What does the fourier transform do

[aaaa202]

Today I found a program, which does Fourier transforms on pictures and tried it on some basic patterns. One of those was a lattice of dots and I have attached this and its Fourier transform to the thread.
I would very much like if someone in basic details could explain what is going on. Why the Fourier transform of the picture looks as it does. Mathematically I understand what there is to know about the Fourier transform and how we represent a function in its frequency domain by writing it as a superposition of sinusoidal functions with different frequencies.
In terms of the mathematical idea behind the FT what is it then, that I observe on the picture? What is my original function (the brightness?) and what does the frequency represent? Please don't hesitate to write a long answer.

 

[vanhees71]

This is more a physics than a mathematics question. However, it's a very interesting one.

A pretty complicated topic in theoretical electromagnetism and optics is the scattering of electromagnetic waves on objects. The most simple approximation is Kirchhoff's scalar theory of diffraction. The idea is close to Huygen's principle, i.e., you shine with light on an opaque screen with some openings (like the one with regular holes shown on the left in your picture). Then from any point of the openings a spherical wave is emitted, and the total wave for an observer is given by the superposition of all these elementary spherical waves. The superposition of the spherical waves leads to partial destruction or amplifying, resulting in characteristic interference patterns.

In a certain limit, the socalled Fraunhofer-diffraction, where you observe the diffraction picture of your object in a distance very far from the extension of the object, the amplitude on this screen is given by the Fourier transform of the object.

Have a look at Wikipedia:

http://en.wikipedia.org/wiki/Fraunhofer_diffraction