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

Eades / Fourier Imaging PASI Santiago, Chile July 200614

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