Gerchberg–Saxton algorithm

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Gerchberg–Saxton (GS) is a simple, widely used method for recovering the missing phase of a wavefront when you know the light’s intensity in two planes that are connected by a Fourier transform. Typically these planes are the image plane and the far-field (diffraction) plane. The idea is that if you know the amplitude in both planes, you can iteratively estimate the phase by moving back and forth between planes with Fourier transforms.

How it works in brief:
- Start with a guess for the Source plane’s phase.
- Build the Source complex field from its amplitude and the guessed phase, then Fourier transform to the Target plane.
- In the Target plane, replace the amplitude with the measured Target amplitude while keeping the new phase.
- Invert back to the Source plane and replace its amplitude with the measured Source amplitude, keeping the current phase.
- Repeat these steps until the phase estimate converges.

Originally developed for electron microscope images and diffraction patterns, GS also works for one-dimensional signals and is widely used to design computer-generated holograms. There are many variants; some implementations begin with a forward Fourier transform of the source distribution.