Abstract

A new deformable mirror influence function based on a Gaussian function is introduced to analyze the fitting capability of a deformable mirror. The modified expressions for both azimuthal and radial directions are presented based on the analysis of the residual error between a measured influence function and a Gaussian influence function. With a simplex search method, we further compare the fitting capability of our proposed influence function to fit the data produced by a Zygo interferometer with that of a Gaussian influence function. The result indicates that the modified Gaussian influence function provides much better performance in data fitting.

©2008 Optical Society of America

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References

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  1. W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
    [Crossref]
  2. J. Alda and G. D. Boreman, “Zernike-based matrix model of deformable mirrors: optimization of aperture size,” Appl. Opt. 32, 2431–2438 (1993).
    [Crossref] [PubMed]
  3. B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
    [Crossref]
  4. D. Redding, S. Basinger, G. Brack, and R. Dekany, “Adaptive optics reconstruction utilizing super-sampled deformable mirror influence functions,” Proc. SPIE 3353, 543–552 (1998).
    [Crossref]
  5. X. J. Rao, N. Ling, and W. H. Jiang, “Experiment of measuring influence function of deformable mirror using digital interferometer,” Acta Opt. Sin. 15, 1447–1451 (1995).
  6. E. D. Li, Y. Dai, H. Y. Wang, and Y. D. Zhang, “Application of eigenmode in the adaptive optics system based on a micromachined membrane deformable mirror,” Appl. Opt. 45, 5651–5656 (2006).
    [Crossref] [PubMed]
  7. A. Menikoff, “Actuator influence functions of active mirrors,” Appl. Opt. 30, 833–838 (1991).
    [Crossref] [PubMed]
  8. L. Arnold, “Influence functions of a thin shallow meniscus-shaped mirror,” Appl. Opt. 362019–2028 (1997).
    [Crossref] [PubMed]
  9. L. Arnold, “Uniform-load and actuator influence functions of a thin or thick annular mirror: application to active mirror support optimization,” Appl. Opt. 35, 1095–1106 (1996).
    [Crossref] [PubMed]
  10. M. A. van Dam, D. L. Mignant, and B. A. Macintosh, “Performance of the Keck Observatory adaptive-optics system,” Appl. Opt. 43, 5458–5467 (2004).
    [Crossref] [PubMed]
  11. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
    [Crossref]

2006 (1)

2004 (1)

1998 (2)

D. Redding, S. Basinger, G. Brack, and R. Dekany, “Adaptive optics reconstruction utilizing super-sampled deformable mirror influence functions,” Proc. SPIE 3353, 543–552 (1998).
[Crossref]

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[Crossref]

1997 (2)

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

L. Arnold, “Influence functions of a thin shallow meniscus-shaped mirror,” Appl. Opt. 362019–2028 (1997).
[Crossref] [PubMed]

1996 (1)

1995 (1)

X. J. Rao, N. Ling, and W. H. Jiang, “Experiment of measuring influence function of deformable mirror using digital interferometer,” Acta Opt. Sin. 15, 1447–1451 (1995).

1993 (1)

1991 (2)

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

A. Menikoff, “Actuator influence functions of active mirrors,” Appl. Opt. 30, 833–838 (1991).
[Crossref] [PubMed]

Alda, J.

Arnold, L.

Basinger, S.

D. Redding, S. Basinger, G. Brack, and R. Dekany, “Adaptive optics reconstruction utilizing super-sampled deformable mirror influence functions,” Proc. SPIE 3353, 543–552 (1998).
[Crossref]

Boreman, G. D.

Brack, G.

D. Redding, S. Basinger, G. Brack, and R. Dekany, “Adaptive optics reconstruction utilizing super-sampled deformable mirror influence functions,” Proc. SPIE 3353, 543–552 (1998).
[Crossref]

Dai, Y.

Dekany, R.

D. Redding, S. Basinger, G. Brack, and R. Dekany, “Adaptive optics reconstruction utilizing super-sampled deformable mirror influence functions,” Proc. SPIE 3353, 543–552 (1998).
[Crossref]

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Ealey, M.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Jiang, W. H.

X. J. Rao, N. Ling, and W. H. Jiang, “Experiment of measuring influence function of deformable mirror using digital interferometer,” Acta Opt. Sin. 15, 1447–1451 (1995).

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[Crossref]

Li, E. D.

Ling, N.

X. J. Rao, N. Ling, and W. H. Jiang, “Experiment of measuring influence function of deformable mirror using digital interferometer,” Acta Opt. Sin. 15, 1447–1451 (1995).

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

Macintosh, B. A.

Menikoff, A.

Mignant, D. L.

Oppenheimer, B. R.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Palmer, D.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Price, T.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Rao, X. J.

X. J. Rao, N. Ling, and W. H. Jiang, “Experiment of measuring influence function of deformable mirror using digital interferometer,” Acta Opt. Sin. 15, 1447–1451 (1995).

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

Redding, D.

D. Redding, S. Basinger, G. Brack, and R. Dekany, “Adaptive optics reconstruction utilizing super-sampled deformable mirror influence functions,” Proc. SPIE 3353, 543–552 (1998).
[Crossref]

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[Crossref]

Shi, F.

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

Sivaramakrishnan, A.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

van Dam, M. A.

Wang, H. Y.

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[Crossref]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[Crossref]

Zhang, Y. D.

Acta Opt. Sin. (1)

X. J. Rao, N. Ling, and W. H. Jiang, “Experiment of measuring influence function of deformable mirror using digital interferometer,” Acta Opt. Sin. 15, 1447–1451 (1995).

Appl. Opt. (6)

Proc. SPIE (3)

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

D. Redding, S. Basinger, G. Brack, and R. Dekany, “Adaptive optics reconstruction utilizing super-sampled deformable mirror influence functions,” Proc. SPIE 3353, 543–552 (1998).
[Crossref]

SIAM J. Optim. (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[Crossref]

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Figures (8)

Fig. 1.
Fig. 1. IF produced by (a) GIF and ZYGO interferometer of (b) hexagonal arrangement DM and (c) square arrangement DM.
Fig. 2.
Fig. 2. Residual error between actual IF and GIF; (a) is the 2D map and (b) is the cross-section.
Fig. 3.
Fig. 3. Cutaway view of the IF of a hexagonal arrangement DM measured by a Zygo interferometer.
Fig. 4.
Fig. 4. /d as a function of the angle.
Fig. 5.
Fig. 5. Residual error after modification in angle direction; (a) is the 2-D map and (b) is the cross-section.
Fig. 6.
Fig. 6. Residual error after modification in angle and radial direction; (a) is the 2-D map and (b) is the cross-section.
Fig. 7.
Fig. 7. actuators arrangement and comparison of fitting capability using GIF and MGIF for different position actuators.
Fig. 8.
Fig. 8. Comparisons of the fitting capability of the square arrangement deformable mirror using GIF and MGIF. (a) and (b) IF calculated by Gaussian function and MGIF, respectively, (c) and (d) 2-D map of the residual error.

Tables (1)

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Table 1. Initial Values Specified for the Simplex Search Method

Equations (6)

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I ( r ) = exp [ ln ( ω ) · ( r d 0 ) α ] ,
d d ¯ = 1 + λ · cos ( 6 θ ) ,
d = d ¯ [ 1 + λ · cos ( 6 θ ) ] .
I ( r , θ ) = exp ( ln ( ω ) { r [ 1 + λ cos ( 6 θ ) ] d ¯ } α ) ·
I ( r , θ ) = exp ( ln ( ω ) { r · [ 1 + λ · cos ( 6 θ ) ] d ¯ } α ) + β · exp { [ ( r d ¯ ) γ ] 2 } ,
rms = 1 MN x = 1 M y = 1 N [ I res ( x , y ) I ¯ res ] 2 ,

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