Shaping self-accelerating Bessel-like optical beams along arbitrary trajectories by magnetic fluid deformable mirror

Wei Xiang; Wu Zhi-Zheng; Cao Zhan; Wang Yuan-Yuan; Dziki Mbemba

doi:10.7498/aps.68.20190063

Abstract

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1. 引　言
2. 基本原理
2.1 镜面轮廓设计
2.2 自加速类贝塞尔光束的生成理论
3. 实验验证
3.1 实验装置设置
3.2 实验结果与分析
4. 总　结

References

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Abstract

With the development of laser technology, the application scope of nondiffracting beams, such as Bessel beams, Mathieu beams, cosine beams, and parabolic beams, which remain invariant along their propagation, continues to expand. During its propagation, the main lobes of these beams tend to bend towards off-axis position, which is called self-accelerating (or self-bending) property. A Bessel-like beam with self-acceleration can realize the propagation of the main lobe along a curved trajectory while maintaining the non-diffraction. Because of the above property, Bessel-like beams have been utilized in various areas such as guiding particles along arbitrarily curved trajectories, self-accelerating beams in nonlinear medium, plasma guidance, and laser-assisted guiding of electric discharges around objects. In this paper, we propose a method of bending the trajectory of Bessel-like beams by using a magnetic fluid deformable mirror (MFDM) instead of traditional spatial light modulator (SLM) and Pancharatnam-Berry (PB) phase manipulation. The MFDM provides a method without pixelation, where all parameters can be rapidly modified for fine-tuning. Furthermore, compared with the conventional deformable mirror, the MFDM has the advantages of a continuous and smooth mirror surface, large shape deformation, low manufacture cost, easy extension, and large inter-actuator stroke. Therefore, it is easy for the MFDM to generate the ideal shape of an axicon. Firstly, according to geometric analysis, the asymmetrical mirror profile to produce a self-accelerating Bessel-like optical beam is proposed. The proposed mirror profile can be used to compensate for the difference in optical path length for each annular slice of the axicon. If a collimated Gaussian beam is incident on the mirror combining the axicon and the asymmetrical mirror profiles, which can obtain Bessel-like beams with arbitrarily curved trajectories. Secondly, the resultant of the self-accelerating Bessel-like beams along parabolic trajectories is validated by the simulation in MATLAB. Finally, a prototype of MFDM consisting of the dual-layer arrays of miniature electromagnetic coils, a Maxwell coil and the magnetic fluid filled in a circular container is fabricated for the experiment. The experimental results show that the Bessel-like beams propagate along the parabolic trajectories, with the MFDM used, and the accuracy of the curved trajectories is verified. The proposed method in this paper opens a new experimental way to the study of Bessel-like beam; the theoretical approach can also be generalized mathematically for other non-paraxial beam propagation.

Keywords:

PACS:

47.65.Cb	(Magnetic fluids and ferrofluids)
85.90.+h	(Other topics in electronic and magnetic devices and microelectronics)
41.85.Ct	(Particle beam shaping, beam splitting)

Authors and contacts

Department of Precision Mechanical Engineering, Shanghai University, Shanghai 200072, China

Corresponding author: Wu Zhi-Zheng, zhizhengwu@shu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 51675321), the Natural Science Foundation of Shanghai, China (Grant No. 15ZR1415800), and the Innovation Program of Shanghai Municipal Education Commission (Grant No. 14ZZ092).

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1. 引　言

1987年, Durnin^[1]得到关于自由空间标量波动方程的一组特解, 其横截面光场分布具有第一类零阶贝塞尔函数形式, 并首次提出“无衍射贝塞尔光束”的概念. 众所周知的无衍射光束还包括马蒂厄(Mathieu)光束^[2]、余弦光束^[3]、抛物线光束^[4]等, 都具有沿传播方向光束强度不变等特性. 近十年来, Siviloglou 和Christodoulides^[5]还提出艾里(Airy)光束, 这是一种新型的近似无衍射光束. 在艾里光束的传播过程中, 它的波包沿着离轴方向发生横向平移, 这个特性就是艾里光束的自加速特性, 也可以看成是自弯曲特性. 除了艾里光束具有自加速特性以外, 贝塞尔光束在传播过程中也可表现出自加速特性^[6], 通过相位调制, 可以在自由空间中生成任意轨迹的类贝塞尔光束^[7]; 零阶类贝塞尔螺旋光束能在局部保持贝塞尔光束的无衍射性和自修复性下实现主瓣沿着螺旋轨迹传播^[8]. 利用这些新型无衍射光束的独特性质, 可以将其应用在沿着任意轨迹操控微粒^[9], 非线性介质中的加速光束^[10], 自弯曲等离子体通道^[11], 以及物体周围的激光辅助引导^[12]等领域.

目前, 用于生成无衍射光束的方法有环缝法^[13]、计算机全息图法^[14]、球差透镜法^[15]、谐振腔法^[16]、轴棱锥法^[17]等. 大多数实验采用的是空间光调制器来弯曲贝塞尔光束的中心光瓣^[18,19], 常用的液晶空间光调制器易造成信号损失或编码错误. 此外, 液晶材料具有非理想化静态特性和动态特性, 会在信号还原过程中造成光谱扭曲. 因此, 具有像素结构的空间光调制器在获取高分辨率成像方面仍然面临着诸多挑战^[20]. 陈欢等^[21]提出了一种基于Pancharatnam-Berry (PB)相位设计制作的超表面平面轴棱锥透镜产生贝塞尔光束的方法. 而本文通过使用磁液变形镜(magnetic fluid deformable mirror, MFDM)代替传统的空间光调制器以及PB相位调控等方法. 基于磁液变形镜的波前校正器能够实时动态校正需要的波前面形, 从而可动态控制零阶类贝塞尔光束的弯曲轨迹. 磁液变形镜本身具有镜面变形幅值大、制造成本低、镜面连续光滑、驱动器易于拓展等优点, 并且克服了传统变形镜驱动器间相对行程较小的缺点, 可生成理想的锥形面^[22]. 本文通过数学几何分析来预测弯曲类贝塞尔光束的轨迹, 再利用磁液变形镜拟合相应面形. 磁液变形镜具有无像素化的特性, 并且可以快速修改参数, 对面形进行微调.

2. 基本原理

2.1 镜面轮廓设计

${{\rm{J}}_0}\left( {{k_r}r} \right)$ 表示零阶贝塞尔光束, 可以采用入射的高斯光束准直后经轴棱锥透射或其等效反射形式来产生^[17], 其中 ${k_r}$ 表示空间频率. 贝塞尔光束在空间中的传播距离取决于轴棱锥的内部锥角 ${\alpha _{ax}}$ , 折射率 ${n_{ax}}$ 和半径 ${R_{ax}}$ .

在轴棱锥上叠加一个可调的镜面轮廓, 叠加后的形状如图1所示. 当准直后的入射光束经过该新合成的面形后, 类贝塞尔光束的中心光瓣将沿着该n阶曲线的轨迹从 ${z_0}$ 弯曲到 ${z_1}$ .

$图 1 经叠加轮廓后弯曲类贝塞尔光束中心光瓣的传播轨迹(红色)\r\nFig. 1. The propagation trajectory of the center lobe of a bended Bessel-like beam through the superimposed profile (in red).$

图 1 经叠加轮廓后弯曲类贝塞尔光束中心光瓣的传播轨迹(红色)

Fig. 1. The propagation trajectory of the center lobe of a bended Bessel-like beam through the superimposed profile (in red).

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如图2(a)所示, 在轴棱锥x-y平面上取一个半径为r的薄圆环切片, 当入射光束经过该薄环后会聚到z轴上的点 ${D_{{\rm{sym}}}}$ . 对于两个任意光束P-D_sym和P′-D_sym, 它们的光程相等. 叠加该薄环上所有光束, 即可在轴上获得相长干涉, 产生零阶类贝塞尔光束的中心. 当类贝塞尔光束的中心光瓣从 ${D_{{\rm{sym}}}}$ 点移动到同一平面 ${z_D}$ 中的 ${D_{{\rm{asym}}}}$ 点时, 为了保证光程P-D_asym和P′-D_asym相等, 需要在轴棱锥上叠加一个镜面轮廓, 它能够补偿轴棱锥每个薄圆环切片上光程的不同. 由图2(b)可知, 薄环和 ${D_{{\rm{asym}}}}$ 点的最长光程为H-D_asym. 假设调整镜在H点的厚度等于0, 就可以通过将薄环上的所有光程与H-D_asym进行比较来计算调整镜面的整体轮廓 $\Delta {\rm{OL}}\left( {r, \theta } \right)$ . 由图2(b)可得 H-D_asym的光程为

$图 2 偏移点几何关系分析　(a) 薄环上任意点与偏移点的光程; (b)偏移点与薄环的最长光程\r\nFig. 2. Off-axis point geometric analysis: (a) The distance between any point on the thin ring and the off-axis point; (b) the longest distance between the off-axis point and the point on the thin ring.$

图 2 偏移点几何关系分析　(a) 薄环上任意点与偏移点的光程; (b)偏移点与薄环的最长光程

Fig. 2. Off-axis point geometric analysis: (a) The distance between any point on the thin ring and the off-axis point; (b) the longest distance between the off-axis point and the point on the thin ring.

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${L_{HD}} = \sqrt {\Delta {D^2} + {r^2}{{\csc }^2}\left( {2{\alpha _{ax}}} \right) + 2r\Delta D}\;,$

(1)

对于薄圆环上半径为r, 角度为 $\theta$ 的任意点P, 可得P-D_asym的光程为

${L_{PD}} = \sqrt {\Delta {D^2} + {r^2}{{\csc }^2}\left( {2{\alpha _{ax}}} \right) + 2r\Delta D\cos \theta }\;,$

(2)

则P-D_asym和H-D_asym之间的光程差表达式为

$\Delta {\rm{OL}}\left( {r, \theta } \right) = {L_{HD}} - {L_{PD}}\;,$

(3)

其中 $\Delta D = {D_{{\rm{sym}}}} - {D_{{\rm{asym}}}}$ .

参数 $\Delta D$ 和类贝塞尔光束的弯曲轨迹有直接的相关性. 实际上, 将 $\Delta D$ 与光束的中心光瓣所遵循的轨迹相关联, 假设沿z轴传播时增加到其n次幂, 可得

$\Delta D = \frac{{{z^n}}}{{4{a_n}}} + {y_0} = \frac{{{r^n}{{\cot }^n}\left( {2{\alpha _{ax}}} \right)}}{{4{a_n}}} + {y_0}\;,$

(4)

其中 ${y_0}$ 为 $z = 0$ 时中心点位置; ${a_n}$ 为轨迹的曲率参数.

将(4)式中的 $\Delta D$ 代入(3)式中, 可以得到如图3所示的连续轮廓. 其中锥角 ${\alpha _{ax}} = {\rm{0}}{\rm{.1}}{{\rm{2}}^ \circ }$ , 折射率 ${n_{dv}} = 1.52$ , 曲率参数 ${a_2} = {\rm{152\; m}}$ , 参数 ${y_0} = {\rm{5 \;{\text{μm}}}}$ , 轴棱锥的半径 ${R_{ax}} = {\rm{1}}{\rm{.5 \;cm}}$ . 本实验中使用的磁液变形镜采用的是该模型的等效反射形式, 此时的光程差 $L\left( {r, \theta } \right)$ 等于(3)式中的 $\Delta {\rm{OL}}\left( {r, \theta } \right)$ 除以2.

$图 3 镜面轮廓\r\nFig. 3. Mirror profile.$

图 3 镜面轮廓

Fig. 3. Mirror profile.

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类似于上述分析, 利用磁液变形镜可以产生任意的镜面轮廓, 从而通过产生期望的非对称相位调制出相应的特殊光束, 如漩涡光束^[23], 厄米-高斯光束^[24], 艾里漩涡光束^[25], 完美漩涡光束^[26]等.

2.2 自加速类贝塞尔光束的生成理论

在柱坐标系下, 高斯光束的场强满足^[27]:

$\begin{split} E(r, z) =\; & \frac{{A_0}{w_0}}{w\left( z \right)}\exp \left( - \frac{{{r^2}}}{{{w^2}\left( z \right)}} \right)\\& \!\!\!\!\! \times\exp \left\{ \!-\! {\rm{i}}\left[ k\left( \frac{r^2}{2R\left( z \right)}\!+\! z \right) \!-\! \varPsi (z) \right]\right\},\end{split}$

(5)

其中 ${A_0}$ 为高斯光束的振幅常量; ${w_0}$ 为高斯光束的束腰; 波数 $k = {2{\text{π}}}/{\lambda}$ ; 高斯光束的束宽 $w\left( z \right) = {w_0}\sqrt {1 + {{\left( {\dfrac{{\lambda z}}{{{\text{π}}w_0^2}}} \right)}^2}}$ ; 高斯光束的等相面曲率半径 $R\left( z \right) = z\bigg( {1 + {{\Big( {\dfrac{{{\text{π}}w_0^2}}{{\lambda z}}} \Big)}^2}} \bigg)$ ; 高斯光束的相位因子 $\varPsi \left( z \right) = {\tan ^{ - 1}}\left( {\dfrac{{\lambda z}}{{{\text{π}}w_0^2}}} \right)$ .

当高斯光束经过上述定义的轴棱锥和可调镜面轮廓共同叠加的镜面时, 其反射后的场强可表示为

$\begin{split} E'\left( {r,\theta ,z} \right) =\, & E (r,z)\exp \{-{\rm i}k [2L(r,\theta) \! \\ & +\! |R_{ax} \!-\! r| \tan (2\alpha _{ax})]\}, \end{split}$

(6)

根据光束传播法(beam propagation method, BPM)^[28], 由(6)式可以得到反射后每个z平面的场强为

$\begin{split} &{E_{\rm{f}}}\left( {r,\theta ,z} \right) \\ = & \exp \left( { - {\rm{i}}kz} \right){\rm{IFFT}} \big\{ \exp \left[ {{\rm i}{\text{π}}\lambda z\left( {v_x^2 + v_y^2} \right)} \right] \\ & \times{\rm{FFT}}\left( {E'\left( {r,\theta ,z} \right)} \right)\big\}\,, \end{split}$

(7)

其中 ${v_x}$ 和 ${v_y}$ 分别为沿着x轴和y轴的空间频率. FFT表示快速傅里叶变换, IFFT表示快速傅里叶逆变换.

当取曲率参数 ${a_2} = {\rm{152 \;m}}$ , 激光的波长 $\lambda = {\rm{635 \;nm}}$ 和束腰 ${w_0}={\rm{ 2 \;mm}}$ , 轴棱锥的锥角 ${\alpha _{ax}} = {0.12^ \circ }$ 和半径 ${R_{ax}} = {\rm{1}}{\rm{.5 \;cm}}$ 时, 可以得到如图4所示的理论上沿着抛物线轨迹的自加速类贝塞尔光束. 随着沿弯曲轨迹传播距离的增加, 类贝塞尔光束的光强仍能呈现多个环状分布, 中心光强为一个亮斑. 当传播距离 $z = 4{\rm{0 \;cm}}$ 时, 自加速类贝塞尔光束的光强分布如图4(b)所示.

$图 4 沿抛物线轨迹的自加速类贝塞尔光束仿真　(a) 数值模拟类贝塞尔光束传输的侧面图; (b) 弯曲轨迹类贝塞尔光束在传播距离$z = 4{\rm{0 \;cm}}$处的光强; (c)—(f) 图(a)中不同传播距离处光强的横截面分布\r\nFig. 4. The simulation of self-accelerating Bessel-like beam along a parabolic trajectory: (a) Numerically simulated side-view propagation of the generated beam; (b) intensity of a bended Bessel-like beam when propagation distance $z = 4{\rm{0 \;cm}}$; (c)−(f) cross-sectional images of Bessel-like beam along different distance.$

图 4 沿抛物线轨迹的自加速类贝塞尔光束仿真　(a) 数值模拟类贝塞尔光束传输的侧面图; (b) 弯曲轨迹类贝塞尔光束在传播距离

$z = 4{\rm{0 \;cm}}$

处的光强; (c)—(f) 图(a)中不同传播距离处光强的横截面分布

Fig. 4. The simulation of self-accelerating Bessel-like beam along a parabolic trajectory: (a) Numerically simulated side-view propagation of the generated beam; (b) intensity of a bended Bessel-like beam when propagation distance

$z = 4{\rm{0 \;cm}}$

; (c)−(f) cross-sectional images of Bessel-like beam along different distance.

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3. 实验验证

3.1 实验装置设置

本实验采用磁液变形镜来产生类贝塞尔光束^[29]. 如图5所示, 实验中使用双层驱动磁液变形镜, 微线圈的位置采用六角形排布, 双层设计不仅实现了行程的增加, 还实现了驱动器密度的增加. 其中上层包含37个微线圈, 尺寸结构较小、排布紧凑, 用于校正面形复杂而幅值较小的高阶像差; 下层包含19个微线圈, 尺寸结构较大、排布稀疏, 用于校正面形简单但幅值较大的低阶像差^[30]. 每个线圈都可以单独驱动, 将该双层线圈置于磁性液体容器下方, 再一同放置在麦克斯韦线圈中, 麦克斯韦线圈可以产生均匀强磁场, 该磁场单独作用时对镜面状态没有影响, 但是和微线圈产生的扰动磁场叠加之后可以使镜面变形量大大增加. 实验中使用的磁性液体来自FerroTec公司, 具有较高的相对磁导率 $\left( {\mu = {\rm{3}}{\rm{.89}}} \right)$ 和较低黏度( $\nu$ = 3 cP, 1 cP = 10^–3 Pa·s)能够更好实现磁液变形镜面形的控制, 同时, 为了提高磁液变形镜镜面的反射率, 在其表面覆盖一层类似液状的银纳米薄膜^[31], 所得表面的反射率大约为80%. 磁液变形镜需在隔振平台上进行操作, 并保持磁性流体的深度在 ${\rm{1\; mm}}$ 左右, 这样就可以最大限度地减少振动对液体表面的影响.

$图 5 磁液变形镜装置　(a) 实物图; (b) 结构示意图\r\nFig. 5. Assembly of the prototype MFDM: (a) Actual diagram; (b) schematic diagram.$

图 5 磁液变形镜装置　(a) 实物图; (b) 结构示意图

Fig. 5. Assembly of the prototype MFDM: (a) Actual diagram; (b) schematic diagram.

下载: 全尺寸图片幻灯片

为了进一步验证基于磁液变形镜来生成轨迹可控的自加速类贝塞尔光束的可行性, 搭建如图6所示的自适应光学实验平台, 该装置由激光光源、双层驱动磁液变形镜、夏克-哈特曼波前传感器、CCD相机、计算机控制平台和反射镜、分束镜、光阑等光学元器件组成. 实验中使用波长为 ${\rm{635 \;nm}}$ 的激光作为实验光源, 经准直后近似为平面波, 1∶15(R1, R2)扩束后经光阑限制再经双层驱动磁液变形镜反射, 然后20∶3 (R3)缩束后经分束镜分为两路光束: 一路由夏克-哈特曼波前传感器(WFS150-5C, THORLABS, USA)接收并用来检测镜面面形, 将波前的分析结果反馈给计算机控制系统, 由控制系统产生变形镜变形所需要的驱动电压, 通过Advantech PCI-1724模拟输出卡驱动磁液变形镜使其镜面变形; 另一路光束由CCD相机(DCU223C, THORLABS, USA)捕获, 获得校正后的波前图像. 本实验中, 利用影响函数对磁液变形镜系统进行解耦, 由设计的分散式PID闭环控制器, 实现对多输入多输出系统每个通道的独立控制, 消除外界干扰, 改善镜面的动态响应性能, 可以实现镜面形状的精确控制.

$图 6 基于波前传感器的磁液变形镜镜面控制实验平台 (a) 光路示意图; (b) 实物图\r\nFig. 6. Layout of the experimental system setup based on the wavefront sensor: (a) Schematic diagram of optical path; (b) actual diagram.$

图 6 基于波前传感器的磁液变形镜镜面控制实验平台 (a) 光路示意图; (b) 实物图

Fig. 6. Layout of the experimental system setup based on the wavefront sensor: (a) Schematic diagram of optical path; (b) actual diagram.

下载: 全尺寸图片幻灯片

3.2 实验结果与分析

图7(a)为夏克-哈特曼波前传感器检测到磁液变形镜初始平面, 可以看出表面存在一定的误差, 其表面变形量PV为 ${\rm{0}}{\rm{.246 \;{\text{μm}}}}$ , RMS误差为 $0.07{\rm{8\; {\text{μm}}}}$ , 对应的精度为 $\lambda {\rm{/8}}$ . 图7(b)为夏克-哈特曼波前传感器采集到的内部锥角为 ${\alpha _{ax}} = {0.12^ \circ }$ 的轴棱锥波前. 为了产生抛物线弯曲轨迹, 取 $n = 2$ 和 ${a_2} = 15{\rm{2 \;m}}$ , 此时得到的镜面轮廓如图7(c)所示. 将图7(b)和图7(c)相结合, 可以得到如图7(d)所示经过调整后的波前轮廓形状, 其表面变形幅值PV为 ${\rm{8}}{\rm{.093\;{\text{μm}}}}$ , RMS误差为 ${\rm{0}}{\rm{.134\; {\text{μm}}}}$ . 所拟合面形的质量主要和驱动器的数目有关, 通过增加驱动器数目可进一步提高面形的拟合精度.

$图 7 波前传感器检测到的波前轮廓(左图: 主视图; 右图: 三维视图)　(a) 初始面形; (b)轴棱锥轮廓; (c) 镜面轮廓; (d) 混合轴棱锥和镜面轮廓后的面形轮廓\r\nFig. 7. Wavefront detected by the wavefront sensor (left: main view; right: 3D view): (a) Initial wavefront; (b) an axicon profile; (c) mirror profile; (d) the combination of both axicon and mirror profile.$

图 7 波前传感器检测到的波前轮廓(左图: 主视图; 右图: 三维视图)　(a) 初始面形; (b)轴棱锥轮廓; (c) 镜面轮廓; (d) 混合轴棱锥和镜面轮廓后的面形轮廓

Fig. 7. Wavefront detected by the wavefront sensor (left: main view; right: 3D view): (a) Initial wavefront; (b) an axicon profile; (c) mirror profile; (d) the combination of both axicon and mirror profile.

下载: 全尺寸图片幻灯片

激光经过图7(d)调整后的磁液变形镜反射, 最终由CCD相机接收到相应的光束. 图8展示了实验上得到的抛物线轨迹的自加速类贝塞尔光束. 由于搭建的光路中各个透镜对(R1, R2, R3)之间的准直及距离存在一定的误差, 以及外部环境振动和驱动器安装误差等干扰存在, 对光束产生一定的影响, 可以看到实验结果中类贝塞尔光束周围产生一些亮条纹. 但由总体实验结果可以看出, 相较于使用空间光调制器来生成轨迹可控的自加速类贝塞尔光束^[6], 基于磁液变形镜产生的自加速类贝塞尔光束仍能够保持相对完整的光瓣, 并且中心光瓣具有较强的光强. 为了进一步验证光束与理想抛物线轨迹的拟合程度, 从 ${\rm{5 \;cm}}$ 到 ${\rm{45 \;cm}}$ , 每隔 ${\rm{5 \;cm}}$ 测量一次轨迹的变化. 由图9可以看出, 实验1中该自加速类贝塞尔光束沿z轴的弯曲轨迹遵循理想轨迹传播. 对弯曲轨迹进行调整只需要修改对应参数即可, 如实验2所示, 当设定新的弯曲轨迹( $\Delta D =$ $\left( {2.632 \times {{10}^{ - 3}}} \right){z^2} + (2.15 \times {10^{ - 4}})z$ )时, 实际测得的类贝塞尔光束的弯曲轨迹仍然沿预定轨迹传播.

$图 8 取四个不同z值的实验结果横截面图\r\nFig. 8. Experimental cross-section profiles for four different values of z.$

图 8 取四个不同z值的实验结果横截面图

Fig. 8. Experimental cross-section profiles for four different values of z.

下载: 全尺寸图片幻灯片

$图 9 理论和实际测得的光束传播轨迹相比较\r\nFig. 9. Comparison between numerical and experimental demonstrations of deflection of the central spot of a bended Bessel-like beam.$

图 9 理论和实际测得的光束传播轨迹相比较

Fig. 9. Comparison between numerical and experimental demonstrations of deflection of the central spot of a bended Bessel-like beam.

下载: 全尺寸图片幻灯片

4. 总　结

介绍了一种基于磁液变形镜来产生自加速类贝塞尔光束的新方法. 根据弯曲轨迹的不同, 通过几何分析得到对应的可调镜面轮廓, 将该轮廓与轴棱锥镜叠加后产生目标轮廓, 再利用磁液变形镜进行拟合. 磁液变形镜具有表面连续光滑、镜面变形幅值大、驱动器间行程大等优点, 提供了一种无像素化的方法产生该目标轮廓, 并且可以快速修改参数进行光束微调. 基于分析的模型, 本文设计了满足自加速类贝塞尔光束弯曲轨迹控制的变形镜结构和参数, 并进行了仿真验证. 最后搭建了基于磁液变形镜的自适应光学系统, 验证了推导模型以及类贝塞尔光束沿预定抛物线轨迹传播的准确性. 实验结果表明, 该磁液变形镜能产生精确的镜面面形并具有良好的激光束波前控制性能, 生成了具有抛物线轨迹传播的类贝塞尔光束并验证了其准确性, 并且可以快速修改参数对光束轨迹进行微调.

References

[1]	Durnin J 1987 J. Opt. Soc. 4 651 Google Scholar
[2]	Hu Y, Nie J, Sun K, Ye Q, Wang L 2017 Opt. Commun. 394 108 Google Scholar
[3]	Nadgaran H, Fallah R 2015 Opt. Commun. 341 160 Google Scholar
[4]	Dolev I, Libster A, Arie A 2012 Appl. Phys. Lett. 101 101109 Google Scholar
[5]	Siviloglou G A, Christodoulides D N 2007 Opt. Lett. 32 979 Google Scholar
[6]	Zhao J, Zhang P, Deng D, Liu J, Gao Y, Chremmos I D, Efremidis N K, Christodoulides D N, Chen Z 2013 Opt. Lett. 38 498 Google Scholar
[7]	Chremmos I D, Chen Z, Christodoulides D N, Efremidis N K 2012 Opt. Lett. 37 5003 Google Scholar
[8]	Jarutis V, Matijošius A, Di Trapani P, Piskarskas A 2009 Opt. Lett. 34 2129 Google Scholar
[9]	Zhao Z, Zang W, Tian J 2016 J. Optics-UK 18 025607 Google Scholar
[10]	Dolev I, Kaminer I, Shapira A, Segev M, Arie A 2012 Phys. Rev. Lett. 108 113903 Google Scholar
[11]	Polynkin P, Kolesik M, Moloney J V, Siviloglou G A, Christodoulides D N 2009 Science 324 229 Google Scholar
[12]	Clerici M, Hu Y, Lassonde P, Milián C, Couairon A, Christodoulides D N, Chen Z, Razzari L, Vidal F, Légaré F, Faccio D, Morandotti R 2015 Sci. Adv. 1 e1400111
[13]	Durnin J, Miceli Jr J J, Eberly J H 1987 Phys. Rev. Lett. 58 1499 Google Scholar
[14]	Vieira T A, Zamboni-Rached M, Gesualdi M R 2014 Opt. Commun. 315 374 Google Scholar
[15]	Herman R M, Wiggins T A 1991 J. Opt. Soc. Am. A 8 932 Google Scholar
[16]	Cox A J, Dibble D C 1992 J. Opt. Soc. Am. A 9 282 Google Scholar
[17]	Dudutis J, GeČys P, RaČiukaitis G 2016 Opt. Express 24 28433 Google Scholar
[18]	Wu G, Wang F, Cai Y 2014 Phys. Rev. A 89 043807 Google Scholar
[19]	赵娟莹, 邓冬梅, 张泽, 刘京郊, 姜东升 2014 物理学报 63 044204 Zhao J Y, Deng D M, Zhang Z, Liu J J, Jiang D S 2014 Acta Phys. Sin. 63 044204
[20]	Chen J S, Jia J, Chu D 2017 Chin. Opt. Lett. 15 100901 Google Scholar
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[22]	Wu Z, Iqbal A, Amara F B 2012 Modeling and Control of Magnetic Fluid Deformable Mirrors for Adaptive Optics Systems (Springer Science & Business Media)
[23]	Allen L, Beijersbergen M W, Spreeuw R J C, Woerdman J P 1992 Phys. Rev. A 45 8185 Google Scholar
[24]	He Y, Liu Z, Liu Y, Zhou J, Ke Y, Luo H, Wen S 2015 Opt. Lett. 40 5506 Google Scholar
[25]	Zhou J, Liu Y, Ke Y, Luo H, Wen S 2015 Opt. Lett. 40 3193 Google Scholar
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Cited By

期刊类型引用(1)

董天为，翁浚溥，孟海螺，吴丽艳. 激光合束技术下光学镜面激光传输损耗的特性研究. 激光杂志. 2020(09): 182-187 .

百度学术

其他类型引用(3)

图 1 经叠加轮廓后弯曲类贝塞尔光束中心光瓣的传播轨迹(红色)

Figure 1. The propagation trajectory of the center lobe of a bended Bessel-like beam through the superimposed profile (in red).

DownLoad: Full-Size Img PowerPoint

图 2 偏移点几何关系分析　(a) 薄环上任意点与偏移点的光程; (b)偏移点与薄环的最长光程

Figure 2. Off-axis point geometric analysis: (a) The distance between any point on the thin ring and the off-axis point; (b) the longest distance between the off-axis point and the point on the thin ring.

DownLoad: Full-Size Img PowerPoint

图 3 镜面轮廓

Figure 3. Mirror profile.

DownLoad: Full-Size Img PowerPoint

图 4 沿抛物线轨迹的自加速类贝塞尔光束仿真　(a) 数值模拟类贝塞尔光束传输的侧面图; (b) 弯曲轨迹类贝塞尔光束在传播距离 $z = 4{\rm{0 \;cm}}$ 处的光强; (c)—(f) 图(a)中不同传播距离处光强的横截面分布

Figure 4. The simulation of self-accelerating Bessel-like beam along a parabolic trajectory: (a) Numerically simulated side-view propagation of the generated beam; (b) intensity of a bended Bessel-like beam when propagation distance $z = 4{\rm{0 \;cm}}$ ; (c)−(f) cross-sectional images of Bessel-like beam along different distance.

DownLoad: Full-Size Img PowerPoint

图 5 磁液变形镜装置　(a) 实物图; (b) 结构示意图

Figure 5. Assembly of the prototype MFDM: (a) Actual diagram; (b) schematic diagram.

DownLoad: Full-Size Img PowerPoint

图 6 基于波前传感器的磁液变形镜镜面控制实验平台 (a) 光路示意图; (b) 实物图

Figure 6. Layout of the experimental system setup based on the wavefront sensor: (a) Schematic diagram of optical path; (b) actual diagram.

DownLoad: Full-Size Img PowerPoint

图 7 波前传感器检测到的波前轮廓(左图: 主视图; 右图: 三维视图)　(a) 初始面形; (b)轴棱锥轮廓; (c) 镜面轮廓; (d) 混合轴棱锥和镜面轮廓后的面形轮廓

Figure 7. Wavefront detected by the wavefront sensor (left: main view; right: 3D view): (a) Initial wavefront; (b) an axicon profile; (c) mirror profile; (d) the combination of both axicon and mirror profile.

DownLoad: Full-Size Img PowerPoint

图 8 取四个不同z值的实验结果横截面图

Figure 8. Experimental cross-section profiles for four different values of z.

DownLoad: Full-Size Img PowerPoint

图 9 理论和实际测得的光束传播轨迹相比较

Figure 9. Comparison between numerical and experimental demonstrations of deflection of the central spot of a bended Bessel-like beam.

DownLoad: Full-Size Img PowerPoint

[1]	Durnin J 1987 J. Opt. Soc. 4 651 Google Scholar
[2]	Hu Y, Nie J, Sun K, Ye Q, Wang L 2017 Opt. Commun. 394 108 Google Scholar
[3]	Nadgaran H, Fallah R 2015 Opt. Commun. 341 160 Google Scholar
[4]	Dolev I, Libster A, Arie A 2012 Appl. Phys. Lett. 101 101109 Google Scholar
[5]	Siviloglou G A, Christodoulides D N 2007 Opt. Lett. 32 979 Google Scholar
[6]	Zhao J, Zhang P, Deng D, Liu J, Gao Y, Chremmos I D, Efremidis N K, Christodoulides D N, Chen Z 2013 Opt. Lett. 38 498 Google Scholar
[7]	Chremmos I D, Chen Z, Christodoulides D N, Efremidis N K 2012 Opt. Lett. 37 5003 Google Scholar
[8]	Jarutis V, Matijošius A, Di Trapani P, Piskarskas A 2009 Opt. Lett. 34 2129 Google Scholar
[9]	Zhao Z, Zang W, Tian J 2016 J. Optics-UK 18 025607 Google Scholar
[10]	Dolev I, Kaminer I, Shapira A, Segev M, Arie A 2012 Phys. Rev. Lett. 108 113903 Google Scholar
[11]	Polynkin P, Kolesik M, Moloney J V, Siviloglou G A, Christodoulides D N 2009 Science 324 229 Google Scholar
[12]	Clerici M, Hu Y, Lassonde P, Milián C, Couairon A, Christodoulides D N, Chen Z, Razzari L, Vidal F, Légaré F, Faccio D, Morandotti R 2015 Sci. Adv. 1 e1400111
[13]	Durnin J, Miceli Jr J J, Eberly J H 1987 Phys. Rev. Lett. 58 1499 Google Scholar
[14]	Vieira T A, Zamboni-Rached M, Gesualdi M R 2014 Opt. Commun. 315 374 Google Scholar
[15]	Herman R M, Wiggins T A 1991 J. Opt. Soc. Am. A 8 932 Google Scholar
[16]	Cox A J, Dibble D C 1992 J. Opt. Soc. Am. A 9 282 Google Scholar
[17]	Dudutis J, GeČys P, RaČiukaitis G 2016 Opt. Express 24 28433 Google Scholar
[18]	Wu G, Wang F, Cai Y 2014 Phys. Rev. A 89 043807 Google Scholar
[19]	赵娟莹, 邓冬梅, 张泽, 刘京郊, 姜东升 2014 物理学报 63 044204 Zhao J Y, Deng D M, Zhang Z, Liu J J, Jiang D S 2014 Acta Phys. Sin. 63 044204
[20]	Chen J S, Jia J, Chu D 2017 Chin. Opt. Lett. 15 100901 Google Scholar
[21]	陈欢, 凌晓辉, 何武光, 李钱光, 易煦农 2017 物理学报 66 044203 Chen H, Ling X H, He W G, Li Q G, Yi X N 2017 Acta Phys. Sin. 66 044203
[22]	Wu Z, Iqbal A, Amara F B 2012 Modeling and Control of Magnetic Fluid Deformable Mirrors for Adaptive Optics Systems (Springer Science & Business Media)
[23]	Allen L, Beijersbergen M W, Spreeuw R J C, Woerdman J P 1992 Phys. Rev. A 45 8185 Google Scholar
[24]	He Y, Liu Z, Liu Y, Zhou J, Ke Y, Luo H, Wen S 2015 Opt. Lett. 40 5506 Google Scholar
[25]	Zhou J, Liu Y, Ke Y, Luo H, Wen S 2015 Opt. Lett. 40 3193 Google Scholar
[26]	Liu Y, Ke Y, Zhou J, Liu Y, Luo H, Wen S, Fan D 2017 Sci. Rep. 7 44096 Google Scholar
[27]	Pampaloni F, Enderlein J 2004 arXiv: physics/0410021
[28]	Kamilov U S, Papadopoulos I N, Shoreh M H, Goy A, Vonesch C, Unser M, Psaltis D 2015 Optica 2 517 Google Scholar
[29]	Brousseau D, Drapeau J, Piché M, Borra E F 2011 Appl. Opt. 50 4005 Google Scholar
[30]	Wu Z, Kong X, Zhang Z, Wu J, Wang T, Liu M 2017 Micromachines 8 72 Google Scholar
[31]	Yen Y T, Lu T Y, Lee Y C, Yu C C, Tsai Y C, Tseng Y C, Chen H L 2014 ACS Appl. Mater. Inter. 6 4292 Google Scholar

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期刊类型引用(1)
1. 董天为，翁浚溥，孟海螺，吴丽艳. 激光合束技术下光学镜面激光传输损耗的特性研究. 激光杂志. 2020(09): 182-187 . 百度学术
其他类型引用(3)

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