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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
26、rmalism of Lagrange in classical Mechanics,and path-integral formalism by Feynman in Quantum Mechanics,which implies that the variation principles may be the Mother Natures fundamental Law.Since we use OPL in the principle,and the time taken for the light passes certain path will be OPL/c0,thus the
27、Fermat principle is also termed least time principle(It is not necessarily a minimum,a stationary point can be minimum,maximum or inflection point,but it is customary to call it least).3-4 Two examples using Snells equation(1).Total Internal Reflection(TIR)-68-itnnInternal reflection it secondary la
28、gs primary (Hechts book,Fig.4.9,4.10 and 4.11)-76-AtAX,.sec0Aii tTotalpriondEEEE ee=+=A is the phase lag between the total wave and primary wave at point A.At BX,this ETotal will act as primary wave and there will be further delay of phase,the secondary wave from BX will have phase lag compared to T
29、otalE,thus generally when we treat the media as continuous:X=,0 for phase Lag.The resulting wave(sum of secondary and primary)would be:0()i kxi tee+0kk=+,0ckk=So the phase velocity is different than that of c even though both primary wave and secondary waves propagate with c,it is caused by the phas
30、e difference between them.(c)Likewise,if0,secondary leads in phase4,(00 means they are along the direction defined byiS?,iP?or more precisely they are in phase withiS?,iP?;siE,piE0 means they are in the reverse direction,out of phase.Same arguments applies for theprE,srE of the reflected andptE,stE
31、of the refracted beams,and their signs relate to the definedrP?,rS?;tP?,tS?respectively as in Fig2.Applying boundary conditions and after straightforward but tedious computation(details skipped here),the Fresnel equations then are:222Pr222cossincoscostan()coscostan()cossin(/)tiitiitiititpPitiitittii
32、tiinnEnnrEnnnnr=+(3-7)where ttiinnn 22r22cossincoscossin()coscossin()cossin()itiiSiittitsSiiittititiinEnnrEnnnr=+(3-8)-81-2coscoscosptiippitiitEntEnn=+(3-9)2cos2cossincoscossin()StiiitsSiiittitEntEnn=+(3-10)ii,ti are related by Snells law.(In the above formula,the first expression is the primary for
33、m derived using boundary conditions of Maxwell equation,the rest are diffrerent variations which can be derived from primary form using Snells law)There are also Irradiance ratio,and Energy ratio which are direct results from(3-7)-(3-10)2In E(as in equation 2-13)Energy=IAreacosIi 2PrrppPiIIrI=,2rssI
34、r=Reflected irradiance 2ttppinItn=2ttssinItn=Refracted irradiance 2ppRr=2ssRr=Reflected energy flow(Reflectance)2coscosttppiinTtn=,2coscosttssiinTtn=Refracted energy flow(Transmittance)From conservation of energy and can be proved by Fresnel Equations:1ppssRTRT+=+=-82-The derived Fresnel equations i
35、nclude all the information needed when we discuss relations between incident and deflected beams,below are some important results from the Fresnel equation.(Focus on the pr,Sr)3-6-1 Amplitude,Brewster angle and Total Internal Reflection The figures show the computation for the coefficients at differ
36、ent incident angles(Figures 4.41;4.42 in Hechts)(1)For the S component,The sr increases monotonically with the incident angle,reaching 1 at glance angle(0st at glance angle),glance angle is90i=?.(2)For the P component,the pr first decrease to 0 as the incident angle increases to B(Brewster angle or
37、polarization angle),then it increases to 1 as angle increases.-83-The Brewster angle 0iBpr=From(3-7),it is easy to see that 0pr=at 90oit+=(tan()it+)sinsincosiitttinnn=tantBinn=(3-11)At this Brewster angle,only S component is reflected and thus the reflected light is linearly polarized,and transmitte
38、d light will be partially polarized due to decreased amount of S.If there are multiple surfaces,the transmitted light will be almost P-polarized.The reflection and transmission is the simplest device to create liner polarized light,such as the Brewster window(a piece of glass aligned at Brewster ang
39、le with the axial of the cavity)in a laser cavity.(3)Total Internal Reflection.For external reflection(itnn),there is a critical angle,where-84-arcsintcinn=,90oti.As ic,pr,sr1 However,pt,st0 (see 3-7,3-8)But energy flow of transmittance:2cos0costtiinTtn=(cos0ti=)This means that there is no energy fl
40、ow into the optical less dense region(tn region),but there are field existing in it.We shall see that this field is created by evanescent wave.3-6-2 Phase shift (Fig 4.44,Hechts)Away from the critical angle in case of total internal reflection,the coefficients,pspsr r t t are real numbers,so the pha
41、se difference here only means the signs change(0 or).At the TIR,the,psr r will become complex number(left as homework to prove)and phase is referred to the argument of the complex number.This is straightforward but I need to clarify the physical meaning of such phase difference.(1)Away from the crit
42、ical angle,the r,st are either+or-,which means the deflected light are either in phase(means+,for r,t)or out of phase().However as we stressed earlier,this+,-are referring to the individual coordinate system we assigned as in Fig.2 of this note,not referring to -85-the original incidentE?.5 We see t
43、hat the deflected light,rE?or tE?are only in phase(along the direction)or out of phase(reversed direction)with respect to incident E?at normal or glance angle(0ii=or 090)where the coordinates axes are parallel or anti-parallel.For other incident angles,the angles betweeniE?andrE?,tE?are generally be
44、tween 0(Hechts Fig 4.45 4.46)Extra comments on phase:For light propagates through interface between two media,thesr,pr,st,pt carries the phase information of reflected and transmitted light with respect to therS,rP,tS,tP direction defined as in Fig2 of this note.At point o(we can define it as origin
45、 point,r?=0)()opii tipioEoE ee=osii tisioEE ee=sposopo=The drawing below summarize given the initial phase difference between S and P components for the incident wave,the phase difference between the S and P components of the Transmitted and 5 This is the price we paid by choosing 3 individual coord
46、inates for the incidence,reflection and transmission.We choose such instead of one x-y-z coordinate to simplify the relations among coefficients,i.e.p component in the incidence only gives p component in reflection and transmission,etc.Such choices simplify the Fresnel equations but complicate the i
47、nterpretation of phase a little bit.-86-Reflected waves can be determined by the transmission and reflection coefficients.Such phase difference between the S,P components will be important to determine the polarization of the field.(2)Examples (Zhaos book p254-255,Fig 10-7,10-8,10-9)The computation
48、for the coefficients of reflection and transmission for the internal and external cases at normal incidence is simplest and the results are illustrated in the figure.-87-It is clear at normal incidence that there is half-wavelength difference(02,)between reflected and incident during external reflec
49、tion(itnn).Between the two reflected beams by the top and bottom surfaces of a media,if n1=n3,i.e.a glass plate in air,between the beams 1(top reflection)and 2(bottom reflection),there are always extra 02(or)difference(We have discussed this earlier in the scattering picture of lights propagation.He
50、re we can make more quantitative derivation using Fresnel equations,please also refer to Fig 10-9 of Zhaos Book).Using Fresnel equations on the reflection coefficients,we can prove(and you are encouraged to do it)that more generally at small incident angle(0i):if n2 is“sandwitched”between n1,n3,i.e.