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1、-230-Chapter 6 Diffraction The topics will be covered in this chapter are:1.Whats Diffraction(Zhaos,chap2 5.1 Hechts 10.1)2.Huygens Fresnel Principle(Kirchhoff equation)(Zhaos chap2 5.2,pg186;Hecht,10.4)3.Fresnel Diffraction(Zhaos chap2 6,pg194-206;Hecht,10.3.1-10.3.5)4.Fraunhoffer Diffraction(Zhaos
2、 chap2 7,pg209-230;Hecht,10.2.1)5.Grating(Zhaos chap4,Hechts,10.2.2-10.2.7)6-1 Whats Diffraction 6-1-1 Definition In Youngs Experiment,we used points to represent the two coherent sources.However in any reality,the sources themselves(aperture or slit in the experiments)have limited sizes,then an ins
3、tant question is:what is the field distribution after such opening:From geometric optics,if p is in the region spanned by angle d/R,then its -231-Ip|Ep|=|Es/(R+r)|.If p is outside the angle d/R,then Ip=0.Is this true?This is not always true as clearly demonstrated in the experiment shown in Figure b
4、elow:From the experimental observation,it is easy to conclude that (1)The distribution of energy on an observing plane I(p)is generally an interference pattern.The energy distribution has highs and lows.(2)Light does not propagate in a rectilinear form,its distribution“spread out”along the direction
5、 wherever it meets restriction.(3)The more restriction,the light seems spread out more.This gives the definition of diffraction.Diffraction(literally or phenomenally):Deviation of light from a rectilinear propagation Diffraction7(physical):interference of waves.(a continuous coherence sources)7 Ther
6、e is little difference from interference.In fact both interference and diffraction arise from superposition of waves.In interference we consider superpose discrete light sources,such as 2 light sources in Youngs:N in F-P case.Here in diffraction,we superpose continuous distribution of light sources
7、as stated in Huygens-Fresnel Principle.-232-The simplest model to treat diffraction is Huygens-Fresnel Principle(H-F-P):On the unobstructed wave front,every point serves as a source of spherical secondary wavelets(same).The complex amplitude of the optical field at any point beyond is a superpositio
8、n of all the wavelets.It is basically Huygens Principle+Superposition of waves.Simple picture of H-F-P in determining E(p)through a hole:a)For.Then of difference in OPL between any two paths,All secondary waves add at P constructively behaves as a point source.b)As increases,the interference at P fr
9、om all the secondary wavelets can either constructive or destructive-diffraction,meaning intensity there may be high or low.c)As such interference would cause light APPEAR to travel in linear form and we get geometric optics prediction.(We shall see that the diffraction arising from the boundary is
10、much smaller comparing to the light in the geometric allowed zone,which is the 0th order diffraction)Simple as its physical picture appear to be,solving the field at p/2ABL/2LAB -233-exactly for an arbitrary setup is actually an extreme complicate and difficult job,except for some simple cases which
11、 we are going to study in this course，namely the Fraunhoffer Diffraction and field of on-axis points in Fresnel diffraction.6-1-2 Fraunhoffer and Fresnel Diffraction To evaluate,the field at p we need to know:(1)field distribution of the secondary wavelets on(diffraction screen)(2)The field at p gen
12、erated by each secondary wavelets by principles of superposition The simple calculation is obtainable if the incoming and outgoing waves are planar wave,and such is Fraunhoffer Diffraction or Far-field diffraction.This requires that both the light source and the observation plane are infinitely dist
13、ant from the diffraction screen,or at least both paraxial and far field requirements are satisfied.This can be easily achieved in experiment by lenses:()E p-234-Then(1)The field Amplitude of the secondary wavelets on is constant.(2)The OPL used in the calculation(we shall explain this in detail late
14、r)only depends linearly on the two aperture variables,such linearity is the mathematical criterion for the Fraunhoffer diffraction.Applying the far-field criterion(in the optical region,far field includes the paraxial)d the size of the (6-1)R is the smaller distance between S and P to.As we shall se
15、e Fraunhoffer diffraction has many important applications and is also instrumental in the Fourier transform optics.Other arrangement does not satisfy(6-1)is called Fresnel diffraction.Fraunhoffer diffraction is only a special case for it.However,the treatment is more involving for the Fresnel type d
16、iffraction.We shall give a brief introduction for the Fresnel diffraction first,the goal is to illustrate:how to use H-F-P,and we only concerns the P on the central axis in this case.We are not going to treat the detailed distribution of the interference,d2dR-235-pattern for the off axis points in F
17、resnel-diffraction case.6-2 Huygens-Fresnel Principle(HFP)and Kirchhoff equation As we stated earlier that the HFP is basically the Huygens principle combined with superposition of the waves(The waves coming from the secondary wavelets).It is Kirchhoff who derived a correct mathematical representati
18、on for the HFP,which we only give out as a result8.For the more detailed derivation of the Kirchhoff theory,please refer to Hechts 10.4 (6-2)8 The theory is called Kirchoff scalar theory and is still an approximation from the rigorous treatment starting from Maxwell equations and correct boundary co
19、nditions.The exact solution to diffraction is far more complicated and Kirchoff theory is accurate enough in many cases.000()_()()(,)()ikropenareaopen areaeU pdU pKFU Q dr=?-236-(we shall give a proof on this later)Here I shall use U(p)as symbol for the field,instead of E(p)because we treat the fiel
20、d as a scalar field,this works when the vector feature of the field can be neglected(such as cases where the polarization all along same direction).Most part in(6-2)is quite intuitive:The field at P would depend on the area of the opening and the surface integral is over the open portion of the diff
21、raction screen;It also depends on the field at the opening and its propagation like a spherical source dictated by Huygens principle;the is called obliquity factor whose origin is not intuitive(like that of HFP)but can be derived with the Kirchhoff scalar theory.(6-2)is going to be the starting poin
22、t for us to deal with diffraction.6-3 Fresnel Diffraction through an open aperture A point source S A screen with aperture What is the field at P,U(P)?iK=001(,)(coscos)2F=+0(,)F -237-(Here P is a point on-axis defined by S and center of)6-3-1 Method of Half-Wavelength(/2)Plate Direct application of(
23、6-2)on this problem for the general setup would result in a nasty integral,the r has to be expressed as function of x,y of the surface area.We shall use an approximation method to evaluate such integral and this is the method of half-wavelength plate.Divide the opening aperture into orbital zone.The
24、 OPL from the edge of each neighboring zones to the field point P differs by/2.(Integral over area reduces to summation of zones)is the contribution to P by the jth/2 plate(zone).To evaluate,we need to first evaluate,the contribution to P by a very small zone on the jth/2 plate.According to the HFP
25、and Kirchhoff equation:(6-3)First the field at the diffraction screen is:(6-4)1()()njjU PUP=()jUp12jjrr=+()jUP()jd UP00()(,)ikrjjedUPKfUdr=00ikRjAUeR=-238-For the spherical surface,the obliquity factor is:(6-5)Next we evaluate the small area element(a belt shown by dashed lines in the figure):Put al
26、l above into(6-3),it becomes:(6-6)(neglects dependence on the/2-plate,treat it as a constant within the/2-plate)001cos1,(,)(1cos)2f=+d22222 sin2sincos()/2()sin()2RdRdRdRRbrR RbrddrR RbRrddrRb =+=+=+()002(,)ikrj PjRdUU Kfe drRb=+ikrjC e dr=11111()(),1,2,.2(1)(1)2(1)jjjjrjjrrikrjrjjikrjjjikrikbjjjikbj
27、UPdUPCe drrCerikjrbjreere=+=0(,)f -239-(6-7)(6-8)The neighboring/2 cancels each other in the contribution to field point P.Absolute Amplitude of P Contribution of 1st,2nd /2-plate.113355241|()|(|)(|).2222222|(6-9)22mmAAAAAAAU PAAAA=+The terms enclosed in the bracket()is close to zero due to the canc
28、ellation between adjacent zones.The signs in the final expression are+if m is an odd number,-if m is even.The expression of|Aj|can be seen from(6-6),(6-7)(6-10)It is depending on,the rest can be treated as constant for the current setup.For the adjacent/2 plate,is almost a constant(the change of is
29、negligible),slowly decreases as the order of/2-plate increases.as j.It reaches 0 as j approaches the other pole of the spherical surface.Based on the above arguments,we conclude:2()(1)(1)jikbjjjjCUPeikA=11()()(1)NNjjjjjU PUPA=1234|()|.U PAAAA=+()002(,)ik R bjA eAKfiRb+=+0(,)f 0(,)f 0(,)f 0(,)f -240-
30、(1)In the case of free propagation(Aperture is infinitely large),(6-8)includes all the sums that cover all the/2-plates on a spherical surface,where the last/2-plate:Then 1()2freeAU P=or 1|()|2freeAU P=(6-11)i.e.The field strength at point P in the free propagation is half the field generated by the
31、 first/2-plate at P.(2)As the aperture size changes,or as we move the P away or towards the aperture,the number of/2-plates contained by the aperture shall vary.If the aperture for point P contains odd number of/2-plates,then the P is bright with 11|()|2|22mfreeAAU PAU+=(if does not change much),I0
32、is intensity for free propagation Intensity will be 4 times higher at point P in the presence of aperture than that of free propagation,which is counter intuitive.Since the energy is conserved(the energy flux through a plane should be same),then there must be a redistribution of energy across the ob
33、servation plane at P.The pattern of Fresnel diffraction by a circular aperture on an observing screen would be rings with bright and dark 01(,)(1cos)002(cos1)mfA=+=0(,)f 0()4I PI-241-circles.If the aperture contains Even number of/2-plates for point P,then is small.For the on-axis point P,its intens
34、ity distribution has been discussed.For the off-axis point,the calculation would be much more involving and will not be treated here.It is enough to know the Bright-Dark ring pattern of the Fresnel Diffraction by a circular aperture.(3)From the derivation,we can also draw conclusion on the coefficie
35、nt K in the Kirchhoff equation(We stated that before and we can prove it now):From(6-11):for free propagation.A1 is given by(6-10),This is indeed what the expression for K.The in the coefficient indicates extra phase difference for the superposition of the secondary wavelets in the HFP.This is exact
36、ly what we discussed that there is an extra phase lag for the secondary wave generated by the array of secondary oscillators(Recall what we discussed for waves propagation 1|22mPAAUiK=()01()2ik R bAAU PeRb+=+()010022(,)(,)1ik R biA eAKfiRbfieK +=+=2ie2-242-in a media;Hecht 4.2.3).6-3-2 Phasor Treatm
37、ent and Vibrational Spirals(curves)What happened if the opening does not contains an integer of/2-plates?For the on-axis point P,such problem can be solved by the phasor method.,m is a real number,not necessarily an integer as in/2-plates case.We still start from HFP:Now we divide the wave front at
38、aperture into many small zones d,each small zone contributes to P with amplitude and phase;And integral U(P)can be estimated then by phasor method.0()2rrm=000()(,)ikrKU PU fe dr=()idA P e000()(,),KdA PfU dk rr=-243-In the figure left I showed the contribution of first/2 plate by dividing the integra
39、l into many(N)phasors.The deviation from a perfect circle is due to the decrease of As,the arrow form a continuous curve(very close to a circle).As the increase of aperture(or as P moves closer),the phasor diagram will be:(The spiral in the above figure is overly exaggerated,for the many lower order
40、 half-wavelength zones,it is very close to a circle)A0 is the field at free propagation.Two-zone-plates in aperture:Three/2-plates in aperture:The figure above over exaggerates the deviation from circle.Because the slow decrease of the obliquity factor,the Z2,Z4 should be very close to O 0(,)f N 110
41、|22OZAA=/22|()|U POZ=?3|()|U POZ=?-244-and Z1,Z3 is about A0 away from O.For an arbitrary phase difference between the center of the aperture and the edge,represented by A on the vibrational curve,with phase difference by;then and can be easily evaluated by treating(approximately)the spiral as a cir
42、cle with radius A0(As long as we can treat as constant)Example:What is|U(P)|for Fresnel diffraction?6-3-3 Circular Obstacle|()|U POA=0(,)f 013rr=0212331|()|2sin()3phasek rU pAA=-245-From/2 plate method:The obstacle blocks l/2 plates,so the contribution to field point P is from(l+1)th/2 plate to(l+m)
43、th:(6-12)since it is the plate with Thus the point P is always bright,with irradiance as if the obstacle does not exist,similar to that of free propagation.This is another counter intuitive result,but confirmed by the experiment,very important in the demonstration of wave characters of light.In term
44、s of vibrational curves,the field at point P over an obstacle is:0l mA+=011(,)(1cos)(1 1)022f=+=1211|.|22|2Pllll mlUAAAAA+=-246-The position of A is determined by L which in turn is determined by the size of obstacle and position of P.6-3-4 Babinet Principle This is a more general result in diffract
45、ion and not limited to the Fresnel diffraction,it is originated from HFP.From HFP:Let Then:(6-13),are transparent parts of screen to add up,for example:For the illustration above,where is free space,then,are said to be“complementary”to each other,this means the transparent region on one exactly matc
46、hes the opaque region on the other and vice 0()|SrrLU PAO=000()()/ikrUPCU Q erd=0ab=+0()()()abUPUPUP+=ab00ab-247-versa.The Babinet principle offers a way to estimate the diffraction pattern of the complementary screens,i.e.if the diffraction by one screen is known;then diffraction by the complementa
47、ry screen is also known.A special case is that if the for a space point P in open space case,then and,the complementary screen generate same intensity and thus same pattern at P.The Fresnel diffraction by the complementary aperture and obstacle offers an illustration to the Babinet Principle:clearly
48、 As Babinet principle stated.6-3-5 Fresnel Zone Plate Imagine for a device as depicted in the picture below,a zone plate for a pair of source and field point P,if all the light passing through the plate 0()0UP=()()abUPUP=()()abIPIP=00ababUAUAUA=?0abAAA+=?-248-are from the odd/2 plates,with the even
49、plates blocked(case b in the figure;or vice versa as case a in the figure).Then there will be no cancellation between the/2 plates,the light field at P would be strongly enhanced and the intensity could be much larger than the free propagation case.The Fresnel zone plate thus has the power to concen
50、trate energy into certain points just as a focusing lens does.It is indeed used as lens in the wavelength region where no material is suitable for conventional lens.As the example of 10 odd/2 plates(or even/2 plates),make light field at P about 20A0,and intensity 400I0.The central question is then g