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

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Presentation on theme: "Eades / Fourier Imaging PASI Santiago, Chile July 2006 1 Fourier Transforms and Images."— Presentation transcript:

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

2 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

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

4 Eades / Fourier Imaging PASI Santiago, Chile July 2006 4

5 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.

6 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.

7 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.

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

9 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

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

11 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

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

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

14 Eades / Fourier Imaging PASI Santiago, Chile July 2006 14

15 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

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

17 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

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

19 Eades / Fourier Imaging PASI Santiago, Chile July 2006 19

20 Eades / Fourier Imaging PASI Santiago, Chile July 2006 20

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

22 Eades / Fourier Imaging PASI Santiago, Chile July 2006 22

23 Eades / Fourier Imaging PASI Santiago, Chile July 2006 23

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

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

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

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

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

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

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

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

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

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

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


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