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Chapter 11: Fraunhofer Diffraction

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Diffraction is Diffraction is interference on the edge -a consequence of the wave nature of light -an interference effect -any deviation from geometrical optics resulting from obstruction of the wavefront

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on the edge of sea

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on the edge of night

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on the edge of dawn

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in the skies

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in the heavens

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on the edge of the shadows

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With and without diffraction

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The double-slit experiment interference explains the fringes - narrow slits or tiny holes -separation is the key parameter -calculate optical path difference diffraction shows how the size/shape of the slits determines the details of the fringe pattern

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Josepf von Fraunhofer (1787-1826)

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-far-field -plane wavefronts at aperture and obserservation -moving the screen changes size but not shape of diffraction pattern Fraunhofer diffraction Next week: Fresnel (near-field) diffraction

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Diffraction from a single slit slit rectangular aperture, length >> width

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Diffraction from a single slit plane waves in - consider superposition of segments of the wavefront arriving at point P - note optical path length differences

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Huygens principle every point on a wavefront may be regarded as a secondary source of wavelets planar wavefront: ctct curved wavefront: In geometrical optics, this region should be dark (rectilinear propagation). Ignore the peripheral and back propagating parts! obstructed wavefront: Not any more!!

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Diffraction from a single slit for each interval ds: Let r = r 0 for wave from center of slit (s=0). Then: where is the difference in path length. -negligible in amplitude factor -important in phase factor E L (field strength) constant for each ds Get total electric field at P by integrating over width of the slit

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Diffraction from a single slit where b is the slit width and Irradiance: After integrating:

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Recall the sinc function 1 for = 0 zeroes occur when sin = 0 i.e. when where m = 1, 2,...

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Recall the sinc function maxima/minima when

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Diffraction from a single slit Central maximum: image of slit angular width hence as slit narrows, central maximum spreads

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Beam spreading angular spread of central maximum independent of distance

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Aperture dimensions determine pattern

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where

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Aperture shape determines pattern

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Irradiance for a circular aperture J 1 ( ) : 1 st order Bessel function where and D is the diameter Friedrich Bessel (1784 1846)

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Irradiance for a circular aperture Central maximum: Airy disk circle of light; image of aperture angular radius hence as aperture closes, disk grows

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How else can we obstruct a wavefront? Any obstacle that produces local amplitude/phase variations create patterns in transmitted light

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Diffractive optical elements (DOEs)

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Phase plates change the spatial profile of the light

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Demo

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Resolution Sharpness of images limited by diffraction Inevitable blur restricts resolution

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Resolution measured from a ground-based telescope, 1978 Pluto Charon

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Resolution http://apod.nasa.gov/apod/ap060624.html measured from the Hubble Space Telescope, 2005

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Rayleighs criterion for just-resolvable images where D is the diameter of the lens

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Imaging system (microscope) - where D is the diameter and f is the focal length of the lens - numerical aperture D/f (typical value 1.2)

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Test it yourself! visual acuity

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Test it yourself!

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Double-slit diffraction considering the slit width and separation

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Double-slit diffraction single-slit diffraction double-slit interference

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Double-slit diffraction

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Multiple-slit diffraction Double-slit diffraction single slit diffraction multiple beam interference single slit diffraction two beam interference

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If the spatial coherence length is less than the slit separation, then the relative phase of the light transmitted through each slit will vary randomly, washing out the fine- scale fringes, and a one-slit pattern will be observed. Fraunhofer diffraction patterns Good spatial coherence Poor spatial coherence Importance of spatial coherence Max

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Imagine using a beam so weak that only one photon passes through the screen at a time. In this case, the photon would seem to pass through only one slit at a time, yielding a one-slit pattern. Which pattern occurs? Possible Fraunhofer diffraction patterns Each photon passes through only one slit Each photon passes through both slits The double slit and quantum mechanics

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Each individual photon goes through both slits! Dimming the incident light: The double slit and quantum mechanics

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How can a particle go through both slits? Nobody knows, and its best if you try not to think about it. Richard Feynman

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Exercises You are encouraged to solve all problems in the textbook (Pedrotti 3 ). The following may be covered in the werkcollege on 12 October 2011: Chapter 11: 1, 3, 4, 10, 12, 13, 22, 27