Abstract

We analyze various scenarios of adaptive wave-front phase-aberration correction in optical-receiver-type systems when inhomogeneties of the wave propagation medium are either distributed along the propagation path or localized in a few thin layers remotely located from the receiver telescope pupil. Phase-aberration compensation is performed with closed-loop control architectures based on decoupled stochastic parallel gradient descent, stochastic parallel gradient descent, and phase conjugation control algorithms. Both receiver system aperture diffraction effects and the effect of wave-front corrector position on phase-aberration compensation efficiency are analyzed.

© 2004 Optical Society of America

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  5. L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. (Oxford) 206, 65–71 (2002).
    [CrossRef]
  6. P. Artal, S. Marcos, R. Navarro, D. Williams, “Odd aberrations and double-pass measurements of retinal image quality,” J. Opt. Soc. Am. A 12, 195–201 (1995).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. B. L. Ellerbroek, F. Rigaut, “Methods for correcting tilt anisoplanatism in laser-guide-star-based multiconjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2539–2547 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  31. M. A. Vorontsov, G. W. Garhart, M. Cohen, G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17, 1440–1453 (2000).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2002

2001

2000

1999

1998

1997

1995

1994

1992

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

L. C. Andrews, “An analytic model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

1991

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]

1977

1975

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).
[CrossRef]

1974

1970

1965

1953

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65, 229–236 (1953).
[CrossRef]

Ageorges, N.

N. Ageorges, C. Dainty, Laser Guide Star Adaptive Optics for Astronomy (Kluwer Academic, Dordrecht, London, 2000).

Aksenov, V.

Albert, O.

L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. (Oxford) 206, 65–71 (2002).
[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]

Andrews, L. C.

L. C. Andrews, “An analytic model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

Arnold, L.

L. Arnold, “Optimized axial support topologies for thin telescope mirrors,” Opt. Eng. 34, 567–574 (1995).
[CrossRef]

Artal, P.

Babcock, H. W.

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65, 229–236 (1953).
[CrossRef]

Backus, S.

Banakh, V.

Barchers, J. D.

Beckers, J. M.

J. M. Beckers, “Detailed compensation of atmospheric seeing using multi-conjugate adaptive optics,” F. J. Roddier, ed., Proc. SPIE1114, 215–217 (1989).
[CrossRef]

Beresnev, L. A.

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]

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]

Buffington, A.

Carhart, G. W.

Cauwenberghs, G.

Chesnokov, S. S.

Cohen, M.

Dainty, C.

N. Ageorges, C. Dainty, Laser Guide Star Adaptive Optics for Astronomy (Kluwer Academic, Dordrecht, London, 2000).

Davletshina, I. V.

Ellerbroek, B. L.

Flicker, C.

Fried, D. L.

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]

Garhart, G. W.

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, New York, 1998).

Hariharan, P.

P. Hariharan, Selected Papers on Interferometry (SPIE Optical Engineering Press, Bellingham, Wash., 1991).

Johnston, D. C.

Just, E. W.

Justh, E. W.

Kapteyn, H.

Koivunen, A. C.

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4, Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).

Krishnaprasad, P. S.

Le Louarn, M.

LeBigot, E. O.

Lee, D. J.

Maginnis, K.

Marchetti, E.

R. Ragazzoni, E. Marchetti, F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

Marcos, S.

Mourou, G.

Muller, R. A.

Murnane, M.

Navarro, R.

Norris, T. B.

L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. (Oxford) 206, 65–71 (2002).
[CrossRef]

O’Meara, T. R.

Pitsianis, N. P.

Plemmons, R. J.

Pruidze, D. V.

Ragazzoni, R.

R. Ragazzoni, E. Marchetti, F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

Ricklin, J. C.

Rigaut, F.

B. L. Ellerbroek, F. Rigaut, “Methods for correcting tilt anisoplanatism in laser-guide-star-based multiconjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2539–2547 (2001).
[CrossRef]

R. Ragazzoni, E. Marchetti, F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

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]

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press, New York, 1999).

Roggemann, M. C.

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]

Russek, U.

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4, Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).

Sarazin, M.

Sherman, L.

L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. (Oxford) 206, 65–71 (2002).
[CrossRef]

Shmalhauzen, V. I.

M. A. Vorontsov, V. I. Shmalhauzen, Principles of Adaptive Optics (Nauka, Moscow, 1985).

Sivokon, V. P.

Smartt, R. N.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).
[CrossRef]

Steel, W. H.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).
[CrossRef]

Tatarskii, V. I.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4, Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).

Tikhomirova, O.

Tokovinin, A.

Tyler, 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]

Van Loan, C.

Vaughn, J. L.

Vdovin, G.

Voelz, D. G.

Vorontsov, M. A.

M. A. Vorontsov, “Decoupled stochastic parallel gradient descent optimization for adaptive optics: integrated approach for wave-front sensor information fusion,” J. Opt. Soc. Am. A 19, 356–368 (2002).
[CrossRef]

M. A. Vorontsov, E. W. Justh, L. A. Beresnev, “Adaptive optics with advanced phase-contrast techniques. I. High-resolution wave-front sensing,” J. Opt. Soc. Am. A 18, 1289–1299 (2001).
[CrossRef]

E. W. Just, M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, P. S. Krishnaprasad, “Adaptive optics with advanced phase-contrast techniques. II. High-resolution wave-front control,” J. Opt. Soc. Am. A 18, 1300–1311 (2001).
[CrossRef]

M. A. Vorontsov, G. W. Garhart, M. Cohen, G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17, 1440–1453 (2000).
[CrossRef]

V. P. Sivokon, M. A. Vorontsov, “High-resolution adaptive phase distortion suppression based solely on intensity information,” J. Opt. Soc. Am. A 16, 2567–2573 (1999).

M. A. Vorontsov, V. P. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. A 15, 2745–2758 (1998).
[CrossRef]

M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, D. G. Voelz, “Adaptive imaging system for phase-distorted extended source/multiple distance objects,” Appl. Opt. 36, 3319–3328 (1997).
[CrossRef] [PubMed]

M. A. Vorontsov, V. I. Shmalhauzen, Principles of Adaptive Optics (Nauka, Moscow, 1985).

Welsh, B. M.

D. C. Johnston, B. M. Welsh, “Analysis of multiconjugate adaptive optics,” J. Opt. Soc. Am. A 11, 394–408 (1994).
[CrossRef]

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC Press, New York, 1998).

Wild, W. J.

Williams, D.

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]

Ye, J. Y.

L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. (Oxford) 206, 65–71 (2002).
[CrossRef]

Young, A. T.

Zeek, E.

Appl. Opt.

Astron. Astrophys.

R. Ragazzoni, E. Marchetti, F. Rigaut, “Modal tomography for adaptive optics,” Astron. Astrophys. 342, L53–L56 (1999).

J. Microsc. (Oxford)

L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. (Oxford) 206, 65–71 (2002).
[CrossRef]

J. Mod. Opt.

L. C. Andrews, “An analytic model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

M. C. Roggemann, A. C. Koivunen, “Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wave-front correction,” J. Opt. Soc. Am. A 17, 53–62 (2000).
[CrossRef]

M. A. Vorontsov, G. W. Garhart, M. Cohen, G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17, 1440–1453 (2000).
[CrossRef]

A. Tokovinin, M. Le Louarn, M. Sarazin, “Isoplanatism in multi-conjugate adaptive optics systems,” J. Opt. Soc. Am. A 17, 1819–1827 (2000).
[CrossRef]

J. D. Barchers, B. L. Ellerbroek, “Improved compensation of turbulence-induced amplitude and phase distortions by means of multiple near-field phase adjustments,” J. Opt. Soc. Am. A 18, 399–411 (2001).
[CrossRef]

M. A. Vorontsov, E. W. Justh, L. A. Beresnev, “Adaptive optics with advanced phase-contrast techniques. I. High-resolution wave-front sensing,” J. Opt. Soc. Am. A 18, 1289–1299 (2001).
[CrossRef]

E. W. Just, M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, P. S. Krishnaprasad, “Adaptive optics with advanced phase-contrast techniques. II. High-resolution wave-front control,” J. Opt. Soc. Am. A 18, 1300–1311 (2001).
[CrossRef]

B. L. Ellerbroek, F. Rigaut, “Methods for correcting tilt anisoplanatism in laser-guide-star-based multiconjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2539–2547 (2001).
[CrossRef]

D. C. Johnston, B. M. Welsh, “Analysis of multiconjugate adaptive optics,” J. Opt. Soc. Am. A 11, 394–408 (1994).
[CrossRef]

B. L. Ellerbroek, C. Van Loan, N. P. Pitsianis, R. J. Plemmons, “Optimizing closed-loop adaptive-optics performance with use of multiple control bandwidths,” J. Opt. Soc. Am. A 11, 2871–2886 (1994).
[CrossRef]

E. O. LeBigot, W. J. Wild, “Theory of branch-point detection and its implementation,” J. Opt. Soc. Am. A 16, 1724–1729 (1999).
[CrossRef]

V. P. Sivokon, M. A. Vorontsov, “High-resolution adaptive phase distortion suppression based solely on intensity information,” J. Opt. Soc. Am. A 16, 2567–2573 (1999).

M. A. Vorontsov, V. P. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. A 15, 2745–2758 (1998).
[CrossRef]

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

P. Artal, S. Marcos, R. Navarro, D. Williams, “Odd aberrations and double-pass measurements of retinal image quality,” J. Opt. Soc. Am. A 12, 195–201 (1995).
[CrossRef]

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

M. A. Vorontsov, “Decoupled stochastic parallel gradient descent optimization for adaptive optics: integrated approach for wave-front sensor information fusion,” J. Opt. Soc. Am. A 19, 356–368 (2002).
[CrossRef]

J. D. Barchers, “Closed-loop stable control of two deformable mirrors for compensation of amplitude and phase fluctuations,” J. Opt. Soc. Am. A 19, 926–945 (2002).
[CrossRef]

J. D. Barchers, D. L. Fried, “Optimal control of laser beams for propagation through a turbulent medium,” J. Opt. Soc. Am. A 19, 1779–1793 (2002).
[CrossRef]

Jpn. J. Appl. Phys.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).
[CrossRef]

Nature

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. Eng.

L. Arnold, “Optimized axial support topologies for thin telescope mirrors,” Opt. Eng. 34, 567–574 (1995).
[CrossRef]

Opt. Lett.

Publ. Astron. Soc. Pac.

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65, 229–236 (1953).
[CrossRef]

Other

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, New York, 1998).

F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press, New York, 1999).

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC Press, New York, 1998).

J. M. Beckers, “Detailed compensation of atmospheric seeing using multi-conjugate adaptive optics,” F. J. Roddier, ed., Proc. SPIE1114, 215–217 (1989).
[CrossRef]

N. Ageorges, C. Dainty, Laser Guide Star Adaptive Optics for Astronomy (Kluwer Academic, Dordrecht, London, 2000).

M. A. Vorontsov, V. I. Shmalhauzen, Principles of Adaptive Optics (Nauka, Moscow, 1985).

P. Hariharan, Selected Papers on Interferometry (SPIE Optical Engineering Press, Bellingham, Wash., 1991).

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4, Wave Propagation through Random Media (Springer-Verlag, Berlin, 1989).

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Figures (10)

Fig. 1
Fig. 1

Adaptive receiver system setting with distant phase-distorting layers. The closed-loop (feedback) system with a single phase corrector operates with a D-SPGD controller and two (near- and far-field) wave-front sensors. The optical relay system and the beam splitter, BS, provide simultaneous re-imaging of the wave-front corrector at the input apertures of both wave-front sensors. The schematic of the near-field wave-front sensor (PDI) is shown at the bottom right. In the adaptive system with the SPGD controller, the near-field wave-front sensor is absent. OR, logical “or.”

Fig. 2
Fig. 2

Reference wave (beam) intensity and phase evolution along the atmospheric propagation path over the distance l after entering a single remote phase screen at z=0. Gray-scale images of the input beam intensity for a Gaussian beam, a, and for a super-Gaussian beam, b, are superimposed on the wave-front corrector aperture. a, Random phase perturbations applied to the wave-front corrector with N=32×32 piston-type actuators represented by gray-scale intensity modulation. c, The remote phase-screen phase perturbation induced into the input wave at z=0 (Dc/r0=3). c–i, Evolution of phase (c–f) and intensity (b, and g–i) of the propagating super-Gaussian beam for different normalized distances lˆ=l/ld (ld=0.5ka2): b and c for lˆ=0, d and g for lˆ=0.005, e and h for lˆ=0.01, and f and i for lˆ=0.05. White arrows point to the centers of phase dislocations (branch points); black arrows point to the boundaries of 2π phase cuts.

Fig. 3
Fig. 3

Aperture-averaged Rytov variance as a function of normalized propagation distance lˆ=l/ld: a, for a single distant phase-distorting layer and b, for ten equally spaced phase screens, with different turbulence strengths characterized by the ratios Dc/r0. Solid curves, super-Gaussian beam with a=1.5Dc; dashed and dotted curves, Gaussian beam with a=1.0Dc and with a=0.5Dc, respectively.

Fig. 4
Fig. 4

Ensemble-averaged Strehl ratio adaptation curves for D-SPGD (solid curves) and SPGD (dotted curves) systems for wave-front correctors with different numbers N of piston-type correcting elements: a, single phase-distorting layer at the receiver telescope pupil; b, distant single phase-distorting layer at the distance l=0.05ld from the telescope pupil; and c, ten phase-distorting layers equally spaced along the distance l=0.05ld. In all cases Dc/r0=6.0 and the control parameters are optimized separately for each N.

Fig. 5
Fig. 5

Strehl ratio 〈St〉 achieved after 40 iterations of the adaptation process versus standard deviation of the phase perturbations introduced by distorting layers for the D-SPGD system with different numbers N of control elements: a, single phase-distorting layer at the pupil of the telescope; b, ten phase-distorting layers equally spaced along the distance l=0.05ld. The control parameters are the same as in Fig. 4.

Fig. 6
Fig. 6

Ensemble-averaged Strehl ratio 〈St〉 achieved after 40 iterations as a function of normalized propagation distance lˆ for different Dc/r0: a, D-SPGD system with a single distant phase screen (solid curves) and multiple distant phase screens (dotted curves); b, high-resolution D-SPGD (solid curves) and phase-conjugation (dotted curves) systems with a single distant phase screen. In a, the averaged stationary Strehl ratios before adaptation, St0, are shown by diamond symbols.

Fig. 7
Fig. 7

Averaged energy-loss factor P/P0 and the metric StP (Strehl ratio 〈St〉 normalized by the energy-loss factor) for the case of a single phase-distorting layer located a distance lˆ from the telescope pupil: a, averaged energy-loss factor P/P0 for a Gaussian beam with a=Dc (solid curves) and for a super-Gaussian beam with a=1.5Dc (dashed curves); b, performance metric StP for the D-SPGD (solid curves) and phase conjugation systems (dotted curves) having N=32×32 control channels and different Dc/r0 values.

Fig. 8
Fig. 8

Compensation of a single distant phase screen by use of the D-SPGD (solid curves) and phase conjugation (dotted curves), and phase reconstruction from interferometric sensor data averaged over the subaperatures (dotted–dashed curves) for different control channel numbers N (number of wave-front corrector actuators): a, Strehl ratio 〈St〉; b, the metric StP versus the normalized propagation distance lˆ for Dc/r0=8.

Fig. 9
Fig. 9

Effect of the wave-front corrector position on phase-aberration compensation efficiency for a single distant phase screen located a distance l=0.05ld from the pupil plane with Dc/r0=8: a, simplified schematic of the adaptive system optical train with the wave-front corrector positioned a distance lC from the conjugate pupil plane; b, c, ensemble-averaged Strehl ratio 〈St〉 versus the normalized corrector displacement distance l^C=lC/lCph for high-resolution (N=128×128) D-SPGD (solid curves) and phase-conjugated (dotted curves) control algorithms for the receiver telescope with infinite aperture, b, and for the telescope having a finite aperture size of Dc coincident with the corrector aperture, c. The aperture radius of the input super-Gaussian beam a=1.5Dc.

Fig. 10
Fig. 10

Effect of wave-front corrector position on phase-aberration compensation efficiency for multiple phase screens equally spaced over the distance l=0.05ld (Dc/r0=8): a, b, ensemble-averaged Strehl ratio 〈St〉 versus normalized displacement distance l^C=lC/lCph for high-resolution (N=128×128) D-SPGD (solid curves) and phase-conjugated (dotted curves) control algorithms for the receiver telescope with infinite aperture, a, and for the telescope aperture coincident with the corrector aperture of size Dc, b. The aperture radius of the input super-Gaussian beam a=1.5 Dc.

Equations (13)

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I¯j=ΩjIδ(r)d2r.
uj(n+1)=uj(n)-γ(n)δI¯j(n)δuj(n),(j=1,, N),
γ(n)=γ0(1-κSt(n)),
J3=j=1NI¯j=ΩCIδ(r)d2r,
uj(n+1)=uj(n)-γ(n)δJf-f(n)δuj(n)(j=1,, N),
2ik Az=2A+2k2n1A,
A(r, z=0)=Iin1/2(r)exp[iφin(r)],
Iin(r)=I0exp[-(|r|2/a2)n],
n1(r, z)=j=1Mδ(z-zj)zj-1zjn1(r, z)dz=-k-1j=1M[δ(z-zj)φj(r)],
GA(q)=2π0.033(1.68/r0)5/3(q2+qA2)-11/6exp(-q2/qa2)×[1+1.802(q/qa)-0.254(q/qa)7/6].
σφ=1Mj=1Mσj21/2,σj2=S-1ΩCφj2(r)d2r,
σI2=1SΩCσ12(r)d2r,
uj=-ΩCφp(r)S0(r-rj)d2r=-Ωjφp(r)d2r,

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