2136 (2016). [CrossRef]  

3. P. A. Nosov, D. E. Piskunov, and A. F. Shirankov, “Laser varifocal system synthesis for longitudinal Gaussian beam shifting,” Appl. Opt. 58(13), 3347–3353 (2019). [CrossRef]  

4. L. Lu, X. Ji, and X. Li, “Influence of oceanic turbulence on propagation characteristics of Gaussian array beams,” Optik (Munich, Ger.) 125(24), 7154–7161 (2014). [CrossRef]  

5. V. G. Arkhipkin and S. A. Myslivets, “Diffraction of vortex Gaussian beams from a two-dimensional Raman-induced grating,” Laser Phys. 31(6), 065401 (2021). [CrossRef]  

6. Y. Zhuang, Q. Yang, P. Wu, et al., “Vortex beam array generated by a volume compound fork grating in lithium niobite,” Results Phys. 24, 104083 (2021). [CrossRef]  

7. W. D. Bilida, J. D. Strohschein, and H. J. J. Seguin, “High-power 24-channel radial array slab rf-excited carbon dioxide laser,” Proc. SPIE 2987, 13–21 (1997). [CrossRef]  

8. H. J. Baker, D. R. Hall, and A. M. Hornby, “Propagation characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32(3), 400–407 (1996). [CrossRef]  

9. B. Lü and H. Ma, “Beam propagation properties of radial laser arrays,” J. Opt. Soc. Am. A 17(11), 2005–2009 (2000). [CrossRef]  

10. K. Cheng, L. Jiao, and X. Zhong, “A comparison of far-field properties of radial noncanonical vortex airy beam arrays and radial noncanonical vortex Gaussian beam arrays,” Opt Commun. 367, 112–122 (2016). [CrossRef]  

11. L. Lu, Z. Wang, and Y. Cai, “Propagation properties of phase-locked radially-polarized vector fields array in turbulent atmosphere,” Opt. Express 29(11), 16833–16844 (2021). [CrossRef]  

12. X. L. Ji and X. Q. Li, “Effective radius of curvature of partially coherent Hermite–Gaussian beams propagating through atmospheric turbulence,” J. Opt. 12(3), 035403 (2010). [CrossRef]  

13. X. L. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18(7), 6922 (2010). [CrossRef]  

14. M. M. Tang and D. M. Zhao, “Regions of spreading of Gaussian array beams propagating through oceanic turbulence,” Appl. Opt. 54(11), 3407 (2015). [CrossRef]  

15. L. Lu, Z. Wang, and J. Zhang, “Average intensity of M×N Gaussian array beams in oceanic turbulence,” Appl. Opt. 54(25), 7500–7507 (2015). [CrossRef]  

16. X. H. Shen, L. B. Kong, and J. T. An, “Beam shaping for two-dimensional laser array by a computational model,” Opt. Lasers. Eng. 169, 107755 (2023). [CrossRef]  

17. J. X. Zhang and J. H. Yuan, “Design and optimization of 1 × 2 and 1 × 4 Y-shaped beam splitters in GaSb-based 2D photonic crystals,” Mod. Phys. Lett B. 36(11), 2250009 (2022). [CrossRef]  

18. K. Ren and X. B. Ren, “Y-shaped beam splitter by graded structure design in a photonic crystal,” Chin. Sci. Bull. 57(11), 1241–1245 (2012). [CrossRef]  

19. W. X. Teo, W. W. Zhu, and J. Gong, “Tunable two-dimensional laser arrays with zero-phase locking,” Phys. Rev. B 105(20), L201402 (2022). [CrossRef]  

20. N Nossir, L Dalil-Essakali, and A Belafhal, “Comparative analysis of some laser beams spreading through oceanic turbulence,” Research Square, rs.3.rs-3750301/v1. 2023. [CrossRef]  

21. M. F. Yousefi, D. Kashani, and A. Mashal, “Analyzing the average intensity distribution and beam width evolution of phase-locked partially coherent radial flat-topped array laser beams in oceanic turbulence,” Laser Phys. 27(2), 026202 (2017). [CrossRef]  

22. Y. Mao, Z. Mei, and J. Gu, “Radial Gaussian-Schell-model array beams in oceanic turbulence,” Appl. Phys. B 123(4), 111–111.7 (2017). [CrossRef]  

23. X. C. Song, Y. G. Jia, S. Q. Wang, et al., “A modified straightforward spectral representation method for accurate and efficient simulation of the stationary non-Gaussian stochastic wave,” Ocean Eng. 205, 107308 (2020). [CrossRef]  

24. T. Shirai and T. Asakura, “Spectral changes of light induced by scattering from spatially random media under the Rytov approximation,” Opt. Soc. Am. A 12(6), 1354–1363 (1995). [CrossRef]  

25. D. Liu, H. Yin, and G. Wang, “Propagation of partially coherent Lorentz–Gauss vortex beam through oceanic turbulence[J],” Appl. Opt. 56(31), 8785–8792 (2017). [CrossRef]  

26. D. Liu and Y. Wang, “Average intensity of a Lorentz beam in oceanic turbulence,” Optik 144, 76–85 (2017). [CrossRef]  

27. X. L. Ji and Z. C. Pu, “Effective Rayleigh range of Gaussian array beams propagating through atmospheric turbulence,” Opt. Commun. 283(20), 3884–3890 (2010). [CrossRef]  

28. E. Q. Zhan and H. Y. Wang, “Calculation of energy attenuation of light propagation through atmosphere and seawater,” Engineering Journal of Wuhan University. 42009), (2009).

29. V. I. Mankovsky and V. I. Haltrin, “Light scattering phase functions measured in waters of Mediterranean Sea,” OCEANS ‘02 MTS 4, 2368–2373 (2002). [CrossRef]  

30. M. H. Smith, P. M. Park, and I. E. Consterdine, “Marine aerosol concentrations and estimated fluxes over the sea,” Q. J. R. Meteorol. Soc. 119(512), 809–824 (1993). [CrossRef]  

31. A Fournier and W T. Reeves, A simple model of ocean waves[C]//Proceedings of the 13th annual conference on Computer graphics and interactive techniques (1986), pp. 75–84.

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All data generated or analyzed during this study are included in this article.

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Figures (11)
Fig. 1.
Fig. 1. GABs transmission from seawater-to-air.

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Fig. 2.
Fig. 2. Diagram of GABs transmitted through seawater-to-air propagation path.

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Fig. 3.
Fig. 3. Schematic of GABs versus distance z with different wind speed (U10 = 39, 35, 31, and 21 m/s).

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Fig. 4.
Fig. 4. Schematic of GABs versus distance z with different turbulence parameter χT, ε, and w.

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Fig. 5.
Fig. 5. 2D and 3D distributions of average intensity of rectangular, T-type, and 1 × 2-type GABs propagating through seawater-to-air path at different propagation path lengths.

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Fig. 6.
Fig. 6. 2D and 3D intensity distributions of rectangular structure GABs at z = 150 m with different values of χT (χT = 10−4 K2/s, 10−6 K2/s, 10−7 K2/s, and 10−10 K2/s, from left to right).

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Fig. 7.
Fig. 7. 2D and 3D intensity distributions of rectangular structure GABs at z = 150 m with different values of w (w = −0.1, −1.5, −2.5, and −4.9, from left to right).

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Fig. 8.
Fig. 8. 2D and 3D intensity distributions of rectangular structure GABs at z = 150 m with different values of ε (ε = 10−3 m2/s3, 10−5 m2/s3, 10−7 m2/s3, and 10−9 m2/s3, from left to right).

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Fig. 9.
Fig. 9. Experimental setup for the beam intensity variation in water-to-air path.

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Fig. 10.
Fig. 10. Schematic of peak intensity versus distance z.

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Fig. 11.
Fig. 11. Schematic of average intensity versus propagation distance z. (z = 0.4, 0.8, 1.2, 1.6, 2.0 m, from left to right)

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Equations (32)

Equations on this page are rendered with MathJax. Learn more.

E ( ℘ ) = { E ( F S ) F 1 , oceanic turbulence E ( F S ) F 1 G 1 , atmospheric turbulence
E ( F S ) ( ρ a , z ) = − i k e x p ( i k z ) 2 π z ∫ ∫ d 2 ρ 0 E 0 ( ρ 0 , 0 ) × exp ⁡ [ i k 2 z ( ρ a − ρ 0 ) 2 ]
F 1 = exp ⁡ ( ψ ( o ) ( ρ a , ρ 1 , z ) )
G 1 = exp ⁡ ( ψ ( a ) ( ρ a , ρ 1 , z ) )
W ( ρ a 1 , ρ a 2 , z ) = E ( ℘ ) ( ρ a 1 , z ) E ( ℘ ) ∗ ( ρ a 2 , z )
I ( o − a ) ( ρ , z ) = W ( ρ , ρ , z )
W ( 0 ) ( ρ 1 , ρ 2 , 0 ) = ∑ N i = 1 ⁡ ∑ N j = 1 ⁡ exp ⁡ [ − ( ρ 1 − r i ) 2 + ( ρ 2 − r j ) 2 w 0 2 ] ,
I ( o − a ) ( ρ , z ) = { ( k 2 π z ) 2 ∫ ∫ d 2 ρ 1 ∫ ∫ d 2 ρ 2 W ( 0 ) ( ρ 1 , ρ 2 , 0 ) × e x p { i k 2 z [ ( ρ − ρ 1 ) 2 − ( ρ − ρ 2 ) 2 ] } × ⟨ exp ⁡ [ ψ ( o c e a n ) ∗ ( ρ , ρ 1 , z ) + ψ ( o c e a n ) ( ρ , ρ 2 , z ) ] ⟩ m , z ≤ L ( k 2 π z ) 2 ∫ ∫ d 2 ρ 1 ∫ ∫ d 2 ρ 2 W ( 0 ) ( ρ 1 , ρ 2 , 0 ) × e x p { i k 2 z [ ( ρ − ρ 1 ) 2 − ( ρ − ρ 2 ) 2 ] } × ⟨ exp ⁡ [ ψ ( o c e a n ) ∗ ( ρ , ρ 1 , L ) + ψ ( o c e a n ) ( ρ , ρ 2 , L ) ] ⟩ m × ⟨ exp ⁡ [ ψ ( a i r ) ∗ ( ρ , ρ 1 , z − L ) + ψ ( a i r ) ( ρ , ρ 2 , z − L ) ] ⟩ m , z > L
exp ⁡ ⟨ [ ψ ( m e d i u m ) ∗ ( ρ , ρ 1 , u α ) + ψ ( m e d i u m ) ( ρ , ρ 2 , u α ) ] ⟩ m = exp ⁡ { − 4 π 2 k 2 u α ∫ 0 1 ∫ 0 ∞ κ Φ n m e d i u m ( κ ) [ 1 − J 0 ( κ ξ | ρ 1 − ρ 2 | ) d κ d ξ ] }
Φ n o c e a n ( κ ) = 0.388 × 10 − 8 ε − 1 / 3 κ − 11 / 3 × [ 1 + 2.35 ( κ η ) 2 / 3 ] χ T w 2 ( w 2 e − A T δ + e − A S δ − 2 w e − A T S δ )
Φ n a i r ( κ ) = 0.033 C n 2 κ − 11 / 3 exp ⁡ [ − ( κ 2 / κ m 2 ) ] ,
J 0 ( κ ξ | ρ 1 − ρ 2 | ) ≈ 1 − 1 4 ( κ ξ | ρ 1 − ρ 2 | ) 2 ,
I α ( o − a ) = { − π 2 k 2 z 3 ∫ 0 ∞ κ 3 Φ n o c e a n ( κ ) d κ , z ≤ L − π 2 k 2 3 ∫ 0 ∞ κ 3 [ L Φ n o c e a n ( κ ) + ( z − L ) Φ n a i r ( κ ) ] d κ , z > L
I ( o − a ) ( ρ , z ) = ( k 2 π z ) 2 ∫ ∫ d 2 ρ 1 ∫ ∫ d 2 ρ 2 ∑ N m = 1 ⁡ ∑ N n = 1 ⁡ exp ⁡ [ − ( ρ 1 − r m ) 2 + ( ρ 2 − r n ) 2 w 0 2 ] × e x p { i k 2 z [ ( ρ − ρ 1 ) 2 − ( ρ − ρ 2 ) 2 ] } × e x p { I α ( o − a ) | ρ 1 − ρ 2 | 2 } .
I ( o − a ) ( ρ , z ) = ( k 2 π z ) 2 ∑ N m = 1 ⁡ ∑ N n = 1 ⁡ ∫ ∫ ∫ ∫ d s x d t x d s y d t y × exp ⁡ [ − 2 s x 2 + t x 2 / 2 − 2 ( x m + x n ) s x + ( x m − x n ) t x + ( x m 2 + x n 2 ) w 0 2 ] × exp ⁡ [ − 2 s y 2 + t y 2 / 2 − 2 ( y m + y n ) s y + ( y m − y n ) t y + ( y m 2 + y n 2 ) w 0 2 ] × exp ⁡ { i k z ( x t x − s x t x + y t y − s y t y ) } × e x p { I α ( o − a ) ( t x 2 + t y 2 ) } .
∫ exp ⁡ ( − p 2 x 2 + q x ) d x = π p exp ⁡ ( q 2 4 p 2 ) ,
I ( o − a ) ( ρ , z )  =  ∑ m = 1 N ∑ n = 1 N ∏ ϑ = x , y exp ⁡ ( A ϑ ϑ 2 + B ϑ ϑ + C ϑ ) D ϑ ,
A ϑ = − 2 α ϑ 3 2 w 0 2 α ϑ 3 2 + 1 w 0 2 + 2 I α ( o − a )
B ϑ = − α ϑ 0 α ϑ 3 2 w 0 2 α ϑ 3 2 + 1 w 0 2 + 2 I α ( o − a i r ) − i α ϑ 1 α ϑ 3 w 0 4 α ϑ 3 2 2 + I α ( o − a ) w 0 2 + 1 2
C ϑ = − α ϑ 0 2 α ϑ 3 2 16 β ϑ 2 + α ϑ 0 2 8 w 0 2 − i α ϑ 0 α ϑ 1 α ϑ 3 4 w 0 2 β ϑ 2 + α ϑ 1 2 4 w 0 4 β ϑ 2 − α ϑ 2 w 0 2
D ϑ = α ϑ 3 2 + 1 w 0 4 + 2 w 0 2 I α ( o − a ) / ( k 2 z )
β ϑ = w 0 2 α ϑ 3 2 2 + 1 2 w 0 2 + I α ( o − a )
α ϑ 0 = − 2 ( ϑ m + ϑ n ) , α ϑ 1 = ϑ m − ϑ n , α ϑ 2 = m ϑ m + n ϑ n , α ϑ 3 = k 2 z
P F = A c A Ω = C , P F ¯ = 1 − A c A Ω = 1 − C
L ( 3 ) w = C L ( 3 ) ′ w + ( 1 − C ) L ( 3 ) ′ ′ w
{ I ( F G ) ( ρ , z ) = I ( o − a ) ( ρ , z ) L ( 1 ) w I ( G H ) ( ρ , z ) = I ( o − a ) ( ρ , H 1 + h ) L ( 2 ) w I ( H I ) ( ρ , z ) = I ( o − a ) ( ρ , z ) L ( 2 ) w L ( 3 ) w ,
F ( h ) = ∫ 0 h 1 2 π σ e − ( h − μ ) 2 2 σ 2 d h .
P A  =  P { I ∗ ( ρ , z ) = I ( F G ) ( ρ , z ) }  =  P { H 1 < z < H 1 + h }  =  F ( z ) − F ( z − h ) ,
P B  =  P { I ∗ ( ρ , z ) = I ( G H ) ( ρ , z ) }  =  F ( z − h ) − F ( z − h − d 0 ) ,
P C  =  P { I ∗ ( ρ , z ) = I ( H I ) ( ρ , z ) }  =  F ( z − H 2 ) − F ( z − h − d 0 ) .
I ∗ ( ρ , z ) ¯ = P A I ( F G ) ( ρ , z ) + P B I ( G H ) ( ρ , z ) + P C I ( H I ) ( ρ , z ) .
I ( ρ , z ) = { I ( o − a ) ( ρ , z ) , 0 ≤ z < H 1 I ∗ ( ρ , z ) ¯ , H 1 ≤ z < H 1 + H 2 I ( o − a ) ( ρ , z ) L ( 2 ) w L ( 3 ) w , H 1 + H 2 ≤ z < H 1 + H 2 + H 3
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