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I am comparing diffraction pattern images and their discrete Fourier transform (DFT). I know that since Fraunhofer diffraction is essentially a Fourier transform, the DFT of the resulting pattern gives me the intensity profile of the wave on the aperture. Well, the DFT does give me the aperture shape, but only the corners! Why is that?

This is the diffraction pattern caused by a green laser passing through an hexagonal aperture.

Diffraction pattern by an hexagonal aperture

This is the DFT of the hexagonal aperture.

FFT of the hexagonal diffraction pattern

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Intuitively this is because of Huygens' principle. You can simulate the diffraction pattern by just summing point sources that are close enough together.Huyens summation

Near the center of the aperture you get an approximate plane wave. The diffraction happens at the aperture boundary. If you do this analytically in the far field, you get (apart from the obliquity factor) and integrand that is just the Green function for a point source. So by doing the integral you're doing a continuous Huygens summation.

Picture by Arne Nordmann (norro) - Own illustration, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=1944668

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