Abstract:Vortex beams with orbital angular momenta with different mode numbers are mutually orthogonal to each other, which makes it possible to improve the information transmission efficiency in space optical communication system. Nevertheless, the implementation of this strategy is limited by the orbital angular momentum crosstalk caused by atmospheric turbulence. Focused Laguerre-Gaussian vortex beams are less affected by atmospheric turbulence due to their lager intensity density. Consequently, focused Laguerre-Gaussian vortex beams can be used as the carriers to reduce the orbit angular momentum crosstalk and increase the channel capacity of information transmission. In this paper, based on the spiral spectrum analysis theory, the analytical expression of spiral spectrum of focused Laguerre Gaussian beam propagating in anisotropic atmospheric turbulence is derived. The influences of turbulence and beam parameters on the received power of focused and unfocused Laguerre Gaussian beam are investigated via numerical calculations. Finally, the multi-phase screen method is used for verificating the simulation. The research findings are as follows. First, with the increase of transmission distance, turbulence intensity and topological charge, the receiving power of orbital angular momentum decreases, that is, the orbital angular momentum crosstalk turns more serious. Second, the larger the turbulence inner-scale, anisotropy index and beam wavelength are, the smaller the orbital angular momentum crosstalk is. Third, when the receiving aperture reaches a certain value, its influence on the orbit angular momentum crosstalk is very small. Fourth, different parameters have different effects on crosstalk, and the orbit angular momentum crosstalk of the focused vortex beam is less than that of the unfocused vortex beam. Therefore, in the vortex optical communication, the focused vortex beams can be used as the signal light to reduce the crosstalk between the orbit angular momentum modes, and thus improving the communication quality. These results have some theoretical reference values for reducing crosstalk in free-space optical communication. Keywords:atmospheric turbulence/ focus vortex beam/ orbital angular momentum crosstalk/ multi-phase screen method
其中, $\rho $, $\phi $分别表示接收平面极坐标中的径向参数和角向参数. $q = 1 + 2{\rm{i}}L/(k{w_0}^2) - L/f$, L为传输距离, L = f. 图1中x表示接收平面上的点离光轴的距离. LG光束聚焦后, 光束尺寸减小, 光强增大, 焦距越短效果越明显. 这是由于聚焦效果与焦距有一定关系, 当瑞利距离远大于焦距f时, 聚焦效果更好. 图 1 聚焦LG光束真空中传输不同距离的光强分布 Figure1. Intensity distribution of the focused LG beam propagating at different distances in vacuum
式中, R为光束的接收孔径, 螺旋谱定义式$P = {C_l} \big/\displaystyle\sum\nolimits_{q = - \infty }^\infty {{C_q}} $表示光束展为不同OAM的螺旋谐波的能量占光束总能量的权重, Cq为各级螺旋谐波的能量. 3.数值分析根据(15)式研究聚焦LG光束、LG光束通过各向异性大气湍流时的螺旋谱特性. 如没有特殊说明, 设置光束和湍流参数如下: λ = 1060 nm, l = 3, p = 0, w0 = 0.02 m, l0 = 0.001 m, L0 = 1.552 m, $C_n^2$=${10^{ - 14}} {m^{ - 2/3}}$, L = 1000 m, μ = 1; α = 11/3. 由图2可得, 随着传输距离增大, 光束螺旋谐波主量不断减小, 相邻的螺旋谐波分量逐渐增大, 轨道角动量发生串扰. 相比LG光束, 聚焦后的LG光束串扰明显较小. 图 2 LG光束(a)、聚焦LG光束(b)在湍流中传输不同距离时的螺旋谱分布 Figure2. Spiral spectrum distribution of LG beam (a) and focused LG beam (b) at different distances in turbulence
图3中轨道角动量模的接收功率Pm表示在发射平面处携带的轨道角动量模l被传输到接收平面处, 接收平面接收到信号的轨道角动量模m = l时的螺旋谱, 即轨道角动量模信号被正确传输的概率. 图 3 不同波束参数下LG、聚焦LG光束在湍流中的接收功率 (a)波长; (b)拓扑荷数; (c)束腰半径; (d)接收孔径 Figure3. Receiving power of LG and focused LG beams in turbulence under different beam parameters: (a) Wavelength; (b) topological charge; (c) waist radius; (d) receiving aperture.
从图3(a)可以看出, 波长越小的光束接收功率越小, OAM发散程度越严重. 同一波长, 聚焦后光束的接收功率增大, 波长越短增大效果越好. 这是因为波长短的波束瑞利距离大, 从而增强了光束的聚焦效果. 图3(b)中随着拓扑荷数的增大接收功率减小, 之后趋于一定值, 即湍流对OAM串扰的影响趋于稳定. 聚焦LG光束的串扰相对较轻, 但随拓扑荷数的增大, 其接收功率逐渐接近非聚焦光束, 聚焦对减小串扰的效果逐渐降低. 图3(c)中, LG光束存在一个最佳的w0值使得接收功率达到最大, 而聚焦LG光束随着w0值的增加其接收功率不断增大. 这是由于聚焦效果随w0值的增大而增强, 大气湍流对光束的影响越小. 图3(d)所示随着接收孔径R的增大OAM发散逐渐增强, 当R达到一定值时, OAM发散趋于稳定. 聚焦光束在R较大时减小串扰的效果较好. 图4所示, 整体来看, 聚焦LG光束的接收功率较大. 但图4(a)中随着传输距离的增大, 聚焦LG光束的接收功率逐渐趋于LG光束. 这是由于随着传输距离增大, 焦距也增大, 当焦距趋于瑞利距离时, 聚焦效果减弱, 光束性质趋于一致. 由图4(b)可知, 接收功率随湍流强度的增大而减小, OAM串扰越严重. 湍流强度较大时, 聚焦后的LG光束对减小串扰的效果较好, 抗湍流效果较明显. 图 4 不同湍流参数下LG、聚焦LG光束在湍流中的接收功率 (a)距离; (b)湍流强度; (c)湍流内尺度; (d)各向异性系数 Figure4. Receiving power of LG and focused LG beams in turbulence under different turbulence parameters: (a) Distance; (b) turbulence intensity; (c) turbulence internal scale; (d) anisotropy coefficient.
图4(c)表明湍流内尺度越小接收功率越小, 随着内尺度不断增大接收功率曲线趋于一定值. 这是由于内尺度较小时, 传输路径上会有更多的湍流涡旋使光强分布更分散. 图4(d)随着各向异性系数的减小, 接收功率逐渐减小. 各向异性系数越小时, 大气湍流强度越大, 此时聚焦对减小串扰的作用较大, 这一结果与图4(b)得出的结论一致. 4.模拟仿真利用多相位屏法进行数值模拟. 相位屏是利用快速傅里叶变换的功率谱反演法产生的, 多相位屏法认为大范围的湍流环境可以通过一层一层的相位屏来模拟, 当光束一层层穿过相位屏就模拟了光束通过真实的湍流环境. 图5采用多相位屏法模拟了聚焦LG光束在大气湍流中传输时的光强分布. 聚焦LG光束受湍流影响光斑逐渐破裂, 相位发生畸变, 受聚焦影响光斑尺寸逐渐减小, 在焦平面处达到最小值. 图 5 多相位屏法模拟聚焦LG光束在湍流中传输示意图 Figure5. Multiphase screen method to simulate the propagation of focused LG beam in turbulence
图6中多相位屏模拟参数如下: 网格数(512 × 512)、尺寸(1.5 m)、间距(100 m). 聚焦LG光束受湍流影响OAM发生串扰, 随传输距离的增大串扰越来越严重; 相比LG光束, 聚焦LG光束的螺旋谐波主量较大, OAM发散程度较轻, 传输距离为1000 m时效果最好. 当传输距离越来越大时, 聚焦LG光束的螺旋谐波主量值越来越接近LG光束, 说明焦距的增大减弱了聚焦效果, 这个结论与前面的理论分析相对应. 图 6 LG光束、聚焦LG光束在大气湍流中传输不同距离时的螺旋谱分布 Figure6. Spiral spectrum distribution of LG and focused LG beams propagation at different distances in atmospheric turbulence