What is Phase Retrieval?

Nearly everyone has heard of the term Holography which comes from the Greek words 'holos' meaning whole and 'graf' meaning message'. Holography was invented by Dennis Gabor in 1947 .

In electromagnetic terms, performing Holography means that you have recorded both the amplitude and the phase of the wave distribution, either directly in the case of radio frequencies with a two channel receiver, or indirectly by creating an interference pattern with an external reference wave and recording the resulting intensity.

Phase retrieval similarly requires only intensity measurements, but with phase retrieval you do not need any type of reference wave or reference channel, and instead you make use of a mathematical relationship between two intensity images. However, in order for this mathematical relationship to be valid there are some preconditions that must be satisfied. For one thing, it is important that there be no reflected waves bouncing around. Secondly, it is important that truncation effects are reduced to a minimum, meaning that the intensity images should be band-limited. In the case of antennas this is carried out with a sampling probe with suitable directivity.

One of the most famous examples of phase retrieval was it´s use to correct the Hubble telescope

Phase retrieval actually has its origins in the fields of electron microscopy, crystallography and astronomical imaging. Pioneering work was carried out in the early 1970s by Gerchberg and Saxton [1], and Misell [2]. In 1982 Fienup  presented the Hybrid Input-Output (HIO) algorithm [3] and gave a detailed comparison of iterative Fourier transform based algorithms.


[1]     Gerchberg, R.W. and Saxton, W.O.,"Phase determination from image and diffraction plane pictures in the electron microscope", Optik, vol. 34, pp. 275-284, 1971.

[2]     D.L. Misell, "A method for the solution of the phase problem in electron microscopy," J. Phys. D: Appl.Phys.,vol. 6, L6-L9, 1973.

[3]     Fienup, J.R., Phase retrieval algorithms: a comparison. Applied Optics   Vol. 21, No. 15 / 1 August 1982. pp 2757-2769.

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