Bit-error rate and average capacity of an absorbent and turbulent underwater wireless communication link with perfect Laguerre-Gauss beam

Hongbin Yang; Qingze Yan; Pan Wang; Lifa Hu; Lifa Hu; Yixin Zhang; Yixin Zhang

doi:10.1364/OE.451981

1. Introduction

In recent years, underwater wireless optical communication system with orbital angular momentum(OAM) information carrier has attracted extensive attention, because OAM mode can form an infinite N qubit basis and provide a new communication link coding scheme with high security, wide bandwidth and low cost [1–9]. However, because the OAM mode is related to the wave-front spatial structure of the carrier, ocean turbulence will inevitably lead to the distortion of the OAM mode and crosstalk between the energy states of the OAM mode [4–9]. Therefore, using different beam structures such as Gaussian Schell model vortex beam [5], Bessel–Gauss beam [6], Hermite-Gauss vortex beam [7], random frozen photons beam [8], partially coherent modified Bessel correlated vortex beam [9], Lommel beam [10], and vortex localized wave [11] to reduce the perturbation of seawater turbulence has become a new trend in this field.

Scalar perfect vortex composed of perfect optical vortex (POV) [12] and perfect Laguerre-Gauss (PLG) beam [13], is a new class of OAM carrying beam. The perfect optical vortex (POV) beam can be generated by experiment, which was first reported by Ostrovsky et al. [12], and their experimental results indicated that the ring radius of POV beam is independent of topological charge. Therefore, POV beam has been widely studied as a new signal source to reduce the influence of seawater turbulence. Such as, through the experiment Karahroudi et al. [14] studied the transmissivity of POV beam in a 2.6 m link of the artificially manufactured underwater environment, discussing the influence of various underwater conditions on the POV beam. Their results show that the temperature difference in the water causes the distortion of POV beam ring. Hu et al. [15] presented a model for the OAM mode carried by POV beam in an anisotropic turbulent seawater using the expansion of plane spiral mode. This model shows that the self-focusing property of the POV beam is beneficial to the propagation of OAM modes. Wang et al. [16] proposed the evolution model of the POV beam and showed the transmission quality of the POV beam in oceanic turbulence is most related to the beam radius, while it is nearly free from the wavelength, topological charge, and radius-thickness ratio. Yang et al. [17] developed a probability of OAM modes carried by POV beam in absorbency ocean turbulence and showed the ring radius of POV beam is dependent of topological charge.

A recent article has demonstrated that the width of such POVs also depends on the topological charge and, in specific regimes, is comparable to regular vortex beams [18]. Consequently, Mendoza-Hernandez et al. [13] proposed PLG beam and point that PLG beam is closer to the ideal POV, maintaining a quasi-constant ring radius and width. However, whether the ring radius of PLG beam in absorptive turbulent seawater is still independent of topological charge. Hence, it is worth to explore the propagation characteristics of OAM mode carried by the PLG beam in the absorptive turbulent seawater.

In this paper, we discuss the effects of weakly turbulent absorbing seawater on bit-error rate (BER) and average capacity of wireless optical communication links with PLG beam in Rytov approximation. In Section 2, theoretical model of PLG beam disturbed by weakly turbulent absorbing seawater is proposed. In Section 3, we gave the theoretical model of the BER and average capacity of optical wireless communication link with PLG beam. The numerical simulations for demonstrating the relationships between BER as well as average capacity of wireless communication links and the parameters of beam or turbulence seawater are given in Section 4, and finally, conclusions are given in Section 5.

2. Perfect Laguerre-Gauss beam in weakly turbulent absorbing seawater

We start by the amplitude of PLG beam at Fourier plane of optical system [13]

(1)$${E_{{m_0}}}({\rho ,\theta ,0} )= {({ - \textrm{i}} )^{{m_0}}}{\left( {\frac{{\sqrt 2 \rho }}{{{w_{{m_0}p}}}}} \right)^{|{{m_0}} |}}\textrm{L}_p^{{m_0}}\left( {\frac{{2{\rho^2}}}{{w_{{m_0}p}^2}}} \right)\exp \left( { - \frac{{{\rho^2}}}{{w_{{m_0}p}^2}}} \right)\exp ({ - \textrm{i}{m_0}\theta } ), $$

and

(2)$${w_{{m_0}p}}\textrm{ = }\frac{{{w_0}}}{{2\sqrt {2p + ({|{{m_0}} |+ 1} )} }}, $$

where $i = \sqrt { - 1}$ is the complex symbol, $\textrm{L}_p^{{m_0}}({\cdot} )$ is the associated Laguerre polynomial with p and m₀ the radial and azimuthal orders, m₀ also indicates the OAM topological charge of the initial OAM mode, $\rho$ is the radial cylindrical coordinate, ${w_0}$ is a beam width parameter at the beam waist plane, $\theta$ is the azimuthally angle.

In cylindrical coordinates, the normalized PLG model at its beam waist plane is given by

(3)$${E_{{m_0}}}({\rho ,\theta ,0} )= {\left( {\frac{{2p!}}{{\pi w_{{m_0}p}^2({|{{m_0}} |+ p} )!}}} \right)^{1/2}}{\left( {\frac{{\sqrt 2 \rho }}{{{w_{{m_0}p}}}}} \right)^{|{{m_0}} |}}\textrm{L}_p^{{m_0}}\left( {\frac{{2{\rho^2}}}{{w_{{m_0}p}^2}}} \right)\exp \left( { - \frac{{{\rho^2}}}{{w_{{m_0}p}^2}}} \right)\exp ({\textrm{i}{m_0}\theta } ). $$

For PLG beam propagation in weakly turbulent absorbing seawater and in cylindrical polar coordinates $({r,\varphi ,z} )$, the complex amplitude at propagation distance $z < < {\textstyle{{4{\pi ^2}w_0^2{{({n_R^2 + n_I^2} )}^2}} \over {3{\lambda ^2}\alpha ({2p + ({|{{m_0}} |+ 1} )} )}}}$ from the source is represented [19]

(4)$${E_m}({r,\varphi ,z} )= {E_{{m_0}}}({r,\varphi ,z} )\exp [{\psi (r,\varphi ,z)} ].$$

Here ψ is the complex phase perturbations caused by seawater turbulence, and ${E_{{m_0}}}({r,\varphi ,z} )$ is given by the Huygens-Fresnel integral of Eq. (1)

(5)$${E_{{m_0}}}({r,\varphi ,z} )= \frac{{{R_{{m_0},}}_p(r,z)}}{{\sqrt {2\pi } }}\exp ({\textrm{i}{m_0}\varphi \textrm{ + i}{k_0}{n_R}z} )\exp [{ - \textrm{i}({2p + |{{m_0}} |+ 1} ){\delta_\textrm{G}} - {k_0}{n_I}z} ], $$

which has OAM eigenvalue ${l_z} = {m_0}\hbar$, the ${R_{{m_0},p}}(r,z)$ is radial basis function which is the field distributions of the PLG beam with p and m₀

(6)$$\begin{aligned} {R_{{m_0},p}}(r,z) &= \frac{1}{{w(z )}}\sqrt {\frac{{4p!}}{{(p + |{{m_0}} |)!}}} {\left( {\frac{{\sqrt 2 r}}{{w(z )}}} \right)^{|{{m_0}} |}}\\ &\times \textrm{L}_p^{{m_0}}\left( {\frac{{2{r^2}}}{{{w^2}(z )}}} \right)\exp \left[ { - \left( {\frac{1}{{{w^2}(z )}} - \frac{{{k_0}{n_I}}}{{2R}}} \right){r^2}} \right]\exp \left( { - \frac{{\textrm{i}{k_0}{n_R}{r^2}}}{{2R}}} \right), \end{aligned}$$

where n_R and n_I are the real part and the imaginary part of the n_oc [11], and the relationship between n_I and the absorption coefficient αof seawater is ${n_I} = \frac{{\lambda \alpha }}{{4\pi }}$ [20];$w(z )= {w_{{m_0}p}}\sqrt {1 + {{({z/{z_R}} )}^2}}$ is the spot size, $R(z )= z[{1 + {{({{z_R}/z} )}^2}} ]$ is the radius of wave front curvature, ${z_R} = {\textstyle{1 \over 2}}{k_0}\sqrt {n_R^2 + n_I^2} w_{{m_0}p}^2$ is the Raleigh range of PLG beam in absorbing seawater and ${\delta _\textrm{G}}$ is the Gouy phase.

3. Bit-error rate and average capacity of the orbital angular momentum link

When PLG beam propagates in turbulent ocean, the beam will be disturbed by refractive index fluctuation, and the OAM signal energy carried by the beam will move to the adjacent OAM energy state to form OAM crosstalk. Therefore this wave can be written as a superposition of new eigenstates m with weight coefficient [2]

(7)$${E_m}({r,\varphi ,z} )= \sum\limits_m {{b_m}({r,z} )\exp ({\textrm{i}m\varphi } )}. $$

In Eq. (7), $m = {m_0}$ represents OAM signal, and $m \ne {m_0}$ represents OAM crosstalk.

The probability distribution of obtaining a measurement of new OAM ${l_z} = m\hbar$ is obtained by summing the probabilities associated with the eigenvalue $m\hbar$,

(8)$$p({m/{m_0}} )= \sum\limits_m {\left\langle {{b_m}({r,z} )b_m^ \ast ({r,z} )} \right\rangle } = \sum\limits_m {\left\langle {{{|{{b_m}({r,z} )} |}^2}} \right\rangle }, $$

where $\left\langle \cdot \right\rangle$ represents the average of seawater turbulence ensemble, * denotes complex conjugate, and $b_p^m(z )$ is given by the basis projections

(9)$$b_p^m(z )= \frac{1}{{ {2\pi } }}\sum\limits_{m ={-} \infty }^\infty {\int_0^{2\pi } {{E_{{m}}}({r,\varphi ,z} )} \exp ({ - \textrm{i}m\varphi } )\textrm{d}\varphi }. $$

Substituting Eq. (9) into Eq. (8), we can rewrite Eq. (8) as

(10)$$p({m/{m_0}} )= \left(\frac{1}{{2\pi }}\right)^2\int_0^{2\pi } {\int_0^{2\pi } {\left\langle {E_m^ \ast ({r^{\prime},\varphi^{\prime},z} ){E_m}({r,\varphi ,z} )} \right\rangle \exp [{ - \textrm{i}m({\varphi - \varphi^{\prime}} )} ]} } \textrm{d}\varphi ^{\prime}\textrm{d}\varphi. $$

The probability of obtaining a measurement of new OAM ${l_z} = m\hbar$ is obtained by integrating over

(11)$$\begin{aligned} P({m/{m_0}}) &= \frac{1}{{2\pi }}\int {\int\!\!\!\int {E_{{m_0}}^ \ast ({r,\varphi^{\prime},z} ){E_{{m_0}}}({r,\varphi ,z} )\left\langle {{e^{\psi ({r,\varphi ,z} )+ {\psi^ \ast }({r,\varphi^{\prime},z} )}}} \right\rangle } } \\ &\times \exp [{ - \textrm{i}m({\varphi - \varphi^{\prime}} )} ]r\textrm{d}r\textrm{d}\varphi ^{\prime}\textrm{d}\varphi , \end{aligned}$$

with [15]

(12)$$\left\langle {\exp [{\psi ({r,\varphi ,z} )+ {\psi^ \ast }({r,\varphi^{\prime},z} )} ]} \right\rangle \textrm{ = }\exp \{{ - [{1 - \cos ({\varphi - \varphi^{\prime}} )} ]2{r^2}/\rho_o^2} \},$$

and

(13)$${\rho _o}\textrm{ = }{\left[ {{\textstyle{1 \over 3}}{\pi^2}k_0^2({n_R^2 + n_I^2} )z\int\limits_0^\infty {{\kappa^3}{\Phi _n}(\kappa )} \textrm{d}\kappa } \right]^{\textrm{ - }1/2}}, $$

where ${\Phi _n}(\kappa )$ is the power spectrum of the fluctuations of the refractive index of seawater in table stratification and isotropic turbulence and is described by the following equation [21]

(14)$$\begin{aligned} {\Phi _n}(\kappa) &= \frac{{0.033C_{\varepsilon {\chi _T}}^2{\kappa ^{ - 11/3}}[1 + 4.6{{(\kappa {l_0})}^{2/3}}]}}{{{{({1 - \varpi } )}^2}{{({{\kappa^2} + \kappa_0^2} )}^{11/6}}}}[{{\varpi^2}\exp ({ - {\kappa^2}/\kappa_T^2} )} \\ &+ \exp ({ - {\kappa^2}/\kappa_S^2} )- 2\varpi \exp ({ - {\kappa^2}/\kappa_{TS}^2} )],\,\,\,\,\,0 < \kappa < \infty , \end{aligned}$$

where $\kappa$ is the scalar spatial wave number of turbulent fluctuation, $C_{\varepsilon {\chi _T}}^2 = 0.809 \times {10^{ - 7}}{\varepsilon ^{ - 1/3}}{\chi _T}{\varpi ^{ - 2}}$${({1 - \varpi } )^2}$ is the structural constant of the salinity- temperature fluctuation of oceanic turbulence, $\varepsilon $ is the rate of dissipation of kinetic energy per unit mass of fluid ranging from ${10^{ - 10}}{m ^2}/{s^3}$ to ${10^{ - 1}}{m ^2}/{s^3}$, ${\chi _T}$ is the dissipation rate of the mean-squared temperature and has the range from ${10^{ - 10}}{K ^2}/s$ to ${10^{ - 2}}{K ^2}/s$, $\varpi$ defines the ratio of temperature and salinity contributions to the refractive index spectrum, which can vary in the interval (−5,0) in the seawaters; in the limit case $\varpi \approx{-} 5$ temperature is dominant in the induction of optical turbulence, while the limit case $\varpi \approx 0$ salinity is dominant in the induction of optical turbulence. ${l_0}$ is a turbulent inner scale, ${\kappa _0} = 2\pi /{L_0}$, ${\kappa _T} = {R_T}/{l_0}$, ${\kappa _S} = {R_S}/{l_0}$, ${\kappa _{TS}} = {R_{TS}}/{l_0}$, ${L_0}$ is a turbulent outer scale, ${R_j} = \sqrt 3 {[{W_j} - {\textstyle{1 \over 3}} + {(9{W_j})^{ - 1}}]^{3/2}}{Q^{\textrm{ - }3/2}}\,\,(j = T,\,S,\,\,TS)$, ${W_j} = {\left\{ {{{\left[ {{\textstyle{{\Pr_j^2{Q^4}} \over {36{\beta^2}}}} - {\textstyle{{{{\Pr }_j}{Q^2}} \over {81\beta }}}} \right]}^{1/2}} - \left[ {{\textstyle{1 \over {27}}} - {\textstyle{{{{\Pr }_j}{Q^2}} \over {6\beta }}}} \right]} \right\}^{1/3}}$, Q is the non-dimensional constant; ${\Pr _T}$ and ${\Pr _S}$ respectively represent the Prandtl number of the temperature and salinity, ${\Pr _{TS}} = 2{\Pr _t}{\Pr _S}/({\Pr _T} + {\Pr _S})$.

Notice the integral relation [19]

(15)$$\int_0^\infty {{\kappa ^{2\mu }}} \frac{{\exp ({ - {\kappa^2}/\kappa_H^2} )}}{{{{({\kappa_0^2 + {\kappa^2}} )}^{11/6}}}}\textrm{d}\kappa = \frac{1}{2}\kappa _0^{2\mu - 8/3}\Gamma \left( {\mu + \frac{1}{2}} \right)\textrm{U}\left( {\mu + \frac{1}{2};\mu - \frac{1}{3};\frac{{\kappa_0^2}}{{\kappa_H^2}}} \right),$$

where $\textrm{U}({\cdot} )$ is the confluent hypergeometric function of the second kind.

By Eqs. (13), (14), and (15), we have the transverse spatial coherence radius of a spherical wave

(16)$$\begin{aligned} {\rho _o} &= \left\{ {\frac{{2.144({n_R^2 + n_I^2} )C_{\varepsilon {\chi_T}}^2z}}{{{\lambda^2}{{({1 - \varpi } )}^2}}}\kappa_0^{1/3}\left[ {{\varpi^2}\textrm{U}\left( {2;\frac{7}{6};\frac{{\kappa_0^2}}{{\kappa_T^2}}} \right) + \textrm{U}\left( {2;\frac{7}{6};\frac{{\kappa_0^2}}{{\kappa_S^2}}} \right)} \right.} \right.\\ &\left. { - 2\varpi \textrm{U}\left( {2;\frac{7}{6};\frac{{\kappa_0^2}}{{\kappa_{TS}^2}}} \right)} \right] + 9.862l_0^{2/3}{\kappa _0}\Gamma \left( {\frac{7}{3}} \right)\frac{{({n_R^2 + n_I^2} )C_{\varepsilon {\chi _T}}^2z}}{{{\lambda ^2}{{({1 - \varpi } )}^2}}}\\ &{\left. {\quad \times \left[ {{\varpi^2}\textrm{U}\left( {\frac{7}{3};\frac{3}{2};\frac{{\kappa_0^2}}{{\kappa_T^2}}} \right) + \textrm{U}\left( {\frac{7}{3};\frac{3}{2};\frac{{\kappa_0^2}}{{\kappa_S^2}}} \right) - 2\varpi \textrm{U}\left( {\frac{7}{3};\frac{3}{2};\frac{{\kappa_0^2}}{{\kappa_{TS}^2}}} \right)} \right]\,} \right\}^{ - 1/2}}, \end{aligned}$$

in here $\Gamma ({\cdot} )$ is gamma function.

Substitute Eq. (5) and Eq. (12) into Eq. (11), we obtain

(17)$$\begin{aligned} P({m/{m_0}}) &= \frac{1}{{{{({2\pi } )}^2}}}\exp ({ - 2{k_0}{n_I}z} )\int {R_{{m_0},p}^ \ast (r,z){R_{{m_0},p}}(r,z)\exp ({ - 2{r^2}/\rho_o^2} )} \\ &\times \left\{ {\int_0^{2\pi } {\int_0^{2\pi } {\exp [{2\cos ({\varphi - \varphi^{\prime}} ){r^2}/\rho_o^2} ]\exp [{ - \textrm{i}({m - {m_0}} )({\varphi - \varphi^{\prime}} )} ]} } \textrm{d}\varphi^{\prime}\textrm{d}\varphi } \right\}r\textrm{d}r. \end{aligned}$$

Consider the integral expression [22]

(18)$$\int_0^{2\pi } {\exp [\tau \cos (\theta - \phi ) - \textrm{i}m\theta ]} \textrm{d}\theta = 2\pi {\textrm{I}_m}(\tau )\exp ( - i m\phi ), $$

where ${\textrm{I}_m}({\cdot} )$ is the Bessel function of second kind with m order. We get the simplified relation of Eq. (17)

(19)$$P({m/{m_0}} )= \exp ({ - 2{k_0}{n_I}z} )\int {R_{{m_0},p}^ \ast (r,z){R_{{m_0},p}}(r,z)\exp ({ - 2{r^2}/\rho_o^2} )} {\textrm{I}_{m - {m_0}}}({2{r^2}/\rho_o^2} )r\textrm{d}r. $$

Substitute Eq. (6) in Eq. (19), then we can further conclude

(20)$$\begin{aligned} P({m/{m_0}}) &= \frac{{4p!\exp ({ - 2{k_0}{n_I}z} )}}{{{w^2}(z )(p + |{{m_0}} |)!}}\int_0^{D/2} {{{\left( {\frac{{2{r^2}}}{{{w^2}(z )}}} \right)}^{|{{m_0}} |}}} \\ &\times \exp \left( { - \frac{{{r^2}}}{{w_{eff}^2}}} \right){\textrm{I}_{m - {m_0}}}\left( {\frac{{2{r^2}}}{{\rho_o^2}}} \right){\left|{\textrm{L}_p^{{m_0}}\left( {\frac{{2{r^2}}}{{{w^2}(z )}}} \right)} \right|^2}r\textrm{d}r, \end{aligned}$$

where ${w_{eff}} = {\left( {\frac{2}{{{w^2}(z )}} + \frac{2}{{\rho_o^2}} - \frac{{{k_0}{n_I}z}}{{R(z )}}} \right)^{ - 1/2}}$ is defined as the effective radius of PLG in the absorbed seawater turbulence.

For $m \ne {m_0}$, $P{({m|{{m_0}} } )_{m \ne {m_0}}}$ is the crosstalk probability of the OAM mode that describes the probability of the photon migrating from the OAM signal mode to the new OAM mode, and $m = {m_0}$, $P{({m|{{m_0}} } )_{m = {m_0}}}$ is the signal probability of the OAM mode.

According to the analysis of literature [23], we define the signal-to- noise ratio (SNR) of OAM mode ${m_0}$ as

(21)$$SNR({m,{m_0}} )= \frac{{P{{({m/{m_0}} )}_{m = {m_0}}}}}{{\sum\limits_{m ={-} \infty }^\infty {P{{({m/{m_0}} )}_{m \ne {m_0}}}\textrm{ + }{N_0}/{P_{TX}}} }}. $$

where ${\sum\limits_{m ={-} \infty }^\infty {P(m/{m_0})} _{m \ne {m_0}}}$ represents the wave power spread from other OAM channels is configured as independent each other. Note: If the SNR of the receiving and transmitting systems $SN{R_0} = {P_{XT}}/{N_0}$ is in decibel, the ${N_0}/{P_{TX}}$ in Eq. (19) needs to be calculated by conversion ${N_0}/{P_{XT}} = {10^{ - \frac{{SN{R_0}(\textrm{db})}}{{10}}}}$.

For signal modulation of on-off keying (OOK), based on (19), the BER of OAM channels is derived as [24]

(22)$${p_m} = \frac{1}{2}\textrm{erfc}\sqrt {\frac{{SNR({m,{m_0}} )}}{2}} ,$$

where $\textrm{erfc}(x )$ denotes the complementary error function.

Now, we analyze the influence of turbulent seawater on the average capacity of optical communication link with PLG beam. Considering that (1) the energy of OAM state carried by PLG beam decreases rapidly with the increase of topological charge m₀ deviation from 0. (2) The photons of OAM signal caused by turbulence mainly transfer to the OAM energy states close to it. Therefore, when the number of channel m of OAM energy state is greater than the topology m₀ of OAM signal state, we can approximate the OAM channel as a symmetric channel N=2m+1. Using the calculation relation of information capacity of multi-level symmetric channel [25], we define the average capacity of optical communication link with N symmetric OAM channels as

(23)$$\begin{array}{l} C = {\log _2}N + ({1 - {p_m}} ){\log _2}({1 - {p_m}} )+ {p_m}{\log _2}\frac{{{p_m}}}{{N - 1}}\\ \;\;\; = {\log _2}N + \left( {1 - \frac{1}{2}\textrm{erfc}\sqrt {\frac{{SNR({m,{m_0}} )}}{2}} } \right){\log _2}\left( {1 - \frac{1}{2}\textrm{erfc}\sqrt {\frac{{SNR({m,{m_0}} )}}{2}} } \right)\\ \;\;\; + \frac{1}{2}\textrm{erfc}\sqrt {\frac{{SNR({m,{m_0}} )}}{2}} {\log _2}\frac{1}{{2({N - 1} )}}\textrm{erfc}\sqrt {\frac{{SNR({m,{m_0}} )}}{2}} . \end{array}$$

where $N ={-} m, - m + 1, \cdots ,0, \cdots m - 1,m.$

4. Numeric analysis

In this section, we will analyze the evolution of BER and average capacity of the wireless optical communication channel (link) with PLG beam in turbulence and the experimental value of the absorbable seawater [20] through numerical simulation.

In the following numerical analysis, we set the calculation parameters as follows, unless otherwise stated. That is, z = 100m, P_Tx/N₀ = 32 dB, m= 9, Pr_T = 0.72, Pr_s = 700, Q = 2.5, ε=10⁻³m²/s³, ϖ= – 4.5, χ_T = 10⁻⁷K²/s, ${w_0}\textrm{ = }0.01\textrm{m}$, η=0.001m, L₀ = 10m, ${m_0} = 1$, p = 0, λ=410nm (n_I = 0.0878 × 10⁻⁹), and D = 0.02m.

Figure 1 shows the receive intensity distribution of PLG beam in turbulent seawater at the (x, y) plane. It can be seen from Fig. 1(a) that with the increase of OAM topological charge, the radius of dark spot and the radius as well as width of bright ring of received intensity also increases, the average signal intensity per unit area of the ring decreases accordingly. Figure 1(b) shows that with the increase of the radial order p, the dark spot radius, bright ring radius and bright ring width decrease accordingly, but the number of bright ring increases and the distribution area of bright rings is enlarged. Thus, with the increase of the radial order p, the average signal intensity per unit area in the receiving aperture decreases. The cross section of light intensity distribution of Fig. 1(a) and Fig. 1(b) are shown in Fig. 1(c) and Fig. 1(d).

Fig. 1. The intensity of the OAM mode carried by perfect optical vortex in xy plane for (a) p = 0, the OAM quantum number m₀ = 1, 3, 5 and 7; (b) m₀ = 1, propagation distance p = 1, 2,3 and 4; (c) two-dimensional graph of light intensity distribution in (a) and (d) two-dimensional graph of light intensity distribution in (c) and (d).

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The average capacity and BER of OAM channels with PLG beam as the function of propagation z for difference wavelengths as well as imaginary index of refraction is shown in Fig. 2, in here, the four wavelengths and their corresponding imaginary refractive index (absorption) are respectively selected as: λ=410 nm (n_I = 0.0878×10⁻⁹), 470 nm (n_I =0.3851×10⁻⁹), 530 nm (n_I =1.7877×10⁻⁹) and 580 nm (n_I =4.1354×10⁻⁹).

Fig. 2. Average capacity and bit-error rate of OAM channels with perfect Laguerre-Gaussian beam versus transmission distance z for difference wavelengths. (a) Average capacity. (b) Bit-error rate.

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Figure 2 shows that the average capacity of OAM channel decreases with the increase of transmission distance, but the change trend of BER is opposite. Clearly, this result is apparently due to the fact that seawater absorption reduces the emission information received by the probe. Figure 2 also shows that the relationship between the average capacity or bit error rate of OAM channel and signal wavelength or virtual refractive index is complex. This phenomenon is caused by the fact that the imaginary refractive index of seawater channel is a nonlinear function of wavelength, and the loss of signal absorption of high imaginary refractive index seawater is greater than that of low imaginary refractive index seawater. However, the turbulence scintillation loss of long wavelength signal is smaller than that of short wavelength signal [19]. The results show that the conclusion that the long-wave signal is beneficial to the signal transmission in the channel obtained from the clean seawater turbulence is not applicable to the actual seawater environment.

In Fig. 3 we report the evolution curves of average capacity and BER of OAM channels with carrier of the PLG versus the initial beam width ${w_0}$ for different received diameter D. From Fig. 3 we can find that the average capacity decreases with increases of the ${w_0}$ and D. This conclusion can be drawn directly from the spatial distribution of receive intensity as shown in Fig. 1. That is, for a given channel parameter, when the radius of transmitting aperture and receiving aperture increases, the dark spot area located in receiving aperture increases, so the average capacity decreases. On the other hand, when the transmitting beam size ${w_0}$ and the received diameter D are larger, the aberration between the beams transmitted in the cylindrical channel formed by the initial and receiving apertures also increases, so that OAM crosstalk increase that causes the received probability of OAM signal decrease [26].

Fig. 3. Average capacity and bit-error rate of OAM channels with perfect Laguerre-Gauss beam versus the initial beam size ${w_0}$ for the difference receiver diameter D. (a) Average capacity. (b)Bit-error rate.

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Figure 4 displays the average capacity and BER of OAM channels with the radial order of PLG p as the variable under the condition in four different topological charges m₀ of OAM mode. The curves in Fig. 4 show that the average capacity of OAM channel decreases with the increase of the radial order of PLG p and the OAM topological charges m₀, but the change trend of BER is opposite. This result can also be explained by the received light intensity distribution of PLG beam through turbulent seawater as shown in Fig. 1. When the radial order of PLG p and the topological charge of the OAM signal mode transmitted increase, the average signal intensity in the receiving aperture domain decreases, as a result, the BER of the system increases and the received signal information decreases. The research results here show that for the underwater communication system using OAM mode as the information carrier, the radial mode order of PLG beam is set as 0, that is, p =0. Furthermore, the information to be transmitted should be loaded on the OAM mode within the small value range.

Fig. 4. Average capacity and bit-error rate of OAM channels with perfect Laguerre-Gauss beam versus the OAM quantum number m₀ for the difference radial order of PLGs p. (a) Average capacity. (b) Bit-error rate.

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In Fig. 5, we investigate the effects of the dissipation rate of kinetic energy per unit mass of fluid ε and dissipation rate of the mean-squared temperature χ_T on the average capacity and BER of OAM channels with PLG beam for the ratio of temperature and salinity contributions to the refractive index spectrum ϖ. It is clear from the Fig. 5 that the average capacity decreases with increasing χ_T and decreasing ε. However, the effect of the χ_T and ε on BER is opposite. The reason for these results is that with the increase of χ_T and the decrease of ε, the turbulence strength of seawater increases, and the distortion of the OAM mode wave front due to turbulence also increases. Of course, the level of OAM crosstalk generated by turbulent channels also increases, and the channel capacity of link with OAM signals decreases. Figure 5 also shows that the disturbance effect of seawater salinity fluctuations on average capacity is higher than that of temperature fluctuations.

Fig. 5. Average capacity and bit-error rate of OAM channels with perfect Laguerre-Gauss beam versus the ratio of temperature and salinity contributions to the refractive index spectrum ϖ for different dissipation rate of kinetic energy per unit mass of fluid ε and dissipation rate of the mean-squared temperature χ_T. (a), (c) average capacity, and (b), (d) bit-error rate.

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In Fig. 6, we give the numerical curve of the average capacity and BER of OAM channels with PLG beam changing as the variation of inner scale and outer scale of seawater turbulence. Figure 6 indicates that the average capacity increases with the increase of inner scale of seawater turbulence, but it decreases with the increase of the outer scale of seawater turbulence, however, the effect of the η and L₀ on BER is opposite. According to the theory of turbulent effects, the inner scale of turbulence mainly produces forward scattering of the transmitting beam, hence, with the increase of the inner scale size of turbulence, the uniform area in the seawater increases, and the wave front distortion of the OAM signal passing through the channel is reduced, that is, the transmittance of the OAM signal is increased. It can also be seen from Fig. 6 that the average capacity of OAM channel increases as the decrease of the outer scale of turbulence. The reason for this result is that the outer scale of turbulence mainly leads to the random deflection of the propagation path of the beam, and the value of the random deflection is proportional to the size of the outer scale of turbulence; therefore, for a channel with a larger turbulence outer scale, the greater the random deflection generated by it, makes the random optical path difference between the sub-beams of the PLG beam that constitute the OAM mode in any plane perpendicular to the optical path also greater, that is, the wave front distortion of the OAM mode is also larger.

Fig. 6. Average capacity and bit-error rate of OAM channels with perfect Laguerre-Gauss beam versus the turbulence’s inner-scale η for the difference outer-scale L₀. (a) Average capacity. (b) Bit-error rate.

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In Fig. 6, we give the numerical curve of the average capacity and BER of OAM channels with PLG beam changing as the variation of inner scale and outer scale of seawater turbulence. Figure 6 indicates that the average capacity increases with the increase of inner scale of seawater turbulence, but it decreases with the increase of the outer scale of seawater turbulence, however, the effect of the η and L₀ on BER is opposite. According to the theory of turbulent effects, the inner scale of turbulence mainly produces forward scattering of the transmitting beam, hence, with the increase of the inner scale size of turbulence, the uniform area in the seawater increases, and the wave front distortion of the OAM signal passing through the channel is reduced, that is, the transmittance of the OAM signal is increased. It can also be seen from Fig. 6 that the average capacity of OAM channel increases as the decrease of the outer scale of turbulence. The reason for this result is that the outer scale of turbulence mainly leads to the random deflection of the propagation path of the beam, and the value of the random deflection is proportional to the size of the outer scale of turbulence; therefore, for a channel with a larger turbulence outer scale, the greater the random deflection generated by it, makes the random optical path difference between the sub-beams of the PLG beam that constitute the OAM mode in any plane perpendicular to the optical path also greater, that is, the wave front distortion of the OAM mode is also larger.

5. Conclusion

In this paper, we focused on the impacts of weakly turbulent absorbing seawater on the average capacity and BER of OAM channels with the PLG carrier using Rytov approximation and the modeling of the average capacity and BER of OAM channels. The main results of this paper are summarized as follows: with the increase of OAM topological charge, the radius of dark spot and bright ring of receive intensity increases, and with the increase of the radial cylindrical coordinate p, the radius of dark spot and bright ring decreases, but the number of dark ring and bright ring increases. The transmission signal with longer wave length is beneficial to signal transmission [10] is no longer suitable for long and weakly turbulent absorbing seawater channel. The average capacity of OAM channel decreases with the increase of transmission distance, the initial beam size and receiver diameter, dissipation rate of the mean-squared temperature, topological charges m₀ of OAM mode, the radial order of PLG p and the outer scale of seawater turbulence. As the rate of dissipation of kinetic energy per unit mass of fluid and inner scale of seawater turbulence increase, the average capacity of OAM channel increases. The variation trend of BER with beam parameters and seawater turbulence parameters is opposite to that of channel capacity.

Funding

National Natural Science Foundation of China (61871202); Jiangsu Province Postgraduate Research and Innovation Plan (SJCX20_0764).

Acknowledgments

We would like to thank the anonymous reviewers for their valuable comments, which significantly improved the presentation of this paper. The authors acknowledge funding from the National Natural Science Foundation of China.

Disclosures

The authors declare no conflicts of interest.

Data availability

All the numerical data in this paper are real and available. Data underlying the results about absorption coefficient n_I presented in this paper are available in Refs. [20].

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Bit-error rate and average capacity of an absorbent and turbulent underwater wireless communication link with perfect Laguerre-Gauss beam

Abstract

1. Introduction

2. Perfect Laguerre-Gauss beam in weakly turbulent absorbing seawater

3. Bit-error rate and average capacity of the orbital angular momentum link

4. Numeric analysis

5. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (23)

Optics Express