Abstract

New control algorithms for an adaptive system with high-resolution piston-type wave-front corrector and optoelectronic feedback that do not require phase information are introduced and analyzed. Numerical simulations of adaptive system performance in the presence of phase distortions described using several common phase fluctuation spectrums demonstrate the effectiveness of this approach. An adaptive optical system having two separate feedback loops for compensation of large- and small-scale phase distortions is studied. Spatiotemporal instabilities that can occur in high-resolution adaptive systems were observed in numerical simulations.

© 1998 Optical Society of America

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References

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  1. R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, 1991).
  2. M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).
  3. G. B. Love, J. S. Fender, S. R. Restaino, “Adaptive wavefront shaping with liquid crystals,” Opt. Photon. News 6 (October), 16–21 (1995).
    [CrossRef]
  4. P. R. Barbier, G. Moddel, “Spatial light modulators: processing light in real time,” Opt. Photon. News 8 (March), 16–21 (1997).
    [CrossRef]
  5. U. Efron, ed., Spatial Light Modulator Technology: Materials, Devices, and Applications (Marcel Dekker, New York, 1995).
  6. J. M. Younse, “Mirrors on a chip,” IEEE Spectr. 30 (November), 27–31 (1993).
    [CrossRef]
  7. M. C. Roggeman, V. M. Bright, B. M. Welsh, S. R. Hick, P. C. Roberts, W. D. Cowan, J. H. Comtois, “Use of micro-electro-mechanical deformable mirrors to control aberrations in optical systems: theoretical and experimental results,” Opt. Eng. 36, 1326–1338 (1997).
    [CrossRef]
  8. J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  9. C. A. Primmerman, T. R. Price, R. A. Humphreys, B. G. Zollars, H. T. Barclay, J. Herrmann, “Atmospheric-compensation experiments in strong-scintillation conditions,” Appl. Opt. 34, 2081–2088 (1995).
    [CrossRef] [PubMed]
  10. J. M. Beckers, “Adaptive optics for astronomy: principles, performance, and applications,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
    [CrossRef]
  11. R. A. Gonsalves, “Nonisoplanatic imaging by phase diversity,” Opt. Lett. 19, 493–495 (1994).
    [CrossRef] [PubMed]
  12. R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberration using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
    [CrossRef]
  13. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
    [CrossRef] [PubMed]
  14. V. Yu. Ivanov, V. P. Sivokon, M. A. Vorontsov, “Phase retrieval from a set of intensity measurements: theory and experiment,” J. Opt. Soc. Am. A 9, 1515–1524 (1992).
    [CrossRef]
  15. A. D. Fisher, C. Warde, “Technique for real-time high-resolution adaptive phase compensation,” Opt. Lett. 87, 353–355 (1983).
    [CrossRef]
  16. D. M. Pepper, C. J. Gaeta, P. V. Mitchell, “Real-time holography, innovative adaptive optics, and compensated optical processors using spatial light modulators,” in Spatial Light Modulator Technology: Materials, Devices, and Applications, U. Efron, ed. (Marcel Dekker, New York, 1995), pp. 585–654.
  17. M. A. Vorontsov, M. E. Kirakosyan, A. V. Larichev, “Correction of phase distortion in a nonlinear interferometer with an optical feedback loop,” Sov. J. Quantum Electron. 21, 105–108 (1991).
    [CrossRef]
  18. M. A. Vorontsov, I. P. Nikolaev, “Nonlinear 2-D feedback optical systems: new approaches for adaptive wavefront correction,” in Atmospheric Propagation and Remote Sensing III, W. A. Flood, W. B. Miller, eds., Proc. SPIE2222, 413–422 (1994).
    [CrossRef]
  19. W. J. Firth, M. A. Vorontsov, “Adaptive phase distortion suppression in a nonlinear system with feedback mirror,” J. Mod. Opt. 40, 1841–1846 (1993).
    [CrossRef]
  20. E. V. Degtiarev, M. A. Vorontsov, “Spatial filtering in nonlinear two-dimensional feedback systems: phase-distortion suppression,” J. Opt. Soc. Am. B 12, 1238–1248 (1995).
    [CrossRef]
  21. L. A. Lugiato, M. S. El. Naschie, eds., Nonlinear Optical Structures, Patterns, Chaos, Vol. 4 of Chaos, Solitons, Fractals (Pergamon, New York, 1994).
  22. M. A. Vorontsov, W. B. Miller, eds., Self-Organization in Optical Systems and Applications in Information Technology (Springer-Verlag, Berlin, 1995).
  23. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  24. 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]
  25. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [CrossRef]
  26. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  27. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1997 (2)

P. R. Barbier, G. Moddel, “Spatial light modulators: processing light in real time,” Opt. Photon. News 8 (March), 16–21 (1997).
[CrossRef]

M. C. Roggeman, V. M. Bright, B. M. Welsh, S. R. Hick, P. C. Roberts, W. D. Cowan, J. H. Comtois, “Use of micro-electro-mechanical deformable mirrors to control aberrations in optical systems: theoretical and experimental results,” Opt. Eng. 36, 1326–1338 (1997).
[CrossRef]

1995 (3)

1994 (1)

1993 (4)

J. M. Beckers, “Adaptive optics for astronomy: principles, performance, and applications,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
[CrossRef]

J. M. Younse, “Mirrors on a chip,” IEEE Spectr. 30 (November), 27–31 (1993).
[CrossRef]

W. J. Firth, M. A. Vorontsov, “Adaptive phase distortion suppression in a nonlinear system with feedback mirror,” J. Mod. Opt. 40, 1841–1846 (1993).
[CrossRef]

J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
[CrossRef] [PubMed]

1992 (3)

1991 (1)

M. A. Vorontsov, M. E. Kirakosyan, A. V. Larichev, “Correction of phase distortion in a nonlinear interferometer with an optical feedback loop,” Sov. J. Quantum Electron. 21, 105–108 (1991).
[CrossRef]

1983 (1)

A. D. Fisher, C. Warde, “Technique for real-time high-resolution adaptive phase compensation,” Opt. Lett. 87, 353–355 (1983).
[CrossRef]

1978 (1)

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

1976 (1)

1965 (1)

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]

Barbier, P. R.

P. R. Barbier, G. Moddel, “Spatial light modulators: processing light in real time,” Opt. Photon. News 8 (March), 16–21 (1997).
[CrossRef]

Barclay, H. T.

Beckers, J. M.

J. M. Beckers, “Adaptive optics for astronomy: principles, performance, and applications,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
[CrossRef]

Bright, V. M.

M. C. Roggeman, V. M. Bright, B. M. Welsh, S. R. Hick, P. C. Roberts, W. D. Cowan, J. H. Comtois, “Use of micro-electro-mechanical deformable mirrors to control aberrations in optical systems: theoretical and experimental results,” Opt. Eng. 36, 1326–1338 (1997).
[CrossRef]

Comtois, J. H.

M. C. Roggeman, V. M. Bright, B. M. Welsh, S. R. Hick, P. C. Roberts, W. D. Cowan, J. H. Comtois, “Use of micro-electro-mechanical deformable mirrors to control aberrations in optical systems: theoretical and experimental results,” Opt. Eng. 36, 1326–1338 (1997).
[CrossRef]

Cowan, W. D.

M. C. Roggeman, V. M. Bright, B. M. Welsh, S. R. Hick, P. C. Roberts, W. D. Cowan, J. H. Comtois, “Use of micro-electro-mechanical deformable mirrors to control aberrations in optical systems: theoretical and experimental results,” Opt. Eng. 36, 1326–1338 (1997).
[CrossRef]

Degtiarev, E. V.

Fender, J. S.

G. B. Love, J. S. Fender, S. R. Restaino, “Adaptive wavefront shaping with liquid crystals,” Opt. Photon. News 6 (October), 16–21 (1995).
[CrossRef]

Fienup, J. R.

Firth, W. J.

W. J. Firth, M. A. Vorontsov, “Adaptive phase distortion suppression in a nonlinear system with feedback mirror,” J. Mod. Opt. 40, 1841–1846 (1993).
[CrossRef]

Fisher, A. D.

A. D. Fisher, C. Warde, “Technique for real-time high-resolution adaptive phase compensation,” Opt. Lett. 87, 353–355 (1983).
[CrossRef]

Fried, D. L.

Gaeta, C. J.

D. M. Pepper, C. J. Gaeta, P. V. Mitchell, “Real-time holography, innovative adaptive optics, and compensated optical processors using spatial light modulators,” in Spatial Light Modulator Technology: Materials, Devices, and Applications, U. Efron, ed. (Marcel Dekker, New York, 1995), pp. 585–654.

Gonsalves, R. A.

Goodman, J. W.

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

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Hardy, J. W.

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Herrmann, J.

Hick, S. R.

M. C. Roggeman, V. M. Bright, B. M. Welsh, S. R. Hick, P. C. Roberts, W. D. Cowan, J. H. Comtois, “Use of micro-electro-mechanical deformable mirrors to control aberrations in optical systems: theoretical and experimental results,” Opt. Eng. 36, 1326–1338 (1997).
[CrossRef]

Humphreys, R. A.

Ivanov, V. Yu.

Kirakosyan, M. E.

M. A. Vorontsov, M. E. Kirakosyan, A. V. Larichev, “Correction of phase distortion in a nonlinear interferometer with an optical feedback loop,” Sov. J. Quantum Electron. 21, 105–108 (1991).
[CrossRef]

Larichev, A. V.

M. A. Vorontsov, M. E. Kirakosyan, A. V. Larichev, “Correction of phase distortion in a nonlinear interferometer with an optical feedback loop,” Sov. J. Quantum Electron. 21, 105–108 (1991).
[CrossRef]

Love, G. B.

G. B. Love, J. S. Fender, S. R. Restaino, “Adaptive wavefront shaping with liquid crystals,” Opt. Photon. News 6 (October), 16–21 (1995).
[CrossRef]

Mitchell, P. V.

D. M. Pepper, C. J. Gaeta, P. V. Mitchell, “Real-time holography, innovative adaptive optics, and compensated optical processors using spatial light modulators,” in Spatial Light Modulator Technology: Materials, Devices, and Applications, U. Efron, ed. (Marcel Dekker, New York, 1995), pp. 585–654.

Moddel, G.

P. R. Barbier, G. Moddel, “Spatial light modulators: processing light in real time,” Opt. Photon. News 8 (March), 16–21 (1997).
[CrossRef]

Nikolaev, I. P.

M. A. Vorontsov, I. P. Nikolaev, “Nonlinear 2-D feedback optical systems: new approaches for adaptive wavefront correction,” in Atmospheric Propagation and Remote Sensing III, W. A. Flood, W. B. Miller, eds., Proc. SPIE2222, 413–422 (1994).
[CrossRef]

Noll, R. J.

Paxman, R. G.

Pepper, D. M.

D. M. Pepper, C. J. Gaeta, P. V. Mitchell, “Real-time holography, innovative adaptive optics, and compensated optical processors using spatial light modulators,” in Spatial Light Modulator Technology: Materials, Devices, and Applications, U. Efron, ed. (Marcel Dekker, New York, 1995), pp. 585–654.

Price, T. R.

Primmerman, C. A.

Restaino, S. R.

G. B. Love, J. S. Fender, S. R. Restaino, “Adaptive wavefront shaping with liquid crystals,” Opt. Photon. News 6 (October), 16–21 (1995).
[CrossRef]

Roberts, P. C.

M. C. Roggeman, V. M. Bright, B. M. Welsh, S. R. Hick, P. C. Roberts, W. D. Cowan, J. H. Comtois, “Use of micro-electro-mechanical deformable mirrors to control aberrations in optical systems: theoretical and experimental results,” Opt. Eng. 36, 1326–1338 (1997).
[CrossRef]

Roggeman, M. C.

M. C. Roggeman, V. M. Bright, B. M. Welsh, S. R. Hick, P. C. Roberts, W. D. Cowan, J. H. Comtois, “Use of micro-electro-mechanical deformable mirrors to control aberrations in optical systems: theoretical and experimental results,” Opt. Eng. 36, 1326–1338 (1997).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Schulz, T. J.

Sivokon, V. P.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, 1991).

Vorontsov, M. A.

E. V. Degtiarev, M. A. Vorontsov, “Spatial filtering in nonlinear two-dimensional feedback systems: phase-distortion suppression,” J. Opt. Soc. Am. B 12, 1238–1248 (1995).
[CrossRef]

W. J. Firth, M. A. Vorontsov, “Adaptive phase distortion suppression in a nonlinear system with feedback mirror,” J. Mod. Opt. 40, 1841–1846 (1993).
[CrossRef]

V. Yu. Ivanov, V. P. Sivokon, M. A. Vorontsov, “Phase retrieval from a set of intensity measurements: theory and experiment,” J. Opt. Soc. Am. A 9, 1515–1524 (1992).
[CrossRef]

M. A. Vorontsov, M. E. Kirakosyan, A. V. Larichev, “Correction of phase distortion in a nonlinear interferometer with an optical feedback loop,” Sov. J. Quantum Electron. 21, 105–108 (1991).
[CrossRef]

M. A. Vorontsov, I. P. Nikolaev, “Nonlinear 2-D feedback optical systems: new approaches for adaptive wavefront correction,” in Atmospheric Propagation and Remote Sensing III, W. A. Flood, W. B. Miller, eds., Proc. SPIE2222, 413–422 (1994).
[CrossRef]

Warde, C.

A. D. Fisher, C. Warde, “Technique for real-time high-resolution adaptive phase compensation,” Opt. Lett. 87, 353–355 (1983).
[CrossRef]

Welsh, B. M.

M. C. Roggeman, V. M. Bright, B. M. Welsh, S. R. Hick, P. C. Roberts, W. D. Cowan, J. H. Comtois, “Use of micro-electro-mechanical deformable mirrors to control aberrations in optical systems: theoretical and experimental results,” Opt. Eng. 36, 1326–1338 (1997).
[CrossRef]

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Younse, J. M.

J. M. Younse, “Mirrors on a chip,” IEEE Spectr. 30 (November), 27–31 (1993).
[CrossRef]

Zollars, B. G.

Annu. Rev. Astron. Astrophys. (1)

J. M. Beckers, “Adaptive optics for astronomy: principles, performance, and applications,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
[CrossRef]

Appl. Opt. (2)

IEEE Spectr. (1)

J. M. Younse, “Mirrors on a chip,” IEEE Spectr. 30 (November), 27–31 (1993).
[CrossRef]

J. Mod. Opt. (2)

W. J. Firth, M. A. Vorontsov, “Adaptive phase distortion suppression in a nonlinear system with feedback mirror,” J. Mod. Opt. 40, 1841–1846 (1993).
[CrossRef]

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. (2)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Eng. (1)

M. C. Roggeman, V. M. Bright, B. M. Welsh, S. R. Hick, P. C. Roberts, W. D. Cowan, J. H. Comtois, “Use of micro-electro-mechanical deformable mirrors to control aberrations in optical systems: theoretical and experimental results,” Opt. Eng. 36, 1326–1338 (1997).
[CrossRef]

Opt. Lett. (2)

R. A. Gonsalves, “Nonisoplanatic imaging by phase diversity,” Opt. Lett. 19, 493–495 (1994).
[CrossRef] [PubMed]

A. D. Fisher, C. Warde, “Technique for real-time high-resolution adaptive phase compensation,” Opt. Lett. 87, 353–355 (1983).
[CrossRef]

Opt. Photon. News (2)

G. B. Love, J. S. Fender, S. R. Restaino, “Adaptive wavefront shaping with liquid crystals,” Opt. Photon. News 6 (October), 16–21 (1995).
[CrossRef]

P. R. Barbier, G. Moddel, “Spatial light modulators: processing light in real time,” Opt. Photon. News 8 (March), 16–21 (1997).
[CrossRef]

Proc. IEEE (1)

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Sov. J. Quantum Electron. (1)

M. A. Vorontsov, M. E. Kirakosyan, A. V. Larichev, “Correction of phase distortion in a nonlinear interferometer with an optical feedback loop,” Sov. J. Quantum Electron. 21, 105–108 (1991).
[CrossRef]

Other (9)

M. A. Vorontsov, I. P. Nikolaev, “Nonlinear 2-D feedback optical systems: new approaches for adaptive wavefront correction,” in Atmospheric Propagation and Remote Sensing III, W. A. Flood, W. B. Miller, eds., Proc. SPIE2222, 413–422 (1994).
[CrossRef]

D. M. Pepper, C. J. Gaeta, P. V. Mitchell, “Real-time holography, innovative adaptive optics, and compensated optical processors using spatial light modulators,” in Spatial Light Modulator Technology: Materials, Devices, and Applications, U. Efron, ed. (Marcel Dekker, New York, 1995), pp. 585–654.

U. Efron, ed., Spatial Light Modulator Technology: Materials, Devices, and Applications (Marcel Dekker, New York, 1995).

R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, 1991).

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

L. A. Lugiato, M. S. El. Naschie, eds., Nonlinear Optical Structures, Patterns, Chaos, Vol. 4 of Chaos, Solitons, Fractals (Pergamon, New York, 1994).

M. A. Vorontsov, W. B. Miller, eds., Self-Organization in Optical Systems and Applications in Information Technology (Springer-Verlag, Berlin, 1995).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

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

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Figures (19)

Fig. 1
Fig. 1

Adaptive system for phase distortion suppression composed of a conventional adaptive optical system (CAOS) and a high-resolution adaptive system with diffractive feedback (ASDF).

Fig. 2
Fig. 2

Optical scheme for a high-resolution adaptive system with diffractive feedback.

Fig. 3
Fig. 3

Optical scheme for a Kerr-slice/feedback-mirror system.

Fig. 4
Fig. 4

Signal processing corresponding to iterative algorithm (7b).

Fig. 5
Fig. 5

Spectral coefficient for phase distortion suppression T as a function of θ for KI0=-8. Solid curves A and C correspond to an adaptive system without spatial filtering. Instability spectral domains are shown in gray. Curves A and B correspond to a system with binary-phase spatial filter (9); curve A corresponds to an adaptive system with a low-pass spectral filter where qcut=q1.

Fig. 6
Fig. 6

Average Strehl ratio St versus number of control channels N=n×n: curve 1, adaptive system based on the phase conjugation algorithm; diffractive feedback adaptive system with (curve 2) binary phase and (curve 3) a low-pass spatial filter (K=-8, α=0.1).

Fig. 7
Fig. 7

Realizations of (a) the distorted phase and [(b)–(d)] the corrected phase for an adaptive system with (b) n=32, (c) n=64, and (d) n=128 (continuous model) control channels. Images (e) and (f) correspond to the spatial spectrums of (a) and (d) (Gaussian spectrum; K=-8, α=0.1, qφ=q1). The spatial filter band width is shown in (f) by the circle qcut=q2.

Fig. 8
Fig. 8

Average spectral energy distribution function E(q) for undistorted (diffraction-limited), distorted (Gaussian spatial spectrum), and adaptively corrected cases for different numbers of control channels N=n×n (K=-8, α=0.1, qcut=q2, qφ=q1).

Fig. 9
Fig. 9

Average spectral energy distribution function E(q) for an adaptive system operated in the presence of phase distortions based on the Andrews power spectrum [D/r0=9]: (curve 1) without correction; with correction of (curve 2) first NL=6 and (curve 3) first NL=15 Zernike polynomials; (curve 4) with correction using diffractive feedback adaptive system only; and with correction with prior removal of (curve 5) first NL=6 and (6) first NL=15 Zernike polynomials (n=128, K=-16, α=0.05, qcut=q2). Curve 7 corresponds to an undistorted wave.

Fig. 10
Fig. 10

Average spectral energy distribution function E(q) for an adaptive system operated in the presence of phase distortions based on the Andrews spectrum (D/r0=9): curve 1, with correction of first NL=15 Zernike polynomials, and with correction using diffractive feedback adaptive system with first NL=15 Zernike polynomials removed for (curve 2) n=16, (curve 3) n=32, and (curve 4) n=128 (K=-16, α=0.05, qcut=q2). Curve 5 corresponds to an undistorted wave.

Fig. 11
Fig. 11

(a) Distorted phase realization based on the Andrews spatial spectrum (D/r0=9); residual phase for (b) a conventional adaptive system with NL=15, (c) a diffractive feedback adaptive system (n=128, K=-16, α=0.05, qcut=q2), and (d) a diffractive feedback adaptive system with first NL=15 Zernike polynomials removed.

Fig. 12
Fig. 12

Strehl ratio St versus feedback coefficient |K| in an adaptive system (NL=6, n=128, and qcut=q2) for a fixed number of iterations m=15. Phase distortion realization is shown in Fig. 15a below.

Fig. 13
Fig. 13

Average Strehl ratio St versus nondimensional parameter ld for a diffractive feedback adaptive system operated in the presence of phase fluctuations based on (curve 1) Andrews and (curve 2) Kolmogorov power spectrums (D/r0=9) with prior removal of first six Zernike polynomials.

Fig. 14
Fig. 14

Phase distortion suppression in the presence of noise: Strehl ratio dynamics in iterative algorithm (7b) for different signal-to-noise ratio S/N (n=128, K=-16, α=0.05, qcut=q2). The distorted phase is shown in Fig. 15(a).

Fig. 15
Fig. 15

(a) Distorted phase realization (Andrews spectrum, D/r0=9) with the first NL=6 Zernike polynomials removed; (b) corrected phase in the presence of noise; and feedback intensities (c) Iδ(m)(r) and (d) I˜δ(m)(r) (S/N=5.1) for n=128, K=-16, α=0.05, and qcut=q2. Images (b)–(d) correspond to m=140.

Fig. 16
Fig. 16

Dependence of the normalized Strehl ratio on input beam intensity modulation parameter μ after m=180 iterations for conventional algorithm based on Eq. (7b) (solid curve) and modified control algorithm (dashed curve); K=-16, α=1/30, and qcut=q2. The normalization parameter St0 corresponds to the Strehl ratio for μ=0. The input beam intensity distribution Iin(r) used in simulations is shown in (a) μ=0.18 and (b) μ=0.33. The distorted phase is shown in Fig. 15(a).

Fig. 17
Fig. 17

Adaptive phase distortion correction in the presence of input beam intensity modulation. Parts (a), (c), and (e) are the residual phase δ(m)(r); (b), (d), and (f) are the feedback control signal νFB(m)(r). Images (a) and (b) correspond to μ=0.18, (c) and (d) correspond to μ=0.33, and (e) and (f) correspond to the modified control algorithm. The phase distortion parameters and algorithm are the same as those in Fig. 16.

Fig. 18
Fig. 18

Traveling-wave instability in a diffractive feedback adaptive system (n=128, K=-16, α=0.05, qcut=q2): (a) distorted phase corresponding to a Kolmogorov spectrum with D/r0=10, (b) feedback intensity distribution Iδ(r) during the adaptation process.

Fig. 19
Fig. 19

2π localized states in a diffractive feedback adaptive system: (a) distorted phase realization for a Gaussian spatial spectrum (St=0.33); (b) steady-state residual phase (St=0.85) for n=128, K=-8, α=0.05, and qcut=q2; (c) phase profiles along the dashed line in (a) and (b).

Tables (1)

Tables Icon

Table 1 Phase Distortion Suppression Efficiency

Equations (31)

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

u(r, t)=j=1Nuj(t)Sj(r),
-2ikAz=2A,0zL,
τnlu(r, t)t=-u(r, t)+d2u(r, t)+KnlIδ(r, t)
τu(r, t)t=-u(r, t)+KvFB(r, t),
vFB(r, t)=h(r-r)[Iδ(r, t)-Iin(r)]d2r.
VFB(q, t)=H(q)F(q, t),
τdudt=-u+KνFB
u(m+1)=(1-α)u(m)+KανFB(m).
T(q)Ψ¯(q)Φ(q)=11-2KI0H(q)sin(q2L/2k),
H(q)=1qq1-1q1<qq20q2<q.
Iin(r)=I0 exp-2rD/2p
G(q)=G0 exp(-2q2/qφ2);
G(q)=2π0.033(1.68/r0)5/3q-11/3;
G(q)=2π0.033(1.68/r0)5/3(q2+qA2)-11/6×exp(-q2/qa2)[1+a1(q/qa)-a2(q/qa)7/6].
E(q)=0q02π|F{Iin1/2(r)exp[iδ¯(r)]}|2q dqdϕ,
I˜δ(m)(r)=Iδ(m)(r)+βξ(m)(r),m=1, 2, 3 ,, 
I˜in(r)=Iin(r)+βξ(0)(r),
S/N=I˜δ(m)(r)d2rβ[ξ(m)(r)-ξ¯]2 d2r1/2,
I˜in(r)=Iin(r)+βI0ξ(r),
μ2=[I˜in(r)-I]2 d2r/I2,I=I˜in(r)d2r
δ(r, t)=δ¯(t)+ψ(r, t).
A0(r)= exp(iqr)A0(q)d2q,
ψ(r, t)= exp(iqr)Ψ(q, t)d2q.
A0(q) * Ψ(q, t)I01/2Ψ(q, t),
A(r, 0, t)=A0(r)exp(iδ¯)[1+iψ(r, t)].
A(q, L, t)=A(q, 0, t)exp(-iσq2),
A(q, L, t)=[A0(q)+iI01/2Ψ(q, t)]exp(iδ¯)exp(-iσq2).
A(r, L, t)=exp(iδ¯)A0(d)(r)+iI01/2× exp(iqr)Ψ(q, t)exp(-iσq2)d2q.
Iδ(r, t)=A02(r)+2I01/2A0(r)× exp(iqr)Ψ(q, t)sin(σq2)d2q.
τΨt+[1-2KI0H(q)sin(σq2)]Ψ=Φ,
Ψ¯(q)=Φ(q)/[1-2KI0H(q)sin(σq2)].

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