Lecture07-FourierTransforms-OpticalDiffraction

Lecture07-FourierTransforms-OpticalDiffraction - EMSE 312...

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Unformatted text preview: EMSE 312 Diffraction Principles Lecture 7 Fourier Transforms Optical Diffraction (cont.) EMSE 312 DIFFRACTION PRINCIPLES EMSE 312 Diffraction Principles Fourier Analysis ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 f exp i exp i 2 1 1 exp i3 exp i3 3 3 1 1 exp i5 exp i5 5 5 etc. = + + + + a n n 1 2 3 4 5 6-6-5-4-3-2-1 EMSE 312 Diffraction Principles Fourier Series Fourier Transforms Fourier Coefficients using ( ) ( ) ( ) ( ) ( ) ( ) ( ) n 2 2 2 2 1 a f e x p i n d 2 1 2 f x exp 2 inkx dx 2 1 f x exp 2 inkx dx a k + + + = = = = x 2 = 2 d d x = ngular Space eal Space EMSE 312 Diffraction Principles Fourier Transforms Basic premise: Take a Fourier Series and stretch the wavelength to infinity The Fourier Coefficient becomes a continuous function as the discrete values for 1/n become infinitely close to each other Continuous function in k-space (reciprocal space) ( ) n a a k ( ) ( ) ( ) ( ) ( ) n 2 2 1 a f e x p i n d 2 1 f x exp 2 inkx dx a k + + = = = a n n 1 2 3 4 5 6-6 -5 -4 -3 -2 -1 EMSE 312 Diffraction Principles Fourier Transforms The Fourier Transform of a single slit: ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) { } ( ) ( ) { } ( ) a / 2 a / 2 a / 2 a / 2 f k f x exp 2 ikx dx 1 exp 2 ikx dx 1 1 exp 2 ikx exp ika exp ika 2 ik 2 ik 1 1 1 exp ika exp ika sin ka k 2 i k + + = = = = = = a 0; x a / 2 f (x) 1; a / 2 x a / 2 x a / 2 < = < x 1 a 2 a 2 Transmission function EMSE 312 Diffraction Principles Fourier Transforms The Fourier Transform of a single slit: for or for or for or for or ( ) sin ka = ( ) ( ) ( ) k k x sin x 1 a lim sin ka lim sin ka a lim a k ka x = = = a 0; x a / 2 f (x) 1; a / 2 x a / 2 a a / 2 < = < x 1 a 2 a 2 Transmission function ka n 2 = k 2 n / a = ( ) sin ka 1 = ( ) sin ka = ( ) sin ka 1 = ka n 2 / 2 = + ka n 2 = + ka n 2 3 / 2 = + ( ) k 2 n 1 / 2 / a = + ( ) k 2 n 1 / a = + ( ) k 2 n 3 / 2 / a = + EMSE 312 Diffraction Principles Fourier Transforms The Fourier Transform of a single slit: for or for or ( ) 2 sin ka = ( ) ( ) ( ) ( ) 2 2 2 2 1 1 I k f k sin ka sin ka k k = = = a 0; x a / 2 f (x) 1; a / 2 x a / 2 a a / 2 < = < x 1 a 2 a 2 Transmission function ka n = k n / a = ( ) 2 sin ka 1 = ka (n 1/ 2) = + ( ) k n 1 / 2 / a = + EMSE 312 Diffraction Principles Fourier Transforms...
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This note was uploaded on 12/01/2011 for the course EMSE 312 taught by Professor Lagerlof,p during the Spring '08 term at Case Western.

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Lecture07-FourierTransforms-OpticalDiffraction - EMSE 312...

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