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Influences of salinity and temperature on propagation of radially polarized rotationally-symmetric power-exponent-phase vortex beams in oceanic turbulence

Youyou Hu, Mei Zhang, Jiantai Dou, Jiang Zhao, and Bo Li

Open Access Open Access

Abstract

In this paper, the propagation properties of radially polarized rotationally-symmetric power-exponent-phase vortex beams (RP-RSPEPVBs) in oceanic turbulence were theoretically and experimentally studied. Based on the extended Huygens-Fresnel diffraction integral and vector beams theories, the theoretical propagation model of RP-RSPEPVBs in the oceanic turbulence was established. Then, the numerical simulations were carried out to study the influences of the propagation distance z, the rate of dissipation of turbulence kinetic energy per unit mass of fluid ε, the temperature-salinity contribution ratio ω, and the dissipation rate of the mean-squared temperature χT on the optical intensity, spectral degree of polarization (DOP) and spectral degree of coherence (DOC) of RP-RSPEPVBs. Further, an experiment setup was demonstrated to confirm the influences of salinity and temperature on propagation of RP-RSPEPVBs in oceanic turbulence. The results showed that increasing salinity, propagation distance, and turbulence intensity, will result in beam diffusion and intensity reduction of the RP-RSPEPVBs, as well as depolarization and decoherence. Contrarily, high temperature mitigated the intensity loss of the RP-RSPEPVBs and the spectral DOP and spectral DOC increased when the turbulence tends to be dominated by temperature. As a vector beam, the RP-RSPEPVB shows well anti-turbulence interference characteristics, which provides a new choice for optical underwater communication and imaging.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

In recent years, a new type of noncanonical vortex beam, with phase wavefront structure of 2π[rem(lφ,2π)/2π]n, has been reported [1], where l is the topological charge, n is the power exponent and φ is the azimuthal angle. Due to the rotationally-symmetric intensity and phase distributions, it’s also named rotationally-symmetric power-exponent-phase vortex beams (RSPEPVBs). Unlike the traditional power-exponent-phase vortex beams [2–6], the intensity distribution of RSPEPVB is fan-blade shape, of which the number is equal to topological charge l [7]. Thus, the RSPEPVBs can be applied for optical trapping of multi-particles and free space optical communication. Meanwhile, with the development of structured beams, cylindrical vector beams (CVBs), with axisymmetric polarization distribution, have triggered extensive attentions [8–10], which can be utilized for laser processing [11–14], particle trapping [15], free-space optical communications [16], super-resolution imaging [17], focus shaping [18], plasmonic trapping [19], atmosphere transmission [20,21], etc. Moreover, radially polarized (RP) RSPEPVBs possess the characteristics of RSPEPVBs and CVBs at the same time [7].

Besides, with the increasing demand of remote sensing [22] and optical communication [23,24] under the seawater, the research of beam propagation in oceanic turbulence is in the ascendant. As an important random medium, the oceanic turbulence, just like the tissue [25,26] and turbulent atmosphere [27–29], will affect the intensity, degree of polarization (DOP), degree of coherence (DOC), detection probability and so on [30–32]. In the turbulence, since the effects of turbulence on the two orthogonally polarized components of RP beams can offset each other, the scintillation index will decrease, which is an effective way to resist turbulent disturbances by manipulating RP beams [28,33]. However, to the best of our knowledge, the propagation of RP-RSPEPVBs in oceanic turbulence is not yet reported.

In this paper, we studied the influences of salinity and temperature on propagation of RP-RSPEPVBs in oceanic turbulence. Based on the extended Huygens-Fresnel diffraction integral and vector beams theories, the theoretical propagation model of RP-RSPEPVBs in the oceanic turbulence was established. Then, the numerical simulations were carried out to study the influences of the propagation distance z, the rate of dissipation of turbulence kinetic energy per unit mass of fluid ε, the temperature-salinity contribution ratio ω, the dissipation rate of the mean-squared temperature χT on the optical intensity, spectral DOP and DOC of RP-RSPEPVBs. Further, an experiment setup was demonstrated to confirm the influences of salinity and temperature on propagation of RP-RSPEPVBs in oceanic turbulence. The study will contribute to the development of underwater optical communication and imaging.

2. Theory

The RSPEPVBs currently studied mainly include Airy RSPEPVBs, Gaussian RSPEPVBs, and Laguerre-Gaussian RSPEPVBs [1,34]. In this paper, we mainly researched the Gaussian RSPEPVBs, of which the electric field in the source plane (z = 0) can be expressed as [7],

$$E(r,\varphi ,0) = {A_{_0}}\exp \left( { - \frac{{\mathop r\nolimits^{_2} }}{{\mathop w\nolimits^{_2} }}} \right)\exp \left\{ {i2\pi \mathop {\left[ {\frac{{rem(l\varphi ,2\pi )}}{{2\pi }}} \right]}\nolimits^{^n} } \right\}\left[ {\frac{r}{w}({\cos \varphi + \sin \varphi } )} \right].$$
where r and φ are radial and azimuthal coordinates respectively. w is the waist width of the beam. A0 represent the amplitude, which is set to 1 for ease of calculation. n and l stand for the power order and topological charge, respectively. rem() is the remainder function.

Then, the second-order coherence and polarization properties of the RP-RSPEPVBs can be characterized by the 2 × 2 cross-spectral density (CSD) matrix [35],

$$W({r_1},{r_2},{\varphi _1},{\varphi _2},0) = \left\langle {{E_i}^\ast ({r_1},{\varphi_1},0){E_j}({r_2},{\varphi_2},0)} \right\rangle ,(i = x,y;j = x,y).$$
where E*(r,φ,0) is the conjugate function of E(r,φ,0), and < > denotes the ensemble average of the realization of the fluctuating medium. By substituting Eq. (1) into Eq. (2), the expression of the CSD matrix for the source plane z = 0 can be represented as,
$$\begin{aligned}W_{ij}(r_1,r_2,\varphi _1,\varphi _2,0) &= \left( {{{r_1r_2} \over {\mathop w\nolimits^2 }}} \right)\exp \left( {-{{r_1^2 + r_2^2 } \over {\mathop w\nolimits^2 }}} \right)\left[ \begin{array}{c}{\cos \varphi _1\cos \varphi _2} \\ {\cos \varphi _1\sin \varphi _2} \\ {\sin \varphi _1\cos \varphi _2} \\{\sin \varphi _1\sin \varphi _2}\end{array}\right] \\ &\times \exp \left\{ {i2\pi \mathop {\left[ {{{rem(l\varphi _1,2\pi )} \over {2\pi }}} \right]}\nolimits^{^n } -i2\pi \mathop {\left[ {{{rem(l\varphi _2,2\pi )} \over {2\pi }}} \right]}\nolimits^{^n } } \right\}.\end{aligned}$$

The Huygens-Fresnel principle is the theoretical basis for the study of beam propagation in classical optics [36]. Taking advantage of the extended Huygens–Fresnel principle, the CSD formula of RP-RSPEPVBs propagating through oceanic turbulence (z > 0) can be expressed as,

$$\begin{aligned}W({\rho _1},{\rho _2},{\theta _1},{\theta _2},z)& = {\left( {\frac{k}{{2\pi z}}} \right)^2}\int\limits_{ - \infty }^{{ + \infty }} {\int\limits_{ - \infty }^{ + \infty } {\int\limits_0^{2\pi } {\int\limits_0^{2\pi } {W({r_1},{r_2},{\varphi _1},{\varphi _2},0)\exp \left\{\! - \frac{{ik}}{{2z}}[{\mathop {({\rho_1} - {r_1})}\nolimits^2 - \mathop {({\rho_2} - {r_2})}\nolimits^2 } ]\!\right\}} } } } \\&\times \left\langle {\exp [{\psi ({{\mathbf{r}}_1},{\boldsymbol{\rho}_1}) + {\psi^\ast }({{\mathbf{r}}_2},{\boldsymbol{\rho}_2})} ]}\right\rangle {r_1}{r_2}d{r_1}d{r_2}d{\varphi _1}d{\varphi _2}.\end{aligned}$$
/where the wave number k = 2π/λ and λ is the wavelength. r1, r2 and ρ1, ρ2 are positions of two points at the source plane and the receiving plane respectively [37]. φ1, φ2 and θ1, θ2 are angles at the source plane and the receiving plane, respectively. Among them, the ensemble average of the oceanic turbulence can be represented as [36],
$$\left\langle {\exp [{\psi ({{\mathbf{r}}_1},{{\boldsymbol{\rho }}_1}) + {\psi^\ast }({{\mathbf{r}}_2},{{\boldsymbol{\rho }}_2})} ]} \right\rangle = \exp \{{ - {k^2}zG[{\mathop {({{\boldsymbol{\rho }}_1} - {{\boldsymbol{\rho }}_2})}\nolimits^2 + ({{\mathbf{r}}_1} - {{\mathbf{r}}_2})({{\boldsymbol{\rho }}_1} - {{\boldsymbol{\rho }}_2}) + \mathop {({{\mathbf{r}}_1} - {{\mathbf{r}}_2})}\nolimits^2 } ]} \},$$
$$G = \frac{{\mathop \pi \nolimits^2 }}{3}\int\limits_0^\infty {\mathop \kappa \nolimits^3 } \varPhi (\kappa )d\kappa .$$

The spatial power spectrum of the refractive index fluctuations of oceanic turbulence, based on the combination of salinity and temperature fluctuation, is given by [38],

$${\varPhi _n}(\kappa ) = 0.388 \times {10^{ - 8}}\mathop \varepsilon \nolimits^{^{ - 1/3}} \mathop \kappa \nolimits^{^{ - 11/3}} [{1 + 2.35\mathop {(\kappa \eta )}\nolimits^{^{2/3}} } ]f(\kappa ,w,{\chi _T}),$$
$$f(\kappa ,w,{\chi _{_T}}) = \frac{{{\chi _T}}}{{\mathop w\nolimits^{_2} }}(\mathop w\nolimits^2 \mathop e\nolimits^{ - {A_T}\delta } + \mathop e\nolimits^{ - {A_S}\delta } - 2w\mathop e\nolimits^{ - {A_{TS}}\delta } ),$$
which is suitable for isotropic and homogenous oceanic turbulence. Where κ is the spatial frequency, which can be defined as the spatial wave number of oceanic turbulence vortex. ε is the rate of dissipation of kinetic energy per unit mass of fluid, and the range of values can be from 10−1m2/s3 to 10−10m2/s3. χT is the dissipation rate of mean-squared temperature ranges with the range of its value from 10−4k2/s to 10−10k2/s. η=10−3m is the Kolmogorov micro scale. δ=8.284(κη)4/3 + 12.978(κη)2, AT = 1.863 × 10−2, AS = 1.9 × 10−4 and ATS = 9.41 × 10−3. ω is the temperature-salinity contribution ratio, with the value varying in the interval [-5,0]. When the turbulence is dominated by temperature or salinity, ω takes -5 or 0. Then, we can obtain that G = 1.2765 × 10−8ω-2ε-1/3η-1/3χT × (47.5708-17.6701ω+6.78335ω2).

Substitute Eq. (3) into Eq. (4), the CSD formula at the plane (z > 0) can be expressed as,

$$\scalebox{0.95}{$\begin{aligned}{W_{ij}}({\rho _1},{\rho _2},{\theta _1},{\theta _2},z) &= {\left( {\frac{k}{{2\pi z}}} \right)^2}\int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {\int\limits_0^{2\pi } {\int\limits_0^{2\pi } {{W_{ij}}({r_1},{r_2},{\varphi _1},{\varphi _2},0)} } } } \exp \left\{ { - \frac{{ik}}{{2z}}[{\mathop {({\rho_1} - {r_1})}\nolimits^2 - \mathop {({\rho_2} - {r_2})}\nolimits^2 } ]} \right\}\\&\times \exp \{{ - {k^2}zG[\mathop {({{\boldsymbol{\rho }}_1} - {{\boldsymbol{\rho }}_2})}\nolimits^2 + ({{\mathbf{r}}_1} - {{\mathbf{r}}_2})({{\boldsymbol{\rho }}_1} - {{\boldsymbol{\rho }}_2}) + \mathop {({{\mathbf{r}}_1} - {{\mathbf{r}}_2})}\nolimits^2 ]} \}{r_1}{r_2}d{r_1}d{r_2}d{\varphi _1}d{\varphi _2}.\end{aligned}$}$$

Besides, the average intensity of the receiving plane is defined as,

$$I({\rho _1},{\rho _2},{\theta _1},{\theta _2},z) = Tr\mathop W\limits^ \leftrightarrow ({\rho _1},{\rho _2},{\theta _1},{\theta _2},z) = {W_{xx}}({\rho _1},{\rho _2},{\theta _1},{\theta _2},z) + {W_{yy}}({\rho _1},{\rho _2},{\theta _1},{\theta _2},z),$$
and the spectral DOP is applied to characterize the polarization degree of the beam [39],
$$P(\rho _1,\rho _2,\theta _1,\theta _2,z) = \sqrt {1-{{4Det\overleftrightarrow W(\rho _1,\rho _2,\theta _1,\theta _2,z)} \over {\mathop {[Tr\overleftrightarrow W(\rho _1,\rho _2,\theta _1,\theta _2,z)]}\nolimits^2 }}} .$$

The unified theory of coherence and polarization proposes that coherence and polarization are interrelated and should be considered together [40]. The spectral DOC for the RP-RSPEPVBs propagating through oceanic turbulence has the following definition [38],

$$\mu (\rho _1,\rho _2,\theta _1,\theta _2,z) = {{Tr\overleftrightarrow W(\rho _1,\rho _2,\theta _1,\theta _2,z)} \over {\sqrt {Tr\overleftrightarrow W(\rho _1,\rho _1,\theta _1,\theta _1,z)Tr\overleftrightarrow W(\rho _2,\rho _2,\theta _2,\theta _2,z)} }}.$$
where Tr and Det represent the trace and determinant of a matrix, respectively.

3. Numerical calculation and analysis

Then, based on the above theoretical propagation model, the numerical calculation was performed by MATLAB R2021b software to explore the effects of beam parameters and oceanic turbulence parameters on optical intensity, spectral DOP and spectral DOC of RP-RSPEPVBs. Except for special explanations, the calculation parameters are set as λ=532nm, χT = 10−8K2/s, ε=10−7m2/s3, ω=-2.5, z = 1m, l = 4, n = 2, and w = 1 mm.

Firstly, the propagation characteristics of RP-RSPEPVBs with different topological charges in oceanic turbulence are given in Fig. 1. It can be found that the intensity petals number of RP-RSPEPVBs equals to the topological charge l, which show a trend of rotation caused by the vortex phase [41]. Moreover, with the increase of propagation distance z, the RP-RSPEPVBs diffused with the intensity weakening, and the larger the topological charge, the more divergent.

 figure: Fig. 1.

Fig. 1. The intensity distribution of the RP-RSPEPVBs propagating in oceanic turbulence with χT = 10−8K2/s, ε=10−7m2/s3, ω=-2.5, and n = 2.

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Then, the impacts of the dissipation rate of the mean-squared temperature χT and the rate of dissipation of turbulence kinetic energy per unit mass of fluid ε are illustrated in Fig. 2. It shows that, with the increase of χT, the RP-RSPEPVBs tend to diverge and the distribution of intensity petals become blurred and the beam width spread larger, as well as the stochastic electromagnetic vortex beam [36]. Because, the bigger the χT, the stronger the turbulence, which causes the intensity to weaken and boost the evolution of optical intensity profile of RP-RSPEPVBs.

 figure: Fig. 2.

Fig. 2. The intensity distribution of the RP-RSPEPVBs propagating in oceanic turbulence with ω=-2.5, l = 4, n = 2, and z = 1m.

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Besides, the effects of the rate of dissipation of turbulence kinetic energy per unit mass of fluid ε and the temperature-salinity contribution ratio ω on RP-RSPEPVBs are shown in Fig. 3. When the turbulence is mainly dominated by salinity (ω approach to 0), the beams gradually diffuse with the intensity decreasing. However, with the decrease of ω, the beams gradually strengthen, which means that the turbulence is mainly dominated by temperature (ω approach to -5). Compared with the temperature dominated turbulence, the RP-RSPEPVBs change more obviously for salinity dominated turbulence, which means that the influence of salinity is greater than temperature. Moreover, when ω and ε approach to -0.5 and 10−10m2/s3, the RP-RSPEPVBs gradually evolve into a Gaussian distribution.

 figure: Fig. 3.

Fig. 3. The intensity distribution of the RP-RSPEPVBs propagating in oceanic turbulence with χT = 10−8K2/s, l = 4, n = 2, and z = 1m.

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In addition, the evolutions of spectral DOP and spectral DOC in oceanic turbulence are important for the applications of RP-RSPEPVBs, which are illustrated in Figs. 4 and 5, respectively.

 figure: Fig. 4.

Fig. 4. The spectral DOP of RP-RSPEPVBs with different turbulence parameters (a) relative strength of turbulent kinetic energy per unit mass of fluid ε, (b) dissipation rate of the mean-squared temperature χT, (c) relative strength of temperature salinity fluctuations ω, (d) propagation distance z.

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 figure: Fig. 5.

Fig. 5. The spectral DOC of RP-RSPEPVBs with different turbulence parameters (a) relative strength of turbulent kinetic energy per unit mass of fluid ε, (b) dissipation rate of the mean-squared temperature χT, (c) relative strength of temperature salinity fluctuations ω, (d) propagation distance z.

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Figure 4 shows that as well as radially polarized beams, the RP-RSPEPVBs gradually evolve from the completely polarized beams into partially polarized beams in the oceanic turbulence [31], which lowers the spectral DOP due to decoherence effect caused by the turbulence. As shown in Figs. 4(a) and 4(b), the spectral DOP decreases with the decrease of ε or the increase of χT, because the strong turbulence will reduce the polarization. Besides, the axial spectral DOP under the dominant condition of salinity (ω approaches to 0) is smaller than that of temperature (ω approaches to -5), and the long propagation distance will negatively result in depolarization, as illustrated in Fig. 4(c) and 4(d) respectively. Even if the optical intensity feature is no longer obvious at longer distances, the spectral DOP is still evident, which verified that the RP-RSPEPVBs have strong anti-turbulence interference ability [28,33].

Similarly, the spectral DOC also decreased with the decrease of ε or the increase of χT, as shown in Figs. 5(a) and 5(b), because the strong turbulence also results in decoherence [42]. Moreover, the spectral DOC is inversely proportional to ω in Figs. 5(c), which varies dramatically when the turbulence is dominated by salinity. Besides, as well as spectral DOP, the long-distance turbulence propagations of RP-RSPEPVBs also lead to the decrease of spectral DOC, as illustrated in Fig. 5(d).

4. Experimental results

In this section, an experimental system of RP-RSPEPVBs propagating in oceanic turbulence was established to verify the numerical calculation results, where the influence of salinity, temperature, and propagation distance in the static water and oceanic turbulence were mainly considered. As shown in Fig. 6, an underwater turbulence simulator, with propagation distance z = 1.2m, was added to the optical path of the generated RP-RSPEPVBs. To make the results more realistic, we used refined sea salt to adjust the salinity and underwater pump to simulate turbulence. Moreover, two window mirrors were utilized to reduce transmitted wavefront distortion of beams.

 figure: Fig. 6.

Fig. 6. Experimental setup diagram of RP-RSPEPVBs propagating in oceanic turbulence. HWP, Half-Wave Plate; PBS, polarization beam splitter; Lenses 1 and 2 constitute a beam expander; SLM, a spatial light modulator; VWP, vortex wave plate; CBP, camera beam profiler.

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To reduce the absorption of water, a 532nm solid-state laser of TEM00 mode was applied as the light source and the combination of half-wave plate (HWP) and the polarization beam splitter (PBS) acted as an attenuator. The attenuated beam is expanded by the expansion system which composed of two lenses (L1 and L2) with the focal lengths of 25.4mm and 100mm, respectively. Furthermore, a spatial light modulator (SLM, HOLOEYES, PLUTO-NIR-011) was applied for the generation of RSPEPVBs. By modulating the phase diagram loaded on the SLM, the Gaussian beams can be converted into RSPEPVBs with different topological charges [1], and the electric field of the RSPEPVBs can be expressed as,

$$E = {A_{_0}}\exp \left( { - \frac{{\mathop r\nolimits^{_2} }}{{\mathop w\nolimits^{_2} }}} \right)\exp \left\{ {i2\pi \mathop {\left[ {\frac{{rem(l\varphi ,2\pi )}}{{2\pi }}} \right]}\nolimits^{^n} } \right\}.$$

Then, a vortex wave plate (VWP) of order m = 1, commonly applied for converting linearly polarized beams or circularly polarized beam into CVBs or vortex beams with transmission efficiency of 96%, was utilized as radial polarizer to convert RSPEPVBs into RP-RSPEPVBs. The effect of the VWP can be demonstrated by the Jones matrix [43,44],

$$J_{VWP} = \left( \begin{array}{cc}{\cos 2\theta } & {\sin 2\theta } \\{\sin 2\theta } & {-\cos 2\theta }\end{array} \right).$$
where θ is direction angle of the fast-axis, which can be expressed by $\theta = {{m\phi } \mathord{/ {\vphantom {{m\phi } 2}} } 2} + \delta$, m is the order of VWP. $\phi$ is the angle between the fast-axis of local positions and 0° fast-axis. δ=σ(m-2)/2, which is a parameter related to the order of the VWP, σ is the angle between the x-axis and the 0° fast-axis. For convenience, we set σ=0. Therefore, the conversion of RSPEPVBs into RP-RSPEPVBs by the VWP of order m = 1 can be described by the following equation,
$$\begin{aligned}E_2 = J_{VWP}\cdot E_1 &= \left[ {\begin{array}{cc} {\cos \phi } & {\sin \phi } \\ {\sin \phi } & {-\cos \phi } \end{array}} \right]\cdot \left[ {\begin{array}{c} 1 \\ 0\end{array}} \right]\cdot A_{_0 }\exp \left( {-\displaystyle{{\mathop r\nolimits^{_2 } } \over {\mathop w\nolimits^{_2 } }}} \right)\exp \left\{ {i2\pi \mathop {\left[ {\displaystyle{{rem(l\varphi ,2\pi )} \over {2\pi }}} \right]}\nolimits^{^n } } \right\}\cdot \\ &= A_{_0 }\exp \left( {-\displaystyle{{\mathop r\nolimits^{_2 } } \over {\mathop w\nolimits^{_2 } }}} \right)\exp \left\{ {i2\pi \mathop {\left[ {\displaystyle{{rem(l\varphi ,2\pi )} \over {2\pi }}} \right]}\nolimits^{^n } } \right\}\left[ {\begin{array}{c} {\cos \phi } \\ {\sin \phi }\end{array}} \right].\end{aligned} $$

Finally, the generated RP-RSPEPVBs is received by the beam analyzer. Fig. 7 shows the phase diagrams on the SLM and intensity distribution of the RP-RSPEPVBs with different topological charges. Ix and Iy are the intensity distribution of x and y components, respectively. It can be seen the blades number of the RP-RSPEPVBs increases with topological charges.

Then, we conducted experiments to study the influences of salinity and temperature on propagation of RP-RSPEPVBs in static water and oceanic turbulence. For convenience, the salinity is denoted by S and the temperature is expressed by T.

 figure: Fig. 7.

Fig. 7. The phase diagrams on the SLM and intensity patterns of generated RP-RSPEPVBs with topological charges of (a1)-(a4) l = 3, (b1)-(b4) l = 4, (c1)-(c4) l = 5, and (d1)-(d4) l = 6.

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The impacts of salinity were shown in Fig. 8. When the beam passed through the water without salinity, no matter in static water or turbulence, the intensity distribution showed obviously intensity petals. However, as the salinity increased, result in the increase of loss, the intensity of RP-RSPEPVBs gradually decreased, which is similar to the simulation results in Fig. 1. Because, increasing salinity is equivalent to increasing the temperature-salinity contribution ratio ω. While the rates of dissipation are related to the gradients of temperature and salinity direction under the classic conditions of oceanic environment [45]. Further, compared with the situation in static water, the intensity petals diffused in the turbulence.

 figure: Fig. 8.

Fig. 8. The intensity distribution of the RP-RSPEPVBs propagating in the ocean with different salinity in(a1)-(d3) static water and (e1)-(h3) turbulence with T = 30℃, and z = 1.2m.

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Besides, the temperature variation changes the refractive index of seawater, which in turn affects the propagation properties of beams. In the experiment, the temperature was controlled by adding a heater to the tank and the effects of temperature variation are given in Fig. 9. As temperature raised, the intensity increased gradually, which is propinquity to the situation of ω tends to -5 in Fig. 1. Moreover, compared with Fig. 8, it can be found that the effect of salinity on the propagation of RP-RSPEPVBs is significantly greater than that of temperature, which is consistent with the simulations.

 figure: Fig. 9.

Fig. 9. The intensity distribution of the RP-RSPEPVBs propagating in the ocean with different temperature in (a1)-(d3) static water and (e1)-(h3) turbulence with S = 1%, and z = 1.2 m.

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Then, the influences of propagation distance were executed by adding reflectors on the ocean turbulence simulator, which were shown in Fig. 10. The RP-RSPEPVB diverged and the intensity significantly decreased as the distance increased, due to the increasing loss, which is consistent with the simulations in Fig. 1.

 figure: Fig. 10.

Fig. 10. The intensity distribution of the RP-RSPEPVBs propagating in (a1)-(d3) static water and (e1)-(h3) turbulence with different distances with S = 1%, and T = 30°C.

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At the same time, the Stokes parameters were used to analyze the polarization characteristics of the RP-RSPEPVBs, and the polarization azimuth (PA) and polarization ellipticity (PE) are directly related to the Stokes parameters [46]. Thus, the Stokes parameters [47], PA and PE [48] can be respectively written as,

$${S_1} = \frac{{I_{\mathop 0\nolimits^\circ }^{\mathop 0\nolimits^\circ } - I_{\mathop {90}\nolimits^\circ }^{\mathop {90}\nolimits^\circ }}}{{I_{\mathop 0\nolimits^\circ }^{\mathop 0\nolimits^\circ } + I_{\mathop {90}\nolimits^\circ }^{\mathop {90}\nolimits^\circ }}},{S_2} = \frac{{I_{\mathop {45}\nolimits^\circ }^{\mathop {45}\nolimits^\circ } - I_{\mathop {135}\nolimits^\circ }^{\mathop {135}\nolimits^\circ }}}{{I_{\mathop {45}\nolimits^\circ }^{\mathop {45}\nolimits^\circ } + I_{\mathop {135}\nolimits^\circ }^{\mathop {135}\nolimits^\circ }}},{S_3} = \frac{{I_{\mathop 0\nolimits^\circ }^{\mathop {135}\nolimits^\circ } - I_{\mathop 0\nolimits^\circ }^{\mathop {45}\nolimits^\circ }}}{{I_{\mathop 0\nolimits^\circ }^{\mathop {135}\nolimits^\circ } + I_{\mathop 0\nolimits^\circ }^{\mathop {45}\nolimits^\circ }}}.$$
$$\psi = \frac{1}{2}\arctan ({S_2}/{S_1}),$$
$$\chi = \frac{1}{2}\arcsin \frac{{{S_3}}}{{\sqrt {\mathop {{S_1}}\nolimits^2 + \mathop {{S_2}}\nolimits^2 + \mathop {{S_3}}\nolimits^2 } }}.$$
where Iij represents the intensity after passing through the quarter-wave plate with the fast-axis angle of i and the linear polarizer with a polarization direction of j. The color polarization camera (LUCID, PHX050S-QC) can directly capture the linear polarization components of the beam in four directions, so it can be used to measure the full Stokes vector of the beam with/without a quarter-wave plate.

Then, the Stokes parameters and two polarization parameters, about the effect of salinity on the spectral DOP, were measured in Fig. 11. By observing the measured Stokes parameters, it can be found that both S1 and S2 consist of four lobes, and S3 = 0 refers to no circular polarization component. After propagating through the oceanic turbulence, the spectral DOP decreased as the increase of salinity. What’s more, there were similar phenomenon in turbulence. Obviously, polarization is more affected by salinity than turbulence. Besides, the polarization singularity clearly after propagating through high salinity water, even though the optical intensity nearly disappeared, which indicates that the RP-RSPEPVBs may enable a more robust communication channel than the scalar RSPEPVBs or fundamental Gaussian channels [32]. Because, the effects of turbulence on the two orthogonally polarized components of RP-RSPEPVBs can offset each other, which reduces the negative effects of turbulence.

 figure: Fig. 11.

Fig. 11. Stokes parameters, ellipticity of polarization PE and azimuth of polarization PA of RP-RSPEPVBs propagating in the oceanic turbulence with different salinity in (a1)-(d4) static water and (e1)-(h4) turbulence..

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5. Conclusions

In summary, the influences of salinity and temperature on the propagation of RP-RSPEPVBs in oceanic turbulence were explored in this paper. The theoretical propagation model of RP-RSPEPVBs in oceanic turbulence was established, and the numerical simulations were performed to study the influences of beam parameters, turbulence parameters and propagation distance on optical intensity, spectral DOP, and spectral DOC of RP-RSPEPVBs. Then, the RP-RSPEPVBs were experimentally generated to research the effects of salinity, temperature, and propagation distance on optical intensity and spectral DOP in static water and turbulence, respectively. The results showed that increasing salinity, propagation distance, and turbulence intensity will result in beam diffusion and intensity reduction of the RP-RSPEPVBs, as well as depolarization and decoherence. Contrarily, high temperature mitigated the intensity loss of the RP-RSPEPVBs and the spectral DOP and spectral DOC increased when the turbulence tends to be dominated by temperature. As a vector beam, the RP-RSPEPVB shows well anti-turbulence interference characteristics, which provides a new choice for optical underwater communication and imaging.

Funding

Jiangsu Provincial Key Research and Development Program (BE2022143); Natural Science Foundation of Jiangsu Province (BK20190953); National Natural Science Foundation of China (62205133).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (11)
Fig. 1.
Fig. 1. The intensity distribution of the RP-RSPEPVBs propagating in oceanic turbulence with χT = 10−8K2/s, ε=10−7m2/s3, ω=-2.5, and n = 2.

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Fig. 2.
Fig. 2. The intensity distribution of the RP-RSPEPVBs propagating in oceanic turbulence with ω=-2.5, l = 4, n = 2, and z = 1m.

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Fig. 3.
Fig. 3. The intensity distribution of the RP-RSPEPVBs propagating in oceanic turbulence with χT = 10−8K2/s, l = 4, n = 2, and z = 1m.

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Fig. 4.
Fig. 4. The spectral DOP of RP-RSPEPVBs with different turbulence parameters (a) relative strength of turbulent kinetic energy per unit mass of fluid ε, (b) dissipation rate of the mean-squared temperature χT, (c) relative strength of temperature salinity fluctuations ω, (d) propagation distance z.

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Fig. 5.
Fig. 5. The spectral DOC of RP-RSPEPVBs with different turbulence parameters (a) relative strength of turbulent kinetic energy per unit mass of fluid ε, (b) dissipation rate of the mean-squared temperature χT, (c) relative strength of temperature salinity fluctuations ω, (d) propagation distance z.

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Fig. 6.
Fig. 6. Experimental setup diagram of RP-RSPEPVBs propagating in oceanic turbulence. HWP, Half-Wave Plate; PBS, polarization beam splitter; Lenses 1 and 2 constitute a beam expander; SLM, a spatial light modulator; VWP, vortex wave plate; CBP, camera beam profiler.

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Fig. 7.
Fig. 7. The phase diagrams on the SLM and intensity patterns of generated RP-RSPEPVBs with topological charges of (a1)-(a4) l = 3, (b1)-(b4) l = 4, (c1)-(c4) l = 5, and (d1)-(d4) l = 6.

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Fig. 8.
Fig. 8. The intensity distribution of the RP-RSPEPVBs propagating in the ocean with different salinity in(a1)-(d3) static water and (e1)-(h3) turbulence with T = 30℃, and z = 1.2m.

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Fig. 9.
Fig. 9. The intensity distribution of the RP-RSPEPVBs propagating in the ocean with different temperature in (a1)-(d3) static water and (e1)-(h3) turbulence with S = 1%, and z = 1.2 m.

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Fig. 10.
Fig. 10. The intensity distribution of the RP-RSPEPVBs propagating in (a1)-(d3) static water and (e1)-(h3) turbulence with different distances with S = 1%, and T = 30°C.

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Fig. 11.
Fig. 11. Stokes parameters, ellipticity of polarization PE and azimuth of polarization PA of RP-RSPEPVBs propagating in the oceanic turbulence with different salinity in (a1)-(d4) static water and (e1)-(h4) turbulence..

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Equations (18)

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$$E(r,\varphi ,0) = {A_{_0}}\exp \left( { - \frac{{\mathop r\nolimits^{_2} }}{{\mathop w\nolimits^{_2} }}} \right)\exp \left\{ {i2\pi \mathop {\left[ {\frac{{rem(l\varphi ,2\pi )}}{{2\pi }}} \right]}\nolimits^{^n} } \right\}\left[ {\frac{r}{w}({\cos \varphi + \sin \varphi } )} \right].$$
$$W({r_1},{r_2},{\varphi _1},{\varphi _2},0) = \left\langle {{E_i}^\ast ({r_1},{\varphi_1},0){E_j}({r_2},{\varphi_2},0)} \right\rangle ,(i = x,y;j = x,y).$$
$$\begin{aligned}W_{ij}(r_1,r_2,\varphi _1,\varphi _2,0) &= \left( {{{r_1r_2} \over {\mathop w\nolimits^2 }}} \right)\exp \left( {-{{r_1^2 + r_2^2 } \over {\mathop w\nolimits^2 }}} \right)\left[ \begin{array}{c}{\cos \varphi _1\cos \varphi _2} \\ {\cos \varphi _1\sin \varphi _2} \\ {\sin \varphi _1\cos \varphi _2} \\{\sin \varphi _1\sin \varphi _2}\end{array}\right] \\ &\times \exp \left\{ {i2\pi \mathop {\left[ {{{rem(l\varphi _1,2\pi )} \over {2\pi }}} \right]}\nolimits^{^n } -i2\pi \mathop {\left[ {{{rem(l\varphi _2,2\pi )} \over {2\pi }}} \right]}\nolimits^{^n } } \right\}.\end{aligned}$$
$$\begin{aligned}W({\rho _1},{\rho _2},{\theta _1},{\theta _2},z)& = {\left( {\frac{k}{{2\pi z}}} \right)^2}\int\limits_{ - \infty }^{{ + \infty }} {\int\limits_{ - \infty }^{ + \infty } {\int\limits_0^{2\pi } {\int\limits_0^{2\pi } {W({r_1},{r_2},{\varphi _1},{\varphi _2},0)\exp \left\{\! - \frac{{ik}}{{2z}}[{\mathop {({\rho_1} - {r_1})}\nolimits^2 - \mathop {({\rho_2} - {r_2})}\nolimits^2 } ]\!\right\}} } } } \\&\times \left\langle {\exp [{\psi ({{\mathbf{r}}_1},{\boldsymbol{\rho}_1}) + {\psi^\ast }({{\mathbf{r}}_2},{\boldsymbol{\rho}_2})} ]}\right\rangle {r_1}{r_2}d{r_1}d{r_2}d{\varphi _1}d{\varphi _2}.\end{aligned}$$
$$\left\langle {\exp [{\psi ({{\mathbf{r}}_1},{{\boldsymbol{\rho }}_1}) + {\psi^\ast }({{\mathbf{r}}_2},{{\boldsymbol{\rho }}_2})} ]} \right\rangle = \exp \{{ - {k^2}zG[{\mathop {({{\boldsymbol{\rho }}_1} - {{\boldsymbol{\rho }}_2})}\nolimits^2 + ({{\mathbf{r}}_1} - {{\mathbf{r}}_2})({{\boldsymbol{\rho }}_1} - {{\boldsymbol{\rho }}_2}) + \mathop {({{\mathbf{r}}_1} - {{\mathbf{r}}_2})}\nolimits^2 } ]} \},$$
$$G = \frac{{\mathop \pi \nolimits^2 }}{3}\int\limits_0^\infty {\mathop \kappa \nolimits^3 } \varPhi (\kappa )d\kappa .$$
$${\varPhi _n}(\kappa ) = 0.388 \times {10^{ - 8}}\mathop \varepsilon \nolimits^{^{ - 1/3}} \mathop \kappa \nolimits^{^{ - 11/3}} [{1 + 2.35\mathop {(\kappa \eta )}\nolimits^{^{2/3}} } ]f(\kappa ,w,{\chi _T}),$$
$$f(\kappa ,w,{\chi _{_T}}) = \frac{{{\chi _T}}}{{\mathop w\nolimits^{_2} }}(\mathop w\nolimits^2 \mathop e\nolimits^{ - {A_T}\delta } + \mathop e\nolimits^{ - {A_S}\delta } - 2w\mathop e\nolimits^{ - {A_{TS}}\delta } ),$$
$$\scalebox{0.95}{$\begin{aligned}{W_{ij}}({\rho _1},{\rho _2},{\theta _1},{\theta _2},z) &= {\left( {\frac{k}{{2\pi z}}} \right)^2}\int\limits_{ - \infty }^{ + \infty } {\int\limits_{ - \infty }^{ + \infty } {\int\limits_0^{2\pi } {\int\limits_0^{2\pi } {{W_{ij}}({r_1},{r_2},{\varphi _1},{\varphi _2},0)} } } } \exp \left\{ { - \frac{{ik}}{{2z}}[{\mathop {({\rho_1} - {r_1})}\nolimits^2 - \mathop {({\rho_2} - {r_2})}\nolimits^2 } ]} \right\}\\&\times \exp \{{ - {k^2}zG[\mathop {({{\boldsymbol{\rho }}_1} - {{\boldsymbol{\rho }}_2})}\nolimits^2 + ({{\mathbf{r}}_1} - {{\mathbf{r}}_2})({{\boldsymbol{\rho }}_1} - {{\boldsymbol{\rho }}_2}) + \mathop {({{\mathbf{r}}_1} - {{\mathbf{r}}_2})}\nolimits^2 ]} \}{r_1}{r_2}d{r_1}d{r_2}d{\varphi _1}d{\varphi _2}.\end{aligned}$}$$
$$I({\rho _1},{\rho _2},{\theta _1},{\theta _2},z) = Tr\mathop W\limits^ \leftrightarrow ({\rho _1},{\rho _2},{\theta _1},{\theta _2},z) = {W_{xx}}({\rho _1},{\rho _2},{\theta _1},{\theta _2},z) + {W_{yy}}({\rho _1},{\rho _2},{\theta _1},{\theta _2},z),$$
$$P(\rho _1,\rho _2,\theta _1,\theta _2,z) = \sqrt {1-{{4Det\overleftrightarrow W(\rho _1,\rho _2,\theta _1,\theta _2,z)} \over {\mathop {[Tr\overleftrightarrow W(\rho _1,\rho _2,\theta _1,\theta _2,z)]}\nolimits^2 }}} .$$
$$\mu (\rho _1,\rho _2,\theta _1,\theta _2,z) = {{Tr\overleftrightarrow W(\rho _1,\rho _2,\theta _1,\theta _2,z)} \over {\sqrt {Tr\overleftrightarrow W(\rho _1,\rho _1,\theta _1,\theta _1,z)Tr\overleftrightarrow W(\rho _2,\rho _2,\theta _2,\theta _2,z)} }}.$$
$$E = {A_{_0}}\exp \left( { - \frac{{\mathop r\nolimits^{_2} }}{{\mathop w\nolimits^{_2} }}} \right)\exp \left\{ {i2\pi \mathop {\left[ {\frac{{rem(l\varphi ,2\pi )}}{{2\pi }}} \right]}\nolimits^{^n} } \right\}.$$
$$J_{VWP} = \left( \begin{array}{cc}{\cos 2\theta } & {\sin 2\theta } \\{\sin 2\theta } & {-\cos 2\theta }\end{array} \right).$$
$$\begin{aligned}E_2 = J_{VWP}\cdot E_1 &= \left[ {\begin{array}{cc} {\cos \phi } & {\sin \phi } \\ {\sin \phi } & {-\cos \phi } \end{array}} \right]\cdot \left[ {\begin{array}{c} 1 \\ 0\end{array}} \right]\cdot A_{_0 }\exp \left( {-\displaystyle{{\mathop r\nolimits^{_2 } } \over {\mathop w\nolimits^{_2 } }}} \right)\exp \left\{ {i2\pi \mathop {\left[ {\displaystyle{{rem(l\varphi ,2\pi )} \over {2\pi }}} \right]}\nolimits^{^n } } \right\}\cdot \\ &= A_{_0 }\exp \left( {-\displaystyle{{\mathop r\nolimits^{_2 } } \over {\mathop w\nolimits^{_2 } }}} \right)\exp \left\{ {i2\pi \mathop {\left[ {\displaystyle{{rem(l\varphi ,2\pi )} \over {2\pi }}} \right]}\nolimits^{^n } } \right\}\left[ {\begin{array}{c} {\cos \phi } \\ {\sin \phi }\end{array}} \right].\end{aligned} $$
$${S_1} = \frac{{I_{\mathop 0\nolimits^\circ }^{\mathop 0\nolimits^\circ } - I_{\mathop {90}\nolimits^\circ }^{\mathop {90}\nolimits^\circ }}}{{I_{\mathop 0\nolimits^\circ }^{\mathop 0\nolimits^\circ } + I_{\mathop {90}\nolimits^\circ }^{\mathop {90}\nolimits^\circ }}},{S_2} = \frac{{I_{\mathop {45}\nolimits^\circ }^{\mathop {45}\nolimits^\circ } - I_{\mathop {135}\nolimits^\circ }^{\mathop {135}\nolimits^\circ }}}{{I_{\mathop {45}\nolimits^\circ }^{\mathop {45}\nolimits^\circ } + I_{\mathop {135}\nolimits^\circ }^{\mathop {135}\nolimits^\circ }}},{S_3} = \frac{{I_{\mathop 0\nolimits^\circ }^{\mathop {135}\nolimits^\circ } - I_{\mathop 0\nolimits^\circ }^{\mathop {45}\nolimits^\circ }}}{{I_{\mathop 0\nolimits^\circ }^{\mathop {135}\nolimits^\circ } + I_{\mathop 0\nolimits^\circ }^{\mathop {45}\nolimits^\circ }}}.$$
$$\psi = \frac{1}{2}\arctan ({S_2}/{S_1}),$$
$$\chi = \frac{1}{2}\arcsin \frac{{{S_3}}}{{\sqrt {\mathop {{S_1}}\nolimits^2 + \mathop {{S_2}}\nolimits^2 + \mathop {{S_3}}\nolimits^2 } }}.$$
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