Download - Fourier Transforms and Images
Eades / Fourier Imaging PASI Santiago, Chile July 20061
Fourier Transforms and Images
Eades / Fourier Imaging PASI Santiago, Chile July 20062
Our aim is to make a connection between diffraction and imaging
- and hence to gain important insights into the process
Eades / Fourier Imaging PASI Santiago, Chile July 20063
What happens to the electrons as they go through the sample?
Eades / Fourier Imaging PASI Santiago, Chile July 20064
Eades / Fourier Imaging PASI Santiago, Chile July 20065
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.
Eades / Fourier Imaging PASI Santiago, Chile July 20066
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.
Eades / Fourier Imaging PASI Santiago, Chile July 20067
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.
Eades / Fourier Imaging PASI Santiago, Chile July 20068
Reimer:Transmission Electron Microscopy
Eades / Fourier Imaging PASI Santiago, Chile July 20069
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
Eades / Fourier Imaging PASI Santiago, Chile July 200610
The Fourier Transform
Fourier series
Fourier transform
dx.iux2exp)u(F)x(f
f t F i t d( ) ( ) exp .
2
F t F n tnn
( ) cos( )
0
2
Eades / Fourier Imaging PASI Santiago, Chile July 200611
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
Eades / Fourier Imaging PASI Santiago, Chile July 200612
Original figure by Thomas Young, courtesy Bradley Carroll
Eades / Fourier Imaging PASI Santiago, Chile July 200613
Bradley Carroll
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Eades / Fourier Imaging PASI Santiago, Chile July 200615
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
Eades / Fourier Imaging PASI Santiago, Chile July 200616
Bragg’s Law
Bragg’s Law
2d sin θ = λ
tells us where there are diffracted beams.
Eades / Fourier Imaging PASI Santiago, Chile July 200617
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
Eades / Fourier Imaging PASI Santiago, Chile July 200618
Sample
Back focal plane
Lens
Image
Eades / Fourier Imaging PASI Santiago, Chile July 200619
Eades / Fourier Imaging PASI Santiago, Chile July 200620
Eades / Fourier Imaging PASI Santiago, Chile July 200621
The Fourier Transform
Fourier series
Fourier transform
dx.iux2exp)u(F)x(f
f t F i t d( ) ( ) exp .
2
F t F n tnn
( ) cos( )
0
2
Eades / Fourier Imaging PASI Santiago, Chile July 200622
Eades / Fourier Imaging PASI Santiago, Chile July 200623
Eades / Fourier Imaging PASI Santiago, Chile July 200624
Optical Transforms Taylor and Lipson 1964
Eades / Fourier Imaging PASI Santiago, Chile July 200625
Convolution theorem
F T f x g x F T f x F T g x
F u G u
. . ( ) ( ) . . ( ) . . ( )
( ) ( )
F T f x g x F u G u. . ( ) ( ) ( ) ( )
Eades / Fourier Imaging PASI Santiago, Chile July 200626
Atlas of Optical Transforms Harburn, Taylor and Welberry 1975
Eades / Fourier Imaging PASI Santiago, Chile July 200627
Atlas of Optical Transforms Harburn, Taylor and Welberry 1975
Eades / Fourier Imaging PASI Santiago, Chile July 200628
Atlas of Optical Transforms Harburn, Taylor and Welberry 1975
Eades / Fourier Imaging PASI Santiago, Chile July 200629
Atlas of Optical Transforms Harburn, Taylor and Welberry 1975
Eades / Fourier Imaging PASI Santiago, Chile July 200630
Optical Transforms Taylor and Lipson 1964
Eades / Fourier Imaging PASI Santiago, Chile July 200631
Optical Transforms Taylor and Lipson 1964
Eades / Fourier Imaging PASI Santiago, Chile July 200632
Optical Transforms Taylor and Lipson 1964
Eades / Fourier Imaging PASI Santiago, Chile July 200633
Atlas of Optical Transforms Harburn, Taylor and Welberry 1975
Eades / Fourier Imaging PASI Santiago, Chile July 200634
Atlas of Optical Transforms Harburn, Taylor and Welberry 1975