Fourier Methods Fraunhofer diffraction = Fourier transform Convolution theorem easy solution to difficult diffraction problems (double slit of finite slit width, diffraction grating) Fourier Methods u p = - i ⇤ η(⇥ i , ⇥ o ) u s (x, y) r e ikr dS Fresnel-Kirchhoff diffraction integral Fraunhofer diffraction in 1D ➙simplifies to β = k sin ⇥ with Note: Us(β) is the Fourier Transform of us(x) The Fraunhofer diffraction pattern is the Fourier transform of the amplitude function leaving the diffracting aperture u p ∝ U s (β )= u s (x)e iβx dx u s (x) Fourier Transform time t and angular frequency ω U (⇥) = ⇥ -⇥ u(t)e iωt dt u(t) = 1 2π ⇥ -⇥ U (⇥)e -iωt d⇥ Fourier transform inverse transform coordinate x and spatial frequency β: U (β ) = ⇥ -⇥ u(x)e iβx dx u(x) = 1 2⇥ ⇥ -⇥ U (β )e -iβx dβ Fourier transform inverse transform (ω,t)→(β,x) Fourier Methods Extension to two dimensions spatial frequencies β x = k sin ⇤ β y = k sin ⇥ [β] = rad / m u p ∝ U (β x , β y )= u s (x, y)e i(β x x+β y y) dxdy
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s ikr Fourier Methods i o...2011/06/08 · Spatial Filtering: Schlieren Photography phase ! amplitude modulation Summary of Lecture 9 Division of wavefront Spatial frequencies and
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