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要想达成以上目标,首先需要从一种可自由曲线塑性的光束出发[16],这种光束的表达式为:
$$E\left( {\xi ,\eta } \right) = \int {\int_{ - \infty }^\infty {H\left( {x,y} \right)\exp \left[ { - 2{{j}}\pi \left( {\xi x + \eta y} \right)} \right]} } {\rm{d}}x{\rm{d}}y$$ (1) 式中:H(x, y)是计算全息图的透过率函数。可以表示为:
$$H(x,y) = \frac{1}{{\displaystyle\int_0^T {\left| {{{c'}_2}\left( t \right)} \right|} {\rm{d}}t}}\int_0^T {{\varphi _i}\left( {x,y,t} \right)\left| {{{c'}_2}\left( t \right)} \right|} {\rm{d}}t$$ (2) 式中:|c'2(t)| = [x'0(t)2 + y'0(t)2]1/2,t∈[0, T = 2π],x0(t)和y0(t)是曲线的参数方程,决定着曲线的形状;φi(x, y, t)为相位项,具体可以表示为:
$$\begin{split} \varphi ( {x,y,t} ) =& \exp \Bigg( \dfrac{i}{{\omega _0^2}}[ {y{x_0}( t ) - x{y_0}(t)}] + \dfrac{{{{i}}\sigma }}{{\omega _0^2}}\int_0^t [ {{x_0}( t ){y'}_0}( t ) - \\ & {y_0}\left( t \right){{x'}_0}\left( t \right) ] {\rm{d}}t \Bigg)\\[-10pt] \end{split}$$ (3) 式中:σ是一个用来控制沿着曲线相位梯度的自由参数;ω0为光束的束腰宽度。当x0(t) = Rcost、 y0(t) = Rsint时该曲线为圆形,且此时计算全息图的透过率函数可以近似认为是一个贝塞尔函数[16]。接下来需要构造摆线,以传统摆线为例,其参数方程可以表示为:
$$ \left\{ \begin{gathered} {x_0}(t) = R × \left[ {m × \cos t + \cos \left( {mt} \right)} \right] \\ {y_0}(t) = R×\left[ {m × \sin t - \sin \left( {mt} \right)} \right] \end{gathered} \right. $$ (4) 其中,R决定了光束的半径,且曲线的尖峰数为m+1。但传统摆线无法对其摆动的幅度进行调整,此时由此生成出的摆线从参数方程的性质出发,传统摆线可以看作是一个半径更小的圆在其内部滚动时其圆周上一定点所画出的轨迹。因此想要改变其摆动幅度,只需要对内部那一个滚动的圆进行调整即可。据此,为传统摆线附加上曲率调控参数n之后可以将公式(4)改为:
$$ \left\{ \begin{gathered} {x_0}(t) = R × \left[ {m × \cos t + n\cos \left( {mt} \right)} \right] \\ {y_0}(t) = R × \left[ {m × \sin t - n\sin \left( {mt} \right)} \right] \end{gathered} \right. $$ (5) 此时n可以控制尖瓣曲线的凹陷程度,m可以调控其尖峰数目。根据公式(5)在R = 0.7 mm模拟生成了不同m,n的曲线及其相位图,如图1所示。图中所有光束的拓扑荷值全部为16。从图中可以观察到,随着n的增大,以顶点为界,顶点之间的曲线凹陷程度越来越大。
图 1 尖峰调控参数m及曲率调控参数n不同时,归一化后的光强及相位。(a1)~(a3)分别为m=2,n=0.25~0.75时的光强分布图;(b1)~(b3)为相应的相位分布图;(c1)~(c3)为m=3,n=0-0.75时的光强分布图;(d1)~(d3)为相应的相位分布图;(e1)~(e3)为m=4-6,n=0.5时的光强分布图;(f1)~(f3)为相应的相位分布图
Figure 1. Normalized light intensity and phase that the peak control parameter m and the curvature control parameter n are different. (a1)-(a3) are the light intensity distribution diagram when m=2 and n=0.25-0.75 respectively; (b1)-(b3) is the corresponding phase distribution diagram; (c1)-(c3) is the light intensity distribution when m=3 and n=0-0.75; (d1)-(d3) is the corresponding phase distribution diagram; (e1)-(e3) is the light intensity distribution when m=4-6 and n=0.5; (f1)-(f3) is the corresponding phase distribution diagram
至此已经生成了一种可以调控其摆动幅度的新型摆线。为了能够更加清晰地看到其在光镊上的应用与通常环形涡旋光束的不同,需要对其梯度力和OAM进行分析。由于任意曲线塑形光束光环宽度很小,这样在计算梯度力和OAM时不利于分析,所以在计算之前需要生成不同环半径,相同拓扑荷值的光束进行叠加,使光束宽度加厚[20]。加厚之后的光强分布如图2(a1)~(a4)所示。在此根据参考文献[21]中的方法对其进行计算,其梯度力计算公式可以表示为:
图 2 尖峰调控参数m=2及曲率调控参数n=0-0.75时,归一化后的光强、梯度力和OAM。(a1)~(a4)分别为 m=2,n=0-0.75时的光强分布图;(b1)~(b4)为(a1)~(a4)中白色虚线框区域相应的梯度力分布图的放大图;(c1)~(c4)为白色虚线框区域相应的OAM密度图的放大图
Figure 2. Normalized light intensity, gradient force and OAM when the peak control parameter m=2 and the curvature control parameter n=0-0.75. (a1)-(a4) are the light intensity distribution diagram when m=2 and n=0-0.75 respectively; (b1)-(b4) is an enlarged version of the gradient force distribution map corresponding to the white dashed frame region in (a1)-(a4); (c1)-(c4) enlarged view of the corresponding OAM density map in the area with the white dotted line
$${F_g}\left( {\xi ,\eta } \right) = \alpha \nabla {\left| {E\left( {\xi ,\eta } \right)} \right|^2}$$ (6) 式中:α是一个与粒子相关的参数;
$\nabla$ 为梯度算子。由于尖峰处最能反映摆线和通常的圆形光束在操纵粒子方面的区别,所以在计算了梯度力之后,只展示了图2(a1)~(a4)中白色虚线框内的放大图,如图2(b1)~(b4)所示,所有放大图的放大倍数均为20倍。可以看到,随着曲率调控参数n的增大,梯度力会逐渐指向尖峰处。OAM的计算公式可以表示为[22]:$$ \boldsymbol{J}(\eta, \xi)=\frac{\varepsilon_{0}}{2 \omega} \boldsymbol{r} \times {\rm{Im}}\left\{\boldsymbol{E}^{*}(\eta, \xi) \cdot[\nabla \boldsymbol{E}(\eta, \xi)]\right\} \\ =\eta \cdot \boldsymbol{P}_{\xi}+\xi \cdot \boldsymbol{P}_{\eta} $$ (7) 式中:ε0为真空介电常量;Im {}为取虚部函数;ω = kc为光的角频率,c为真空中的光速,k为波数;P为坡印廷矢量。其计算公式为[21]:
$$\boldsymbol{P}(\eta, \xi)=\frac{\rm{\varepsilon}_{0}}{4 \omega} {\rm{Im}}\left\{\boldsymbol{E}^{*}(\eta, \xi) \cdot[\nabla \boldsymbol{E}(\eta, \xi)]\right\}$$ (8) 最终白色虚线框内的OAM计算结果放大图如图2(c1)~(c4)所示。与梯度力不同,因为从始至终拓扑荷值并没有发生变化,所以OAM密度在曲线上的分布依旧是均匀的,不存在尖端处OAM增大的现象。而且在几何模型中,OAM所产生的横向力相比于梯度力是微不足道的[21]。所以结合梯度力和OAM进行总体分析,理论上可以判断出随着n的增大粒子在尖峰处运动会变得越来越困难直至无法运动,从而将粒子捕获在尖峰处,之后为光束附加上初始相位差φ0使其旋转即可完成等间距旋转。
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下面将通过上述光镊实验装置对摆线光束操纵微粒特性进行研究。为了突出在生物学中的应用,文中采用了酵母菌细胞进行操纵。首先选用半径R=0.7 mm、尖峰调控参数m=2,曲率调控参数n=0.01时的摆线光束使酵母菌细胞进行旋转,之后观察到细胞每旋转一周之后将曲率调控参数调大,每次增长0.01,直至粒子无法旋转。随后将曲率调控参数n重新调回0.01,粒子重新开始旋转。图4截取了视频2其中一圈的图像作为展示,可以清楚地看到粒子再次沿着光束进行逆时针旋转。该次实验捕获的酵母菌细胞直径约为6.3 μm。实验结果充分证明了摆线光束可以完成粒子运动时的启停工作。
图 4 尖峰调控参数m=2、曲率调控参数n=0.02时的摆线光束操纵粒子进行旋转
Figure 4. When the peak control parameter m=2 and the curvature control parameter n=0.02, the cycloid beam manipulates the particle to rotate
接下来,选用半径R = 0.7 mm,尖峰调控参数m = 2,曲率调控参数n = 0.75时的摆线光束,其拓扑荷值为40。分别捕获了直径为5.25 μm、5.98 μm和5.36 μm的三个酵母菌细胞进行旋转运动,并且通过调整初始相位差φ0的变化速度可以控制旋转的速度逐渐加快。图5展示了从视频2中截取的部分图像,图中三个粒子恰好旋转一周。为了证明摆线光束可以完成等间距旋转粒子,分别标记了粒子1、2、3,且设粒子1和粒子2之间间距为S1,粒子2和粒子3之间的间距为S2,粒子1和粒子3之间的间距为S3。在其运动过程中每隔0.5 s进行一次记录,然后测量S1、S2、S3的大小,在实验中相机所拍摄的图片由于被物镜放大100倍,其像素尺寸为45 nm×45 nm。最终得到它们的平均值和标准误差分别为:
$$ \left\{ \begin{gathered} \overline {{S_1}} = 10.157 \;\text{μ}{\rm{m}} \\ \overline {{S_2}} = 9.877\;\text{μ}{\rm{m}} \\ \overline {{S_3}} = 10.397\;\text{μ}{\rm{m}} \end{gathered} \right. $$ (9) $$ \left\{ \begin{gathered} E{r_1} = 96.971\;{\rm{nm}} \\ E{r_2} = 113.990\;{\rm{nm}} \\ E{r_3} = 72.847\;{\rm{nm}} \end{gathered} \right. $$ (10) 平均值误差是由于生物细胞往往不是一个标准的球体,所以被捕获的位置会有细微的差别。同时由于粒子在液体中会受到布朗运动的影响,从而对结果造成一定的误差,在实验中对布朗运动进行了测量,发现单粒子被光点捕获后的1 min内平均位移为69.77 nm。但即使如此,该实验标准误差依旧维持在纳米级。因此,可以确定摆线光束的等间距操纵微粒是成立的,并且该实验仅以三个尖峰的光束作为例证,通过调整不同的尖峰调控参数m = 2和不同的半径R即可得到不同数量、不同间距的摆线光束,以完成不同领域的应用需求。
Equal spacing control of particle via cycloidal beam (Invited)
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摘要: 轨道角动量(OAM)的发现为光镊的研究开辟了新的道路。但具有OAM的光束在操纵微粒时,由于生物细胞不可能大小形状完全相同,所以当其进行旋转等操作时,粒子运动速度的不均匀会导致粒子之间的间距不可控。针对该问题,首先通过任意曲线塑形技术,并为传统摆线公式附加曲率调控参数提出了一种调控模式丰富的摆线光束,理论分析了该光束的OAM和梯度力,证明了解决上述问题的可能性。最后在实验中实现了粒子在运动过程中的启停,且成功操纵三个粒子进行等间距旋转,实验测得三个微粒在整个旋转过程中间距变化的误差可以维持在纳米级。这项工作为未来光捕获和操纵多种微粒在其他领域的应用铺平了道路,特别是在生物科学领域。Abstract: The discovery of orbital angular momentum (OAM) opens up a new way for the study of optical tweezers. However, the size and shape of biological cells cannot be exactly the same, when the beam with OAM manipulates the particles. So, the uneven velocity of the particles will lead to uncontrollable spacing between the particles when it carries out operations such as rotation. To solve the problem, a cycloid beam with rich regulation modes was proposed by using an arbitrary curve shaping technique and adding curvature control parameters to the traditional cycloid formula. The OAM and gradient force of the cycloid beam was theoretically analyzed, and the possibility of solving the problem was theoretically analyzed. Finally, the start and stop of particles in the process of motion were realized in the experiment, and the three particles were successfully manipulated to rotate at the same distance. The experimental results show that the error of the distance variation of the three particles during the whole rotation process can be maintained at the nanometer level. The work paves the way for future applications of light to capture and manipulate a variety of particles in other fields, particularly in the biological sciences.
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图 1 尖峰调控参数m及曲率调控参数n不同时,归一化后的光强及相位。(a1)~(a3)分别为m=2,n=0.25~0.75时的光强分布图;(b1)~(b3)为相应的相位分布图;(c1)~(c3)为m=3,n=0-0.75时的光强分布图;(d1)~(d3)为相应的相位分布图;(e1)~(e3)为m=4-6,n=0.5时的光强分布图;(f1)~(f3)为相应的相位分布图
Figure 1. Normalized light intensity and phase that the peak control parameter m and the curvature control parameter n are different. (a1)-(a3) are the light intensity distribution diagram when m=2 and n=0.25-0.75 respectively; (b1)-(b3) is the corresponding phase distribution diagram; (c1)-(c3) is the light intensity distribution when m=3 and n=0-0.75; (d1)-(d3) is the corresponding phase distribution diagram; (e1)-(e3) is the light intensity distribution when m=4-6 and n=0.5; (f1)-(f3) is the corresponding phase distribution diagram
图 2 尖峰调控参数m=2及曲率调控参数n=0-0.75时,归一化后的光强、梯度力和OAM。(a1)~(a4)分别为 m=2,n=0-0.75时的光强分布图;(b1)~(b4)为(a1)~(a4)中白色虚线框区域相应的梯度力分布图的放大图;(c1)~(c4)为白色虚线框区域相应的OAM密度图的放大图
Figure 2. Normalized light intensity, gradient force and OAM when the peak control parameter m=2 and the curvature control parameter n=0-0.75. (a1)-(a4) are the light intensity distribution diagram when m=2 and n=0-0.75 respectively; (b1)-(b4) is an enlarged version of the gradient force distribution map corresponding to the white dashed frame region in (a1)-(a4); (c1)-(c4) enlarged view of the corresponding OAM density map in the area with the white dotted line
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