Gerchberg–Saxton algorithm: Difference between revisions

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{{Short description|Algorithm for phase retrieval}}

[[Image:GS-diagram.png|thumb|360px|The Gerchberg-Saxton Algortithm]]

[[File:CGHGerchberg-Saxton XmasTreealgorithm.jpg|thumb|400px|The replay field of a computer generated hologram generated by the Gerchberg-Saxton algorithm. The `star'FT is the zero-order diffractionFourier peaktransform.]]

The '''Gerchberg–Saxton (GS) algorithm''' is an iterative [[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|archive-url=https://web.archive.org/web/20160328053000/http://www.u.arizona.edu/~ppoon/GerchbergandSaxton1972.pdf|url-status=dead|archive-date=March 28, 2016|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 wavefront propagation between these two planes is given by the [[Fourier transform]]. The original paper by Gerchberg and Saxton considered image and diffraction pattern of a sample acquired in an electron microscope.

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 '''Gerchberg–Saxton (GS) algorithm''' is an iterative [[algorithmpseudocode]] forbelow retrievingperforms the phaseGS ofalgorithm ato pairobtain ofa light distributions (or any other mathematically validphase distribution) relatedfor via athe propagatingplane function"Source", such asthat theits [[Fourier transform]], ifwould theirhave intensitiesthe atamplitude theirdistribution respectiveof optical planesthe areplane known"Target".

The Gerchberg-Saxton algorithm is one of the most prevalent methods used to create [[computer-generated hologram]]s.<ref>{{Cite journal |last=Memmolo |first=Pasquale |last2=Miccio |first2=Lisa |last3=Merola |first3=Francesco |last4=Paciello |first4=Antonio |last5=Embrione |first5=Valerio |last6=Fusco |first6=Sabato |last7=Ferraro |first7=Pietro |last8=Antonio Netti |first8=Paolo |date=2014-01-01 |title=Investigation on specific solutions of Gerchberg–Saxton algorithm |url=https://www.sciencedirect.com/science/article/pii/S0143816613001942 |journal=Optics and Lasers in Engineering |volume=52 |pages=206–211 |doi=10.1016/j.optlaseng.2013.06.008 |issn=0143-8166|url-access=subscription }}</ref>

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.

==Pseudo-codePseudocode algorithm==

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 [[pseudo-code]] 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.

==Pseudo-code algorithm==

'''Let:'''

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IFT – inverse Fourier transform

''i'' – the imaginary unit, √&minus;1 (square root of &minus;1)

exp – exponential function (exp(x) = ''e''^''x'')

Target and Source be the Target and Source Amplitude planes respectively

A, B, C & D be complex planes with the same dimension as Target and Source

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for real ''x'', amplitude(''x'') = |''x''|

Phase – Phase extracting function:

e.g. Phase(z) = arctan(y / x)

'''end Let'''

'''Gerchberg–Saxton Algorithmalgorithm''' Gerchberg–Saxton(Source, Target, Retrieved_Phase) '''is'''

A := IFT(Target)

'''while''' error criterion is not satisfied

B := Amplitude(Source) *× exp(i* × Phase(A))

C := FT(B)

D := Amplitude(Target) *× exp(i* × Phase(C))

A := IFT(D)

'''end while'''

Retrieved_Phase = Phase(A)

'''end Gerchberg–Saxton Algorithm'''

This is just one of the many ways to implement the GS algorithm. Aside from optimizations, others may start by performing a forward Fourier Transformtransform to the source distribution.

==See also==

* [[Phase retrieval]]

* [[Fourier optics]]

* [[Holography]]

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==References==

{{reflist}}

* R. W. Gerchberg and W. O. Saxton, “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==

* Dr W. Owen Saxton's pages [http://www-hrem.msm.cam.ac.uk/people/saxton/] {{Webarchive|url=https://web.archive.org/web/20080613024950/http://www-hrem.msm.cam.ac.uk/people/saxton/ |date=2008-06-13 }}, [https://www.murrayedwards.cam.ac.uk/fellows/dr-w-owen-saxton]

* [http://www.ysbl.york.ac.uk/~cowtan/fourier/coeff.html Graphical explanatory material by Kevin Cowtan]

*[http://www-hrem.msm.cam.ac.uk/people/saxton/ Dr W. Owen Saxton's page]

* [http://www.optics.rochester.edu/workgroups/fienup/index.html Applications and publications on phase retrieval from the University of Rochester, Institute of Optics]

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

* MATLAB GS algorithms [https://ch.mathworks.com/matlabcentral/fileexchange/68647-gerchberg-saxton-phase-retrieval-algorithm/], [https://ch.mathworks.com/matlabcentral/fileexchange/65979-gerchberg-saxton-algorithm]

{{DEFAULTSORT:Gerchberg-Saxton algorithm}}

[[Category:Digital signal processing]]