By Keigo Iizuka; John Wiley & Sons, Inc
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Additional info for Elements of Photonics, Volume I : In Free Space and Special Media
We will demonstrate that the diffraction pattern can be elegantly expressed by the Fourier transform of the source. Let E xi , yi represent the field at point P on the screen placed a distance zi away from the source field E x0 , y0 . The distributed source E x0 , y0 is considered as an ensemble of point sources. Each point source radiates a spherical wave. The field at the observation point P is comprised of contributions from an ensemble of fields radiated from all the point sources. 26 where E x0 , y0 is the magnitude of the point source located at x0 , y0 and r is the distance between x0 , y0 and xi , yi .
96) is shown in Fig. 11b. The similarity theorem for the Fourier–Bessel transform does not follow Eq. 98 a The circle function and its Fourier transform are shown in Figs. 11b, respectively. 100 where ri is the radial coordinate in the plane of the diffraction pattern. This diffraction pattern was first derived by Sir George Biddell Airy and is referred to as the Airy pattern. The diffraction photograph is shown in Fig. 11c. 5 Delta Function The delta function υ x is conveniently made to represent a point source .
Next, the Fourier transform of the shah function will be derived. 120 nD 1 where an D 1 T T/2 υxe j2 n/T x dx D 1 T/2 and where T is the period of the delta functions and is unity. Thus, Eq. 121 nD 1 Whenever x is an integer, x becomes infinite, so that Eq. 121) constitutes an array of delta functions spaced by unity. 121) is an alternate expression for the shah function. Thus, using Eqs. 122b af x is a very special function in that the Fourier transform is the same as the function itself. 7 Diffraction from an Infinite Array of Similar Apertures with Regular Spacing Making use of the shah function, the diffraction pattern will be calculated for a one-dimensional array of slits such as shown in Fig.
Elements of Photonics, Volume I : In Free Space and Special Media by Keigo Iizuka; John Wiley & Sons, Inc