are you familiar with <PERSON> diffraction? This is what is physically happening in your first picture as the light passes through the lens. The idea is that it is the new light field that is rigorously the 2D Fourier transform of the original one. However, in the end, you only measure intensity, i.e. the (square) amplitude.
Therefore, the first measurement is the amplitude of the field, the second measurement is the amplitude of the Fourier transform of the field. Combining both you can extract information on the phase of the initial field. Just to make it clear, this does not correspond to the amplitude of the Fourier transform of the amplitude of the initial field as you seem to suggest in the second picture (it would not even give any relevant information as you noted yourself).
Hope this helps and please tell me if there are any mistakes.
(Edit following <PERSON>' comment)
I'll write explicitly the math behind my explication. Physically, your light is the EM field, a vector field, but in your case, you approximate it by a scalar field. 780
Since it is regulated by a real linear second order in time differential equation, just as for the harmonic oscillator, you condense the field and its derivative by a single complex field $U(x,y)$ (complex amplitude). The intensity is therefore (up to a multiplicative constant) $I(x,y) = |U(x,y)|^2$ and the phase $\phi(x,y) = \arg(U(x,y))$.
Physically, you only measure $I$ with a photodetector, but you want $U$, so $\phi$. The trick is to transform $U$ and measure the new intensity in order to tease out information of $\phi$. The Fraunhofer diffraction transforms $U$ to $\mathcal F U$, the Fourier transform on a complex function. The second measurement is therefore of $|\mathcal F U|^2$ which is indirectly related to $\phi$. Heuristically, $U$ has $2$ real degrees of freedoms (per point), and you have two observables (per points), so you "should" be able to solve the equations to retrieve $U$. The Gerchberg-Saxton algorithm does this solve in a self-consistent loop.
To summarise, unlike image processing where the signals are real, in optics you are interested in complex signals. 976