Gerchberg–Saxton algorithm

<h2 id="pseudocode-algorithm">Pseudocode algorithm</h2>
Let:
 FT – forward Fourier transform
 IFT – inverse Fourier transform
 <i>i</i> – the imaginary unit, √−1 (square root of −1)
 exp – exponential function (exp(x) = <i>e</i><i>x</i>)
 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
 Amplitude – Amplitude-extracting function:
   e.g. for complex <i>z</i> = <i>x</i> + <i>iy</i>, amplitude(<i>z</i>) = sqrt(<i>x</i>·<i>x</i> + <i>y</i>·<i>y</i>)
       for real <i>x</i>, amplitude(<i>x</i>) = |<i>x</i>|
 Phase – Phase extracting function:
   e.g. Phase(z) = arctan(y / x)
end Let

algorithm 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)

<p>This is just one of the many ways to implement the GS algorithm.  Aside from optimizations, others may start by performing a forward Fourier transform to the source distribution.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Phase_retrieval/GBEB9vrD">Phase retrieval</a></li>
<li><a href="/facts/Fourier_optics/H4bVRGYx">Fourier optics</a></li>
<li><a href="/facts/Holography/4J1gW0M6">Holography</a></li>
<li><a href="/facts/Adaptive-additive_algorithm/ABTDvM1D">Adaptive-additive algorithm</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li>Dr W. Owen Saxton's pages <a href="http://www-hrem.msm.cam.ac.uk/people/saxton/">[1]</a> <a href="https://web.archive.org/web/20080613024950/http://www-hrem.msm.cam.ac.uk/people/saxton/">Archived</a> 2008-06-13 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a>, <a href="https://www.murrayedwards.cam.ac.uk/fellows/dr-w-owen-saxton">[2]</a></li>
<li><a href="http://www.optics.rochester.edu/workgroups/fienup/index.html">Applications and publications on phase retrieval from the University of Rochester, Institute of Optics</a></li>
<li><a href="https://www.creatgraphy.com/05/2020/allgemein/lightmodulation-gerchberg-saxton-algorithmus-gsa-660/">A Python-Script of the GS by Dominik Doellerer</a></li>
<li>MATLAB GS algorithms <a href="https://ch.mathworks.com/matlabcentral/fileexchange/68647-gerchberg-saxton-phase-retrieval-algorithm/">[3]</a>, <a href="https://ch.mathworks.com/matlabcentral/fileexchange/65979-gerchberg-saxton-algorithm">[4]</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Gerchberg, R. W.; Saxton, W. O. (1972). "A practical algorithm for the determination of the phase from image and diffraction plane pictures" (PDF). Optik. 35: 237–246. Archived from the original (PDF) on March 28, 2016. <a href="https://web.archive.org/web/20160328053000/http://www.u.arizona.edu/~ppoon/GerchbergandSaxton1972.pdf" target="_blank">https://web.archive.org/web/20160328053000/http://www.u.arizona.edu/~ppoon/GerchbergandSaxton1972.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Memmolo, Pasquale; Miccio, Lisa; Merola, Francesco; Paciello, Antonio; Embrione, Valerio; Fusco, Sabato; Ferraro, Pietro; Antonio Netti, Paolo (2014-01-01). "Investigation on specific solutions of Gerchberg–Saxton algorithm". Optics and Lasers in Engineering. 52: 206–211. doi:10.1016/j.optlaseng.2013.06.008. ISSN 0143-8166. <a href="https://www.sciencedirect.com/science/article/pii/S0143816613001942" target="_blank">https://www.sciencedirect.com/science/article/pii/S0143816613001942</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Gerchberg–Saxton algorithm open-in-new

Gerchberg–Saxton algorithm