论文总字数：25846字

摘 要

在激光出现之后，人们对激光的热情未曾减少，对它的探索已经出现在了生活与科技的方方面面。为了更高效的利用激光的优点，对激光光场的调控已经成了研究的热点。其中，由于光的偏振在近几年才得到人们的广泛关注，导致人们对偏振光的研究尚未成熟。矢量光场是一种偏振态对随空间分布而变化的光场。与标量光场不同，矢量光场具有更为特殊的性质，在光学微加工与光场的空间衍化方面具有重要的作用。此外，由于激光的光场对于粒子有力的作用，人们便根据此研究出了光镊。与传统的力学镊子不同，光镊由于不会压迫粒子形变从而对粒子产生损伤，以及可以捕获小瑞利粒子而得到了人们的广泛关注，并逐渐应用到光学微操作与光学捕获等方面。本论文围绕着两种求解矢量光场的焦场分布的方法与光力的计算展开理论研究，主要内容如下：

1. 介绍了计算光场的焦场分布的传统方法，理查德沃尔夫矢量衍射理论。重点介绍了其理论推导并说明其物理意义。此外还给出了光场聚焦模型，并且使用该方法计算了径向偏振光的径向分量与纵向分量的分布。
2. 介绍了一种新型的计算光场的分布的方法，快速傅里叶变换方法。重点介绍了其原理，并通过该方法与理查德沃尔夫矢量衍射理论计算同种入射光场的焦场的结果的对比，探究其可靠性，分析误差来源，并介绍该方法的优缺点和应用价值。
3. 介绍了光场中粒子所受力的表达式。探究了光场中粒子所受力的种类，求解其数学表达式。重点介绍了现存的求解光力时使用的物理模型，比较其优缺点与适用范围，分析最适合所例举事例的物理模型。
4. 将快速傅里叶变换与偶极子模型结合，从理论上分析了各种光场的焦场捕获粒子是的光力分布。通过模拟求解了径向偏振光、角向偏振光、圆偏振光焦场对金粒子的力的分布。并且探讨了偏振态、自旋方向等对粒子受力的影响。

关键词：矢量光场，光镊，偏振态，光力

Abstract

After the appearance of laser, people's enthusiasm for laser has not decreased, and its exploration has appeared in all aspects of life and technology. In order to utilize the advantages of laser more efficiently, the regulation of laser light field has become a hot topic and because the polarization of light has only been widely concerned in recent years, the research on polarized light has not been mature. Vector optical field is a kind of optical field whose polarized states changes with the spatial distribution. Unlike scalar optical field, vector optical field has more special properties and plays an important role in optical micro-machining and spatial evolution of optical field. In addition, the optical tweezers have been developed based on the powerful effect of laser light field on particles. Optical tweezers is different with the traditional mechanics, due to optical tweezers do not compress things to deformation and then damage the particles, and can capture small Rayleigh particles, so they have obtained the widespread attention, and gradually applied to the optical micro operations and optical trapping, and so on. We focuses on two methods for solving the focal field distribution of vector optical field and the calculating of optical force. The main contents are as follows:

1. We introduce the traditional method of calculating the focal field distribution of optical field, Richard wolff’s vector diffraction theory. The theoretical derivation and its physical meaning are introduced emphatically.The optical field focusing model is also presented, and the distribution of radial component and longitudinal component of radial polarized light is calculated by using this method.
2. A new method for calculating the distribution of light field, fast Fourier transform, is introduced. We mainly introduced the principle, and through preparing this method with Richard wolff vector diffraction theory to calculate the same incident light field of focal field to verify its reliability, then we analysed of error sources, and introduced the advantages and disadvantages of the method and application value.
3. The expression of the force exerted by particles in a light field is introduced. The types of forces on particles in a light field are investigated and their mathematical expressions are obtained. In this paper, the existing physical model for solving the optical force is introduced, its advantages and disadvantages are compared with its application range, and the physical model that is most suitable for the example is analyzed.
4. Combining the fast Fourier transform with the dipole model, the optical force distribution of the focal field particles captured by various optical fields is theoretically analyzed. The force distribution of radial polarized light, angular polarized light and circular polarized light focal field on gold particles was solved by simulation. The effects of polarization state and spin direction on the force of particles are also discussed.

KEY WORDS: Vector field, Optical tweezers, Optical force, State of Polarization

目 录

摘 要 I

Abstract III

第一章 绪论 1

1.1 光场的基本性质 1

1.2 矢量光场的研究背景及意义 2

1.3 本论文的主要内容 3

第二章 理查德沃尔夫矢量衍射理论 5

2.1 数学描述 5

2.2 径向偏振光紧聚焦特性 10

2.3 角向偏振光紧聚焦特性 13

2.4 本章小结 13

第三章 快速傅里叶变换 14

3.1 Debye近似 14

3.2 数值实现 17

3.2.1 采样条件 17

3.2.2 采样间隔 18

3.3 典型实例 18

3.3.1 1.20NA水浸物镜 19

3.3.2 1.45NA油浸物镜 20

3.4 结论 21

第四章 光力计算 23

4.1 光力的组成 23

4.2 光力模型 24

4.3 应用 25

4.3.1 偏振态对光力的影响 25

致谢 27

参考文献 28

绪论

光学是一门我们既熟悉又陌生的学科。早在公元前400年的时候，人们就已经发现并总结光学现象，并将之应用于生活之中。影现象的纸质记录首次出现在《墨经》中，除此之外，这本书中还通过立竿见影和针孔成像等事例总结出了光的直线传播特性，并且严肃规范的记录了在凹面镜、凸面镜和平面镜中的成像关系。而到了17世纪，人们通过科技的发展研究出了望远镜和显微镜，而这两个光学元件的广泛应用在一定程度上也促进了几何光学的进一步向前发展。之后，斯涅耳和笛卡尔探究发现了光的反射定律和折射定律。紧接着，以牛顿为代表的物理学家提出的光的微粒说和以惠更斯为首的物理学家提出的光的波动说，进行了一场旷日持久的争论。到了19世纪初，波动光学说由于惠更斯-菲涅尔原理的提出完美解释了光的干涉和衍射现象而得以初步形成，随后，菲涅尔又观察到了光的偏振和偏振光的干涉现象，自此人们开始注意到光现象与电磁学现象之间的联系。接着，麦克斯韦理论的提出，使得人们更加深刻的认识到光的物理性质。而20世纪初，爱因斯坦将普朗克的量子理论运用到光电效应之中，完美的解释了运动物体的光学现象。由于人们对光的不断探索，光学学科的发展经历了从几何光学，到波动光学，再到光的量子性的过程，光学进入了一个崭新的时期。

自1958年，汤斯根据稀土晶体在用氖照射时会发出明亮且始终聚拢的强光这一现象提出了激光的概念之后，激光就由于其良好的单色性、方向性、相干性和能量高度集中等特点，广泛的应用于通信、工业、科技以及医疗领域，成了20世纪人类最伟大的发明之一。尽管如此，激光器的潜能还未被挖掘完毕。为了提高激光的性能，更好的使用激光这只利器，人们探索出可以通过调节激光的光场来提高对于激光的控制。

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