賀鋒濤，房 偉，張建磊*，楊 祎，杜 迎，張 斌

漢克-貝塞爾光束在各向異性海洋湍流中軌道角動量傳輸特性分析

賀鋒濤1，房 偉1，張建磊1*，楊 祎1，杜 迎1，張 斌2

1西安郵電大學(xué)電子工程學(xué)院，陜西 西安 710121；2中國船舶重工集團第705研究所，水下信息與控制重點實驗室，陜西 西安 710077

基于Rytov近似理論，分析了各向異性海洋湍流中漢克-貝塞爾(HB)光束的交叉譜密度，研究了軌道角動量(OAM)模式探測概率、串?dāng)_概率及HB光束的螺旋相位譜，建立了各向異性海洋湍流中OAM模式探測概率模型。結(jié)果表明，HB光束在各向異性海洋湍流環(huán)境中發(fā)射OAM模式的探測概率高于在各向同性海洋湍流環(huán)境中的探測概率。并且隨著各向異性因子的增大，海洋湍流對發(fā)射OAM模式探測概率的影響減小，串?dāng)_模式的探測概率也隨之下降。

各向異性海洋湍流；漢克-貝塞爾光束；軌道角動量；螺旋相位譜；光學(xué)渦旋

1 引 言

渦旋光束因為攜帶有軌道角動量(orbital angular momentum，OAM)，且OAM具有相互正交性，因此利用OAM空域復(fù)用技術(shù)可以提高光通信信道容量。近年來，渦旋光束在無線光通信中得到了研究人員的廣泛關(guān)注。Ren等[1]通過復(fù)用4個攜帶不同OAM模式的綠光實現(xiàn)了40 Gbit/s的鏈路。但光束在海洋中傳輸時，海洋湍流對光束的傳輸特性產(chǎn)生影響，導(dǎo)致相位畸變、模式串?dāng)_[2-3]。

基于Nikishov[4]建立的海洋湍流折射率起伏空間功率譜，Cheng等[5]研究了Laguerre-Gaussian(LG)光束在各向同性海洋湍流中的傳輸特性，分析了各向同性海洋湍流對LG光束軌道角動量模式探測概率的影響。Yin等[6]研究了Hankel-Bessel(HB)光束在各向同性海洋環(huán)境中螺旋相位譜受湍流的影響。此外，其他學(xué)者對艾利光束[7]、部分相干LG光束[8]在各向同性海洋湍流中的傳輸特性進行了研究，以及疊加光束在海洋湍流中的抗干擾特性[9-11]。但是上述研究都是基于各向同性的海洋湍流環(huán)境，實際上海洋湍流環(huán)境由于地球自轉(zhuǎn)的原因是各向異性的[12]，Huang等[13]分析了各向異性海洋湍流中光束質(zhì)量、平均光強，討論了光束初始相干度與抗湍流干擾之間的關(guān)系，Chen等[14]研究了在各向異性海洋湍流中部分相干修正貝塞爾(partially coherent modified Bessel correlated，PCMBC)光束OAM模式與各向異性因子的關(guān)系，Li等[15]研究了Hermite-Gaussian(HG)光束OAM在各向異性海洋湍流的傳輸特性；此外HB光束具有無衍射特性，即通過一定傳輸距離后中心光斑、光強分布保持不變，通過障礙物后可以重建橫向強度分布[16-18]，因此研究HB光束在各向異性海洋湍流的傳輸特性對海洋環(huán)境無線光通信鏈路有重要意義。目前，關(guān)于各向異性海洋湍流中HB光束軌道角動量的傳輸特性的研究還未見報道。

本文首先基于Rytov近似理論推導(dǎo)了各向異性海洋湍流中HB光束的交叉譜密度；數(shù)值模擬分析了在各向異性海洋湍流和各向同性海洋湍流下HB光束發(fā)射OAM模式探測概率隨傳輸距離的變化；然后計算了HB光束在各向異性海洋湍流中的螺旋相位譜；分析討論了在不同各向異性因子下，平衡參數(shù)、溫度方差耗散率、動能耗散率與OAM模式探測概率的關(guān)系。

2 理論分析

2.1 各向異性海洋湍流模型

在Markov[13]近似下，各向異性海洋湍流中折射率波動空間譜模型為

2.2 HB光束在各向異性海洋湍流中的傳輸特性

HB光束在自由空間中傳輸?shù)膹?fù)振幅為[5]

在弱湍流起伏區(qū)[19]，經(jīng)過海洋湍流后的HB光束復(fù)振幅為[5]

利用Rytov相位結(jié)構(gòu)函數(shù)二次近似可得：

將式(1)代入式(6)，可得相干長度為

2.3 螺旋相位譜分析

HB光束在海洋中傳輸時，由于各向異性海洋湍流的影響，會使發(fā)射OAM模式的能量擴散到其他OAM模式上，產(chǎn)生新的OAM模式，這種現(xiàn)象稱為模式串?dāng)_，會致使在接收端檢測到的發(fā)射OAM模式概率降低。此時，忽略各向異性海洋湍流引起的光束擴展，通過對HB光束基模的疊加可得到接收端HB光束，即：

將式(5)代入式(9)可得：

將式(2)代入式(10)，并利用積分關(guān)系[20]：

OAM模式概率密度的解析表達式為

當(dāng)HB光束軌道角動量模式為時，接收處的螺旋諧波能量為

3 數(shù)值模擬分析

圖1 OAM模式為1的HB光束在不同的各向異性海洋湍流中，OAM模式的探測概率隨傳輸距離z的變化曲線

圖2 OAM發(fā)射模式l0=5，傳輸距離為z=50 m，HB光束的螺旋相位譜

圖3 各向異性因子x分別為1、3、6，不同的w時，發(fā)射OAM模式的探測概率隨z的變化曲線

圖4 各向異性因子x分別為1、3、6，不同的e時，發(fā)射OAM模式的探測概率隨z的變化曲線

圖5 各向異性因子x分別為1、3、6，不同的cT時，發(fā)射OAM模式的探測概率隨z的曲線變化

從圖3、4、5中可以觀察到，探測概率隨著平衡參數(shù)、溫度耗散率T的增大而減小，隨著動能耗散率的增大而增大。更重要的是，當(dāng)、、T一定時，隨著各向異性因子的增大，發(fā)射OAM模式探測概率明顯增大。這表明HB光束在各向異性海洋環(huán)境中傳輸受湍流的影響明顯小于在各向同性的海洋中傳輸所受到的影響，并且各向異性因子越大，發(fā)射OAM模式的探測概率越大，海洋湍流產(chǎn)生的模式串?dāng)_越小。

4 結(jié) 論

本文首先介紹了各向異性海洋湍流的折射率空間譜模型，在此基礎(chǔ)上推導(dǎo)了在各向異性海洋湍流中HB光束的空間相干長度，分析得到各向異性海洋湍流中HB光束交叉譜密度，從而得到各向異性海洋湍流中HB光束OAM模式探測概率數(shù)學(xué)模型；數(shù)值模擬了在各向異性海洋湍流下HB光束OAM模式探測概率、串?dāng)_概率以及螺旋相位譜分布，并驗證了各向異性海洋湍流譜中=1時，湍流對HB光束的影響與各向同性湍流譜的結(jié)果一致。結(jié)果表明，隨著溫度方差耗散率、平衡參數(shù)的增加，以及動能耗散率的減小，接收端模式串?dāng)_加重，發(fā)射OAM模式的探測概率減小，螺旋相位譜擴展嚴重；進一步發(fā)現(xiàn)，隨著各向異性因子的增大，海洋湍流對HB光束的模式串?dāng)_影響減小，發(fā)射OAM模式的探測概率和螺旋相位譜的擴展有顯著的改善。本研究結(jié)果為海洋無線光通信系統(tǒng)的性能估計提供一定參考價值。

[1] Ren Y X, Li L, Wang Z,.Orbital angular momentum-based space division multiplexing for high-capacity underwater optical communications[J]., 2016, 6(8): 33306.

[2] Baykal Y. Intensity fluctuations of multimode laser beams in underwater medium[J]., 2015, 32(4): 593?598.

[3] Wu Y Q, Zhang Y X, Li Y,.Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence[J]., 2016, 371: 59?66.

[4] Nikishov V V, Nikishov V I. Spectrum of turbulent fluctuations of the sea-water refraction index[J]., 2000, 27(1): 82–98.

[5] Cheng M J, Guo L X, Li J T,.Propagation of an optical vortex carried by a partially coherent Laguerre-Gaussian beam in turbulent ocean[J]., 2016, 55(17): 4642?4648.

[6] Yin X L, Guo Y L, Yan H,.Analysis of orbital angular momentum spectra of Hankel-Bessel beams in channels with oceanic turbulence[J]., 2018, 67(11): 114201.

尹霄麗, 郭翊麟, 閆浩, 等. 漢克-貝塞爾光束在海洋湍流信道中的螺旋相位譜分析[J]. 物理學(xué)報, 2018, 67(11): 114201.

[7] Liu H L, Hu Z H, Xia J,.Generation of non-diffracting beam and its application[J]., 2018, 67(21): 214204.

劉會龍, 胡總?cè)A, 夏菁, 等. 無衍射光束的產(chǎn)生及其應(yīng)用[J]. 物理學(xué)報, 2018, 67(21): 214204.

[8] Lu L, Ji X L, Baykal Y. Wave structure function and spatial coherence radius of plane and spherical waves propagating through oceanic turbulence[J]., 2014, 22(22): 27112?27122.

[9] Kotlyar V V, Kovalev A A, Soifer V A. Superpositions of asymmetrical Bessel beams[J]., 2015, 32(6): 1046?1052.

[10] Yousefi M, Golmohammady S, Mashal A,. Analyzing the propagation behavior of scintillation index and bit error rate of a partially coherent flat-topped laser beam in oceanic turbulence[J]., 2015, 32(11): 1982?1992.

[11] Ke X Z, Xu J Y. Interference and detection of vortex beams with orbital angular momentum[J]., 2016, 43(9): 192?197.

柯熙政, 胥俊宇. 渦旋光束軌道角動量干涉及檢測的研究[J]. 中國激光, 2016, 43(9): 192?197.

[12] Li Y, Zhang Y X, Zhu Y,. Effects of anisotropic turbulence on average polarizability of Gaussian Schell-model quantized beams through ocean link[J]., 2016, 55(19): 5234?5239.

[13] Huang X W, Deng Z X, Shi X H,.Average intensity and beam quality of optical coherence lattices in oceanic turbulence with anisotropy[J]., 2018, 26(4): 4786?4797.

[14] Chen M Y, Zhang Y X. Effects of anisotropic oceanic turbulence on the propagation of the OAM mode of a partially coherent modified Bessel correlated vortex beam[J]., 2019, 29(4): 694?705.

[15] Li Y, Yu L, Zhang Y X. Influence of anisotropic turbulence on the orbital angular momentum modes of Hermite-Gaussian vortex beam in the ocean[J]., 2017, 25(11): 12203?12215.

[16] Wu G H, Tong C M, Cheng M J,.Superimposed orbital angular momentum mode of multiple Hankel-Bessel beam propagation in anisotropic non-Kolmogorov turbulence[J]., 2016, 14(8): 080102.

[17] Zhu Y, Liu X J, Gao J,.Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence[J]., 2014, 22(7): 7765?7772.

[18] Doster T, Watnik A T. Laguerre-Gauss and Bessel-Gauss beams propagation through turbulence: analysis of channel efficiency[J]., 2016, 55(36): 10239?10246.

[19] Yan X, Guo L X, Cheng M J,. Probability density of orbital angular momentum mode of autofocusing Airy beam carrying power-exponent-phase vortex through weak anisotropic atmosphere turbulence[J]., 2017, 25(13): 15286?15298.

[20] Jeeffrey A, Zwillinger D.[M]. 7th ed. Beijing: Academic Press, 2007: 485?509.

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

賀鋒濤，房偉，張建磊，等. 漢克-貝塞爾光束在各向異性海洋湍流中軌道角動量傳輸特性分析[J]. 光電工程，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

國家自然科學(xué)基金資助項目(61805199)；國防科技創(chuàng)新特區(qū)項目(18-H 863-01-ZT-001-004-02)；陜西省自然科學(xué)基金資助項目(2018JQ6065)；水下信息與控制國家重點實驗室項目資助的課題(XK-01-61-KS-0176)

賀鋒濤(1974-)，男，博士，副教授，主要從事水下無線光通信、激光高分辨成像及激光散斑傳感檢測的研究。 E-mail：hefengtao@xupt.edu.cn

張建磊(1988-)，男，博士，主要從事三維成像與顯示技術(shù)的研究。E-mail：zhangjianlei@xupt.edu.cn