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2维有限能量Airy光束加载涡旋后,在初始平面(z=0),场分布为[10, 24-25]:
$ \begin{aligned} E(x, y, 0) &=f_{\mathrm{A}}\left(\frac{x}{w_x}\right) \exp \left(\frac{a x}{w_x}\right) f_{\mathrm{A}}\left(\frac{y}{w_y}\right) \exp \left(\frac{a y}{w_y}\right) \times \\ & {\left[\left(x-x_{\mathrm{d}}\right)+\mathrm{i}\left(y-y_{\mathrm{d}}\right)\right]^l } \end{aligned} $
(1) 式中,wx和wy为x和y方向横向尺度比例参数;a为指数截断因子,其大小决定了光束衰减快慢;xd和yd为原始位置,l为拓扑荷数;fA(·)即为Airy函数:
$ f_{\mathrm{A}}(x)=\frac{1}{2 {\rm{ \mathsf{ π} }}} \int_{-\infty}^{\infty} \exp \left(-\frac{\mathrm{i} u^3}{3}-\mathrm{i} x u\right) \mathrm{d} u $
(2) 当Airy涡旋光束通过NIM时,根据Collins公式得到Airy涡旋光束通过NIM的传输动力学方程:
$ \begin{gathered} E(x, y, z)=\frac{i k_0}{2 {\rm{ \mathsf{ π} }} B} \iint E\left(x_0, y_0, 0\right) \times \\ \exp \left\{-\frac{i k_0}{2 B}\left[A\left(x_0^2+y_0^2\right)-2\left(x x_0+y y_0\right)+\right.\right. \\ \left.\left.D\left(x^2+y^2\right)\right]\right\} \mathrm{d} x_0 \mathrm{~d} y_0 \end{gathered} $
(3) 式中,x0和y0为输入平面的横坐标和纵坐标,$k_0=\frac{2 {\rm{ \mathsf{ π} }}}{\lambda} $是真空中光的波数;A, B, D分别为ABCD矩阵光学的传输矩阵元。
由于拓扑荷数对光束的传输特性没有明显影响[26],为便于研究,选取单位拓扑荷数。把(1)式代入(3)式,选取涡旋的拓扑荷数l=1,经积分整理可得:
$ E(x, y, z)=\frac{B}{k_0 A^2} \exp [Q(x, y, z)]\left(P_1+P_2+P_3\right) $
(4) 式中,
$ \begin{gathered} Q(x, y, z)= \\ \frac{a}{A}\left(\frac{x-2 x_c}{w_x}+\frac{y-2 y_c}{w_y}\right)+\left(\frac{-\mathrm{i} k_0 D}{2 B}+\frac{\mathrm{i} k_0}{2 A B}\right) \times \\ \left(x^2+y^2\right)+\mathrm{i}\left[\frac{B^3}{12 A^3 k_0^3}\left(\frac{1}{w_x^6}+\frac{1}{w_y^6}\right)-\right. \\ \frac{a^2 B}{2 A k_0}\left(\frac{1}{w_x^2}+\frac{1}{w_y^2}\right)- \\ \left.\frac{B}{2 A^2 k_0}\left(\frac{x}{w_x^3}+\frac{y}{w_y^3}\right)\right] \end{gathered} $
(5) $ \begin{gathered} P_1=\frac{k_0}{B} f_{\mathrm{A}}\left(\frac{x-x_{\mathrm{c}}}{A w_x}-\frac{\mathrm{i} a B}{A k_0 w_x^2}\right) f_{\mathrm{A}}\left(\frac{y-y_{\mathrm{c}}}{A w_y}-\frac{\mathrm{i} a B}{A k_0 w_y^2}\right) \times \\ \quad\left[x-A x_{\mathrm{d}}-2 x_{\mathrm{c}}+\mathrm{i}\left(y-A y_{\mathrm{d}}-2 y_{\mathrm{c}}\right)\right] \end{gathered} $
(6) $ \begin{aligned} P_2=& \frac{-\mathrm{i}}{w_x}\left[f_{\mathrm{A}}^{\prime}\left(\frac{x-x_{\mathrm{c}}}{A w_x}-\frac{\mathrm{i} a B}{A k_0 w_x^2}\right)+\right.\\ &\left.a f_{\mathrm{A}}\left(\frac{x-x_{\mathrm{c}}}{A w_x}-\frac{\mathrm{i} a B}{A k_0 w_x^2}\right)\right] \times \\ & f_{\mathrm{A}}\left(\frac{y-y_{\mathrm{c}}}{A w_y}-\frac{\mathrm{i} a B}{A k_0 w_y^2}\right) \end{aligned} $
(7) $ \begin{gathered} P_3=\frac{1}{w_y}\left[f_{\mathrm{A}}{ }^{\prime}\left(\frac{y-y_{\mathrm{c}}}{A w_y}-\frac{\mathrm{i} a B}{A k_0 w_y^2}\right)+\right. \\ \left.a f_{\mathrm{A}}\left(\frac{y-y_{\mathrm{c}}}{A w_y}-\frac{\mathrm{i} a B}{A k_0 w_y^2}\right)\right] f_{\mathrm{A}}\left(\frac{x-x_{\mathrm{c}}}{A w_x}-\frac{\mathrm{i} a B}{A k_0 w_x^2}\right) \end{gathered} $
(8) 式中, P2和P3为Airy光束的复振幅,P1为Airy光束加载涡旋后的结果,涡旋的中心可从(6)式得到:(Axd+2 xc,Ayd+ 2yc); fA′(·)为Airy函数的导数;xc= $\frac{B^2}{4 A k_0{ }^2 w_x{ }^3} $, yc = $ \frac{B^2}{4 A k_0{ }^2 w_y{ }^3}$, 为Airy光束主峰在x和y方向中心位置。
Airy涡旋光束通过NIM介质的几何结构示意图如图 1所示。
根据矩阵光学理论可知,当Airy涡旋光束通过NIM时,光学传输系统的ABCD矩阵为:
$ \left(\begin{array}{ll} A & B \\ C & D \end{array}\right)=\left(\begin{array}{ll} 1 & \frac{z}{n} \\ 0 & 1 \end{array}\right) $
(9) 式中,n为NIM的折射率。由于Airy涡旋光束的主峰是自加速偏转的,偏转加速度与wx, wy和λ有关。此外,涡旋位置也随着传输距离增加而偏转。在某一特定传输距离处,Airy涡旋光束的涡旋位置和Airy光束主峰位置重叠,此时:
$ \left\{\begin{array}{l} x_{\mathrm{c}}=A x_{\mathrm{d}}+2 x_{\mathrm{c}} \\ y_{\mathrm{c}}=A x_{\mathrm{d}}+2 y_{\mathrm{c}} \end{array}\right. $
(10) 根据(9)式和(10)式,涡旋与主峰重叠时的z为:
$ z_0=2|n| k_0 w_x \sqrt{-w_x x_{\mathrm{d}}} $
(11)
艾里涡旋光束通过负折射率介质的传输特性
Study on propagation properties of vortex Airy beams through negative index medium
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摘要: 为了研究Airy涡旋光束通过负折射率介质(NIM)的传输动力学特性, 利用Collins公式推导出了Airy涡旋光束通过NIM的传输动力学方程, 并用该方程研究了Airy涡旋光束在NIM中的光强、涡旋、相位等传输特性。结果表明, 通过调节NIM的参数可实现对Airy涡旋光束主峰位置、涡旋位置、主峰与涡旋的重叠位置和光强的控制。光束在NIM中的特性研究在光学显微操控和光学分选等领域具有潜在价值。Abstract: In order to investigate the propagation dynamics of vortex Airy beams passing through the negative index medium(NIM), the propagation dynamics equation was obtained based on the Collins formula. The intensity, vortex, and phase were studied by using the equation. The results show that it is possible to controlling the center lobe, superimposition position, and the intensity by adjusting the parameters of the negative index medium. All these properties of the propagation of the beam in NIM may have applications in areas such as optical micromanipulation and optical sorting.
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Key words:
- physical optics /
- vortex Airy beams /
- negative index medium /
- Collins formula /
- intensity /
- propagation properties
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