1. Introduction

Advances in underwater communication network construction, submarine remote control and resource telemetry have greatly advanced the research of underwater wireless optical communication, especially the research of underwater vortex carrier optical communication. However, for the underwater wireless vortex carrier optical communication working in water, especially seawater, the most difficult problems need to be solved are signal distortion caused by turbulence and signal energy reduction caused by absorption [1,2]. Therefore, up to now, many research results on the effect of oceanic turbulence and beam parameters on the transmitting of the vortex beam have been published [3–18]. These works reported in detail the influences of ocean turbulence on the average light intensity [3–6], light intensity transmittance [7], the received signal probability [2,8–14] and crosstalk probability of vortex modes [15], beam wander [16], and capacity [17,18] of turbulent channels with vortex beam as carrier, all the results provide a theoretical basis for weakening the negative effects of oceanic turbulence. However, these studies do not take into account the effect of seawater absorption on signal transmission, and, seawater absorption is inevitable. Therefore, the main motivation of this study is to analyze the disturbance characteristics of turbulent seawater on the signal transmission law in the presence of seawater absorption. It is noted that the finite energy “freeze wave” (FW) has the advantage of resisting absorption and diffraction of the medium and has no undesirable longitudinal period [19,20]. The finite-energy FW is formed by the suitable superposition of the co-propagating equal-order Bessel-Gauss (BG) beams with same frequency [21], and was realized in experiment by holographic setup in non-absorbing [22,23] or absorbing homogeneous media [24,25]. However, as the finite energy FW is used as the signal information carrier of underwater wireless optical communication system, whether the ability of this beam to resist absorption and diffraction can be maintained under the disturbance of seawater turbulence. Therefore, the absorption and diffraction resistance of finite-energy frozen waves in turbulent ocean is a topic worthy of discussion.

In this paper, considering that seawater is one of the turbulence and absorption media [26], we extend finite energy FW from a uniform absorption medium to turbulent absorbed seawater and call it a random frozen photons (RFP) beam and utilize the statistical moment method to determine the control function of longitudinal light intensity profile. By the control function of longitudinal light intensity profile, we present a model of the RFP beam in absorbed as well as turbulent seawater and analyze the freezing evolution characteristics of the RFP beam as the parameters of beam and channel of turbulent ocean.

2. Random frozen photons in a turbulent ocean

Seawater is an absorption medium, so the refractive index of seawater ${n_{oc}}$ should be described by complex refractive index, that is ${n_{oc}} = {\bar{n}_R} + \textrm{i}{\bar{n}_I}$. In cylindrical coordinates $({r,\theta ,z} )$, by eikonal approximation and superimposing 2M+1 BG beams of the same frequency as well as same order in different propagation angles, the amplitude of the FW in seawater can be describe by [21]

Note: (1) ${\beta _{mR}} = Q + \textrm{2}\pi m/L$, and ${\beta _{mI}} = ({Q + \textrm{2}\pi m/L} ){{{{\bar{n}}_I}} \mathord{\left/ {\vphantom {{{{\bar{n}}_I}} {{{\bar{n}}_R}}}} \right.} {{{\bar{n}}_R}}}$ where Q is a constant that determines the radius of cylindrical surface ${r_\textrm{0}}$ via the first solution of ${ {[{\partial /({\partial r} )} ][{{\textrm{J}_{{l_\textrm{0}}}}({{\alpha_m}r} )} ]} |_{r = {r_0}}} = 0$ when ${l_\textrm{0}} \ne 0$, for ${l_\textrm{0}} = 1$, ${r_0} = {{1.841} \mathord{\left/ {\vphantom {{1.841} {\sqrt {{k^2} - \beta_m^2} }}} \right.} {\sqrt {{k^2} - \beta _m^2} }}$ [20]; (2) To ensure that the beam only propagates forward, ${\beta _{mR}}$ should be from 0 to $2\pi \sqrt {\bar{n}_R^2 - \bar{n}_I^2} /\lambda$ [21].

Furthermore, it is also noted that turbulent seawater is a random medium, and the random property of light in turbulent seawater can be described by statistical method [27]. Therefore, when the FW beam propagates in a turbulent ocean of weak signal fluctuation, FW is a random FW and is called a random frozen photons (RFP) beam. For RFP beam propagation in a turbulent ocean of weak signal fluctuation and by the Rytov approximation [28], we describe the amplitude of the RFP beam by the following function

3. Signal intensity of a random frozen photons beam

For the RFP beam transmits in a turbulent ocean, we must use the method proposed in literature [28] to determine the superposition coefficient of the RFP beam in turbulent seawater. That is, from the statistical moment theory, the control function of light intensity longitudinal profile [19] of the RFP beam on a cylindrical surface of radius $r = {r_0}$ for $0 \le z \le L$ will is defined by the following equation

Besides, for the light propagation in the “window wavelength” region of a turbulent ocean, the transmission distance $z \ll {z_R}$ that means $w_z^2 = w_0^2[{1 + {{({{z \mathord{\left/ {\vphantom {z {{z_R}}}} \right.} {{z_R}}}} )}^2}} ]\approx w_0^2$, and ${\textrm{J}_{{l_0}}}[{{{{\alpha_m}r} \mathord{\left/ {\vphantom {{{\alpha_m}r} {({1 + \textrm{i}{z \mathord{\left/ {\vphantom {z {{z_R}}}} \right.} {{z_R}}}} )}}} \right.} {({1 + \textrm{i}{z \mathord{\left/ {\vphantom {z {{z_R}}}} \right.} {{z_R}}}} )}}} ]\approx {\textrm{J}_{{l_0}}}({{\alpha_m}r} )$. As a result, we rewrite Eq. (3) as

Now, the statistical averaging control function of light intensity longitudinal profile can be expressed as

Further, for moderate-absorption medium, ${\bar{n}_I} \ll {\bar{n}_R}$, so ${k_I} \ll {k_R}$, and when ${r_0} \ll z$, we have

Then, the statistical averaging control function of light intensity longitudinal profile can be simplified as

In Eq. (9), ${\rho _{oc}}$ is the coherence length of a plane wave propagating in the turbulent ocean and is given by [29]

By the integral expression of the confluent hypergeometric function of the second kind $\textrm{U}({a;b;c} )$ [31]

Substituting Eq. (9) and Eq. (14) into Eq. (8), we have analytic approximate relation of the statistical averaging control function of light intensity longitudinal profile

Moreover, for the RFP beam propagating in a turbulent ocean, $Q \sim {k_r} > > \pi ({m + n} )/L$, ${\beta _{mR}}$ is approximately equal to ${\bar{\beta }_R}$ [21], that is, ${\beta _{mR}}$ is given by

By above analysis, the Eq. (15) can be simplified as

Multiplying the factor $\exp [{ - \textrm{i}{{\textrm{2}\pi } \mathord{\left/ {\vphantom {{\textrm{2}\pi } L}} \right.} L}({m^{\prime} - n^{\prime}} )z} ]$ and integrating the z on both sides of the above formula, we have the second moment of random complex weighting factors for each Bessel-Gaussian beam in the superposition $\left\langle {\tilde{A}_n^ \ast {{\tilde{A}}_m}} \right\rangle$

To eliminate unwanted longitudinal periodicity of light intensity [20], by $\exp \left\{ {\left[ {\frac{{2({k_R^2 - k_I^2} )}}{{w_\textrm{0}^\textrm{2}{{({k_R^2 + k_I^2} )}^2}}}\left( {\alpha_0^2 - \frac{{4r_0^2}}{{w_\textrm{0}^\textrm{4}}}} \right)} \right]{z^2}} \right\} = \sum\limits_{p = 0}^\infty {\frac{{{{(z )}^{2p}}}}{{p!}}{{\left[ {\frac{{2({k_R^2 - k_I^2} )}}{{w_\textrm{0}^\textrm{2}{{({k_R^2 + k_I^2} )}^2}}}\left( {\alpha_0^2 - \frac{{4r_0^2}}{{w_\textrm{0}^\textrm{4}}}} \right)} \right]}^p}}$, we have another new result proposed and derived in this paper

Further, for $F(z )= {F_0} = \textrm{constant,}{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{0} \le z \le L$, and by the integral formula [31]

Now, the light intensity of the RFP beam in a turbulent ocean is simplified as

4. Effects of a turbulent ocean and beam parameters on light transmittance

In this section, we analyze the diffracting- and absorption- resistance of the RFP beam in turbulent and absorbent seawater by numerically calculation. In the following discussion, unless otherwise specified, the parameters are set as: $\lambda = 410\textrm{nm}$, ${m_0} = 1$, ${w_0} = 0.5{\mathop{\rm m}\nolimits}$, ${r_0} = 4.9806 \times {10^{ - 4}}{\mathop{\rm m}\nolimits}$, $r = {r_0}$, $L = 350{\mathop{\rm m}\nolimits}$, $M = 18$, $Q = 0.9999999838{k_r}$, $\varpi ={-} 1$, $\eta = 0.001{\mathop{\rm m}\nolimits}$, ${L_0} = 10{\mathop{\rm m}\nolimits}$, ${C_1} = 2.35$, $q = 2.5$, $\gamma = 0.72$, ${\Pr _T} = 7$, $\textrm{P}{\textrm{r}_S} = 700$, ${\bar{n}_r} = 1.34$, ${\bar{n}_\textrm{i}} = 1.0114 \times {10^{ - 10}}$, ${\Lambda _{{m_0}}} = \textrm{1/}0.5819$, and $C_m^2 = {10^{ - 16}}{{\mathop{\rm m}\nolimits} ^{{{ - 2} \mathord{\left/ {\vphantom {{ - 2} 3}} \right.} 3}}}$. Besides, $|{\textrm{F}(z)} |{{\kern 1pt} ^2}$ is designed as $|{\textrm{F}(z)} |{{\kern 1pt} ^2} = 1,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{for}\;\textrm{0} \le z < L$.

Since the RFP beam is constructed by superimposing 2M+1 BG beams, it is necessary to study the effect of superimposing number of beams M on signal transmittance ${T_I}$. The beam waist ${w_0}$ and OAM quantum numbers ${l_0}$ are tentatively set at 0.2m and 1, respectively, and for given other parameters and constitute conditions of frozen beams [19,20], the maximum superimposing numbers is M=18. For comparison, we plot ${T_I}$ of the BG beam under different r and z in Fig. 1(a) and plot ${T_I}$ of the RFP beam under different r, z and M in Figs. 1(b), 1(c), and 1(d). Figures 1(a) and 1(b) show that, as we expected, the evolution of the ${T_I}$ of the BG beam and single RFP beam over z and r is almost the same: In the transmission direction, the ${T_I}$ of the BG beam and the RFP beam for M=0 both decrease with the increase of the z, which does not reach what we expected, that is, frozen transmittance; in the radial direction, they are both periodic, which means the transverse light intensity distribution of the RFP beam for M=0 is always light and shade rings with dark hollow center, and the brightness of the halo gradually decreases along the radial direction. But, from Figs. 1(b) and 1(c), we can see that as the superimposing number of the RFP beam increases, the ${T_I}$ is getting closer and closer to freezing in z direction, and the dark ring part of the ${T_I}$ in r direction gradually disappears. This can be explained by the redistribution of energy caused by the superposition of multiple BG beams. Therefore, the larger M is good for forming the RFP beam with a stable and continuous transmittance.

 

Fig. 1. Transmittance ${T_I}$ of a random frozen photons beam transmitting in a turbulent ocean versus the propagation z and radial coordinate r for the different superimposing numbers M.

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To optimize the beam parameters of the RFP beam, in Fig. 2, we show the influences of different OAM quantum numbers ${l_0}$ and beam waists ${w_0}$ on the ${T_I}$ of the RFP beam in a turbulent ocean. Figures 2(a), 2(b), 2(c) and 2(d) show that, for given beam parameters $M = 18$ and ${l_0} = 1$, as ${w_0}$ increases, the ${T_I}$ tends to stabilize and and stays at 0.5 when ${w_0}=0.5\textrm{m}$ within the propagation distance 300m, that is, the freezing of the light transmittance is achieved. In addition, for given beam parameters $M = 18$ and ${w_0} = 0.5\textrm{m}$, as ${l_0}$ increases, ${T_I}$ loses its freezing properties in z direction, and in r direction the signal strength outside the bright ring gradually disappears. Therefore, the optimal light source parameters of the RFP beam can obtain by choosing larger M and ${w_0}$, as well as smaller ${l_0}$.

 

Fig. 2. Transmittance ${T_I}$ of a random frozen photons beam transmitting in a turbulent ocean versus the propagation z and radial coordinate r for the different beam waists ${w_0}$ in (a), (b), (c), (d) and for different OAM quantum numbers ${l_0}$ in (c), (e), (f).

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Now, we focus on the effect of the equivalent “temperature structure” constants $C_\textrm{m}^\textrm{2}$ on the ${T_I}$ of RFP beam transmitting in a turbulent ocean. As showed in Figs. 3(a), 3(b) and 3(c), as the $C_\textrm{m}^\textrm{2}$ increases from $\textrm{1}{\textrm{0}^{ - 18}}{\textrm{m}^{ - 2/3}}$ to $\textrm{1}{\textrm{0}^{ - 16}}{\textrm{m}^{ - 2/3}}$ in weak turbulence, ${T_I}$ of the RFP beam is freezing. But when $C_\textrm{m}^\textrm{2}$ increases to $\textrm{1}{\textrm{0}^{ - 14}}{\textrm{m}^{ - 2/3}}$ in moderate turbulence, the ${T_I}$ in r direction disappears, and the ${T_I}$ in z direction also decreases significantly. Although the transmittance decreases with the increase of turbulent strength, nevertheless, the ${T_I}$ in z direction still is freezing, which implies that the change of turbulent strength can only affect the value of the ${T_I}$ of the RFP beam, but cannot alter its freezing properties in the transmission direction. Besides, in Fig. 3(d) we also study the influence of the dissipation rate of the mean-squared temperature ${\chi _T}$ and the rate of dissipation of kinetic energy per unit mass of fluid $\varepsilon$ on the ${T_I}$ of the RFP beam at $z = 30\textrm{m}$, and $z = 330\textrm{m}$. Fig. 3(d) shows that the ${T_I}$ increases with the increase of the $\varepsilon$ and the decrease of ${\chi _T}$, which is because the larger $\varepsilon$ or smaller ${\chi _T}$ corresponds to the weaker ocean turbulence.

 

Fig. 3. Transmittance ${T_I}$ of a random frozen photons beam in a turbulent ocean versus the propagation distance z and radial coordinate r for different equivalent “temperature structure” constants in (a), (b), (c) and versus the dissipation rate of the mean-squared temperature ${\chi _T}$ and the rate of dissipation of kinetic energy per unit mass of fluid $\varepsilon$ at $z\textrm{ = 30}$ and 330 m in (d).

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Figure 4 gives the influences of the different ratio of temperature and salinity $\varpi $ on the ${T_I}$ of the RFP beam transmitting in a turbulent ocean. From Figs. 4(a), 4(b) and 4(c), we can see that, as the $\varpi $ increases from -5, -1, to -0.5 that corresponds to temperature fluctuation dominating, common effect of temperature and salty fluctuations, and salty fluctuation dominating, the transmittance distributions in these three states are almost the same. Further, as the $\varpi $ increases to -0.1, the oceanic turbulence almost completely arises from salinity fluctuation, in this case, the ${T_I}$ in r direction outside the bright ring disappears, and the ${T_I}$ in z direction also decreases significantly. Therefore, we can draw a conclusion that ${T_I}$ of the RFP beam in a turbulent ocean will significantly decrease when the turbulence is almost entirely dominated by salinity fluctuation, but the transmittance freezing will not be destroyed.

 

Fig. 4. Light transmittance ${T_I}$ of a random frozen photons beam transmitting in a turbulent ocean versus the propagation z and radial coordinate r for the different ratio of temperature and salinity $\varpi $.

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Figure 5 gives the influences of different inner scales ${\eta _0}$ and outer scales ${\eta _c}$ of turbulence on the ${T_I}$ of the RFP beam in a turbulent ocean. From Fig. 5, we can see that as the ${\eta _0}$ increases from 1mm to 50mm for ${\eta _c} = 10\textrm{m}$, and the ${\eta _c}$ increases from 10m to 50m for ${\eta _0} = 50\textrm{mm}$, the transmittance distribution is almost constant, which means that the effects of the inner and outer scales of oceanic turbulence on the transmittance ${T_I}$ of the RFP beam is small enough to ignore.

 

Fig. 5. Light transmittance ${T_I}$ of a random frozen photons beam transmitting in a turbulent ocean versus the propagation z and radial coordinate r for the different inner scales ${\eta _0}$ and outer scales ${\eta _c}$ of turbulence.

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5. Conclusions

A random frozen photons beam is the superposition of multiple Bessel-Gaussian beams with light intensity longitudinal profile controllable superposition coefficient. So transmittance profile of a random frozen photons beam can be arbitrarily tailored along the beam’s axis of propagation. In this paper, we extended the theory of finite energy FW in uniform absorption medium to turbulent seawater and constructed the transmittance of the RFP beam in ocean turbulence. Our study shows that the RFP beam has better transmittance freezing characteristics than a BG beam. The RFP beam with optimized number of superposition sub beams M, beam waist ${w_0}$ and OAM quantum number ${l_0}$ can maintain freezing characteristics in a fairly long turbulent transport channel for seawater, and this freezing characteristics will not be destroyed by the variation of turbulent parameters $C_\textrm{m}^\textrm{2}$, ${\chi _T}$, $\varepsilon$, and $\varpi $. Besides, like other beam [2–18], the salinity fluctuation affects the transmittance of RFP beam more than the temperature fluctuation does, and, the influences of ${\eta _0}$ and ${\eta _c}$ of oceanic turbulence on the transmittance of the RFP beam are negligible. The conclusion of this paper is that, although the RFP beam is obtained by shaping the BG beam, the RFP beam will freeze transmission in over a longer turbulent ocean link than that of the BG beam.