Fourier transforms, convolution, and Fraunhofer diffraction

[marnobingo665]

I've been exposed to this notion in multiple classes (namely math and physics) but can't find any details about how one would actually calculate something using this principle: Diffraction in optics is closely related to Fourier transforms and finding the Fraunhofer diffraction of an aperture composed of simple shapes is equivalent to finding the Fourier transform of the convolution of those shapes.

For example a professor told us that the FT of a circle is a 1st order Bessel function and that the FT of a rectangle is a sinc function. She demonstrated finding the Fraunhofer diffraction of the following aperture: (please see first reply)

where white represents an opening and black represents and opaque region. She said it is equivalent to multiplying a circular aperture (f1) with a single opaque slit (f2) and that it would come out to be a thick band with a black dot (missing light) in the middle and thinner bands on both sides. But she never worked through how she used FTs to achieve that result: Where in this problem do you apply the concept of convolution, and what does it even mean to "take the FT of a rectangle"? How would I do this numerically instead of just sketching out intuitively what it looks like?

 

[marnobingo665]

Image attached, couldn't figure out how to upload in original post!

 

[RPinPA]

Yes, this is true. Some relevant keywords: Fourier optics, spatial filtering, spatial Fourier Transform. There is a close connection between optics and Fourier transforms.
The 2-D Fourier Transform just involves doing a FT in each direction. For a digital FT, I think all you're doing is a FT along each row, and then a FT along each column.

Here's a page with the relevant equations (look for the formulas with the double summations)
http://fourier.eng.hmc.edu/e101/lectures/Image_Processing/node6.html

Note that $\sum_{n=0}^{N-1}\sum_{m=0}^{M-1} f_{mn} e^{-j 2\pi(\frac {mk} {M} + \frac {nl} {N})}$ = $\sum_{n=0}^{N-1} e^{-j 2\pi \frac {nl} {N}} \sum_{m=0}^{M-1} f_{mn} e^{-j 2\pi \frac {mk} {M}}$ which I believe allows you to calculate it by rows and columns in the way I described.