By J. M. Cowley

The 1st version of this hugely profitable e-book seemed in 1975 and developed from lecture notes for sessions in actual optics, diffraction physics and electron microscopy given to complex undergraduate and graduate scholars. The e-book bargains with electron diffraction and diffraction from disordered or imperfect crystals and hired an strategy utilizing the Fourier rework from the start rather than as an extension of a Fourier sequence treatment.This 3rd revised variation is a significantly rewritten and up to date model which now comprises all very important advancements that have taken position in recent times.

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CHAPTER 2 Fourier transforms and convolutions 2. I. i. Delta-functions and discontinuities In t h i s c h a p t e r we p r o v i d e the m a t h e m a t i c a l b a c k g r o u n d for m u c h of w h a t f o l l o w s . M o s t c o n s i d e r a t i o n s of k i n e m a t i c a l d i f f r a c t i o n i n v o l v e the use of the F o u r i e r t r a n s f o r m in one f o r m or a n o t h e r . One of the m o s t i m p o r t a n t p r o p e r t i e s of the F o u r i e r t r a n s f o r m i n v o l v e s the c o n c e p t of a C o n v o l u t i o n , or C o n v o l u t i o n i n t e g r a l , a n d for b o t h t h i s a n d the F o u r i e r t r a n s f o r m it is c o n v e n i e n t to use d e l t a - f u n c t i o n s .

7. 22 A / ' I I ' " / l I/a Fig. 7. The diffraction pattern from a row of circular holes equally spaced in an opaque screen. The widths of the lines are made to suggest their intensities. 3 F O U R I E R TRANSFORMS AND DIFFRACTION: EXAMPLES 47 and the Fourier t r a n s f o r m with respect to y is unity. Hence the Fourier t r a n s f o r m in x and y is a set of straight lines parallel to the v axis and equally spaced at intervals of a -I in the u direction. This distribution will be m o d u l a t e d by m u l t i p l y i n g by the Fourier t r a n s f o r m of O(x,y), namely (ru42/2) J1 (nAU) / (nAU) , where A is the diameter of the holes and U = (u2+v2) I/2 .

FOURIER TRANSFORMS Row circular of AND CONVOLUTIONS CH. 2 holes The e x t e n s i o n to two d i m e n s i o n s a l l o w s the a b o v e r e s u l t s to be a p p l i e d to c a l c u l a t e the d i f f r a c t i o n p a t t e r n s f r o m m a n y s i m p l e t w o - d i m e n s i o n a l d i s t r i b u t i o n s . However, in u s i n g d e l t a f u n c t i o n s a n d c o n v o l u t i o n s it is i m p o r t a n t to d i s t i n g u i s h c a r e f u l l y the d e l t a functions, a n d c o n v o l u t i o n s in one a n d two d i m e n s i o n s .