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空心光束的理论模型可由两束不同束宽(w0和w0η分别为两束光束的束宽,且0 < η < 1)的平顶光束之差构成。那么,空心光束的复振幅可写为:
$ \begin{array}{c} U_{0}\left(\boldsymbol{r}^{\prime}, 0\right)=\sum\limits_{m=1}^{M} \alpha_{m} \exp \left[-\left(m p_{m} \frac{x^{\prime 2}}{w_{0}{}^{2}}\right)\right] \sum\limits_{n=1}^{N} \alpha_{n} \times \\ \exp \left[-\left(n p_{n} \frac{y^{\prime 2}}{w_{0}^{2}}\right)\right]-\sum\limits_{m=1}^{M} \alpha_{m} \exp \left[-\left(m p_{m} \frac{x^{\prime 2}}{\eta^{2} w_{0}{}^{2}}\right)\right] \times \\ \sum\limits_{n=1}^{N} \alpha_{n} \exp \left[-\left(n p_{n} \frac{y^{\prime 2}}{\eta^{2} w_{0}{}^{2}}\right)\right] \end{array} $
(1) 式中,η为光束遮拦比;M, N为光束阶数; αm, αn, pm和pn的表达式详见参考文献[3]。
则空心光束在z=0处的光强表达式为[3]:
$ \begin{array}{c} I_{0}\left(\boldsymbol{r}_{1}{ }^{\prime}, \boldsymbol{r}_{2}, 0\right)=U_{0}{ }^{*}\left(\boldsymbol{r}_{1}{ }^{\prime}, 0\right) U_{0}\left(\boldsymbol{r}_{2}{ }^{\prime}, 0\right)= \\ I_{01}\left(\boldsymbol{r}_{1}{ }^{\prime}, \boldsymbol{r}_{2}{ }^{\prime}, 0\right)+I_{02}\left(\boldsymbol{r}_{1}{ }^{\prime}, \boldsymbol{r}_{2}{ }^{\prime}, 0\right)- \\ I_{03}\left(\boldsymbol{r}_{1}{ }^{\prime}, \boldsymbol{r}_{2}{ }^{\prime}, 0\right)-I_{04}\left(\boldsymbol{r}_{1}{ }^{\prime}, \boldsymbol{r}_{2}{ }^{\prime}, 0\right) \end{array} $
(2) 式中,U0是环状光束的复振幅,U0*是U0的共轭;r1′和r2′分别是U0函数、U0*函数中z=0处垂直于传输方向平面上的空间位置。其中,
$ \begin{array}{c} I_{01}\left(\boldsymbol{r}_{1}{ }^{\prime}, \boldsymbol{r}_{2}{ }^{\prime}, 0\right)=\sum\limits_{m=1}^{M} \sum\limits_{m^{\prime}=1}^{M} \alpha_{m} \alpha_{m^{\prime}} \times \\ \exp \left[-\left(m p_{m} \frac{x_{1}{ }^{\prime 2}}{w_{0}{ }^{2}}+m^{\prime} p_{m^{\prime}} \frac{x_{2}{ }^{\prime 2}}{w_{0}{ }^{2}}\right)\right] \times \\ \sum\limits_{n=1}^{N} \sum\limits_{n^{\prime}=1}^{N} \alpha_{n} \alpha_{n^{\prime}} \exp \left[-\left(n p_{n} \frac{y_{1}{ }^{\prime 2}}{w_{0}{ }^{2}}+n^{\prime} p_{n^{\prime}} \frac{y_{2}{ }^{\prime 2}}{w_{0}{ }^{2}}\right)\right] \end{array} $
(3) $ \begin{array}{c} I_{03}\left(\boldsymbol{r}_{1}{ }^{\prime}, \boldsymbol{r}_{2}{ }^{\prime}, 0\right)=\sum\limits_{m=1}^{M} \sum\limits_{m^{\prime}=1}^{M} \alpha_{m} \alpha_{m^{\prime}} \times \\ \exp \left[-\left(m p_{m} \frac{x_{1}{ }^{\prime 2}}{w_{0}{ }^{2}}+m^{\prime} p_{m^{\prime}} \frac{x_{2}{ }^{\prime 2}}{\eta^{2} w_{0}{ }^{2}}\right)\right] \times \\ \sum\limits_{n=1}^{N} \sum\limits_{n^{\prime}=1}^{N} \alpha_{n} \alpha_{n^{\prime}} \exp \left[-\left(n p_{n} \frac{y_{1}{ }^{\prime 2}}{w_{0}{ }^{2}}+n^{\prime} p_{n^{\prime}} \frac{y_{2}{ }^{\prime 2}}{\eta^{2} w_{0}{ }^{2}}\right)\right] \end{array} $
(4) 式中,x1′和y1′是r1′中的直角坐标;x2′和y2′是r2′中的直角坐标;m′,n′是分别对应x2′和y2′的取值阶数。将(3)式中的w0换为ηw0,得到I02(r1′, r2′, 0);将(4)式中的w0与ηw0互换,得到I04(r1′, r2′, 0)。
为展示空心光束在z=0处光强分布等高线图,可令(3)式中r1′=r2′,并选取光束参量w0=0.02m,η=0.6,M=N=4, 见图 1。
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在光束的传输路径上,光束的二阶矩宽度定义为[3]:
$ w^{2}(z)=\frac{2 \iint \boldsymbol{r}^{2} I(\boldsymbol{r}, z) \mathrm{d}^{2} \boldsymbol{r}}{\iint I(\boldsymbol{r}, z) \mathrm{d}^{2} \boldsymbol{r}} $
(5) 式中,I(r, z)为空心光束传输路径上z处的光强分布,r为传输路径上的空间位置。利用广义惠更斯-菲涅耳公式,I(r, z)可表示为:
$ \begin{array}{*{20}{c}} {I(\mathit{\boldsymbol{r}}, z) = {{\left( {\frac{k}{{2{\rm{ \mathit{ π} }}z}}} \right)}^2}\iint {{{\rm{d}}^2}} {\mathit{\boldsymbol{r}}_1}^\prime \iint {{{\rm{d}}^2}} {\mathit{\boldsymbol{r}}_2}^\prime {I_0}\left( {{\mathit{\boldsymbol{r}}_1}^\prime , {\mathit{\boldsymbol{r}}_2}^\prime , 0} \right) \times }\\ {\exp \left\{ {\left( {\frac{{{\rm{i}}k}}{{2z}}} \right)\left[ {\left( {{\mathit{\boldsymbol{r}}_1}^{\prime 2} - {\mathit{\boldsymbol{r}}_2}^{\prime 2}} \right) - 2\left( {\mathit{\boldsymbol{r}} \cdot {\mathit{\boldsymbol{r}}_1}^\prime - \mathit{\boldsymbol{r}} \cdot {\mathit{\boldsymbol{r}}_2}^\prime } \right)} \right]} \right\} \times }\\ {{{\left\langle {\exp \left[ {{\psi ^*}\left( {\mathit{\boldsymbol{r}}, {\mathit{\boldsymbol{r}}_1}^\prime , z} \right) + \psi \left( {\mathit{\boldsymbol{r}}, {\mathit{\boldsymbol{r}}_2}^\prime , z} \right)} \right]} \right\rangle }_{\rm{m}}}} \end{array} $
(6) 式中,波数k=2π/λ; ψ(r, r2′, z)是湍流介质的复相位结构函数,ψ*(r,r1′, z)是其共轭函数。湍流系综平均〈〉m表示为[4-5]:
$ \begin{array}{c} \left\langle\exp \left[\psi^{*}\left(\boldsymbol{r}, \boldsymbol{r}_{1}{}^{\prime}, z\right)+\psi\left(\boldsymbol{r}, \boldsymbol{r}_{2}{}^{\prime}, z\right)\right]\right\rangle_{\mathrm{m}}= \\ \exp \left\{-4 {\rm{ \mathit{ π}}}^{2} k^{2} z \int_{0}^{1} \int_{0}^{\infty} k \varPhi_{n}(\kappa, \alpha) \times\right. \\ \left.\left[1-\mathrm{J}_{0}\left(\kappa \xi\left|\boldsymbol{r}_{2}{}^{\prime}-\boldsymbol{r}_{1}{}^{\prime}\right|\right)\right] \mathrm{d} \kappa \mathrm{d} \xi\right\} \end{array} $
(7) 式中,J0()为零阶贝塞尔函数,κ表示空间频率,ξ为传输路径参量,κm为低空间截止频率, κ0为高空间截止频率。
根据非Kolmogorov湍流统计[6], Φn(κ,α)=$F(\alpha )\tilde C_n^2\exp \left[ { - \left( {\frac{{{\kappa ^2}}}{{{\kappa _m}^2}}} \right)} \right]/{\left( {{\kappa ^2} + {\kappa _0}^2} \right)^{\alpha /2}}$,其中, $\tilde C_n^2$为折射率结构参量;高空间截止频率${\kappa _0} = \frac{{2{\rm{ \mathsf{ π} }}}}{{{L_0}}}$,低空间截止频率${\kappa _m} = \frac{{t(\alpha )}}{{{l_0}}}$(L0,l0分别为湍流的外尺度和内尺度);F(α)=Γ(α-1)×$\frac{{\cos \left( {\alpha {\rm{ \mathsf{ π} /2}}} \right)}}{{4{{\rm{ \mathsf{ π} }}^2}}}$,$t(\alpha ) = {\left[ {\Gamma \left( {5 - \frac{\alpha }{2}} \right) \cdot F(\alpha ) \cdot \frac{2}{{3{\rm{ \mathsf{ π} }}}}} \right]^{1/(\alpha - 5)}}$,其中Γ(·)为伽马函数,α表示湍流广义指数。
综合(2)式、(6)式及(7)式,并代入(5)式,采用积分变换法,并经繁琐的积分运算,可求得空心光束传输于非Kolmogorov湍流中的二阶矩宽度解析式:
$ w^{2}(z)=w_{1}{ }^{2}+w_{2}{ }^{2} z^{2}+T z^{3} $
(8) 其中,
$ \begin{aligned} w_{1}{ }^{2}=& w_{0}{ }^{2}\left\{\sum _ { m = 1 } ^ { M } \sum _ { m ^ { \prime } = 1 } ^ { M } \sum _ { n = 1 } ^ { N } \sum _ { n ^ { \prime } = 1 } ^ { N } \left[\frac{P}{\sqrt{Q_{1} Q_{2}}}\left(\frac{1}{Q_{1}}+\frac{1}{Q_{2}}\right)\left(1+\eta^{4}\right)-\right.\right.\\ &\left.\left.\frac{P \eta^{4}}{\sqrt{Q_{3} Q_{4}}}\left(\frac{1}{Q_{3}}+\frac{1}{Q_{4}}\right)-\frac{P \eta^{4}}{\sqrt{Q_{5} Q_{6}}}\left(\frac{1}{Q_{5}}+\frac{1}{Q_{6}}\right)\right]\right\} C^{-1} \end{aligned} $
(9) $ \begin{array}{c} w_{2}{}^{2}=\left\{\sum \limits_ { m = 1 } ^ { M } \sum \limits_ { m ^ { \prime } = 1 } ^ { M } \sum \limits_ { n = 1 } ^ { N } \sum \limits_ { n ^ { \prime } = 1 } ^ { N } \frac { 4 } { w _ { 0 } ^ { 2 } k ^ { 2 } } \left[\frac{2 P}{\sqrt{Q_{1} Q_{2}}}\left(\frac{R_{1}}{Q_{1}}+\frac{R_{2}}{Q_{2}}\right)-\right.\right. \\ \left.\left.\frac{P \eta^{2}}{\sqrt{Q_{3} Q_{4}}}\left(\frac{R_{1}}{Q_{3}}+\frac{R_{2}}{Q_{4}}\right)-\frac{P \eta^{2}}{\sqrt{Q_{5} Q_{6}}}\left(\frac{R_{1}}{Q_{5}}+\frac{R_{2}}{Q_{6}}\right)\right]\right\} C^{-1} \end{array} $
(10) $ \begin{array}{c} C=\sum\limits_{m=1}^{M} \sum\limits_{m^{\prime}=1}^{M} \sum\limits_{n=1}^{N} \sum\limits_{n^{\prime}=1}^{N}\left\{\frac{P}{\sqrt{Q_{1} Q_{2}}}\left(1+\eta^{2}\right)-\right. \\ \left.\left(\frac{P}{\sqrt{Q_{3} Q_{4}}}+\frac{P}{\sqrt{Q_{5} Q_{6}}}\right) \eta^{2}\right\} \end{array} $
(11) $ T=\frac{2 {\rm{ \mathit{ π}}}^{2} z^{3}}{3} \int_{0}^{\infty} \kappa^{3} \varPhi_{n}(\kappa, \alpha) \mathrm{d} \kappa $
(12) 式中,P=αmαm′αnαn′,Q1=mpm+m′pm′,Q3=ηmpm+m′pm′,Q5=mpm+ηm′pm′,R1=(mpm)(m′pm′)。若将m及m′分别改为n和n′,即得Q2, Q4, Q6和R2。(8)式中,湍流项T与湍流广义指数α、内尺度l0及外尺度L0相关,即α, l0和L0的取值将影响空心光束的二阶矩宽度。
瑞利区间的定义:光束的横截面积扩展达z=0处两倍时的传输距离[17],即:
$ w^{2}\left(z_{\mathrm{R}}\right)=w_{1}{ }^{2}+w_{2}{ }^{2} z^{2}+T z^{3}=2 w_{1}{ }^{2} $
(13) 求解可得:
$ z_{\mathrm{R}}=\frac{1}{3 T}\left(D+\frac{w_{2}{ }^{4}}{D}-w_{2}{ }^{2}\right) $
(14) 式中,$D = {\left[ {\frac{3}{2}T{{\left( {81w_1^4{T^2} - 12w_1^2w_2^6} \right)}^{1/2}} - w_2^6 + \frac{{27}}{2}w_1^2{T^2}} \right]^{1/3}}$。
另一方面,湍流距离表示为光束的横截面积因受湍流影响而扩展10%及90%的传输距离[18],分别用zt, 1和zt, 2表示:
$ \frac{w^{2}\left(z_{\mathrm{t}}\right)-w_{\mathrm{f}}{ }^{2}\left(z_{\mathrm{t}}\right)}{w^{2}\left(z_{\mathrm{t}}\right)}=\mu $
(15) 式中,wf为自由空间中的束宽,wf2=w12+w22z2。求解三次方程,可得到空心光束传输于非Kolmogorov湍流中的湍流距离zt解析表达式:
$ z_{\mathrm{t}}=\frac{\mu^{2} w_{2}{ }^{4}+\mu w_{2}{ }^{2} J+J^{2}}{3(1-\mu) J T} $
(16) 式中,
$ \begin{array}{c} J=\left\{3(1- \mu) T \frac{\left[81 \mu^{2}(1-\mu)^{2} w_{1}{ }^{4} T^{2}+12 w_{1}{ }^{2} w_{2}{ }^{6}\right]^{1 / 2}}{2}+\right.\\ \left.\frac{27 \mu(1-\mu)^{2} w_{1}{ }^{2} T^{2}}{2}+\mu^{3} w_{2}{ }^{6}\right\}^{1 / 3} \end{array} $
(17) 显然,当μ=10%时,(16)式表示zt, 1;当μ=90%时,(16)式表示zt, 2。
应当指出的是,光束的传输路径可被zt,1和zt, 2划分为3个区域:(1)区域Ⅰ:0 < z≤zt, 1,湍流介质对光束扩展所构成的影响甚小,可忽略,扩展主要由空间衍射造成;(2)区域Ⅱ:zt, 1 < z < zt, 2,空间衍射和湍流均对光束的扩展造成一定影响;(3)区域Ⅲ:z≥zt, 2,湍流为光束扩展的主要影响因素,空间衍射对其的影响可以忽略。
空心光束在非Kolmogorov湍流传输路径上的区域分割
Region division of hollow beams in non-Kolmogorov turbulent path
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摘要: 为了研究空心光束在非Kolmogorov湍流传输路径上的区域范围与各参量之间的关系及不同区域内光束的扩展情况, 采用广义惠更斯-菲涅耳原理推导了空心光束传输于非Kolmogorov湍流中的二阶矩宽度、瑞利区间及湍流距离的解析式, 并利用湍流距离把传输路径分割为3个区域进行数值分析。结果表明, 区域Ⅰ、区域Ⅱ的长度及区域Ⅲ的起始点都随湍流广义指数α的增大而先减小再增大(当α=3.11时出现一个极小值), 且随遮拦比η和光束阶数M(及N)的增大而增大; M(及N)取值较小时(M(及N) < 3), 湍流在瑞利区间内对光束扩展造成的影响不能忽略, M(及N)和η越大, 越容易忽略湍流在瑞利区间内对光束扩展所构成的影响; 光束在传输路径上依次进入区域Ⅰ、区域Ⅱ及区域Ⅲ, 其光束扩展逐渐变得更加剧烈, 且随着M(及N)和η的增加, 区域Ⅱ长度和区域Ⅲ的起始点相较于区域Ⅰ的长度增加更为显著。该研究结果为空心光束传输于湍流中的相关应用提供了参考。
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关键词:
- 大气光学 /
- 非Kolmogorov湍流 /
- 湍流距离 /
- 空心光束 /
- 遮拦比
Abstract: To study the relationship between the region range and the parameters of hollow beam in the non-Kolmogorov turbulence propagation path and the beam expansion in different regions, the expressions for the mean-squared width, Rayleigh range, and turbulence distance of hollow beams propagating through non-Kolmogorov turbulence were given by using the extended Huygens-Fresnel principle, and the propagation path was divided into three regions by using the turbulence distance for numerical analysis. The results show that the length of region Ⅰ and region Ⅱ and the starting point of region Ⅲ decrease first and then increase with the increasing of the turbulence generalized exponent parameter α (There is a minimal value, when α=3.11), and increase with the increasing of obscure ratio η and beam orders M(and N). When the value of M(and N)is small(M(and N) < 3), the effect of turbulence on beam spread in Rayleigh range can not be ignored. The larger M(and N)and η is, the easier it is to ignore the effect of turbulence on beam spread in Rayleigh range. In the transmission path, the beam enters area Ⅰ, area Ⅱ and area Ⅲ in turn, and then expands more and more violently. With the increasing of M(and N)and η, the length of region Ⅱ and the starting point of region Ⅲ increase more significantly than the length of region Ⅰ. The results provide a reference for the application of hollow beam propagation in turbulence.-
Key words:
- atmospheric optics /
- non-Kolmogorov turbulence /
- turbulence distance /
- hollow beams /
- obscure ratio
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