Gerchberg–Saxton algorithm: Difference between revisions

Revision as of 11:55, 29 August 2019

Revision as of 10:42, 18 December 2019

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[[File:GS-diagram.png|thumb|360px|The Gerchberg-SaxtonGerchberg–Saxton Algorithm]]

[[File:CGH XmasTree.jpg|thumb|The replay field of a computer generated hologram generated by the Gerchberg-SaxtonGerchberg–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.

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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-codepseudocode]] 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-codePseudocode algorithm==

'''Let:'''

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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 Transform to the source distribution.