第2讲几何光学和眼睛(1).ppt

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1、第一章第一章 几何光学几何光学(参见书第一章内容参见书第一章内容)Geometrical OpticsGeo“几何”徐光启和利玛窦几何原本中译本的一个伟大贡献在于确定了研究图形的这一学科中文名称为“几何”，并确定了几何学中一些基本术语的译名。“几何”的原文是“geometria”，徐光启和利玛窦在翻译时，取“geo”的音为“几何”，而“几何”二字中文原意又有“衡量大小”的意思。用“几何”译“geometria”，音义兼顾，确是神来之笔。几何学中最基本的一些术语，如点、线、直线、平行线、角、三角形和四边形等中文译名，都是这个译本定下来的。这些译名一直流传到今天，且东渡日本等国，影响深远。（徐光启

2、(15621633)，字子先，上海吴淞人。上海人h一般不发音，所以geo里虽然没有h也给加上了，类似的还有：Neon=霓虹灯）Ray Optics BasicsPostulates(基本假设基本假设)of Ray OpticsLight travels in form of raysAn optical medium is characterized by the refractive index ntime it takes light to travel distance d:d/c=nd/cvacoptical path:ndAn inhomogeneous optical medium

3、:n(r)r=(x,y,z)time taken by the light to travel from A to B is proportional to optical path length Fermats Principle:optical rays travel between two points A,B on the path where the optical path length is an extremumin most cases a minimumlight travels the path of shortest timehomogeneous medium:lig

4、ht travels on straight linesOptical path length and time delay 光程与时差光程与时差QP的传播时间：QPMN介质中光速用真空光速和折射率代替：光程的另一角度认识 光经历Q和P两点的光程等于传播时间乘以真空光速。Optical path descriptionSuppose we have known Snells law already,in a stepped index stack,we haveNotice thatorWe deriveOptical path equationThe second derivative wi

5、ll be:From the equation,we may derive the optical path trace其中利用了path equationExample:gradient index fiberFor the index of a fiber as(in radial direction 径向)In the case of Substituting in to the path equation,we haveVery much like a harmonic oscillator equation!Then the trace will be like a wave as

6、a function of x!Solution Oscillation periodncnfSolutionDependent on the angleFor small incident angle Oscillationperiod is approximated as Gradient indexNatural index gradientIndex above the sea The higher is the aptitudes,the smaller the index isHowever in the continent,if the surface temperature i

7、s very high,the index distribution may be inversedThe higher the index is larger!Mirage 海市蜃楼海市蜃楼海市蜃楼海市蜃楼Gradient indexArtificial gradient indexGradient index fiberIntense laser resulted index gradientSelf-focusing,Self-defocusingThermal lensingGeometric optics of rainbowGeometric optics of rainbow费马

8、原理费马原理 Fermat principle*原理表述*数学表达*导出反射定律、折射定律*费马原理与成像*费马原理的评述 光程的概念对几何光学的重要意义体现在费马原理中。几何光学的基础是前面所提到三个实验定律，费马却用光程的概念高度概括地把它们归结成一个统一的原理。17世纪的一位法国数学家，提出了一个数学难题，使得后来的数学家一筹莫展，这个人就是费马(16011665)。这道题是这样的：当n2时，xn+yn=zn没有正整数解。在数学上这称为“费马大定理”。为了获得它的一个肯定的或者否定的证明，历史上几次悬赏征求答案，一代又一代最优秀的数学家都曾研究过，但是300多年过去了，至今既未获得最终证

9、明，也未被推翻。即使用现代的电子计算机也只能证明：当n小于等于4100万时，费马大定理是正确的。由于当时费马声称他已解决了这个问题，但是他没有公布结果，于是留下数学难题中少有的千古之谜。费马生于法国南部，在大学里学的是法律，以后以律师为职业，并被推举为议员。费马的业余时间全用来读书，哲学、文学、历史、法律样样都读。30岁时迷恋上数学，直到他64岁病逝，一生中有许多伟大的发现。不过，他极少公开发表论文、著作，主要通过与友人通信透露他的思想。在他死后，由儿子通过整理他的笔记和批注挖掘他的思想。好在费马有个“不动笔墨不读书”的习惯，凡是他读过的书，都有他的圈圈点点，勾勾画画，页边还有他的评论。他利用

10、公务之余钻研数学，并且成果累累。后世数学家从他的诸多猜想和大胆创造中受益非浅，赞誉他为“业余数学家之王”。费马对数学的贡献包括：与笛卡尔共同创立了解析几何；创造了作曲线切线的方法，被微积分发明人之一牛顿奉为微积分的思想先驱；通过提出有价值的猜想，指明了关于整数的理论数论的发展方向。他还研究了掷骰子赌博的输赢规律，从而成为古典概率论的奠基人之一。QPDescription 表述：平稳值的三种基本的含义：极小值常见情况，常数成像系统的物像关系极大值个别现象光线沿光程为平稳值光线沿光程为平稳值(stationary)的路径而传播的路径而传播(书上书上)。Optical rays travel bet

11、ween two points A,B on the path where the optical path length is an extremum,in most cases a minimumlight travels the path of shortest time（英语表达）英语表达）SSAny difference?费马原理的数学表达式 mathematical expression路径积分 integration of optical path length：是路径（l）的函数，平稳值要求变分为零路径（l）的值就有极大、极小和常数这三种情况。极大：？极小：折射反射定律常数：成

12、像变分值为极大和常数情况 maximum or constantABD(切点)DADB ADBBut ADB violent the reflection lawLight takes the longest path!极大:常数:费马原理与三个实验定律费马原理与三个实验定律:experiment verification1、光在均匀介质中直线传播(直线距离最短)2、反射定律（反射角入射角）3、折射定律（life-guard problem）QQMMP要点：反射光线在入射面，反射角等于入射角，光程最短。M”(1)救生员应该在最短的时间内到达被救者(2)在海滩上和在海水里奔跑速度不一样，因此要选

13、择最佳入水地点(3)计算路径（时间）并变分为零，得到折射定律！3、折射定律 Life guard problem 救生员问题救生员问题(1)折射光线在入射面内，方法和反射定律推导一样。(2)入射角和折射角的关系；QMP的光程：根据费马原理，L对x的一阶导数为零；3、费马原理推导折射定律（contd)费马原理推导折射定律（contd)导出折射定律：因为：费马原理与成像费马原理与成像等光程与成像等光程与成像：其中：严格等光程 严格成像，近似等光程 近似成像，非等光程 不成像，费马原理应用例题-导出单球面折射傍轴成像公式等光程：傍轴条件：单球面，界面两单球面，界面两边折射率不同边折射率不同不是半径代

14、入等光程：值得指出的两点：1、单球面傍轴成像公式的获得没有使用折射定律，结果和利用折射定律所得结果一致。2、傍轴条件：0,the lens deflects rays toward the axis.f 0If f 0,the lens deflects rays away from the axis.f 0R2 0R1 0A lens focuses parallel rays to a point one focal length away.ffAt the focal plane,all rays converge to the z axis(xout=0)independent of

15、input position.Parallel rays at a different angle focus at a different xout.A lens followed by propagation by one focal length:Assume all input rays have qin=0For all rays xout=0!Types of lenses 典型透镜典型透镜Lens nomenclature 透镜术语Which type of lens to use(and how to orient it)depends on the aberrations a

16、nd application.双凸透镜 平凸透镜 凹凸透镜双凹透镜平凹透镜半月型透镜Ray matrix for a curved Mirror 反射镜矩阵反射镜矩阵Like a lens,a curved mirror will focus a beam.Its focal length is R/2.Note that a flat mirror has R=and hence an identity ray matrix.Consider a mirror with radius of curvature,R,with its optic axis perpendicular to th

17、e mirror:qinqoutxin=xoutq1qsRzq1Miscellaneous matrices 矩阵元总结矩阵元总结空间传播矩阵空间传播矩阵透镜变换矩阵透镜变换矩阵反射镜镜变换矩阵反射镜镜变换矩阵曲面变换矩阵曲面变换矩阵平面界面变换矩阵平面界面变换矩阵Consecutive lenses透镜组合透镜组合f1f2Suppose we have two lenses right next to each other(with no space in between).So two consecutive lenses act as one whose focal length is

18、computed by the“resistive sum.”（类似电阻并联定律）As a result,we define a measure of inverse lens focal length,the diopter(屈光度）or the power 1 diopter=1 m-1IfA system images an object when B=0.成像条件：矩阵元成像条件：矩阵元 B=0When B=0,all rays from a point xin arrive at a point xout,independent of angle.xout=A xinWhen B=0

19、,A is the magnification.与入射角无关The Lens Law透镜成像定律透镜成像定律From the object to the image,we have:1)A distance do2)A lens of focal length f3)A distance di1/M根据成像条件，令B0像距物距透镜The Lens Law(contd)透镜成像定律透镜成像定律whereMore on matrix optics:thick lens and principle planePrimary principle plane 第一主平面Secondary princip

20、le plane 第二主平面V1H1V2H2d2n1n2n2例题：例题：用矩阵法求解直径27mm,折射率1.54的玻璃球的焦距和主平面位置已知：n1=n3=1,n2=1.54,r1=13.5,r2=-13.5,d2=27利用：n2Homework 0909241.Wandas world Your goldfish Wanda lives in a sphere of water(refractive index n=1.3,radius R=-20cm).At one instance,Wanda has wandered to the center of here water world(

21、see Fig.1 below).Model Wanda as a stick perpendicular to the optical axis and the water sphere as a thick lens.You may ignore the effect of the glass container of Wandas worlda)Where is Wandas image formed?b)Is the image real or virtual?c)Is the image erect or inverted?d)What is the magnification?Rw

22、aterairWandaFig.1Homework 090924(Contd)2.A parallel ray bundle of width a1 is incident from the left on a two-lens system composed of two lenses L1(focal length f1)and L2(focal length f2)as shown in Figure 2.What should the separation between the two lenses be in order for a parallel ray bundle to e

23、merge from the system?What is the width of this outgoing ray bundle?L1L2f2f1Fig.2a1？Homework 090924(Contd)3.Work out the system matrix for the composite element shown in Figure 3 and use it to answer the following questions.a)What is the optical power of this composite element?b)If a plane wave is i

24、ncident from the left,where will it focus?c)This system is used to image an object at infinity.Is the image real or virtual?Fig.3Homework 090924(Contd)4.A thin bi-convex lens of index 1.5 is know to have focal length of 50cm in air when immersed in a transparent liquid medium,the focal length is mea

25、sured to be 250cm.What is the refractive index n of the liquid?5.Youd like to look through a lens at your pet Kitter and see it standing right side up shrunk to 1/3 its normal height.If the absolute value of the focal length is f,determine what kind of lens is needed(i.e.positive or negative)as well

26、 as the object and image distances in term of f.Mirror curvatures matter in lasers.Laser Cavities 激光谐振腔激光谐振腔Two flat mirrors,the“flat-flat”laser cavity,is difficult to align and maintain aligned.Two concave curved mirrors,the“stable”laser cavity,is easy to align and maintain aligned.Two convex mirro

27、rs,the“unstable”laser cavity,is impossible to align!An unstable cavity(or“resonator”)can work if you do it properly!In fact,it produces a large beam,useful for high-power lasers,which must have large beams.Unstable Resonators 非稳定谐振腔非稳定谐振腔The mirror curvatures determine the beam size,which,for a stable resonator,is small(100 mm to 1 mm).An unstable resonator can have a very large beam.But the gain must be high.And the beam has a hole in it.