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The Gerchberg–Saxton (GS) algorithm is an iterative
Applications and publications on phase retrieval from the University of Rochester, Institute of Optics
A Python-Script of the GS by Dominik Doellerer
* MATLAB GS algorithm
{{DEFAULTSORT:Gerchberg-Saxton algorithm Digital signal processing Physical optics Articles with example pseudocode
The Gerchberg–Saxton (GS) algorithm is an iterative phase retrieval
Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex spectrum F(k), of amplitude , F (k), , and phase \psi(k):
::F(k) = , F(k), e^ =\int_^ f(x)\ e^\,dx
where ''x'' is an ''M''-dimensional spat ...
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
for retrieving the phase of a complex-valued wavefront from two intensity measurements acquired in two different planes. 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
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
. 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 pseudocode
In computer science, pseudocode is a description of the steps in an algorithm using a mix of conventions of programming languages (like assignment operator, conditional operator, loop) with informal, usually self-explanatory, notation of actio ...
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".
The Gerchberg-Saxton algorithm is one of the most prevalent methods used to create computer-generated hologram
Computer-generated holography (CGH) is a technique that uses computer algorithms to generate holograms. It involves generating holographic interference patterns. A computer-generated hologram can be displayed on a dynamic holographic display, or i ...
s.
Pseudocode algorithm
Let: FT – forward Fourier transform IFT – inverse Fourier transform ''i'' – the imaginary unit, √−1 (square root of −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 Amplitude – Amplitude-extracting function: e.g. for complex ''z'' = ''x'' + ''iy'', amplitude(''z'') = sqrt(''x''·''x'' + ''y''·''y'') for real ''x'', amplitude(''x'') = , ''x'', 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) 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.See also
*Phase retrieval
Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex spectrum F(k), of amplitude , F (k), , and phase \psi(k):
::F(k) = , F(k), e^ =\int_^ f(x)\ e^\,dx
where ''x'' is an ''M''-dimensional spat ...
* Fourier optics
Fourier optics is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination, or '' superposition'', of plane waves. It has some parallels to the Huygens–Fresnel pr ...
* Holography
Holography is a technique that allows a wavefront to be recorded and later reconstructed. It is best known as a method of generating three-dimensional images, and has a wide range of other uses, including data storage, microscopy, and interfe ...
* Adaptive-additive algorithm
References
External links
* Dr W. Owen Saxton's pageApplications and publications on phase retrieval from the University of Rochester, Institute of Optics
A Python-Script of the GS by Dominik Doellerer
* MATLAB GS algorithm
{{DEFAULTSORT:Gerchberg-Saxton algorithm Digital signal processing Physical optics Articles with example pseudocode