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Chapter 11: Fraunhofer Diffraction Diffraction is… - a consequence of the wave nature of light - an interference effect - any deviation from geometrical optics resulting from obstruction of the wavefront Diffraction is… interference on the edge …on the edge of sea …on the edge of night …on the edge of dawn …in the skies …in the heavens …on the edge of the shadows …on the edge of the shadows With and without diffraction The double-slit experiment interference explains the fringes -narrow slits or tiny holes -separation is the key parameter -calculate optical path difference D diffraction shows how the size/shape of the slits determines the details of the fringe pattern Josepf von Fraunhofer (1787-1826) Fraunhofer diffraction - far-field - plane wavefronts at aperture and obserservation - moving the screen changes size but not shape of diffraction pattern Next week: Fresnel (near-field) diffraction Diffraction from a single slit slit rectangular aperture, length >> width Diffraction from a single slit plane waves in - consider superposition of segments of the wavefront arriving at point P - note optical path length differences D Huygens’ principle every point on a wavefront may be regarded as a secondary source of wavelets curved wavefront: planar wavefront: c Dt Not any more!! obstructed wavefront: In geometrical optics, this region should be dark (rectilinear propagation). Ignore the peripheral and back propagating parts! Diffraction from a single slit for each interval ds: E L ds dEP r i ( kr t ) e Let r = r0 for wave from center of slit (s=0). Then: E L ds d E P r0 D i ( k r0 D t ) e where D is the difference in path length. -negligible in amplitude factor -important in phase factor EL (field strength) constant for each ds Get total electric field at P by integrating over width of the slit Diffraction from a single slit After integrating: EP E L b sin r0 e i ( kr 0 t ) where b is the slit width and 2 E0 Irradiance: 0c I 2 1 2 0c E Lb E0 2 r0 sin 2 2 I0 I I 0 sinc 2 2 kb sin Recall the sinc function 1 for = 0 sinc sin zeroes occur when sin = 0 i.e. when 1 kb sin m 2 where m = ±1, ±2, ... Recall the sinc function sinc sin maxima/minima when d sin d cos sin cos sin 0 2 sin cos tan Diffraction from a single slit I I 0 sinc 2 Central maximum: image of slit angular width D 2 b hence as slit narrows, central maximum spreads Beam spreading angular spread of central maximum independent of distance Aperture dimensions determine pattern Aperture dimensions determine pattern I I 0 sinc 2 sinc 2 where kb ka sin 2 2 sin Aperture shape determines pattern Irradiance for a circular aperture 2 J 1 I I 0 2 J1(): 1st order Bessel function Friedrich Bessel (1784 – 1846) where 1 kD sin 2 and D is the diameter Irradiance for a circular aperture Central maximum: Airy disk circle of light; “image” of aperture angular radius D 1/ 2 1 . 22 D hence as aperture closes, disk grows How else can we obstruct a wavefront? Any obstacle that produces local amplitude/phase variations create patterns in transmitted light Diffractive optical elements (DOEs) Diffractive optical elements (DOEs) Phase plates change the spatial profile of the light Demo Resolution Sharpness of images limited by diffraction Inevitable blur restricts resolution Resolution measured from a ground-based telescope, 1978 Charon Pluto Resolution measured from the Hubble Space Telescope, 2005 http://apod.nasa.gov/apod/ap060624.html Rayleigh’s criterion for just-resolvable images D min 1 . 22 D where D is the diameter of the lens Imaging system (microscope) x min f D min 1 . 22 f D - where D is the diameter and f is the focal length of the lens - numerical aperture D/f (typical value 1.2) x min Test it yourself! visual acuity Test it yourself! Double-slit diffraction considering the slit width and separation Double-slit diffraction I 4 I 0 sinc 2 1 2 cos 2 1 2 single-slit diffraction double-slit interference kb sin ka sin Double-slit diffraction I 4 I 0 sinc 2 cos 2 Double-slit diffraction Double-slit diffraction sin 2 cos 4 I 0 2 IP single slit two beam diffraction interference Multiple-slit diffraction IP sin I 0 single slit diffraction 2 sin N sin( ) 2 multiple beam interference Importance of spatial coherence 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 finescale fringes, and a one-slit pattern will be observed. Max Fraunhofer diffraction patterns Good spatial coherence Poor spatial coherence The double slit and quantum mechanics 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 Dimming the incident light: Each individual photon goes through both slits! How can a particle go through both slits? “Nobody knows, and it’s best if you try not to think about it.” Richard Feynman Exercises You are encouraged to solve all problems in the textbook (Pedrotti3). The following may be covered in the werkcollege on 12 October 2011: Chapter 11: 1, 3, 4, 10, 12, 13, 22, 27