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根据维格纳分布函数的二阶矩理论,部分相干月牙形光束在源场的交叉谱密度(cross-spectral density,CSD)函数形式为[18]:
$ \begin{array}{c}{W^{(0)}\left(\boldsymbol{r}_{1}, \boldsymbol{r}_{2}\right)=\tau^{*}\left(\boldsymbol{r}_{1}\right) \tau\left(\boldsymbol{r}_{2}\right) \int p_{1}\left(\boldsymbol{v}_{1}\right) \times} \\ {\exp \left[-\mathrm{i} k\left(\boldsymbol{r}_{2}-\boldsymbol{r}_{1}\right) \cdot \boldsymbol{v}_{1}\right] \mathrm{d}^{2} \boldsymbol{v}_{1}}\end{array} $
(1) 式中,k是波数,r1与r2是传输过程中两个随机的位置矢量,p1(v1)是任意参量v1≡(vx, vy)值的非负函数,τ(r)具有高斯分布[10]:
$ \tau(\boldsymbol{r})=\exp \left(-\frac{\boldsymbol{r}^{2}}{2 \sigma_{0}^{2}}\right) $
(2) 式中,σ0为光腰宽度,p1(v1)为极坐标形式下的可分离函数[18]:
$ p_{1}\left(\boldsymbol{v}_{1}\right)=\frac{k^{2} \delta^{2}\left(k \delta \boldsymbol{v}_{1}\right)^{2 n}}{2^{n} n ! \pi} \exp \left(-\frac{k^{2} \delta^{2} \boldsymbol{v}_{1}^{2}}{2}\right) \cos ^{2}\left(\frac{\boldsymbol{\theta}}{2}\right) $
(3) 式中,λ是波长,n为光束阶数, θ=arctan(vy/vx)表示极坐标中向量v1的方位角, δ为相干长度。
基于广义惠更斯-菲涅耳原理,在z>0的半空间中,光束从源平面处传输到接收平面处的谱密度(spectral density,SD)函数可表示成[10]:
$ \begin{array}{*{20}{c}} {W(\mathit{\boldsymbol{\rho }}, \mathit{\boldsymbol{\rho }}, z) = \frac{1}{{{\lambda ^2}{z^2}}}\smallint \smallint W^{(0)} \left( {{\mathit{\boldsymbol{r}}_1}, {\mathit{\boldsymbol{r}}_2}} \right) \times }\\ {\exp \left( { - \frac{{{\rm{i}}k}}{{2z}}{\mathit{\boldsymbol{r}}_1}^2 + \frac{{{\rm{i}}k}}{z}{\mathit{\boldsymbol{r}}_1} \cdot \mathit{\boldsymbol{\rho }}} \right) \times }\\ {\exp \left( {\frac{{{\rm{i}}k}}{{2z}}{\mathit{\boldsymbol{r}}_2}^2 + \frac{{{\rm{i}}k}}{z}{\mathit{\boldsymbol{r}}_2} \cdot \mathit{\boldsymbol{\rho }}} \right)\left\langle {\exp \left[ {{\mathit{\Psi }^*}\left( {{\mathit{\boldsymbol{r}}_1}, \mathit{\boldsymbol{\rho }}, z} \right) + } \right.} \right.}\\ {{{\left. {\mathit{\Psi }\left( {{\mathit{\boldsymbol{r}}_2}, \mathit{\boldsymbol{\rho }}, z} \right)]} \right\rangle }_m}{{\rm{d}}^2}{\mathit{\boldsymbol{r}}_1}{{\rm{d}}^2}{\mathit{\boldsymbol{r}}_2}} \end{array} $
(4) 式中,ρ是接收平面处某一位置矢量。带有下标m的角括号表示空间-频域中相干理论电场的综合平均,Ψ(r2, ρ, z)是由介质折射率的随机分布引起的复杂相位扰动,Ψ*(r1, ρ, z)表示复共轭,其中各向同性湍流扰动项[10]:
$ \begin{array}{C} {\left\langle {\exp \left[ {{\mathit{\Psi }^*}\left( {{\mathit{\boldsymbol{r}}_1}, \mathit{\boldsymbol{\rho }}, z} \right) + \mathit{\Psi }\left( {{\mathit{\boldsymbol{r}}_2}, \mathit{\boldsymbol{\rho }}, z} \right)} \right]} \right\rangle _m} = \\ \exp \left[ { - \frac{{{{\left( {{\mathit{\boldsymbol{r}}_2} - {\mathit{\boldsymbol{r}}_1}} \right)}^2}}}{{{\rho _0}^2}}} \right] \end{array} $
(5) 式中,ρ02=3/(2π2k2Tz)为空间相干半径,取决于湍流功率谱的选择。
湍流函数T可以表示为:
$ T = \int_0^\infty {{\kappa ^3}} {\mathit{\Phi }_n}(\kappa ){\rm{d}}\kappa $
(6) 湍流谱为:
$ \begin{array}{*{20}{c}} {{\mathit{\Phi }_n}(\kappa ) = A(\alpha ){{\tilde C}_n}^2{\zeta ^2}{{\left( {{\zeta ^2}{\kappa _{xy}}^2 + {\kappa _z}^2 + {\kappa _0}^2} \right)}^{ - \frac{\alpha }{2}}} \times }\\ {\quad \exp \left( { - \frac{{{\zeta ^2}{\kappa _{xy}}^2 + {\kappa _z}^2}}{{{\kappa _m}^2}}} \right), (\kappa > 0, 3 < \alpha < 5)} \end{array} $
(7) 式中,$ \kappa=\sqrt{\zeta^{2}\left({\kappa_x}^{2}+{\kappa_y}^{2}\right)+{\kappa_z}^{2}}=\sqrt{\zeta^{2} {\kappa_x y}^{2}+{\kappa_z}^{2}}$,κ0=2π/L0, κm=C(α)/l0,l0是湍流内尺度,L0是湍流外尺度,$ {\tilde C_n}^2 = \beta {C_n}^2$是结构常数,α为湍流参量,β为量纲常数,ζ为各向异性参量。
$ \begin{array}{c}{C(\alpha)=\left\{\pi A(\alpha) \Gamma\left(\frac{3}{2}-\frac{\alpha}{2}\right)\left(\frac{3-\alpha}{3}\right)\right\}^{\frac{1}{\alpha-5}}}, \\ {(3<\alpha<5)}\end{array} $
(8) $ A(\alpha)=\frac{\Gamma(\alpha-1)}{4 \pi^{2}} \cos \left(\alpha \frac{\pi}{2}\right), (3<\alpha<5) $
(9) 式中,Γ为伽马函数。
把(7)式代入(6)式,得:
$ \begin{array}{*{20}{l}} {T = \int_\infty ^0 {{\kappa ^3}} {\mathit{\Phi }_n}(\kappa ){\rm{d}}\kappa = {\zeta ^{2 - \alpha }}\frac{{A(\alpha )}}{{2(\alpha - 2)}}\frac{{{C_n}^2}}{{{\zeta ^2}}} \times }\\ {\left[ {\frac{{{\kappa _m}^{2 - \alpha }}}{{{\zeta ^2}}}\beta \exp \left( {\frac{{{\kappa _0}^2}}{{{\kappa _m}^2}}} \right)\Gamma \left( {2 - \frac{\alpha }{2}, \frac{{{\kappa _0}^2}}{{{\kappa _m}^2}}} \right) - 2\frac{{{\kappa _m}^{4 - \alpha }}}{{{\zeta ^2}}}} \right]} \end{array} $
(10) $ \beta=2 \frac{{\kappa_0}^{2}}{\zeta^{2}}-2 \frac{{\kappa_m}^{2}}{\zeta^{2}}+\alpha \frac{{\kappa_m}^{2}}{\zeta^{2}} $
(11) (5) 式等号右边可表示为:
$ \begin{array}{c}{\exp \left[-\frac{\left(\boldsymbol{r}_{2}-\boldsymbol{r}_{1}\right)^{2}}{{\rho_0}^{2}}\right]=\int p_{2}\left(\boldsymbol{v}_{2}\right) \times} \\ {\exp \left[-\mathrm{i} k\left(\boldsymbol{r}_{2}-\boldsymbol{r}_{1}\right) \cdot \boldsymbol{v}_{2}\right] \mathrm{d}^{2} \boldsymbol{v}_{2}}\end{array} $
(12) 式中,p2(v2)=k2ρ02exp(-k2ρ02v22/2)/(2π)。
引入下列代换式:
$ \left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{r}}_{\rm{d}}} = {\mathit{\boldsymbol{r}}_2} - {\mathit{\boldsymbol{r}}_1}}\\ {{\mathit{\boldsymbol{r}}_{\rm{s}}} = \left( {{\mathit{\boldsymbol{r}}_2} + {\mathit{\boldsymbol{r}}_1}} \right)/2} \end{array}} \right. $
(13) 将(1)式、(2)式、(3)式、(5)式、(10)式、(12)式以及(13)式带入(4)式中,经过积分运算得:
$ \begin{array}{c}{\langle I(\boldsymbol{\rho}, z)\rangle=\frac{1}{\lambda^{2} z^{2}} \iiint A^{*}\left(\boldsymbol{r}_{\mathrm{s}}-\frac{\boldsymbol{r}_{\mathrm{d}}}{2}\right) A\left(\boldsymbol{r}_{\mathrm{s}}+\frac{\boldsymbol{r}_{\mathrm{d}}}{2}\right) \times} \\ {p_{1}\left(\boldsymbol{v}_{1}\right) p_{2}\left(\boldsymbol{v}_{2}\right) \exp \left(-\frac{\mathrm{i} k}{z} \boldsymbol{r}_{\mathrm{d}} \cdot \boldsymbol{\rho}\right) \times} \\ {\quad \exp \left[-\mathrm{i} k \boldsymbol{r}_{\mathrm{d}} \cdot\left(\boldsymbol{v}_{1}+\boldsymbol{v}_{2}\right)\right] \mathrm{d}^{2} \boldsymbol{r}_{\mathrm{s}} \mathrm{d}^{2} \boldsymbol{r}_{\mathrm{d}} \mathrm{d}^{2} \boldsymbol{v}_{1} \mathrm{d}^{2} \boldsymbol{v}_{2}}\end{array} $
(14) 式中,A被定义为A(r)=τ(r)exp[ikr2/(2z)],将A以及A*分别表示成它们的傅里叶变换形式:
$ \left\{ {\begin{array}{*{20}{l}} {A\left( {{\mathit{\boldsymbol{r}}_{\rm{s}}} + \frac{{{\mathit{\boldsymbol{r}}_{\rm{d}}}}}{2}} \right) = {{\left( {\frac{k}{{2\pi }}} \right)}^2}\int {\tilde A\left( {{\mathit{\boldsymbol{u}}_1}} \right) \times } }\\ {\exp \left[ {{\rm{i}}k{\mathit{\boldsymbol{u}}_1} \cdot \left( {{\mathit{\boldsymbol{r}}_{\rm{s}}} + \frac{{{\mathit{\boldsymbol{r}}_{\rm{d}}}}}{2}} \right)} \right]{{\rm{d}}^2}{\mathit{\boldsymbol{u}}_1}}\\ {{A^*}\left( {{\mathit{\boldsymbol{r}}_{\rm{s}}} - \frac{{{\mathit{\boldsymbol{r}}_{\rm{d}}}}}{2}} \right) = {{\left( {\frac{k}{{2\pi }}} \right)}^2}\int {{{\tilde A}^*}} \left( {{\mathit{\boldsymbol{u}}_2}} \right) \times }\\ {\exp \left[ { - {\rm{i}}k{\mathit{\boldsymbol{u}}_2} \cdot \left( {{\mathit{\boldsymbol{r}}_{\rm{s}}} - \frac{{{\mathit{\boldsymbol{r}}_{\rm{d}}}}}{2}} \right)} \right]{{\rm{d}}^2}{\mathit{\boldsymbol{u}}_2}} \end{array}} \right. $
(15) 有u1=u2=u, 将(15)式带入(14)式,并对rs, rd与u积分可得:
$ \begin{aligned}\langle I(\boldsymbol{\rho}, z)\rangle &=\frac{1}{\lambda^{2} z^{2}} \iiiint\left|\tilde{A}\left(\boldsymbol{v}_{1}+\boldsymbol{v}_{2}+\frac{\boldsymbol{\rho}}{z}\right)\right|^{2} \times \\ & p_{1}\left(\boldsymbol{v}_{1}\right) p_{2}\left(\boldsymbol{v}_{2}\right) \mathrm{d}^{2} \boldsymbol{v}_{1} \mathrm{d}^{2} \boldsymbol{v}_{2} \end{aligned} $
(16) 式中,$\tilde{A}(\boldsymbol{u})=\int A(\boldsymbol{r}) \exp (-\mathrm{i} k \boldsymbol{r} \cdot \boldsymbol{u}) \mathrm{d} \boldsymbol{r} $。
将p1(v1), p2(v2)与$\tilde{A}(\boldsymbol{u}) $带入(16)式,并对v2积分可得:
$ \begin{aligned}\langle I(\boldsymbol{\rho}, z)\rangle &=\frac{1}{F(z)} \frac{k^{2} {\boldsymbol{\rho}_0}^{2}}{2 M_{1}} \frac{k^{2} \delta^{2}(k \delta)^{2 n}}{2^{n} n ! \pi} \exp \left(-\frac{M_{2}}{{\sigma_0}^{2}} \boldsymbol{\rho}^{2}\right) \times \\ & \int {\boldsymbol{v}_1}^{2 n} \exp \left[-\left(\frac{M_{2} z^{2}}{{\sigma_0}^{2}}+\frac{k^{2} \delta^{2}}{2}\right) {\boldsymbol{v}_1}^{2}\right] \times \\ & \exp \left(-\frac{2 M_{2} z}{{\sigma_0}^{2}} \boldsymbol{\rho} \cdot \boldsymbol{v}_{1}\right) \cos ^{2}\left(\frac{\theta}{2}\right) \mathrm{d}^{2} \boldsymbol{v}_{1} \end{aligned} $
(17) 式中,$F(z)=1+\frac{z^{2}}{k^{2} {\sigma_0}^{4}}, M_{1}=\frac{z^{2}}{{\sigma_0}^{2} F(z)}+\frac{k^{2} {\rho_0}^{2}}{2} $$ M_2=\frac{1}{F(z)}\left[1-\frac{z^{2}}{{\sigma_0}^{2} F(z) M_{1}}\right]$。
将(17)式由直角坐标转化到极坐标, 对v1与θ积分,将得到在各向异性湍流中传输的月牙形光束的平均光强的解析表达式:
$ \begin{array}{*{20}{c}} {\langle I(\mathit{\boldsymbol{\rho }}, z)\rangle = \frac{1}{{F(z)}}\frac{{{k^2}{\rho _0}^2}}{{2{M_1}}}\frac{{{k^2}{\delta ^2}{{(k\delta )}^{2n}}}}{{{2^n}n!}} \times }\\ {{\exp \left( { - \frac{{{M_2}}}{{{\sigma _0}^2}}{\mathit{\boldsymbol{\rho }}^2}} \right)\exp ( - t)\left\{ {\frac{1}{2}n!{{\left( {\frac{{2{\sigma _0}^2}}{{2{M_2}{z^2} + {k^2}{\sigma _0}^2{\delta ^2}}}} \right)}^{n + 1}}{{\rm{L}}_1} - } \right.}} \\ {\left. {\frac{1}{2}\left( {n - \frac{1}{2}} \right)!\frac{{{M_2}z}}{{{\sigma _0}^2}}\mathit{\boldsymbol{\rho }}{{\left( {\frac{{2{\sigma _0}^2}}{{2{M_2}{z^2} + {k^2}{\sigma _0}^2{\delta ^2}}}} \right)}^{n + 3/2}}{{\rm{L}}_2}\cos \phi } \right\}} \end{array} $
(18) 式中,Ln0表示广义拉盖尔函数,L1=Ln0(t), L2=Ln-1/21(t), t=-2M22z2ρ2/[σ02(2M2z2+k2σ02δ2)]。
对(17)式做计算,得出光束在传输横截面上方均根束宽为:
$ {W_{{\rm{LT}}}}(z) = \frac{{{\sigma _0}}}{z}\sqrt {\frac{1}{{{M_2}}}} = \sqrt {\frac{1}{{{k^2}{\sigma _0}^2}} + \frac{{{\sigma _0}^2}}{{{z^2}}} + \frac{{4{\pi ^2}Tz}}{3}} $
(19) 从(19)式可知,方程右侧平方根第1项表示光束在自由空间中传输时由σ0所引起的衍射;第2项表示光束在自由空间中传输时只与初始参量和传播距离有关,且其与传输距离的平方成反比;第3项表示各向异性湍流引起的衍射。在湍流函数T=0的自由空间中,w(z)由无穷大降至一常数1/(kσ0)。当光束的传输距离很短时,光束受到各向异性湍流引起的衍射可以忽略不计;随着传输距离的增加,由各向异性湍流引起的衍射项将起着主导作用,因为(19)式中第3项值与z成正比,而第2项的值与z的平方成反比。
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ANDREWS和PHILIPS基于几何光学近似和Rytov近似,引入高斯滤波函数得到漂移模型为[19]:
$ \begin{aligned}\left\langle {r_\mathrm{c}}^{2}\right\rangle &= 4 \pi k^{2} W_{\mathrm{FS}}^{2}(L) \int_{0}^{L} \int_{0}^{\infty} \kappa \mathit{\Phi}_{n}(\kappa) H_{\mathrm{LS}}(\kappa, z) \times \\ &\left\{1-\exp \left[-\frac{\mathit{\Lambda} L \kappa^{2}(1-z / L)^{2}}{k}\right]\right\} \mathrm{d} \kappa \mathrm{d} z \end{aligned} $
(20) 式中,
$ \left\{ {\begin{array}{*{20}{l}} {{H_{{\rm{LS}}}}(\kappa , z) = \exp \left[ { - {\kappa ^2}{W_{{\rm{LT}}}}^2(z)} \right]}\\ {\mathit{\Lambda } = \frac{{2L}}{{k{W_{{\rm{FS}}}}^2(L)}}}\\ {1 - \exp \left[ { - \frac{{\mathit{\Lambda }L{\kappa ^2}{{(1 - z/L)}^2}}}{k}} \right] \simeq \frac{{\mathit{\Lambda }L{\kappa ^2}{{(1 - z/L)}^2}}}{k}} \end{array}} \right. $
(21) 式中,L为发射端和接收端之间的距离,WFS(L)表示自由空间下的光束宽度,HLS(κ, z)是大标量滤波函数,WLT(z)表示湍流下的光束宽度,Λ表示无量纲参量,则:
$ \begin{array}{*{20}{c}} {\left\langle {{r_{\rm{c}}}^2} \right\rangle = 4\pi k{W_{{\rm{FS}}}}^2(L)A(\alpha )\overline {{{\tilde C}_n}^2} \mathit{\Lambda }L{\zeta ^{2 - \alpha }} \times }\\ {\int_0^\infty {{\kappa ^3}} {{\left( {{{\tilde \kappa }_0}^2 + {\kappa ^2}} \right)}^{ - \frac{\alpha }{2}}}\exp \left[ { - {\kappa ^2}\left( {\frac{1}{{{{\tilde \kappa }_m}^2}} + } \right.} \right.}\\ {\left. {\left. {{W_{{\rm{LT}}}}^2(z)} \right)} \right]{\rm{d}}\kappa \int_0^L {{{\left( {1 - \frac{z}{L}} \right)}^2}} {\rm{d}}z} \end{array} $
(22) 将(7)式代入(22)式,经过计算得:
$ \begin{array}{*{20}{c}} {\left\langle {{r_{\rm{c}}}^2} \right\rangle = \frac{{8\pi {L^2}A(\alpha )\overline {{{\tilde C}_n}^2} {\zeta ^{2 - \alpha }}}}{{2( - 2 + \alpha )}} \times }\\ {\Gamma \left[ {2 - \frac{\alpha }{2}, {\kappa _0}^2\left( {\frac{{\rm{d}}}{{{\zeta ^2}}} + \frac{1}{{{\kappa _m}^2}}} \right)} \right]\int_0^L {{{(1 - z/L)}^2}} {\rm{d}}z \times }\\ {\left\{ { - 2{{\tilde \kappa }_0}^{4 - \alpha } + {{\tilde \kappa }_m}^2\exp \left[ {{\kappa _0}^2\left( {\frac{d}{{{\zeta ^2}}} + \frac{1}{{{\kappa _m}^2}}} \right)} \right]{{\left( {d + \frac{1}{{{{\tilde \kappa }_m}^2}}} \right)}^{\alpha /2 - 2}} \times } \right.}\\ {\left. {\left( {\frac{{2d{\kappa _0}^2}}{{{\kappa _m}^2}} + \frac{{2{\kappa _0}^2}}{{{{\tilde \kappa }_m}^2{\kappa _m}^2}} + \frac{{ - 2 + \alpha }}{{{{\tilde \kappa }_m}^2}}} \right)} \right\}} \end{array} $
(23) 式中, d=WLT2(z), κm=C(α)/l0, κ0=2π/L0,$ {\tilde{C}_n}^{2}=\beta {C_n}^{2}, \beta=2 {\tilde{\kappa}_0}^{2}-2 {\tilde{\kappa}_m}^{2}+\alpha {\tilde{\kappa}_m}^{2}, {\tilde{\kappa}_0}=\kappa_{0} / \zeta, {\tilde{\kappa}_m}=\kappa_{m} / \zeta, r_{\mathrm{c}}$即表示部分相干月牙形光束在非K谱中的光束漂移。
部分相干月牙形光束在非Kolmogorov谱中的漂移
Beam wander of a partially coherent crescent-like beam in non-Kolmogorov turbulence
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摘要: 为了探究部分相干月牙形光束在非Kolmogorov谱中漂移的演化规律, 采用拓展Huygens-Fresnel原理, 得到了相应的解析表达式, 并运用MATLAB进行了数值模拟。结果表明, 在非Kolmogorov谱中, 部分相干月牙形光束的漂移分别随着各向异性参量的增大、湍流内尺度的增大、湍流外尺度的减小、结构常数的减小而降低; 与各向同性湍流相比, 各向异性湍流对漂移的影响较小; 月牙形光束的最大光强位置的离轴距离分别随着波长、光束阶数的增大而增大, 随着相干长度的增大而减小。月牙形光束由于最大光强位置的离轴特性, 有利于绕过障碍物传输, 所得结论对实际光通信有一定参考价值。Abstract: In order to investigate the evolution of beam wander of partially coherent crescent beams in non-Kolmogorov turbulence, the extended Huygens-Fresnel principle was used and the corresponding analytical expressions were obtained. Numerical simulation was carried out by using MATLAB. The results show that, in the non-Kolmogorov turbulence, beam wander of partially coherent crescent-like beams decreases with the increase of anisotropic parameters, the increase of turbulent inner scale, the decrease of turbulent outer scale and the decrease of structural constants respectively. Compared with isotropic turbulence, anisotropic turbulence has little effect on beam wander. Off-axis distance of the maximum intensity position of crescent-like beam increases with the increase of wavelength and beam order respectively. It decreases with the increase of coherence length. Off-axis characteristic at the position of the maximum intensity is beneficial for crescent-like beams to transmit around obstacles. The obtained conclusions have some reference value for practical optical communication.
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