fourier transforms and images n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Fourier Transforms and Images PowerPoint Presentation
Download Presentation
Fourier Transforms and Images

Loading in 2 Seconds...

play fullscreen
1 / 34

Fourier Transforms and Images - PowerPoint PPT Presentation


  • 99 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Fourier Transforms and Images' - brock


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide2
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
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.

slide6
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.

slide7
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.

slide8

Reimer:

Transmission Electron Microscopy

slide9
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
The Fourier Transform

Fourier series

Fourier transform

slide11
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

slide15
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

Bragg’s Law

2d sin θ = λ

tells us where there are diffracted beams.

what does a lens do
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

slide18
Sample

Back focal plane

Lens

Image

the fourier transform1
The Fourier Transform

Fourier series

Fourier transform

slide24

Optical Transforms

Taylor and Lipson 1964

slide30

Optical Transforms

Taylor and Lipson 1964

slide31

Optical Transforms

Taylor and Lipson 1964

slide32

Optical Transforms

Taylor and Lipson 1964