Abstract

The multi-perturbation stochastic parallel gradient descent (SPGD) method for adaptive optics is presented in this work. The method is based on a new architecture. The incoming beam with distorted wavefront is split into N sub-beams. Each sub-beam is modulated by a wavefront corrector and its performance metric is measured subsequently. Adaptive system based on the multi-perturbation SPGD can operate in two modes – the fast descent mode and the modal basis updating mode. Control methods of the two operation modes are given. Experiments were carried out to prove the effectiveness of the proposed method. Analysis as well as experimental results showed that the two operation modes of the multi-perturbation SPGD enhance the conventional SPGD in different ways. The fast descent mode provides faster convergence than the conventional SPGD. The modal basis updating mode can optimize the modal basis set for SPGD with global coupling.

© 2015 Optical Society of America

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References

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  1. M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
    [Crossref] [PubMed]
  2. A. J. Wright, S. P. Poland, J. M. Girkin, C. W. Freudiger, C. L. Evans, and X. S. Xie, “Adaptive optics for enhanced signal in CARS microscopy,” Opt. Express 15(26), 18209–18219 (2007).
    [Crossref] [PubMed]
  3. P. N. Marsh, D. Burns, and J. M. Girkin, “Practical implementation of adaptive optics in multiphoton microscopy,” Opt. Express 11(10), 1123–1130 (2003).
    [Crossref] [PubMed]
  4. W. Lubeigt, G. Valentine, J. Girkin, E. Bente, and D. Burns, “Active transverse mode control and optimization of an all-solid-state laser using an intracavity adaptive-optic mirror,” Opt. Express 10(13), 550–555 (2002).
    [Crossref] [PubMed]
  5. M. Booth, “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. Express 14(4), 1339–1352 (2006).
    [Crossref] [PubMed]
  6. R. El-Agmy, H. Bulte, A. H. Greenaway, and D. Reid, “Adaptive beam profile control using a simulated annealing algorithm,” Opt. Express 13(16), 6085–6091 (2005).
    [Crossref] [PubMed]
  7. E. Theofanidou, L. Wilson, W. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236(1–3), 145–150 (2004).
    [Crossref]
  8. F. Gonte, A. Courteville, and R. Dandliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. 41(5), 1073–1076 (2002).
    [Crossref]
  9. M. Vorontsov and V. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. A 15(10), 2745–2758 (1998).
    [Crossref]
  10. M. A. Vorontsov, G. W. Carhart, M. Cohen, and G. Cauwenberghs, “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am. A 17(8), 1440–1453 (2000).
    [Crossref] [PubMed]
  11. 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(2), 356–368 (2002).
    [Crossref] [PubMed]
  12. M. A. Vorontsov and G. W. Carhart, “Adaptive wavefront control with asynchronous stochastic parallel gradient descent clusters,” J. Opt. Soc. Am. A 23(10), 2613–2622 (2006).
    [Crossref] [PubMed]
  13. R. Conan, C. Bradley, P. Hampton, O. Keskin, A. Hilton, and C. Blain, “Distributed modal command for a two-deformable-mirror adaptive optics system,” Appl. Opt. 46(20), 4329–4340 (2007).
    [Crossref] [PubMed]
  14. G. Dai, “Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen–Loève functions,” J. Opt. Soc. Am. A 12(10), 2182–2193 (1995).
    [Crossref]
  15. R. Hufnagel and N. Stanley, “Modulation transfer function associated with image transmission through turbulent media,” J. Opt. Soc. Am. 54(1), 52–61 (1964).
    [Crossref]
  16. D. Hutt, “Modeling and measurements of atmospheric optical turbulence over land,” Opt. Eng. 38(8), 1288–1295 (1999).
    [Crossref]
  17. L. Cui, B. Xue, L. Cao, S. Zheng, W. Xue, X. Bai, X. Cao, and F. Zhou, “Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence,” Opt. Express 19(18), 16872–16884 (2011).
    [Crossref] [PubMed]
  18. J. Cang and X. Liu, “Scintillation index and performance analysis of wireless optical links over non-Kolmogorov weak turbulence based on generalized atmospheric spectral model,” Opt. Express 19(20), 19067–19077 (2011).
    [Crossref] [PubMed]
  19. V. Mahajan, “Strehl ratio for primary aberrations in terms of their aberration variance,” J. Opt. Soc. Am. 73(6), 860–861 (1983).
    [Crossref]
  20. P. Piatrou and M. Roggemann, “Beaconless stochastic parallel gradient descent laser beam control: numerical experiments,” Appl. Opt. 46(27), 6831–6842 (2007).
    [Crossref] [PubMed]
  21. M. J. Booth, “Adaptive optics in microscopy,” Philos. Trans. Royal Soc. A 365(1861), 2829–2843 (2007).
    [Crossref] [PubMed]
  22. R. M. Rennie, D. A. Duffin, and E. J. Jumper, “Characterization and aero-optic correction of a forced two-dimensional weakly compressible shear layer,” AIAA J. 46(11), 2787–2795 (2008).
    [Crossref]
  23. D. G. Pérez and G. Funes, “Beam wandering statistics of twin thin laser beam propagation under generalized atmospheric conditions,” Opt. Express 20(25), 27766–27780 (2012).
    [Crossref] [PubMed]
  24. R. Yuan, J. Sun, T. Luo, X. Wu, C. Wang, and Y. Fu, “Simulation study on light propagation in an anisotropic turbulence field of entrainment zone,” Opt. Express 22(11), 13427–13437 (2014).
    [Crossref] [PubMed]
  25. J. Dietiker and K. Hoffmann, “Predicting wall pressure fluctuation over a backward-facing step using detached eddy simulation,” J. Aircr. 46(6), 2115–2120 (2009).
    [Crossref]
  26. E. Jumper and E. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37(3), 299–339 (2001).
    [Crossref]
  27. T. Bui-Thanh, M. Damodaran, and K. Willcox, “Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition,” AIAA J. 42(8), 1505–1516 (2004).
    [Crossref]
  28. A. Thayil and M. Booth, “Self calibration of sensorless adaptive optical microscopes,” J. Euro. Opt. Soc. Rapid Pub. 6, 11045 (2011).
    [Crossref]

2014 (1)

2012 (1)

2011 (3)

2009 (1)

J. Dietiker and K. Hoffmann, “Predicting wall pressure fluctuation over a backward-facing step using detached eddy simulation,” J. Aircr. 46(6), 2115–2120 (2009).
[Crossref]

2008 (1)

R. M. Rennie, D. A. Duffin, and E. J. Jumper, “Characterization and aero-optic correction of a forced two-dimensional weakly compressible shear layer,” AIAA J. 46(11), 2787–2795 (2008).
[Crossref]

2007 (4)

2006 (2)

2005 (1)

2004 (2)

E. Theofanidou, L. Wilson, W. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236(1–3), 145–150 (2004).
[Crossref]

T. Bui-Thanh, M. Damodaran, and K. Willcox, “Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition,” AIAA J. 42(8), 1505–1516 (2004).
[Crossref]

2003 (1)

2002 (4)

W. Lubeigt, G. Valentine, J. Girkin, E. Bente, and D. Burns, “Active transverse mode control and optimization of an all-solid-state laser using an intracavity adaptive-optic mirror,” Opt. Express 10(13), 550–555 (2002).
[Crossref] [PubMed]

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[Crossref] [PubMed]

F. Gonte, A. Courteville, and R. Dandliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. 41(5), 1073–1076 (2002).
[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(2), 356–368 (2002).
[Crossref] [PubMed]

2001 (1)

E. Jumper and E. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37(3), 299–339 (2001).
[Crossref]

2000 (1)

1999 (1)

D. Hutt, “Modeling and measurements of atmospheric optical turbulence over land,” Opt. Eng. 38(8), 1288–1295 (1999).
[Crossref]

1998 (1)

1995 (1)

1983 (1)

1964 (1)

Arlt, J.

E. Theofanidou, L. Wilson, W. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236(1–3), 145–150 (2004).
[Crossref]

Bai, X.

Bente, E.

Blain, C.

Booth, M.

A. Thayil and M. Booth, “Self calibration of sensorless adaptive optical microscopes,” J. Euro. Opt. Soc. Rapid Pub. 6, 11045 (2011).
[Crossref]

M. Booth, “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. Express 14(4), 1339–1352 (2006).
[Crossref] [PubMed]

Booth, M. J.

M. J. Booth, “Adaptive optics in microscopy,” Philos. Trans. Royal Soc. A 365(1861), 2829–2843 (2007).
[Crossref] [PubMed]

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[Crossref] [PubMed]

Bradley, C.

Bui-Thanh, T.

T. Bui-Thanh, M. Damodaran, and K. Willcox, “Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition,” AIAA J. 42(8), 1505–1516 (2004).
[Crossref]

Bulte, H.

Burns, D.

Cang, J.

Cao, L.

Cao, X.

Carhart, G. W.

Cauwenberghs, G.

Cohen, M.

Conan, R.

Courteville, A.

F. Gonte, A. Courteville, and R. Dandliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. 41(5), 1073–1076 (2002).
[Crossref]

Cui, L.

Dai, G.

Damodaran, M.

T. Bui-Thanh, M. Damodaran, and K. Willcox, “Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition,” AIAA J. 42(8), 1505–1516 (2004).
[Crossref]

Dandliker, R.

F. Gonte, A. Courteville, and R. Dandliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. 41(5), 1073–1076 (2002).
[Crossref]

Dietiker, J.

J. Dietiker and K. Hoffmann, “Predicting wall pressure fluctuation over a backward-facing step using detached eddy simulation,” J. Aircr. 46(6), 2115–2120 (2009).
[Crossref]

Duffin, D. A.

R. M. Rennie, D. A. Duffin, and E. J. Jumper, “Characterization and aero-optic correction of a forced two-dimensional weakly compressible shear layer,” AIAA J. 46(11), 2787–2795 (2008).
[Crossref]

El-Agmy, R.

Evans, C. L.

Fitzgerald, E.

E. Jumper and E. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37(3), 299–339 (2001).
[Crossref]

Freudiger, C. W.

Fu, Y.

Funes, G.

Girkin, J.

Girkin, J. M.

Gonte, F.

F. Gonte, A. Courteville, and R. Dandliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. 41(5), 1073–1076 (2002).
[Crossref]

Greenaway, A. H.

Hampton, P.

Hilton, A.

Hoffmann, K.

J. Dietiker and K. Hoffmann, “Predicting wall pressure fluctuation over a backward-facing step using detached eddy simulation,” J. Aircr. 46(6), 2115–2120 (2009).
[Crossref]

Hossack, W.

E. Theofanidou, L. Wilson, W. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236(1–3), 145–150 (2004).
[Crossref]

Hufnagel, R.

Hutt, D.

D. Hutt, “Modeling and measurements of atmospheric optical turbulence over land,” Opt. Eng. 38(8), 1288–1295 (1999).
[Crossref]

Jumper, E.

E. Jumper and E. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37(3), 299–339 (2001).
[Crossref]

Jumper, E. J.

R. M. Rennie, D. A. Duffin, and E. J. Jumper, “Characterization and aero-optic correction of a forced two-dimensional weakly compressible shear layer,” AIAA J. 46(11), 2787–2795 (2008).
[Crossref]

Juskaitis, R.

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[Crossref] [PubMed]

Keskin, O.

Liu, X.

Lubeigt, W.

Luo, T.

Mahajan, V.

Marsh, P. N.

Neil, M. A. A.

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[Crossref] [PubMed]

Pérez, D. G.

Piatrou, P.

Poland, S. P.

Reid, D.

Rennie, R. M.

R. M. Rennie, D. A. Duffin, and E. J. Jumper, “Characterization and aero-optic correction of a forced two-dimensional weakly compressible shear layer,” AIAA J. 46(11), 2787–2795 (2008).
[Crossref]

Roggemann, M.

Sivokon, V.

Stanley, N.

Sun, J.

Thayil, A.

A. Thayil and M. Booth, “Self calibration of sensorless adaptive optical microscopes,” J. Euro. Opt. Soc. Rapid Pub. 6, 11045 (2011).
[Crossref]

Theofanidou, E.

E. Theofanidou, L. Wilson, W. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236(1–3), 145–150 (2004).
[Crossref]

Valentine, G.

Vorontsov, M.

Vorontsov, M. A.

Wang, C.

Willcox, K.

T. Bui-Thanh, M. Damodaran, and K. Willcox, “Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition,” AIAA J. 42(8), 1505–1516 (2004).
[Crossref]

Wilson, L.

E. Theofanidou, L. Wilson, W. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236(1–3), 145–150 (2004).
[Crossref]

Wilson, T.

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[Crossref] [PubMed]

Wright, A. J.

Wu, X.

Xie, X. S.

Xue, B.

Xue, W.

Yuan, R.

Zheng, S.

Zhou, F.

AIAA J. (2)

R. M. Rennie, D. A. Duffin, and E. J. Jumper, “Characterization and aero-optic correction of a forced two-dimensional weakly compressible shear layer,” AIAA J. 46(11), 2787–2795 (2008).
[Crossref]

T. Bui-Thanh, M. Damodaran, and K. Willcox, “Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition,” AIAA J. 42(8), 1505–1516 (2004).
[Crossref]

Appl. Opt. (2)

J. Aircr. (1)

J. Dietiker and K. Hoffmann, “Predicting wall pressure fluctuation over a backward-facing step using detached eddy simulation,” J. Aircr. 46(6), 2115–2120 (2009).
[Crossref]

J. Euro. Opt. Soc. Rapid Pub. (1)

A. Thayil and M. Booth, “Self calibration of sensorless adaptive optical microscopes,” J. Euro. Opt. Soc. Rapid Pub. 6, 11045 (2011).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

E. Theofanidou, L. Wilson, W. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. 236(1–3), 145–150 (2004).
[Crossref]

Opt. Eng. (2)

F. Gonte, A. Courteville, and R. Dandliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. 41(5), 1073–1076 (2002).
[Crossref]

D. Hutt, “Modeling and measurements of atmospheric optical turbulence over land,” Opt. Eng. 38(8), 1288–1295 (1999).
[Crossref]

Opt. Express (9)

L. Cui, B. Xue, L. Cao, S. Zheng, W. Xue, X. Bai, X. Cao, and F. Zhou, “Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence,” Opt. Express 19(18), 16872–16884 (2011).
[Crossref] [PubMed]

J. Cang and X. Liu, “Scintillation index and performance analysis of wireless optical links over non-Kolmogorov weak turbulence based on generalized atmospheric spectral model,” Opt. Express 19(20), 19067–19077 (2011).
[Crossref] [PubMed]

A. J. Wright, S. P. Poland, J. M. Girkin, C. W. Freudiger, C. L. Evans, and X. S. Xie, “Adaptive optics for enhanced signal in CARS microscopy,” Opt. Express 15(26), 18209–18219 (2007).
[Crossref] [PubMed]

P. N. Marsh, D. Burns, and J. M. Girkin, “Practical implementation of adaptive optics in multiphoton microscopy,” Opt. Express 11(10), 1123–1130 (2003).
[Crossref] [PubMed]

W. Lubeigt, G. Valentine, J. Girkin, E. Bente, and D. Burns, “Active transverse mode control and optimization of an all-solid-state laser using an intracavity adaptive-optic mirror,” Opt. Express 10(13), 550–555 (2002).
[Crossref] [PubMed]

M. Booth, “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. Express 14(4), 1339–1352 (2006).
[Crossref] [PubMed]

R. El-Agmy, H. Bulte, A. H. Greenaway, and D. Reid, “Adaptive beam profile control using a simulated annealing algorithm,” Opt. Express 13(16), 6085–6091 (2005).
[Crossref] [PubMed]

D. G. Pérez and G. Funes, “Beam wandering statistics of twin thin laser beam propagation under generalized atmospheric conditions,” Opt. Express 20(25), 27766–27780 (2012).
[Crossref] [PubMed]

R. Yuan, J. Sun, T. Luo, X. Wu, C. Wang, and Y. Fu, “Simulation study on light propagation in an anisotropic turbulence field of entrainment zone,” Opt. Express 22(11), 13427–13437 (2014).
[Crossref] [PubMed]

Philos. Trans. Royal Soc. A (1)

M. J. Booth, “Adaptive optics in microscopy,” Philos. Trans. Royal Soc. A 365(1861), 2829–2843 (2007).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. U.S.A. 99(9), 5788–5792 (2002).
[Crossref] [PubMed]

Prog. Aerosp. Sci. (1)

E. Jumper and E. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37(3), 299–339 (2001).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Schematic of adaptive system based on multi-perturbation SPGD. The incoming beam with distortion is split into N sub-beams. BSn represent beam splitter in the nth sub-beam. WCn indicate wave corrector in the nth sub-beam. (n = 1, 2 … N.).
Fig. 2
Fig. 2 Schematic of the experimental setup. Distortion is added to the plane wave by the spatial light modulator SLMdistortion. OBJ: objective lens. CL: collimating lens. BS: beam splitter. Wave corrector WC1 and WC2 are also spatial light modulators. f = 400mm is the focal length of the Fourier lenses. The pinhole and photodetector of the first sub-beam can be replaced by the CCD on the translation stage.
Fig. 3
Fig. 3 Gray-scale image of the Kolmogorov turbulence phase-distortion.
Fig. 4
Fig. 4 Typical focal spots of light wave with Kolmogorov turbulence phase-distortion (a) before correction, (b) after 100 correction cycles by using the fast descent mode of the multi-perturbation SPGD, and (c) after 100 correction cycles by using the conventional SPGD. Insets: the intensity profile along the top-left to bottom-right diagonal line.
Fig. 5
Fig. 5 Curve of averaged quality metric versus iteration number.
Fig. 6
Fig. 6 Histograms for the beam quality metric optimization process with (a) the fast descent mode of the multi-perturbation SPGD and (b) the conventional SPGD.
Fig. 7
Fig. 7 Obtained candidate modal basis sets corresponding to (a) Kolmogorov turbulence with von Karman spectrum, (b) non-Kolmogorov turbulence with power spectrum exponent α= 4.5, (c) anisotropic Kolmogorov turbulence with anisotropy factor ε= 1, and (d) back-facing step flow with Reynold number Re = 37,000.
Fig. 8
Fig. 8 Typical modal basis updating process. Upper half: Quality metric and modal basis choice of WC1. Lower half: Quality metric and modal basis choice of WC2. The phase distortion of the incoming beam is changed every 100 rounds, and the basis sets of wavefront correctors are updated every 300 rounds. The vertical dashed lines indicate time points when the basis sets of wavefront correctors are updated. Red font refers to the basis set with better convergence performance.

Equations (5)

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

δ u = n=1 N sign(δ J n ) δ u n n=1 N sign(δ J n ) δ u n σ.
δ J =J( u ( i ) +δ u )J( u ( i ) )=J( u ( i ) ) n=1 N sign(δ J n ) δ u n n=1 N sign(δ J n ) δ u n σ+O( σ ), = n=1 N | δ J n | n=1 N sign(δ J n ) δ u n σ+O( σ )
u n = u ( i+1 ) = u ( i ) μδ J δ u , n=1,...,N.
| δ J | n=1 N | δ J n | n=1 N sign(δ J n ) δ u n σ= N | δJ | c σ N δ u n = N | δJ | c .
Φ n ( k x , k y )=0.033 C n 2 ( k x 2 + k y 2 +4 π 2 L 0 2 ) 11/6 ,

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