电湿效应无机械运动变焦光学系统的研究

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摘要

变焦光学系统是照相机、摄像机等许多光学器件中的关键元件，传统变焦光

学系统一般由两片或更多的固定透镜组合而成，通常是利用电机和齿轮等机械装

置来调节透镜间的相对位置来实现变焦。但由于存在机械可动部件，此类镜头变

焦速度慢、价格昂贵、坚固性差，而且不易在诸如内窥镜、拍照手机等便携式微

小型装置中实现。因此研发寿命长、可靠性高且便于微小型化的变焦光学系统具

有非常重要的应用价值和巨大的市场前景。

随着研究的不断深入，多种基于不同机制的微型变焦透镜结构相继被提出，

基于介质上电湿效应的双液体变焦透镜以其突出的性能而越来越受到人们的重

视。这种通过外加电压改变液面曲率来实现变焦的透镜无机械可动部件、结构紧

凑小巧、寿命长、功耗小，具有广阔的应用前景。本文在双液体变焦透镜的基础

上提出新型的无机械运动变焦光学系统，开展的主要研究工作和获得的结论如下：

基于热力学第一定律和热力学第二定律所引出的赫姆霍兹函数，利用赫姆霍

兹自由能最小化的方法推导了适应于圆柱结构和圆锥结构双液体变焦透镜的杨氏

方程，结果显示当圆锥结构双液体变焦透镜的顶角为零时，适应于圆锥结构双液

体变焦透镜的杨氏方程即可过渡到适应于圆柱结构双液体变焦透镜的杨氏方程，

其形式与传统的杨氏方程是相当一致的，都是将三相系统的接触角与三相系统中

的两两相之间的界面张力联系起来，主要的区别在于传统的杨氏方程仅仅适应于

气液固三相系统，而本文推导的杨氏方程将传统的杨氏方程推广到了双液体变焦

透镜这样的液液固三相系统。

基于界面化学所给出液体界面内外压力差与液体界面曲率半径的关系即拉普

拉斯方程，讨论了决定双液体变焦透镜中两液体界面面型的形状方程，并给出了

两种液体密度相同的情况下形状方程的解析解和两种液体密度不同的情况形状方

程的数值解，结果表明当组成双液体变焦透镜的两种液体的密度一样时，两种液

体所形成的界面总是为球面。

设计了四种不同结构的无机械运动变焦光学系统，对这四种结构的无机械运

动变焦系统的变焦以及调焦原理做了详细的说明，基于以上两个方面的理论基础，

深入分析了这四个系统在满足变焦光学系统两个基本要求时其相关参数必须适合

的条件，并给出了相应的关系表达式，在此基础上通过数值逼近的办法计算了每

一个系统在外加电压使其变焦的同时又能保持像面位置不动的情况下系统可以达

到的变焦范围。结果表明前两个无机械运动变焦系统虽然在结构上相对简单，但

它们的变焦范围相对后两个系统来说更窄一些，并且它们在外加电压的作用下所

能达到的焦距值相对较长；而后两种结构的无机械运动变焦光学系统的变焦范围

相对前面两个系统来说要更宽一些，并且系统的焦距值可以根据实际需要通过选

择不同焦距值的固定透镜满足相应的要求。

对双液体变焦透镜进行了一些相关的实验研究，其中包括在制备双液体变焦

透镜时要涉及到的防水涂层、液体材料以及容器尺寸等因素对双液体透镜的影响；

外加电压对蒸镀有透明导电层的平板玻璃上液滴形状的影响，这些相关的实验将

为以后制备性能较好的双液体变焦透镜以及无机械运动变焦系统提供材料选择等

方面的参考。

关键词：表面张力电湿效应接触角变焦透镜机械运动

III

ABSTRACT

A variable-focus optical system is a key element in many optical devices, such as

cameras, video cameras etc. Traditional variable-focus optical systems are usually

composed of two or more fixed lenses, which change the focal length by modulating the

relative position of fixed lenses with mechanical equipments, like motor and gear. Such

mechanical equipments make traditional variable-focus optical systems low response

velocity, very expensive, and fragile. In addition they are also not convenient to be

implemented in small spaces, such as in endoscopes and mobile phone cameras.

Thereby, the development of variable-focus optical systems with long life, high stability

and small volume has very important values of application and great potential of

market.

At present, many kinds of miniature variable-focus lenses are researched one after

the other. The double-liquid variable-focus lens based on electrowetting on dielectrics is

attracting many researchers’ attentions because of its preferable performance. Such a

lens, which varies the focal length by changing the curvature of liquid interface, has no

mechanical elements. The compact volume, long life and low consumption make it wide

prospect of application. This paper provides a new-type of variable-focus optical system

without mechanical movements on the basis of double-liquid variable-focus lenses. The

main research contents and important results for the paper are given as follows.

According to Helmholtz function derived by the first and second law of

thermodynamics, explicit calculations and detailed analyses upon extended Young-type

equations respectively for the cylindrical and conical double-liquid lens are given by a

Helmholtz energy minimization method. When the apex angle of the conical

double-liquid lens is zero, the extended Young-type equation for the conical

double-liquid lens is exactly that for the cylindrical double-liquid lens. The Young-type

equations, which relate the cosine of the equilibrium contact angle and three interfacial

tensions for a liquid-liquid-solid system are similar to the traditional Young equation

that relates the equal parameters for a vapor-liquid-solid system.

A differential equation describing the interface shape of two liquids in

double-liquid variable-focus lenses is set up grounded on the Laplace equation relating

the pressure difference cross a liquid interface to the curvature of the liquid interface.

An analytical solution when two liquids have same density and a numerical solution

when two liquids have different density for the differential equation are deduced in

detail. The results show that the interface shape of two liquids is completely spherical

when the density difference of two liquids is zero.

Four kinds of variable-focus optical systems without mechanical movements are

provided. The structure and principle of these systems are introduced. Based on the

theoretical analyses on double-liquid variable-focus lenses, detailed calculations are

presented to predict how the parameters for each system are related to meet the basic

requirements of zoom lenses. Corrsponding relational expressions are given and solved

so that we can obtain the zoom ratio of each system. For the first two systems, their

structures are relatively simple, but they have smaller zoom range and longer focal

length. For the last two systems, the zoom range can be larger and their focal lengths

can be adjusted by choosing different fixed lens with different focal length.

Some basic experiments related with double-liquid variable-focus lens are

performed. How the hydrophobic layer, liquid material and chamber volume influence

the lens and how the applied voltage changes the shape of a liquid drop on a glass plate

are primarily discussed. Such experiments will supply references to fabricating

double-liquid variable-focus lenses and variable-focus optical system without

mechanical movement with good property.

Key Word：Surface tension, Electrowetting, Contact angle, variable-

focus lens，Mechanical movements

中文摘要

ABSTRACT

第一章绪论.....................................................................................................................1

§1.1 变焦光学系统的发展..........................................................................................1

§1.2 双液体变焦透镜的发展及研究现状..................................................................4

§1.3 研究意义、目的和内容安排..............................................................................5

§1.4 论文的创新点......................................................................................................6

第二章双液体变焦透镜的物理和化学基础.................................................................7

§2.1 热力学理论基础.................................................................................................7

§2.1.1 热力学第一定律..........................................................................................7

§2.1.2 热力学第二定律........................................................................................10

§2.1.3 热力学的基本方程....................................................................................14

§2.2 界面化学理论基础...........................................................................................15

§2.2.1 表面和界面...............................................................................................15

§2.2.2 表面张力和表面能...................................................................................16

§2.2.3 浸润现象与杨氏方程...............................................................................18

§2.2.4 拉普拉斯方程........................................................................................... 19

§2.3 流体静力学基础................................................................................................22

§2.4 本章小结............................................................................................................24

第三章双液体变焦透镜的基本理论分析...................................................................25

§3.1 双液体变焦透镜的结构和原理........................................................................25

§3.1.1 电湿效应....................................................................................................25

§3.1.2 双液体变焦透镜的结构............................................................................26

§3.1.3 双液体变焦透镜的原理............................................................................27

§3.2 双液体变焦透镜液体界面面型的分析............................................................28

§3.2.1 双液体界面面型的形状方程....................................................................28

§3.2.2 液体的密度差为零时双液体界面面型形状方程的解析解....................29

§3.2.3 液体的密度差不为零时双液体界面面型的形状方程及其数值解........31

§3.3 适应于液液固三相系统的杨氏方程................................................................34

§3.3.1 适应于圆柱结构双液体变焦透镜的杨氏方程........................................35

§3.3.2 适应于圆锥结构双液体变焦透镜的杨氏方程........................................39

第四章基于双液体变焦透镜的无机械运动变焦光学系统.......................................43

§4.1 组分为两个双液体变焦透镜的无机械运动变焦系统....................................43

§4.2 两个液体界面在同一圆柱容器中的无机械运动变焦系统............................52

§4.3 固定透镜和圆柱双液体变焦透镜组成的无机械运动变焦系统....................61

§4.4 固定透镜和圆锥双液体变焦透镜组成的无机械运动变焦系统....................71

§4.5 本章小结............................................................................................................80

第五章双液体变焦透镜的相关实验研究...................................................................81

§5.1 防水涂层对双液体透镜的影响........................................................................81

§5.2 液体材料对双液体透镜的影响........................................................................84

§5.3 容器的尺寸对双液体透镜的影响....................................................................85

§5.4 外加电压对平板玻璃片上液滴形状的影响....................................................86

第六章总结与展望.......................................................................................................88

参考文献.........................................................................................................................90

在读期间公开发表的论文和承担科研项目及取得成果.............................................95

致谢.............................................................................................................................96

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摘要：

I摘要变焦光学系统是照相机、摄像机等许多光学器件中的关键元件，传统变焦光学系统一般由两片或更多的固定透镜组合而成，通常是利用电机和齿轮等机械装置来调节透镜间的相对位置来实现变焦。但由于存在机械可动部件，此类镜头变焦速度慢、价格昂贵、坚固性差，而且不易在诸如内窥镜、拍照手机等便携式微小型装置中实现。因此研发寿命长、可靠性高且便于微小型化的变焦光学系统具有非常重要的应用价值和巨大的市场前景。随着研究的不断深入，多种基于不同机制的微型变焦透镜结构相继被提出，基于介质上电湿效应的双液体变焦透镜以其突出的性能而越来越受到人们的重视。这种通过外加电压改变液面曲率来实现变焦的透镜无机械可动部件、结构紧凑小巧...

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