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1、-58-Chapter 3 The propagation of Light(Hecht,chap 4,Zhao,chap1.(1)-(3),chap3,(10)We focus on what are most relevant to geometric optics:Reflection and Refraction and Transmission in this chapter.We are not going to treat absorption and dispersion in detail here(Please refer to Hechts 3.5 or Zhaos Vo
2、l2,P228-244 for more details on this)We shall first give out the Phenomenal Rules(That is rules based on macroscopic observations)that govern the propagation of light through media;(Zhaos 1.1-1.3)Then we are going to take a deeper look from microscopic point of view,such as scattering process betwee
3、n light and particles,to further understand the physical basis for such rules.Finally,we are going to use Maxwell equation to prove the results and derive Fresnel equations that give quantitative results on reflection and -59-refraction.3-1 Treatise Based on Macroscopic Observations(1)Light travels
4、in straight path,in homogeneous isotropic media.Homogeneous:The media is evenly distributed,density is a constant.Isotropic:The physical property is not directional dependent.In inhomogeneous or anisotropic media,such as air with temperature gradient,etc.,light path can be bent,which can generate ph
5、enomenon known as Mirage.There are also cases that when light meets something that on the size of the wave length of light,deviation from straight path occurs,we are going to treat this behavior in topics of diffraction later.(Here,strictly speaking,we treat wave length of light is much smaller than
6、 the sizes in concern,or0)(2)Reflection and Refraction:When light goes from one media to another,at the interface:-60-Reflection:11ii=;12nn Internal reflection Refraction:1122sinsinnini=Snells law (3-1)All the“rays”are in the same plane.plane of incidence Rays:For conveniently showing the propagatio
7、n of light,we use rays to represent waves.A ray is a line in space corresponding to the direction of flow of energy.In the isotropic media,rays are simply in the direction of wave vectork?,which are orthogonal trajectories of the wave front(perpendicular to the equal-phase plane).Changes of light up
8、on refraction (Hecht,pg.102)(a)Changes direction of propagation Snells law.1122sinsinnini=(b)Changes the cross section of the beam of plane wave,thus varies the -61-irradiance I(energy/Area)11cosSABi=22cosSABi=2211coscosSiSi=(c)Changes phase velocity of light cn=If is same(think why?),the wave lengt
9、h in a media will depend onn,0n=0 vacuum wave length (3-2)3-2 Huygenss Principle Every point on a propagating wave front serves as a source of spherical secondary wavelet,such that the wave front at some later time is the envelope of these wavelets.(Fig 4-26 in Hechts Book or Fig2-2 in Zhaos Book)-6
10、2-The secondary wavelets in Huygenss principles are imaginary points on the wave front(virtual wavelets).These secondary wavelets have the same freq.and same phase velocity of the propagating wave in media.The Huygenss principle is a very useful tool,and can be used to derive the rules of reflection
11、 and refraction(It is basically some simple geometry and the details are in Zhaos Book,pg.26-30).We are going to see that this principle is highly useful to construct the wave front,estimate the phase distribution over space etc.when we treat the diffraction.(The Huygenss Fresnel principle)As to the
12、 physical basis for the principle,when light interacts with the atoms/molecules of the media,the secondary wavelets can be viewed as the radiation from the forced dipole oscillation plus(superpose)the primary wave.However in vacuum,the secondary wavelets in the principle have to be viewed as imagina
13、ry points,so:Treat the principle as a useful tool rather than the real physical existence would be most proper.-63-3-3 Fermats Principle3 Important Concept:Optical Path Length(OPL)It is the effective(equivalent)distance that light traveled at speed of0C,the vacuum speed.For propagating wave,the phas
14、e difference between two pointsQ,Pdepends on distance of QP and the wave length(or2k=).In different media,the is different(0n=).The same distance may correspond to different phase change.To eliminate this inconvenience,we introduce the concept of optical path length(OPL).Defined as:OPL is:()Pi iQiQP
15、nlndl=(3-3)()QP is related to the phase difference between Q and P for a wave.3 The proof of this and its association with least action principle in classical mechanics is more demanding,please refer to Goldsteins“Classical Mechanics”2nd edition,chap.10(10-8)for details.Or Born and Wolfs“Principles
16、of Optics”chap.3.-64-0QiQEE e=0PiPEE e=02()QPpQQP=(3-4)Thus we take the consideration on refractive index into()QP,the OPL and treat light in any media travels at speed 0C,or wave length of 0 when calculate its phase difference.Fermats Principle (Variational principle).A light ray goes from point Q
17、to point P must traverse an optical path length that is stationary with respect to variations of the path:()0PQQPndl=(3-5)This principle can also be used to prove the rules of deflection and refraction.(The light travels in straight path in homogeneous,isotropic media is obvious with Fermats princip
18、le).(Optional)The physical significance underline the principle is discussed in Hechts Book(Fig4.34,4.35,also see pg.139-140,Fig.4.68).Basically,it has to be understood from superposition of waves(Interference).In the figure below:the vertical axis is the optical path length;the horizontal axis is s
19、ome parameter,specifying the path(here is -65-a simplification where the path can be specified by one variable)The paths around the stationary paths(specified by points A,O,B)have almost equal OPL and thus when superposed,will interfere constructively.The paths away from stationary will arrive P out
20、-of-phase with each other and therefore tend to cancel.A more concrete example is reflection by mirror:given the initial and final points(S,P),the position on the mirror(say x)will specify the path.The contribution(represented by phasors)around the stationary path(group-I)and that from another regio
21、n is shown in the figure(b).-66-Another way of seeing this,borrowed half-wavelength plate method used in diffraction(we shall come to this again).Within each half-wavelength zone,the contributions are phasors(the little arrows in the figure,representing the probability amplitude of light taking some
22、 particular path)with phase difference between 22 and add up sort of constructively to give none zero result(such as bigger arrow 1 and 2 above);between half-wavelength zones they are out of phase and cancel each other.The total contributions will be summation of all these zones and that will result
23、 in many cancellations.The important property of the stationary point(around zone 1)is that the contribution of this zone around the stationary is the largest(recall the definition of stationary).In 1contribution from zone 12contribution from zone 2-67-the figure above,the line segment of zone 1 is
24、the longest and there will be more states(represented by little arrow phasors)and thus larger resultant contribution.The contributions from other regions(say zone 2 and 3 etc.)have similar amount but cancelled each other.So the summation will show the contribution from zone 1 only and the light appe
25、ars to take paths around stationary point.(The above reasoning is not a rigorous proof,but used to see the superposition principle behind the Fermats principle,and the approach is idea of Feynman path integral in quantum).The least OPL(stationary OPL)of Fermats principle triggers the least action Fo
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