光学光学光学光学 (5).pdf
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1、-153-Wave Optics Interference,Interferometer,Coherence Diffraction-Fourier Optics Fundamental principle:Superposition of waves Central problem:(the kernel question)Phase distribution over a wave front.We shall first look at interference(Hecht chap7.1-7.2,Zhaos Chap2 section 1-4),focusing on the prin
2、ciple of superposition illustrating the importance of phase relation in interference,and the evaluation of phase distribution over a wavefront,the using Youngs Experiment as example.Later we shall look into some interferometer devices(Zhaos,chap3;Hecht 9.3,9.4,9.6)and the important concept of cohere
3、nce.Chapter 5 Interference,Interferometer and Coherence Lets start this important chapter with some examples to illustrate the phenomena we call interference,which is the hallmark behavior of wave.-154-In both cases,the total intensity(energy)from contribution of two paths is not simple summation of
4、 the energy of each path:12III+,such phenomenon is called interference,and we shall see it arises from superposition of waves(not the superposition of irradiance or intensity).Whether the superposition of waves displays interference or not depends on the coherence among component waves.5-1 Superposi
5、tion of Waves This is a fundamental postulate like that superposition of forces and the fields(here the wave is just E-M field anyway).The math view of the -155-principle of superposition is originated from the wave equation discussed earlier.22221uUVt=If the v is independent of U then:iiiUaU=is the
6、 solution of the wave equation provided iU is the solution,ia is a coefficient.The field(or waves)shall be added to get the total field(wave),and the system response to the field will be treated linear as long as the field is not too strong,this is the domain of linear optics.5-1-1 Superposition pri
7、nciple in linear optics A general light wave can be written as:()0ipi tEE ee=where 0E the amplitude;()p the spatial phase(including the initial phase);t the oscillation with time or temporal phase.For plane wave:0()pk r=+?spherical wave:0()pkr=+However,common detectors only detect the irradiance(or
8、intensity)of the light,which we know:220TIEE=Since we are interested in the relative irradiance in most cases,and we shall let:(as if chosen proper unit)220IEE=(5-1)-156-Our common sense sometimes misleadingly makes us to perceive that in the cases of multiple light sources,the resultant irradiance
9、is iiII=This is the case for illumination by incoherent sources such as light bulbs,candles,etc.However,iiII=is generally wrong,as demonstrated by the two examples at the beginning of the chapter.It is the summation of the field E,not I that is physically correct,as stated in the principle of superp
10、osition:Suppose we have many different field sources contributing to a point p,then from superposition principle,the total field at p is:()0()jjipitpjjjjEEpE ee=?(5-2)We shall first study simpler case where different components have same frequency and in such casej=,the intensity can be easily compu
11、ted with(5-1):22000|()()2cos()()jkjjkjkjkjjkjIEEEEEEpp=+(5-4)sintan()cosojjjojjjEpE=(5-5)(5-4)is just(5-3),a result of superposition;the(5-5)can be relatively easily derived by phasor method.-158-The phasor method in superposition of waves usually offers an intuitive and speedy solution of waves wit
12、h same frequency and parallel E0.5-1-3 Standing Waves (Hecht,7.1.4)An interesting puzzle:recalling the reflection by a surface at normal incident angle in discussion of Fresnel equations:In the example above when tinn(external reflection),the reflected lights pE and B?field are shown in the figure.F
13、rom the Fresnel equation we know ipE and rpE are reversed in direction(so,irB B?are in same direction,since irk andk?reversed,andEBk for traveling -159-E-M waves).So it seems if we superpose the incident and reflecting waves,irEEE=+?diminishes,while irBBB=+?increases.Recall that for traveling waves,
14、EcB=there seems existing an apparent paradox.The reason is that the superposition of two traveling waves travelling in opposite directions will result in a standing wave,not a traveling wave.The general solution to the wave equation takes form of 12(,)()()x tC f xvtC g xvt=+So when two waves travel
15、in opposite direction,the summation also satisfies the wave equation.For this case,the resultant wave will be a standing wave.(I follow the Hechts derivation and use sines for wave;if you use cosines or Euler formula,same results)For the wave propagates along-x:0sin()IIIEEkxt=+The reflected wave pro
16、pagates along+x:0sin()RRREEkxt=+The Reflecting Mirror at x=0,the boundary condition at x=0,requires 0IREEE=+=Lets consider the simple case where r=1 or00IREE=,we can set 0I=(which is arbitrary anyway,we always have the freedom to set the initial phase of ONE wave as zero),and the(0)00IRRxEE=+=and th
17、e superposition of wave is:00sin()sin()2sincosEEkxtkxtEkxt=+=(5-6)-160-(5-6)is the expression for a standing wave.It is standing in a sense that the spatial distribution sinkx,does not move with time(the nodes and the antinodes are fixed in space,in contrast to the traveling wave);the amplitude chan
18、ges with time as coswt(see figure 7.10,7.11 in Hechts book and Fig 7.14 shows a setup to verify the existence of the standing wave).For an open end standing wave(i.e.only has boundary condition at one end,such as waves reflected by one mirror as shown):,the resulting standing wavesincoskxt,there is
19、no restriction on the value of k,it can take any values.Another important case is the light field in a cavity:such as the two mirror cavity separated by L.The boundary condition would be(0)0E xxL=,the resultant stable wave is still in the form ofsincoskxt,but with restriction on k.where 22,0,1,2(57)
20、mLmmk=This is directly from the boundary condition E(x=L)=0klm=.The reason that I wrote it in form of(5-7)is that it represents a clear physical -161-picture,as I shall demonstrate below.From the superposition of wave,the field inside the cavity bounded by M1,M2 is the superposition of the incident
21、wave and the reflected ones(multiple times)by M1and M2.Another equivalent way to look at this 2-Mirror,L length cavity is taking it as circle with 2L periphery,the waves travels along in circle.The only stable ones are the waves that after one round passage overlap exactly with the original ones i.e
22、.02,0,1,2_2LmmornLm=in terms of Optical Path length.These waves are standing waves;for those do not satisfy(5-7)the superposition will result a null field(E=0).For length=L cavity,the standing waves of first 3 lowest m(called modes)are:The standing waves have form of(5-6),and then the expression for
23、 the B -162-field can be derived from Maxwell equation:cossinBEBkxtt Thus what are nodes for the E field(sinkx=0),are antinodes for B(coskx=1);what are antinodes for E are nodes for B.This is the relation for,E B?of standing waves,and this explains the puzzle stated at the beginning of this section.
24、Not only in the light(such as light field in a cavity,which is fundamental in laser),standing wave also plays crucial roles in other waves,such as sound waves.Most music instruments are based on it.Also if you ever tried to entertain your friends by inhaling helium(He)gas,your sound would be much hi
25、gher pitched(high frequency).Can you explain this using what you have learned?5-1-4 The addition of waves of different frequency (Hecht,7.2)(a)Beats Simple case:two waves with different frequency,travel in same direction,parallel E field,same 0E?10=+20=10kkk=+20kkk=(true for+2min121 212122()OOIIII I
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