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基于涡旋光束的模型,PCAVB在源平面上的电场分布可以表示为[13]:
$ \begin{array}{l} \varepsilon \left( {{x_0}, {y_0};0} \right) = \frac{{{\varepsilon _0}}}{{{\sigma _0}^\mathit{p}}}{\left( {\frac{{{x_0}^2 + {y_0}^2}}{{{\sigma _0}^2}}} \right)^l} \times \\ \;\;\;{\rm{exp}}\left( { - \frac{{{x_0}^2 + {y_0}^2}}{{{\sigma _0}^2}}} \right){({x_0} - {\rm{i}}{y_0})^p} \end{array} $
(1) 式中, ε0为常数,σ0为高斯光束的束宽,l表示光束阶数,x0和y0表示入射面上任意位置坐标, p表示光场拓扑荷数。当p在一个周期内,光束相位从0变化到2πp,拓扑荷数在传输过程中不会发生改变,且一般取整数值,理论上该值的取值没有上限[14-15]。涡旋光束在源平面(z=0)上的2阶统计特性可以用一个2×2交叉谱密度函数矩(cross spectral density matrix,CSDM)表示[16-17]:
$ \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\mathit{\boldsymbol{W}}({\mathit{\boldsymbol{r}}_{01}}, {\mathit{\boldsymbol{r}}_{02}};0) = \\ \left[ \begin{array}{l} {\mathit{\boldsymbol{W}}_{xx}}({\mathit{\boldsymbol{r}}_{01}}, {\mathit{\boldsymbol{r}}_{02}};0)\;\;\;{\mathit{\boldsymbol{W}}_{xy}}({\mathit{\boldsymbol{r}}_{01}}, {\mathit{\boldsymbol{r}}_{02}};0)\\ {\mathit{\boldsymbol{W}}_{yx}}({\mathit{\boldsymbol{r}}_{01}}, {\mathit{\boldsymbol{r}}_{02}};0)\;\;\;{\mathit{\boldsymbol{W}}_{yy}}({\mathit{\boldsymbol{r}}_{01}}, {\mathit{\boldsymbol{r}}_{02}};0) \end{array} \right] \end{array} $
(2) 式中, r01和r02是源平面(z=0)上的任意两个位置矢量,Wab(r01, r02; 0)=〈εa*(r01; 0)εb(r02; 0)〉m,(a, b=x, y), εx和εy表示在x和y方向上的电场分量,*为复共轭,〈·〉m表示系综平均。为了简化运算,可以假设PCAVB的场强εx和εy不相关,即Wxy(r01, r02; 0)=Wyx(r01, r02; 0)=0。
这里对角元素Wxx(r01, r02; 0), Wyy(r01, r02; 0)可以表示为[13]:
$ \begin{array}{l} \;\;{\mathit{\boldsymbol{W}}_{aa}}\left( {{\mathit{\boldsymbol{r}}_{01}}, {\mathit{\boldsymbol{r}}_{02}};0} \right){\rm{ = }}{\varepsilon _a}^2{\left( {\frac{{{\mathit{\boldsymbol{r}}_{01}}^2{\mathit{\boldsymbol{r}}_{02}}^2}}{{{\sigma _0}^4}}} \right)^{l + \frac{p}{2}}} \times \\ \;\;\;\;\;{\rm{exp}}\left( { - \frac{{{\mathit{\boldsymbol{r}}_{01}}^2 + {\mathit{\boldsymbol{r}}_{02}}^2}}{{{\mathit{\boldsymbol{\sigma }}_0}^2}}} \right){\rm{exp}}[{\rm{i}}p({\varphi _{02}} - {\varphi _{01}})] \times \\ {\rm{exp}}\left[ { - \frac{{{\mathit{\boldsymbol{r}}_{01}}^2 + {\mathit{\boldsymbol{r}}_{02}}^2 - 2\left| {{\mathit{\boldsymbol{r}}_{01}}} \right|\left| {{\mathit{\boldsymbol{r}}_{02}}} \right|{\rm{cos}}\left( {{\varphi _{02}} - {\varphi _{01}}} \right)}}{{{\delta _{aa}}^2}}} \right], \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {a{\rm{ = }}x, y} \right) \end{array} $
(3) 式中,φ01和φ02表示方位坐标; δxx和δyy分别表示x和y方向上的相干长度,相干长度越大,光束相干性越好; 波数k=2π/λ,λ为波长。研究表明,部分相干光在湍流中的传输性能优于完全相干光[18],部分相干光的相干长度在一定范围内减小时,能够有效减小光束相位波前发生变化,进而使光束漂移下降。相干长度太大,激光的相干性越优异,使得激光在大气湍流中的传输不易发散,但容易出现光斑,使得传输质量降低;相干长度太小,光束相干性差,会使激光在大气湍流传输中极易发散,有效传输距离大大降低。
利用扩展的惠更斯-菲涅耳原理的近轴形式,PCAVB经过大气湍流后在接收平面上的CSDM可以表示为[19-22]:
$ \begin{array}{*{20}{l}} {\mathit{\boldsymbol{W}}({\mathit{\boldsymbol{r}}_1},{\mathit{\boldsymbol{r}}_2};z) = {{\left( {\frac{1}{{\lambda z}}} \right)}^2}\iint {\mathit{\boldsymbol{W}}({\mathit{\boldsymbol{r}}_{01}},{\mathit{\boldsymbol{r}}_{02}};0) \times }} \\ {\;\;\;{\text{exp}}\left\{ {\frac{{{\text{i}}k}}{{2z}}} \right.[({\mathit{\boldsymbol{r}}_{01}}^2 - {\mathit{\boldsymbol{r}}_{02}}^2) + ({\mathit{\boldsymbol{r}}_1}^2 - {\mathit{\boldsymbol{r}}_2}^2) - } \\ {2({\mathit{\boldsymbol{r}}_{01}} \cdot {\mathit{\boldsymbol{r}}_1} - {\mathit{\boldsymbol{r}}_{02}} \cdot {\mathit{\boldsymbol{r}}_2})] - 4{{\rm{\pi }}^2}{k^2}z\int_0^1 {{\text{d}}\chi } \int_0^\infty {\{ 1 - } } \\ {\;\;\;\;\;{{\text{J}}_0}\left[ {\kappa \left| \chi \right.\left. {\left( {{\mathit{\boldsymbol{r}}_{01}} - {\mathit{\boldsymbol{r}}_{02}}} \right)} \right|} \right]\} {\mathit{\Phi }_n}\left( \kappa \right)\left. {\kappa {\text{d}}\kappa } \right\}} \end{array} $
(4) 式中,J0(·)是零阶贝塞尔函数,r1和r2是接收面上的两个位置矢量,Φn(κ)为湍流的空间功率谱函数,z为传输总长度,κ是空间波数的大小,χ=h′/zcosγ,h′为传输高度变量,γ为天顶角。令r1=r2=r,将(3)式代入(4)式可得PCAVB在湍流中传输时接收面交叉谱密度函数的解析式[13]:
$ \begin{array}{l} {\mathit{\boldsymbol{W}}_{aa}}\left( {\mathit{\boldsymbol{r}};z} \right) = \frac{{{\varepsilon _a}^2{k^2}}}{{4{z^2}{\sigma _0}^{4l + 2p}}} \times \\ \left\{ {\sum\limits_{h = 0}^\infty {\sum\limits_{q = 0}^\infty {\frac{{(2 - {\delta _{h, 0}}){v_a}^{2h + 2q + p}{{({t_a}{u_a})}^{ - s}}{{[\Gamma \left( s \right)]}^2}}}{{{{\left[ {\left( {2h} \right)!} \right]}^2}q!\Gamma \left( {q + p + 2h + 1} \right)}}} } } \right.\cdot\\ \;\;\;\;\;\;\;{\left( {\frac{{{k^2}{\mathit{\boldsymbol{r}}^2}}}{{4{z^2}}}} \right)^{2h}}{}_{_1}{F_1}\left( {s;2h + 1; - \frac{{{k^2}{\mathit{\boldsymbol{r}}^2}}}{{4{t_a}{z^2}}}} \right)\cdot\\ \;\;\;\;\;\;\;\;\;\;\;{\;_1}{F_1}\left. {\left( {s;2h + 1; - \frac{{{k^2}{\mathit{\boldsymbol{r}}^2}}}{{4{u_a}{z^2}}}} \right)} \right\} \end{array} $
(5) 式中, Γ(·)是伽马函数,1F1(·)是库默尔函数。h=0时,δh, 0=1;h≠0时,δh, 0=0。
$ s = l + p + q + 2h + 1{\rm{ }} $
(6) $ {u_a} = \frac{1}{{{\sigma _0}^2}} + \frac{1}{{{f_0}^2}} + \frac{1}{{{\delta _{aa}}^2}} - \frac{{{\rm{i}}k}}{{2z}}{\rm{ }} $
(7) $ {t_a} = \frac{1}{{{\sigma _0}^2}} + \frac{1}{{{f_0}^2}} + \frac{1}{{{\delta _{aa}}^2}} + \frac{{{\rm{i}}k}}{{2z}} $
(8) $ {v_a} = \frac{1}{{{f_0}^2}} + \frac{1}{{{\delta _{aa}}^2}} $
(9) 式中,a=x, y。球面波在接收面上的空间相干长度表示为[23]:
$ {f_0} = {\left[ {\frac{1}{3}{{\rm{ \mathit{ π} }}^2}{k^2}z\int_0^\infty {{\kappa ^3}{\mathit{\Phi }_n}\left( \kappa \right){\rm{d}}\kappa } } \right]^{ - \frac{1}{2}}} $
(10) PCAVB的平均光强可以表示为[13]:
$ \mathit{\boldsymbol{I}}\left( {r, z} \right) = {\mathit{\boldsymbol{W}}_{xx}}\left( {\mathit{\boldsymbol{r}};z} \right) + {\mathit{\boldsymbol{W}}_{yy}}\left( {\mathit{\boldsymbol{r}};z} \right) $
(11) $ \begin{array}{l} \;\;\;\;\;\mathit{\boldsymbol{I}}\left( {\mathit{\boldsymbol{r}}, z} \right) = \frac{{{\varepsilon _x}^2{k^2}}}{{4{z^2}{\sigma _0}^{4l + 2p}}} \times \\ \left\{ {\sum\limits_{h = 0}^\infty {\sum\limits_{q = 0}^\infty {\frac{{(2 - {\delta _{h, 0}}){v_x}^{2h + 2q + p}{{({t_x}{u_x})}^{ - s}}{{[\Gamma \left( s \right)]}^2}}}{{{{\left[ {\left( {2h} \right)!} \right]}^2}q!\Gamma \left( {q + p + 2h + 1} \right)}}} } } \right.\cdot\\ \;\;\;\;\;\;\;{\left( {\frac{{{k^2}{\mathit{\boldsymbol{r}}^2}}}{{4{z^2}}}} \right)^{2h}}{}_{_1}{F_1}\left( {s;2h + 1; - \frac{{{k^2}{\mathit{\boldsymbol{r}}^2}}}{{4{t_x}{z^2}}}} \right)\cdot\\ \;\;\;{\;_1}{F_1}\left. {\left( {s;2h + 1; - \frac{{{k^2}{\mathit{\boldsymbol{r}}^2}}}{{4{u_x}{z^2}}}} \right)} \right\} + \frac{{{\varepsilon _y}^2{k^2}}}{{4{z^2}{\sigma _0}^{4l + 2p}}} \times \\ \left\{ {\sum\limits_{h = 0}^\infty {\sum\limits_{q = 0}^\infty {\frac{{(2 - {\delta _{h, 0}}){v_y}^{2h + 2q + p}{{({t_y}{u_y})}^{ - s}}{{[\Gamma \left( s \right)]}^2}}}{{{{\left[ {\left( {2h} \right)!} \right]}^2}q!\Gamma \left( {q + p + 2h + 1} \right)}}} } } \right.\cdot\\ \;\;\;\;\;\;{\left( {\frac{{{k^2}{\mathit{\boldsymbol{r}}^2}}}{{4{z^2}}}} \right)^{2h}}{}_{_1}{F_1}\left( {s;2h + 1; - \frac{{{k^2}{\mathit{\boldsymbol{r}}^2}}}{{4{t_y}{z^2}}}} \right)\cdot\\ \left. {\;\;\;\;\;\;\;\;\;\;\;\;{\;_1}{F_1}\left( {s;2h + 1; - \frac{{{k^2}{\mathit{\boldsymbol{r}}^2}}}{{4{u_y}{z^2}}}} \right)} \right\} \end{array} $
(12) 初始平面上PCAVB的偏振度可表示为[13]:
$ {P^*} = \left| {\frac{{{\varepsilon _{0, x}}^2 - {\varepsilon _{0, y}}^2}}{{{\varepsilon _{0, x}}^2 + {\varepsilon _{0, y}}^2}}} \right| $
(13) 在各向异性湍流中,PCAVB的均方根空间宽度可定义为[25]:
$ {W_z} = \sqrt {\frac{{\iint {{\mathit{\boldsymbol{r}}^{\text{2}}} \cdot }I\left( {\mathit{\boldsymbol{r}},z} \right){{\text{d}}^2}\mathit{\boldsymbol{r}}}}{{\iint {I\left( {\mathit{\boldsymbol{r}},z} \right){{\text{d}}^2}\mathit{\boldsymbol{r}}}}}} $
(14) 将(12)式代入(14)式可得[13]:
$ \begin{array}{l} {W_z}^2 = \frac{{2l + p + 1}}{2}{\sigma _0}^2 + \left\{ {\frac{2}{{{k^2}{\sigma _0}^2}}} \right.\left[ {\frac{{{p^2}}}{{2l + p}}} \right. + 1 + \\ \;\;\;\frac{{2{\sigma _0}^2}}{{{\varepsilon _{0, x}}^2 + {\varepsilon _{0, y}}^2}}\left. {\left. {\left( {\frac{{{\varepsilon _{0x}}^2}}{{{\delta _{xx}}^2}} + \frac{{{\varepsilon _{0y}}^2}}{{{\delta _{yy}}^2}}} \right)} \right]} \right\}{z^2} + {T_a}{z^3} \end{array} $
(15) 式中,Ta为各向异性大气湍流量[20]:
$ {T_a} = \frac{4}{3}{{\rm{ \mathit{ π} }}^2}\int_0^\infty {{\kappa ^3}{\mathit{\Phi }_n}(\kappa ){\rm{d}}\kappa } $
(16) 引入各向异性功率谱模拟大气湍流,在各向异性湍流中,功率谱的形式由下式给出[12, 26]:
$ \begin{array}{l} {\mathit{\Phi }_n}\left( \kappa \right) = \frac{{A\left( \alpha \right){C_n}^2{\xi ^2}{\rm{exp}}( - {\kappa ^2}/{\kappa _m}^2)}}{{{{\left( {{\kappa ^2} + {\kappa _0}^2} \right)}^{\alpha /2}}}}\\ \;\;\;\;\;\;\;\;\left( {0 \le \kappa < \infty , 3 < \alpha < 4} \right) \end{array} $
(17) 式中,κ=ξ2(κx2+κy2)=ξ2κxy2, ξ≥1,Cn2是单位为m3-α的广义折射率结构常数,表示大气湍流的强弱,Cn2越大,大气湍流越强,α是广义指数参量, ξ是各向异性因子,用于描述大气湍流功率谱尺度,κm=c(α)/l0,κ0=2π/L0, l0与L0是湍流的内尺度和外尺度。湍流的一种与广义指数参量α有关的结构函数A(α)和c(α)分别如下所示:
$ A\left( \alpha \right) = \frac{1}{{4{{\rm{ \mathit{ π} }}^2}}}\Gamma (\alpha - 1){\rm{cos}}\left( {\frac{{{\rm{ \mathit{ π} }}\alpha }}{2}} \right) $
(18) $ c\left( \alpha \right) = {\left[ {\frac{{2{\rm{ \mathit{ π} }}}}{3}\Gamma \left( {\frac{{5 - \alpha }}{2}} \right)A\left( \alpha \right)} \right]^{\frac{1}{{\alpha - 5}}}} $
(19) 将(16)式~(19)式整理合并,得到:
$ \begin{array}{l} {T_a} = \frac{{2{{\rm{ \mathit{ π} }}^2}A\left( \alpha \right){C_n}^2}}{{3\left( {\alpha - 2} \right){\xi ^2}}}\left\{ {{\kappa _{\rm{m}}}^{2 - \alpha }} \right.\left[ {2{\kappa _0}^2 + \left( {\alpha - 2} \right){\kappa _m}^2} \right] \times \\ \;\;\;\;\;\;\;\;\;{\rm{exp}}\left( {\frac{{{\kappa _0}^2}}{{{\kappa _{\rm{m}}}^2}}} \right)\Gamma \left( {2 - \frac{\alpha }{2}, \frac{{{\kappa _0}^2}}{{{\kappa _{\rm{m}}}^2}}} \right) - \left. {2{\kappa _0}^{4 - \alpha }} \right\} \end{array} $
(20) 将(15)式、(20)式整合:
$ \begin{array}{l} {W_z}^2 = \frac{{2l + p + 1}}{2}{\sigma _0}^2 + \left\{ {\frac{2}{{{k^2}{\sigma _0}^2}}} \right.\left[ {\frac{{{p^2}}}{{2l + p}}} \right. + 1 + \\ 2{\sigma _0}^2\left. {\left. {\left( {\frac{1}{{{\delta _{xx}}^2}}\cdot\frac{{1 + {P^*}}}{2} + \frac{1}{{{\delta _{yy}}^2}}\cdot\frac{{1 - {P^*}}}{2}} \right)} \right]} \right\}{z^2} + \\ \frac{{2{{\rm{ \mathit{ π} }}^2}A\left( \alpha \right){C_n}^2}}{{3\left( {\alpha - 2} \right){\xi ^2}}}\left\{ {{\kappa _{\rm{m}}}^{2 - \alpha }} \right.[2{\kappa _0}^2 + \left( {\alpha - 2} \right){\kappa _{\rm{m}}}^2] \times \\ \;\;\;\;\;{\rm{exp}}\left( {\frac{{{\kappa _0}^2}}{{{\kappa _{\rm{m}}}^2}}} \right)\Gamma \left( {2 - \frac{\alpha }{2}, \frac{{{\kappa _0}^2}}{{{\kappa _{\rm{m}}}^2}}} \right) - \left. {2{\kappa _0}^{4\alpha }} \right\}{z^3} \end{array} $
(21) 用几何光学近似与Rytov近似解,可以得到湍流条件下光束漂移的二阶矩模型[27]:
$ \begin{array}{l} {\rho _{\rm{c}}}^2 = 4{{\rm{ \mathit{ π} }}^2}{\kappa ^2}{W_{FS}}^2\int_0^z {\int_0^\infty {\kappa {\mathit{\Phi }_n}\left( \kappa \right){\rm{exp}}( - {\kappa ^2}{W_z}^2) \times } } \\ \;\;\;\;\;\;\left\{ {\left. {1 - {\rm{exp}}\left[ { - \left. {\frac{{2{z^2}{\kappa ^2}{{\left( {1 - z\prime /z} \right)}^2}}}{{{k^2}{W_{FS}}^2}}} \right]} \right.} \right\}} \right.{\rm{d}}\kappa {\rm{d}}z\prime \end{array} $
(22) 式中,z′代表在传输总长度z的范围内,源平面到截取点之间的距离,WFS为光束在自由空间中的束宽扩展,利用光束在湍流中传输的漂移模型公式可得[27]:
$ \begin{array}{l} {\rho _c}^2 = \frac{{4{{\rm{ \mathit{ π} }}^2}A\left( \alpha \right){C_n}^2{\kappa _0}^{ - \alpha }}}{{\left( {\alpha - 2} \right){\xi ^2}}}{z^2}{\int_0^{\rm{z}} {\left( {1 - \frac{{z\prime }}{2}} \right)} ^2} \times \\ \;\;\;\;\left\{ { - 2{\kappa _0}^4 + {\kappa _0}^\alpha {\kappa _{\rm{m}}}^2{{({\kappa _{\rm{m}}}^{ - 2} + {W_z}^2)}^{\alpha /2}} \times } \right.\\ \;\;\;[2{\kappa _0}^2({\kappa _{\rm{m}}}^2{W_z}^2) + 1 + \left( {\alpha - 2} \right){\kappa _{\rm{m}}}^2] \times \\ {({\kappa _{\rm{m}}}^2{W_z}^2 + 1)^{ - 2}}{\rm{exp}}\left( {\frac{{{\kappa _0}^2}}{{{\kappa _{\rm{m}}}^2}} + {\kappa _0}^2{W_z}^2} \right) \times \\ \;\;\;\;\Gamma \left( {2 - \frac{\alpha }{2}, \frac{{{\kappa _0}^2}}{2}{\kappa _{\rm{m}}}^2 + {\kappa _0}^2{W_z}^2} \right){\rm{d}}z\prime \end{array} $
(23) 式中,Γ(·, ·)是不完全伽马函数,光束的均方根漂移与相对漂移的表达式为[28]:
$ \left\{ \begin{array}{l} {B_{{\rm{w, RMS}}}} = {({\rho _{\rm{c}}}^2)^{1/2}}\\ {B_{{\rm{w, r}}}} = {\left( {\frac{{{\rho _{\rm{c}}}^2}}{{{W_z}^2}}} \right)^{1/2}} \end{array} \right. $
(24) 光束的均方根漂移表达出光束偏离中心的真实距离,相对漂移能够直观反映出光束漂移与光束扩展之间的关系。最新的实验研究表明[29],用本文中的方法推导出的光束漂移理论模型适用于任意部分相干光束在大气湍流中传输的漂移,且理论结果与实验数据吻合。本文中采用的部分相干光束为反常涡旋光束也适用此模型。
反常涡旋光束在各向异性大气湍流中的漂移
Wander of anomalous vortex beams propagating through
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摘要: 为了研究部分相干反常涡旋光束在各向异性大气湍流中的漂移特性,基于各向异性非Kolmogorov大气湍流谱模型,利用几何光学近似与Rytov近似解得到了部分相干反常涡旋光束(PCAVB)在各向异性湍流中传输的光束漂移解析式,并对该漂移解析式进行数值模拟。研究了拓扑荷、相干长度、各向异性因子、折射率结构常数等光束参量及湍流参量对PCAVB在湍流中的均方根光束漂移与相对光束漂移的影响。结果表明,相干性越好的反常涡旋光束,会使光束在各向异性湍流中的漂移现象变得更严重;随拓扑荷数的增加,可以一定程度降低湍流对光束漂移的影响;湍流随各向异性因子的增大,对部分相干反常涡旋光束的漂移影响明显减弱;广义指数参量在3~3.3范围增长时,对PCAVB均方根漂移与相对漂移影响最大,而在3.3~4之间增长时,对光束均方根漂移与相对漂移影响逐渐减弱。此研究对反常涡旋光束在各向异性大气湍流中的传输提供了一种理论模型参考。Abstract: In order to study the wander characteristics of partially coherent anomalous vortex beams in anisotropic atmospheric turbulence, based on the anisotropic non-Kolmogorov atmospheric turbulence spectrum model, the analytical expression of the beam wander of partially coherent anomalous vortex beam (PCAVB) propagating in anisotropic turbulence was obtained by using the geometrical optics approximation and Rytov approximation, and the numerical simulation was carried out at the same time. The effects of beam parameters and turbulence parameters such as topological charge, coherence length, anisotropy parameters, refractive index structure constant on root-mean-square beam wander, and relative beam wander of PCAVB in turbulence were studied. The results show that the more coherent the anomalous vortex beam is, the more serious the wander of the beam in the anisotropic turbulence will be. With the increase of topological charge, the influence of turbulence on beam wander can be reduced to some extent. The influence of turbulence on the wander of partially coherent anomalous vortex beams is obviously reduced with the increase of anisotropic factor. When the generalized exponential parameter increases in the range of 3~3.3, it has the largest influence on PCAVB root-mean-square wander and relative wander, while when it increases in the range of 3.3~4, the influence on beam root-mean-square wander and relative wander gradually weakens. This study provides a theoretical reference model for the propagation of anomalous vortex beam in anisotropic atmospheric turbulence.
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Key words:
- atmospheric optics /
- anomalous vortex beam /
- anisotropic atmospheric turbulence /
- wander
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Figure 1. In the anisotropic turbulence, the root mean square wander Bw, RMS and relative wander Bw, r of PCAVB vary with each parameter
a, b—λ=632.8nm, σ0=10mm, ξ=3, δxx=5mm, δyy=10mm, P*=0, p=1, l=1, l0=10mm, L0=50m, α=11/3 c, d—λ=632.8nm, σ0=10mm, δxx=10mm, δyy=20mm, P*=1, p=1, l=1, l0=10mm, L0=10m, Cn2=10-13m3-α, α=3.4 e, f— =632.8nm, σ0=2mm, ξ=2, δxx=5mm, δyy=10mm, P*=0, l=1, l0=10mm, L0=50m, α=11/3, Cn2=10-13m3-α g, h—λ=632.8nm, σ0=10mm, ξ=3, P*=0, p=1, l=1, l0=10mm, L0=10m, α=11/3, Cn2=10-14m3-α
Figure 2. In the anisotropic turbulence, the root mean square wander Bw, RMS and relative wander Bw, r of PCAVB vary with the generalized exponential parameters α
a, b—λ=632.8nm, σ0=5mm, ξ=2, δxx=10mm, δyy=20mm, P*=0, l=1, l0=10mm, L0=10m, z=15km, Cn2=5×10-14m3-α c, d—λ=632.8nm, σ0=5mm, p=1, δxx=10mm, δyy=20mm, P*=0, l=1, l0=10mm, L0=10m, z=15km, Cn2=5×10-14m3-α e, f—λ=632.8nm, σ0=5mm, ξ=2, p=1, P*=0, l=1, l0=10mm, L0=10m, z=15km, Cn2=5×10-14m3-α
Figure 3. In the anisotropic turbulence, the root mean square wander Bw, RMS and relative wander Bw, r of PCAVB vary with the transmission distance z, anisotropic factor ξ, coherence length δxx, δyy
a, b—λ=632.8nm, σ0=10mm, δxx=5mm, δyy=10mm, P*=0, p=1, l=1, l0=10mm, L0=50m, Cn2=10-13m3-α, α=11/3 c, d—λ=632.8nm, z=15km, σ0=10mm, ξ=3, P*=0, p=1, l=1, l0=10mm, L0=10m, α=11/3, Cn2=10-14m3-α
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