Fourier Transforms and Images

Eades / Fourier Imaging PASI Santiago, Chile July 20061

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



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


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 where d and are related by

2d sin 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





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




Optical Transforms Taylor and Lipson 1964


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. . ( ) ( ) ( ) ( )


Atlas of Optical Transforms Harburn, Taylor and Welberry 1975

















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Fourier Transforms and Images