Geometrical Optics

1、Part One Geometrical Optics1 Properties of Light1.1 The Rectilinear Propagation of Light直线的直线的If the image screen is moved closer to the pinhole screen, the image will be proportionately smaller, whereas if it is moved farther away, the image will be proportionately larger. Excellent sharp photograp

2、hs of stationary objects can be made with this arrangement.Photograph taken with a pinhole camera. Plate distance 9.5cm; Panchromatic film; exposure 3.0min; square hole=0.33mm.1.2 The Speed of LightThe speed of light is finite and that it has an approximate value of8300,000/3 10/km sm s The ancient

3、astronomers believed that light traveled with an infinite speed.-Galileos experiment.Experimental arrangement described by the French physicist Fizeau, with which he determined the speed of light in air in 1849.720 teethElectromagnetic waves of all wavelengths, from X rays at one end of the spectrum

4、 to the longest radio waves, are believed to travel with exactly the same speed in a vacuum.The most generally accepted value of this universal constant is82.997925 10/m s1.3 The Speed of Light in Stationary MatterFoucaults apparatus for determining the speed of light in water.Over 40 years later th

5、e American physicist Michelson measured the speed of light in air and water.For water he found the value of 225,000km/s, which is just three-fourths the speed in a vacuum.The speed of light in air at normal temperature and pressure is about 87km/s less than in a vacuum, or82.99706 10/m sThe index of

6、 refraction, or refractive index, of any optical medium is defined as the ratio between the speed of light in a vacuum and the speed of light in the medium:1.4 The Refractive Index （折射率折射率）cn refractive indexspeed in vacuumspeed in mediumUsing the speeds given before, we obtain the following values

7、for the refractive indices:1.520n For glass:1.333n For water:1.000n For air:Accurate determination of the refractive index of air at standard temperature (00C) and pressure (760mmHg) give1.000292n The refractive indices of the most commonly used optical glasses range from 1.52 to 1.72.The optical de

8、nsity of any transparent medium is a measure of its refractive index. A medium with a relatively high refractive index is said to have a high optical density, while one with a low index is said to have a low optical density.透明的透明的Also, if n = 0 = 0 i.e. light stops in its track ! See:P. Ball, Nature

9、, January 8, 2002D. Philips et al. Nature 409409, 490-493 (2001)C. Liu et al. Physical Review Letters 8888, 23602 (2002)1.5 Optical Path （光程光程）The path d of a ray of light in any medium is given by the product velocity times time:cdttnorndctThe product nd is called the optical path:nd The optical pa

10、th represents the distance light travels in a vacuum in the same time it travels a distance d in the medium.If a light ray travels through a series of optical media of thickness d, d,d,and refractive indices n, n, n, the total optical path is just the sum of the separate values:ndn dn d The optical

11、path through a series of optical media.Optical path lengthn1n4n2n5nmn3SPOptical path length Transit time from S to Pmiiisnct11miiisnOPL1PSdssnOPL)(PSdsvcOPLSame for all rays1.6 Laws of Reflection and RefractionWhenever a ray of light is incident on the boundary separating two different media, part o

12、f the ray is reflected back into the first medium and the remainder is refracted (bent in its path) as it enters the second medium. The directions taken by these rays can best be described by two well-established laws of nature.反射和折射定律反射和折射定律入射入射边界边界Reflection and refraction at the boundary separati

13、ng two media with refractive indices n and n,respectively.The law of reflection:1)The incident ray, the normal, and the reflected ray all lie in the same plane (the plane of incidence), which is perpendicular to the interface separating the two media.2)The angle at which the incident ray strikes the

14、 interface is exactly equal to the angle the reflected ray makes with the same interface.Angle of incidence = angle of reflection分界面分界面法线法线Heros principle and reflectionABAORO Hero (150BC-250AD) asserted that the path taken by light in going from some point A to a point B via a reflecting surface is

15、 the shortest possible oneThe law of refraction:1)The refracted ray lies in the plane of incidence and on the opposite side of the normal.2)The sine of the angle of incidence and the sine of the angle of refraction bear a constant ratio one to the other, for all angles of incidence:constsinsin对边对边正弦

16、正弦In addition, the constant is found to have exactly the ratio of the refractive indices of the two media n and n . Hence we can writewhich can be written in the symmetrical formsinsinnnsinsinnn对称的对称的The ratio n/n is often called the relative index and Snells law can be writtenWhen the angles of inc

17、idence and refraction are very small, a good approximation is obtained by setting the sines of angles equal to the angles themselves, obtainingsinsinrnn n1.7 Graphical Construction for RefractionThe angle of deviationTo prove that this construction follows Snells law exactly, we apply the law of sin

18、es to the triangle ORP:Since sin()sin ,ORn OPnsubstitution gives directly）sin(sinOPORsinsinnn代入，置换代入，置换1.8 The Principle of ReversibilityIf a reflected or refracted ray is reversed in direction, it will retrace its original path.Since reversibility holds at each reflecting and refracting surface, it

19、 holds also for even the most complicated light paths.可逆性可逆性反向反向折回折回复杂的复杂的1.9 Fermats Principle dndnndOne can also define an optical path in a medium of continuously varying refractive index by replacing the summation by an integral. The paths of the rays are then curved, and Snells law of refractio

20、n loses its meaning.We shall now consider Fermats principle, which is applicable to any type of variation of n and hence contains within it the laws of reflection and refraction as well:The path taken by a light ray in going from one point to another through any set of media is such as to render its

21、 optical path equal, in the first approximation, to other paths closely adjacent to the actual one. The actual path between two points taken by a beam of light is the one that is traversed in the least time.Fermat himself stated: other form of Fermats Principle: light, in going from point Q to Q, tr

22、averses the route having the smallest optical path length.Fermats principle applied to reflection at a plane surface.Consider finally the optical properties of an ellipsoidal(椭球的) reflector, as shown below:Fermats principle applied to an elliptical reflector.Graphs of optical paths involving reflect

23、ion illustrating conditions for (a) maximum, (b) stationary, and (c) minimum light paths. The Modern formulation of Fermats Principle Consider a function f(x), 0df xdx Fermats Principle in its modern form reads: a light ray in going from point Q to Point Q must traverse an optical path length that i

24、s stationary with respect to variations of that path.To prove the law of refraction from Fermats principleGeometry of a refracted ray used in illustrating Fermats principle.The length of the optical path between the point Q in the upper medium of index n and another point Q in the lower medium of in

25、dex n passing through any point A on the surface isWhere d and d represent the distances QA and AQ, respectively.Now if we let h and h represent perpendicular distances to the surface and p the total length of the x axis intercepted by these perpendiculars, we obtaindnnd 截取截取222)(hphd22 2 xhd thus2/

26、122 2/122)()(xhnhphnAccording to Fermats principle, must be a minimum or a maximum (or in general stationary) for the actual path.By differentiating the above Equation with respect to x and setting the result equal to zero, we obtain求导，微分求导，微分02)(21)22()(212/122 2/122xxhnxphphndxdwhich gives2/122 2/

27、122)()(xhxnhphxpnor simplydxndxpnsinsinnn Fermat and Mirages One can see the Sun after it has actually passed below the horizon.地平线地平线Formula of Gladstone and Dale1nPT11nT1.10 Color Dispersion （色散色散） Dispersion corresponds to the phenomenon （现象现象）whereby the index of refraction of a medium is frequency (wavelength, color) dependent.It is customary in the exact specification of indices of refraction to use the particular colors corresponding to certain dark lines in the spectrum of the sun the Fraunhofer lines, which are designated by the letters A,B,C,The d