贺锋涛，房 伟，张建磊*，杨 祎，杜 迎，张 斌

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

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。

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

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

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]

在弱湍流起伏区[19]，经过海洋湍流后的HB光束复振幅为[5]

利用Rytov相位结构函数二次近似可得：

将式(1)代入式(6)，可得相干长度为

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：

将式(5)代入式(9)可得：

将式(2)代入式(10)，并利用积分关系[20]：

OAM模式概率密度的解析表达式为

当HB光束轨道角动量模式为时，接收处的螺旋谐波能量为

3 数值模拟分析

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

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

图3 各向异性因子x分别为1、3、6，不同的w时，发射OAM模式的探测概率随z的变化曲线

图4 各向异性因子x分别为1、3、6，不同的e时，发射OAM模式的探测概率随z的变化曲线

图5 各向异性因子x分别为1、3、6，不同的cT时，发射OAM模式的探测概率随z的曲线变化

从图3、4、5中可以观察到，探测概率随着平衡参数、温度耗散率T的增大而减小，随着动能耗散率的增大而增大。更重要的是，当、、T一定时，随着各向异性因子的增大，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=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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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

1School of Electronic Engineering, Xi¢an University of Posts and Telecommunications, Xi¢an, Shaanxi 710121, China;2Key Laboratory of Underwater Information and Control, China Shipbuilding Industry Corporation 705 Research Institute, Xi¢an, Shaanxi 710077, China

OAM spectra of HB vortex beam for0=5 with propagation distance=50 m

Overview:The orbital angular momentum (OAM) is carried to Hank-Bessel (HB) vortex beam, and the HB vortex beam has non-diffracting nature and self-focusing properties, for instance, it does not change without diffracting propagation. Lateral intensity distribution can be reconstructed when the HB beam encounter obstacles. With the development of underwater wireless optical communication (UOWC) technology, the OAM-carrying beam is used to study high-capacity and ultra-high-speed underwater wireless optical communication. Different OAM modes are orthogonal to each other, and the channel capacity of the underwater wireless optical communication link can be improved by using the orbital angular momentum spatial multiplexing technique. Consequently, HB vortex beams can be used as the carriers to increase the channel capacity of information transmission. However, due to the rotation of the earth, the OAM mode crosstalk of the vortex beam is caused by the anisotropic ocean turbulence when the beam is transmitted in ocean. The effects include beam point jitter, intensity and phase fluctuation and damage beam pattern. Thereby, the detection probability of transmitting OAM is reduced, and the error rate of the underwater wireless optical communication link is increased. Therefore, in this paper, the spiral phase spectrum of the HB vortex beam in an anisotropic ocean turbulent channel is studied. Firstly, based on the Rytov approximation theory, the cross-spectral density of HB beams in anisotropic ocean turbulence is analyzed, and the influence of anisotropic ocean turbulence on HB beam propagation is studied. An OAM crosstalk model of HB beam in anisotropic ocean turbulence is established by analyzing the spiral phase spectrum of HB beams in anisotropy ocean turbulence. The relationship between mode crosstalk and equilibrium parameters, temperature variance dissipation rate, dynamic energy dissipation rate is discussed, and compared with the transmission characteristics of HB beams in isotropic 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 increases 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. In the same way, with the increase of the temperature variance dissipation rate and the equilibrium parameter, and the decrease of the dynamic energy dissipation rate, the spatial coherence length in oceanic turbulence decreases is increased. Moreover, OAM mode detection probability, the crosstalk probability and the spiral phase spectrum of the HB beam are more negatively affected by ocean turbulence dominated by saliniy fluctuations.

Citation: He F T, Fang W, Zhang J L,Analysis of the transmission characteristics of Hank-Bessel beam in anisotropic ocean turbulence[J]., 2020, 47(6): 190591

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

1School of Electronic Engineering, Xi¢an University of Posts and Telecommunications, Xi¢an, Shaanxi 710121, China;2Key Laboratory of Underwater Information and Control, China Shipbuilding Industry Corporation 705 Research Institute, Xi¢an, Shaanxi 710077, China

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.

anisotropic ocean turbulence; Hank-Bessel beam; orbital angular momentum; orbital angular momentum spectrum; optical vortex

O439；P401

A

10.12086/oee.2020.190591

: He F T, Fang W, Zhang J L,. Analysis of the transmission characteristics of Hank-Bessel beam in anisotropic ocean turbulence[J]., 2020,47(6): 190591

贺锋涛，房伟，张建磊，等. 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[J]. 光电工程，2020，47(6): 190591

Supported by National Natural Science Foundation of China (61805199), the National Defense Innovation Special Zone Project of Science and Technology of China (18-H 863-01-ZT-001-004-02), the National Natural Science Foundation of Shaanxi (2018JQ6065), and National Key Laboratory Project of Underwater Information and Control ( XK-01-61-KS-0176).

* E-mail: zhangjianlei@xupt.edu.cn

2019-09-30；

2019-12-20

国家自然科学基金资助项目(61805199)；国防科技创新特区项目(18-H 863-01-ZT-001-004-02)；陕西省自然科学基金资助项目(2018JQ6065)；水下信息与控制国家重点实验室项目资助的课题(XK-01-61-KS-0176)

贺锋涛(1974-)，男，博士，副教授，主要从事水下无线光通信、激光高分辨成像及激光散斑传感检测的研究。 E-mail：hefengtao@xupt.edu.cn

张建磊(1988-)，男，博士，主要从事三维成像与显示技术的研究。E-mail：zhangjianlei@xupt.edu.cn