Physics 2102 Jonathan Dowling Lecture 27 THU 29 APR 2010 Ch 36 Diffraction Radar The Smaller The Wavelength the Better The Targeting Resolution X band 10cm Ka band 1cm K band 2cm Laser 1 m Angles of the Secondary Maxima The diffraction minima are precisely at the angles where sin p a and p so that sin 0 However the diffraction maxima are not quite at the angles where sin p a and p so that sin 1 0 05 0 04 0 03 1 633 nm a 0 2 mm sin I I max 2 0 02 0 01 2 0 005 3 0 01 p p a Max 1 0 00475 0 00453 2 0 00791 0 00778 3 0 01108 0 01099 4 0 01424 0 01417 5 0 01741 0 01735 4 0 015 5 0 02 radians 0 025 0 03 To find the maxima one must look near sin p a for places where the slope of the diffraction pattern goes to zero i e where d sin 2 d 0 This is a transcendental equation that must be solved numerically The table gives the Max solutions Note that Max p a Example Diffraction of a laser through a slit 1 2 cm Light from a helium neon laser 633 nm passes through a narrow slit and is seen on a screen 2 0 m behind the slit The first minimum of the diffraction pattern is observed to be located 1 2 cm from the central maximum How wide is the slit 1 y1 0 012 m 0 0060 rad L 2 00 m 6 33 10 7 m 4 a 1 06 10 m 0 106 mm 3 sin 1 1 6 00 10 rad Width of a Single Slit Diffraction Pattern y1 0 y1 y2 y3 w p L yp a w 2 L a p 1 2 3 positions of dark fringes width of diffraction peak from min to min 1m 1000m 10 m 1m You are doing 137 mph on I 10 and you pass a little old lady doing 55mph when a cop Located 1km away fires his radar gun which has a 10 cm opening Can he tell you from the L O L if the gun Is X band What about Laser Laser band 1 m X band 10cm w 2 L 2 0 1m 1000m 2000m a 0 1m No Beam is much wider than car separation too wide to tell who is who w 2 L 2 0 000001m 1000m 0 02m a 0 1m Yes Beam width is much less than car separation Exercise 1 2 Two single slit diffraction patterns are shown The distance from the slit to the screen is the same in both cases Which of the following could be true a The slit width a is the same for both 1 2 b The slit width a is the same for both 1 2 c The wavelength is the same for both width a1 a2 d The slit width and wavelength is the same for both p1 p2 e The slit width and wavelength is the same for both p1 p2 Combined Diffraction and Interference So far we have treated diffraction and interference independently However in a two slit system both phenomena should be present together sin 2 I 2slit 4 I1slit cos a a y sin L d d y sin L 2 d a a Interference Only degrees Diffraction Only degrees Both degrees Notice that when d a is an integer diffraction minima will fall on top of missing interference maxima Diffraction Gratings A device 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 Fig 36 17 Fig 36 18 d sin m for m 0 1 2 maxima lines 36 11 Circular Apertures Single slit of aperture a Hole of diameter D When light passes through a circular aperture instead of a vertical slit the diffraction pattern is modified by the 2D geometry The minima occur at about 1 22 D instead of a Otherwise the behavior is the same including the spread of the diffraction pattern with decreasing aperture The Rayleigh Criterion The Rayleigh Resolution Criterion says that the minimum separation to separate two objects is to have the diffraction peak of one at the diffraction minimum of the other i e 1 22 D Example The Hubble Space Telescope has a mirror diameter of 4 m leading to excellent resolution of close lying objects For light with wavelength of 500 nm the angular resolution of the Hubble is 1 53 x 10 7 radians Example A spy satellite in a 200km low Earth orbit is imaging the Earth in the visible wavelength of 500nm How big a diameter telescope does it need to read a newspaper over your shoulder from Outer Space Example Solution 1 22 D The smaller the wavelength or the bigger the telescope opening the better the angular resolution Letters on a newspaper are about x 10mm apart Orbit altitude R 200km D is telescope diameter R Formula x R R 1 22 D D R 1 22 x x 200x103m 1 22x500x10 9m 10X10 3m 12 2m Los Angeles from Space Corona Declassified Spy Photo Circa 1960 s Some Professor s House Google Maps
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