Gerchberg–Saxton algorithm: Difference between revisions

Revision as of 17:14, 26 August 2020

Revision as of 16:57, 25 January 2021

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[[File:GSGerchberg-diagramSaxton algorithm.pngjpg|thumb|360px400px|Schematic view of the error reductionGerchberg-Saxton algorithm for [[iterative phase retrieval]], -FT ais generalizationFourier of the Gerchberg-Saxton algorithmtransform.|alt=]]

The '''Gerchberg–Saxton (GS) algorithm''' is an iterative [[algorithm]] phase retrieval algorithm for retrieving the phase of a complex-valued wavefront from two intensity measurements acquired in two different planes.<ref>{{Cite journal|last=Gerchberg|first=R. W.|last2=Saxton|first2=W. O.|date=1972|title=A practical algorithm for the determination of the phase from image and diffraction plane pictures|url=http://www.u.arizona.edu/~ppoon/GerchbergandSaxton1972.pdf|journal=Optik|language=EN|volume=35|pages=237–246}}</ref> Typically, the two planes are: the image plane and the far field (diffraction) plane, and the wavfront propagation between these two planes is given by the [[Fourier transform]]. The original paper by Gerchberg and Saxton considered image and diffraction pattern of sample acquired in an electron microscope.

[[File:CGH XmasTree.jpg|thumb|The replay field of a computer generated hologram generated by the Gerchberg–Saxton algorithm. The 'star' is the zero-order diffraction peak.]]

The '''Gerchberg–Saxton (GS) algorithm''' is an iterative [[algorithm]] for retrieving the phase of a pair of light distributions (or any other mathematically valid distribution) related via a propagating function, such as the [[Fourier transform]], if their intensities at their respective optical planes are known.

It is often necessary to know only the phase distribution from one of the planes, since the phase distribution on the other plane can be obtained by performing a Fourier transform on the plane whose phase is known. Although often used for two-dimensional signals, the GS algorithm is also valid for one-dimensional signals.

The paper by R. W. Gerchberg and W. O. Saxton on this algorithm is entitled “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” and was published in ''Optik'' (35, 237–246 1972).

The [[pseudocode]] below performs the GS algorithm to obtain a phase distribution for the plane, Source, such that its Fourier transform would have the amplitude distribution of the plane, Target.

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==See also==

* [[Phase retrieval]]

* [[Fourier optics]]

* [[Holography]]

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==References==

{{reflist}}

* R. W. Gerchberg and W. O. Saxton, "[http://antoine.wojdyla.fr/assets/archive/gerchberg_saxton1972.pdf A practical algorithm for the determination of the phase from image and diffraction plane pictures],” Optik 35, 237 (1972)

*{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 19.5.2. Deterministic Constraints: Projections onto Convex Sets | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=1011}}

==External links==

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* [https://slmtoolbox.neocities.org/ SLM ToolBox]. A [[freeware]] Windows program for calculating holograms and displaying them on a [[spatial light modulator]]

* [https://www.creatgraphy.com/05/2020/allgemein/lightmodulation-gerchberg-saxton-algorithmus-gsa-660/ A Python-Script of the GS by Dominik Doellerer]

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