﻿ 完美涡旋光的自由空间传播特性
 光学仪器  2022, Vol. 44 Issue (5): 69-76 PDF

1. 上海理工大学 上海市现代光学系统重点实验室，上海 200093;
2. 上海理工大学 光电信息与计算机工程学院，上海 200093

Free space propagation characteristics of perfect vortex beam
DAI Mengting1,2, GENG Tao1,2
1. Shanghai Key Laboratory of Modern Optical System, University of Shanghai for Science and Technology, Shanghai 200093, China;
2. School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: The ideal perfect vortex beam is a special beam whose intensity distribution does not change with the change of the topological charge. Compared with ordinary vortex beam, it can greatly improve the application efficiency in particle manipulation and optical fiber transmission. In order to explore the free space propagation characteristics of the perfect vortex beam, this article uses Hank transformation to calculate and analyze the effects of the topological charge, the initial surface ring radius and the ring width on its diffraction characteristics in detail, it is found that the perfect vortex beam does not have the characteristic of non-diffraction, and the halo will broaden with the increase of the diffraction distance and gradually transform to the Bessel function. When the radius of the initial surface increases or the width of the ring decreases, the diffraction effect increases, and the effect of the ring width is greater than the ring radius. Compared with the previous two cases, the topological charge has less influence on the diffraction effect. The research in this paper is expected to provide a useful theoretical reference for the further application of the perfect vortex beam.
Key words: perfect vortex beam    topological charge    Fourier transform    diffraction

1 理　论

Ostrovsky等[17]提出了一种光强分布不依赖于拓扑荷数的理想PVB，其可以表示为

 $PV\left( {r,\theta}\right) \propto {\text{δ}}\left( {r - R} \right)\exp \left( {{\rm{i}}l\theta } \right)$ (1)

 $F\left( {\rho ,\phi } \right) \propto {J_l}\left( {{k_r}\rho } \right)\exp \left( {{\rm{i}}l\phi } \right)$ (2)

 ${F_G}\left( {\rho ,\phi } \right) \propto {J_l}\left( {{k_r}\rho } \right)\exp \left( { - \frac{{{\rho ^2}}}{{w_g^2}}} \right)\exp \left( {{\rm{i}}l\phi } \right)$ (3)

 图 1 产生PVB的实验光路图 Figure 1 Experimental light path for generating PVB

 $u\left( {r,\theta ,z = 0} \right) \propto \exp \left( { - \frac{{{r^2} + {R^2}}}{{{T^2}}}} \right){I_l}\left( {\frac{{2Rr}}{{{T^2}}}} \right)\exp \left( {{\rm{i}}l\theta } \right)$ (4)

 图 2 参数 $R = 3\;{\rm mm}$ ， $T = 0.04\;{\rm{mm}}$ 保持不变，拓扑荷数l不同的PVB的光强和相位分布 Figure 2 The normalized light intensity and phase distribution of PVBs with different topological chargesl with the parameter $R = 3\;{\rm mm}$ and $T = 0.04\;{\rm{mm}}$ unchanged

 $\begin{split} & u\left( {r,\theta ,z} \right) = 2 {\text{π}} \exp \left( {{\rm{i}}l\theta } \right)\int {g\left( \rho \right){J_l}\left( {2 {\text{π}}\rho r} \right)}\cdot \\ & \exp \left( {2{\rm{i}} {\text{π}} z\sqrt {{\lambda ^{ - 2}} - {\rho ^2}} } \right)\rho {\rm{d}}\rho \end{split}$ (5)

2 模拟分析

PVB的复振幅分布可由式（4）表示，但其中含有第一类l阶修正贝塞尔函数 ${I_l}\left( \cdot \right)$ ，因此不易于计算和理论分析。文献[18]通过理论分析指出，当式（4）中 $R/T \gg 1$ 时，式（4）可以近似简化为

 $u\left( {r,\theta ,z = 0} \right) \sim \exp \left( { - \frac{{{{\left( {r - R} \right)}^2}}}{{{T^2}}}} \right)\exp \left( {{\rm{i}}l\theta } \right)$ (6)

 图 3 $R{\text{ = }}3\;{\rm mm}$ ， $l = {\text{31}}$ 时，式（4）和式（6）计算的归一化径向光强分布 Figure 3 The normalized radial intensity distribution calculated by equation（4）and equation（6） with $R{\text{ = }}3\;{\rm mm}$ and $l = {\text{31}}$

 图 4 $R{\text{ = }}3\;{\rm mm}$ 时，不同拓扑荷数PVB的 ${W_0}/T$ 与 $R/T$ 的变化关系 Figure 4 The relationship between ${W_0}/T$ and $R/T$ of PVB with different topological loads when $R{\text{ = }}3\;{\rm mm}$

 图 5 参数 $R{\text{ = 2}}\;{\rm{mm}}$ ， $T = 0.056\;{\rm{mm}}$ ， $l = 15$ 的PVB在不同截面上的径向光强分布 Figure 5 Radial light intensity distribution of PVB with parameters $R{\text{ = 2}}\;{\rm mm}$ , $T = 0.056\;{\rm mm}$ and $l = 15$ on different sections

 图 6 不同参数的PVB在 $z = 200\;{\rm mm}$ 处的环后环宽度 ${W_z}$ 与初始面环宽度 ${W_0}$ 的比值 ${W_{z}}/{W_0}$ Figure 6 The ratio of the back ring width Wz to the initial face ring width W0 at $z = 200\;{\rm mm}$ for PVB with different parameters ${W_z}/{W_0}$

 图 7 $R = 2\;{\rm mm}$ ， $T = 0.056\;{\rm mm}$ 时，不同拓扑荷数l的PVB的传播距离z与 ${W_z}/{W_0}$ 的关系 Figure 7 The relationship between the propagation distance z and ${W_z}/{W_0}$ of PVB with different topological chargesl when $R = 2\;{\rm mm}$ and $T = 0.056\;{\rm mm}$
3 结　论

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