光电工程  2020, Vol. 47 Issue (6): 190591      DOI: 10.12086/oee.2020.190591     
汉克-贝塞尔光束在各向异性海洋湍流中轨道角动量传输特性分析
贺锋涛1 , 房伟1 , 张建磊1 , 杨祎1 , 杜迎1 , 张斌2     
1. 西安邮电大学电子工程学院,陕西 西安 710121;
2. 中国船舶重工集团第705研究所,水下信息与控制重点实验室,陕西 西安 710077
摘要:基于Rytov近似理论,分析了各向异性海洋湍流中汉克-贝塞尔(HB)光束的交叉谱密度,研究了轨道角动量(OAM)模式探测概率、串扰概率及HB光束的螺旋相位谱,建立了各向异性海洋湍流中OAM模式探测概率模型。结果表明,HB光束在各向异性海洋湍流环境中发射OAM模式的探测概率高于在各向同性海洋湍流环境中的探测概率。并且随着各向异性因子的增大,海洋湍流对发射OAM模式探测概率的影响减小,串扰模式的探测概率也随之下降。
关键词各向异性海洋湍流    汉克-贝塞尔光束    轨道角动量    螺旋相位谱    光学涡旋    
Analysis of the transmission characteristics of Hank-Bessel beam in anisotropic ocean turbulence
He Fengtao1, Fang Wei1, Zhang Jianlei1, Yang Yi1, Du Ying1, Zhang Bin2     
1. School of Electronic Engineering, Xi'an University of Posts and Telecommunications, Xi'an, Shaanxi 710121, China;
2. Key Laboratory of Underwater Information and Control, China Shipbuilding Industry Corporation 705 Research Institute, Xi'an, Shaanxi 710077, China
Abstract: Based on the Rytov approximation theory, we analyze the cross-spectral density of Hankel-Bessel (HB) beams in anisotropic ocean turbulence. In this paper, we study the orbital angular momentum (OAM) mode detection probability, the crosstalk probability and the spiral phase spectrum of the HB beam, and establish the OAM mode detection probability model in anisotropic ocean turbulence. The results show that the detection probability of the emission mode is decreased and the spiral phase spectrum is expanded due to the ocean turbulence. Furthermore, with the increase of anisotropy factor, the influence of ocean turbulence on the detection probability of HB beam becomes smaller. Meanwhile, with the increase of the temperature variance dissipation rate and the equilibrium parameter, and the decrease of the dynamic energy dissipation rate, the influence of ocean turbulence on the orbital angular momentum transmission is increased.
Keywords: anisotropic ocean turbulence    Hank-Bessel beam    orbital angular momentum    orbital angular momentum spectrum    optical vortex    

1 引言

涡旋光束因为携带有轨道角动量(orbital angular momentum,OAM),且OAM具有相互正交性,因此利用OAM空域复用技术可以提高光通信信道容量。近年来,涡旋光束在无线光通信中得到了研究人员的广泛关注。Ren等[1]通过复用4个携带不同OAM模式的绿光实现了40 Gbit/s的链路。但光束在海洋中传输时,海洋湍流对光束的传输特性产生影响,导致相位畸变、模式串扰[2-3]

基于Nikishov[4]建立的海洋湍流折射率起伏空间功率谱,Cheng等[5]研究了Laguerre-Gaussian(LG)光束在各向同性海洋湍流中的传输特性,分析了各向同性海洋湍流对LG光束轨道角动量模式探测概率的影响。Yin等[6]研究了Hankel-Bessel(HB)光束在各向同性海洋环境中螺旋相位谱受湍流的影响。此外,其他学者对艾利光束[7]、部分相干LG光束[8]在各向同性海洋湍流中的传输特性进行了研究,以及叠加光束在海洋湍流中的抗干扰特性[9-11]。但是上述研究都是基于各向同性的海洋湍流环境,实际上海洋湍流环境由于地球自转的原因是各向异性的[12],Huang等[13]分析了各向异性海洋湍流中光束质量、平均光强,讨论了光束初始相干度与抗湍流干扰之间的关系,Chen等[14]研究了在各向异性海洋湍流中部分相干修正贝塞尔(partially coherent modified Bessel correlated,PCMBC)光束OAM模式与各向异性因子的关系,Li等[15]研究了Hermite-Gaussian(HG)光束OAM在各向异性海洋湍流的传输特性;此外HB光束具有无衍射特性,即通过一定传输距离后中心光斑、光强分布保持不变,通过障碍物后可以重建横向强度分布[16-18],因此研究HB光束在各向异性海洋湍流的传输特性对海洋环境无线光通信链路有重要意义。目前,关于各向异性海洋湍流中HB光束轨道角动量的传输特性的研究还未见报道。

本文首先基于Rytov近似理论推导了各向异性海洋湍流中HB光束的交叉谱密度;数值模拟分析了在各向异性海洋湍流和各向同性海洋湍流下HB光束发射OAM模式探测概率随传输距离的变化;然后计算了HB光束在各向异性海洋湍流中的螺旋相位谱;分析讨论了在不同各向异性因子下,平衡参数、温度方差耗散率、动能耗散率与OAM模式探测概率的关系。

2 理论分析 2.1 各向异性海洋湍流模型

在Markov[13]近似下,各向异性海洋湍流中折射率波动空间谱模型为

$ \begin{array}{*{20}{l}} {{\varphi _{{\rm{an}}}}({\kappa ^\prime }) = 0.388 \times {{10}^{ - 8}}{\varepsilon ^{ - 1/3}}\xi {\chi _{\rm{T}}}{{({\kappa ^\prime })}^{ - 11/3}}[1 + 2.35{{({\kappa ^\prime }\eta )}^{2/3}}]}\\ {{\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} \times [{\rm{exp}}( - {A_{\rm{T}}}\delta ) + {\omega ^{ - 2}}{\rm{exp}}({A_{\rm{S}}}\delta ) - {\omega ^{ - 1}}{\rm{exp}}({A_{{{\rm{T}}_{\rm{s}}}}}\delta )], } \end{array} $ (1)

其中:ξ为各向异性因子,当ξ=1时,海洋湍流为各向同性;${\kappa ^\prime } = \xi \sqrt {\kappa _x^2 + \kappa _y^2} $表示空间频率;ε是单位质量湍流动能耗散率,取值范围是10-1m2/s3~10-10 m2/s3χT是温度方差耗散率,取值范围是10-4K2/s~10-10 K2/s;η是柯尔莫哥洛夫(Kolmogorov)尺度;ω是温度盐度平衡参数,表征温度与盐度在功率谱中的比值,取值范围是-5~0,当ω=-5时表示海洋湍流完全受温度影响,当ω=0时表示海洋湍流完全受盐度影响。

$ \begin{array}{*{20}{c}} {{A_{\rm{T}}} = 1.863 \times {{10}^{ - 2}};\quad {A_{\rm{S}}} = 1.9 \times {{10}^{ - 4}};\quad {A_{{{\rm{T}}_{\rm{s}}}}} = 9.41 \times {{10}^{ - 3}};}\\ {\delta = 8.284{{({\kappa ^\prime }\eta )}^{4/3}} + 12.978{{({\kappa ^\prime }\eta )}^2}}。\end{array} $
2.2 HB光束在各向异性海洋湍流中的传输特性

HB光束在自由空间中传输的复振幅为[5]

$ \begin{array}{*{20}{l}} {M(\mathit{\boldsymbol{\rho }}, z) = {{\rm{i}}^{3{l_0} + 1}}{l_0}!{A_0}\sqrt {\frac{\pi }{{2kz}}} {\rm{exp}}\left[ {{\rm{i}}(kz - \frac{{\pi {l_0}}}{4} - \frac{\pi }{4}) + {\rm{i}}{l_0}\varphi } \right]}\\ {{\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} \times {{\rm{J}}_{{l_0}/2}}(k{\rho ^2}/4z), } \end{array} $ (2)

其中:k=2π/λλ是波长,l0是HB光束的OAM模式数,A0是光功率常数,Jm(x)是m阶第一类贝塞尔函数,ρ=(ρ, φ)是源平面二维位置矢量。

在弱湍流起伏区[19],经过海洋湍流后的HB光束复振幅为[5]

$ M(\mathit{\boldsymbol{\rho }}, z) = {M_{{l_0}}}(\mathit{\boldsymbol{\rho }}, z){\rm{exp}}[{\psi _1}(\mathit{\boldsymbol{\rho }}, z)], $ (3)

其中ψ1(ρ, z)是各向异性海洋湍流对HB光束产生的相位扰动。在各向异性海洋湍流环境中,HB光束的交叉谱密度为[5]

$ \begin{array}{*{20}{l}} {W(\mathit{\boldsymbol{\rho }}, \mathit{\boldsymbol{\rho }}, z) = \langle M(\mathit{\boldsymbol{\rho }}, z)M({\mathit{\boldsymbol{\rho }}^\prime }, z)\rangle }\\ {\quad \approx {M_{{l_0}}}(\mathit{\boldsymbol{\rho }}, z)M_{{l_0}}^*(\mathit{\boldsymbol{\rho }}, z) \times \langle {\rm{exp}}[{\psi _1}(\mathit{\boldsymbol{\rho }}, z) + \psi _1^*(\mathit{\boldsymbol{\rho }}, z)]\rangle , } \end{array} $ (4)

其中:“*”为复数共轭符号,〈·〉是对各向异性海洋湍流的系综平均[15]

利用Rytov相位结构函数二次近似可得:

$ \begin{array}{*{20}{l}} {\langle {\rm{exp}}[{\psi _1}(\mathit{\boldsymbol{\rho }}, z) + \psi _1^*(\mathit{\boldsymbol{\rho }}, z)]}\\ {\quad \approx {\rm{exp}}[ - ({\rho ^2} + {\rho ^{\prime 2}} - 2\rho {\rho ^\prime }{\rm{cos}}(\varphi - {\varphi ^\prime }))/{\rho _{{\rm{oc - }}\xi }}], } \end{array} $ (5)

其中ρoc-ξ是各向异性海洋湍流球面波的空间相干长度,其表达式为[13]

$ {\rho _{{\rm{oc - }}\xi }} = {\left[ {\frac{{{\pi ^2}kz}}{3}{\xi ^{ - 4}}\int_0^\infty {{\kappa ^3}} {\varphi _{{\rm{an}}}}(\kappa ){\rm{d}}\kappa } \right]^{ - 1/2}}。$ (6)

将式(1)代入式(6),可得相干长度为

$ \begin{array}{*{20}{l}} {{\rho _{{\rm{oc - }}\xi }} = \xi |\omega |{{[1.802 \times {{10}^{ - 7}}{k^2}z{{(\varepsilon \eta )}^{ - 1/3}}]}^{ - 1/2}}}\\ { \cdot {{[{\chi _{\rm{T}}}(0.483{\omega ^2} - 0.835\omega + 3.380)]}^{ - 1/2}}}。\end{array} $ (7)
2.3 螺旋相位谱分析

HB光束在海洋中传输时,由于各向异性海洋湍流的影响,会使发射OAM模式的能量扩散到其他OAM模式上,产生新的OAM模式,这种现象称为模式串扰,会致使在接收端检测到的发射OAM模式概率降低。此时,忽略各向异性海洋湍流引起的光束扩展,通过对HB光束基模的叠加可得到接收端HB光束,即:

$ M(\mathit{\boldsymbol{\rho }}, z) = \sum\nolimits_{l = - \infty }^\infty {{a_l}(\mathit{\boldsymbol{\rho }}, z){\rm{exp}}({\rm{i}}l\varphi )} , $ (8)

其中al(ρ, z)为不同模式的系数。对al(ρ, z)求系综平均可得到HB光束OAM模式概率密度为

$ \begin{array}{*{20}{l}} {\langle |{a_l}(\mathit{\boldsymbol{\rho }}, z){|^2}\rangle = \frac{1}{{4{\pi ^2}}}\int_0^{2\pi } {\int_0^{2\pi } {{M_{{l_0}}}} } (\rho , \varphi , 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} \cdot M_{{l_0}}^*({\rho ^\prime }, {\varphi ^\prime }, z){\rm{exp}}[ - {\rm{i}}l(\varphi - {\varphi ^\prime })]}\\ {{\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} \cdot [\langle {\rm{exp}}[{\psi _1}(\rho , \varphi , z) + \psi _1^*({\rho ^\prime }, {\varphi ^\prime }, z)]\rangle ]{\rm{d}}\varphi {\rm{d}}{\varphi ^\prime }}。\end{array} $ (9)

将式(5)代入式(9)可得:

$ \begin{array}{*{20}{c}} {\langle |{a_l}(\mathit{\boldsymbol{\rho }}, z){|^2}\rangle = \frac{1}{{4{\pi ^2}}}\int_0^{2\pi } {\int_0^{2\pi } {{M_{{l_0}}}} } (\rho , \varphi , z)M_{{l_0}}^*({\rho ^\prime }, {\varphi ^\prime }, z)}\\ { \cdot {\rm{exp}}[ - ({\rho ^2} + {\rho ^{\prime 2}} - 2\rho {\rho ^\prime }{\rm{cos}}(\varphi - {\varphi ^\prime }))/{\rho _{{\rm{oc - }}\xi }}]}\\ { \cdot {\rm{exp}}[ - {\rm{i}}l(\varphi - {\varphi ^\prime })]{\rm{d}}\varphi {\rm{d}}{\varphi ^\prime }}。\end{array} $ (10)

将式(2)代入式(10),并利用积分关系[20]

$ \begin{array}{*{20}{l}} {\int_0^{2\pi } {{\rm{exp}}} [ - {\rm{i}}n {\varphi _1} + \eta {\rm{cos}}({\varphi _1} - {\varphi _2})]{\rm{d}}{\varphi _1}}\\ {\quad {\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} = 2\pi {\rm{exp}}( - {\rm{i}}n {\varphi _2}){I_n}(\eta )}。\end{array} $ (11)

OAM模式概率密度的解析表达式为

$ \begin{array}{*{20}{l}} {\langle |{a_l}(\mathit{\boldsymbol{\rho }}, z){|^2}\rangle = \frac{\pi }{{2kz}}{{({l_0}!{A_0})}^2}{{\left| {{{\rm{J}}_{{l_0}/2}}\left( {\frac{{k{\rho ^2}}}{{4z}}} \right)} \right|}^2}}\\ {{\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} \cdot {\rm{exp}}\left( { - \frac{{2{\rho ^2}}}{{\rho _{{\rm{oc}} - \xi }^2}}} \right){{\rm{I}}_{l - {l_0}}}\left( {\frac{{2{\rho ^2}}}{{\rho _{{\rm{oc}} - \xi }^2}}} \right)}。\end{array} $ (12)

式(11)和式(12)中,In(η)是n阶第一类修正贝塞尔函数。

当HB光束轨道角动量模式为l时,接收处的螺旋谐波能量为

$ \begin{array}{*{20}{l}} {{E_l}(z) = \int_0^R {\frac{\pi }{{2kz}}} {{({l_0}!{A_0})}^2}{{\left| {{{\rm{J}}_{{I_0}/2}}\left( {\frac{{k{\rho ^2}}}{{4z}}} \right)} \right|}^2}}\\ {{\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} \cdot {\rm{exp}}\left( { - \frac{{2{\rho ^2}}}{{\rho _{{\rm{oc}} - \xi }^2}}} \right){{\rm{I}}_{l - {l_0}}}\left( {\frac{{2{\rho ^2}}}{{\rho _{{\rm{oc}} - \xi }^2}}} \right)\rho {\rm{d}}\rho , } \end{array} $ (13)

其中R为光束的接收孔径。模式数为l的OAM探测概率Pl(z)为

$ {P_l}(z) = \frac{{{E_l}(z)}}{{\sum\limits_{m = - \infty }^{m = \infty } {{E_m}} (z)}}, $ (14)

其中$\sum\nolimits_{m = - \infty }^{m = \infty } {{E_m}(z)} $为所有OAM模式在接收端的能量总和。当l=l0Pl0(z)是发射模式l0的探测概率。

3 数值模拟分析

基于上述理论,数值模拟了各向异性海洋湍流中HB光束发射OAM模式探测概率、串扰模式概率和螺旋相位谱的分布变化。仿真中的参数设置为:取波长λ=532 nm,Kolmogorov尺度η=10-3 m,光功率A0=10,接收孔径R=3 cm,传输距离z=50 m。

图 1给出了HB光束发射OAM模式以及串扰模式探测概率,随传输距离z的变化曲线。其中,HB光束发射OAM模式l0=1,海洋湍流参数为:温度方差耗散率χT=10-7K-2/s,湍流动能耗散率ε=10-3 m2/s,平衡参数ω=-4。图 1(a)为基于各向同性海洋湍流折射率功率谱得到的OAM模式探测概率,图 1(b)1(c)1(d)为各向异性海洋湍流折射率功率谱得到的OAM模式探测概率,各向异性因子ξ分别取1、2、3。图 1表明,随着传输距离z的增加,HB光束发射OAM模式的探测概率逐渐下降,而串扰模式探测概率逐渐增加。可以看出,图 1(a)的结果与图 1(b)结果相同,即当各向异性因子ξ=1时,各向异性海洋湍流转化为各向同性湍流。图 1可以发现发射OAM模式在各向同性海洋湍流中的探测概率明显小于在各向异性海洋湍流中的探测概率。这表明各向同性海洋湍洋湍流对HB光束OAM模式的影响更为严重,并且在各向异性海洋湍流中,随着各向异性因子的增大,湍流导致的模式串扰减小,发射OAM模式的探测概率随距离增大时下降得更为缓慢。

图 1 OAM模式为1的HB光束在不同的各向异性海洋湍流中,OAM模式的探测概率随传输距离z的变化曲线 Fig. 1 The variation of the detecting probability of OAM modes with the transmission distance of HB beam in different anisotropic ocean turbulence

图 2给出了OAM模式l0=5的HB光束,在传输距离z=50 m处,温度方差耗散率χT=10-7K-2/s,湍流动能耗散率ε=10-3 m2/s,平衡参数ω=-4,不同各向异性因子下HB光束的螺旋相位谱,可以更清楚地看出,随着各向异性因子增大,接收端探测到的发射OAM模式的概率越大,串扰模式的探测概率越小,HB光束受湍流的影响越小,如接收端OAM模式l0=5在ξ=3时的探测概率明显大于ξ=1时的探测概率。

图 2 OAM发射模式l0=5,传输距离为z=50 m,HB光束的螺旋相位谱 Fig. 2 OAM spectra of HB vortex beam for l0=5 with propagation distance z=50 m

图 345分析了不同各向异性因子下OAM模式l0=1的HB光束的探测概率随不同平衡参数ω、温度方差耗散率χT、湍流动能耗散率ε的变化曲线。图 3为温度耗散率χT=10-7K-2/s,动能耗散率ε=10-3 m2/s,平衡参数ω取-4.5、-3.5和-2.5时,发射OAM模式的探测概率随传输距离z的变化。图 3(a)3(b)分别是基于各向同性海洋湍流折射率功率谱和各向异性海洋湍流折射率得到的发射OAM模式探测概率,图 3(b)中当各向异性因子取ξ=1时,与图 3(a)中发射OAM模式探测概率曲线一致,随着ω增大,OAM模式探测概率越小。平衡参数ω定义为温度与盐度对海洋湍流变化贡献的比值,ω的值越接近0,盐度对湍流的贡献越大,此时ρoc-ξ值很小,海洋湍流导致的模式串扰也更严重。所以,盐度波动为主的海洋湍流对HB光束的影响更大。更重要的是,图 3(b)表明,ω的值恒定时,随着各向异性因子的增大,OAM模式的探测概率明显增大。

图 3 各向异性因子ξ分别为1、3、6,不同的ω时,发射OAM模式的探测概率随z的变化曲线 Fig. 3 Detection probability of launch OAM mode against z for different ξ and ω

图 4 各向异性因子ξ分别为1、3、6,不同的ε时,发射OAM模式的探测概率随z的变化曲线 Fig. 4 Detection probability of launch OAM mode against z for different ξ and ε

图 5 各向异性因子ξ分别为1、3、6,不同的χT时,发射OAM模式的探测概率随z的曲线变化 Fig. 5 Detection probability of launch OAM mode against z for different ξ and χT

图 4为温度耗散率χT=10-7K-2/s,平衡参数ω=-4,动能耗散率ε取10-1 m2/s-3和10-2 m2/s-3时,发射OAM模式的探测概率随传输距离z的变化,图 4(a)4(b)分别是基于各向同性海洋湍流折射率功率谱和各向异性海洋湍流折射率得到的发射OAM模式探测概率,同样可以看出,各向异性因子ξ=1时,与基于各向同性海洋湍流功率谱得到的结果相同,发射OAM模式的探测概率随着动能耗散率的增大而增大。因为χT一定时,ε的取值越大,ρoc-ξ的值也越大,发射OAM模式的探测概率也就越高。同时也可以得到,当ε恒定时,随着各向异性因子的增大,OAM模式探测概率受海洋湍流的影响显著减小。

图 5为平衡参数ω=-4,动能耗散率10-3 m2/s-3,温度耗散率χT取5×10-7 K2/s和1×10-6 K2/s时,在不同各向异性因子海洋湍流中发射OAM模式的探测概率随传输距离z的变化。图 5(a)5(b)分别是基于各向同性海洋湍流折射率功率谱和各向异性海洋湍流折射率得到的发射OAM模式探测概率。可以看出,各向异性因子恒定时,发射OAM模式的探测概率随着温度方差耗散率的增大而减小。原因是ε一定时,χT的取值越大,ρoc-ξ的值反而越小,海洋湍流对HB光束的带来的负面影响越严重,因此模式串扰越严重,发射OAM模式的探测概率也就越小。更为重要的是,改变各向异性因子对OAM模式探测概率的影响更大。

图 345中可以观察到,探测概率随着平衡参数ω、温度耗散率χT的增大而减小,随着动能耗散率ε的增大而增大。更重要的是,当ωεχT一定时,随着各向异性因子ξ的增大,发射OAM模式探测概率明显增大。这表明HB光束在各向异性海洋环境中传输受湍流的影响明显小于在各向同性的海洋中传输所受到的影响,并且各向异性因子越大,发射OAM模式的探测概率越大,海洋湍流产生的模式串扰越小。

4 结论

本文首先介绍了各向异性海洋湍流的折射率空间谱模型,在此基础上推导了在各向异性海洋湍流中HB光束的空间相干长度,分析得到各向异性海洋湍流中HB光束交叉谱密度,从而得到各向异性海洋湍流中HB光束OAM模式探测概率数学模型;数值模拟了在各向异性海洋湍流下HB光束OAM模式探测概率、串扰概率以及螺旋相位谱分布,并验证了各向异性海洋湍流谱中ξ=1时,湍流对HB光束的影响与各向同性湍流谱的结果一致。结果表明,随着温度方差耗散率、平衡参数的增加,以及动能耗散率的减小,接收端模式串扰加重,发射OAM模式的探测概率减小,螺旋相位谱扩展严重;进一步发现,随着各向异性因子的增大,海洋湍流对HB光束的模式串扰影响减小,发射OAM模式的探测概率和螺旋相位谱的扩展有显著的改善。本研究结果为海洋无线光通信系统的性能估计提供一定参考价值。

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