# Oversampling and phase retrieval in Fraunhofer diffraction

I am a Grade 11 student and am working on a physics research about Fraunhofer diffraction. I have 2 very basic questions (maybe kind of unprofessional). Hope someone can help.

1.Suppose a beam of coherent, monochromatic, parallel light penetrates an object, and a diffraction pattern forms on the far-field screen. As long as the oversampling ratio satisfies certain conditions, we can use the diffraction pattern to uniquely reconstruct the original object, using phase retrieval program. Is that the case?
If so, where I can download a phase retrieval program for Fraunhofer diffraction? I want a program that I can simply input an image of a diffraction pattern, and the program can generate the image of the original object. Is that possible? Or I can program it on MATLAB?

2. What is "oversampling to a diffraction pattern"? And the oversampling ratio? Is it that surround the original object (electron density region) with empty space (no-density region); the area of the empty space measures the extent of oversampling? Or, use different resolution to record the same diffraction pattern?
I read some articles found on Google Scholar, but I am a little confused.

Thanks for the help!

## Answers and Replies

UltrafastPED
Science Advisor
Gold Member
The 2D diffraction pattern can be transformed into an image via the 2D Fourier transform. This is available in MATLAB.

Consider the transmission electron microscope (TEM):

1. In diffraction mode the screen shows the 2D diffraction pattern of a crystal
2. When switched to imaging mode the screen shows the crystal

The "switching" is simply the introduction of an additional magnetic lens, which performs the Fourier transform

See http://micron.ucr.edu/public/manuals/Tem-intro.pdf [Broken]

Additional:

It is not possible to retrieve the precise phase without further information. This is known as "The Phase Problem". In the case of simple diffraction patterns there are only a few possibilities, and all can be tried. In the case of a lens, the phase information has not been lost; it is present in the travelling wave.

See https://en.wikipedia.org/wiki/Phase_retrieval
and http://digitus.itk.ppke.hu/~matyi/optika/Phase_Diversity/AO82_PRComparison1.pdf [Broken]

This is a very mathematical subject.

Last edited by a moderator:
1 person
The 2D diffraction pattern can be transformed into an image via the 2D Fourier transform. This is available in MATLAB.

Consider the transmission electron microscope (TEM):

1. In diffraction mode the screen shows the 2D diffraction pattern of a crystal
2. When switched to imaging mode the screen shows the crystal

The "switching" is simply the introduction of an additional magnetic lens, which performs the Fourier transform

See http://micron.ucr.edu/public/manuals/Tem-intro.pdf [Broken]

Thanks for your answer. Maybe I did not explain my problems very clearly.

I guess you used the TEM equipment to show that the diffraction pattern can be Fourier transformed back to the original image. Yet I think that we cannot directly inverse Fourier transform the diffraction pattern because it is only a graph showing the intensity of light; the phase information is lost in Fraunhofer diffraction. Therefore we have to use iterative algorithms to retrieve the phase before we can get the image.

The problem is in what situations the phase can be uniquely retrieved. Some people say that certain oversampling conditions should be satisfied. My main problems are:

1. What conditions have to be satisfied to make the phase retrieval possible?
2. If the phase retrieval condition is satisfied, how can we retrieve the phase and get the image? (i.e. the algorithm)

Please explain more about it if you are willing to. Thanks a lot!

Last edited by a moderator:
UltrafastPED
Science Advisor
Gold Member
See additions to #2.

It's been a while since I did any of this but look for papers by Fineup and Gerchberg-Saxton.