optical tweezers.doc

我们在几何光学模型下计算其在微米级的介质球上的作用力大小，也被称为“光镊”。这可作为描述激光阱作用于活体细胞及其细胞器的操作系统的简单模型。梯度力和散射力被定义为在几何光学条件下的多光束复合形态。计算使用输入强度分布为TEM00和TEM01模和小球整个截面折射率各异的受力情况。强大的均匀势阱可能与力的变量因子小于2球体横截面有关。对于10MW功率的激光和1.2相对折射率计算其中梯度阱最弱的方向俘获力量可达1.2*10 -6达因dyne（向后）。结果表明，良好的俘获要求高收敛光束来自高数值孔径的目标。给出一个对照，使用明场或差分干涉光对比光学相衬光学阱单光束梯度辐射压力光阱。INTRODUCTIONThis paper gives a detailed description of the trapping of micron-sized dielectric spheres can serve as first simple models of living cells in biological trapping experiments and also as basic particles in physical trapping experiments. Optical trapping of small particles by the forces of laser radiation pressure has been used for about 20 years in the physical sciences for the manipulation and study of micron and submicron dielectric particles and even individual atoms. These techniques have also been extended more recently to biological particles.本文给出了一种微米大小的介质球俘获的详细描述，可以作为活细胞的第一个简单的模型生物俘获实验和基本粒子物理俘获实验。利用激光辐射压的作用力对微小粒子的光学俘获，已经被应用于微米和亚微米介质颗粒甚至单个原子的操纵和研究等物理学。这些技术最近也扩展得到了生物粒子。The basic forces of radiation pressure acting on dielectric particles and atoms are known. For dielectric spheres large compared with the wavelength, one is in the geometric optics regime and can thus use simple ray optics in the derivation of the radiation pressure force from the scattering of incident light momentum. This approach was used to calculate the forces for the original trapping experiments on micron-sized dielectric spheres. These early traps were either all optical two-beam traps or single beam levitation traps which required gravity or electrostatic forces for their stability. For particles in the Rayleigh regime where the size is much less than the wavelength the particle acts as a simple dipole. The forces on a dipole divides itself naturally into two component pointing in the direction of the intensity gradient of the light.辐射压作用于介质粒子和原子的基本作用力，是已知的。对于介质球尺寸大于波长，是在几何光学模型下，因此可以使用简单的射线光学，从入射光动量的散射辐射压力的推导。这种近似常用来计算原微米大小的介质球俘获实验的作用力。这些早期的势阱，无论所有的光学双光束势阱或单光束浮置势阱，不考虑重力或静电引力对于其稳定性的影响。对于颗粒大小比波长在瑞利制度作为一个简单的偶极子的粒子行为。偶极子的力量本身的自然划分成两个组件指向在光的强度梯度方向。The single-beam gradient trap, sometimes referred to as “optical tweezers,” was originally designed for Rayleigh particles. It consists of a single strongly focused laser beam. Conceptually and practically it is one of the simplest laser traps. Its stability in the Rayleigh regime is the result of the gradient force pulling particles toward the high focus of the beam over the scattering force trying to push particles away from the focus in the direction of the incident light. Subsequently it was found experimentally that single-beam gradient traps could also trap and manipulate micron-sized and a variety of biological particles, including living cells and organelles within living cells. Best results were obtained using infrared trapping beams to reduced optical damage. The trap in these biological applications was built into a standard high resolution microscope in which one uses the same high numerical aperture (NA) microscope objective for both trapping and viewing. The micro-manipulative abilities of single-beam gradient traps are finding use in a variety of experiments in the biological sciences. Experiments have been performed in the trapping of viruses and bacteria; the manipulation of yeast cells, blood cells, protozoa, and various algae and plant cells; the measurement of the compliance of bacterial flagella; internal cell surgery; manipulation of chromosomes; trapping and force measurement on sperm cells; and recently, observations on the force of motor molecules driving mitochondrion and latex spheres along microtubules. Optical techniques have also been used for cell sorting.单光束梯度阱，有时也被称为“光镊”，最初是设计用于瑞利粒子。它由一个单一的高度聚焦的激光束。在理论上和实践上，它是最简单的激光阱之一。一方面，梯度力拉动粒子朝光束聚焦中心移动，另一方面，散射力又试图推动粒子远离入射光聚焦中心，这样最终导致粒子在瑞利模型下达到一种平衡。随后实验发现，单光束梯度阱也可以捕获和操纵微米大小的各种各样的生物粒子，包括活体细胞和活细胞内细胞器。最好的结果是，使用红外俘获光束，以减少光损伤。这些生物学上应用的阱，需要一个标准的高分辨率显微镜，其中有一个用来俘获和观察的高数值孔径（NA）的物镜。单光束梯度阱的微操控能力现已被用于各种生物科学实验中。实验已经进行俘获病毒和细菌，酵母细胞，血细胞，原生动物以及各种藻类和植物细胞；内部细胞手术；细菌鞭毛的可塑性测量；染色体的操纵；俘获精子细胞及其力的测量；最近，观察电机分子受力推动管线粒体和乳胶沿微移动。光学技术也已用于细胞分类。Qualitative descriptions of the operation of the single-beam gradient trap in the ray optics regime have already been given. In Fig. 1 taken from reference 26, the action of the trap on a dielectric sphere is described in terms of the converging beam, under the simplifying assumption of zero surface reflection. In this approximation the forces Fa and Fb are entirely due to refraction and are shown pointing in the direction of the momentum change. One sees that for arbitrary displacements of the sphere origin 0 from the focus f that the vector sum of Fa and Fb gives a net restoring force F directed back to the focus, and the trap is stable. In this paper we quantify the above qualitative picture of trap. We show how to define the gradient and scattering force on a sphere in a natural way for beams of arbitrary shape. One can then describe trapping in the ray optics regime in the same terms as in the Rayleigh regime. Result are given for the trapping forces over the entire cross-section of the sphere. The forces are calculated for input beams with various TEM00 and TEM*01 mode intensity profiles at the input aperture of high numerical aperture trapping objective of NA =1.25. The results confirm the qualitative observation that good trapping beam with high convergence angle. One can design traps in which the trapping forces vary at most by a factor of 1.8 over the cross-section of the sphere with trapping forces as high as Q=0.30 where the force F is given in terms of the dimensionless factor Q in the expression F=Q (n1p/c). P is the incident power and n1P/c is the incident momentum per second in a medium of index of refraction n1. There has been a previous calculation of single-beam gradient trapping forces on spheres in the geometrical optics limit by Wright et al. , over a limited portion of the sphere, which gives much poorer results. They find trapping forces of Q=0.055 in the above units which vary over the sphere cross-section by more than an order of magnitude.已获得单光束梯度阱在几何光学模型下的作用的定性描述。如图1 【参考26】，在表面零反射的简化假设下，电介质领域的势阱作用是描述会聚光束。在这种近似下，Fa和Fb的力完全是由于折射产生并指向动量变化的方向。可以看到，对于从原点O到焦点f的任意位移，Fa和Fb的矢量和给出了一个指向焦点的合回复力，这个势阱可稳定存在。在本文中，我们量化上述阱的定性图像。对于任意形状的光束，我们展示如何以自然而然的方式定义在尺寸远大于波长的小球上的梯度力和散射力。然后，在相同条件下，可以像瑞利模型一样描绘在射线光学模型下的俘获过程。整个球体截面的俘获力已经给出。在输入光束强度分布如TEM00和TEM*01模式下，和输入孔径高达1.25的情况下计算俘获目标的作用力，结果证实定性观察，效果较好的俘获应具有较好的会聚角。可以设计势阱俘获力大多数不同的因素在俘获力为Q领域的截面1.8= 0.30是无量纲因子Q在力F表达式f= Q（n1p/ C）。P是入射功率，n1P/c为折射率n1的介质每秒入射的动量。Wright等人曾做过在几何光学极限下小球有限部分上的单光束梯度势阱力，所得到的结果差很多。他们发现，在上述情况下，截面上Q=0.055的俘获力的相差超过一个数量级。LIGHT FORCES IN THE RAY OPTICS REGIMEIn the ray optics or geometrical optics regime one decomposes the total light beam into individual rays, each with appropriate intensity, direction, and state of polarization, which propagate in straight lines in media of uniform refractive index. Each ray has the characteristics of a plane wave of zero wavelength which can change directions when it reflects, refracts, and changes polarization at dielectric interfaces according to the usual Fresnel formulas. In this regime diffractive effects are neglected (see Chapter 3 of reference 28).在射线光学或几何光学的模型下，一束总的光束分解成独立的光束，每束在均匀折射率介质中直线传播的光，具有相应的强度、方向、偏振态。按照菲涅尔公式，当它在介质表面发生反射，折射，及改变偏振态时，每个入射线表现出可以改变方向零波长平面波的特征。在这种情况下衍射效应可以忽略（见参考第3章28）。The simple ray optics model of the single-beam gradient trap used here for calculating the trapping forces on a sphere of diameter is illustrated in Fig.2. The trap consists of an incident parallel beam of arbitrary mode structure and polarization which enters a high NA microscope objective and is focused ray-by-ray to dimensionless focal point f. Fig.2 shows the case where f is located along the Z axis of the sphere. The maximum convergence angle for rays at the edge of the input aperture of a high NA objective lens such as the Leitz PLAPO 1.25W (E. Leitze, Inc., Wetzlar, Germany) or the Zeiss PLAN NEOFLUAR 63/1.2W water immersion objectives (Carl Zeiss, Inc., Thornwood, NY), for example, is max=70. Computation of the total force on the sphere consists of summing the contributions of each beam ray entering the aperture at radius r with respect to the beam axis and angle with respect the Y axis. The effect of neglecting the finite size of the actual beam focus, which can approach the limit of /2n1 (see reference 29), is negligible for spheres much larger than . The point focus description of the convergent beam in which the ray directions and momentum continue in straight lines through the focus gives the correct incident polarization and momentum for each ray. The rays then reflect and refract at the surface of the sphere giving rise to the light forces.单光束梯度阱的简单射线光学模型在这里用于计算一个球体直径远大于时的俘获力量，如图2显示。这个阱包括一个以任意模结构和偏振态的平行光射入高数值孔径的显微物镜并全部聚焦于无穷近处的焦点f。图2表明其中f是位于小球的Z轴上。像莱茨PLAPO1.25W（E. 莱茨，德国-韦茨拉尔公司）或蔡司PLAN NEOFLUAR 63/1.2W水浸物镜（卡尔蔡司公司，Thornwood，纽约州）这样的高数值孔径物镜镜头，位于其入射孔径边沿的光束的最大收敛角，比如max=70。对于小球所受到的总作用力的计算，主要考虑在进入光圈在光轴周围半径为r内并与Y轴夹角的每束光的作用力的合力。对于小球尺寸远大于波长，可以忽略光束聚焦的大小的限制，这就使得可以趋近于/2n1的衍射极限（见参考29）。汇聚光在通过焦点时方向及动量保持不变，这种点聚焦的表述对每束光的入射偏振态和动量也是正确的。光线在小球表面的折射和反射也导致光力的改变。The model of Wright et al. (27) tries to describe the single-beam gradient trap in terms of both wave and ray optics. It uses the TEM00 Gaussian mode beam propagation formula to describe the focused trapping beam and takes the ray directions of the individual rays to be perpendicular to the Gaussian beam phase fronts. Since the curvature of the phase fronts vary considerably along the beam, the ray directions also change, from values as high as 30or more with respect to the beam axis in the far-field, to 0at the beam focus. This is physically incorrect. It implies that rays can change their direction in a uniform medium, which is contrary to geometrical optics. It also implies that the momentum of the beam can change in a uniform medium without interacting with a material object, which violates the conservation of light momentum. The constancy of the light momentum and ray direction for a Gaussian beam can be seen in another way. If one resolves a Gaussian beam into an equivalent angular distribution of plane waves (see Section 11.4.2 of reference 28) one sees that these plane waves can propagation with no momentum or direction changes right through the focus. Another important point is that the Gaussian beam propagation formula is strictly correct only for transversely polarized beams in the limit of small far-field diffraction angles , where =/0 (0 being the focus spot radius). This formula therefore provides a poor description of high convergence beams used in good traps. The proper wave description of a highly convergent beam is much more complex than the Gaussian beam formula. It involves strong axial electric field components at the focus (from the edge rays) and requires use of the vector wave equation as opposed to the scalar wave equation used for Gaussian beams.赖特等人的模型（27）致力于用波动光学和射线光学来描述单光束梯度阱。它使用的TEM00模高斯光束传输公式来描述汇聚俘获光和单光束的方向垂直于高斯光束相位波前。由于相位波前的曲率沿着光束变化很大，光束方向也随之变化，从值在远场光轴处高达30以上，到光束焦点处的0。这是从物理上就不正确。这意味着，在一个相同的介质中，光线可以改变自己的方向，这违背几何光学的。这也意味着，光束的动量，可以在均匀介质发生改变而不与物质发生作用，这违反了光动量保护原理。高斯光束的光动量及方向的恒定可以用另外的方式去看待。如果能讲高斯光分解成角分布等价的平面波（见参考28，11.4.2节），那么就可以知道这些平面波可以无动量、方向改变地通过焦点。另外很重要的一点，高斯光束传播公式，严格的定义在小远场衍射角内才是正确的，其中=/0（0为焦点半径）。因此，这个公式对高收敛光束在好的势阱中的使用的描述不合适。适当的高度会聚束波介绍高斯光束公式复杂得多。它涉及到焦点（从边缘射线）强大的轴向电场分量，要求对高斯光束使用矢量波方程而不是标量波动方程的。 Apart from the major differences near the focus, the model of Wright et al. (27) should be fairly close to the ray optics model used here in the far-field of the trapping beam. The principal distinction between the two calculations, however, is the use by Wright et al. of beams with relatively small convergence angle. They calculate forces for beams with spot sizes W0=0.5, 0.6, and 0.7m, which implies values of of 29, 24, and 21,respectively. Therefore, these are beams having relatively small convergence angles compared with convergence angles of max=70which are available from a high NA objective.除了重大分歧的焦点附近，赖特等人的模型（27）应相当接近射线光学模型用于诱捕远场光束。然而，两种计算方法之间的主要区别是由Wright等的使用。光束衔接的角度比较小。他们计算梁的力量与光斑尺寸W0= 0.5，0.6，0.7m，这意味着29日，24日和21，分别的值。因此，这些都是相对较小的衔接角度的光束相max衔接的角度=70，这是从高NA的目标。Consider first the force due to a single ray of power P hitting a dielectric sphere at an angle of incidence with incident momentum per second of n1P/c (see Fig. 3). The total force on the sphere is the sum of contributions due to the reflected ray of power PR and the infinite number of emergent refracted rays of successively decreasing power PT2, PT2R, PT2Rn, . The quantities R and T are the Fresnel reflection and transmission coefficients of the surface at . The net force acting through the origin O can be broken into Fz and Fy components as given by Roosen and co-workers (3, 22)(see Appendix for a sketch of the derivation). Where and r are the angles of incidence and refraction. These formulas sum over all scattered rays and are therefore exact. The forces are polarization dependent since R and T are different for rays polarized perpendicular or parallel to the plane of incidence.首先考虑由于功率P单光束以入射角与每秒入射击中一个介质球的力，产生的动量n1P/ c（见图3）。小球上的总力是由功率为PR反射光和无数次功率陆续减少PT2, PT2R, PT2Rn自然折射光作用的合成而来的。数值R和T为呈角的表面菲涅尔反射和透射系数。Roosen和同事给出通过原点O的合力可以分解成Fz和Fy分量（322）（见附录派生的草图）。这里和r入射及反射角。这个公式只有在加上所有散射光之后才是准确的。对于垂直于入射平面和平行于入射平面的光线，其与偏振方向相关的R和T也不一样。In Eq.1 we denote the Fz component pointing in the direction of the incident ray as the scattering force component Fs for this single ray. Similarly, in Eq.2 we denote the Fy component pointing in the direction perpendicular to the ray as the gradient force component Fg for the ray. For beams of complex shape such as the highly convergent beams used in the single-beam gradient trap, we define the scattering and gradient forces of the beam as the vector sums of the scattering and gradient force contributions of the individual rays of the beam. Fig.2 B depicts the direction of the scattering force component and gradient force component of a single ray of the convergent beam striking the sphere at angle. One can show that the gradient force as defined above is conservative. This follows from the fact that Fg, the gradient force for a ray, can be expressed solely as a function of , the radial distance from the ray to the particle. This implies that the integral of the work done on a particle in going around an arbitrary closed path can be expressed as an integral of Fg()d which is clearly zero. If the gradient force for an arbitrary collection of rays is conservative. Thus the conservative property of the gradient force as defined in the geometric optics regime is the same as in the Rayleigh regime. The work done by the scattering force, however, is always path dependent and is not conservative in any regime. As will be seen, these new definitions of gradient and scattering force for beams of more complex shape allow us to describe the operation of the gradient trap in the same manner in both the geometrical optics and Rayleigh regimes.在等式1中，我们用指向入射光线方向的Fz表示单光束散射力分量Fs。同样，在等式2中，我们用与入射光线方向垂直的Fz分量表示单光束梯度力分量Fs。对于复合型光束，如单光束梯度阱中使用的高度收敛的光束，我们定义光束的散射和梯度力为的各单光束的散射力和梯度力的矢量和。图2B所示单收敛光以角入射小球后的梯度力及散射力分量的方向。可以证明上述定义的梯度力是保守的。光梯度力Fg，可以由从光束到小球的距离的函数唯一得表出，由此得出上述结论。这意味着，在粒子周围所做的功的积分可以表示成Fg（）d的的任意封闭路径上的积分，这显然是零。如果梯度力对于任意汇聚光是恒定的，那么，在几何光学模型下定义的梯度力守恒与在瑞利模型下是一致的。然而，散射力所做的功与路径有关，在任何模型下都不恒定。我们将会看到，在更复杂的光束下的关于梯度力和散射力的新定义，使我们在几何光学模型和瑞利模型下用同样地方法描述梯度力的作用。To get a feeling for the magnitudes of the forces, we calculate the scattering force Fs, the gradient force Fg, and the absolute magnitude of the total force Fmag=(Fs2+Fg2)1/2 as a function of the angle of incidence using Eqs.1 and 2. We consider as a typical example the case of a circularly polarized ray hitting a sphere of effective index of refraction n=1.2. The force for such a circularly polarized ray is the average of the forces for rays polarized perpendicular and parallel to the plane of incidence. The effective index of a particle is defined as the index of particle n2 divided by the index of the surrounding medium n1; that is, n=n2/n1. A polystyrene sphere in water has n=1.6/1.33=1.2. Fig.4 shows the results for the forces Fs, Fg, and Fmag versus expressed in terms of the dimensionless factors Qs, Qg, and Qmag=(Qs2+Qg2)1/2, where F=Q*n1P/c . The quantity n1P/c is the incident momentum per second of a ray of power P in a medium of index of refraction n1(19, 31). Recall that the maximum radiation pressure force derivable from a ray of momentum per second n1P/c corresponds to Q=2 for the case of a ray reflected perpendicularly from a totally reflecting mirror. One sees that for n=1.2 a maximum gradient force of Qmax as high as 0.5 is generated for rays at angles of gmax. The corresponding value of scattering force Qs at gmax is also listed. The fact that Qs continues to grow relative to Qgmax as n increases indicates potential difficulties in achieving good gradient traps at high n.要得到作用力的定量认识，我们需要根据等式1、2计算的散射力Fs、梯度力Fg、总作用力的绝对值Fmag=（FS 2+ FG 2）1 / 2对于的入射角的函数。我们以圆偏振光入射有效折射率n为1.2的小球为例，这样一个圆偏振光的力是偏振方向垂直于入射面和平行于入射面的光的作用力的平均。一个粒子的有效折射率是指粒子折射率n2除以指数周围介质折射率n1;，即n = n2/n1。在水中的聚苯乙烯小球n=1.6/1.33=1.2。图4表示与力Fs、Fg和Fmag对应的无量纲因子Qs、Qg及Qmag与的关系曲线，其中，F= Q* n1P/ C。数值n1P/ C是功率为P的光在折射指数为n1的介质上每秒产生的动量（19，31）。通过对于光在全反射镜上的垂直反射，推导对于Q=2时，每秒n1P/c光动量的最大辐射光压。可以看到，对于n=1.2，入射角gmax使得Qmax高达0.5的光产生最大梯度力。还列出了相应的散射力Qs在gmax的值。事实上，随着n的增加Qs值相对Qgmax继续增大，表明在高n值时，实现良好的梯度阱存在潜在的困难。FORCE OF THE GRADIENT TRAP ON SPHERESTrap focus along Z axisConsider the computation of the force of a gradient trap on a sphere when the focus f of the trapping beam is located along the Z axis at a distance S above the center of the sphere at O, as shown in Fig.2. The total force on the sphere, for an axially-symmetric plane-polarized input trapping beam, is clearly independent of the direction of polarization by symmetry considerations. It can therefore be assumed for conv