Fourier Transforms and Images

1 / 34

# Fourier Transforms and Images - PowerPoint PPT Presentation

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.

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

## PowerPoint Slideshow about 'Fourier Transforms and Images' - brock

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

### Fourier Transforms and Images

- and hence to gain important insights into the process

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.

Therefore, we must be able to find a way of linking the descriptions. The link is the Fourier Transform.

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

The Fourier Transform

Fourier series

Fourier transform

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

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

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

The Fourier Transform

Fourier series

Fourier transform

Optical Transforms

Taylor and Lipson 1964

Optical Transforms

Taylor and Lipson 1964

Optical Transforms

Taylor and Lipson 1964

Optical Transforms

Taylor and Lipson 1964