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2-D Airy光束阵列如图 1所示。由4个单独的2-D Airy光束组成,其中间隔距离2c[21]表示2-D Airy光束阵列的光束间距。
以点(c, c)为中心的2-D Airy光束的有限能量函数方程为[1]:
$ \begin{gathered} \phi\left(s_x, s_y, \xi\right)=\phi\left(s_x, \xi\right) \times \phi\left(s_y, \xi\right)= \\ A\left[s_x-\left(\frac{\xi}{2}\right)^2+\mathrm{i} \alpha \xi\right] \times A\left[s_y-\left(\frac{\xi}{2}\right)^2+\mathrm{i} \alpha \xi\right] \times \\ \exp \left[\alpha s_x-\frac{\alpha \xi^2}{2}-\mathrm{i}\left(\frac{\xi^3}{12}\right)+\mathrm{i}\left(\frac{\alpha^2 \xi}{2}\right)+\mathrm{i}\left(\frac{s_x \xi}{2}\right)+\right. \\ \left.\alpha s_y-\frac{\alpha \xi^2}{2}-\mathrm{i}\left(\frac{\xi^3}{12}\right)+\mathrm{i}\left(\frac{\alpha^2 \xi}{2}\right)+\mathrm{i}\left(\frac{s_y \xi}{2}\right)\right] \end{gathered} $
(1) 式中: ϕ()为电场包络;A[·]为Airy函数;sx=(x+c)/x0和sy=(y+c)/y0为无量纲的横向坐标;x0和y0为横向尺度因子;ξ=z/(kx02)为归一化的传输距离,k=2π/λ为波束,λ为真空中波长;α为衰减因子,且0 < α ≪ 1。
因此,根据式(1),基于旋转矩阵理论[22]生成2-D Airy光束阵列,其函数方程为:
$ \begin{gathered} \mathit{\Phi }\left(s_x, s_y, \xi\right)=\phi\left(s_x, s_y, \xi\right)+\phi\left(s_x, -s_y, \xi\right)+ \\ \phi\left(-s_x, s_y, \xi\right)+\phi\left(-s_x, -s_y, \xi\right) \end{gathered} $
(2) 通过高斯光束在频域的位移原理,实现2-D Airy光束阵列焦点处强度的调控。为了便于理解高斯光束位移的理论,首先考虑1-D Airy光束生成高斯光束的位移,根据式(1)对于1-D有限能量Airy光束在频域的傅里叶谱[1]可以表示为: exp(-αω2)exp[i(ω3-3α2ω-iα3)/3],其中ω是频域归一化的波数。在频域上移动高斯光束,其傅里叶谱变为exp[-α(ω-ωG)2]×exp[i(ω3-3α2ω-iα3)/3],而ωG表示高斯光束在频域的归一化位移。因此产生的1-D有限能量Airy光束新的电场包络ϕ1(sx, z)[15]为:
$ \begin{gathered} \phi_1\left(s_x, \xi\right)=f\left(s_x, \xi\right) A\left[s_x-\left(\frac{\xi}{2}\right)^2+\right. \\ \left.\mathrm{i} \alpha\left(\xi-2 \omega_{\mathrm{G}}\right)\right] \exp \left(-\alpha \omega_{\mathrm{G}}{ }^2-\mathrm{i} 2 \alpha^2 \omega_{\mathrm{G}}\right) \end{gathered} $
(3) $ \begin{gathered} f\left(s_x, \xi\right)=\exp \left[\alpha s_x+\mathrm{i} s_x \frac{\xi}{2}+\right. \\ \left.\left(\mathrm{i} \frac{\alpha^2}{2}+\alpha \omega_{\mathrm{G}}\right) \xi+\left(\frac{-\alpha}{2}\right) \xi^2-\mathrm{i} \frac{\xi^3}{12}\right] \end{gathered} $
(4) 从式(3)可以看出, 1-D有限能量Airy光束新的峰值强度位置为ξ-2ωG=0,当ωG=0时,在z=0处光强最大;当高斯光束的位移ωG≠0时,对峰值强度的位置产生变化,所以在频域移动高斯光束对1-D Airy光束的强度分布会产生影响。
类似的,2-D情况下,高斯光束位移后的2-D有限能量Airy光束新的电场包络为ϕ1(sx, z)×ϕ1(sy, z),高斯光束在(x, y)方向的位移表示为(DG, x, DG, y),其中DG, x和DG, y分别为高斯光束在x和y方向的归一化位移。2-D Airy光束峰值光束强度位置为$\xi-\sqrt{2} D_{\mathrm{G}}=0$,同样的高斯光束的位移也会影响2-D Airy光束强度分布。根据式(2)以及式(3),高斯光束位移后的新2-D Airy光束阵列函数为:
$ \begin{gathered} \mathit{\Phi }_1\left(s_x, s_y, \xi\right)=\phi_1\left(s_x, s_y, \xi\right)+\phi_1\left(s_x, -s_y, \xi\right)+ \\ \phi_1\left(-s_x, s_y, \xi\right)+\phi_1\left(-s_x, -s_y, \xi\right) \end{gathered} $
(5)
2维Airy光束阵列强度的调控技术研究
Research on the technique of regulating the intensity of 2-D Airy beam array
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摘要: 为了控制2-D Airy光束阵列在焦点的强度, 利用高斯光束在频域移动可以进行调控的原理, 对其在大气湍流中的效果进行了仿真。结果表明, 通过高斯光束在频域的位移, 实现了2-D Airy光束阵列焦点处强度从0.85增强到1.1, 且操作灵活方便, 不需要通过重复编码相位图增加光束数目就可增强光束在焦点的强度; 在中等湍流强度下, 光束强度可从0.85增强到1.03。该研究对激光在大气中抵抗大气湍流、提高激光通信的质量, 具有一定的参考意义。Abstract: In order to control the intensity of the 2-D Airy beam array at the focal point, a simulation study was conducted using the principle of modulation of the Gaussian beam moving in the frequency domain. The results show that the intensity enhancement from 0.85 to 1.1 at the focal point of the 2-D Airy beam array is achieved by shifting the Gaussian beam in the frequency domain, and the operation is flexible and convenient without increasing the number of beams by repeatedly encoding the phase diaphragm to enhance the intensity of the beam at the focal point, and its effect in atmospheric turbulence is simulated, beam intensity enhancement from 0.85 to 1.03 at moderate turbulence intensity. The modulation is of research significance for the laser to resist atmospheric turbulence in the atmosphere and improve the quality of laser communication.
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Key words:
- physical optics /
- Airy beam /
- intensity modulation /
- beam array
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