1. Fourier Transforms and Images

2. Our aim is to make a connection between diffraction and imaging
- and hence to gain important insights into the process

3. What happens to the electrons as they go through the sample?

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. 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. 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. Reimer:
Transmission Electron Microscopy

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. The Fourier Transform
Fourier series
Fourier transform

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. Original figure by Thomas Young, courtesy Bradley Carroll

13. Bradley Carroll

15. This analysis tells us that a sine modulation - produced by the sample - with a period d, will produce scattered beams at angles q, where d and q are related by
2d sin q = l
we have seen this before

16. Bragg’s Law Bragg’s Law 2d sin θ = λ
tells us where there are diffracted beams.

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. Sample
Back focal plane
Lens
Image

21. The Fourier Transform
Fourier series
Fourier transform

24. Optical Transforms
Taylor and Lipson 1964

25. Convolution theorem

26. Atlas of Optical Transforms Harburn, Taylor and Welberry 1975

27. Atlas of Optical Transforms Harburn, Taylor and Welberry 1975

28. Atlas of Optical Transforms Harburn, Taylor and Welberry 1975

29. Atlas of Optical Transforms Harburn, Taylor and Welberry 1975

30. Optical Transforms
Taylor and Lipson 1964

31. Optical Transforms
Taylor and Lipson 1964

32. Optical Transforms
Taylor and Lipson 1964

33. Atlas of Optical Transforms Harburn, Taylor and Welberry 1975

34. Atlas of Optical Transforms Harburn, Taylor and Welberry 1975