30. Diffraction and the Fourier Transform

30. Diffraction and the Fourier Transform Diffraction examples Diffraction by an edge Arago spot The far-field Fraunhofer Diffraction Some examples Simeon Poisson Francois Arago (1781 - 1840) (1786 - 1853). Reminder: Fresnel-Kirchoff Diffraction aperture (x0, y0). Coordinates: (x1, y1). the plane of the aperture: x1, y1 r01 z 2 x0 x1 y0 y1 . 2 2. the plane of observation: x0, y0. z (a distance z downstream). observation region x12 2 x0 x1 y12 2 y0 y1 . E x0 , y0 . exp jk . 2 z . 2 z Aperture x1 , y1 E x1 , y1 dx1dy1.. Quadratics in the exponent: a messy integral A plane wave incident on a sharp edge Fresnel Diffraction integral, in one dimension: .. 1 x x 2 . E ( x0 ) exp jkz exp jk 0 1 dx1. j z 2 z . x1 0. geometrical exact Diffraction result shadow Intensity The irradiance exactly at the edge is 25% of the value far from the edge. position x0. Diffraction by an Edge Light passing by an edge Electrons passing by an edge An interesting manifestation of Diffraction effects: The Spot of Arago If a beam encounters a circular stop , it develops a hole, which fills in as it propagates and diffracts: r r Eincident E(r,0) E(r,z).

spherical wave’s point of origin to a point on the central axis of the disc is the same. Constructive interference on the center line! In 1818, Poisson used Fresnel’s theory to predict this phenomenon. He regarded this as proof that Fresnel’s wave theory was nonsense, and that light must be a particle and not a wave. But almost immediately,

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