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图1为大气湍流中空间光-光纤耦合示意图,激光束在大气湍流中传输一段距离z后,入射到耦合透镜的表面,透镜将由于大气湍流影响而畸变的光束汇聚,并耦合进光纤内部。
图 1 大气湍流中空间光-光纤耦合示意图
Figure 1. Schematic diagram of spatial light-fiber coupling in turbulent atmosphere
设定一部分相干光束经过大气湍流之后,通过一焦距为f的透镜耦合进单模光纤。标定光源面坐标为
$r \equiv (x,y)$ ,透镜端(接收端)坐标为$\rho \equiv \left( {{\rho _x},{\rho _y}} \right)$ 。在文中,将耦合效率ηc定义为耦合进光纤的平均光功率和入射在光学系统接收平面内的平均光功率的比值[17-18],即$$\left\langle {{\eta _{\rm c}}} \right\rangle = \left\langle {{P_{\rm c}}} \right\rangle /\left\langle {{P_{\rm a}}} \right\rangle ,$$ (1) 其中
$\left\langle {{P_{\rm c}}} \right\rangle$ 和$\left\langle {{P_{\rm a}}} \right\rangle$ 具体表达式分别为:$$ \begin{split} {P_{\rm c}} = & \displaystyle\int {\int_A {W\left( {{\rho _1},{\rho _2},{\textit{z}}} \right)F_A^*\left( {{\rho _1}} \right){F_A}\left( {{\rho _2}} \right)} } \times\\ & \exp \left[ { - \dfrac{1}{{{w^2}}}\left( {\rho _1^2 + \rho _2^2} \right)} \right]{\rm {d^2}}{\rho _1}{{\rm {d^2}}}{\rho _2}, \\ \end{split} $$ (2) $${P_{\rm a}} = \int_A {W\left( {\rho ,\rho ,{\textit{z}}} \right)} \exp \left( { - \frac{{2{\rho ^2}}}{{{w^2}}}} \right){d^2}\rho ,$$ (3) 式中:
$W\left( {{\rho _1},{\rho _2},{\textit{z}}} \right)$ 为光束经过大气湍流传输至z处(接收端)的交叉谱密度函数;${w^2} = {{{D^2}}/ 8}$ ,D为接收孔径直径;${F_A}\left( \rho \right)$ 为接收孔径平面处的模场分布,其表达式为$${F_A}\left( \rho \right) = \frac{1}{{{w_{\rm a}}}}\sqrt {\frac{2}{\pi }} \exp \left( { - \frac{{{\rho ^2}}}{{{w_{\rm a}}^2}}} \right),$$ (4) 式中:
${w_{\rm a}} = \lambda f/\left( {\pi {w_j}} \right)$ ,为孔径平面上的有效模宽,λ为波长,wj为光纤模场半径,f为耦合透镜的焦距。若需求得光束经过大气湍流之后的耦合效率,则首先需要得到光束在接收端z处的交叉谱密度函数表达式。因此,下面将介绍研究对象-非均匀关联光束在接收端的交叉谱密度函数。
在空间-频率域中,标量部分相干光束可以用交叉谱密度函数来描述。近年来,交叉谱密度函数作为一种常用的方法用来研究准单色部分相干场,源平面处场的交叉谱密度函数定义为两点间的相关函数[19]:
$$W\left( {{{{r}}_1},{{{r}}_2}} \right) = \left\langle {{E^*}\left( {{{{r}}_1}} \right)E\left( {{{{r}}_2}} \right)} \right\rangle ,$$ (5) 式中:E(r)表示垂直于传输方向的电场涨落,角括号表示系综平均。
2007年,Gori指出:交叉谱密度函数必须满足以下非负正定条件,才能成为数学上可实现的真实函数[6],
$$W\left( {{{{r}}_1},{{{r}}_2}} \right) = \int {p\left( v \right)} {H^*}\left( {{{{r}}_1},v} \right)H\left( {{{{r}}_2},v} \right)dv,$$ (6) 式中:H(r,v)为任意函数;p(v)为非负函数。从上式可以看出,通过选择合适的H(r,v)和p(v)函数,可以构建物理上可实现的交叉光谱密度函数及具有特殊相干结构的部分相干光束模型。
为得到非均匀关联光束,在公式(6)中,笔者设[7]
$$p\left( v \right) = {\left( {4\pi } \right)^{ - 1/2}}\left( {\frac{2}{a}} \right)\exp \left( { - \frac{{{v^2}}}{{{a^2}}}} \right),$$ (7) $$H\left( {{{r}},v} \right) = \exp \left( { - \frac{{{{{r}}^2}}}{{w_0^2}}} \right)\exp \left( { - ikv{{{r}}^2}} \right),$$ (8) 将公式(7)、(8)代入公式(6),经过积分运算得到非均匀关联光束在光源处的交叉谱密度函数,即,
$$W\left( {{{{r}}_1},{{{r}}_2}} \right) = \exp \left( { - \frac{{{{r}}_1^2 + {{r}}_2^2}}{{w_0^2}}} \right)\exp \left[ { - \frac{{{{\left( {{{r}}_1^2 - {{r}}_2^2} \right)}^2}}}{{r_{ \rm c}^4}}} \right],$$ (9) 式中:w0为光束束腰;rc=(2/ka)(1/2)为光束相干长度,k=2π/λ为波数,a为正实常数。
下面,笔者将推导非均匀关联光束经过大气湍流后的交叉光谱密度函数。基于广义惠更斯-菲涅尔衍射积分公式[2],部分相干光束经过弱湍流大气传输,其交叉光谱密度函数可以表示为:
$$ \begin{split} W\left( {{{{\rho }}_1},{{{\rho }}_2}} \right) =& \dfrac{1}{{{\lambda ^2}{{\textit{z}}^2}}} \displaystyle\int {\int {\int {\int_{ - \infty }^\infty {W\left( {{{{r}}_1},{{{r}}_2}} \right)} } } } \times\\ &\exp \left[ { - \dfrac{{ik}}{{2{\textit{z}}}}{{\left( {{{{r}}_1} - {{{\rho }}_1}} \right)}^2} + \dfrac{{ik}}{{2{\textit{z}}}}{{\left( {{{{r}}_2} - {{{\rho }}_2}} \right)}^2}} \right] \times\\ & \left\langle {\exp \left[ {\varPsi \left( {{{{r}}_1},{{{\rho }}_1}} \right) + {\varPsi ^*}\left( {{{{r}}_2},{{{\rho }}_2}} \right)} \right]} \right\rangle {\rm d^2}{{{r}}_1}{\rm d^2}{{{r}}_2}, \end{split} $$ (10) 式中:Ψ为随机介质的折射率波动引起的复杂相位扰动。上式中的统计平均表达式为[2]:
$$\begin{array}{l} \left\langle {\exp \left[ {\varPsi \left( {{{{r}}_1},{{{\rho }}_1}} \right) + {\varPsi ^ * }\left( {{{{r}}_2},{{{\rho }}_2}} \right)} \right]} \right\rangle {\rm{ = }} \exp \bigg\{ - 4{\pi ^2}{k^2}{\textit{z}} \displaystyle\int_0^1 \displaystyle\int_0^\infty {{\kappa }}{\varPhi _n}\left( {{\kappa }} \right) \cdot \\ \left\{ {1 - {J_0}\left[ {\left| {\left( {1 - \xi } \right){{P}} + \xi {{Q}}} \right|{{\kappa }}} \right]} \right\} {\rm {d^2}}{{\kappa }}{\rm d}\xi \bigg\}, \end{array} $$ (11) 其中,P=ρ1−ρ2, Q=r1−r2, J0为零阶贝塞尔函数,可以近似表示为:
$${J_0}\left[ {\left| {\left( {1 - \xi } \right){{P}} + \xi {{Q}}} \right|{{\kappa }}} \right] \sim {\rm{1}} - \frac{1}{4}{\left[ {\left( {1 - \xi } \right){{P}} + \xi {{Q}}} \right]^2}{{{\kappa }}^2}.$$ (12) 将公式(12)代入到公式(11),得到:
$$ \begin{array}{l} \left\langle {\exp \left[ {\varPsi \left( {{ r_1},{ \rho _1}} \right) + {\varPsi ^*}\left( {{ r_2},{ \rho _2}} \right)} \right]} \right\rangle = \exp \Biggr\{ { - \dfrac{{{\pi ^2}{k^2}{\textit{z}}T}}{3}\left[ {{{\left( {{ \rho _1} - { \rho _2}} \right)}^2} + } \right.} \\ {\left. {\left( {{ \rho _1} - { \rho _2}} \right) \cdot \left( {{ r_1} - { r_2}} \right) + {{\left( {{ r_1} - { r_2}} \right)}^2}} \right]} \Biggr\} ,\\ \end{array} $$ (13) 式中:
$T = \displaystyle\int_0^\infty {{\kappa ^3}\varPhi \left( \kappa \right)\rm d\kappa }$ ,$\varPhi \left( \kappa \right)$ 表示湍流大气的折射率起伏强度谱。笔者选择van Karman湍流谱[20-21]进行接下来的研究,即$$\varPhi \left( \kappa \right) = A\left( \alpha \right)C_n^2{\left( {{\kappa ^2} + \kappa _0^2} \right)^{ - \alpha /2}}\exp \left( { - {{{\kappa ^2}} / {\kappa _m^2}}} \right),$$ (14) 基于van Karman谱,公式(13)中的T可以表示如下:
$$\begin{array}{l} T = \dfrac{{A(\alpha )}}{{2(\alpha - 2)}}C_n^2 \times \left[ {\beta \kappa _m^{2 - \alpha }\exp \left( {\dfrac{{\kappa _0^2}}{{\kappa _m^2}}} \right){\varGamma _1}\left( {2 - \dfrac{\alpha }{2},\dfrac{{\kappa _0^2}}{{\kappa _m^2}}} \right) - 2\kappa _0^{4 - \alpha }} \right], \\ \;\;\left( {3 < \alpha < 4} \right), \\ \end{array} $$ (15) 式中:
$C_n^2$ 为折射率起伏结构常数;Γ1为伽马函数;$\;\beta = 2\kappa _0^2 - 2\kappa _m^2 + \alpha \kappa _m^2$ ;κ0=2π/L0,L0为湍流的外尺度; κm=5.92/ l0,l0为湍流的内尺度。将公式(6)~(8)代入到公式(10),为了使积分运算简化,交换积分次序后公式(10)等效为:
$$W\left( {{{{\rho}} _1},{{{\rho}} _2},{\textit{z}}} \right) = \int {p\left( v \right)P\left( {{{{\rho}} _1},{{{\rho}} _2},v,{\textit{z}}} \right)} {\rm d}v,$$ (16) 其中,
$$ \begin{split} P\left( {{ \rho _1},{ \rho _2},v,{\textit{z}}} \right) = &{\left( {\dfrac{k}{{2\pi {\textit{z}}}}} \right)^2}\int {\int\limits_{ - \infty }^\infty {H_0^*\left( {{ r_1},v} \right){H_0}\left( {{ r_2},v} \right)} } \times \hfill \\ & \exp \Biggr\{ { - \frac{{{\pi ^2}{k^2}{\textit{z}}T}}{3}} \left[ {{{\left( {{ \rho _1} - { \rho _2}} \right)}^2} + \left( {{ \rho _1} - { \rho _2}} \right) \cdot } \right. \\ & {\left. {\left( {{ r_1} - { r_2}} \right) + {{\left( {{ r_1} - { r_2}} \right)}^2}} \right]} \Biggr\} \times \exp \Biggr[ { - \frac{{ik}}{{2{\textit{z}}}}} \\ & {\left[ {{{\left( {{ r_1} - { \rho _1}} \right)}^2} - {{\left( {{ r_2} - { \rho _2}} \right)}^2}} \right]} \Biggr]{\rm d^2}{r_1}{\rm d^2}{r_2} .\\ \end{split} $$ (17) 将公式(8)代入上式,经过繁琐积分之后,得到:
$$ \begin{split} P\left( {{{{\rho }}_1},{{{\rho }}_2},v,{\textit{z}}} \right) =& \dfrac{{w_0^2}}{{2w_{\textit{z}}^2}}\exp \left[ { - \dfrac{{ik}}{{2{\textit{z}}}}\left( {{{\rho }}_1^2 - {{\rho }}_2^2} \right)} \right] \times \exp \Biggl[ - \Biggl( \dfrac{{w_0^2{k^2}}}{{8{{\textit{z}}^2}}} +\\ & \dfrac{1}{3}{\pi ^2}{k^2}{\textit{z}}T \Biggl){{( {{{{\rho }}_1} - {{{\rho }}_2}} )}^2} \Biggl] \times {\rm{exp}} \Biggl\{ - \dfrac{1}{{w_{\textit{z}}^2}}\Biggl| - i\Biggl[ \dfrac{{kw_0^2}}{{4{\textit{z}}}}( {1- }\\ & {2v{\textit{z}}} ) - \dfrac{1}{3}{\pi ^2}k{{\textit{z}}^2}T \Biggl]( {{{{\rho }}_1} - {{{\rho }}_2}} ) +\Biggl ( {\dfrac{{{{{\rho }}_1} + {{{\rho }}_2}}}{2}} \Biggl) \Biggl|^2 \Biggl\} \end{split} $$ (18) 其中,
$$w_{\textit{z}}^2 = \dfrac{{w_0^2}}{2}{\left( {1 - 2v{\textit{z}}} \right)^2} + {\left( {\dfrac{{\sqrt 2 {\textit{z}}}}{{k{w_0}}}} \right)^2} + \dfrac{{4{\pi ^2}{{\textit{z}}^3}}}{3}T.$$ (19) 至此,笔者根据公式(16)和公式(18)可以计算非均匀关联光束传输经过大气湍流后在接收面上的交叉光谱密度函数。
基于公式(16)和公式(18),并结合公式(1)~(4),经过积分运算之后得到非均匀关联光束经过大气湍流之后的耦合效率:
$$\left\langle {{\eta _{\rm c}}} \right\rangle = \dfrac{{ \displaystyle\int {p\left( v \right)} \pi {w_0}^2/\left( {{w_{\textit{z}}}^2{w_{\rm a}}^{\rm{2}}{C_{\textit{z}}}} \right){\rm d}v}}{{ \displaystyle\int {p\left( v \right)} \pi {w_0}^2{w^2}/\left( {2{w^2} + 4{w_{\textit{z}}}^2} \right){\rm d}v}},$$ (20) 其中
$$ \begin{split} {C_{\textit{z}}} =& {\left( {{A_{\textit{z}}} + \dfrac{{{B_{\textit{z}}}^2}}{{{w_{\textit{z}}}^2}} + \dfrac{1}{{4{w_{\textit{z}}}^2}} + \dfrac{1}{{{w_a}^{\rm{2}}}} + \dfrac{1}{{{w^2}}}} \right)^2} +\\ & {\left( {\dfrac{k}{{2{\textit{z}}}}} \right)^2} - {\left( {{A_{\textit{z}}} + \dfrac{{{B_{\textit{z}}}^2}}{{{w_{\textit{z}}}^2}} - \dfrac{1}{{4{w_{\textit{z}}}^2}}} \right)^2}, \end{split} $$ (21) $$\begin{array}{l} {A_{\textit{z}}} = \dfrac{{w_0^2{k^2}}}{{8{{\textit{z}}^2}}} + \dfrac{1}{3}{\pi ^2}{k^2}{\textit{z}}T; \\ {B_{\textit{z}}} = \dfrac{{kw_0^2}}{{4{\textit{z}}}}\left( {1 - 2v{\textit{z}}} \right) - \dfrac{1}{3}{\pi ^2}k{{\textit{z}}^2}T. \\ \end{array} $$ (22) 根据公式(7)和(20)可以计算非均匀关联光束经过大气湍流传输后在接收面上的光纤耦合效率。
Coupling efficiency of non-uniformly correlated beams into a single-mode fiber in turbulence (Invited)
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摘要: 研究了非均匀关联光束经过大气湍流后的光纤耦合效率。研究结果表明,大气湍流中非均匀关联光束的光纤耦合效率优于传统高斯谢尔模光束;且调控光束相干长度可以提高光纤耦合效率;针对不同传输距离的情况,可以通过调控光束相干长度实现耦合效率的最优化。同时,文中还讨论了光源参数:束腰和波长;耦合透镜参数:接收孔径和焦距;湍流强度对光纤耦合效率的影响。该研究成果证明光场相干结构调控技术在提升光纤耦合效率中的应用,在自由空间激光通信研究领域存在重要价值。Abstract: The coupling efficiency of non-uniformly correlated beams through atmospheric turbulence was studied. The results show that the fiber coupling efficiency of such beams is higher than that of the traditional Gaussian Schell-model beam; and the regulation of the coherence length of such beams can improve the fiber coupling efficiency; for different transmission distances, the optimization of the coupling efficiency can be achieved by adjusting the coherence length of such beams. Moreover, the effect of the light source parameters: beam waist and wavelength; coupling lens parameters: received aperture and focal length; turbulence intensity on the coupling efficiency of optical fiber was also discussed. The results show that the application of optical field correlation structure manipulation technology in improving the coupling efficiency of optical fiber has important value in the field of free space optical communication.
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Key words:
- atmosphere turbulent /
- correlation structure /
- coupling efficiency
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