Fourier Analysis's relation to diffraction

[know]

My advisor recently asked me a series of questions that made me realize I don't fully understand how the Fourier transform of, let's say series of vertical lines, produces a diffraction pattern like those seen in a TEM. To get a better understanding I re-read Hecht's short little chapter on Fourier series, integrals, and their corresponding graphs in k-space. I understand that Fourier series allow us to write any equation in terms of a series of sines and cosines, and that calculating the related coefficients tells us how each frequency of each of the sines and cosines is weighted into the representation of the equation. I'm not, however, seeing how taking the Fourier series, Fourier transform, or diffraction patterns are related. Could anyone help with the missing link?

 

[Ben Niehoff]

It only works in the far-field approximation. Represent the incoming wave as

$$E_0 e^{i\vec k \cdot \vec x}$$

To find the total amplitude at a screen a large distance away, what do you have to do? You have to integrate the above thing over the shape of your aperture.

Which, if you notice, is the same thing as taking a 2-dimensional Fourier transform.

 

[PhilDSP]

A Fourier transform of a particular function is the procedure that produces a weighted set of values taken out of the Fourier Series that is equivalent to the function when summed. The Fourier Series is the "basis" for the representation of the function.

It's very useful to break a function or wave into separate frequency components because any dispersive function will have a different time skew for each frequency.

 

[f95toli]

Just to add to what Ben Niehoff has already written, the connection simply comes from the fact that the integrals/sums have you to calculate when working with diffraction just happens to be identical to the integrals/sums you do in Fourier analysis.
Of course one could argue that there is a deeper connection but from a practical point of view you might as well just see it as a lucky "coincidence".

This type of of integral is quite common in physics, a few years ago I "accidentally" derived the same relation when working on something completely different (no connection to light).
Fortunately I realized what is was and it saved me a lot of time since it meant I could use an FFT in my numerical simulations.