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Polarization changes of partially-coherent airy-gaussian beams in a slanted turbulent atmosphere

Cheng Ke Lu Gang Zhu Bo-yuan Shu Ling-yun

程科, 卢刚, 朱博源, 舒凌云. 斜程湍流大气中部分相干艾里光束的偏振特性研究[J]. 中国光学. doi: 10.37188/CO.2020-0095
引用本文: 程科, 卢刚, 朱博源, 舒凌云. 斜程湍流大气中部分相干艾里光束的偏振特性研究[J]. 中国光学. doi: 10.37188/CO.2020-0095
Cheng Ke, Lu Gang, Zhu Bo-yuan, Shu Ling-yun. Polarization changes of partially-coherent airy-gaussian beams in a slanted turbulent atmosphere[J]. Chinese Optics. doi: 10.37188/CO.2020-0095
Citation: Cheng Ke, Lu Gang, Zhu Bo-yuan, Shu Ling-yun. Polarization changes of partially-coherent airy-gaussian beams in a slanted turbulent atmosphere[J]. Chinese Optics. doi: 10.37188/CO.2020-0095

斜程湍流大气中部分相干艾里光束的偏振特性研究

doi: 10.37188/CO.2020-0095
详细信息
  • 中图分类号: TN929.1

Polarization changes of partially-coherent airy-gaussian beams in a slanted turbulent atmosphere

More Information
    Author Bio:

    Cheng Ke (1979—), M.Sc, Professor, College of Optoelectronic Engineering, Chengdu University of Information Technology. His research interests are in propagation and high-power laser control. E-mail: ck@cuit.edu.cn

    Corresponding author: ck@cuit.edu.cn
  • 摘要: 偏振是激光通信中保密编码的重要参数,研究斜程湍流大气中的偏振特性对激光通信具有重要意义。利用广义惠更斯-菲涅尔原理和偏振-相干统一理论,推导了无衍射的部分相干艾里高斯光束在斜程湍流大气传输中的偏振度解析式,详细研究了湍流参数、相干长度、天顶角、截断因子和分布因子对偏振度的影响。研究结果表明:与水平湍流相比,光束在斜程湍流下恢复到初始偏振需要更长的传输距离。天顶角、接收高度、截断因子、分布因子越大和相干长度越小时,光束偏振度峰值也越大。高相干性的高斯光束比艾里光束更易于保持偏振度不变。无衍射艾里光束中选取合适的光学参数更有利于信息传输与编码,本文结果对激光大气通信领域有着潜在的应用价值。
  • Figure  1.  The DoP of a partially coherent Airy-Gaussian beam passing horizontally through a turbulent atmosphere over propagation distance.

    Figure  2.  Change of the DoP in partially coherent Airy-Gaussian beams in a slanted turbulence atmosphere over the zenith angle with varying parameters. The fixed parameters are C0=1.7×10-14m2/3 and z=10km.

    Figure  3.  Change in DoP for partially coherent Airy-Gaussian beams over propagation distance z for different coherence lengths, where solid and dashed lines show a horizontal and slanted path, respectively. The other parameters are a=0.05, b=20,H=10000m and Cn2=10-14m2/3.

    Figure  4.  Change in DoP for partially coherent Airy-Gaussian beams over propagation distance z for different truncation or distribution factors, where the solid and dash lines show the horizontal and slanted paths, respectively. The fixed parameters are (a) b=20, H=10000m and Cn2=10-14m2/3; (b) a=0.05, H=10000m and Cn2=10-14m2/3.

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Polarization changes of partially-coherent airy-gaussian beams in a slanted turbulent atmosphere

doi: 10.37188/CO.2020-0095
    通讯作者: ck@cuit.edu.cn
  • 中图分类号: TN929.1

摘要: 偏振是激光通信中保密编码的重要参数,研究斜程湍流大气中的偏振特性对激光通信具有重要意义。利用广义惠更斯-菲涅尔原理和偏振-相干统一理论,推导了无衍射的部分相干艾里高斯光束在斜程湍流大气传输中的偏振度解析式,详细研究了湍流参数、相干长度、天顶角、截断因子和分布因子对偏振度的影响。研究结果表明:与水平湍流相比,光束在斜程湍流下恢复到初始偏振需要更长的传输距离。天顶角、接收高度、截断因子、分布因子越大和相干长度越小时,光束偏振度峰值也越大。高相干性的高斯光束比艾里光束更易于保持偏振度不变。无衍射艾里光束中选取合适的光学参数更有利于信息传输与编码,本文结果对激光大气通信领域有着潜在的应用价值。

English Abstract

程科, 卢刚, 朱博源, 舒凌云. 斜程湍流大气中部分相干艾里光束的偏振特性研究[J]. 中国光学. doi: 10.37188/CO.2020-0095
引用本文: 程科, 卢刚, 朱博源, 舒凌云. 斜程湍流大气中部分相干艾里光束的偏振特性研究[J]. 中国光学. doi: 10.37188/CO.2020-0095
Cheng Ke, Lu Gang, Zhu Bo-yuan, Shu Ling-yun. Polarization changes of partially-coherent airy-gaussian beams in a slanted turbulent atmosphere[J]. Chinese Optics. doi: 10.37188/CO.2020-0095
Citation: Cheng Ke, Lu Gang, Zhu Bo-yuan, Shu Ling-yun. Polarization changes of partially-coherent airy-gaussian beams in a slanted turbulent atmosphere[J]. Chinese Optics. doi: 10.37188/CO.2020-0095
    • The polarization properties of fully coherent and partially coherent beams in the turbulent atmosphere have been extensively studied owing to their potential applications in atmospheric communication and laser detection[1-8]. This is especially true for the polarization changes of partially coherent beams passing through different media, which can be described using the unified theory of coherence and polarization[1, 2]. The polarization characteristic is an important parameter of a laser beam, which is used to code information in signal transmissions[9]. Tremendous effort has been devoted to elucidate the effect of atmospheric turbulence, coherence length and aperture diffraction on the polarization changes or spectral properties of different laser beams[10-12], and to explore the condition of polarization maintenance in turbulent atmospheres[3]. Likewise, propagation and controlling of a diffraction-free Airy beam in many linear media have been extensively investigated, owing to its self-healing and self-bending properties[13-17]. Furthermore, Airy beams modulated by a Gaussian factor, Hermite function, Bessel wave or vortex core have also been proposed from theoretical and experimental perspectives[18-25]. For example, the Airy-Gaussian beam can be regarded as an Airy beam limited by a Gaussian factor. Bandres et al.[18] described the propagation of a generalized Airy-Gaussian beam in ABCD optical systems. The propagation properties or momentum evolution of an Airy-Gaussian beam in quadratic-index medium, Kerr medium, uniaxial crystals and left-handed materials were further studied by Deng et al.[19, 21-23]. The effect of coherence length, the truncation parameter and propagation distance on polarization maintenance in an Airy beam in a horizontally turbulent atmosphere was numerically analyzed by Yang et al.[26]. However, in a slanted turbulent atmosphere the competitive relationship between a Gaussian factor and an Airy function, and the influence of coherence and zenith angle on the DoP are worthy of further investigation.

      The main purpose of this paper is to explore the polarization changes of partially a coherent Airy-Gaussian beam passing through a slanted turbulent atmosphere that results from the propagation of the actual light wave in optical communication being a slanted rather than horizontal. The dependence of polarization maintenance on the turbulent parameter, coherence length, zenith angle, truncation factor and distribution factor is further analyzed, and the comparison Is made for the DoP between Gaussian and non-diffraction Airy beams. The findings in this paper provide the possibility for laser detection, laser imaging radar and atmospheric communication using a partially coherent Airy-Gaussian beam.

    • Assume that the electric field of an Airy-Gaussian beam at z=0 in the is expressed as a Cartesian coordinate as[21]

      $$E(x,y,0) = \prod\limits_{\chi = x,y} {Ai\left(\dfrac{\chi }{{{w_0}}}\right)\exp \left(\dfrac{{a\chi }}{{{w_0}}}\right)} \exp \left( { - \dfrac{{{\chi ^2}}}{{b{w_0}^2}}} \right),$$ (1)

      where Ai is the Airy function, w0 represents the transverse scale of the Airy beam and a denotes a positive exponential truncation factor. The b parameter represents the distribution factor, which can cause the Airy-Gaussian beam to evolve into a pure Airy beam when it is larger, or a Gaussian beam when it is smaller.

      According to the expression for the electric field in Eq. (1), the cross-spectral density function of partially coherent Airy-Gaussian beams in the plane of z=0 can be described by[27]

      $$\begin{split}&W({x_1},{x_2},{y_1},{y_2},0) \\ & \;\;\;\; = E\left( {{x_1},{y_1},0} \right){E^ * }\left( {{x_2},{y_2},0} \right)\mu \left( {{x_1} - {x_2},{y_1} - {y_2}} \right),\end{split}$$ (2)

      where μ(●) is spatial coherence and can be expressed by the Schell-model correlator

      $$\mu \left( {{x_1} - {x_2},{y_1} - {y_2}} \right) = \exp \left[ { - \dfrac{{{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2}}}{{2{\sigma ^2}}}} \right]$$ (3)

      With σ being the coherence length. Based on the extended Huygens-Fresnel principle[27], the cross-spectral density function in the receiving plane of partially coherent Airy-Gaussian beams propagating in a turbulent atmosphere can be expressed as

      $$ {\begin{array}{l} W({x'_1},{x'_2},{y'_1},{y'_2},z)\\ \quad= {\left( {\dfrac{k}{{2\pi z}}} \right)^2}\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {W({x_1},{x_2},{y_1},{y_2},0)d{x_1}d{x_2}d{y_1}d{y_2}} } } } \\ \quad \times \exp \left\{ {\left. {\dfrac{{ik}}{{2z}}\left[ {{{\left( {{{x'}_1} - {x_1}} \right)}^2} - {{\left( {{{x'}_2} - {x_2}} \right)}^2}} \right]} \right\}} \right.{\left\langle {\exp \left[ {\psi \left( {{x_1},{{x'}_1}} \right) + {\psi ^ * }\left( {{x_2},{{x'}_2}} \right)} \right]} \right\rangle _m}\\ \quad \times \exp \left\{ {\left. {\dfrac{{ik}}{{2z}}\left[ {{{\left( {{{y'}_1} - {y_1}} \right)}^2} - {{\left( {{{y'}_2} - {y_2}} \right)}^2}} \right]} \right\}} \right.{\left\langle {\exp \left[ {\psi \left( {{y_1},{{y'}_1}} \right) + {\psi ^ * }\left( {{y_2},{{y'}_2}} \right)} \right]} \right\rangle _m}, \end{array}}$$ (4)

      where k is the wavenumber related to the wavelength λ and k=2π/λ, ψ(●) is the phase function depending on the properties of the medium, <>m specifies the ensemble average of the turbulent medium, and is given by[28]

      $$ {\begin{array}{l} {\left\langle {\exp \left[ {\psi \left( {{x_1},{{x'}_1}} \right) + {\psi ^ * }\left( {{x_2},{{x'}_2}} \right)} \right]} \right\rangle _m} \approx \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\exp \left[ { - \dfrac{{{{\left( {{{x'}_1} - {{x'}_2}} \right)}^2} + \left( {{{x'}_1} - {{x'}_2}} \right)\left( {{x_1} - {x_2}} \right) + {{\left( {{x_1} - {x_2}} \right)}^2}}}{{\rho _0^2}}} \right], \end{array}}$$ (5)
      $$ {\begin{array}{l} {\left\langle {\exp \left[ {\psi \left( {{y_1},{{y'}_1}} \right) + {\psi ^ * }\left( {{y_2},{{y'}_2}} \right)} \right]} \right\rangle _m} \approx\\ \;\;\;\;\;\;\;\;\;\;\;\;\; \exp \left[ { - \dfrac{{{{\left( {{{y'}_1} - {{y'}_2}} \right)}^2} + \left( {{{y'}_1} - {{y'}_2}} \right)\left( {{y_1} - {y_2}} \right) + {{\left( {{y_1} - {y_2}} \right)}^2}}}{{\rho _0^2}}} \right] \end{array}}$$ (6)

      with

      $${\rho _0} = {[1.46{k^2}\sec \left( \theta \right){\int\limits_0^H {C_n^2\left( h \right)\left( {1 - \frac{h}{H}} \right)} ^{5/3}}dh]^{ - 3/5}}.$$ (7)

      Here, Cn2(h) is the altitude-dependent structure constant and h shows the altitude from the ground. Assume that H is the vertical height between the source plane and the receiving plane, θ is the zenith angle, and L and z can be written as L=Hsec(θ) and z=hsec(θ).

      For the case of a slanted turbulent atmosphere, the structure constant of Cn2(h) is related to the altitude h. Here, we utilize the ITU-R model[29] to describe the altitude-dependent structure constant. For example,

      $$ \begin{array}{l} C_n^2\left( h \right) = 8.148 \times {10^{ - 56}}{V^2}{h^{10}}\exp \left( { - h/1000} \right)\\ \;\;\;\;\;\;\;\;+ 2.7 \times {10^{ - 16}}\exp \left( { - h/1500} \right) + {C_0}\exp \left( { - h/100} \right) \end{array}$$ (8)

      with the wind speed along the vertical path of V=(vg2+30.69vg+348.91)1/2, where vg is the ground wind speed and C0 is the nominal value at ground level (a typical value is 1.7×10-14m2/3).

      Based on the method of variable separation, the cross-spectral density function $W({x'_1},{x'_2},{y'_1},{y'_2},z)$ in Eq. (4) can be decomposed into two independent terms

      $$W({x'_1},{x'_2},{y'_1},{y'_2},z) = {W_x}\left( {{{x'}_1},{{x'}_2},z} \right){W_y}\left( {{{y'}_1},{{y'}_2},z} \right).$$ (9)

      From Eq. (9) one can see that the law of propagation of ${W_x}\left( {{{x'}_1},{{x'}_2},z} \right)$ is the same as that of ${W_y}\left( {{{y'}_1},{{y'}_2},z} \right)$, where the cross-spectral density function ${W_x}\left( {{{x'}_1},{{x'}_2},z} \right)$ is written as

      $$ {\begin{array}{l} {W_x}\left( {{{x'}_1},{{x'}_2},z} \right) \\ \quad= \dfrac{k}{{2\pi z}}\iint\limits_\infty {d{x_1}d{x_2}\prod\limits_{\chi = {x_1},{x_{_2}}} {Ai\left(\dfrac{\chi }{{{w_0}}}\right)\exp \left(\dfrac{{a\chi }}{{{w_0}}}\right)} \exp \left( { - \dfrac{{{\chi ^2}}}{{b{w_0}^2}}} \right)\exp \left[ { - \dfrac{{{{({x_1} - {x_2})}^2}}}{{2{\sigma ^2}}}} \right]}\\ \quad\times \exp \left\{ {\left. {\dfrac{{ik}}{{2z}}\left[ {{{\left( {{{x'}_1} - {x_1}} \right)}^2} - {{\left( {{{x'}_2} - {x_2}} \right)}^2}} \right]} \right\}} \right.\\ \quad \times \exp \left[ { - \dfrac{{{{\left( {{{x'}_1} - {{x'}_2}} \right)}^2} + \left( {{{x'}_1} - {{x'}_2}} \right)\left( {{x_1} - {x_2}} \right) + {{\left( {{x_1} - {x_2}} \right)}^2}}}{{\rho _0^2}}} \right], \end{array}} $$ (10)

      The averaged intensity in the x-dimension is obtained by using ${x'_1} = {x'_2} = x'$ and the following formulas[30]:

      $$Ai\left( x \right) = \dfrac{1}{{2\pi }}\int\limits_{ - \infty }^\infty {\exp \left[ {i\left( {\dfrac{{{s^3}}}{3} + xs} \right)} \right]} ds,$$ (11)
      $$\begin{split}&Ai\left( {{x_1}} \right)Ai\left( {{x_2}} \right) \\ &\quad= \dfrac{1}{{{2^{1/3}}\pi }}\int\limits_{ - \infty }^\infty {Ai\left[ {{2^{2/3}}\left( {{t^2} + \dfrac{{{x_1} + {x_2}}}{{2w}}} \right)} \right]} \exp \left[ {i\dfrac{{\left( {{x_1} - {x_2}} \right)t}}{w}} \right]dt,\end{split}$$ (12)
      $$\int\limits_{ - \infty }^\infty {\exp \left( { - {p^2}{x^2} \pm qx} \right)dx = \exp \left( {\dfrac{{{q^2}}}{{4{p^2}}}} \right)} \dfrac{{\sqrt \pi }}{p},$$ (13)
      $$\begin{split}& \int\limits_{ - \infty }^\infty {\exp \left[ {i\left( {\dfrac{{{t^3}}}{3} + a{t^2} + bt} \right)} \right]dt }\\ &\quad= 2\pi \exp \left[ {i\left( {\dfrac{2}{3}{a^3} - ab} \right)} \right]Ai\left( {b - {a^2}} \right),\end{split}$$ (14)

      After tedious integral calculations, the averaged intensity in the x-dimension is obtained as

      $$ \begin{split} {W_x}\left( {x',x',z} \right) =& {2^{4/3}}{\pi ^2}A\exp \left[ {i\left( {\dfrac{2}{3}E_1^3 - {E_1}{E_3}} \right)} \right]Ai\left( {{E_3} - E_1^2} \right)\\ &\times \exp \left[ {i\left( {\dfrac{2}{3}E_2^3 - {E_2}{E_4}} \right)} \right]Ai\left( {{E_4} - E_2^2} \right), \end{split} $$ (15)

      where

      $$\begin{split}&A = \dfrac{k}{{{2^{7/3}}{\pi ^3}z}}\sqrt {\dfrac{{\pi b{w_0}^2z}}{{2B}}} \exp \left( {\dfrac{{{a^2}b}}{2} + \dfrac{{c_1^2}}{{4B}}} \right),\\ &B = \dfrac{1}{{2b{w_0}^2}} + \dfrac{1}{{2{\sigma ^2}}} + \dfrac{1}{{\rho _0^2}} + \dfrac{{b{w_0}^2{k^2}}}{{8{z^2}}},\end{split}$$ (16)
      $$\begin{split} &{E_{1,2}} = \dfrac{i}{{16B{w_0}^2}} + \dfrac{{bi}}{8} - \dfrac{{i{b^2}{k^2}{w_0}^2}}{{64B{z^2}}} \pm \dfrac{{bk}}{{16Bz}},\\ &{E_{3,4}} = \dfrac{{ab}}{2} \mp \dfrac{{{c_1}}}{{4B{w_0}}} + \frac{{ib{c_1}k{w_0}}}{{8Bz}}, \end{split}$$ (17)
      $${c_1} = \dfrac{{abik{w_0} - 2ikx'}}{{2z}}.$$ (18)

      E1 and E4 of the “±” in the Eq. (17) are positive, and all other cases are negative. Similarly, the average intensity in y-dimension can be also derived by letting ${y'_1} = {y'_2} = y'$ in ${W_y}\left( {{{y'}_1},{{y'}_2},z} \right)$. Therefore the average intensity of partially coherent Airy-Gaussian beams is obtained as

      $$I\left( {x',y',z} \right) = {W_x}\left( {x',x',z} \right){W_y}\left( {y',y',z} \right).$$ (19)

      We focus our attention on the degree of polarization P(xʹ,yʹ,z) of partially coherent Airy-Gaussian beams[1]

      $$P\left( {x',y',z} \right) = \sqrt {1 - \frac{{4Det\left[ {\hat W\left( {x',y',z} \right)} \right]}}{{{{\left\{ {Tr\left[ {\hat W\left( {x',y',z} \right)} \right]} \right\}}^2}}}} ,$$ (20)

      where Det and Tr are the determinant and trace of the matrix $\hat W\left( {x',y',z} \right)$. The cross-spectral density matrix for a partially coherent Airy-Gaussian beam is expressed by

      $$\hat W\left( {x',y',z} \right) = \left[ {\begin{array}{*{20}{c}} {{W_{xx}}\left( {x',y',z} \right)}&{{W_{xy}}\left( {x',y',z} \right)} \\ {{W_{yx}}\left( {x',y',z} \right)}&{{W_{yy}}\left( {x',y',z} \right)} \end{array}} \right],$$ (21)

      where the non-diagonal elements are arranged as zero, i.e., Wxy(xʹ,yʹ,z)=Wyx(xʹ,yʹ,z)=0. Therefore, the expression of the degree of polarization can be simplified as

      $$P\left( {x',y',z} \right) = \frac{{\left| {{W_{xx}}\left( {x',y',z} \right) - {W_{yy}}\left( {x',y',z} \right)} \right|}}{{{W_{xx}}\left( {x',y',z} \right) + {W_{yy}}\left( {x',y',z} \right)}}.$$ (22)

      The derivation of cross-spectral density Wij(xʹ,yʹ,z) is similar to that of Eqs. (10)-(18):

      $$ {\begin{array}{l} {W_{ij}}\left( {x',y',z} \right) = {2^{8/3}}{\pi ^4}{A_{11ij}}{A_{12ij}}\exp \left[ {i\left( {\frac{2}{3}E_{11ij}^3 - {E_{11ij}}{E_{31ij}}} \right)} \right]Ai\left( {{E_{31ij}} - E_{11ij}^2} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times \exp \left[ {i\left( {\frac{2}{3}E_{21ij}^3 - {E_{21ij}}{E_{41ij}}} \right)} \right]Ai\left( {{E_{41ij}} - E_{21ij}^2} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times \exp \left[ {i\left( {\frac{2}{3}E_{12ij}^3 - {E_{12ij}}{E_{32ij}}} \right)} \right]Ai\left( {{E_{32ij}} - E_{12ij}^2} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times \exp \left[ {i\left( {\frac{2}{3}E_{22ij}^3 - {E_{22ij}}{E_{42ij}}} \right)} \right]Ai\left( {{E_{42ij}} - E_{22ij}^2} \right). \end{array}} $$ (23)

      where

      $$ {A_{11ij}} = A\exp \left( {\dfrac{{{a^2}b}}{2} + \dfrac{{c_1^2}}{{4{B_{ij}}}}} \right),\;{A_{12ij}} = A\exp \left( {\dfrac{{{a^2}b}}{2} + \dfrac{{c_2^2}}{{4{B_{ij}}}}} \right), $$ (24)
      $$\begin{split} &{E_{11ij}} = \dfrac{i}{{16{B_{ij}}{w_0}^2}} + \dfrac{{bi}}{8} + \dfrac{{i{b^2}{k^2}{w_0}^2}}{{64{B_{ij}}{z^2}}} + \dfrac{{bk}}{{16{B_{ij}}z}}, \\ &{E_{31ij}} = \dfrac{{ab}}{2} - \dfrac{{{c_1}}}{{4{B_{ij}}{w_0}}} + \dfrac{{ib{c_1}k{w_0}}}{{8{B_{ij}}z}}, \end{split}$$ (25)
      $$\begin{split} &{E_{12ij}} = \frac{i}{{16{B_{ij}}{w_0}^2}} + \frac{{bi}}{8} - \frac{{i{b^2}{k^2}{w_0}^2}}{{64{B_{ij}}{z^2}}} + \frac{{bk}}{{16{B_{ij}}z}}, \\ &{E_{32ij}} = \frac{{ab}}{2} - \frac{{{c_2}}}{{4{B_{ij}}{w_0}}} + \frac{{ib{c_2}k{w_0}}}{{8{B_{ij}}z}}, \end{split}$$ (26)
      $$\begin{split} &{E_{21ij}} = \frac{i}{{16{B_{ij}}{w_0}^2}} + \frac{{bi}}{8} + \frac{{i{b^2}{k^2}{w_0}^2}}{{64{B_{ij}}{z^2}}} - \frac{{bk}}{{16{B_{ij}}z}},\\ &{E_{42ij}} = \frac{{ab}}{2} + \frac{{{c_2}}}{{4{B_{ij}}{w_0}}} + \frac{{ib{c_2}k{w_0}}}{{8{B_{ij}}z}}, \end{split}$$ (27)
      $$\begin{split} &{E_{22ij}} = \frac{i}{{16{B_{ij}}{w_0}^2}} + \frac{{bi}}{8} - \frac{{i{b^2}{k^2}{w_0}^2}}{{64{B_{ij}}{z^2}}} - \frac{{bk}}{{16{B_{ij}}z}},\\ &{E_{41ij}} = \frac{{ab}}{2} + \frac{{{c_1}}}{{4{B_{ij}}{w_0}}} + \frac{{ib{c_1}k{w_0}}}{{8{B_{ij}}z}}, \end{split}$$ (28)
      $$ {c_2} = \frac{{abik{w_0} - 2iky'}}{{2z}}, \;{B_{ij}} = \frac{1}{{2b{w_0}^2}} + \frac{1}{{2\sigma _{ij}^2}} + \frac{1}{{\rho _0^2}} + \frac{{b{w_0}^2{k^2}}}{{8{z^2}}}. $$ (29)

      By substituting Eqs. (24)-(29) into Eq. (23), one can find that the DoP for partially coherent Airy-Gaussian beams depends on the truncation factor, distribution factor, zenith angle and coherence length. Details will be shown through numerical examples in the following sections.

    • Based on the analytical expression of the DoP in section 2, numerical calculations are performed to illustrate the polarization changes in a turbulent atmosphere. The calculation parameters a=0.1, b=10, λ=633 nm, σx=1 cm, σy=2 cm, θ=60˚ and H=500 m remain fixed unless otherwise stated.

      Fig. 1 describes the DoP of partially coherent Airy-Gaussian beams passing horizontally through a turbulent atmosphere for different refractive index structure parameters. One can see that the DoP firstly increases and then decreases during propagation in a turbulent atmosphere. Most notably, if the partially coherent Airy-Gaussian beam travels a sufficiently long distance in a turbulent atmosphere, it becomes completely unpolarized, yet in free space, the polarization maintains a value of P=0.85 after a long propagation distance. In comparison, a weaker turbulent parameter over a smaller distance can acquire a larger peak of polarization and has greater difficulty restoring the initial polarization at the source plane. Polarization changes can be considered sensitive to different turbulence values and coherence lengths, meaning that polarization changes with spatial coherence properties when a laser propagates in a turbulent atmosphere. Maintaining consistent polarization in stronger turbulence is important for laser communications.

      Figure 1.  The DoP of a partially coherent Airy-Gaussian beam passing horizontally through a turbulent atmosphere over propagation distance.

      The effect of the zenith angle on the DoP of partially coherent Airy-Gaussian beams with varying parameters is further shown in Fig. 2. It is clearly seen that an increase of the DoP is accompanied by an increase in zenith angle. A higher receiving height, a larger truncation factor or a lower coherence length can lead to an increase in the DoP. There exist critical values θc for different distribution factors. For example, the critical value of θc=53˚ in Fig.2(c), the larger distribution factors acquire the smaller polarization where θ<θc, while the opposite is true when θ>θc. As shown in b=1 in Fig. 2(c) and σx=4 cm and σy=8 cm in Fig. 2(d), these changes in the DoP are both small from θ=0 to 80˚, which indicates that a smaller distribution factor or a higher coherence length is beneficial for reducing the effect of zenith angle on polarization.

      Figure 2.  Change of the DoP in partially coherent Airy-Gaussian beams in a slanted turbulence atmosphere over the zenith angle with varying parameters. The fixed parameters are C0=1.7×10-14m2/3 and z=10km.

      Fig. 3 describes the dependence of the DoP for partially coherent Airy-Gaussian beams on coherence length for slanted and horizontal atmospheric turbulence. For the slanted path, the DoP also increases and then decreases to a stable value, where this stable value can be maintained until it finally decreases to zero for sufficiently long distances. For example, the polarization in the slanted path increases to a peak value of P=0.49 at z=1.9 km and maintains the value of P=0.42 with an increase in propagation distance, as shown in σx=1 cm and σy=2 cm in Fig. 3. Likewise, the horizontal path has its peak polarization at 1.26 km. The slanted path can have stronger polarization in comparison to the horizontal path. The change in the DoP for a larger coherence is less than that of a smaller coherence length, and compared with a slanted path, the change in the DoP in the horizontal path is also small, which indicates that the beam with a higher coherence or horizontal path is more applicable for maintaining its initial polarization in optical signal transmission.

      Figure 3.  Change in DoP for partially coherent Airy-Gaussian beams over propagation distance z for different coherence lengths, where solid and dashed lines show a horizontal and slanted path, respectively. The other parameters are a=0.05, b=20,H=10000m and Cn2=10-14m2/3.

      The dependence of the DoP for partially coherent Airy-Gaussian beams on truncation factor and the distribution factor for slanted and horizontal atmospheric turbulence is also plotted in Figs. 4 (a) and (b), respectively. It can be seen that a larger truncation factor or distribution factor can increase the maximal polarization of the Airy-Gaussian beams, which also leads to a larger difference between turbulence polarization and its initial polarization. Compared with a horizontal path, a slanted path at its maximal value needs a longer distance. For example, the propagation distances are z=1.25 km in a=0.01 and z=2.35 km in a=0.2 of Fig. 4(a), respectively. From Fig. 4(b) one can see that the polarization in slanted turbulence is sensitive to the distribution factor. A larger distribution factor can greatly increase the peak of polarization, and also results in the destabilization of polarization during propagation. A lower distribution factor causes an Airy-Gaussian beam to evolve into a Gaussian beam, which indicates that the polarization maintenance of a Gaussian beam in a horizontally turbulent atmosphere has an advantage over that of a pure Airy beam. The physical reason may be associated with the optical properties of Airy beams, such as their main-lobe and side-lobe structures, their self-healing abilities, the fact that they are diffraction-free, and their relatively complex spatial structures and propagation properties when compared to those of a Gaussian beam.

      Figure 4.  Change in DoP for partially coherent Airy-Gaussian beams over propagation distance z for different truncation or distribution factors, where the solid and dash lines show the horizontal and slanted paths, respectively. The fixed parameters are (a) b=20, H=10000m and Cn2=10-14m2/3; (b) a=0.05, H=10000m and Cn2=10-14m2/3.

    • Based on the extended Huygens-Fresnel principle, the analytical expressions of cross-spectral density for partially coherent Airy-Gaussian beams propagating in a turbulent atmosphere is derived and used to investigate the effects of the turbulent parameter, zenith angle, truncation factor, distribution factor and propagation path on the degree of polarization in the unified theory of coherence and polarization. It has been shown that a weaker turbulent parameter keeps polarization invariance less effectively in comparison to a stronger turbulent parameter. An increase in zenith angle, receiving height, truncation factor and a decreases in coherence length can increase the degree of polarization, and a smaller distribution factor or a higher coherence length is beneficial to reducing the effect of zenith angle on the polarization. The analysis shows that polarization maintenance in a Gaussian beam with higher coherence in horizontally turbulent atmospheres has greater advantages to that of a pure Airy beam when analyzed for polarization invariance. The results of this paper may be useful for studies in atmospheric communication and environment detection, and show that optical information transmission and encoding can be achieved by selecting appropriate parameters.

    • This work was supported by the Sichuan Science and Technology Program (No.2020YJ0431)

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