Abstract

We introduce a coupled system of nonlinear partial differential equations (PDEs) to model a particular microelectromechanical systems (MEMS) deformable mirror consisting of a continuous facesheet coupled with electrostatic plate-driven actuators. To make the problem computationally tractable, we reduced the system to a single linear PDE for the facesheet coupled with nonlinear algebraic constraints for each of the actuators. We also introduce a nonlinearly constrained quadratic optimization problem for open-loop control of the MEMS mirror. Numerical simulation and control results are presented, and shortcomings of the model are discussed.

© 2006 Optical Society of America

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  1. R. Dekany, M. Britton, D. Gavel, B. L. Ellerbroek, G. Herriot, C. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advances in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).
  2. C. L. Hom, P. D. Dean, and S. R. Winzer, "Simulating electrostrictive deformable mirrors: I. Nonlinear static analysis," Smart Mater. Struct. 8, 691-699 (1999).
    [CrossRef]
  3. C. L. Hom, P. D. Dean, and S. R. Winzer, "Simulating electrostrictive deformable mirrors: II. Nonlinear dynamic analysis," Smart Mater. Struct. 8, 700-708 (1999).
    [CrossRef]
  4. T. G. Bifano, R. K. Mali, J. K. Dorton, J. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, "Continuous-membrane surface-micromachined silicon deformable mirror," Opt. Eng. (Bellingham) 36, 1354-1360 (1997).
    [CrossRef]
  5. R. K. Mali, N. Vandelli, and M. N. Horenstein, "Development of microelectromechanical deformable mirrors for phase modulation of light," Opt. Eng. (Bellingham) 36, 542-548 (1997).
    [CrossRef]
  6. T. G. Bifano, J. Perreault, P. Bierden, and C. Dimas, "Micromachined deformable mirrors for adaptive optics," in High-resolution Wavefront Control: Methods, Devices, and Applications IV, J.D.Gonglewski, M.A.Vorontsov, M.T.Gruneisen, S.R.Restaino, and R.K.Tyson, eds., Proc. SPIE 4825, 10-13 (2002).
  7. J. Perreault, T. G. Bifano, B. M. Levine, and M. N. Horenstein, "Adaptive optics correction using microelectromechanical deformable mirrors," Opt. Eng. (Bellingham) 41, 561-566 (2002).
    [CrossRef]
  8. B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).
  9. L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3rd ed., Vol. 7 of Course of Theoretical Physics (Pergamon, 1986).
  10. P. Dorato, C. T. Abdallah, and V. Cerone, Linear Quadratic Control: An Introduction (Krieger, 2000).
  11. O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems: Theory and Computation (Society for Industrial and Applied Mathematics, 2001).
    [CrossRef]
  12. Y. Saad, Iterative Methods for Sparse Linear Systems (Society for Industrial and Applied Mathematics, 2003).
    [CrossRef]
  13. R. C. Smith, Smart Material Systems (Society for Industrial and Applied Mathematics, 2005).
    [CrossRef]
  14. C. R. Vogel, Computational Methods for Inverse Problems (Society for Industrial and Applied Mathematics, 2002).
    [CrossRef]

2005

R. C. Smith, Smart Material Systems (Society for Industrial and Applied Mathematics, 2005).
[CrossRef]

2004

R. Dekany, M. Britton, D. Gavel, B. L. Ellerbroek, G. Herriot, C. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advances in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

2003

Y. Saad, Iterative Methods for Sparse Linear Systems (Society for Industrial and Applied Mathematics, 2003).
[CrossRef]

2002

C. R. Vogel, Computational Methods for Inverse Problems (Society for Industrial and Applied Mathematics, 2002).
[CrossRef]

T. G. Bifano, J. Perreault, P. Bierden, and C. Dimas, "Micromachined deformable mirrors for adaptive optics," in High-resolution Wavefront Control: Methods, Devices, and Applications IV, J.D.Gonglewski, M.A.Vorontsov, M.T.Gruneisen, S.R.Restaino, and R.K.Tyson, eds., Proc. SPIE 4825, 10-13 (2002).

J. Perreault, T. G. Bifano, B. M. Levine, and M. N. Horenstein, "Adaptive optics correction using microelectromechanical deformable mirrors," Opt. Eng. (Bellingham) 41, 561-566 (2002).
[CrossRef]

2001

O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems: Theory and Computation (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

2000

P. Dorato, C. T. Abdallah, and V. Cerone, Linear Quadratic Control: An Introduction (Krieger, 2000).

1999

C. L. Hom, P. D. Dean, and S. R. Winzer, "Simulating electrostrictive deformable mirrors: I. Nonlinear static analysis," Smart Mater. Struct. 8, 691-699 (1999).
[CrossRef]

C. L. Hom, P. D. Dean, and S. R. Winzer, "Simulating electrostrictive deformable mirrors: II. Nonlinear dynamic analysis," Smart Mater. Struct. 8, 700-708 (1999).
[CrossRef]

1997

T. G. Bifano, R. K. Mali, J. K. Dorton, J. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, "Continuous-membrane surface-micromachined silicon deformable mirror," Opt. Eng. (Bellingham) 36, 1354-1360 (1997).
[CrossRef]

R. K. Mali, N. Vandelli, and M. N. Horenstein, "Development of microelectromechanical deformable mirrors for phase modulation of light," Opt. Eng. (Bellingham) 36, 542-548 (1997).
[CrossRef]

1986

L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3rd ed., Vol. 7 of Course of Theoretical Physics (Pergamon, 1986).

Abdallah, C. T.

P. Dorato, C. T. Abdallah, and V. Cerone, Linear Quadratic Control: An Introduction (Krieger, 2000).

Axelsson, O.

O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems: Theory and Computation (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

Barker, V. A.

O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems: Theory and Computation (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

Bauman, B.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Bierden, P.

T. G. Bifano, J. Perreault, P. Bierden, and C. Dimas, "Micromachined deformable mirrors for adaptive optics," in High-resolution Wavefront Control: Methods, Devices, and Applications IV, J.D.Gonglewski, M.A.Vorontsov, M.T.Gruneisen, S.R.Restaino, and R.K.Tyson, eds., Proc. SPIE 4825, 10-13 (2002).

Bifano, T. G.

T. G. Bifano, J. Perreault, P. Bierden, and C. Dimas, "Micromachined deformable mirrors for adaptive optics," in High-resolution Wavefront Control: Methods, Devices, and Applications IV, J.D.Gonglewski, M.A.Vorontsov, M.T.Gruneisen, S.R.Restaino, and R.K.Tyson, eds., Proc. SPIE 4825, 10-13 (2002).

J. Perreault, T. G. Bifano, B. M. Levine, and M. N. Horenstein, "Adaptive optics correction using microelectromechanical deformable mirrors," Opt. Eng. (Bellingham) 41, 561-566 (2002).
[CrossRef]

T. G. Bifano, R. K. Mali, J. K. Dorton, J. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, "Continuous-membrane surface-micromachined silicon deformable mirror," Opt. Eng. (Bellingham) 36, 1354-1360 (1997).
[CrossRef]

Britton, M.

R. Dekany, M. Britton, D. Gavel, B. L. Ellerbroek, G. Herriot, C. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advances in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).

Castanon, D. A.

T. G. Bifano, R. K. Mali, J. K. Dorton, J. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, "Continuous-membrane surface-micromachined silicon deformable mirror," Opt. Eng. (Bellingham) 36, 1354-1360 (1997).
[CrossRef]

Cerone, V.

P. Dorato, C. T. Abdallah, and V. Cerone, Linear Quadratic Control: An Introduction (Krieger, 2000).

Dean, P. D.

C. L. Hom, P. D. Dean, and S. R. Winzer, "Simulating electrostrictive deformable mirrors: I. Nonlinear static analysis," Smart Mater. Struct. 8, 691-699 (1999).
[CrossRef]

C. L. Hom, P. D. Dean, and S. R. Winzer, "Simulating electrostrictive deformable mirrors: II. Nonlinear dynamic analysis," Smart Mater. Struct. 8, 700-708 (1999).
[CrossRef]

Dekany, R.

R. Dekany, M. Britton, D. Gavel, B. L. Ellerbroek, G. Herriot, C. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advances in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).

Dillion, D.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Dimas, C.

T. G. Bifano, J. Perreault, P. Bierden, and C. Dimas, "Micromachined deformable mirrors for adaptive optics," in High-resolution Wavefront Control: Methods, Devices, and Applications IV, J.D.Gonglewski, M.A.Vorontsov, M.T.Gruneisen, S.R.Restaino, and R.K.Tyson, eds., Proc. SPIE 4825, 10-13 (2002).

Dorato, P.

P. Dorato, C. T. Abdallah, and V. Cerone, Linear Quadratic Control: An Introduction (Krieger, 2000).

Dorton, J. K.

T. G. Bifano, R. K. Mali, J. K. Dorton, J. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, "Continuous-membrane surface-micromachined silicon deformable mirror," Opt. Eng. (Bellingham) 36, 1354-1360 (1997).
[CrossRef]

Ellerbroek, B. L.

R. Dekany, M. Britton, D. Gavel, B. L. Ellerbroek, G. Herriot, C. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advances in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).

Evans, J. Wilhelmsen

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Gavel, D.

R. Dekany, M. Britton, D. Gavel, B. L. Ellerbroek, G. Herriot, C. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advances in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).

Gavel, D. T.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Graham, J. R.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Green, J. J.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Herriot, G.

R. Dekany, M. Britton, D. Gavel, B. L. Ellerbroek, G. Herriot, C. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advances in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).

Hom, C. L.

C. L. Hom, P. D. Dean, and S. R. Winzer, "Simulating electrostrictive deformable mirrors: I. Nonlinear static analysis," Smart Mater. Struct. 8, 691-699 (1999).
[CrossRef]

C. L. Hom, P. D. Dean, and S. R. Winzer, "Simulating electrostrictive deformable mirrors: II. Nonlinear dynamic analysis," Smart Mater. Struct. 8, 700-708 (1999).
[CrossRef]

Horenstein, M. N.

J. Perreault, T. G. Bifano, B. M. Levine, and M. N. Horenstein, "Adaptive optics correction using microelectromechanical deformable mirrors," Opt. Eng. (Bellingham) 41, 561-566 (2002).
[CrossRef]

R. K. Mali, N. Vandelli, and M. N. Horenstein, "Development of microelectromechanical deformable mirrors for phase modulation of light," Opt. Eng. (Bellingham) 36, 542-548 (1997).
[CrossRef]

T. G. Bifano, R. K. Mali, J. K. Dorton, J. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, "Continuous-membrane surface-micromachined silicon deformable mirror," Opt. Eng. (Bellingham) 36, 1354-1360 (1997).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3rd ed., Vol. 7 of Course of Theoretical Physics (Pergamon, 1986).

Levine, B. M.

J. Perreault, T. G. Bifano, B. M. Levine, and M. N. Horenstein, "Adaptive optics correction using microelectromechanical deformable mirrors," Opt. Eng. (Bellingham) 41, 561-566 (2002).
[CrossRef]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3rd ed., Vol. 7 of Course of Theoretical Physics (Pergamon, 1986).

Lloyd, J. P.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Lockwood, C.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Macintosh, B. A.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Makidon, R. B.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Mali, R. K.

T. G. Bifano, R. K. Mali, J. K. Dorton, J. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, "Continuous-membrane surface-micromachined silicon deformable mirror," Opt. Eng. (Bellingham) 36, 1354-1360 (1997).
[CrossRef]

R. K. Mali, N. Vandelli, and M. N. Horenstein, "Development of microelectromechanical deformable mirrors for phase modulation of light," Opt. Eng. (Bellingham) 36, 542-548 (1997).
[CrossRef]

Max, C.

R. Dekany, M. Britton, D. Gavel, B. L. Ellerbroek, G. Herriot, C. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advances in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).

Olivier, S.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Palmer, D.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Perreault, J.

J. Perreault, T. G. Bifano, B. M. Levine, and M. N. Horenstein, "Adaptive optics correction using microelectromechanical deformable mirrors," Opt. Eng. (Bellingham) 41, 561-566 (2002).
[CrossRef]

T. G. Bifano, J. Perreault, P. Bierden, and C. Dimas, "Micromachined deformable mirrors for adaptive optics," in High-resolution Wavefront Control: Methods, Devices, and Applications IV, J.D.Gonglewski, M.A.Vorontsov, M.T.Gruneisen, S.R.Restaino, and R.K.Tyson, eds., Proc. SPIE 4825, 10-13 (2002).

T. G. Bifano, R. K. Mali, J. K. Dorton, J. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, "Continuous-membrane surface-micromachined silicon deformable mirror," Opt. Eng. (Bellingham) 36, 1354-1360 (1997).
[CrossRef]

Perrin, M. D.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Poyneer, L.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Saad, Y.

Y. Saad, Iterative Methods for Sparse Linear Systems (Society for Industrial and Applied Mathematics, 2003).
[CrossRef]

Severson, S.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Sheinis, A. I.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Sivaramakrishnan, A.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Smith, R. C.

R. C. Smith, Smart Material Systems (Society for Industrial and Applied Mathematics, 2005).
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Sommargren, G.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Soummer, R.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Troy, M.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Vandelli, N.

T. G. Bifano, R. K. Mali, J. K. Dorton, J. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, "Continuous-membrane surface-micromachined silicon deformable mirror," Opt. Eng. (Bellingham) 36, 1354-1360 (1997).
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R. K. Mali, N. Vandelli, and M. N. Horenstein, "Development of microelectromechanical deformable mirrors for phase modulation of light," Opt. Eng. (Bellingham) 36, 542-548 (1997).
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Veran, J.-P.

R. Dekany, M. Britton, D. Gavel, B. L. Ellerbroek, G. Herriot, C. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advances in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).

Vogel, C. R.

C. R. Vogel, Computational Methods for Inverse Problems (Society for Industrial and Applied Mathematics, 2002).
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B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Winzer, S. R.

C. L. Hom, P. D. Dean, and S. R. Winzer, "Simulating electrostrictive deformable mirrors: I. Nonlinear static analysis," Smart Mater. Struct. 8, 691-699 (1999).
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C. L. Hom, P. D. Dean, and S. R. Winzer, "Simulating electrostrictive deformable mirrors: II. Nonlinear dynamic analysis," Smart Mater. Struct. 8, 700-708 (1999).
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Wishnow, E.

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

Opt. Eng. (Bellingham)

J. Perreault, T. G. Bifano, B. M. Levine, and M. N. Horenstein, "Adaptive optics correction using microelectromechanical deformable mirrors," Opt. Eng. (Bellingham) 41, 561-566 (2002).
[CrossRef]

T. G. Bifano, R. K. Mali, J. K. Dorton, J. Perreault, N. Vandelli, M. N. Horenstein, and D. A. Castanon, "Continuous-membrane surface-micromachined silicon deformable mirror," Opt. Eng. (Bellingham) 36, 1354-1360 (1997).
[CrossRef]

R. K. Mali, N. Vandelli, and M. N. Horenstein, "Development of microelectromechanical deformable mirrors for phase modulation of light," Opt. Eng. (Bellingham) 36, 542-548 (1997).
[CrossRef]

Smart Mater. Struct.

C. L. Hom, P. D. Dean, and S. R. Winzer, "Simulating electrostrictive deformable mirrors: I. Nonlinear static analysis," Smart Mater. Struct. 8, 691-699 (1999).
[CrossRef]

C. L. Hom, P. D. Dean, and S. R. Winzer, "Simulating electrostrictive deformable mirrors: II. Nonlinear dynamic analysis," Smart Mater. Struct. 8, 700-708 (1999).
[CrossRef]

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T. G. Bifano, J. Perreault, P. Bierden, and C. Dimas, "Micromachined deformable mirrors for adaptive optics," in High-resolution Wavefront Control: Methods, Devices, and Applications IV, J.D.Gonglewski, M.A.Vorontsov, M.T.Gruneisen, S.R.Restaino, and R.K.Tyson, eds., Proc. SPIE 4825, 10-13 (2002).

R. Dekany, M. Britton, D. Gavel, B. L. Ellerbroek, G. Herriot, C. Max, and J.-P. Veran, "Adaptive optics requirements definition for TMT," in Advances in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 879-890 (2004).

B. A. Macintosh, B. Bauman, J. Wilhelmsen Evans, J. R. Graham, C. Lockwood, L. Poyneer, D. Dillion, D. T. Gavel, J. J. Green, J. P. Lloyd, R. B. Makidon, S. Olivier, D. Palmer, M. D. Perrin, S. Severson, A. I. Sheinis, A. Sivaramakrishnan, G. Sommargren, R. Soummer, M. Troy, J. K. Wallace, and E. Wishnow, "Extreme Adaptive Optics Planet Imager: overview and status," in Advancements in Adaptive Optics, D.B.Calia, B.L.Ellerbroek, and R.Ragazzoni, eds., Proc. SPIE 5490, 359-369 (2004).

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Y. Saad, Iterative Methods for Sparse Linear Systems (Society for Industrial and Applied Mathematics, 2003).
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R. C. Smith, Smart Material Systems (Society for Industrial and Applied Mathematics, 2005).
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C. R. Vogel, Computational Methods for Inverse Problems (Society for Industrial and Applied Mathematics, 2002).
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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the BMC MEMS DM. Downward displacement of the mirror facesheet corresponds to positive w, and downward displacement of the actuator plate corresponds to positive z i . The gap size g is the distance between the actuator plate and the base substate.

Fig. 2
Fig. 2

Solutions to the actuator equation (6) for (a) the parameter pair ( ρ i = 20 , ν = 50 ) and (b) the pair ( ρ i = 10 , ν = 50 ). Positive (upward) displacement of the actuator plate in this figure corresponds to downward displacement in Fig. 1.

Fig. 3
Fig. 3

Actuator post displacement plotted against parameters ρ and ν.

Fig. 4
Fig. 4

Test case 1. Constant facesheet displacement, or piston, is plotted against parameter ν.

Fig. 5
Fig. 5

Test case 2. Facesheet response to a pair of poked actuators. (a) Mesh plot of displacement due to a single poked actuator; (b) cross section of mesh plot in (a); (c) cross sections of the differential displacement from two poked actuators and the sum of the differential displacements from the individual poked actuators. Circles in subplots (b) and (c) indicate actuator locations.

Fig. 6
Fig. 6

Test case 3. Facesheet response for a poked and released actuator. Solid curve, cross section of DM displacement w 1 when a single actuator is poked from zero bias. Dashed curve, cross section of displacement w 2 when a single actuator is released from positive bias. The bias voltage for w 2 matches the poke voltage for w 1 .

Fig. 7
Fig. 7

Open-loop control simulation results. (a) Mesh plot of target displacement ϕ; (b) gray-scale intensity plot of residual w k ϕ after k = 1 iterations; (c) residual w k ϕ after k = 5 iterations; (d) a semilog plot of norm of the wavefront error, w k ϕ , and a plot of the norm of the gradient of the cost functional in Eq. (34).

Equations (38)

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D fs 2 w = i = 1 n a p i δ ( x x i ) ,
4 w = def 4 w x 4 + 2 4 w x 2 y 2 + 4 w y 4 .
D a 4 z i = p i δ ( x 0 ) + ϵ 0 ϵ r V i 2 ( g z i ( x ) ) 2 .
w ( x i ) = z i ( 0 ) , i = 1 , , n a .
z ̃ i = z i g , x ̃ = x l .
4 z ̃ i = ρ i δ ( x ̃ 0 ) + ν i ( 1 z ̃ i ( x ̃ ) ) 2 ,
ρ i = l 4 p i g D a ,
ν i = ϵ 0 ϵ r l 4 V i 2 g 3 D a .
w ̃ = w g , x ̃ = x L ,
4 w ̃ + σ i = 1 n a ρ i δ ( x ̃ x ̃ i ) = 0 ,
σ = ( L l ) 4 D a D fs .
w ̃ ( x ̃ i ) = z ̃ i ( 0 ) , i = 1 , , n a .
4 w + σ i = 1 n a ρ i δ ( x x i ) = 0 ,
w ( x i ) = f ( ρ i , ν i ) , i = 1 , , n a .
w ( x ) = k = 1 N w k b k ( x ) ,
k = 1 N ( 2 b j 2 b k d x ) w k + σ i = 1 n a ρ i b j ( x i ) = 0 , j = 1 , , N .
k = 1 N w k b k ( x i ) = f ( ρ i , ν i ) , i = 1 , , n a .
G ( w , ρ ) = [ G 1 ( w , ρ ) G 2 ( w , ρ ) ] = def [ A w + σ B ρ B T w f ( ρ , ν ) ] = [ 0 0 ] .
B j i = b j ( x i ) , j = 1 , , N , i = 1 , , n a .
( w k + 1 , ρ k + 1 ) = ( w k , ρ k ) G ( w k , ρ k ) 1 G ( w k , ρ k ) ,
k = 0 , 1 , .
G ( w , ρ ) = [ A σ B B T D ] .
[ D ] i i = f ρ i , i = 1 , , n a .
[ A σ B B T D ] [ Δ w Δ ρ ] = [ G 1 G 2 ] ,
Δ ρ = D 1 ( B T Δ w + G 2 ) ,
( A + σ B D 1 B T ) Δ w = G 1 σ B D 1 G 2 .
Δ w i ( x , y ) = def w i ( x , y ) w bias , i = 1 , 2 .
a ̂ = arg min a { ( H a ϕ ) T Q ( H a ϕ ) + a T R a }
= ( H T Q H + R ) 1 H T Q ϕ .
f ( ρ , ν ) f ( ρ bias , ν bias ) + D bias Δ ρ + E bias a .
[ D bias ] i i = f i ρ i ( ρ i , bias , ν i , bias ) , [ E bias ] i i = f i ν i ( ρ i , bias , ν i , bias ) ,
i = 1 , , n act .
A ( w bias + Δ w ) + σ B ( ρ bias + Δ ρ ) = 0 ,
B T ( ρ bias + Δ ρ ) f ( ρ bias , ν bias ) D bias Δ ρ = E bias a .
( A + σ B D bias 1 B T ) Δ w = σ B D bias 1 E bias a .
H = σ ( A + σ B D bias 1 B T ) 1 B D bias 1 E bias .
ϕ ( x , y ) = w bias + 0.6 e 15 r 2 ( ( x 0.5 ) + ( y 0.5 ) )
J ( ν ) = 0 1 0 1 ( w ( x , y ) ϕ ( x , y ) ) 2 d x d y + ν R ν ,

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