I'm from an image processing background and am working my way into optics. I'm working on phase retrieval problems, trying to understand the Gerchberg-Saxton algorithm. I found this video which is really great, but which made me realize I probably had some misconceptions regarding some of the physics going on. I would like to clear these up before going further, so apologies for the probably very confused question and maybe approximate vocabulary.
The setup in the video is as follows : some field (with amplitude and phase) is imaged on an image plane, where we get the image of the intensity of the field, but not the phase, and we would like to know that phase. So the field is also projected "in the far field", to get the image in the "Fourier plane".
I get the gist of the rest of the algorithm pretty well, which involves comparing images in the image plane and the Fourier plane, enforcing constraints on the amplitude to iteratively correct the phase.
but what I don't get - and I think this is probably my lack of knowledge regarding Fourier optics speaking - is what truly is the "Fourier transform" discussed here. From my image processing background, the Fourier transform decomposes an image, i.e. a real signal, function of space, into an amplitude and phase image in frequency space. Something like :
So, I first thought that the Fourier transform discussed here was decomposition happening between the "image plane"'s intensity and the "Fourier plane"'s intensity + amplitude, ie the red arrows I've added to the figure.
However, that doesn't seem right, because the phase information seems to be taken into account during the Fourier transform. But then, what does the transform indicated by the following red arrow mean ? The field has both phase and amplitude information : is it in frequency space ? does it carry the same information as the phase and amplitude of the Fourier transform (ie the two right images on the bottom row) ? What does it mean to take the Fourier transform of a such a signal ?