Fourier Transforms and Images
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Transcript Fourier Transforms and Images
Fourier Transforms and
Images
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
Our aim is to make a connection between
diffraction and imaging
- and hence to gain important insights into
the process
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
What happens to the electrons as
they go through the sample?
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
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
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PASI Santiago, Chile July 2006
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
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PASI Santiago, Chile July 2006
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
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PASI Santiago, Chile July 2006
Reimer:
Transmission
Electron
Microscopy
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
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
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PASI Santiago, Chile July 2006
The Fourier Transform
Fourier series
F (t ) Fn cos(2nt )
n0
Fourier transform
f (t )
F ( ) exp 2it. d
f (x) F(u) exp2iux.dx
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
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
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PASI Santiago, Chile July 2006
Original figure by Thomas Young, courtesy Bradley Carroll
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
Bradley Carroll
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
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
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
Bragg’s Law
Bragg’s Law
2d sin θ = λ
tells us where there are diffracted beams.
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
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
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PASI Santiago, Chile July 2006
Sample
Lens
Back focal plane
Image
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PASI Santiago, Chile July 2006
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PASI Santiago, Chile July 2006
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
The Fourier Transform
Fourier series
F (t ) Fn cos(2nt )
n0
Fourier transform
f (t )
F ( ) exp 2it. d
f (x) F(u) exp2iux.dx
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
Optical Transforms
Taylor and Lipson 1964
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PASI Santiago, Chile July 2006
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
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PASI Santiago, Chile July 2006
Atlas of Optical Transforms
Eades / Fourier Imaging
Harburn, Taylor and Welberry 1975
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PASI Santiago, Chile July 2006
Atlas of Optical Transforms
Eades / Fourier Imaging
Harburn, Taylor and Welberry 1975
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PASI Santiago, Chile July 2006
Atlas of Optical Transforms
Eades / Fourier Imaging
Harburn, Taylor and Welberry 1975
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PASI Santiago, Chile July 2006
Atlas of Optical Transforms
Eades / Fourier Imaging
Harburn, Taylor and Welberry 1975
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PASI Santiago, Chile July 2006
Optical Transforms
Taylor and Lipson 1964
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
Optical Transforms
Taylor and Lipson 1964
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
Optical Transforms
Taylor and Lipson 1964
Eades / Fourier Imaging
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PASI Santiago, Chile July 2006
Atlas of Optical Transforms
Eades / Fourier Imaging
Harburn, Taylor and Welberry 1975
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PASI Santiago, Chile July 2006
Atlas of Optical Transforms
Eades / Fourier Imaging
Harburn, Taylor and Welberry 1975
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PASI Santiago, Chile July 2006