Abstract

An approach for wave-front sensing using reflected laser light from a rough object is proposed. Light from a single laser beam is split into two beams, and the beams are launched from spatially separated apertures to illuminate an object. The reflected laser light is measured in the pupil plane of a receive telescope and in a plane conjugate to the object. By modulation of one of the two illuminator beams, the intensity pattern associated with each beam, as well as the field cross product of the two beams, is measured in each plane. A phase-retrieval algorithm is formulated by using projections onto constraint sets to recover the complex field associated with each illuminator. The algorithm is found to converge rapidly to the correct solution, particularly when compared with the convergence rates of more conventional phase-retrieval approaches. The new algorithm exhibits excellent performance in strong scintillation and is very tolerant to noise, exhibiting only a very small noise gain.

© 2003 Optical Society of America

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2002 (8)

J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. 41, 1012–1021 (2002).
[CrossRef] [PubMed]

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2002).
[CrossRef]

J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of a shearing interferometer in strong scintillation in the absence of additive measurement noise,” Appl. Opt. 41, 3674–3684 (2002).
[CrossRef] [PubMed]

J. D. Barchers, T. A. Rhoadarmer, “Evaluation of phase-shifting approaches for a point-diffraction interferometer with the mutual coherence function,” Appl. Opt. 41, 7499–7509 (2002).
[CrossRef]

R. B. Holmes, K. Hughes, P. Fairchild, B. Spivey, A. Smith, “Description and simulation of an active imaging technique utilizing two speckle fields: root reconstructors,” J. Opt. Soc. Am. A 19, 444–457 (2002).
[CrossRef]

R. B. Holmes, K. Hughes, P. Fairchild, B. Spivey, A. Smith, “Description and simulation of an active imaging technique utilizing two speckle fields: iterative reconstructors,” J. Opt. Soc. Am. A 19, 458–471 (2002).
[CrossRef]

H. H. Bauschke, P. L. Combettes, D. R. Luke, “Phase retrieval and error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
[CrossRef]

J. D. Barchers, “Application of the parallel generalized projection algorithm to the control of two finite-resolution deformable mirrors for scintillation compensation,” J. Opt. Soc. Am. A 19, 54–63 (2002).
[CrossRef]

2001 (1)

2000 (1)

1999 (1)

1998 (2)

D. L. Fried, “Scaling laws for propagation through turbulence,” J. Atmos. Ocean. Opt. 11, 982–990 (1998).

D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

1992 (2)

1991 (3)

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature 353, 141–143 (1991).
[CrossRef]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide star,” Nature 353, 144–146 (1991).
[CrossRef]

C. C. Wackerman, A. E. Yagle, “Use of Fourier domain real-plane zeros to overcome phase retrieval stagnation,” J. Opt. Soc. Am. A 8, 1898–1904 (1991).
[CrossRef]

1990 (1)

1987 (1)

1986 (1)

1984 (1)

1982 (2)

J. R. Fienup, “Phase-retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

1979 (1)

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

1977 (3)

1975 (1)

1974 (1)

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

1973 (3)

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D Appl. Phys. 6, L6–L9 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations,” J. Phys. D Appl. Phys. 6, 2200–2216 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: II. Sources of error,” J. Phys. D Appl. Phys. 6, 2217–2225 (1973).
[CrossRef]

1972 (2)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane images,” Optik (Stuttgart) 35, 225–246 (1972).

D. Y. Gezari, A. Labeyrie, R. V. Stachnik, “Speckle interferometry: diffraction-limited measurements of nine stars with the 200-inch telescope,” Astrophys. J. 173, L1–L5 (1972).
[CrossRef]

1971 (1)

R. V. Shack, B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

1967 (1)

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point in convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Allen, J. G.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Ameer, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide star,” Nature 353, 144–146 (1991).
[CrossRef]

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Baliunar, S. L.

J. C. Shelton, T. G. Schneider, D. McKenna, S. L. Baliunar, “First tests of the Cassegrain adaptive optics system of the Mount Wilson 100-inch telescope,” in Adaptive Optical Systems and Technologies, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 72–77 (1995).
[CrossRef]

Barchers, J. D.

Barclay, H. T.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature 353, 141–143 (1991).
[CrossRef]

Bauschke, H. H.

Boeke, B. R.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide star,” Nature 353, 144–146 (1991).
[CrossRef]

Brown, J. M.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Browne, S. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide star,” Nature 353, 144–146 (1991).
[CrossRef]

Bruck, Y. M.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Christou, J. C.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Combettes, P. L.

Duncan, T. S.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Eager, R. J.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Ealey, M. A.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Ellerbroek, B. L.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Fairchild, P.

Fienup, J. R.

Fried, D. L.

J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of a shearing interferometer in strong scintillation in the absence of additive measurement noise,” Appl. Opt. 41, 3674–3684 (2002).
[CrossRef] [PubMed]

J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. 41, 1012–1021 (2002).
[CrossRef] [PubMed]

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2002).
[CrossRef]

D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
[CrossRef]

D. L. Fried, “Scaling laws for propagation through turbulence,” J. Atmos. Ocean. Opt. 11, 982–990 (1998).

D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
[CrossRef] [PubMed]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide star,” Nature 353, 144–146 (1991).
[CrossRef]

D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–374 (1977).
[CrossRef]

D. L. Fried, P.O. Box 680, Moss Landing,Calif. 95038 (personal communication, 2000).

Fugate, R. Q.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide star,” Nature 353, 144–146 (1991).
[CrossRef]

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane images,” Optik (Stuttgart) 35, 225–246 (1972).

Gezari, D. Y.

D. Y. Gezari, A. Labeyrie, R. V. Stachnik, “Speckle interferometry: diffraction-limited measurements of nine stars with the 200-inch telescope,” Astrophys. J. 173, L1–L5 (1972).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gubin, L. G.

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point in convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Gurkan, D.

Hardy, J.

Holmes, R. B.

Hughes, K.

Jones, G. W.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Knox, K. T.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

Koliopouosis, C.

Kotzer, T.

Kowalczyk, A. M.

Kuhns, R. M.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Labeyrie, A.

D. Y. Gezari, A. Labeyrie, R. V. Stachnik, “Speckle interferometry: diffraction-limited measurements of nine stars with the 200-inch telescope,” Astrophys. J. 173, L1–L5 (1972).
[CrossRef]

Lee, D. J.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Lefebvre, J.

Levi, A.

Link, D. J.

J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of a shearing interferometer in strong scintillation in the absence of additive measurement noise,” Appl. Opt. 41, 3674–3684 (2002).
[CrossRef] [PubMed]

J. D. Barchers, D. L. Fried, D. J. Link, “Evaluation of the performance of Hartmann sensors in strong scintillation,” Appl. Opt. 41, 1012–1021 (2002).
[CrossRef] [PubMed]

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Lowrey, W. H.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Luke, D. R.

McKenna, D.

J. C. Shelton, T. G. Schneider, D. McKenna, S. L. Baliunar, “First tests of the Cassegrain adaptive optics system of the Mount Wilson 100-inch telescope,” in Adaptive Optical Systems and Technologies, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 72–77 (1995).
[CrossRef]

Misell, D. L.

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D Appl. Phys. 6, L6–L9 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations,” J. Phys. D Appl. Phys. 6, 2200–2216 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: II. Sources of error,” J. Phys. D Appl. Phys. 6, 2217–2225 (1973).
[CrossRef]

Murphy, D. V.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature 353, 141–143 (1991).
[CrossRef]

O’Meara, T. R.

Oliker, M. D.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Page, D. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature 353, 141–143 (1991).
[CrossRef]

Paxman, R. G.

Platt, B. C.

R. V. Shack, B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

Polyak, B. T.

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point in convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Primmerman, C. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature 353, 141–143 (1991).
[CrossRef]

Raik, E. V.

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point in convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Rhoadarmer, T. A.

Roberts, P. H.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide star,” Nature 353, 144–146 (1991).
[CrossRef]

Rosen, J.

Ruane, R. E.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide star,” Nature 353, 144–146 (1991).
[CrossRef]

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane images,” Optik (Stuttgart) 35, 225–246 (1972).

Schneider, T. G.

J. C. Shelton, T. G. Schneider, D. McKenna, S. L. Baliunar, “First tests of the Cassegrain adaptive optics system of the Mount Wilson 100-inch telescope,” in Adaptive Optical Systems and Technologies, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 72–77 (1995).
[CrossRef]

Schulz, T. J.

Shack, R. V.

R. V. Shack, B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

Shamir, J.

Shelton, J. C.

J. C. Shelton, T. G. Schneider, D. McKenna, S. L. Baliunar, “First tests of the Cassegrain adaptive optics system of the Mount Wilson 100-inch telescope,” in Adaptive Optical Systems and Technologies, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 72–77 (1995).
[CrossRef]

Smith, A.

Sodin, L. G.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Spinhirne, J. M.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Spivey, B.

Stachnik, R. V.

D. Y. Gezari, A. Labeyrie, R. V. Stachnik, “Speckle interferometry: diffraction-limited measurements of nine stars with the 200-inch telescope,” Astrophys. J. 173, L1–L5 (1972).
[CrossRef]

Stark, H.

Swindle, D. W.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Thompson, B. J.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

Tyler, G. A.

G. A. Tyler, “Reconstruction and assessment of the least-squares and slope discrepancy components of the phase,” J. Opt. Soc. Am. A 17, 1828–1839 (2000).
[CrossRef]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide star,” Nature 353, 144–146 (1991).
[CrossRef]

Vaughn, J. L.

Voas, J. K.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Von Neumann, J.

J. Von Neumann, Functional Operators, Vol. II of Annals Mathematical Studies (Princeton U. Press, Princeton, N.J., 1950).

Wackerman, C. C.

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

Wild, W. J.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Wilson, K. B.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Wopat, L. M.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide star,” Nature 353, 144–146 (1991).
[CrossRef]

Wyant, J. C.

Wynia, J. L.

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

Yagle, A. E.

Yang, Y.

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

Zollars, B. G.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature 353, 141–143 (1991).
[CrossRef]

Appl. Opt. (8)

Astrophys. J. (2)

D. Y. Gezari, A. Labeyrie, R. V. Stachnik, “Speckle interferometry: diffraction-limited measurements of nine stars with the 200-inch telescope,” Astrophys. J. 173, L1–L5 (1972).
[CrossRef]

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

IEEE Trans. Med. Imaging (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

J. Atmos. Ocean. Opt. (1)

D. L. Fried, “Scaling laws for propagation through turbulence,” J. Atmos. Ocean. Opt. 11, 982–990 (1998).

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (15)

R. B. Holmes, K. Hughes, P. Fairchild, B. Spivey, A. Smith, “Description and simulation of an active imaging technique utilizing two speckle fields: root reconstructors,” J. Opt. Soc. Am. A 19, 444–457 (2002).
[CrossRef]

R. B. Holmes, K. Hughes, P. Fairchild, B. Spivey, A. Smith, “Description and simulation of an active imaging technique utilizing two speckle fields: iterative reconstructors,” J. Opt. Soc. Am. A 19, 458–471 (2002).
[CrossRef]

D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
[CrossRef]

G. A. Tyler, “Reconstruction and assessment of the least-squares and slope discrepancy components of the phase,” J. Opt. Soc. Am. A 17, 1828–1839 (2000).
[CrossRef]

A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
[CrossRef]

H. H. Bauschke, P. L. Combettes, D. R. Luke, “Phase retrieval and error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
[CrossRef]

J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
[CrossRef]

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of it Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
[CrossRef]

C. C. Wackerman, A. E. Yagle, “Use of Fourier domain real-plane zeros to overcome phase retrieval stagnation,” J. Opt. Soc. Am. A 8, 1898–1904 (1991).
[CrossRef]

C. C. Wackerman, A. E. Yagle, “Phase retrieval and estimation with use of real-plane zeros,” J. Opt. Soc. Am. A 11, 2016–2026 (1994).
[CrossRef]

R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
[CrossRef]

J. R. Fienup, A. M. Kowalczyk, “Phase retrieval for a complex-valued object by using a low-resolution image,” J. Opt. Soc. Am. A 7, 450–458 (1990).
[CrossRef]

J. D. Barchers, “Evaluation of the impact of finite resolution effects on scintillation compensation using two deformable mirrors,” J. Opt. Soc. Am. A 18, 3098–3109 (2001).
[CrossRef]

J. D. Barchers, “Application of the parallel generalized projection algorithm to the control of two finite-resolution deformable mirrors for scintillation compensation,” J. Opt. Soc. Am. A 19, 54–63 (2002).
[CrossRef]

H. Stark, Y. Yang, D. Gurkan, “Factors affecting convergence in the design of diffractive optics by iterative vector-space methods,” J. Opt. Soc. Am. A 16, 149–159 (1999).
[CrossRef]

J. Phys. D Appl. Phys. (3)

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D Appl. Phys. 6, L6–L9 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations,” J. Phys. D Appl. Phys. 6, 2200–2216 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: II. Sources of error,” J. Phys. D Appl. Phys. 6, 2217–2225 (1973).
[CrossRef]

Nature (2)

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature 353, 141–143 (1991).
[CrossRef]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric wave-front distortion using scattered light from a laser guide star,” Nature 353, 144–146 (1991).
[CrossRef]

Opt. Commun. (2)

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2002).
[CrossRef]

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane images,” Optik (Stuttgart) 35, 225–246 (1972).

USSR Comput. Math. Math. Phys. (1)

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point in convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Other (6)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

D. L. Fried, P.O. Box 680, Moss Landing,Calif. 95038 (personal communication, 2000).

J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, D. J. Link, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, K. B. Wilson, J. L. Wynia, W. J. Wild, “The Starfire Optical Range 3.5-m telescope adaptive optical system,” in Adaptive Optical Systems and Technologies, D. Bonaccini, ed., Proc. SPIE3353, 22–33 (1998).
[CrossRef]

J. C. Shelton, T. G. Schneider, D. McKenna, S. L. Baliunar, “First tests of the Cassegrain adaptive optics system of the Mount Wilson 100-inch telescope,” in Adaptive Optical Systems and Technologies, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 72–77 (1995).
[CrossRef]

H. Stark, Y. Yang, Vector Space Projections (Wiley, New York, 1998).

J. Von Neumann, Functional Operators, Vol. II of Annals Mathematical Studies (Princeton U. Press, Princeton, N.J., 1950).

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Figures (6)

Fig. 1
Fig. 1

Illuminator configuration: Light from a single laser is split and used to illuminate an object. The resultant reflected interference pattern is received at a large-aperture telescope. The phase of one of the beams is modulated so that the intensity associated with each beam and the field cross product of the two beams can be measured in the pupil and object planes.

Fig. 2
Fig. 2

Illustration of the SPR algorithm, which, like the Gerchberg–Saxton algorithm, iterates between the pupil and object planes, enforcing an amplitude constraint at each iteration step.

Fig. 3
Fig. 3

Illustration of the DPR algorithm, which iterates between the pupil and object planes, at each step enforcing an amplitude constraint on each beam as well as a constraint on the phase difference between the two beams.

Fig. 4
Fig. 4

Example comparison of the performance of the DPR, SPR, and m-SPR algorithms using illumination scheme 1. The Strehl (ratio) as a function of iteration number for the three algorithms for both illuminator beams. The DPR algorithm converges rapidly to a value of unit Strehl. For one of the illuminator beams, the SPR algorithms obtain the correct phase profile. However, for the other illuminator beam, the SPR algorithms fail to converge to the correct phase profile.

Fig. 5
Fig. 5

Evaluation of the performance of the DPR, SPR, and m-SPR algorithms for three illumination schemes. In each chart, two bars are shown for each algorithm and illumination scheme. Left bar, variable of interest for the first complex field; right bar, variable of interest for the second complex field. (a) Probability of successful reconstruction, (b) average number of iterations to converge for those that achieved a successful reconstruction, and (c) average Strehl ratio. The DPR rapidly reconstructs the complex field 100% of the time, whereas the SPR algorithms achieve only a roughly 50% probability of successful reconstruction, and at a much slower convergence rate. The performance of the m-SPR algorithm is slightly worse than that of the SPR, indicating that augmentation of spatial filters to the iteration process has not improved algorithm performance.

Fig. 6
Fig. 6

Effect of measurement noise on the estimation accuracy of the DPR algorithm. The effect of measurement noise is found to be well approximated by the formula Strehl=exp(-1.225/SNR2).

Tables (1)

Tables Icon

Table 1 Performance of the DPR as a Function of Rytov Number

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

PCjx=arg minPCjxCj PCjx-x.
J(xk)=PC1xk-xk+PC2xk-xk.
Tz[]=F-1{F[]exp(iπλzκ¯2)}.
C1{V(r¯)L2||V(r¯)|=A0(r¯)}.
PC1[V(r¯)]=A0(r¯)exp{i arg[V(r¯)]}.
C2{V(r¯)L2||Tz[V(r¯)]|=Az(r¯z)}.
C˜2{V˜(r¯z)L2||V˜(r¯z)|=Az(r¯z)}.
PC˜2[V˜(r¯z)]=Az(r¯z)exp{i arg[V˜(r¯z)]}.
PC˜2[V˜(r¯z)]-V˜(r¯z)=Tz*{PC˜2[V˜(r¯z)]-V˜(r¯z)}=Tz*{PC˜2[V˜(r¯z)]}-V˜(r¯).
PC2[V(r¯)]=Tz*{Az(r¯z)exp[iθ(r¯z)]},
Tz,α[]=F-1F[]expiπλzκ¯2-πα22κ¯2,
I=A0(r¯)exp[iϕ0(r¯)], Tz*{Az(r¯z)exp[iϕz(r¯z)]}.
ϕ0(r¯)=arg(Tz*{Az(r¯z)exp[iϕz(r¯z)]}),
ϕz(r¯z)=arg(Tz{A0(r¯)exp[iϕ0(r¯)]}).
ϕ0(r¯)=arg(Tz,α*{Az(r¯z)exp[iϕz(r¯z)]}),
ϕz(r¯z)=arg(Tz,α{A0(r¯)exp[iϕ0(r¯)]}),
Iα=A0(r¯)exp[iϕ0(r¯)], Tz,α*{Az(r¯z)exp[iϕz(r¯z)]}.
[V1(r¯), V2(r¯)]2=V1(r¯)2+V2(r¯)2.
D1{[V1(r¯), V2(r¯)]L2||V1(r¯)|=A1(r¯); |V2(r¯)|=A2(r¯);andarg[V1(r¯)V2*(r¯)]=ϕ˜12(r¯)}.
W1(r¯)=A1(r¯)exp[iϕ1(r¯)],
W2(r¯)=A2(r¯)exp[iϕ2(r¯)],
W2(r¯)=A2(r¯)exp{i[ϕ1(r¯)-ϕ˜12(r¯)]}.
PD1[V1(r¯), V2(r¯)]
=argminPD1[V1(r¯),V2(r¯)]D1 PD1[V1(r¯), V2(r¯)]
-[V1(r¯), V2(r¯)].
ϕ1(r¯)=argminϕ^1(r¯)|A1(r¯)exp[iϕ^1(r¯)]-V1(r¯)|2+|A2(r¯)exp{i[ϕ^1(r¯)-ϕ˜12(r¯)]}-V2(r¯)|2.
Jϕ=R2dr¯×{A1(r¯)exp[iϕ^1(r¯)]-V1(r¯)}{A1(r¯)exp[iϕ^1(r¯)]-V1(r¯)}*+{A2(r¯)exp{i[ϕ^1(r¯)-ϕ˜12(r¯)]}-V2(r¯)}×{A2(r¯)exp{i[ϕ^1(r¯)-ϕ˜12(r¯)]}-V2(r¯)}*
=R2drA¯12(r¯)+|V1(r¯)|2+A22(r¯)+|V2(r¯)|2-V1*(r¯)A1(r¯)exp[iϕ^1(r¯)]-V1(r¯)A1(r¯)exp[-iϕ^1(r¯)]-V2*(r¯)A2(r¯)exp{i[ϕ^1(r¯)-ϕ˜12(r¯)]}-V2(r¯)A2(r¯)exp{-i[ϕ^1(r¯)-ϕ˜12(r¯)]}.
Jϕϕ^1(r¯)=-iV1*(r¯)A1(r¯)exp[iϕ^1(r¯)]+iV1(r¯)A1(r¯)exp[-iϕ^1(r¯)]-iV2*(r¯)A2(r¯)exp{i[ϕ^1(r¯)-ϕ˜12(r¯)]}+iV2(r¯)A2(r¯)exp{-i[ϕ^1(r¯)-ϕ˜12(r¯)]}.
exp[-iϕ^1(r¯)]{V1(r¯)A1(r¯)+V2*(r¯)A2(r¯)exp[iϕ˜12(r¯)]}
=exp[iϕ^1(r¯)]{V1*(r¯)A1(r¯)+V2*(r¯)A2(r¯)exp[-iϕ˜12(r¯)]}.
ϕ1(r¯)=arg{A1(r¯)V1(r¯)+A2(r¯)V2(r¯)exp[iϕ˜12(r¯)]}.
D2{[V1(r¯), V2(r¯)]L2||Tz[V1(r¯)]|=A1(r¯z); |Tz[V2(r¯)]|=A2(r¯z);andarg{Tz[V1(r¯)]Tz*[V2*(r¯)]}=ϕ˜12(r¯z)}.
X1(r¯z)=A1(r¯z)exp[iϕ1(r¯z)],
X2(r¯z)=A2(r¯z)exp{i[ϕ1(r¯z)-ϕ˜12(r¯z)]},
ϕ1(r¯z)=arg{A1(r¯z)V1(r¯z)+A2(r¯z)V2(r¯z)exp[iϕ˜12(r¯z)]}.
X1(r¯)=Tz*[X1(r¯z)],
X2(r¯)=Tz*[X2(r¯z)].
Strehl(U^j)=drU¯j(r¯)U^j*(r¯)2dr¯|Uj(r¯)|2dr¯|U^j(r¯)|2,

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