Abstract

A method is introduced for predicting control voltages that will generate a prescribed surface shape on a MEMS deformable mirror. The algorithm is based upon an analytical elastic model of the mirror membrane and an empirical electromechanical model of its actuators. It is computationally simple and inherently fast. Shapes at the limit of achievable mirror spatial frequencies with up to 1.5μm amplitudes have been achieved with less than 15nm rms error.

© 2007 Optical Society of America

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References

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  1. M. E. Motamedi, MOEMS-Micro-Opto-Electro-Mechanical Systems (SPIE Press, 2005).
  2. K. M. Morzinski, J. W. Evans, S. Severson, B. Macintosh, D. Dillon, D. Gavel, C. Max, and D. Palmer, "Characterizing the potential of MEMS deformable mirrors for astronomical adaptive optics," Proc. SPIE 6272, 627221 (2006).
    [CrossRef]
  3. E. J. Fernández and P. Artal, "Membrane deformable mirror for adaptive optics: performance limits in visual optics," Opt. Express 11, 1056-1069 (2003).
    [CrossRef] [PubMed]
  4. Y. Zhou and T. Bifano, "Characterization of contour shapes achievable with a MEMS deformable mirror," Proc. SPIE 6113, 123-130 (2006).
  5. F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press, 1999), pp. 77-80.
  6. J. M. Beckers, "Adaptive optics for astronomy: principles, performance, and applications," Annu. Rev. Astron. Astrophys. 31, 13-62 (1993).
    [CrossRef]
  7. J. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998).
  8. J. W. Evans, B. Macintosh, L. Poyneer, K. Morzinski, S. Severson, D. Dillon, D. Gavel, and L. Reza, "Demonstrating sub-nm closed loop MEMS flattening," Opt. Express 14, 5558 (2006).
    [CrossRef] [PubMed]
  9. D. T. Gavel, "Adaptive optics control strategies for extremely large telescopes," Proc. SPIE 4494, 215-220 (2002).
    [CrossRef]
  10. Y. Zhou and T. Bifano, "Adaptive optics using a MEMS deformable mirror," Proc. SPIE 6018, 350-356 (2005).
  11. C. R. Vogel and Q. Yang, "Modeling, simulation, and open-loop control of a continuous facesheet MEMS deformable mirror," J. Opt. Soc. Am. A 23, 1074-1081 (2006).
    [CrossRef]
  12. K. M. Morzinski, K. B. W. Harpsee, D. T. Gavel, and S. M. Ammons, "The open loop control of MEMS: modeling and experimental results," Proc. SPIE 6467, 64670G (2007).
    [CrossRef]
  13. M. H. Miller, J. A. Perrault, G. G. Parker, B. P. Bettig, and T. G. Bifano, "Simple models for piston-type micromirror behavior," J. Micromech. Microeng. 16, 303-313 (2006).
    [CrossRef]
  14. T. G. Bifano, R. Mali, J. Perreault, K. Dorton, N. Vandelli, M. Horentein, and D. Castanon, "Continuous membrane, surface micromachined silicon deformable mirror," Opt. Eng. (Bellingham) 36, 1354-1360 (1997).
    [CrossRef]
  15. T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, "Microelectromechanical deformable mirrors," IEEE J. Sel. Top. Quantum Electron. 5, 83-90 (1999).
    [CrossRef]
  16. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1976), pp. 79-104 and 378-428.
  17. A. Papavasiliou, S. Olivier, T. Barbee, C. Walton, and M. Cohn, "MEMS actuated deformable mirror," Proc. SPIE 6113, 190-199 (2006).
  18. S. D. Senturia, Microsystem Design (Springer Science, 2001), p. 134.
  19. R. J. Roark and W. C. Young, Formulas for Stress and Strain (McGraw-Hill, 1982), p. 97.

2007 (1)

K. M. Morzinski, K. B. W. Harpsee, D. T. Gavel, and S. M. Ammons, "The open loop control of MEMS: modeling and experimental results," Proc. SPIE 6467, 64670G (2007).
[CrossRef]

2006 (6)

M. H. Miller, J. A. Perrault, G. G. Parker, B. P. Bettig, and T. G. Bifano, "Simple models for piston-type micromirror behavior," J. Micromech. Microeng. 16, 303-313 (2006).
[CrossRef]

A. Papavasiliou, S. Olivier, T. Barbee, C. Walton, and M. Cohn, "MEMS actuated deformable mirror," Proc. SPIE 6113, 190-199 (2006).

K. M. Morzinski, J. W. Evans, S. Severson, B. Macintosh, D. Dillon, D. Gavel, C. Max, and D. Palmer, "Characterizing the potential of MEMS deformable mirrors for astronomical adaptive optics," Proc. SPIE 6272, 627221 (2006).
[CrossRef]

Y. Zhou and T. Bifano, "Characterization of contour shapes achievable with a MEMS deformable mirror," Proc. SPIE 6113, 123-130 (2006).

J. W. Evans, B. Macintosh, L. Poyneer, K. Morzinski, S. Severson, D. Dillon, D. Gavel, and L. Reza, "Demonstrating sub-nm closed loop MEMS flattening," Opt. Express 14, 5558 (2006).
[CrossRef] [PubMed]

C. R. Vogel and Q. Yang, "Modeling, simulation, and open-loop control of a continuous facesheet MEMS deformable mirror," J. Opt. Soc. Am. A 23, 1074-1081 (2006).
[CrossRef]

2005 (1)

Y. Zhou and T. Bifano, "Adaptive optics using a MEMS deformable mirror," Proc. SPIE 6018, 350-356 (2005).

2003 (1)

2002 (1)

D. T. Gavel, "Adaptive optics control strategies for extremely large telescopes," Proc. SPIE 4494, 215-220 (2002).
[CrossRef]

1999 (1)

T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, "Microelectromechanical deformable mirrors," IEEE J. Sel. Top. Quantum Electron. 5, 83-90 (1999).
[CrossRef]

1997 (1)

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

1993 (1)

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

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]

IEEE J. Sel. Top. Quantum Electron. (1)

T. G. Bifano, J. Perreault, R. K. Mali, and M. N. Horenstein, "Microelectromechanical deformable mirrors," IEEE J. Sel. Top. Quantum Electron. 5, 83-90 (1999).
[CrossRef]

J. Micromech. Microeng. (1)

M. H. Miller, J. A. Perrault, G. G. Parker, B. P. Bettig, and T. G. Bifano, "Simple models for piston-type micromirror behavior," J. Micromech. Microeng. 16, 303-313 (2006).
[CrossRef]

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

Opt. Eng. (Bellingham) (1)

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

Opt. Express (2)

Proc. SPIE (6)

A. Papavasiliou, S. Olivier, T. Barbee, C. Walton, and M. Cohn, "MEMS actuated deformable mirror," Proc. SPIE 6113, 190-199 (2006).

Y. Zhou and T. Bifano, "Characterization of contour shapes achievable with a MEMS deformable mirror," Proc. SPIE 6113, 123-130 (2006).

K. M. Morzinski, J. W. Evans, S. Severson, B. Macintosh, D. Dillon, D. Gavel, C. Max, and D. Palmer, "Characterizing the potential of MEMS deformable mirrors for astronomical adaptive optics," Proc. SPIE 6272, 627221 (2006).
[CrossRef]

D. T. Gavel, "Adaptive optics control strategies for extremely large telescopes," Proc. SPIE 4494, 215-220 (2002).
[CrossRef]

Y. Zhou and T. Bifano, "Adaptive optics using a MEMS deformable mirror," Proc. SPIE 6018, 350-356 (2005).

K. M. Morzinski, K. B. W. Harpsee, D. T. Gavel, and S. M. Ammons, "The open loop control of MEMS: modeling and experimental results," Proc. SPIE 6467, 64670G (2007).
[CrossRef]

Other (6)

F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press, 1999), pp. 77-80.

J. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998).

M. E. Motamedi, MOEMS-Micro-Opto-Electro-Mechanical Systems (SPIE Press, 2005).

S. D. Senturia, Microsystem Design (Springer Science, 2001), p. 134.

R. J. Roark and W. C. Young, Formulas for Stress and Strain (McGraw-Hill, 1982), p. 97.

S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1976), pp. 79-104 and 378-428.

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Figures (9)

Fig. 1
Fig. 1

Continuous facesheet MEMS DM technology.

Fig. 2
Fig. 2

Cross section of an actuated DM subaperture (top) and free body diagram for its post (bottom). F M is the mirror force, F A is the actuator restoring force, and F E is the applied electrostatic force.

Fig. 3
Fig. 3

Estimating mirror forces F M [ i ] from generalized surface-normal mirror load q ( x , y ) .

Fig. 4
Fig. 4

Simplified parallel-plate and linear-spring actuator model.

Fig. 5
Fig. 5

(Left) Shape used to calibrate DM. Identical voltages are applied to a ring of actuators to vary F M at the central actuator. (Right) w p ( V ) for the central actuator is measured for different ring voltages to produce the calibration dataset spanning { w p , F M , V } . One-hundred data points from a single central actuator of the 12 × 12 DM were collected and fit to a surface of the same functional form as Eq. (6). Surface fitting error was 4 V rms.

Fig. 6
Fig. 6

Demonstration of DM open-loop control. The difference between the actual and predicted voltages for the deflected actuator was less than the calibration surface fitting error, 4 V rms.

Fig. 7
Fig. 7

(Left) Open-loop voltage prediction for a 1.5 μ m amplitude ( 185 V ) stripe pattern. (Center) Predicted voltage map and (right) residual errors between desired and predicted shapes, with 111 nm peak-to-valley and 13.5 nm rms error.

Fig. 8
Fig. 8

(Left) Open-loop and closed-loop voltage maps for ideal focus shift Zernike. (Top center) Closed-loop voltage map and (top right) residual errors between desired and achieved shapes ( 206 nm peak-to-valley and 36.7 nm rms). (Bottom center) Open-loop voltage map and (bottom right) residual errors between desired and predicted shapes ( 228 nm peak-to-valley and 41 nm rms).

Fig. 9
Fig. 9

Actuator mechanical stiffness under central point load. The agreement between the empirically derived actuator stiffness (squares) and the TriboIndenter measured stiffness (circles) suggests the analytical model used for describing plate deformation is accurate. Actuator stiffness is also compared to a first-order analytical fixed-fixed beam approximation (dashed line), which does not account for actuator stretching behavior.

Equations (7)

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4 w ( x , y ) = q ( x , y ) D ,
D = E h 3 12 ( 1 ν 2 ) ,
4 w ( x , y ) = q ( x , y ) D + 6 h 2 { [ ( w ( x , y ) x ) 2 2 w ( x , y ) x 2 ] + [ ( w ( x , y ) y ) 2 2 w ( x , y ) y 2 ] } .
F E = ϵ 0 A V 2 2 ( g 0 w p ) 2 ,
F A = k A w p .
V i = 2 ( g 0 w p , i ) 2 ( k A w p , i + F M , i ) ϵ 0 A ,
F M = F A + F E .

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