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Chapter 31 Diffraction and Polarization1.Diffraction of Light2.Diffraction Gratings3.X-Ray Diffraction4.Polarized LightWhen the slip wider,light travels in straight line.31-1.Diffraction of Light(P703)1.Diffraction(衍射衍射)by a single slitscreenshadowscreenWhen the slip gets narrow,the light produce a diffraction pattern on the screen.When monochromatic(单色单色)light from a distant source passes through a narrow slit or obstacle produces a diffraction pattern(光能绕过障碍物的边缘传播光能绕过障碍物的边缘传播形成具有形成具有明暗相间的衍射图样明暗相间的衍射图样)is called diffraction of light(P703)(P703).Condition of diffraction:the width of slit(dimension of barrier)is similar to the wavelength a 波传到的任何一点都是子波的波源，各波传到的任何一点都是子波的波源，各次级次级子波子波在空间某点的相干叠加，就决定了该点波的强度。在空间某点的相干叠加，就决定了该点波的强度。Hugens-Fresnels principleRecalling:The method discussing location of diffraction pattern for a single-slit diffraction is called fresnel half-wave-zone method(半波带半波带).2.Locating the diffraction bands(P704)*S f f aL LABslit aEO PPath difference of AP&BP:screenscreenLens doesnt produce additional light-path difference.Two half-wave-zones can be gotten,and P is first dark fringe(out of phase pairs rays)In a single-slit diffraction experiment,dark fringes are produced where the path length differences(a sin )between the top and bottom rays are equal to ,2,3,.central bright(中央明纹中央明纹)first dark fringe(ii)(i)first bright fringeaP 屏幕屏幕screenscreen(iii)Conclusion:opposite to the condition of interference patternAngular-width(角宽度角宽度):For central bright fringe:Linear-width:when a 1:半角宽度半角宽度 half of the central width.其他明纹宽度为中央明纹宽度的一半其他明纹宽度为中央明纹宽度的一半For 1-order dark fringe,Similarly,for 2-order dark fringe,Position of 1th dark fringe一级暗纹坐标一级暗纹坐标Position of 2th dark fringeWidth of each order of bright fringes:Solution:A slit 1.00 mm wide is illuminated by light of =589 nm.We see a diffraction pattern on a screen 3.00 m away.What is the distance between the first two diffraction minima on the same side of central diffraction maximum?The separation of the two minima is:Using the condition of minima of diffraction:andExample:3.Intensity in single-slit diffraction(P705)(a)The central maximum;(b)Between center and first minimum;(c)The first minimum;(d)Near secondary maximum.See Fig.31-8 in P706当当 增加增加,半波带数增加半波带数增加,未被抵消的未被抵消的半波带面积减少半波带面积减少,所以光强所以光强I I 变小变小.i.e.,I 31-2.Diffraction by a Circular ApertureDiffraction pattern of a circular aperture(P710):Basic arrangement of experimentS*d L 1 fBased on theoretical calculation,the first minimum of diffraction pattern of a circular aperture of diameter d(radius r)locates(第第一级暗环的衍射角满足一级暗环的衍射角满足):Angular-radius of central maximum(艾里斑艾里斑):If d ,diffraction cant be seen仅当通光孔径足够大时仅当通光孔径足够大时,艾里斑才可能很小艾里斑才可能很小.占入射光总占入射光总光强的光强的84%d d AiryAiry spot getting smallThe angular separation of the two point sources is such that the central maximum of the diffraction pattern of one source is centered on the first minimum of the diffraction pattern of the other a condition called Rayleighs criterion for resolvability.瑞利判据瑞利判据：如果一个点光源的衍射图像的中央最亮处刚：如果一个点光源的衍射图像的中央最亮处刚好与另一个点光源的衍射图像第一个最暗处相重合好与另一个点光源的衍射图像第一个最暗处相重合,认认为这两个点光源恰好能为这一光学仪器所分辨。为这两个点光源恰好能为这一光学仪器所分辨。Resolvability(分辨分辨)&Rayleighs criterion(判据判据)(P710):It can be seen from the text given in page 710 thatd d Increasing Increasing R RThe condition of two objects are barely resolved by this criterion of an angle separation R(最小分辨角最小分辨角):For small angle,(Rayleighs criterion)Resolvability(分辨本领分辨本领):s1d*s2 R RIt is said that a camera in a spy-satellite can distinguish clearly the number of cars licence ruuning on the road.Solution:An angle separation R To distinguish(要识别要识别)the distance of 5cm between numbers on a licence from the height of 160km,what should the circular aperture of diameter d be?(Assume =500nm).Example:31-3.Diffraction by a double slit(P708)No diffraction:ad fLensIConsidering the diffraction of light.设双缝缝宽均为设双缝缝宽均为a，在在夫琅禾费衍射下，每缝夫琅禾费衍射下，每缝衍射图样位置相重叠衍射图样位置相重叠difference of interference&diffractionThink of:II00The plot of Fig.31-10c suggests the interference pattern for two actual slits.That plot was constructed by using the curve of Fig.31-10b as an envelope on the intensity plot in Fig.31-10a.The positions of the fringes are not changed;only the intensities are affected.31-4.Diffraction Gratings(衍射光栅衍射光栅)()(P713)One of the most useful tools in the study of light and of objects that emit and absorb light is the diffraction grating.abP O1.Characteristic parameters(1)The width of a slit:a(2)Separation between slits:dIt is also called grating spacing 光栅常数光栅常数:d=a+b(10-4 10-6m)2.The diffraction pattern of gratingsDiffraction-pattern of gratings is the superposition of single-slit diffraction and N-slit interference.If N rulings occupy a total width w,thend=w/N(3)For grating,the slits on it are often called rulings(划线划线)or lines.光栅公式光栅公式(i)If the path length difference between adjacent rays is an integer number of wavelengths,i.e.(m=0,1,2,)principal maximaThe important difference between a double-slit and multiple-slit diffraction pattern is that bright maxima are much sharper&narrower for grating.two-slitsix-slitP714(ii)Missing order will be emerged,ifd sin =m andi.e.a sin =m Central bright fringes will vanish which orders accord with d/a being integer number(级次为级次为d d/a a的整数倍的那些主极大消失的整数倍的那些主极大消失).).There are 500 rulings in 1cm on a grating,the wavelength =589.3nm.How many fringes can be seen if light perpendicularly incident?From (a+b)sin =m and|90So mmax=3,i,e.The 3rdorder can be seen,in all 7 fringes(0,1,2,3).Solution:Example:Light of wavelength 600nm incident normally on a diffraction grating.The 2th-order bright fringe occur at angle given by sin=0.20,and the 4th-order maxima are missing.(a)What is the separation between adjacent slits?(b)What is the slit-width of this grating?(c)How many orders of intensity maxima can be observed by this grating.Answer:So mmax=10,the 10th-order cant be seen,in all 15 fringes(i,e.0,1,2,3,5,6,7,9).missing orders:Example:31-5.X-Ray Diffraction(P718)W.K.Rontgen 伦琴伦琴(18451923)(German experimental phys.)discovered X-ray in 1895.X-rays are electromagnetic radiation whose wavelengths are of the order of 1 (=10-10 m).Compare this with a wavelength of 550 nm.A standard optical diffraction grating can not be used to discriminate between different wavelengths in the X-ray wavelength range.The experiment done by M.V.Laue(18791960)(German Phys.)in 1912 for a crystalline solid:The experiment done by W.L.Bragg(British phys.)in 1913.Interplanar spacing晶面间距晶面间距d and reflection angle掠射角掠射角 COBACOBA =AC+CB=2dsin 2d sin =k，k=1,2,3,.Braggs lawThe path difference between the layers of adjacent atoms(上下相邻原子层反射线的光程差上下相邻原子层反射线的光程差):For intensity maxima of X-ray diffraction:Where the angle is called Bragg angle.X-ray diffraction is a powerful tool for studying both X-ray spectra and the arrangement of atoms in crystals 用于测波长或测晶体的晶格常数用于测波长或测晶体的晶格常数.31-6.Polarization of Light(P720)Polarization of light proofs that light is transverse wave(光的偏振证明了光的横波性光的偏振证明了光的横波性).Head-on view of the plane of oscillation没有优势方向没有优势方向自然光的分解自然光的分解source of light are polarized randomly or unpolarized.The electromagnetic waves emitted by any common自然光的表示法：自然光的表示法：E播播传传方方向向振振动动面面光矢光矢 1.Plane-polarized light光矢量在屏平面内光矢量在屏平面内光矢量与屏平面斜交光矢量与屏平面斜交光矢量与屏平面垂直光矢量与屏平面垂直 Denoting method:Its field oscillations are in a fixed plane(direction)光矢量只在一个固定平面内沿一个固定方向振光矢量只在一个固定平面内沿一个固定方向振动的光叫动的光叫线线偏振光偏振光.2.Partially polarized light(部分部分偏振光偏振光)Its field oscillations are neither completely random nor parallel to a single axis(the electric fields oscillating along one direction have greater possibility than those oscillating along other ones).部分偏振光可分解为两束振动方向相互垂直部分偏振光可分解为两束振动方向相互垂直的、不等幅的、不相干的线偏振光的、不等幅的、不相干的线偏振光.平行板面的光振动较强平行板面的光振动较强垂直板面的光振动较强垂直板面的光振动较强3.Cosine-squared rule(malus law)Polarizing sheet(polaroid filters)偏振片偏振片:We can transform unpolarized visible light into polarized light by sending it through a polarizing sheet(polaroid),were invented in 1932 by Edwin Land when he was an under graduate student.If a beam of plane-polarized light strikes a polaroid whose axis is at an angle to the incident polazization direction,the beam will emerge plane-polarized parallel to the polaroid axis and its amplitude will be reduced by cos as Fig.shown Malus law(马吕斯定律马吕斯定律)(721):I0I=EE022=cosE02E022 cosEE0=Malus law extinction消光消光MNote:各偏振片只允许和自己的偏各偏振片只允许和自己的偏振化方向相同的偏振光透过。振化方向相同的偏振光透过。Conceptual example:Unpolarized light falls on two crossed polaroids(axes at 900),no light passes through.What happens if a third polaroid,with axis at 450 to each of the other two,is placed between them?Example 31-10 of P722:Solution:4.Polarization by reflection(P723)Unpolarized light can be polarized,either partially or totally,by reflection.Light reflected from the smooth surface of water in a lake is partially polarized parallel to the surface.unpolarizedpartially polarizedLook at P723!(a)Without polarizer;(b)using a polarizer which is adjusted to absorb most of the polarized light reflected from the waters surface,and any fish to be seen more readily.P Brewster Brewster 或或 起偏角起偏角i0 0 +r0 0=90=90 from Brewsters lawWe haveThe amount of polazization in the reflected beam depends on the angle,varying from no polazization at normal incident to 100 percent polazization at an angle known as the polarizing angle,P.Solution:The index of refraction of glass to the air(玻璃对玻璃对空气空气)is .Find the angle of Brewsters.Example:
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