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OPTICS  光学

Optical pulling at macroscopic distances
宏观距离的光学拉力

Abstract  抽象

Xiao Li 1 ∗ 1 ∗ ^(1**){ }^{\mathbf{1 *}}, Jun Chen 2 ∗ 2 ∗ ^(2**){ }^{2 *}, Zhifang Lin 3 , 4 , J a c k N g 1 , 5 † 3 , 4 , J a c k N g 1 , 5 † ^(3,4,Jack)Ng^(1,5†){ }^{\mathbf{3 , 4}, ~ J a c k ~} \mathbf{N g}^{1,5 \dagger} Optical tractor beams, proposed in 2011 and experimentally demonstrated soon after, offer the ability to pull particles against light propagation. It has attracted much research and public interest. Yet, its limited microscopicscale range severely restricts its applicability. The dilemma is that a long-range Bessel beam, the most accessible beam for optical traction, has a small half-cone angle, θ 0 θ 0 theta_(0)\boldsymbol{\theta}_{0}, making pulling difficult. Here, by simultaneously using several novel and compatible mechanisms, including transverse isotropy, Snell’s law, antireflection coatings (or impedance-matched metamaterials), and light interference, we overcome this dilemma and achieve long-range optical pulling at θ 0 ≈ 1 ∘ θ 0 ≈ 1 ∘ theta_(0)~~1^(@)\theta_{0} \approx 1^{\circ}. The range is estimated to be 14 cm when using ∼ 1 W ∼ 1 W ∼1W\sim 1 \mathrm{~W} of laser power. Thus, macroscopic optical pulling can be realized in a medium or in a vacuum, with good tolerance of the half-cone angle and the frequency of the light.
Xiao Li 1 ∗ 1 ∗ ^(1**){ }^{\mathbf{1 *}} ， Jun Chen 2 ∗ 2 ∗ ^(2**){ }^{2 *} ， Zhifang Lin 3 , 4 , J a c k N g 1 , 5 † 3 , 4 , J a c k N g 1 , 5 † ^(3,4,Jack)Ng^(1,5†){ }^{\mathbf{3 , 4}, ~ J a c k ~} \mathbf{N g}^{1,5 \dagger} 光学牵引光束于 2011 年提出，并在不久后进行了实验演示，它提供了拉动粒子对抗光传播的能力。它吸引了许多研究和公众的兴趣。然而，其有限的微观范围严重限制了它的适用性。困境在于，远距离贝塞尔光束（最容易接近的光学牵引光束）具有较小的半锥角， θ 0 θ 0 theta_(0)\boldsymbol{\theta}_{0} 这使得拉动变得困难。在这里，通过同时使用几种新颖且兼容的机制，包括横向各向同性、斯涅尔定律、抗反射涂层（或阻抗匹配的超材料）和光干涉，我们克服了这一困境并实现了远距离光拉力。 θ 0 ≈ 1 ∘ θ 0 ≈ 1 ∘ theta_(0)~~1^(@)\theta_{0} \approx 1^{\circ} 使用 ∼ 1 W ∼ 1 W ∼1W\sim 1 \mathrm{~W} 激光功率时，范围估计为 14 厘米。因此，可以在介质或真空中实现宏观光学牵引，对半锥角和光频率具有良好的耐受性。

INTRODUCTION  介绍

An optical force is well known for its ability to “trap” or “propel” microscopic particles (1-26). A counterintuitive phenomenon was discovered in 2011 (27-31): By redirecting the incident photons forward, the lightinduced Lorentz force can pull distant objects backward against the propagation of light. This phenomenon, now termed the optical pulling force (OPF) or optical tractor beam (OTB), is emerging as a reality in laboratories ( 32 , 33 ) ( 32 , 33 ) (32,33)(32,33) and offering new functionality in optical micromanipulation. Yet, its short micrometer-scale working range remains an obstacle that limits its practical applications.
众所周知，光学力能够“捕获”或“推动”微观粒子 （1-26）。2011 年发现了一个违反直觉的现象 （27-31）：通过向前重定向入射光子，光诱导的洛伦兹力可以将远处的物体向后拉，以对抗光的传播。这种现象现在被称为光学拉力 （OPF） 或光学牵引光束 （OTB），正在实验室中成为现实， ( 32 , 33 ) ( 32 , 33 ) (32,33)(32,33) 并在光学显微作中提供新功能。然而，其较短的微米级工作范围仍然是限制其实际应用的障碍。
A photon propagating in the direction θ 0 θ 0 theta_(0)\theta_{0} and elastically scattered into angle θ θ theta\theta by a particle will transfer to the particle a forward momentum of
沿该方向 θ 0 θ 0 theta_(0)\theta_{0} 传播并被粒子弹性散射成一定角度 θ θ theta\theta 的光子将向粒子传递
Δ p z = ℏ k ( cos ⁡ θ 0 − cos ⁡ θ ) Δ p z = ℏ k cos ⁡ θ 0 − cos ⁡ θ Deltap_(z)=ℏk(cos theta_(0)-cos theta)\Delta p_{z}=\hbar k\left(\cos \theta_{0}-\cos \theta\right)
where the first and second terms are the initial and final momentum of the photon, respectively. The particle first intercepts the photon, resulting in an extinction force, which is given by the first term in Eq. 1 . It then re-emits the photon, leading to a recoil force, given by the second term (29). Because the extinction force is positively definite, to generate the OPF, the recoil force must be negative and stronger than the extinction force. Equation 1 contributes to pulling when θ < θ 0 θ < θ 0 theta < theta_(0)\theta<\theta_{0} and pushing when θ > θ 0 θ > θ 0 theta > theta_(0)\theta>\theta_{0}. Inducing the OPF is equivalent to squeezing most of the scattered light into a solid cone defined by θ < θ 0 θ < θ 0 theta < theta_(0)\theta<\theta_{0}, which is hereafter referred to as the pulling cone. Thus, in general, a small θ 0 θ 0 theta_(0)\theta_{0} is unfavorable for the OPF. In the literature, the minimum θ 0 θ 0 theta_(0)\theta_{0} reported for the OPF is ∼ 50 ∘ ∼ 50 ∘ ∼50^(@)\sim 50^{\circ} in simulations using a Bessel beam (34) and ∼ 86 ∘ ∼ 86 ∘ ∼86^(@)\sim 86^{\circ} in experiments that essentially used a pair of plane waves (32). In the hypothetical case of a " 2 D " (two-dimensional) system, simulations found the OPF at ∼ 42 ∘ ( 28 ) ∼ 42 ∘ ( 28 ) ∼42^(@)(28)\sim 42^{\circ}(28). The values of θ 0 θ 0 theta_(0)\theta_{0} achieved in the literature are schematically summarized in Fig. 1F.
其中，第一项和第二项分别是光子的初始动量和最终动量。粒子首先拦截光子，产生消光力，由方程 1 中的第一项给出。然后它重新发射光子，产生第二项 （29） 给出的反冲力。因为消光力是正定的，所以要产生 OPF，反冲力必须是负的并且比消光力强。方程 1 有助于 pull when θ < θ 0 θ < θ 0 theta < theta_(0)\theta<\theta_{0} 和 push when θ > θ 0 θ > θ 0 theta > theta_(0)\theta>\theta_{0} 。诱导 OPF 相当于将大部分散射光挤压成一个由 θ < θ 0 θ < θ 0 theta < theta_(0)\theta<\theta_{0} 定义的实心圆锥体，以下简称为拉锥体。因此，一般来说，小 θ 0 θ 0 theta_(0)\theta_{0} 对 OPF 不利。在文献中，OPF 报告的最小值 θ 0 θ 0 theta_(0)\theta_{0} 是在 ∼ 50 ∘ ∼ 50 ∘ ∼50^(@)\sim 50^{\circ} 使用贝塞尔光束 （34） 的模拟和 ∼ 86 ∘ ∼ 86 ∘ ∼86^(@)\sim 86^{\circ} 基本上使用一对平面波的实验 （32） 中。在“2 D”（二维）系统的假设情况下，模拟发现 OPF 为 ∼ 42 ∘ ( 28 ) ∼ 42 ∘ ( 28 ) ∼42^(@)(28)\sim 42^{\circ}(28) 。图 1F 对文献中 θ 0 θ 0 theta_(0)\theta_{0} 实现的值进行了示意性总结。
Ideal Bessel beams have an infinite cross section and energy. Therefore, they do not exist in reality. One could produce an approximate
理想的贝塞尔梁具有无限的横截面和能量。因此，它们在现实中不存在。可以生成一个近似的
Bessel beam, which is nondiffracting over a distance long compared to the Rayleigh range of a Gaussian beam. The range of a Bessel beam, as produced by an axicon, is roughly given by (35)
贝塞尔光束，与高斯光束的瑞利范围相比，它在很长的距离内不发生衍射。由锥透镜产生的贝塞尔梁的范围大致由（35）给出
Range ( θ 0 ) ≈ ω 0 / tan ⁡ ( θ 0 ) Range θ 0 ≈ ω 0 / tan ⁡ θ 0 Range(theta_(0))~~omega_(0)//tan(theta_(0))\operatorname{Range}\left(\theta_{0}\right) \approx \omega_{0} / \tan \left(\theta_{0}\right)
where ω 0 ω 0 omega_(0)\omega_{0} is the waist of the incident beam that illuminates the input side of the axicon, which is assumed to be 2.5 mm , as in the experiment of (36). To extend its range, it is impractical to adopt a very large ω 0 ω 0 omega_(0)\omega_{0}, which also requires a large axicon. One may then take the alternative route of decreasing θ 0 θ 0 theta_(0)\theta_{0}. However, this causes a practical dilemma in optical pulling: Although the range of the beam is short for large θ 0 θ 0 theta_(0)\theta_{0} (35), it is difficult to achieve the OPF if θ 0 θ 0 theta_(0)\theta_{0} is small (29). We note that a Bessel beam or other propagation invariant beam can also be produced by other approaches (35, 37-39).
其中 ω 0 ω 0 omega_(0)\omega_{0} 是照亮锥透镜输入侧的入射光束的束腰，假设为 2.5 mm，如 （36） 的实验所示。为了扩展其范围，采用非常大 ω 0 ω 0 omega_(0)\omega_{0} 的 是不切实际的，这也需要一个大的 axicon。然后，人们可以采取递减 θ 0 θ 0 theta_(0)\theta_{0} 的替代路线。然而，这在光学拉力中造成了一个实际的困境：虽然光束的范围短于大 θ 0 θ 0 theta_(0)\theta_{0} （35），但如果 θ 0 θ 0 theta_(0)\theta_{0} 很小 （29），则很难实现 OPF。我们注意到贝塞尔光束或其他传播不变光束也可以通过其他方法产生 （35， 37-39）。
Here, we completely overcome this dilemma in terms of the range and half-cone angle by enabling OPF at small θ 0 θ 0 theta_(0)\theta_{0} down to 1 ∘ 1 ∘ 1^(@)1^{\circ}, which is approximately two orders of magnitude smaller than the previously reported angle of ∼ 50 ∘ ∼ 50 ∘ ∼50^(@)\sim 50^{\circ} in 3D (34), as illustrated in Fig. 1F. We achieved this by eliminating diffraction in the azimuthal direction by transverse isotropy (Fig. 1, A and B), eliminating reflection by using antireflection coatings (ARCs) (Fig. 1C), restricting the diffracted wave to propagate near the edge of the pulling cone by using cylindrical geometry to enforce Snell’s law (Fig. 1D), and finally collimating the diffracted light by interference to induce OPF (Fig. 1E). For θ 0 = 1 ∘ θ 0 = 1 ∘ theta_(0)=1^(@)\theta_{0}=1^{\circ}, the range of the Bessel beam could be > 14 cm > 14 cm > 14cm>14 \mathrm{~cm} according to Eq. 2 . We note that, to produce a long-range Bessel beam with θ 0 = 1 ∘ θ 0 = 1 ∘ theta_(0)=1^(@)\theta_{0}=1^{\circ}, the available laser power is also a limiting factor. The total power for the central ring of the Bessel beam is 0.3 W for θ 0 = 1 ∘ θ 0 = 1 ∘ theta_(0)=1^(@)\theta_{0}=1^{\circ}. The total power needed will be of this order of magnitude. Other than this constraint in power, however, long-range pulling is not forbidden by physical laws.
在这里，我们完全克服了范围和半锥角方面的这一困境，使 OPF 在小 θ 0 θ 0 theta_(0)\theta_{0} 到 1 ∘ 1 ∘ 1^(@)1^{\circ} ，这比之前报道的 3D 角度 ∼ 50 ∘ ∼ 50 ∘ ∼50^(@)\sim 50^{\circ} （34） 大约小两个数量级，如图 1F 所示。我们通过横向各向同性消除方位角方向的衍射（图 1、A 和 B），通过使用增透膜 （ARC） 消除反射（图 1C），通过使用圆柱形几何来执行斯涅尔定律（图 1D），最后通过干涉准直衍射光以感应 OPF（图 1E），从而实现这一目标。对于 θ 0 = 1 ∘ θ 0 = 1 ∘ theta_(0)=1^(@)\theta_{0}=1^{\circ} ，贝塞尔梁的范围可以 > 14 cm > 14 cm > 14cm>14 \mathrm{~cm} 按照方程 2 。我们注意到，为了产生长 θ 0 = 1 ∘ θ 0 = 1 ∘ theta_(0)=1^(@)\theta_{0}=1^{\circ} 距离贝塞尔光束，可用的激光功率也是一个限制因素。贝塞尔梁中心环的总功率为 0.3 W。 θ 0 = 1 ∘ θ 0 = 1 ∘ theta_(0)=1^(@)\theta_{0}=1^{\circ} 所需的总功率将达到这个数量级。然而，除了这种功率限制之外，物理定律并不禁止远程拉扯。
Other interesting optical forces, such as optical lift (40) and lateral forces perpendicular to the light propagation direction (41-45), have also been reported. Tractor beams not based solely on the Lorentz force are also of immense interest. They have been proposed by using, for example, photophoretic forces ( 46 , 47 ) ( 46 , 47 ) (46,47)(46,47), acoustic forces ( 48 − 50 ) ( 48 − 50 ) (48-50)(48-50), and even matter waves (51), in addition to optical beams (52-59). The tractor beam is a highly counterintuitive construct, although it violates no fundamental physical law. Most tractor beams, even those relying exclusively on lasers, require a medium to function, such as air or water. Some, such as an acoustic tractor beam, require a medium for the wave itself to propagate. Others, such as the photophoretic tractor beam, work by light-induced thermal forces that cannot exist without a medium.
其他有趣的光学力，例如光学升力 （40） 和垂直于光传播方向的侧向力 （41-45），也有报道。不仅仅基于洛伦兹力的牵引梁也引起了极大的兴趣。例如，除了光束 （52-59） 之外，还通过使用光泳力 ( 46 , 47 ) ( 46 , 47 ) (46,47)(46,47) 、声力 ( 48 − 50 ) ( 48 − 50 ) (48-50)(48-50) 甚至物质波 （51） 来提出它们。牵引光束是一种非常违反直觉的结构，尽管它没有违反任何基本物理定律。大多数牵引光束，即使是那些完全依赖激光的光束，都需要介质才能发挥作用，例如空气或水。有些，例如声学牵引光束，需要一种介质来使波本身传播。其他的，如光泳牵引光束，是通过光诱导的热力工作，如果没有介质，这些热力就无法存在。
Fig. 1. Illustration of the series of mechanisms applied, in turn, to achieve the OPF. The yellow particle is illuminated by a transversely isotropic Bessel beam, which propagates in the direction of the gray arrow, and its constituting k k kk vectors lie on the cone formed by the revolution of the blue arrows about the partial axis. The green layer represents the ARC, and the semitransparent black cone is the pulling cone. In (A), the scattered fields are represented by the pink arrows. In (B) to (E), the scattered fields are represented by the cone formed by the revolution of the red arrows about the partial axis. (A) Scattering by a general particle. Light is scattered everywhere. (B) Introducing transverse isotropy to eliminate the diffraction in the azimuthal direction. © Introducing ARC to eliminate reflection. (D) Snell’s law approximately aligns the scattered field to the edge of the pulling cone. (E) The diffracted light is collimated by the interference. (F) Halfcone angles achieved in the literature.
图 1.依次用于实现 OPF 的一系列机制的图示。黄色粒子由横向各向同性贝塞尔光束照射，该光束沿灰色箭头的方向传播，其构成 k k kk 向量位于蓝色箭头绕部分轴旋转形成的圆锥体上。绿色图层代表 ARC，半透明的黑色圆锥体是拉锥体。在 （A） 中，分散的字段由粉红色箭头表示。在 （B） 到 （E） 中，散射场由红色箭头绕偏轴旋转形成的圆锥体表示。（A） 一般粒子的散射。光线到处散射。（B） 引入横向各向同性以消除方位角方向的衍射。© 引入 ARC 以消除反射。（D） 斯涅尔定律将散射场大致与拉锥的边缘对齐。（E） 衍射光涉准直。（F） 文献中实现的半锥角。
Here, we stress that the OTB we designed operates on the OPF alone, and it can work in a vacuum. This is an important step toward remote sampling in a vacuum, which is desirable in, for example, space exploration; light can both push and pull a handle to collect samples from a remote site.
在这里，我们强调我们设计的 OTB 仅在 OPF 上运行，并且可以在真空中工作。这是朝着在真空中远程采样迈出的重要一步，这在太空探索等应用中是可取的;Light 可以推拉手柄以从远程站点收集样本。

RESULTS  结果

Pulling enhanced by transverse isotropy
横向各向同性增强拉力

Consider a hypothetical 2D system that consists of a slab and a beam that are translationally invariant along one direction. By symmetry, light diffraction is limited to within the scattering plane, so the degrees of freedom for the light to diffract are reduced from 2 to 1 . Then, among all possible scattering angles, a fraction of θ 0 / π θ 0 / π theta_(0)//pi\theta_{0} / \pi contributes to pulling, which is significantly better than the fraction of sin 2 ⁡ θ 0 / 2 sin 2 ⁡ θ 0 / 2 sin^(2)theta_(0)//2\sin ^{2} \theta_{0} / 2 in a general 3 D system at small θ 0 θ 0 theta_(0)\theta_{0}. One may thus intuitively expect the OPF to be much more achievable in 2D in general. At θ 0 = 1 ∘ θ 0 = 1 ∘ theta_(0)=1^(@)\theta_{0}=1^{\circ}, the pulling cone spans 5.6 × 10 − 3 5.6 × 10 − 3 5.6 xx10^(-3)5.6 \times 10^{-3} out of all solid angles, which is more than 70 times larger than that of a 3 D system, which is 7.6 × 10 − 5 7.6 × 10 − 5 7.6 xx10^(-5)7.6 \times 10^{-5}. However, the fraction is very small in either case; thus, pulling at θ 0 ≈ 1 ∘ θ 0 ≈ 1 ∘ theta_(0)~~1^(@)\theta_{0} \approx 1^{\circ} is not easy at all. Moreover, 2D systems, if not somewhat fictive, are very long in one dimension; thus, they are quite heavy and difficult to manipulated by light.
考虑一个假设的二维系统，它由一个板和一个梁组成，它们沿一个方向平移不变。通过对称性，光衍射仅限于散射平面内，因此光发生衍射的自由度从 2 减少到 1 。然后，在所有可能的散射角中，一小部分 θ 0 / π θ 0 / π theta_(0)//pi\theta_{0} / \pi 有助于拉扯，这明显优于一般三维系统中小 θ 0 θ 0 theta_(0)\theta_{0} 的分数 sin 2 ⁡ θ 0 / 2 sin 2 ⁡ θ 0 / 2 sin^(2)theta_(0)//2\sin ^{2} \theta_{0} / 2 。因此，人们可以直观地期望 OPF 在 2D 中通常更容易实现。在 θ 0 = 1 ∘ θ 0 = 1 ∘ theta_(0)=1^(@)\theta_{0}=1^{\circ} 处，拉锥跨 5.6 × 10 − 3 5.6 × 10 − 3 5.6 xx10^(-3)5.6 \times 10^{-3} 度为所有立体角，比 3 D 系统的 大 70 多倍，即 7.6 × 10 − 5 7.6 × 10 − 5 7.6 xx10^(-5)7.6 \times 10^{-5} 。但是，在任何一种情况下，分数都非常小;因此，拉扯一点 θ 0 ≈ 1 ∘ θ 0 ≈ 1 ∘ theta_(0)~~1^(@)\theta_{0} \approx 1^{\circ} 都不容易。此外，二维系统，如果不是有点虚构的话，在一维中是很长的;因此，它们很重，很难被光纵。
Here, we note that a system with transverse isotropy shares the advantages of a 2D system but not its disadvantages. Because a system consists of a particle and a beam that have transverse isotropy, the Poynting vectors of the incident and scattered fields have no azimuthal component; therefore, diffraction in the azimuthal direction is forbidden. Consequently, light propagating in the plane defined by a constant ϕ ϕ phi\phi remains on that plane after being scattered. This, in a sense, mimics 2D diffraction.
在这里，我们注意到具有横向各向同性的系统具有二维系统的优点，但没有缺点。由于系统由具有横向各向同性的粒子和光束组成，因此入射场和散射场的坡印廷矢量没有方位角分量;因此，禁止在方位角方向发生衍射。因此，在由常数 ϕ ϕ phi\phi 定义的平面中传播的光在散射后仍保留在该平面上。从某种意义上说，这模拟了 2D 衍射。
The OPF is typically realized using a propagation-invariant beam, which, ideally, does not diffract as the beam propagates. Here, if not otherwise specified, we use the m = 0 m = 0 m=0m=0 azimuthally polarized Bessel beam
OPF 通常使用传播不变光束来实现，理想情况下，光束不会随着光束的传播而发生衍射。在这里，如果没有特别说明，我们使用 m = 0 m = 0 m=0m=0 方位偏振贝塞尔光束
E = − E 0 i exp ⁡ ( i k 0 cos ⁡ θ 0 z ) J 0 ′ ( k 0 sin ⁡ θ 0 ρ ) sin ⁡ θ 0 ϕ ^ E = − E 0 i exp ⁡ i k 0 cos ⁡ θ 0 z J 0 ′ k 0 sin ⁡ θ 0 ρ sin ⁡ θ 0 ϕ ^ E=-E_(0)i exp(ik_(0)cos theta_(0)z)((J_(0)^(')(k_(0)sin theta_(0)rho))/(sin theta_(0))) hat(phi)\mathbf{E}=-E_{0} i \exp \left(i k_{0} \cos \theta_{0} z\right) \frac{J_{0}^{\prime}\left(k_{0} \sin \theta_{0} \rho\right)}{\sin \theta_{0}} \hat{\phi}
where k 0 k 0 k_(0)k_{0} is the background wave number, ( ρ , ϕ , z ) ( ρ , ϕ , z ) (rho,phi,z)(\rho, \phi, z) are the cylindrical coordinates, J 0 ′ J 0 ′ J_(0)^(')J_{0}^{\prime} is the derivative of the zeroth-order Bessel function with respect to its argument, and the fixed half-cone angle θ 0 θ 0 theta_(0)\theta_{0} denotes the angle between the beam propagation direction ( z ^ ) ( z ^ ) ( hat(z))(\hat{z}) and the wave vectors of the Fourier components of Eq. 3. Throughout this paper, unless explicitly stated, the incident wavelength is fixed at 532 nm . The normalization of the Bessel beam’s intensity is described in Materials and Methods.
其中 k 0 k 0 k_(0)k_{0} 是背景波数， ( ρ , ϕ , z ) ( ρ , ϕ , z ) (rho,phi,z)(\rho, \phi, z) 是圆柱坐标， J 0 ′ J 0 ′ J_(0)^(')J_{0}^{\prime} 是零阶贝塞尔函数相对于其参数的导数，固定的半锥角 θ 0 θ 0 theta_(0)\theta_{0} 表示光束传播方向 ( z ^ ) ( z ^ ) ( hat(z))(\hat{z}) 与方程 3 的傅里叶分量的波矢之间的角度。在本文中，除非明确说明，否则入射波长固定为 532 nm 。贝塞尔光束强度的归一化在 材料和方法 中进行了描述。
Figure 2 plots the OPF acting on a spherical particle illuminated by different concentric beams, including the transversely isotropic m = 0 m = 0 m=0m=0 azimuthally polarized Bessel beam given in Eq. 3 (Fig. 2A), a transversely isotropic m = 0 m = 0 m=0m=0 radially polarized Bessel beam (Fig. 2B)
图 2 绘制了作用在由不同同心光束照射的球形粒子上的 OPF，包括方程 3 中给出的横向各向同性 m = 0 m = 0 m=0m=0 方位偏振贝塞尔光束（图 2A），横向各向同性 m = 0 m = 0 m=0m=0 径向偏振贝塞尔光束（图 2B）
E = E 0 exp ⁡ ( i k 0 cos ⁡ θ 0 z ) ( i cos ⁡ θ 0 J 0 ′ ( k 0 sin ⁡ θ 0 ρ ) sin ⁡ θ 0 ρ ^ + J 0 ′ ( k 0 sin ⁡ θ 0 ρ ) z ^ ) E = E 0 exp ⁡ i k 0 cos ⁡ θ 0 z i cos ⁡ θ 0 J 0 ′ k 0 sin ⁡ θ 0 ρ sin ⁡ θ 0 ρ ^ + J 0 ′ k 0 sin ⁡ θ 0 ρ z ^ {:[E=],[qquadE_(0)exp(ik_(0)cos theta_(0)z)(i(cos theta_(0)J_(0)^(')(k_(0)sin theta_(0)rho))/(sin theta_(0))( hat(rho))+J_(0)^(')(k_(0)sin theta_(0)rho)( hat(z)))]:}\begin{aligned} & \mathbf{E}= \\ & \qquad E_{0} \exp \left(i k_{0} \cos \theta_{0} z\right)\left(i \frac{\cos \theta_{0} J_{0}^{\prime}\left(k_{0} \sin \theta_{0} \rho\right)}{\sin \theta_{0}} \hat{\boldsymbol{\rho}}+J_{0}^{\prime}\left(k_{0} \sin \theta_{0} \rho\right) \hat{\mathbf{z}}\right) \end{aligned}
an m = 0 m = 0 m=0m=0 linearly polarized Bessel beam (Fig. 2C)
m = 0 m = 0 m=0m=0 线偏振贝塞尔光束（图 2C）
E = E 0 exp ⁡ ( i k 0 cos ⁡ θ 0 z ) ( cos ⁡ θ 0 J 0 ( k 0 sin ⁡ θ 0 ρ ) y ^ − i y ρ sin ⁡ θ 0 J 0 ′ ( k 0 sin ⁡ θ 0 ρ ) z ^ ) E = E 0 exp ⁡ i k 0 cos ⁡ θ 0 z cos ⁡ θ 0 J 0 k 0 sin ⁡ θ 0 ρ y ^ − i y ρ sin ⁡ θ 0 J 0 ′ k 0 sin ⁡ θ 0 ρ z ^ {:[E=E_(0)exp(ik_(0)cos theta_(0)z)(cos theta_(0)J_(0)(k_(0)sin theta_(0)rho)( hat(y)):}],[{:-i(y)/( rho)sin theta_(0)J_(0)^(')(k_(0)sin theta_(0)rho)( hat(z)))]:}\begin{gathered} \mathbf{E}=E_{0} \exp \left(i k_{0} \cos \theta_{0} z\right)\left(\cos \theta_{0} J_{0}\left(k_{0} \sin \theta_{0} \rho\right) \hat{\mathbf{y}}\right. \\ \left.-i \frac{y}{\rho} \sin \theta_{0} J_{0}^{\prime}\left(k_{0} \sin \theta_{0} \rho\right) \hat{\mathbf{z}}\right) \end{gathered}
and a pair of copropagating plane waves (Fig. 2D)
和一对共传播的平面波（图 2D）
E = E 0 x ^ ( e i k 0 sin ⁡ θ 0 y + e − i k 0 sin ⁡ θ 0 y ) e i k 0 cos ⁡ θ 0 z E = E 0 x ^ e i k 0 sin ⁡ θ 0 y + e − i k 0 sin ⁡ θ 0 y e i k 0 cos ⁡ θ 0 z E=E_(0) hat(x)(e^(ik_(0)sin theta_(0)y)+e^(-ik_(0)sin theta_(0)y))e^(ik_(0)cos theta_(0)z)\mathbf{E}=E_{0} \hat{\mathbf{x}}\left(e^{i k_{0} \sin \theta_{0} y}+e^{-i k_{0} \sin \theta_{0} y}\right) e^{i k_{0} \cos \theta_{0} z}
Fig. 2. Pulling enhanced by transversely isotropic beam. The blue color indicates the phase space region where the OPF exists. The insets are schematic illustrations of a sphere ( n p = 1.6 n p = 1.6 n_(p)=1.6n_{\mathrm{p}}=1.6 ) in water ( n b = 1.33 n b = 1.33 n_(b)=1.33n_{\mathrm{b}}=1.33 ) illuminated by a different beam. (A) m = 0 m = 0 m=0m=0 azimuthally polarized Bessel beam (transversely isotropic) (see Eq. 3). (B) m = 0 m = 0 m=0m=0 radially polarized Bessel beam (transversely isotropic) (see Eq. 4). © m = 0 m = 0 m=0m=0 linearly polarized Bessel beam (nontransversely isotropic) (see Eq. 5). (D) Propagation-invariant beam formed by a pair of plane waves (nontransversely isotropic) (see Eq. 6).
图 2.横向各向同性梁增强了拉力。蓝色表示 OPF 所在的相空间区域。插图是水 （ n p = 1.6 n p = 1.6 n_(p)=1.6n_{\mathrm{p}}=1.6 n b = 1.33 n b = 1.33 n_(b)=1.33n_{\mathrm{b}}=1.33 ） 中的球体 （ ） 由不同光束照射的示意图。（A） m = 0 m = 0 m=0m=0 方位极化贝塞尔光束（横向各向同性）（见方程 3）。（B） m = 0 m = 0 m=0m=0 径向偏振贝塞尔光束（横向各向同性）（见方程 4）。© m = 0 m = 0 m=0m=0 线偏振贝塞尔光束（非横向各向同性）（见方程 5）。（D） 由一对平面波（非横向各向同性）形成的传播不变光束（见方程 6）。
In terms of inducing the OPF, the azimuthally polarized beam given in Eq. 3, which is transversely isotropic, outperformed all others. The radially polarized beam was also transversely isotropic, but it was still expected to be less effective in pulling. The electromagnetic boundary conditions required the electromagnetic fields for the azimuthal polarization to be continuous across the particle boundary, resulting in less reflection, and thus, it had a stronger OPF. However, the electric field for the radial polarization was discontinuous, leading to stronger reflection; thus, it had a weaker OPF. That is, the difference in performance between the radial and azimuthal polarization was due to the asymmetry in permittivity and permeability distribution. We note that Fig. 2 also suggests that OPF depends not only on the type of beam but also on its polarization.
在感应 OPF 方面，方程 3 中给出的方位偏振光束是横向各向同性的，其性能优于所有其他光束。径向偏振光束也是横向各向同性的，但预计它的拉力仍然较差。电磁边界条件要求方位极化的电磁场在粒子边界上连续，从而减少反射，因此具有更强的 OPF。然而，径向极化的电场是不连续的，导致更强的反射;因此，它的 OPF 较弱。也就是说，径向极化和方位极化之间的性能差异是由于介电常数和磁导率分布的不对称性造成的。我们注意到，图 2 还表明 OPF 不仅取决于光束的类型，还取决于其偏振。
Figure 3 compares the optical force acting on a dielectric cylinder and two rectangular dielectric blocks with different aspect ratios. The incident beam was the Bessel beam given in Eq. 3. All particles had the same volume and length and were made from the same material. The OPF diminished with increasing aspect ratio of the rectangular blocks (red and blue dashed curves) as the particle shape deviated from transverse isotropy. A highly symmetric system places more restrictions on the light propagation; therefore, the light has a lesser degree of freedom to diffract, which is favorable for the OPF where strong forward scattering is required.
图 3 比较了作用在介电圆柱体和两个具有不同纵横比的矩形介电块上的光力。入射光束是方程 3 中给出的贝塞尔光束。所有颗粒具有相同的体积和长度，并且由相同的材料制成。当粒子形状偏离横向各向同性时，OPF 随着矩形块（红色和蓝色虚线）纵横比的增加而减小。高度对称的系统对光的传播施加了更多限制;因此，光的衍射自由度较小，这有利于需要强前向散射的 OPF。
Figures 2 and 3 highlight the importance of transverse isotropy in optical pulling. Although the magnitude of the pulling force was smaller than that of the pushing force, they were of the same order of magnitude. We also calculated the OPF acting on a spheroid (see section S1) induced by the Bessel beam given in Eq. 3. Its performance in the OPF was even better than that of a bare cylinder. The reflection from the flat ends of the cylinder was directly backward, generating strong
图 2 和图 3 强调了横向各向同性在光拉力中的重要性。虽然拉力的大小小于推力的大小，但它们是同一个数量级。我们还计算了方程 3 中给出的贝塞尔光束诱导的作用在球体（参见 S1 节）上的 OPF。它在 OPF 中的性能甚至比裸圆柱体的性能还要好。圆柱体平端的反射是直接向后的，产生强烈的
forward forces, whereas that of the spheroid did not. The good performance of the spheroid further confirms the important role of transverse isotropy in optical pulling. For reasons to be discussed later, we still preferred working with the cylinder.
向前的力，而椭球体的力则没有。球体的良好性能进一步证实了横向各向同性在光牵引中的重要作用。出于稍后讨论的原因，我们仍然更喜欢使用圆柱体。

Pulling enhanced by Snell's law
斯涅尔定律增强的拉力

We went beyond transverse isotropy and considered a dielectric microcylinder of diameter D D DD and length L L LL, coated without (Fig. 4A) and with (Fig. 4D) ARC. To induce the OPF at a small θ 0 θ 0 theta_(0)\theta_{0}, the particle must squeeze most of the scattered light into the tiny pulling cone. Apparently, this would not be possible if the scattering was strong, as light would be scattered everywhere. In Fig. 4B, the phase space region with the OPF is highlighted in blue for a bare glass cylinder without ARC in water and illuminated by the Bessel beam given in Eq. 3. Because the microcylinder had straight parallel sides and flat ends, light impinging on it reflected and diffracted approximately according to Snell’s law but with some slight deviation due to diffraction by the particle with finite size and finite curvature. Accordingly, by Snell’s law, all light was scattered into an angle near the edge of the pulling cone, i.e., θ = θ 0 + Δ θ θ = θ 0 + Δ θ theta=theta_(0)+Delta theta\theta=\theta_{0}+\Delta \theta, where the small Δ θ Δ θ Delta theta\Delta \theta was induced by diffraction. It turns out that if one further excites the fundamental waveguide mode (FWM) by using a particle with an appropriate size, then Δ θ < 0 Δ θ < 0 Delta theta < 0\Delta \theta<0, which induces the OPF, as explained in the next section. Here, at each θ 0 θ 0 theta_(0)\theta_{0}, we chose a diameter D D DD such that the FWM was excited. The relationship between D D DD and θ 0 θ 0 theta_(0)\theta_{0} is plotted in Fig. 4G, which can be written down analytically for small θ 0 θ 0 theta_(0)\theta_{0}
我们超越了横向各向同性，考虑了一个直径 D D DD 和长度 L L LL 的介电微圆柱体，没有（图 4A）和（图 4D）ARC 涂层。为了在小 θ 0 θ 0 theta_(0)\theta_{0} 处诱导 OPF，粒子必须将大部分散射光挤压到微小的拉锥中。显然，如果散射很强，这是不可能的，因为光会到处散射。在图 4B 中，对于水中没有 ARC 的裸玻璃圆柱体，带有 OPF 的相空间区域以蓝色突出显示，并由方程 3 中给出的贝塞尔光束照亮。由于微圆柱体具有笔直的平行侧面和平坦的末端，因此照射在其上的光大约根据斯涅尔定律进行反射和衍射，但由于具有有限尺寸和有限曲率的粒子的衍射而产生一些轻微的偏差。因此，根据斯涅尔定律，所有光都散射到拉锥边缘附近的一个角度，即 θ = θ 0 + Δ θ θ = θ 0 + Δ θ theta=theta_(0)+Delta theta\theta=\theta_{0}+\Delta \theta ，其中小 Δ θ Δ θ Delta theta\Delta \theta 光是由衍射感应的。事实证明，如果通过使用具有适当大小的粒子进一步激发基波导模式 （FWM），那么 Δ θ < 0 Δ θ < 0 Delta theta < 0\Delta \theta<0 会感应出 OPF，如下一节所述。在这里，在每个 θ 0 θ 0 theta_(0)\theta_{0} 上，我们都选择一个直径 D D DD ，以便 FWM 被激发。和 θ 0 θ 0 theta_(0)\theta_{0} 之间的关系 D D DD 如图 4G 所示，对于小 θ 0 θ 0 theta_(0)\theta_{0}
D ( θ 0 ) ≈ 2 z 1 / k b θ 0 D θ 0 ≈ 2 z 1 / k b θ 0 D(theta_(0))~~2z_(1)//k_(b)theta_(0)D\left(\theta_{0}\right) \approx 2 z_{1} / k_{b} \theta_{0}
Here, z 1 z 1 z_(1)z_{1} is the first zero of the first-order Bessel function. Inducing the OPF at small θ 0 θ 0 theta_(0)\theta_{0} is possible only with a large D D DD owing to the diffraction limit. Further details on the FWM and the derivation of Eq. 7 can be found in section S2. We must stress that the excitation of the FWM is by no means a requirement for the OPF, although it can be an advantage sometimes. The FWM can be effectively excited only for θ 0 θ 0 theta_(0)\theta_{0} less than some critical value, and the OPF for θ 0 θ 0 theta_(0)\theta_{0} outside this range is not plotted on the figures. The OPF exists for θ 0 > 30 ∘ θ 0 > 30 ∘ theta_(0) > 30^(@)\theta_{0}>30^{\circ}. A larger θ 0 θ 0 theta_(0)\theta_{0} has a larger pulling cone; therefore, it is more likely for the scattered light to induce the OPF. When θ 0 θ 0 theta_(0)\theta_{0} is small ( < 30 ∘ ) < 30 ∘ ( < 30^(@))\left(<30^{\circ}\right), the OPF almost disappears completely because of the strong reflection by the flat end of the cylinder. The optical force versus L L LL when θ 0 = 2 ∘ θ 0 = 2 ∘ theta_(0)=2^(@)\theta_{0}=2^{\circ} is plotted in Fig. 4C. We observed some periodically spaced peaks, which can be attributed to the Fabry-Perot resonances.
这里， z 1 z 1 z_(1)z_{1} 是一阶 Bessel 函数的第一个零。由于衍射极限，只有在大 D D DD OPF 的情况下才能在小 θ 0 θ 0 theta_(0)\theta_{0} OPF 下诱导。有关 FWM 和方程 7 推导的更多详细信息，请参见 S2 节。我们必须强调，FWM 的激励绝不是 OPF 的要求，尽管它有时可能是一个优势。FWM 只能在小于某个临界值时 θ 0 θ 0 theta_(0)\theta_{0} 被有效激发，并且超出此范围的 θ 0 θ 0 theta_(0)\theta_{0} OPF 不会绘制在图上。OPF 存在 θ 0 > 30 ∘ θ 0 > 30 ∘ theta_(0) > 30^(@)\theta_{0}>30^{\circ} 。较大的 θ 0 θ 0 theta_(0)\theta_{0} 具有较大的拉锥体;因此，散射光更有可能诱导 OPF。当较小时 θ 0 θ 0 theta_(0)\theta_{0} ( < 30 ∘ ) < 30 ∘ ( < 30^(@))\left(<30^{\circ}\right) ，由于圆柱体平坦端的强烈反射，OPF 几乎完全消失。光力与时间 θ 0 = 2 ∘ θ 0 = 2 ∘ theta_(0)=2^(@)\theta_{0}=2^{\circ} 的关系 L L LL 如图 4C 所示。我们观察到一些周期性间隔的峰，这可以归因于法布里-珀罗共振。

Pulling enhanced by ARCs  ARC 增强的拉取

To further improve the OPF, we noted that backscattering generated a strong forward force. For particles with flat ends, such as a cylinder, ARCs may be applied on its ends to reduce or even completely eliminate reflection. We noted that ARC was also adopted in ( 28 , 60 − 62 ) ( 28 , 60 − 62 ) (28,60-62)(28,60-62) to improve the performance of the optical force and the OPF. Our ARCs were characterized by a refractive index
为了进一步改进 OPF，我们注意到反向散射产生了强大的向前力。对于具有平端的粒子（如圆柱体），可以在其末端应用 ARC 以减少甚至完全消除反射。我们注意到 ARC 也被用于 ( 28 , 60 − 62 ) ( 28 , 60 − 62 ) (28,60-62)(28,60-62) 提高光学力和 OPF 的性能。我们的 ARC 以折射率为特征
n ARC = n p n b n ARC = n p n b n_(ARC)=sqrt(n_(p)n_(b))n_{\mathrm{ARC}}=\sqrt{n_{\mathrm{p}} n_{\mathrm{b}}}
and a thickness  和 a 厚度
d ( θ 0 ) = λ cos ⁡ θ ARC 4 n ARC − 4 n b sin ⁡ θ 0 sin ⁡ θ ARC d θ 0 = λ cos ⁡ θ ARC 4 n ARC − 4 n b sin ⁡ θ 0 sin ⁡ θ ARC d(theta_(0))=(lambda cos theta_(ARC))/(4n_(ARC)-4n_(b)sin theta_(0)sin theta_(ARC))d\left(\theta_{0}\right)=\frac{\lambda \cos \theta_{\mathrm{ARC}}}{4 n_{\mathrm{ARC}}-4 n_{\mathrm{b}} \sin \theta_{0} \sin \theta_{\mathrm{ARC}}}
  1. 1 1 ^(1){ }^{1} Department of Physics, Hong Kong Baptist University, Hong Kong, China. 2 2 ^(2){ }^{2} Institute of Theoretical Physics and Collaborative Innovation Center of Extreme Optics, Shanxi University, Shanxi, China. 3 3 ^(3){ }^{3} State Key Laboratory of Surface Physics, Key Laboratory of Micro and Nano Photonic Structures, and Department of Physics, Fudan University, Shanghai, China. 4 4 ^(4){ }^{4} Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, China. 5 5 ^(5){ }^{5} Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Hong Kong, China.
    1 1 ^(1){ }^{1} 香港浸会大学物理系，中国香港。 2 2 ^(2){ }^{2} 山西大学理论物理研究所与极限光学协同创新中心，中国山西。 3 3 ^(3){ }^{3} 复旦大学表面物理国家重点实验室、微纳米光子结构重点实验室、物理系，中国上海。 4 4 ^(4){ }^{4} 南京大学先进微结构协同创新中心，中国南京。 5 5 ^(5){ }^{5} 香港浸会大学计算与理论研究所，中国香港。
    *These authors contributed equally to this work.
    *这些作者对这项工作做出了同等贡献。
    †Corresponding author. Email: jacktfng@hkbu.edu.hk
    †通讯作者。电子邮件：jacktfng@hkbu.edu.hk