Abstract

By analyzing the Poisson equation describing the static behavior of membrane and bimorph deformable mirrors and biharmonic equation describing the continuous facesheet mirror with push-pull actuators, we found that to achieve a high quality correction of low-order aberrations these mirrors should have sufficient number of actuators positioned outside the correction aperture. In particular, any deformable mirror described by the Poisson equation requires at least two actuators to be placed outside the working aperture per period of the azimuthal aberration of the highest expected order. Any deformable mirror described by the biharmonic equation, such as a continuous facesheet mirror with push-pull actuators, requires at least four actuators to be placed outside the working aperture per period of the azimuthal aberration of the highest expected order, and these actuators should not be positioned on a single circle.

© 2008 Optical Society of America

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References

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  1. H. Babcock, "The possibility of compensating astronomical seeing," Publications of the Astronomical Society of the Pacific 65, 229 (1953).
    [CrossRef]
  2. V. Linnick, "On the principal possibility of the reduction of the influence of the atmosphere on the image of a star," Optics and Spectroscopy (USSR) 3, 401 (1957). English translation in F. Merkle (Ed.), "Active and adaptive optics," ESO Conference 48, 535 (1994).
  3. R. P. Grosso and M. Yellin, "Membrane mirror as an adaptive optical element," J. Opt. Soc. Am. 67, 399 (1977).
    [CrossRef]
  4. G. Vdovin and P Sarro, "Flexible mirror micromachined in silicon," Appl. Opt. 34, 2968 (1996).
    [CrossRef]
  5. E. Steinhaus and S. G. Lipson, "Bimorph piezoelectric flexible mirror," J. Opt. Soc. Am. 69, 478-481 (1979).
    [CrossRef]
  6. C. Schwartz, E. Ribak, and S. G. Lipson, "Bimorph adaptive mirrors and curvature sensing," J. Opt. Soc. Am. A 11, 895 (1994).
    [CrossRef]
  7. R. H. Freeman and J. E. Pearson, "Deformable mirrors for all seasons and reasons," Appl. Opt. 21, 580 (1982).
    [CrossRef] [PubMed]
  8. B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, "Investigating a Xin? tics Inc. deformable mirror," Proc. SPIE 3126, 569-579 (1997).
    [CrossRef]
  9. T. Bifano, "Mems wavefront correctors: Electromechanical theory and recent performance advances," In "Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis," Topical Meetings on CD-ROM, Optical Society of America, page ATuD1 (2007).
  10. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, (McGraw-Hill, 1953).
  11. M. Born and E. Wolf, Principles of Optics, (Pergamon Press, 1993).
  12. M. Loktev, D.W. De Lima Monteiro, and G. Vdovin, "Comparison study of the performance of piston, thin plate and membrane mirrors for correction of turbulence-induced phase distortions," Opt. Commun. 192, 91 (2001).
    [CrossRef]

2001

M. Loktev, D.W. De Lima Monteiro, and G. Vdovin, "Comparison study of the performance of piston, thin plate and membrane mirrors for correction of turbulence-induced phase distortions," Opt. Commun. 192, 91 (2001).
[CrossRef]

1997

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, "Investigating a Xin? tics Inc. deformable mirror," Proc. SPIE 3126, 569-579 (1997).
[CrossRef]

1996

1994

1982

1979

1977

1953

H. Babcock, "The possibility of compensating astronomical seeing," Publications of the Astronomical Society of the Pacific 65, 229 (1953).
[CrossRef]

Babcock, H.

H. Babcock, "The possibility of compensating astronomical seeing," Publications of the Astronomical Society of the Pacific 65, 229 (1953).
[CrossRef]

De Lima Monteiro, D.W.

M. Loktev, D.W. De Lima Monteiro, and G. Vdovin, "Comparison study of the performance of piston, thin plate and membrane mirrors for correction of turbulence-induced phase distortions," Opt. Commun. 192, 91 (2001).
[CrossRef]

Dekany, R.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, "Investigating a Xin? tics 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 Xin? tics Inc. deformable mirror," Proc. SPIE 3126, 569-579 (1997).
[CrossRef]

Freeman, R. H.

Grosso, R. P.

Lipson, S. G.

Loktev, M.

M. Loktev, D.W. De Lima Monteiro, and G. Vdovin, "Comparison study of the performance of piston, thin plate and membrane mirrors for correction of turbulence-induced phase distortions," Opt. Commun. 192, 91 (2001).
[CrossRef]

Oppenheimer, B. R.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, "Investigating a Xin? tics 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 Xin? tics Inc. deformable mirror," Proc. SPIE 3126, 569-579 (1997).
[CrossRef]

Pearson, J. E.

Price, T.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, "Investigating a Xin? tics Inc. deformable mirror," Proc. SPIE 3126, 569-579 (1997).
[CrossRef]

Ribak, E.

Sarro, P

Schwartz, C.

Sivaramakrishnan, A.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, "Investigating a Xin? tics Inc. deformable mirror," Proc. SPIE 3126, 569-579 (1997).
[CrossRef]

Steinhaus, E.

Vdovin, G.

M. Loktev, D.W. De Lima Monteiro, and G. Vdovin, "Comparison study of the performance of piston, thin plate and membrane mirrors for correction of turbulence-induced phase distortions," Opt. Commun. 192, 91 (2001).
[CrossRef]

G. Vdovin and P Sarro, "Flexible mirror micromachined in silicon," Appl. Opt. 34, 2968 (1996).
[CrossRef]

Yellin, M.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

M. Loktev, D.W. De Lima Monteiro, and G. Vdovin, "Comparison study of the performance of piston, thin plate and membrane mirrors for correction of turbulence-induced phase distortions," Opt. Commun. 192, 91 (2001).
[CrossRef]

Proc. SPIE

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, "Investigating a Xin? tics Inc. deformable mirror," Proc. SPIE 3126, 569-579 (1997).
[CrossRef]

Publications of the Astronomical Society of the Pacific

H. Babcock, "The possibility of compensating astronomical seeing," Publications of the Astronomical Society of the Pacific 65, 229 (1953).
[CrossRef]

Other

V. Linnick, "On the principal possibility of the reduction of the influence of the atmosphere on the image of a star," Optics and Spectroscopy (USSR) 3, 401 (1957). English translation in F. Merkle (Ed.), "Active and adaptive optics," ESO Conference 48, 535 (1994).

T. Bifano, "Mems wavefront correctors: Electromechanical theory and recent performance advances," In "Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis," Topical Meetings on CD-ROM, Optical Society of America, page ATuD1 (2007).

S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, (McGraw-Hill, 1953).

M. Born and E. Wolf, Principles of Optics, (Pergamon Press, 1993).

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Figures (4)

Fig. 1.
Fig. 1.

Example of the DM modes that cannot be produced exactly without the actuators positioned outside the correction area: Z7 7 (left) for both membrane and thin plate DM and Z3 5 (right) for a thin-plate DM only.

Fig. 2.
Fig. 2.

Residual rms error for a 37-ch DM described by (2) for low order Zernike polynomials Znm=Zm n for three different values of the correction aperture R. The actuator geometry and the correction apertures of the DM are shown in the inset.

Fig. 3.
Fig. 3.

Residual rms error for a 37-ch membrane deformable mirror described by Eq. 1 for low order Zernike polynomials Znm=Zm n for three different values of the correction aperture R. The actuator geometry and the correction apertures of the DM are shown in the inset.

Fig. 4.
Fig. 4.

Calculated and experimental residual rms error of compensation of coma with a 37ch DM with a geometry shown in the inset in Fig. 1. The difference between the theory and the experiment is explained by the fact that the maximum achievable precision in the experiment is limited by the feedback loop error.

Tables (1)

Tables Icon

Table 1. Zernike polynomials tree, the terms satisfying to the condition Eq. 3 are printed as bold capital: Z m n ; the terms satisfying to both conditions Eq. 3 and Eq. 4 are printed as regular capital: Zm n ; all other terms are printed as regular: zm n .

Equations (15)

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

Δ φ ( x , y ) = p ( x , y ) T ,
Δ 2 φ ( x , y ) = p ( x , y ) D ,
Δ φ ( x , y ) = 0 ,
Δ 2 φ ( x , y ) = 0 ,
K tot ( N 2 + 3 N + 2 ) 2 .
K ext 2 N + 1 ,
K ext 4 N - 2 .
Δf= 1 ρ ρ ( ρ f ρ )+ 1 ρ 2 2 f θ 2 ,
Δ ρ sin n cos n θ = n 2 ρ sin n 2 cos n θ n 2 ρ sin n 2 cos n θ = 0
Δ ρ sin n cos ( n 2 ) θ = n 2 ρ sin n 2 cos ( n 2 ) θ ( n 2 ) 2 ρ sin n 2 cos ( n 2 ) θ = 4 ( n 1 ) ρ sin n 2 cos ( n 2 ) θ ,
Δ 2 ρ sin n cos n θ = 0
Δ 2 ρ sin n cos ( n 2 ) θ = 0.
Δ Z n ± n ( ρ , θ ) = 0
Δ 2 Z n n ( ρ , θ ) = 0 ,
Δ 2 Z n ± ( n 2 ) ( ρ , θ ) = 0 .

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