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Diffraction around an edge and through an aperture Huygens’ principle is consistent with diffraction: The larger the wavelength with respect to the aperture size, the more significant the diffraction Young’s Double Slit Experiment If light is a wave, interference effects will be seen, where one part of wavefront can interact with another part. One way to study this is to do a double-slit experiment: Two-Source Interference Pattern Nodal points – blue dots Anti-nodal points – red dots Young’s Double Slit Experiment If light is a wave, there should be an interference pattern Path Difference In a double-slit experiment, what 1) there is no difference path difference have the waves 2) half a wavelength from each slit traveled to give a 3) one wavelength minimum at the indicated 4) three wavelengths position? 5) more than three wavelengths Intensity Path Difference 1) there is no difference In a double-slit experiment, what 2) half a wavelength path difference have the waves 3) one wavelength from each slit traveled to give a minimum at the indicated 4) three wavelengths position? 5) more than three wavelengths 2 3 Intensity For Destructive Interference = 1/2 , 3/2 , 5/2 , 7/2 , … 3/2 5/2 7/2 /2 = (m + 1/2) Diffraction by a Single Slit or Disk bright dark Note: There are bright and dark fringes beyond the shadow. These resemble interference fringes of a double slit. Diffraction by a Single Slit or Disk Wavelets from the center and edge of the slit Contrary to expectation, an interference pattern is observed because the wave Dark Band (Interference of front approaching slit, D, is itself a D Y wavelets) source of wavelets. It is these individual D D wavelets that will interfere destructively Path difference = m/2 at the part of the screen where the dark band is observed. Wave front going through the slit is made up of many wavelets. Now from the diagram the angle to the X first dark band is given by: sinD = (/2) / (D/2) = / D If D is small we can use….. similar triangles, for small angles Therefore for any dark band sin D = tan D = Y / X sin D = m / D so….. Y/X = m/D m = 1,2,3……. or…. Y =mX/D m = 1,2,3……. Diffraction by a Single Slit or Disk If we look at the intensity of the bright bands we find that they are not the same. The brightest band is in the center and the bands seem to get dimmer as the angle increases. The diagram shows how the intensity changes with angle. You can see that as sin approaches / D the intensity falls to zero as expected from our theory. Also note that the second band is much less intense than the central Practice Problem band and is half as wide. Light of wavelength 750 nm passes through a slit 1.0 x 10-3 mm wide. How wide is the central maximum a) in degrees and b) in centimeters on a screen 20 cm away? = 750 x 10-9 m D = 1.0 x 10-6 m X = 0.2 m away m=1 Width of central maximum is 2 or 2Y a) Width = 2 = 2 sin-1 (m / D) = 2 sin-1 (1) (750 x 10-9m) / (1.0 x 10-6 m ) = 97.20 b) Width = 2Y = 2 X tan D = 2 (0.2) tan 48.590 = 0.45 m Check Your Understanding The diffraction pattern below arises 1) narrow the slit from a single slit. If we would like 2) widen the slit to sharpen the pattern, i.e., make 3) enlarge the screen the central bright spot narrower, what should we do to the slit width? 4) close off the slit Check Your Understanding The diffraction pattern below arises 1) narrow the slit from a single slit. If we would like 2) widen the slit to sharpen the pattern, i.e., make 3) enlarge the screen the central bright spot narrower, what should we do to the slit width? 4) close off the slit The angle at which one finds the first minimum is: sin = /d The central bright spot can be d narrowed by having a smaller angle. This in turn is accomplished by widening the slit. Check Your Understanding Blue light of wavelength passes through a single slit of width d and 1) d/4 forms a diffraction pattern on a screen. 2) d/2 If the blue light is replaced by red light 3) no change needed of wavelength 2, the original diffraction 4) 2 d pattern can be reproduced if the slit width is changed to: 5) 4 d Check Your Understanding Blue light of wavelength passes through a single slit of width d and 1) d/4 forms a diffraction pattern on a screen. 2) d/2 If the blue light is replaced by red light 3) no change needed of wavelength 2, the original diffraction 4) 2 d pattern can be reproduced if the slit width is changed to: 5) 4 d d sin = m (minima) If 2 then we must have d 2d for sin to remain unchanged d (and thus give the same diffraction pattern). When violet light of wavelength 415 nm falls on a single slit, it creates a central diffraction peak that is 9.20 cm wide on a screen that is 2.55 m away. How wide is the slit? = 415 x 10-9 m Y = 4.6 x 10-2 m X = 2.55 m away m=1 D=? Because X is much bigger than Y, D is small, so we can use….. Y =mX/D m = 1,2,3……. D = mX/Y = (1) (415 x 10-9 m) (2.55 m) / (4.66 x 10-2 m) D = 2.30 x 10-5 m = 2.30 x 10-2 mm Rayleigh Criterion for Resolution of Two Diffracted Images Two images are said to be just resolved when the central maximum of one image falls on the first minimum of the diffraction pattern of the other image. From single slit diffraction sinD = / D If << d, sinD ≈ D min = / D Where min is in radians The apertures of microscopes and telescopes are circular. Analysis shows… min = 1.22 / D Check Your Understanding An optical telescope has a 21 cm 1) bigger mirror diameter. To give the same 2) smaller angular resolution as an optical 3) the same telescope the effective diameter of a radio telescope would have to be…. 4) doesn’t matter Check Your Understanding An optical telescope has a 21 cm 1) bigger mirror diameter. To give the same 2) smaller angular resolution as an optical 3) the same telescope the effective diameter of a radio telescope would have to be…. 4) doesn’t matter The minimum angle of resolution is given by: min = 1.22 /d Radio waves have a much longer wavelength, so to give the same minimum angle of resolution the dish diameter has to be much larger Check Your Understanding A scientist wants to observe finer 1) Make the objective bigger detail on a specimen under a 2) Use smaller wavelengths microscope. They could…. 3) Use oil between the specimen and the objective 4) All of the above Check Your Understanding A scientist wants to observe finer 1) Make the objective bigger detail on a specimen under a 2) Use smaller wavelengths microscope. They could…. 3) Use oil between the specimen and the objective 4) All of the above The minimum angle of resolution is Spherical and chromatic given by: aberration however limit min = 1.22 /d the size of the objective, d. Also UV is absorbed To make min smaller you would have by glass. to make the objective lens bigger or Electrons can be used use a smaller wavelength. Oil has a because of their wave higher n than air so n = air / n. properties (E = h f) Two of Jupiter’s largest moons (it has 16 or more) are separated by a distance of 3.0 x 106 km at a time when the planet is 3.3 x 108 km from Earth. a) Would a person be able to resolve both moons with the unaided eye, assuming night-time pupil diameter of 7.5 mm? b) What minimum diameter of mirror would be needed in a radio telescope? Use: light = 550 nm, radio = 21 cm and s = r = 550 x 10-9 m s = 3.0 x 109 m r = 3.3 x 1011 m d = 7.5 x 10-3 m dmin = ? =s/r = (3.0 x 109 m / (3.3 x 1011 m) = 0.009 rad a) min = 1.22 / d = 1.22 (550 x 10-9 m / (7.5 x 10-3 m) = 0.000089 rad Yes > min , so a person would be able to resolve both moons b) dmin = 1.22 / = 1.22 (21 x 10-2 m) / 0.009 = 28 m Now you can see why big dishes are needed for radio telescopes A camera on a spy satellite orbiting at 200 km has a diameter of 35 cm. What is the smallest distance this camera can resolve on the surface of the earth? Use: light = 550 nm, and s = r = 550 x 10-9 m s=? r = 200 x 103 m d = 35 x 10-2 m min = 1.22 / d = 1.22 (550 x 10-9 m / (35 x 10-2 m) = 1.92 x 10-6 rad s = r min = (200 x 103 m)(1.92 x 10-6 ) = 0.38 m (38 cm) This size of lens would not be good enough to read license plate numbers but good enough to resolve individual people.

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posted: | 11/9/2011 |

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