Fourier Transforms and Images
Our aim is to make a connection between diffraction and imaging - and hence to gain important insights into the process
What happens to the electrons as they go through the sample?
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.
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.
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.
Reimer: Transmission Electron Microscopy
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
The Fourier Transform Fourier series Fourier transform
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
Original figure by Thomas Young, courtesy Bradley Carroll
Bradley Carroll
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
Bragg’s Law Bragg’s Law 2d sin θ = λ tells us where there are diffracted beams.
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
Sample Back focal plane Lens Image
The Fourier Transform Fourier series Fourier transform
Optical Transforms Taylor and Lipson 1964
Convolution theorem
Atlas of Optical Transforms Harburn, Taylor and Welberry 1975
Atlas of Optical Transforms Harburn, Taylor and Welberry 1975
Atlas of Optical Transforms Harburn, Taylor and Welberry 1975
Atlas of Optical Transforms Harburn, Taylor and Welberry 1975
Optical Transforms Taylor and Lipson 1964
Optical Transforms Taylor and Lipson 1964
Optical Transforms Taylor and Lipson 1964
Atlas of Optical Transforms Harburn, Taylor and Welberry 1975
Atlas of Optical Transforms Harburn, Taylor and Welberry 1975