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在柱坐标系和近轴近似下,Airy涡旋光束的电场解析表达式为[23-24]:
$ \begin{gathered} E_{0}(r, \varphi, z)=-\frac{\mathrm{i} k}{z} w_{0}\left(r_{0}-w_{0} \beta^{2}\right) \mathrm{J}_{l_{0}}\left(\frac{k r_{0} r}{z}\right) \times \\ \exp \left(\mathrm{i} k \frac{r^{2}}{2 z}+\frac{\beta^{3}}{3}+\mathrm{i} l_{0} \varphi\right) \end{gathered} $
(1) 式中,柱坐标(r, φ, z)分别表示光束的径向位置、角向位置和传输距离; k=2π/λ是波数,λ为波长; w0与主Airy波瓣的宽度有关,r0表示主亮环的半径,β为截断指数,l0是OAM的拓扑荷数; Jl0()表示阶数为l0的第1类贝塞尔函数。
基于拓展的惠更斯-菲涅耳原理,当传输距离为z时,部分相干Airy涡旋光束的交叉谱密度函数可以表示为[25-26]:
$ \begin{gathered} W\left(\boldsymbol{\rho}_{1}, \boldsymbol{\rho}_{2}, z\right)=\left(\frac{k}{2 {\rm{ \mathsf{ π} }} z}\right)^{2} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} W\left({\boldsymbol{\rho}^{\prime}}_{1}, \boldsymbol{\rho}_{2}{}^{\prime}, 0\right) \times \\ \exp \left\{-\frac{\mathrm{i} k}{2 z}\left[\left(\boldsymbol{\rho}_{1}-\boldsymbol{\rho}_{1}{}^{\prime}\right)^{2}-\left(\boldsymbol{\rho}_{2}-\boldsymbol{\rho}_{2}{}^{\prime}\right)^{2}\right]\right\} \times \\ \left\langle\exp \left[\psi\left(\boldsymbol{\rho}_{1}{}^{\prime}, \boldsymbol{\rho}_{1}\right)+\psi^{*}\left(\boldsymbol{\rho}_{2}{}^{\prime}, \boldsymbol{\rho}_{2}\right)\right]\right\rangle \mathrm{d}^{2} \boldsymbol{\rho}_{1}{}^{\prime} \mathrm{d}^{2} \boldsymbol{\rho}_{2}{}^{\prime} \end{gathered} $
(2) 式中,ρ1≈(r, φ)和ρ2≈(r′, φ′)为两个位于输出面的任意横向位置矢量,ρ1′和ρ2′为两个位于源平面的任意横向位置矢量; W(ρ1′, ρ2′, 0)是部分相干Airy涡旋光束在源平面的交叉谱密度函数; 〈〉为系综平均,*表示复共轭,ψ表示球面波在湍流中由源平面传输到输出面时复相位的随机项。
利用Rytov近似,部分相干Airy涡旋光束在输出面的交叉谱密度函数可以表示为[27]:
$ \begin{aligned} W\left(\boldsymbol{\rho}_{1}, \boldsymbol{\rho}_{2}, z\right)& \approx E_{0}\left(\boldsymbol{\rho}_{1}, z\right) E_{0}^{*}\left(\boldsymbol{\rho}_{2}, z\right) \mu\left(\boldsymbol{\rho}_{1}, \boldsymbol{\rho}_{2}\right) \times \\ &\left\langle\exp \left[\psi\left(\boldsymbol{\rho}_{1}, z\right)+\psi\left(\boldsymbol{\rho}_{2}, z\right)\right]\right\rangle \end{aligned} $
(3) 式中,μ(ρ1, ρ2)是光谱的相干度,它的高斯形式可以表示为[28]:
$ \begin{aligned} &\mu\left(\boldsymbol{\rho}_{1}, \boldsymbol{\rho}_{2}\right)=\exp \left[-\frac{\left(\boldsymbol{\rho}_{1}-\boldsymbol{\rho}_{2}\right)^{2}}{\sigma_{0}{ }^{2}}\right]= \\ &\exp \left[-\frac{r^{2}+r^{\prime 2}-2 r r^{\prime} \cos \left(\varphi-\varphi^{\prime}\right)}{\sigma_{0}{ }^{2}}\right] \end{aligned} $
(4) 式中,σ0是光束源的空间相干宽度。
(3) 式最后一项为大气湍流引起的相位波动,可以表示为[29]:
$ \begin{gathered} \left\langle\exp \left[\psi\left(\boldsymbol{\rho}_{1}, z\right)+\psi\left(\boldsymbol{\rho}_{2}, z\right)\right]\right\rangle= \\ \exp \left\{-4 {\rm{ \mathsf{ π} }}^{2} k^{2} z \int_{0}^{1} \int_{0}^{\infty} \kappa \varPhi_{n}(\kappa, \alpha)[1-\right. \\ \left.\left.\mathrm{J}_{0}\left(\kappa \xi\left|\boldsymbol{\rho}_{1}-\boldsymbol{\rho}_{2}\right|\right)\right] \mathrm{d} {\kappa} \mathrm{d} \xi\right\} \end{gathered} $
(5) 式中,0阶贝塞尔函数J0可以近似为[30]:
$ \mathrm{J}_{0}\left(\kappa \xi\left|\boldsymbol{\rho}_{1}-\boldsymbol{\rho}_{2}\right|\right) \approx 1-\frac{\left(\kappa \xi\left|\boldsymbol{\rho}_{1}-\boldsymbol{\rho}_{2}\right|\right)^{2}}{4} $
(6) 式中,κ为2维空间频率;Φn(κ, α)是湍流介质的折射率起伏有效功率谱函数;ξ是归一化距离变量,ξ=1-z/L,L为沿z轴从发射机的发射孔到接收机的传输距离。
根据非Kolmogorov湍流理论,受湍流的内外尺度效应的影响,湍流有效功率谱可以表示为:
$ \begin{array}{c} \varPhi_{n}(\kappa, \alpha)=A(\alpha) C_{n}{}^{2} \kappa^{-\alpha}\left[\exp \left(-\frac{\kappa^{2}}{\kappa_{n, x}{}^{2}}\right)-\right. \\ \left.\exp \left(-\frac{\kappa^{2}}{\kappa_{m, n, x}{}^{2}}\right)+\frac{\kappa^{\alpha}}{\left(\kappa^{2}+\kappa_{y}{}^{2}\right)^{\frac{\alpha}{2}}}\right] \end{array} $
(7) 式中,α是湍流谱幂指数,Cn2是单位为m3-α的湍流折射率起伏结构常数。
$ A(\alpha)=\frac{\Gamma(\alpha-1)}{4 {\rm{ \mathsf{ π} }}^{2}} \cos \left(\frac{{\rm{ \mathsf{ π} }}}{2} \alpha\right) $
(8) 式中,Γ()为伽马函数。
$ \left\{\begin{array}{l} \frac{1}{\kappa_{n, x}{}^{2}}=\frac{1}{\kappa_{n}{}^{2}}+\frac{1}{\kappa_{x}{}^{2}} \\ \frac{1}{\kappa_{m, n, x}{}^{2}}=\frac{1}{\kappa_{m}{}^{2}}+\frac{1}{\kappa_{n}{}^{2}}+\frac{1}{\kappa_{x}{}^{2}} \end{array}\right. $
(9) 式中,κn=c(α)/n0, κm=8π/m0分别表示湍流涡流内尺度和外尺度对应的空间频率, n0和m0分别为湍流内尺度和外尺度。
$ c(\alpha)=\left[\frac{2 {\rm{ \mathsf{ π} }}}{3} A(\alpha) \Gamma\left(\frac{5-\alpha}{2}\right)\right]^{1 /(\alpha-5)} $
(10) ${\kappa _x} = \sqrt {k{\eta _x}/z} , \;{\kappa _y} = \sqrt {k{\eta _y}/z} $分别是大尺度和小尺度滤波函数的空间截止频率。
参量ηx, ηy可以表示为[31]:
$ \left\{\begin{array}{l} \eta_{x}=\frac{1}{1+f_{x}(\alpha) \sigma_{\mathrm{R}}{}^{4 /(\alpha-2)}}\left[\frac{7.35 \beta(\alpha)}{\Gamma(3-\alpha / 2)}\right]^{2 /(6-\alpha)} \\ f_{x}(\alpha)=[1.02 r(\alpha) I(\alpha)]^{2 /(\alpha-6)} \\ I(\alpha)=(\alpha-1)^{\frac{6-\alpha}{\alpha-2}} \frac{\Gamma^{2}(\alpha-3)}{\Gamma(2 \alpha-6)} \\ r(\alpha)=\frac{1}{\alpha-2}{2} ^\frac{(3-\alpha)(\alpha-10)}{\alpha-2}\left[-\frac{\Gamma(1-\alpha / 2)}{\Gamma(\alpha / 2)}\right]^{\frac{\alpha-6}{\alpha-2}} \times \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \Gamma\left(\frac{6-\alpha}{\alpha-2}\right)[\beta(\alpha)]^{\frac{8-2 \alpha}{\alpha-2}} \end{array}\right. $
(11) $ \left\{\begin{array}{ll} \eta_{y}= {[0.06375(\alpha-2) \beta(\alpha)]^{2 /(2-\alpha)} \times} \\ \ \ \ \ \ \ \ \ \ {\left[1+f_{y}(\alpha) \sigma_{\mathrm{R}}{}^{4 /(\alpha-2)}\right]} \\ f_{y}(\alpha) =(\ln 2 / 0.51)^{2 /(2-\alpha)} \end{array}\right. $
(12) $ \left\{\begin{array}{l} \beta(\alpha)=-4 \Gamma\left(1-\frac{\alpha}{2}\right) \sin \left(\frac{{\rm{ \mathsf{ π} }} \alpha}{4}\right) \frac{\Gamma^{2}(\alpha / 2)}{\Gamma(\alpha)} \\ \sigma_{\mathrm{R}}{}^{2}=\beta(\alpha) A(\alpha) C_{n}^{2} {\rm{ \mathsf{ π} }}^{2} k^{3-\alpha / 2} z^{\alpha / 2} \end{array}\right. $
(13) 基于Rytov相位结构函数的二次项近似,(5)式可以表示为[32]:
$ \begin{gathered} \left\langle\exp \left|\psi(r, \varphi, z)+\psi^{*}\left(r^{\prime}, \varphi^{\prime}, z\right)\right|\right\rangle \approx \\ \exp \left\{-\frac{1}{3} {\rm{ \mathsf{ π} }}^{2} k^{2} z \times\left[r^{2}+r^{\prime 2}-2 r r^{\prime} \cos \left(\varphi-\varphi^{\prime}\right)\right] \times\right. \\ \left.\int_{0}^{\infty} \kappa^{3} \varPhi_{n}(\kappa, \alpha) \mathrm{d} \kappa\right\} \approx \\ \exp \left\{-\left[r^{2}+r^{\prime 2}-2 r r^{\prime} \cos \left(\varphi-\varphi^{\prime}\right)\right] / \rho_{0}^{2}\right\} \end{gathered} $
(14) 式中, ρ0为球面波在非Kolmogorov湍流中的空间相干长度,形式可以表示为:
$ \rho_{0}=\left[\frac{1}{3} {\rm{ \mathsf{ π} }}^{2} k^{2} z \int_{0}^{\infty} \kappa^{3} \varPhi_{n}(\kappa, \alpha) \mathrm{d} \kappa\right]^{-\frac{1}{2}} $
(15) 当部分相干Airy涡旋光束在大气湍流中传播时,受到大气折射率起伏变化的影响,湍流引起的累积效应假定为OAM光束的相位扰动,导致涡旋模式偏离了OAM的原始本征态。携带轨道角动量的部分相干Airy涡旋光束的复振幅表达式为:
$ E(r, \varphi, z)=E_{0}(r, \varphi, z) \exp [\psi(r, \varphi, z)] $
(16) 为了获得新的涡旋模式分量的权值,将部分相干Airy涡旋光束分解为携带相位因子exp(ilφ)螺旋谐波的叠加[33]:
$ E(r, \varphi, z)=\frac{1}{\sqrt{2 {\rm{ \mathsf{ π} }}}} \sum\limits_{l=-\infty}^{\infty} \bar{\omega}_{l}(r, z) \exp (\mathrm{i} l \varphi) $
(17) 式中,l是基于螺旋谱理论分解后的拓扑荷数。
展开系数:
$ \bar{\omega}_{l}(r, z)=\frac{1}{\sqrt{2 {\rm{ \mathsf{ π} }}}} \int_{0}^{2 {\rm{ \mathsf{ π} }}} E(r, \varphi, z) \exp (-\mathrm{i} l \varphi) \mathrm{d} \varphi $
(18) 携带轨道角动量的部分相干Airy涡旋光束通过大气湍流在(r, z)位置的模式强度概率密度可以用|ωl(r, z)|2的系综平均表示:
$ \begin{gathered} \left\langle\left|\bar{\omega}_{l}(r, z)\right|^{2}\right\rangle=\frac{1}{2 {\rm{ \mathsf{ π} }}} \int_{0}^{2 {\rm{ \mathsf{ π} }}} \int_{0}^{2 {\rm{ \mathsf{ π} }}} E_{0}(r, \varphi, z) \times \\ E_{0}{ }^{*}\left(r^{\prime}, \varphi^{\prime}, z\right) \mu(r, \varphi) \mu^{\prime}\left(r, \varphi^{\prime}\right) \times \\ \left\langle\exp \left|\psi(r, \varphi, z)+\psi^{*}\left(r^{\prime}, \varphi^{\prime}, z\right)\right|\right\rangle \times \\ \exp \left[-\mathrm{i} l\left(\varphi-\varphi^{\prime}\right)\right] \mathrm{d} \varphi \mathrm{d} \varphi^{\prime} \end{gathered} $
(19) 结合(1)式、(4)式、(14)式,部分相干Airy涡旋光束在非Kolmogorov湍流中传播的模式强度概率密度的表达式:
$ \begin{gathered} \left\langle\left|\bar{\omega}_{l}(r, z)\right|^{2}\right\rangle=\frac{1}{2 {\rm{ \mathsf{ π} }}} \int_{0}^{2 {\rm{ \mathsf{ π} }}} \int_{0}^{2 {\rm{ \mathsf{ π} }}}-\frac{\mathrm{i} k}{z} w_{0}\left(r_{0}-w_{0} \beta^{2}\right) \mathrm{J}_{l_{0}}\left(\frac{k r_{0} r}{z}\right) \times \\ \exp \left(\mathrm{i} k \frac{r^{2}}{2 z}+\frac{\beta^{3}}{3}+\mathrm{i} l_{0} \varphi\right) \frac{\mathrm{i} k}{z} w_{0}\left(r_{0}-w_{0} \beta^{2}\right) \times \\ \mathrm{J}_{l_{0}}\left(k r_{0} r^{\prime} / z\right) \exp \left(-\mathrm{i} k r^{\prime 2} /(2 z)+\beta^{3} / 3-\mathrm{i} l_{0} \varphi^{\prime}\right) \times \\ \exp \left\{-\left[r^{2}+r^{\prime 2}-2 r r^{\prime} \cos \left(\varphi-\varphi^{\prime}\right)\right] / \sigma_{0}{}^{2}\right\} \times \\ \exp \left\{-\left[r^{2}+r^{\prime 2}-2 r r^{\prime} \cos \left(\varphi-\varphi^{\prime}\right)\right] / \rho_{0}{}^{2}\right\} \times \\ \exp \left[-\mathrm{i} l\left(\varphi-\varphi^{\prime}\right)\right] \mathrm{d} \varphi \mathrm{d} \varphi^{\prime} \end{gathered} $
(20) 根据积分表达式:
$ \begin{gathered} \int_{0}^{2 {\rm{ \mathsf{ π} }}} \exp \left[-\mathrm{i} l_{\varphi}+\xi \cos \left(\varphi-\varphi^{\prime}\right)\right] \mathrm{d} \varphi= \\ 2 {\rm{ \mathsf{ π} }} \exp \left(-\mathrm{i} l \varphi^{\prime}\right) \mathrm{I}_{l}(\xi) \end{gathered} $
(21) 可得:
$ \begin{gathered} \left\langle\left|\bar{\omega}_{l}(r, z)\right|^{2}\right\rangle=2 {\rm{ \mathsf{ π} }}\left(k^{2} / z^{2}\right) w_{0}^{2}\left(r_{0}-w_{0} \beta^{2}\right)^{2} \times \\ \left|\mathrm{J}_{l_{0}}\left(k r_{0} r / z\right)\right|^{2} \exp \left[2 \beta^{3} / 3-\left(1 / \sigma_{0}{}^{2}+\right.\right. \\ \left.\left.1 / \rho_{0}{}^{2}\right)\left(2 r^{2}\right)\right] \mathrm{I}_{l-l_{0}}\left[\left(1 / \sigma_{0}{}^{2}+1 / \rho_{0}{}^{2}\right)\left(2 r^{2}\right)\right] \end{gathered} $
(22) 式中,Il和Il-l0分别表示阶数为l和l-l0的第1类修正贝塞尔函数。假设l-l0=Δl,则(22)式可写成
$ \begin{gathered} \left\langle\left|\bar{\omega}_{\Delta l}(r, z)\right|^{2}\right\rangle=2 {\rm{ \mathsf{ π} }}\left(k^{2} / z^{2}\right) w_{0}{}^{2}\left(r_{0}-w_{0} \beta^{2}\right)^{2} \times \\ \left|\mathrm{J}_{l_{0}}\left(k r_{0} r / z\right)\right|^{2} \exp \left[2 \beta^{3} / 3-\left(1 / \sigma_{0}^{2}+\right.\right. \\ \left.\left.1 / \rho_{0}{}^{2}\right)\left(2 r^{2}\right)\right] \mathrm{I}_{\Delta l}\left[\left(1 / \sigma_{0}{}^{2}+1 / \rho_{0}{}^{2}\right)\left(2 r^{2}\right)\right] \end{gathered} $
(23) 当Δl=0时,即在接收平面上探测到的光子的OAM模式等于发射的光子的OAM模式,〈|ωΔl=0(r, z)|2〉表示部分相干Airy涡旋光束的MPD;当Δl≠0时,即接收到的OAM模式与发射的OAM模式不相等,说明光束在大气湍流中传播时,涡旋光束的螺旋谱被分散,无法保持其本征态,跃迁到相邻的轨道角动量模式上,〈|ωΔl≠0(r, z)|2〉表示CPD[34]。
对于直径为D的有限孔径接收机, 涡旋模式的螺旋频谱分布可以表示为:
$ P_{\Delta l}=\frac{\int_{0}^{D / 2}\left\langle\left|\bar{\omega}_{\Delta l}(r, z)\right|^{2}\right\rangle r \mathrm{~d} r}{\sum\limits_{l=-\infty}^{\infty} \int_{0}^{D / 2}\left\langle\left|\bar{\omega}_{\Delta l}(r, z)\right|^{2}\right\rangle r \mathrm{~d} r} $
(24)
部分相干Airy涡旋光束在非Kolmogorov谱中的模态强度
Modal intensity of partially coherent Airy vortex beams in non-Kolmogorov turbulence
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摘要: 为了探究部分相干Airy涡旋光束在非Kolmogorov谱中模态强度的演化规律,基于广义的Huygens-Fresnel原理和Rytov近似理论,推导了部分相干Airy涡旋光束的轨道角动量模态概率的解析式。结合MATLAB的数值模拟,研究了部分相干Airy涡旋光束在非Kolmogorov谱湍流大气中传输时湍流参量和波束参量与涡旋模态强度的关系,对部分相干Airy涡旋光束的相干宽度在传输过程中对模态强度的影响进行了理论分析。结果表明,选取拓扑荷数较小、主亮环半径较大、波长较长的部分相干Airy涡旋光束能有效减缓湍流效应的影响,减小强湍流中模态间的串扰;较大的湍流谱幂指数和较小的探测器孔径直径能提高部分相干Airy涡旋光束的模态强度;与完全相干涡旋光束相比,部分相干涡旋光束具有较强的湍流阻力,在大气湍流中能有更好的传输性能,相干性较差会导致螺旋谱分布弥散。这些结果对自由空间光通信的研究具有一定的参考价值。
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关键词:
- 大气光学 /
- 非Kolmogorov谱 /
- 惠更斯-菲涅耳原理 /
- 部分相干Airy涡旋光束 /
- 误比特率
Abstract: In order to study the modal intensity of partially coherent Airy vortex beams in the non-Kolmogorov turbulence, based on the generalized Huygens-Fresnel principle and the Rytov approximation theory, the analytical expressions of modal probability of partially coherent Airy vortex beams carrying orbital angular momentum were derived and the numerical simulation was carried out with MATLAB. The influence of turbulent parameters and beam parameters on the intensity of the vortex mode when partially coherent Airy vortex beams propagate in a non-Kolmogorov turbulence were investigated, the influence of the coherence width of partially coherent Airy vortex beams on the modal intensity during transmission was theoretically studied. The results indicate that partially coherent Airy vortex beams with a smaller topological charge, larger main ring radius and longer wavelength can effectively mitigate the influence of turbulence effect and reduce the crosstalk between modes in strong turbulence; larger non-Kolmogorov spectrum parameter and smaller detector aperture diameter can improve the modal intensity of partially coherent Airy vortex beams. Furthermore, compared with fully coherent vortex beams, partially coherent vortex beams have stronger turbulence resistance and better transmission performance in atmospheric turbulence. While poor coherence will lead to dispersion of spiral spectrum. The research results provide reference for the application of partially coherent Airy vortex beams in free space optical communication. -
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