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UF PHY 2049 - Diffraction

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DiffractionDiffraction and the Wave Theory of LightSlide 3Slide 4Intensity in Single-Slit Diffraction, QualitativelySlide 6Proof of Eqs. 36-5 and 36-6Diffraction by a Circular ApertureResolvabilityDiffraction by a Double SlitDiffraction GratingsWidth of LinesWidth of Lines, cont’dGrating SpectroscopeOptically Variable GraphicsGratings: Dispersion and Resolving PowerProof of Eq. 36-30Proof of Eq. 36-32Dispersion and Resolving Power ComparedX-Ray DiffractionX-Ray Diffraction, cont’dSlide 22hittSlide 241Chapter 36 In Chapter 35, we saw how light beams passing through different slits can interfere with each other and how a beam after passing through a single slit flares-diffracts- in Young's experiment. Diffraction through a single slit or past either a narrow obstacle or an edge produces rich interference patterns. The physics of diffraction plays an important role in many scientific and engineering fields.In this chapter we explain diffraction using the wave nature of light and discuss several applications of diffraction in science and technology. Diffraction36-2Diffraction Pattern from a single narrow slit.Diffraction and the Wave Theory of Light36-CentralmaximumSide or secondarymaximaLightFresnel Bright Spot.BrightspotLightThese patterns cannot be explained using geometrical optics (Ch. 34)!3When the path length difference between rays r1 and r2 is /2, the two rays will be out of phase when they reach P1 on the screen, resulting in destructive interference at P1. The path length difference is the distance from the starting point of r2 at the center of the slit to point b.For D>>a, the path length difference between rays r1 and r2 is (a/2) sin .36-Fig. 36-4Diffraction by a Single Slit: Locating the Minima4Repeat previous analysis for pairs of rays, each separated by a vertical distance of a/2 at the slit.Setting path length difference to /2 for each pair of rays, we obtain the first dark fringes at:36-Fig. 36-5Diffraction by a Single Slit: Locating the Minima, Cont'd(first minimum)sin sin2 2aalq q l= � =For second minimum, divide slit into 4 zones of equal widths a/4 (separation between pairs of rays). Destructive interference occurs when the path length difference for each pair is /2.(second minimum)sin sin 24 2aalq q l= � =Dividing the slit into increasingly larger even numbers of zones, we can find higher order minima:(minima-dark fringes)sin , for 1, 2,3a m mq l= = K5Fig. 36-6To obtain the locations of the minima, the slit was equally divided into N zones, each with width x. Each zone acts as a source of Huygens wavelets. Now these zones can be superimposed at the screen to obtain the intensity as function of , the angle to the central axis.To find the net electric field E (intensity  E2) at point P on the screen, we need the phase relationships among the wavelets arriving from different zones:Intensity in Single-Slit Diffraction, Qualitatively36-phase path length2difference differencepl� � � �� �=� � � �� �� �� � � �( )2sinxpf ql� �D = D� �� �N=18= 0 small1st min.1st sidemax.6Here we will show that the intensity at the screen due to a single slit is:36-Fig. 36-7Intensity in Single-Slit Diffraction, Quantitatively( )2sin (36-5)mI Iaqa� �=� �� �1where sin (36-6)2apa f ql= =, for 1, 2,3m ma p= = KIn Eq. 36-5, minima occur when:sin , for 1, 2,3or sin , for 1,2,3 (minima-dark fringes)am ma m mpp qlq l= == =KKIf we put this into Eq. 36-6 we find:7If we divide slit into infinitesimally wide zones x, the arc of the phasors approaches the arc of a circle. The length of the arc is Em.  is the difference in phase between the infinitesimal vectors at the left and right ends of the arc. is also the angle between the 2 radii marked R.Proof of Eqs. 36-5 and 36-636-Fig. 36-812sin2ERqf =The dash line bisecting f forms two triangles, where:mERf =In radian measure:1212sinmEEqff=Solving the previous 2 equations for E one obtains:( )( )222sin mm mIEI II Eqqaqa� �= � =� �� �The intensity at the screen is therefore:( )2sinapf ql� �=� �� � is related to the path length difference across the entire slit:8Diffraction by a Circular Aperture36-Distant point source, e,g., starlensImage is not a point, as expected from geometrical optics! Diffraction is responsible for this image patterndLightaLightasin 1.22 (1st min.- circ. aperture)dlq =sin 1.22 (1st min.- single slit)alq =9Rayleigh’s Criterion: two point sources are barely resolvable if their angular separation R results in the central maximum of the diffraction pattern of one source’s image is centered on the first minimum of the diffraction pattern of the other source’s image.Resolvability36-Fig. 36-10 small1sin 1.22 1.22 (Rayleigh's criterion)RRd dql lq-� �=� �� ��10Double slit experiment described in Ch. 35 where assumed that the slit width a<<. What if this is not the case? Diffraction by a Double Slit36-Fig. 36-14Two vanishingly narrow slits a<<Single slit a~Two Single slits a~( )( )22sincos (double slit)mI Iaq ba� �=� �� �sindpb ql=sinapa ql=11Device with N slits (rulings) can be used to manipulate light, such as separate different wavelengths of light that are contained in a single beam. How does a diffraction grating affect monochromatic light?Diffraction Gratings36-Fig. 36-17Fig. 36-18sin for 0,1, 2 (maxima-lines)d m mq l= = K12Width of Lines36-Fig. 36-19Fig. 36-20The ability of the diffraction grating to resolve (separate) different wavlength depends on the width of the lines (maxima)13In this course, a sound wave is roughly defined as any longitudinal wave (particles moving along the direction of wave propagation).Width of Lines, cont’d36-Fig. 36-21hwsin , sinhw hwNd q l q qD = D �Dhw (half width of central line)NdlqD =hw (half width of line at )cosNdlq qqD =14Separates different wavelengths (colors) of light into distinct diffraction linesGrating Spectroscope36-Fig. 36-22Fig. 36-2315Gratings embedded in device send out hundreds or even thousands of diffraction orders to produce virtual images that vary with viewing angle. Complicated to design and extremely difficult to counterfeit, so makes an excellent security graphic. Optically Variable Graphics36-Fig. 36-2516Dispersion: the angular spreading of different wavelengths by a gratingGratings: Dispersion and Resolving


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