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Cross posted from MSE.

I have a mathematical problem arising from a physics application, which I feel must have been solved before, but I don't know the terminology associated with it. I am looking for references. Briefly, the problem is this:

Given an input function $f$ and a desired output function $g$, find a real-valued function $\phi$ such that the modulus of the Fourier transform $\left|\mathcal{F}\left\{fe^{i\phi}\right\}\right|$ is as close as possible to $|g|$ (with respect to some norm--say $L^2$).

(In my particular case, all functions are defined on a compact subset of $\mathbb{R}^2$, in case it matters.)

In practice, the input function $f$ is an electric field, the phase function $\phi$ is provided by a "spatial light modulator", and the magnitude of the Fourier transform gives the output intensity of a light field, which you want to have a specified form.

I'm interested in both abstract results and algorithmic solutions to this problem.

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    $\begingroup$ Is $\phi$ completely arbitrary? If so, then you can just choose $\phi := -i\log( \mathcal{F}^{-1}(g) / f)$ and have exact equality in some cases. And if $f$ has zeros, you can approximate $f$ arbitrarily close by some (say Schwartz) function without zeros. $\endgroup$ – Johannes Hahn Feb 7 at 21:40
  • $\begingroup$ @JohannesHahn: I don't think you can get close usually: the $L^2$ norm of $fe^{i\phi}$ is independent of $\phi$, so if this isn't close to the norm of $g$ to start with, then there's nothing you can do. $\endgroup$ – Christian Remling Feb 7 at 22:52
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    $\begingroup$ @JohannesHahn $\phi$ is real (a pure phase), which I believe precludes an exact solution in general. I've edited this into the problem statement for clarity. $\endgroup$ – Yly Feb 8 at 1:58
  • $\begingroup$ Given $f,G \in L^2$ you want to find $e^{i \phi},e^{i \varphi}$ minimizing $\| fe^{i \phi}-\mathcal{F}^{-1}[G e^{i \varphi}]\|^2$, so that $\varphi = arg( \mathcal{F}[e^{i \phi}f])-arg(G)$. You can then look at $\partial_t \| \mathcal{F}[fe^{i (\phi+t\psi)}]-G e^{i arg(\mathcal{F}[e^{i (\phi+t \psi)}f])-arg(G)}\|^2]|_{t= 0}$ $\endgroup$ – reuns Feb 8 at 8:50

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