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矢量部分相干光在源平面上可以用2×2的交叉谱密度矩阵来表示[23]:
$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{W}}_0}\left( {{{\mathit{\boldsymbol{\rho '}}}_1},{{\rho '}_2};\omega } \right) \equiv }\\ {\left[ {\begin{array}{*{20}{l}} {{W_{xx,0}}\left( {{{\mathit{\boldsymbol{\rho '}}}_1},{{\mathit{\boldsymbol{\rho '}}}_2};\omega } \right)}&{{W_{xy,0}}\left( {{{\mathit{\boldsymbol{\rho '}}}_1},{{\mathit{\boldsymbol{\rho '}}}_2};\omega } \right)}\\ {{W_{yx,0}}\left( {{{\mathit{\boldsymbol{\rho '}}}_1},{{\mathit{\boldsymbol{\rho '}}}_2};\omega } \right)}&{{W_{yy,0}}\left( {{{\mathit{\boldsymbol{\rho '}}}_1},{{\mathit{\boldsymbol{\rho '}}}_2};\omega } \right)} \end{array}} \right]} \end{array} $
(1) 式中,ρ′1, ρ′2表示在源平面上任意两点的位置矢量,ω为角频率。
矢量椭圆多高斯-谢尔模型光束在z=0平面处的交叉谱密度矩阵可以表示为:
$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{W}}_{\alpha \beta }}\left( {{{\mathit{\boldsymbol{\rho '}}}_1},{{\mathit{\boldsymbol{\rho '}}}_2};0} \right) = {A_\alpha }{A_\beta }{B_{\alpha \beta }}\frac{1}{{{C_0}}}\sum\limits_{n = 1}^n {\left[ {\begin{array}{*{20}{c}} N\\ n \end{array}} \right]\frac{{{{\left( { - 1} \right)}^{n - 1}}}}{n}} \times }\\ {\exp \left[ { - \frac{{{{\left( {{{\mathit{\boldsymbol{\rho '}}}_1} - {{\mathit{\boldsymbol{\rho '}}}_2}} \right)}^2}}}{{2n\delta _{\alpha \beta }^2}}} \right]\exp \left( { - \frac{{\mathit{\boldsymbol{\rho '}}_1^2}}{{4\sigma _\alpha ^2}} - \frac{{\mathit{\boldsymbol{\rho '}}_2^2}}{{4\sigma _\beta ^2}}} \right),}\\ {\left( {\alpha = x,y;\beta = x,y} \right)} \end{array} $
(2) 式中, Aα和Aβ为常数,与位置无关,σα和σβ表示沿着x或y方向上的谱密度均方根宽度,δαβ表示空间相干长度,n为光束阶数(n=1, …, N),归一化因子。其中参量Bαβ满足以下条件:
$ \left\{ \begin{array}{l} {B_{\alpha \beta }} \equiv 1,\left( {\alpha = \beta } \right)\\ {B_{\alpha \beta }} \le 1,\left( {\alpha \ne \beta } \right)\\ {B_{\alpha \beta }} = {B_{\beta \alpha }}^ * \end{array} \right. $
(3) 式中,*表示复共轭。矢量椭圆多高斯-谢尔模光束在大气湍流中传输时的交叉谱密度W(ρ, ρd; z)满足拓展的惠更斯-菲涅耳原理[24]:
$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{W}}\left( {\mathit{\boldsymbol{\rho }},{\mathit{\boldsymbol{\rho }}_{\rm{d}}};z} \right) = {{\left( {\frac{k}{{2{\rm{ \mathsf{ π} }}z}}} \right)}^2}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\mathit{\boldsymbol{W}}\left( {\mathit{\boldsymbol{\rho '}},{{\mathit{\boldsymbol{\rho '}}}_{\rm{d}}};0} \right)} } } } \times }\\ {\exp \left[ {\frac{{{\rm{i}}k}}{z}\left( {\mathit{\boldsymbol{\rho }} - \mathit{\boldsymbol{\rho '}}} \right) \cdot \left( {{\mathit{\boldsymbol{\rho }}_{\rm{d}}} - {{\mathit{\boldsymbol{\rho '}}}_{\rm{d}}}} \right) - } \right.}\\ {\left. {\mathit{\boldsymbol{H}}\left( {{\mathit{\boldsymbol{\rho }}_{\rm{d}}},{{\mathit{\boldsymbol{\rho '}}}_{\rm{d}}};z} \right)} \right]{{\rm{d}}^2}\mathit{\boldsymbol{\rho '}}{{\rm{d}}^2}{{\mathit{\boldsymbol{\rho '}}}_{\rm{d}}}} \end{array} $
(4) 式中,波数k=2π/λ,λ为光束波长。上式中用到的和差矢量公式如下:
$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{\rho '}} = \frac{{{{\mathit{\boldsymbol{\rho '}}}_1} + {{\mathit{\boldsymbol{\rho '}}}_2}}}{2},{{\mathit{\boldsymbol{\rho '}}}_{\rm{d}}} = {{\mathit{\boldsymbol{\rho '}}}_1} - {{\mathit{\boldsymbol{\rho '}}}_2},}\\ {\mathit{\boldsymbol{\rho }} = \frac{{{\mathit{\boldsymbol{\rho }}_1} + {\mathit{\boldsymbol{\rho }}_2}}}{2},{\mathit{\boldsymbol{\rho }}_{\rm{d}}} = {\mathit{\boldsymbol{\rho }}_1} - {\mathit{\boldsymbol{\rho }}_2}} \end{array} $
(5) 式中,ρ1和ρ2为接收面上的任意两点位置矢量,即:
$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{W}}_{\alpha \beta }}\left( {{{\mathit{\boldsymbol{\rho '}}}_1},{{\mathit{\boldsymbol{\rho '}}}_2};0} \right) = {\mathit{\boldsymbol{W}}_{\alpha \beta }}\left( {\mathit{\boldsymbol{\rho '}},{{\mathit{\boldsymbol{\rho '}}}_{\rm{d}}};0} \right) = }\\ {{\mathit{\boldsymbol{W}}_{\alpha \beta }}\left( {\mathit{\boldsymbol{\rho '}} + \frac{{{{\mathit{\boldsymbol{\rho '}}}_{\rm{d}}}}}{2},\mathit{\boldsymbol{\rho '}} - \frac{{{{\mathit{\boldsymbol{\rho '}}}_{\rm{d}}}}}{2};0} \right)} \end{array} $
(6) $ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{H}}\left( {{\mathit{\boldsymbol{\rho }}_{\rm{d}}},{{\mathit{\boldsymbol{\rho '}}}_{\rm{d}}};z} \right) = 4{{\rm{ \mathsf{ π} }}^2}{k^2}z\int_0^1 {{\rm{d}}\xi \int_0^\infty {\left[ {1 - } \right.} } }\\ {\left. {{{\rm{J}}_0}\left( {\kappa \left| {{{\mathit{\boldsymbol{\rho '}}}_{\rm{d}}}\xi + \left( {1 - \xi } \right){\mathit{\boldsymbol{\rho }}_{\rm{d}}}} \right|} \right)} \right]{\mathit{\Phi }_n}\left( \kappa \right)\kappa {\rm{d}}\kappa } \end{array} $
(7) 式中,ξ表示一个积分项,取值范围0 < ξ < 1。另外,功率谱函数为:
$ \begin{array}{*{20}{c}} {T = \int_0^\infty {{\rm{d}}\kappa '{{\kappa '}^3}{{\mathit{\Phi '}}_n}} \left( {\kappa '} \right)/\left( {{\mu _x}{\mu _y}} \right) = }\\ {\frac{{{\mu _z}A\left( v \right)}}{{2\left( {v - 2} \right)}}\tilde C_n^2\left[ {\beta \kappa _m^{2 - v}\exp \left( {\frac{{\kappa _0^2}}{{\kappa _m^2}}} \right) \times } \right.}\\ {\left. {{{\rm{\Gamma }}_1}\left( {2 - \frac{v}{2},\frac{{\kappa _0^2}}{{\kappa _m^2}}} \right) - 2\kappa _0^{4 - v}} \right]} \end{array} $
(8) 式中,A(υ)=Γ(υ-1)cos(υπ/2)/(4π2),β=2κ02-2κm2+υκm2,υ为广义指数参量,κ0=2π/L0,κm=c(υ)/l0,L0为外尺度,l0为内尺度,$c\left( \upsilon \right) = {\left[ {\Gamma \left( {5 - \frac{\upsilon }{2}} \right)A\left( \upsilon \right)\frac{2}{3}{\rm{ \mathsf{ π} }}} \right]^{\frac{1}{{\upsilon - 5}}}}$,J0为零阶贝塞尔函数,Φn′表示折射率波动的1维功率谱,κ′为2维空间频率大小,Γ( )为伽马函数,Γ1( )是不完全伽马函数,$\tilde C_n^2$为大气湍流结构常数,μx, μy表示两个横方向上的各向异性因子,μz是传播方向的各向异性因子。
由(5)式~(7)式且$\mathit{\boldsymbol{\rho }}\mathit{'} = \mathit{\boldsymbol{\rho }}\mathit{''}, {\mathit{\boldsymbol{\rho }}_{\rm{d}}}\mathit{'} = {\mathit{\boldsymbol{\rho }}_{\rm{d}}} + \frac{z}{k}{\kappa _{\rm{d}}}$,可将(4)式化简为[24]:
$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{W}}\left( {\mathit{\boldsymbol{\rho }},{\mathit{\boldsymbol{\rho }}_{\rm{d}}};z} \right) = {{\left( {\frac{1}{{2{\rm{ \mathsf{ π} }}}}} \right)}^2}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\mathit{\boldsymbol{W}}\left( {\mathit{\boldsymbol{\rho ''}},{\mathit{\boldsymbol{\rho }}_{\rm{d}}} + \frac{z}{k}{\kappa _{\rm{d}}};0} \right)} } } } \times }\\ {\exp \left[ { - {\rm{i}}\mathit{\boldsymbol{\rho }}{\mathit{\boldsymbol{\kappa }}_{\rm{d}}} + {\rm{i}}\mathit{\boldsymbol{\rho ''}}{\mathit{\boldsymbol{\kappa }}_{\rm{d}}} - } \right.}\\ {\left. {\mathit{\boldsymbol{H}}\left( {{\mathit{\boldsymbol{\rho }}_{\rm{d}}},{\mathit{\boldsymbol{\rho }}_{\rm{d}}} + \frac{z}{k}{\mathit{\boldsymbol{\kappa }}_{\rm{d}}};z} \right)} \right]{\rm{d}}\mathit{\boldsymbol{\rho ''}}{{\rm{d}}^2}{\mathit{\boldsymbol{\kappa }}_{\rm{d}}}} \end{array} $
(9) 式中,κd≡(κd, x κd, y)为空间频率域的位置向量,其中矢量椭圆多高斯-谢尔模光束在源平面的交叉谱密度函数为:
$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{W}}\left( {\mathit{\boldsymbol{\rho ''}},{\mathit{\boldsymbol{\rho }}_{\rm{d}}} + \frac{z}{k}{\mathit{\boldsymbol{\kappa }}_{\rm{d}}};0} \right) = {A_\alpha }{A_\beta }{B_{a\beta }}\frac{1}{{{C_0}}}\sum\limits_{n = 1}^N {\left[ {\begin{array}{*{20}{c}} N\\ n \end{array}} \right]} \frac{{{{( - 1)}^{n - 1}}}}{n} \times }\\ {\exp \left\{ { - \frac{{{{\left[ {\mathit{\boldsymbol{\rho ''}} + \frac{1}{2}\left( {{\mathit{\boldsymbol{\rho }}_{\rm{d}}} + z{\mathit{\boldsymbol{\kappa }}_{\rm{d}}}/k} \right)} \right]}^2}}}{{4\sigma _\alpha ^2}} - } \right.}\\ {\left. {\frac{{{{\left[ {\mathit{\boldsymbol{\rho ''}} + \frac{1}{2}\left( {{\mathit{\boldsymbol{\rho }}_{\rm{d}}} + z{\mathit{\boldsymbol{\kappa }}_{\rm{d}}}/k} \right)} \right]}^2}}}{{4\sigma _\beta ^2}} - \frac{{{{\left( {{\mathit{\boldsymbol{\rho }}_{\rm{d}}} + z{\mathit{\boldsymbol{\kappa }}_{\rm{d}}}/k} \right)}^2}}}{{2n{\delta _{\alpha \beta }}^2}}} \right\} = }\\ {A_x^2\frac{1}{{{C_0}}}\sum\limits_{n = 1}^N {\left[ {\begin{array}{*{20}{c}} N\\ n \end{array}} \right]} \frac{{{{\left( { - 1} \right)}^{n - 1}}}}{n}\exp \left[ { - \frac{1}{{4\sigma _x^2}}{{\left( {\mathit{\boldsymbol{\rho ''}} + \frac{{{{\left( {{\mathit{\boldsymbol{\rho }}_{\rm{d}}} + z{\mathit{\boldsymbol{\kappa }}_{\rm{d}}}/k} \right)}^2}}}{2}} \right)}^2} - } \right.}\\ {\left. {\frac{1}{{4\sigma _x^2}}{{\left( {\mathit{\boldsymbol{\rho ''}} + \frac{{{\mathit{\boldsymbol{\rho }}_{\rm{d}}} + z{\mathit{\boldsymbol{\kappa }}_{\rm{d}}}/k}}{2}} \right)}^2} - \frac{{{{\left( {{\mathit{\boldsymbol{\rho }}_{\rm{d}}} + z{\mathit{\boldsymbol{\kappa }}_{\rm{d}}}/k} \right)}^2}}}{{2n{\delta _{xx}}^2}}} \right] + }\\ {A_y^2\frac{1}{{{C_0}}}\sum\limits_{n = 1}^N {\left[ {\begin{array}{*{20}{c}} N\\ n \end{array}} \right]} \frac{{{{\left( { - 1} \right)}^{n - 1}}}}{n}\exp \left[ { - \frac{1}{{4\sigma _y^2}}{{\left( {\mathit{\boldsymbol{\rho ''}} + \frac{{{{\left( {{\mathit{\boldsymbol{\rho }}_{\rm{d}}} + z{\mathit{\boldsymbol{\kappa }}_{\rm{d}}}/k} \right)}^2}}}{2}} \right)}^2} - } \right.}\\ {\left. {\frac{1}{{4\sigma _y^2}}{{\left( {\mathit{\boldsymbol{\rho ''}} + \frac{{{\mathit{\boldsymbol{\rho }}_{\rm{d}}} + z{\mathit{\boldsymbol{\kappa }}_{\rm{d}}}/k}}{2}} \right)}^2} - \frac{{{{\left( {{\mathit{\boldsymbol{\rho }}_{\rm{d}}} + z{\mathit{\boldsymbol{\kappa }}_{\rm{d}}}/k} \right)}^2}}}{{2n{\delta _{yy}}^2}}} \right]} \end{array} $
(10) 根据维格纳分布函数定义[24]:
$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{h}}\left( {\mathit{\boldsymbol{\rho }},\mathit{\boldsymbol{\theta }},z} \right) = {{\left( {\frac{k}{{2{\rm{ \mathsf{ π} }}}}} \right)}^2}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\mathit{\boldsymbol{W}}\left( {\mathit{\boldsymbol{\rho }},{\mathit{\boldsymbol{\rho }}_{\rm{d}}};z} \right)} } \times }\\ {\exp \left( { - {\rm{i}}k\mathit{\boldsymbol{\theta }} \cdot {\mathit{\boldsymbol{\rho }}_{\rm{d}}}} \right){{\rm{d}}^2}{\mathit{\boldsymbol{\rho }}_{\rm{d}}}} \end{array} $
(11) 式中,θ≡(θx, θy)表示这个矢量沿z方向的角度,kθx和kθy分别是沿x轴和y轴方向的波矢分量。那么可得:
$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{h}}\left( {\mathit{\boldsymbol{\rho }},\mathit{\boldsymbol{\theta }},z} \right) = {\mathit{\boldsymbol{h}}_{xx}}\left( {\mathit{\boldsymbol{\rho }},\mathit{\boldsymbol{\theta }},z} \right) + {\mathit{\boldsymbol{h}}_{yy}}\left( {\mathit{\boldsymbol{\rho }},\mathit{\boldsymbol{\theta }},z} \right) = }\\ {A_x^2\frac{{\sigma _x^2{k^2}}}{{8{{\rm{ \mathsf{ π} }}^3}}}\frac{1}{{{C_0}}}\sum\limits_{n = 1}^N {\left[ {\begin{array}{*{20}{c}} N\\ n \end{array}} \right]} \frac{{{{\left( { - 1} \right)}^{n - 1}}}}{n}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\exp } } } } \left[ { - {a_{xx}}\mathit{\boldsymbol{\kappa }}_{\rm{d}}^2 - } \right.}\\ {\frac{{2z}}{k}{b_{xx}}{\mathit{\boldsymbol{\rho }}_{\rm{d}}}{\mathit{\boldsymbol{\kappa }}_{\rm{d}}} - {\rm{i}}\mathit{\boldsymbol{\rho }}{\mathit{\boldsymbol{\kappa }}_{\rm{d}}} - \left( {{b_{xx}} + \frac{{{{\rm{ \mathsf{ π} }}^2}{k^2}zT}}{{2{\mu _x}}}} \right) \times }\\ {\left. {\mathit{\boldsymbol{\rho }}_{\rm{d}}^2 - {\rm{i}}k\mathit{\boldsymbol{\theta }} \cdot {\mathit{\boldsymbol{\rho }}_{\rm{d}}}} \right]{{\rm{d}}^2}{\mathit{\boldsymbol{\kappa }}_{\rm{d}}}{{\rm{d}}^2}{\mathit{\boldsymbol{\rho }}_{\rm{d}}} + }\\ {A_y^2\frac{{\sigma _y^2{k^2}}}{{8{{\rm{ \mathsf{ π} }}^3}}}\frac{1}{{{C_0}}}\sum\limits_{n = 1}^N {\left[ {\begin{array}{*{20}{c}} N\\ n \end{array}} \right]} \frac{{{{\left( { - 1} \right)}^{n - 1}}}}{n}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\exp } } } } \left[ { - {a_{yy}}\mathit{\boldsymbol{\kappa }}_{\rm{d}}^2 - } \right.}\\ {\frac{{2z}}{k}{b_{yy}}{\mathit{\boldsymbol{\rho }}_{\rm{d}}}{\mathit{\boldsymbol{\kappa }}_{\rm{d}}} - {\rm{i}}\mathit{\boldsymbol{\rho }}{\mathit{\boldsymbol{\kappa }}_{\rm{d}}} - }\\ {\left. {\left( {{b_{yy}} + \frac{{{{\rm{ \mathsf{ π} }}^2}{k^2}zT}}{{2{\mu _y}}}} \right)\mathit{\boldsymbol{\rho }}_{\rm{d}}^2 - {\rm{i}}k\mathit{\boldsymbol{\theta }} \cdot {\mathit{\boldsymbol{\rho }}_{\rm{d}}}} \right]{{\rm{d}}^2}{\mathit{\boldsymbol{\kappa }}_{\rm{d}}}{{\rm{d}}^2}{\mathit{\boldsymbol{\rho }}_{\rm{d}}}} \end{array} $
(12) 其中设置的参量有:
$ \begin{array}{*{20}{c}} {\left\{ {\begin{array}{*{20}{l}} {{a_{\alpha \alpha }} = \left( {\frac{1}{{8\sigma _\alpha ^2}} + \frac{1}{{2n\delta _{\alpha \alpha }^2}} + \frac{{{{\rm{ \mathsf{ π} }}^2}{k^2}zT}}{{3{\mu _\alpha }}}} \right)\frac{{{z^2}}}{{{k^2}}} + \frac{{\sigma _\alpha ^2}}{2}}\\ {{b_{\alpha \alpha }} = \frac{1}{{8\sigma _\alpha ^2}} + \frac{1}{{2n\delta _{\alpha \alpha }^2}} + \frac{{{{\rm{ \mathsf{ π} }}^2}{k^2}zT}}{{2{\mu _\alpha }}}} \end{array},} \right.}\\ {\left( {\alpha = x,y} \right)} \end{array} $
(13) 利用的积分公式[24]:
$ \int_{ - \infty }^\infty {\exp } \left( { - {s^2}{x^2} \pm qx} \right){\rm{d}}x = \frac{{\sqrt {\rm{ \mathsf{ π} }} }}{s}\exp \left( {\frac{{{q^2}}}{{4{s^2}}}} \right) $
(14) 通过数值计算,得出矢量椭圆多高斯-谢尔模光束的传输质量因子[24]:
$ \begin{array}{*{20}{c}} {{M^2}\left( z \right) = k{{\left( {\left\langle {{\mathit{\boldsymbol{\rho }}^2}} \right\rangle \left\langle {{\mathit{\boldsymbol{\theta }}^2}} \right\rangle - {{\left\langle {\mathit{\boldsymbol{\rho }} \cdot \mathit{\boldsymbol{\theta }}} \right\rangle }^2}} \right)}^{\frac{1}{2}}} = }\\ {k\left[ {\left( {\left\langle {{x^2}} \right\rangle + \left\langle {{y^2}} \right\rangle } \right)\left( {\left\langle {\theta _x^2} \right\rangle + \left\langle {\theta _y^2} \right\rangle } \right)} \right. - }\\ {{{\left. {{{\left( {\left\langle {x{\theta _x}} \right\rangle + \left\langle {y{\theta _y}} \right\rangle } \right)}^2}} \right]}^{\frac{1}{2}}}} \end{array} $
(15) 光束的n1+n2+m1+m2阶维格纳分布函数定义为[24]:
$ \begin{array}{*{20}{c}} {\left\langle {{x^{{n_1}}}{y^{{n_2}}}\theta _x^{{m_1}}\theta _y^{{m_2}}} \right\rangle = }\\ {\frac{1}{I}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {{x^{{n_1}}}} } {y^{{n_2}}}\theta _x^{{m_1}}\theta _y^{{m_2}}\mathit{\boldsymbol{h}}\left( {\mathit{\boldsymbol{\rho }},\mathit{\boldsymbol{\theta }},z} \right){{\rm{d}}^2}\mathit{\boldsymbol{\rho }}{{\rm{d}}^2}\mathit{\boldsymbol{\theta }}} \end{array} $
(16) 式中,光束的强度分布可以表示为:
$ I = \int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\mathit{\boldsymbol{h}}\left( {\mathit{\boldsymbol{\rho }},\mathit{\boldsymbol{\theta }},z} \right){{\rm{d}}^2}\mathit{\boldsymbol{\rho }}{{\rm{d}}^2}\mathit{\boldsymbol{\theta }}} } } } $
(17) 由(7)式~(17)式可得:
$ I = 2{\rm{ \mathsf{ π} }}\left( {A_x^2\sigma _x^2 + A_y^2\sigma _y^2} \right) $
(18) $ \begin{array}{*{20}{c}} {\left\langle {{\mathit{\boldsymbol{\rho }}^2}} \right\rangle = \frac{1}{{{C_0}}}\sum\limits_{n = 1}^N {\left[ {\begin{array}{*{20}{c}} N\\ n \end{array}} \right]\frac{{{{\left( { - 1} \right)}^{n - 1}}}}{n}\frac{{8{\rm{ \mathsf{ π} }}}}{I}} \times }\\ {\left( {A_x^2{a_{xx}}\sigma _x^2 + A_y^2{a_{yy}}\sigma _y^2} \right)} \end{array} $
(19) $ \begin{array}{*{20}{c}} {\left\langle {{\mathit{\boldsymbol{\theta }}^2}} \right\rangle = \frac{1}{{{C_0}}}\sum\limits_{n = 1}^N {\left[ {\begin{array}{*{20}{c}} N\\ n \end{array}} \right]\frac{{{{\left( { - 1} \right)}^{n - 1}}}}{n}\frac{{2{\rm{ \mathsf{ π} }}}}{I}} \left[ {\sigma _x^2A_x^2 \times } \right.}\\ {\left. {\left( {\frac{4}{{{k^2}}}{b_{xx}} + \frac{{2{{\rm{ \mathsf{ π} }}^2}zT}}{{{\mu _x}}}} \right) + \sigma _y^2A_y^2\left( {\frac{4}{{{k^2}}}{b_{yy}} + \frac{{2{{\rm{ \mathsf{ π} }}^2}zT}}{{{\mu _y}}}} \right)} \right]} \end{array} $
(20) $ \begin{array}{*{20}{c}} {\langle \mathit{\boldsymbol{\rho }} \cdot \mathit{\boldsymbol{\theta }}\rangle = \frac{1}{{{C_0}}}\sum\limits_{n = 1}^N {\left[ {\begin{array}{*{20}{c}} N\\ n \end{array}} \right]} \frac{{{{\left( { - 1} \right)}^{n - 1}}}}{n}\frac{{8{\rm{ \mathsf{ π} }}}}{I}\frac{z}{{{k^2}}} \times }\\ {\left( {{b_{xx}}\sigma _x^2A_x^2 + {b_{yy}}\sigma _y^2A_y^2} \right)} \end{array} $
(21) 那么可得:
$ \begin{array}{*{20}{c}} {{M^2}\left( z \right) = k{{\left( {\left\langle {{\mathit{\boldsymbol{\rho }}^2}} \right\rangle \left\langle {{\mathit{\boldsymbol{\theta }}^2}} \right\rangle - {{\left\langle {\mathit{\boldsymbol{\rho }} \cdot \mathit{\boldsymbol{\theta }}} \right\rangle }^2}} \right)}^{\frac{1}{2}}} = }\\ {\frac{1}{{{C_0}}}\sum\limits_{n = 1}^N {\left[ {\begin{array}{*{20}{c}} N\\ n \end{array}} \right]} \frac{{{{\left( { - 1} \right)}^{n - 1}}}}{n}k \times }\\ {\left\{ {\frac{{8{\rm{ \mathsf{ π} }}}}{I}\left( {A_x^2{a_{xx}}\sigma _x^2 + A_y^2{a_{yy}}\sigma _y^2} \right) \times } \right.}\\ {\frac{{2{\rm{ \mathsf{ π} }}}}{I}\left[ {\sigma _x^2A_x^2\left( {\frac{4}{{{k^2}}}{b_{xx}} + \frac{{2{{\rm{ \mathsf{ π} }}^2}zT}}{{{\mu _x}}}} \right) + \sigma _y^2A_y^2\left( {\frac{4}{{{k^2}}}{b_{yy}} + \frac{{2{{\rm{ \mathsf{ π} }}^2}zT}}{{{\mu _y}}}} \right)} \right] - }\\ {{{\left. {{{\left[ {\frac{{8{\rm{ \mathsf{ π} }}}}{I}\frac{z}{{{k^2}}}\left( {{b_{xx}}\sigma _x^2A_x^2 + {b_{yy}}\sigma _y^2A_y^2} \right)} \right]}^2}} \right\}}^{1/2}}} \end{array} $
(22)
EEMGSM光束在各向异性湍流中的光束质量因子研究
M2 factor of electromagnetic elliptic multi-Gaussian- Schell mode beam in anisotropic turbulence
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摘要: 为了探究矢量椭圆多高斯-谢尔模(EEMGSM)光束的传输特性, 基于广义惠更斯-菲涅耳原理和维格纳分布函数的二阶矩理论, 理论推导了EEMGSM光束在各向异性湍流中传输的质量因子解析表达式。通过数值计算和分析, 探究了初始偏振度、初始相干度、腰宽、波长和湍流结构常数等与质量因子的变化规律。结果表明, EEMGSM光束的质量因子随着阶数、波长和腰宽的增大而减小; 随着初始相干度和湍流结构常数的减小而减小; 在相同条件下, 具有大初始偏振度的EEMGSM光束的光束质量受各向异性的影响小于具有小初始偏振度的EEMGSM光束的光束质量; 且在相同条件下, EEMGSM光束的质量因子比标量椭圆高斯-谢尔模(EGSM)光束小, 即EEMGSM光束具有缓解各向异性湍流影响的优点。所得结论对于自由空间光通信的研究具有一定的理论参考价值。Abstract: In order to investigate the propagation characteristics of electromagnetic elliptical multi-Gaussian-Schell mode (EEMGSM) beams, based on the generalized Huygens-Fresnel principle and the second moment theory of Wigner distribution function, the analytical expression of M2 factor of EEMGSM beams propagating in anisotropic turbulence was derived theoretically. Through numerical calculation and analysis, the variations of initial polarization, initial coherence, waist width, wavelength and turbulence structure constant with M2 factor were investigated. The results show that, M2 factor of EEMGSM beam decreases with the increase of order, wavelength and waist width. It decreases with the decrease of initial coherence and turbulence structure constant. Under the same conditions, the beam quality of EEMGSM beams with large initial polarization is less affected by anisotropy than that of EEMGSM beams with small initial polarization. Under the same conditions, the quality factor of EEMGSM beam is smaller than that of scalar elliptical Gaussian-Schell mode (EGSM) beam. EEMGSM beam has the advantage of alleviating the influence of anisotropic turbulence. The study has certain theoretical reference value for the research of free space optical communication.
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