The reconstruction of discontinuous light-phase functions is of major importance in adaptive optics. An efficient and simple algorithm that can reconstruct large arrays of phases from phase differences is presented. We prove that the algorithm yields a perfect result in the absence of noise, and we describe the function that it maximizes. We suggest a method that makes use of the reconstructed phase to measure the position of branch points. A simulation of the reconstruction of a 33×33 phase array is presented.

© 1998 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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1997 (1)

1995 (1)

1992 (1)

1980 (1)

1977 (2)

Barclay, H. T.

Burden, R. L.

R. L. Burden and J. D. Faires, Numerical Analysis, 4th ed. (PWS-Kent, Boston, Mass., 1988).

Faires, J. D.

R. L. Burden and J. D. Faires, Numerical Analysis, 4th ed. (PWS-Kent, Boston, Mass., 1988).

Fried, D. L.

Herrmann, J.

Hudgin, R. H.

Humphreys, R. A.

Price, T. R.

Primmerman, C. A.

Szeto, R. K.-H.

Vaughan, J. L.

Zollars, B. G.

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Figures (2)

Fig. 1
Fig. 1

Illustration of the definition of the II matrix in the case of Hudgin discretization. Complex numbers are represented by arrows. As depicted, the calculation of the 44th component of IIu consists of a summation of four terms; the argument of IIu44 can be interpreted as a weighted average of four phase estimates obtained by means of the neighboring phases arguj and phase differences d44,j, where j11, 43, 45, 77.

Fig. 2
Fig. 2

(a) A phase function associated with the complex light amplitude Ux, ydef̲̲x+iy. The central point is called a branch point—the origin of a phase discontinuity. The 2π discontinuities are created by the noncontinuity of the argument function. (b), (c) Reconstruction of 33×33 phases of U from simulated phase-gradient measurements, to which a noise of 20% was added. (d) The automatic branch-point detection that we propose in this Letter works perfectly with this example (the precision parameter of the algorithm was =1%); the graph displays the curl of the corrected phase differences defined in the text.

Equations (3)

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IIujdef̲̲k neighbor of jwj,k expidj,kuk,
uIIu=2j,kwj,k cosϕj-ϕk-dj,kujuk,
ddef̲̲d+Aϕ-d mod 2π,