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大气湍流理论指出湍流整体在时间变化与空间分布上都是不规则的,其折射率由于湍流的随机运动而不断变化,使得光束在自由空间传输时受到不规则影响,发生光束展宽、光强闪烁等一系列畸变,造成光束传输性能的下降。
根据Kolmogorov湍流理论[20],大气湍流内部的大尺度涡旋与小尺度涡旋不断进行着能量的传递,使大气湍流保持不断运动的状态。当光束传输经过大尺度湍流涡旋时,由于折射作用,在传输过程中光束会发生到达角起伏,在接收面上产生不规则的浮动,造成接收光信号的不稳定,进一步加剧湍流引起的闪烁效应,使整个系统的传输性能发生严重下降。在光束经过小尺度湍流涡旋后,由于湍流效应与衍射效应共同作用,使光束在自由空间衍射展宽的基础之上进一步发生展宽,导致光斑发生严重畸变,改变光束原本的光强分布,降低传输质量。
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OAM光束是一种拥有螺旋相位波前的特殊光束,光束中心存在相位奇点,奇点处光强为零。在近轴近似下,亥姆霍兹方程可以表示为:
$$ \frac{1}{\rho }\left( {\rho \frac{{\partial E}}{{\partial r}}} \right) + \frac{1}{{{\rho ^2}}}\frac{{{\partial ^2}E}}{{\partial {\varphi ^2}}} + 2ik\frac{{\partial E}}{{\partial {\textit{z}}}} = 0 $$ (1) 对公式(1)进行求解就可以得到其解为拉盖尔-高斯模。在柱坐标系下,原点处的拉盖尔-高斯光束复振幅的表示为[21]:
$$ \begin{split} {U^l}(r,\theta ,{\textit{z}}) =& \sqrt {\dfrac{{2p!}}{{\pi (\left| l \right| + p)!}}} \dfrac{1}{{w({\textit{z}})}}\exp \left[ { - \dfrac{{{r^2}}}{{{w^2}({\textit{z}})}}} \right] \times \\ & {\left(\dfrac{{r\sqrt 2 }}{{w({\textit{z}})}}\right)^{\left| l \right|}}L_p^{\left| l \right|}\left[ { - \dfrac{{2{r^2}}}{{{w^2}({\textit{z}})}}} \right]\exp ( - il\theta ) \end{split} $$ (2) 式中:z为水平的传输距离;
$w\left( {\textit{z}} \right) = $ $ {w_0}\sqrt {1 + {{({\textit{z}}/{{\textit{z}}_R})}^2}} $ ,为拉盖尔-高斯光束的束腰宽度,w0为初始束腰宽度,${{\textit{z}}_R} = \pi {\omega _0}/\lambda $ 为瑞利距离;r 为光束的径向距离;p为该光束的径向指数;$L_p^{\left| l \right|}$ 为缔合拉盖尔多项式; l为光束的拓扑荷数;θ为光束方位角。 $$ L_p^{\left| l \right|}(x) = \sum\limits_{k = 0}^p {{{( - 1)}^k}} C_{p + \left| l \right|}^{p - k}\frac{{{x^k}}}{{k!}} $$ (3) 将公式(3)代入公式(2)中,对其进行化简就得到了拉盖尔-高斯光束在束腰位置的归一化方程[21]:
$$ \begin{split} {U^{\left| l \right|}}(r,\theta ) =& \sqrt {\dfrac{2}{\pi }} \dfrac{1}{{{w_0}}}\exp \left( { - \dfrac{{{r^2}}}{{w_0^2}}} \right) \times \\ &\sqrt {\dfrac{1}{{\left| l \right|!}}} \left(\dfrac{{r\sqrt 2 }}{{{w_0}}}\right)\exp ( - il\theta ) \end{split} $$ (4) 光束沿着z轴在大气湍流中传输时,在Rytov近似条件下,基于惠更斯-菲涅耳原理,光束传输到z处时的光场表达式为[15]:
$$ \begin{split} U(\rho ,{\textit{z}}) =& - \dfrac{i}{{\lambda {\textit{z}}}}\exp (ik{\textit{z}})\iint_s {U(r,\theta )}\exp \left[ {\dfrac{{ik}}{{2{\textit{z}}}}\left( {\rho - r} \right)} \right] \times \\ & \exp \left[ {\psi \left( {r,\rho ,{\textit{z}}} \right)} \right]r{\rm d}r{\rm d}\theta \end{split} $$ (5) 式中:U(r,θ)为光束源平面的光场;λ为光束的波长;k=2π/λ,为波数;ψ(r, ρ, z)为大气湍流对光束复相位的干扰;r表示源平面的位置矢量;ρ为输出位置的位置矢量。
根据光束在z处的光场表达式可以求解出输出平面的光强分布。当OAM光束在大气湍流中传输至z处时,其光强分布的表达式为[15]:
$$ \begin{split} {{I}}\left( {\rho ,{\textit{z}}} \right) =& \dfrac{{{k^2}}}{{{{\textit{z}}^2}}}\mathop \sum \limits_{n = - \infty }^\infty \mathop \int \limits_0^\infty \mathop \int \limits_0^\infty {\rm exp}\left[ { - \left( {\dfrac{1}{{\omega _0^2}} + \dfrac{1}{{\rho _0^2}}} \right)} \right]\left( {r_1^2 + r_2^2} \right) \times \\ & {\rm exp}\left[ { - \dfrac{{{{i}}k}}{{2{\textit{z}}}}\left( {r_1^2 - r_2^2} \right)} \right]{{{J}}^n}\left( {\dfrac{{k\rho {r_1}}}{{\textit{z}}}} \right) \times \\ & {{{J}}^n}\left( {\dfrac{{k\rho {r_2}}}{{\textit{z}}}} \right){{{I}}_{n + m}}\left( {\dfrac{{2{r_1}{r_2}}}{{\rho _0^2}}} \right){r_1}{\kern 1pt} {r_2}{\kern 1pt} {\rm{d}}{r_1}{\kern 1pt} {\rm{d}}{r_2} \end{split} $$ (6) 式中:r1和r2分别表示源平面中任意两个点位置矢量的模;
${{{J}}^n}$ 为n阶贝塞耳函数;$ \; {\rho }_{0}={\left(0.545{C}_{n}^{2}{k}^{2}{\textit{z}}\right)}^{-3/5} $ ,代表球面波传输经过大气湍流介质后的大气相干长度,$C_n^2$ 为大气折射率结构常数,在湍流理论中用来表示大气湍流的强弱;In+m是n+m阶修正贝塞耳函数。根据公式(6),大气湍流对传输性能的影响主要分为两种因素:一种是光束初始因素,如光束束腰半径、光束的波长、光束相干度等;另一种是传输介质因素,包括传播距离、湍流强度、湍流的内尺度与外尺度等参数。
光束的展宽通常使用归一化光束展宽比y进行表征,即光束在发射端的光斑宽度Rt与接收端的光斑宽度Rr的比值:
$$ y = \frac{{{R_{\rm t}}}}{{{R_{\rm r}}}} $$ (7) 由于OAM光束具有独特的光强分布,对于光束光斑尺寸的传统描述已经不适用,OAM光束的光束宽度定义为,光束横截面上为光强最大的圆周[22],其表达式如下:
$$ R = \sqrt {2\left| l \right|} \sigma \sqrt {1 + \frac{{2{{\textit{z}}^2}}}{{{k^2}{\sigma ^4}}}} $$ (8) 公式(8)表明,光束展宽的程度受到传输介质因素与光束初始因素的影响。
对于自由空间光通信来说,指向偏差是一个非常重要的性能参数,代表了整个系统的稳定性。光束质心的表达式[23]为:
$$ \hat x = \dfrac{{\displaystyle\sum\limits_{i = {p_1}}^{{p_2}} {\displaystyle\sum\limits_{j = {q_1}}^{{q_2}} {i \cdot } f(i,j)} }}{{\displaystyle\sum\limits_{i = {p_1}}^{{p_2}} {\displaystyle\sum\limits_{j = {q_1}}^{{q_2}} {f(i,j)} } }}\quad \hat y = \dfrac{{\displaystyle\sum\limits_{i = {p_1}}^{{p_2}} {\displaystyle\sum\limits_{j = {q_1}}^{{q_2}} {j \cdot } f(i,j)} }}{{\displaystyle\sum\limits_{i = {p_1}}^{{p_2}} {\displaystyle\sum\limits_{j = {q_1}}^{{q_2}} {f(i,j)} } }} $$ (9) 式中:(p1, q1)为CCD左上角坐标;(p2, q2)为CCD右下角坐标;f(i,j)为在(i,j)位置处像素的灰度值;(
$\hat x$ ,$\hat y$ )即为光束的质心。将质心坐标与原点坐标做差求出质心与原点的偏移量,从而得到指向偏差散布圆的半径为:$$ {r_{\rm c}} = \sqrt {{{(\hat x - \bar x)}^2} + {{(\hat y - \bar y)}^2}} $$ (10) 式中:(
$\bar x$ ,$\bar y$ )为光束在无湍流时的原点坐标。 -
当光束在大气湍流信道传输时,会受到湍流效应与衍射效应的共同影响使光斑发生畸变,导致接收端的光束发生展宽。文中搭建了一条大气湍流信道中的传输链路,使用拓扑荷数为3的OAM光束与高斯光束作为载波在链路中进行传输,图3所示为高斯光束与OAM光束经过大气湍流信道后的光斑,图3(a)和(b)分别为高斯光束经过无湍流信道与湍流信道,图3(c)和(d)分别为OAM光束经过无湍流信道与湍流信道。由图3可知,OAM光束与高斯光束经过大气湍流信道后,两种光束均发生了不同程度的畸变,导致传输质量的严重下降,因此OAM光束与高斯光束在大气湍流信道中的传输特性是至关重要的。
图 3 高斯光束与OAM光束经过无湍流链路与湍流链路的光斑图
Figure 3. Pattern of Gaussian beam and OAM beam passing through link without turbulence and link with turbulence
文中采用归一化光束展宽比表述经过大气湍流后光束的展宽程度。根据公式(7),光束展宽比为1时,代表传输后的光束没有发生光束展宽,相反,光束展宽比的数值越小说明光束在传输后的展宽程度越大。图4(a)~(d)所示是OAM光束与高斯光束通过
$\Delta T$ 为0 ℃、80 ℃、160 ℃和240 ℃的大气湍流信道后的光束展宽比曲线。实验结果表明,在$\Delta T$ 为0 ℃时,即无湍流条件下,OAM光束与高斯光束的光束展宽比为0.95和0.97,都接近1,但高斯光束的光束展宽比大于OAM光束的光束展宽比,可见在无湍流条件下,OAM光束的展宽程度大于高斯光束。当$\Delta T$ 分别为80 ℃、160 ℃ 和240 ℃时,OAM光束的光束展宽比分别为0.91、0.88和0.84,高斯光束的光束展宽比分别为0.88、0.82和0.76,高斯光束的光束展宽比相比同等条件下OAM光束的光束展宽比降低了3.4%,7.3%和10.5%。这是由于OAM光束具有独特的螺旋相位,这一特性使得OAM光束在大气湍流中传播时能够抵抗湍流效应引起的衍射[28],从而减少光束的展宽。实验结果与参考文献[15]的研究结果一致,证实了大气湍流信道下,高斯光束的光束展宽程度要大于OAM光束的展宽程度,并且随着湍流强度的增加,高斯光束的展宽程度远大于OAM光束,说明OAM光束在大气湍流信道中传输时受到湍流的影响更小。 -
在大气湍流信道中,受湍流效应的影响光束的传播路径发生不规则的偏折,造成指向偏差。指向偏差会增加光束的闪烁,加剧功率抖动,甚至导致光斑溢出探测器的接收孔径,从而降低系统的传输性能。图5所示为OAM光束与高斯光束通过不同强度的大气湍流信道后的指向偏差。指向偏差表示为光束中心相对于接收面中心的偏移,根据公式(9)对光束的质心进行计算,再将质心坐标与原点坐标做差即可求出质心与原点的偏移量,利用公式(10)计算得到指向偏差散布圆的半径。由于大气湍流的不确定性,指向偏差在直角坐标系的两个轴向上的分布是相互独立的。图5(a)~(c)是OAM光束在大气湍流装置的
$\Delta T$ 为80 ℃、160 ℃和240 ℃条件下传输后的指向偏差;图5(d)~(f)是高斯光束在大气湍流装置的$\Delta T$ 为80 ℃、160 ℃和240 ℃ 条件下传输后的指向偏差。实验数据表明,随着湍流强度的增加,两种光束在接收面中心附近出现的次数逐渐减少,而在远离接收面中心位置出现的次数越来越多且分布范围越来越大;在大气湍流信道中高斯光束的分布范围明显比OAM光束的分布范围更广,且在接收面中心附近的出现的次数更少,同时随着湍流强度的增强,高斯光束与OAM光束指向偏差的差距逐渐增大。这说明OAM光束在湍流信道中更不易受到湍流影响产生指向偏差,进一步证明了OAM光束受到湍流的影响更小,比高斯光束有着更好的抵抗湍流能力。图6所示为在通过不同强度的湍流信道后OAM光束与高斯光束指向偏差的散布圆直径。在大气湍流装置的$\Delta T$ 为80 ℃、160 ℃和240 ℃时,OAM光束的散布圆直径分别是3.69 mm、4.71 mm和6.07 mm,高斯光束的散布圆直径分别是4.16 mm、5.8 mm和8.72 mm,与高斯光束相比,OAM光束指向偏差的散布圆直径分别减少12.7%、23.1%和30.4%。这表明在湍流信道中传输后,OAM光束的散布圆直径远小于高斯光束,进而可以推断出OAM光束具有更好的抑制湍流影响的能力。 -
光束经过大气湍流信道传输后,在接收端的光强分布会随着空间分布与时间先后不断地发生变化,表现为接收功率的抖动,这种现象主要是由于光束传播路径上折射率的不断变化。接收端功率抖动的幅度决定了空间光通信系统的稳定性与传输质量。在实验中笔者在测量功率抖动时使用Thorlabs公司的PM100功率计,该功率计可实时测量0.1 s间隔的光束的接收功率,为减少测量误差,每次测量取3000个功率点即5 min的测量时间,在测得功率抖动的数据后,笔者对这些数据的方差进行了计算,功率抖动的方差可描述接收功率的抖动幅度,方差公式表示为:
$$ {S^2} = \frac{{{{(M - {x_1})}^2} + {{(M - {x_2})}^2} + {{(M - {x_3})}^2} \cdots + {{(M - {x_n})}^2}}}{n} $$ (11) 式中:x1,x2,···,xn为测得的功率抖动散点;M为功率抖动的平均数;n为测得的功率抖动散点的数量。
图7所示为不同强度的湍流信道中OAM光束与高斯光束传输后的功率抖动,图7(a)~(d)是OAM光束在大气湍流装置的
$\Delta T$ 为0 ℃、80 ℃、160 ℃和240 ℃条件下传输后的功率抖动;图7(e)~(h)是高斯光束在大气湍流装置的$\Delta T$ 为0 ℃、80 ℃、160 ℃和240 ℃ 条件下传输后的功率抖动。由图可知,在湍流信道中OAM光束的抖动幅度明显小于高斯光束,且随着湍流强度的增加,两种光束的差距不断增大。图8为 OAM光束与高斯光束通过不同强度的湍流后功率抖动的方差变化曲线,OAM光束与高斯光束在温度差为 0 ℃的条件下,功率抖动方差分别为0.026和0.023,这表明,在无湍流情况下,OAM光束的功率抖动略大于高斯光束。经$\Delta T$ 为 80 ℃的大气湍流信道后,功率抖动分别为0.067和0.094;经$\Delta T$ 为160 ℃的大气湍流信道后,功率抖动分别为0.12和0.182;经$\Delta T$ 为240 ℃的大气湍流信道后,功率抖动分别为0.201和0.331。两种光束在经过大气湍流信道之后,OAM光束的功率抖动明显小于高斯光束,说明OAM光束对湍流有更好的抵抗能力,相比高斯光束受到大气湍流的影响更小。 -
OAM光束与高斯光束通过不同强度的湍流后的误码率曲线如图9所示。在温度差为0 ℃时,即无湍流情况下,误码率为3.8×10−3(FEC极限)时,OAM光束和高斯光束传输后的灵敏度分别为−32.06 dBm和−32.76 dBm,说明在无湍流条件下,OAM光束载波传输系统的误码率低于高斯光束载波传输系统,这是由于在无湍流时OAM光束的光束展宽和功率抖动导致灵敏度低于高斯光束。在
$\Delta T$ 为80 ℃的大气湍流装置后,OAM光束载波传输系统和高斯光束载波传输系统的灵敏度分别为−31.59 dBm和−31.31 dBm;在$\Delta T$ 为160 ℃的大气湍流装置后,OAM光束载波传输系统和高斯光束载波传输系统的灵敏度分别为−30.57 dBm和−29.04 dBm;在$\Delta T$ 为240 ℃的大气湍流装置后,OAM光束载波传输系统和高斯光束载波传输系统的灵敏度分别为−28.97 dBm和−26.47 dBm。在经过大气湍流信道之后,OAM光束载波传输系统的灵敏度高于高斯光束载波传输系统的灵敏度,这说明OAM光束载波受到湍流的影响更小,有着更好抑制大气湍流影响的能力。
Experimental research on transmission performance on OAM beam and Gaussian beam in atmospheric turbulence channel
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摘要: 通过实验研究了大气湍流信道下中轨道角动量(OAM)光束与高斯光束的传输性能,并将两种光束的传输性能进行对比。实验以加载调制信号的OAM光束以及高斯光束为传输载波,分别测量了在大气湍流信道下两种光束的光束展宽、指向偏差、功率抖动以及误码率。实验结果表明:在大气湍流信道中传输后,OAM光束相比高斯光束,光束展宽减少10.5%,功率抖动的方差下降0.13,指向偏差的散布圆直径减小30.4%,并且光束中心更集中于接收面中心;OAM光束载波系统的最低探测灵敏度达到−28.97 dBm,相比高斯光束提升2.5 dB。实验结果证明了在大气湍流信道传输中,OAM光束比高斯光束受到湍流的影响更小,并且随着湍流强度的增加,OAM光束恶化程度远小于高斯光束。实验的结论为大气湍流信道下自由空间激光通信的发展和应用提供了参考。Abstract: The transmission performance of orbital angular momentum(OAM) beam and Gaussian beam in atmospheric turbulent channel was studied and compared experimentally. The beam spreading, pointing error, power jitter and bit error rate of the two beams were measured respectively, the OAM beam loaded with modulated signals and the Gaussian beam were as carrier. The experimental results show that in the atmospheric turbulence channel, compared with Gaussian beam, beam spreading ratio of the OAM beams reduces by 10.5%, variance of power jitter falls 0.13, dispersion circle diameter of the pointing error decreases by 30.4%, and more focused on the distribution of the center of the beam axis. The lowest detection sensitivity of the OAM beam carrier system reaches −28.97 dBm, which improved 2.5 dB compared with that of Gaussian beam. The experimental results verify that OAM beam is less affected by turbulence than Gaussian beam in atmospheric turbulence channel, and the deterioration of beam quality of OAM beam is much less than Gaussian beam with the increase of turbulence intensity. The results of the experiment provide a reference for the development and application of free space laser communication in atmospheric turbulent channel.
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