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Eades / Fourier Imaging PASI Santiago, Chile July 2006 1 Fourier Transforms and Images

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 2 Our aim is to make a connection between diffraction and imaging - and hence to gain important insights into the process

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 3 What happens to the electrons as they go through the sample?

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 4

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 5 What happens to the electrons a) The electrons in the incident beam are scattered into diffracted beams. b) The phase of the electrons is changed as they go through the sample. They have a different kinetic energy in the sample, this changes the wavelength, which in turn changes the phase.

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 6 The two descriptions are alternative descriptions of the same thing. Therefore, we must be able to find a way of linking the descriptions. The link is the Fourier Transform.

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 7 A function can be thought of as made up by adding sine waves. A well-known example is the Fourier series. To make a periodic function add up sine waves with wavelengths equal to the period divided by an integer.

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 8 Reimer: Transmission Electron Microscopy

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 9 The Fourier Transform The same idea as the Fourier series but the function is not periodic, so all wavelengths of sine waves are needed to make the function

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 10 The Fourier Transform Fourier series Fourier transform

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 11 So think of the change made to the electron wave by the sample as a sum of sine waves. But each sine wave term in the sum of waves is equivalent to two plane waves at different angles This can be seen from considering the Young's slits experiment - two waves in different directions make a wave with a sine modulation

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 12 Original figure by Thomas Young, courtesy Bradley Carroll

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 13 Bradley Carroll

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 14

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 15 This analysis tells us that a sine modulation - produced by the sample - with a period d, will produce scattered beams at angles where d and are related by 2d sin we have seen this before

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 16 Bragg’s Law 2d sin θ = λ tells us where there are diffracted beams.

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 17 What does a lens do? A lens brings electrons in the same direction at the sample to the same point in the focal plane Direction at the sample corresponds to position in the diffraction pattern - and vice versa

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 18 Sample Back focal plane Lens Image

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 20

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 21 The Fourier Transform Fourier series Fourier transform

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 22

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 23

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 24 Optical Transforms Taylor and Lipson 1964

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 25 Convolution theorem

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 26 Atlas of Optical Transforms Harburn, Taylor and Welberry 1975

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 27 Atlas of Optical Transforms Harburn, Taylor and Welberry 1975

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 28 Atlas of Optical Transforms Harburn, Taylor and Welberry 1975

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 29 Atlas of Optical Transforms Harburn, Taylor and Welberry 1975

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 30 Optical Transforms Taylor and Lipson 1964

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 31 Optical Transforms Taylor and Lipson 1964

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 32 Optical Transforms Taylor and Lipson 1964

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 33 Atlas of Optical Transforms Harburn, Taylor and Welberry 1975

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Eades / Fourier Imaging PASI Santiago, Chile July 2006 34 Atlas of Optical Transforms Harburn, Taylor and Welberry 1975

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