Abstract

The transient response of a deformable mirror to be used in a closed-loop adaptive-optics imaging system is modeled and evaluated. A theoretical model is developed that describes the motion of the mirror membrane. This allows an adaptive control to achieve reduced overshoot and short settling time. Applicability of the model is tested on a mirao 52-e electromagnetic deformable mirror using a specially designed high-speed adaptive-optics test bench. This test bench permits precise mirror motion measurements up to 10kHz.

© 2010 Optical Society of America

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  1. N. Devaney, D. Coburn, C. Coleman, C. Dainty, E. Dalimier, T. Farrell, D. Lara, D. Mackey, and R. Mackey, “Characterisation of MEMs mirrors for use in atmospheric and ocular wavefront correction,” Proc. SPIE 6888, 688802 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. D. A. Horsley, H. Park, S. P. Laut, and J. S. Werner, “Characterization of a bimorph deformable mirror using stroboscopic phase-shifting interferometry,” Sens. Actuators A 134, 221–230 (2007).
    [CrossRef]
  6. E. J. Fernandez, L. Vabre, B. Hermann, A. Unterhuber, B. Povazay, and W. Drexler, “Adaptive optics with a magnetic deformable mirror: applications in the human eye,” Opt. Express 14, 8900–8917 (2006).
    [CrossRef] [PubMed]
  7. International Electrotechnical Commission standards on “Safety requirements for electrical equipment for measurement, control, and laboratory use,” http://www.iec.ch/.
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    [CrossRef]
  9. S. E. Winters, J. H. Chung, and S. A. Velinsky, “Dynamic modeling and control of a deformable mirror,” Mech. Des. Struct. Mach. 32, 195–213 (2004).
    [CrossRef]
  10. L. Ljung and T. Glad, Modeling of Dynamic Systems (Prentice-Hall, 1994).
  11. F. Gustafsson, L. Ljung, and M. Millnert, Signalbehandling (Studentlitteratur, 2001).
  12. E. Odlund, M. Navarro, E. Lavergne, F. Martins, X. Levecq, and A. Dubra, “Optimization of the temporal performance of a deformable mirror for use in ophthalmic applications,” Proc. SPIE 7139, 713912 (2008).
    [CrossRef]
  13. E. Walter and L. Pronzato, Identification of Parametric Models from Experimental Data (Springer-Verlag, 1997).
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    [CrossRef]
  15. C. Correia, H.-F. Raynaud, C. Kulcsár, and J.-M. Conan, “On the optimal reconstruction and control of adaptive optical systems with mirror dynamics,” J. Opt. Soc. Am. A 27, 333–349 (2010).
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2010 (1)

2009 (1)

2008 (2)

E. Odlund, M. Navarro, E. Lavergne, F. Martins, X. Levecq, and A. Dubra, “Optimization of the temporal performance of a deformable mirror for use in ophthalmic applications,” Proc. SPIE 7139, 713912 (2008).
[CrossRef]

N. Devaney, D. Coburn, C. Coleman, C. Dainty, E. Dalimier, T. Farrell, D. Lara, D. Mackey, and R. Mackey, “Characterisation of MEMs mirrors for use in atmospheric and ocular wavefront correction,” Proc. SPIE 6888, 688802 (2008).
[CrossRef]

2007 (1)

D. A. Horsley, H. Park, S. P. Laut, and J. S. Werner, “Characterization of a bimorph deformable mirror using stroboscopic phase-shifting interferometry,” Sens. Actuators A 134, 221–230 (2007).
[CrossRef]

2006 (1)

2005 (2)

2004 (1)

S. E. Winters, J. H. Chung, and S. A. Velinsky, “Dynamic modeling and control of a deformable mirror,” Mech. Des. Struct. Mach. 32, 195–213 (2004).
[CrossRef]

2001 (1)

F. Gustafsson, L. Ljung, and M. Millnert, Signalbehandling (Studentlitteratur, 2001).

2000 (1)

M. R. Hart, R. A. Conant, K. Y. Lau, and R. S. Muller, “Stroboscopic interferometer system for dynamic MEMS characterization,” IEEE J. Microelectromech. Syst. 9, 409–417(2000).
[CrossRef]

1999 (1)

C. L. Hom, “Simulating electrostrictive deformable mirror: II. Nonlinear dynamic analysis,” Smart Mater. Struct. 8, 700–708(1999).
[CrossRef]

1997 (1)

E. Walter and L. Pronzato, Identification of Parametric Models from Experimental Data (Springer-Verlag, 1997).

1994 (1)

L. Ljung and T. Glad, Modeling of Dynamic Systems (Prentice-Hall, 1994).

Booth, M.

Chung, J. H.

S. E. Winters, J. H. Chung, and S. A. Velinsky, “Dynamic modeling and control of a deformable mirror,” Mech. Des. Struct. Mach. 32, 195–213 (2004).
[CrossRef]

Coburn, D.

N. Devaney, D. Coburn, C. Coleman, C. Dainty, E. Dalimier, T. Farrell, D. Lara, D. Mackey, and R. Mackey, “Characterisation of MEMs mirrors for use in atmospheric and ocular wavefront correction,” Proc. SPIE 6888, 688802 (2008).
[CrossRef]

Coleman, C.

N. Devaney, D. Coburn, C. Coleman, C. Dainty, E. Dalimier, T. Farrell, D. Lara, D. Mackey, and R. Mackey, “Characterisation of MEMs mirrors for use in atmospheric and ocular wavefront correction,” Proc. SPIE 6888, 688802 (2008).
[CrossRef]

Conan, J.-M.

Conant, R. A.

M. R. Hart, R. A. Conant, K. Y. Lau, and R. S. Muller, “Stroboscopic interferometer system for dynamic MEMS characterization,” IEEE J. Microelectromech. Syst. 9, 409–417(2000).
[CrossRef]

Correia, C.

Dainty, C.

N. Devaney, D. Coburn, C. Coleman, C. Dainty, E. Dalimier, T. Farrell, D. Lara, D. Mackey, and R. Mackey, “Characterisation of MEMs mirrors for use in atmospheric and ocular wavefront correction,” Proc. SPIE 6888, 688802 (2008).
[CrossRef]

E. Dalimier and C. Dainty, “Comparative analysis of deformable mirrors for ocular adaptive optics,” Opt. Express 13, 4275–4285 (2005).
[CrossRef] [PubMed]

Dalimier, E.

N. Devaney, D. Coburn, C. Coleman, C. Dainty, E. Dalimier, T. Farrell, D. Lara, D. Mackey, and R. Mackey, “Characterisation of MEMs mirrors for use in atmospheric and ocular wavefront correction,” Proc. SPIE 6888, 688802 (2008).
[CrossRef]

E. Dalimier and C. Dainty, “Comparative analysis of deformable mirrors for ocular adaptive optics,” Opt. Express 13, 4275–4285 (2005).
[CrossRef] [PubMed]

Devaney, N.

N. Devaney, D. Coburn, C. Coleman, C. Dainty, E. Dalimier, T. Farrell, D. Lara, D. Mackey, and R. Mackey, “Characterisation of MEMs mirrors for use in atmospheric and ocular wavefront correction,” Proc. SPIE 6888, 688802 (2008).
[CrossRef]

Drexler, W.

Dubra, A.

E. Odlund, M. Navarro, E. Lavergne, F. Martins, X. Levecq, and A. Dubra, “Optimization of the temporal performance of a deformable mirror for use in ophthalmic applications,” Proc. SPIE 7139, 713912 (2008).
[CrossRef]

Farrell, T.

N. Devaney, D. Coburn, C. Coleman, C. Dainty, E. Dalimier, T. Farrell, D. Lara, D. Mackey, and R. Mackey, “Characterisation of MEMs mirrors for use in atmospheric and ocular wavefront correction,” Proc. SPIE 6888, 688802 (2008).
[CrossRef]

Fernandez, E. J.

Glad, T.

L. Ljung and T. Glad, Modeling of Dynamic Systems (Prentice-Hall, 1994).

Gustafsson, F.

F. Gustafsson, L. Ljung, and M. Millnert, Signalbehandling (Studentlitteratur, 2001).

Hart, M. R.

M. R. Hart, R. A. Conant, K. Y. Lau, and R. S. Muller, “Stroboscopic interferometer system for dynamic MEMS characterization,” IEEE J. Microelectromech. Syst. 9, 409–417(2000).
[CrossRef]

Hermann, B.

Hom, C. L.

C. L. Hom, “Simulating electrostrictive deformable mirror: II. Nonlinear dynamic analysis,” Smart Mater. Struct. 8, 700–708(1999).
[CrossRef]

Horsley, D. A.

D. A. Horsley, H. Park, S. P. Laut, and J. S. Werner, “Characterization of a bimorph deformable mirror using stroboscopic phase-shifting interferometry,” Sens. Actuators A 134, 221–230 (2007).
[CrossRef]

Kawata, S.

Kulcsár, C.

Lara, D.

N. Devaney, D. Coburn, C. Coleman, C. Dainty, E. Dalimier, T. Farrell, D. Lara, D. Mackey, and R. Mackey, “Characterisation of MEMs mirrors for use in atmospheric and ocular wavefront correction,” Proc. SPIE 6888, 688802 (2008).
[CrossRef]

Lau, K. Y.

M. R. Hart, R. A. Conant, K. Y. Lau, and R. S. Muller, “Stroboscopic interferometer system for dynamic MEMS characterization,” IEEE J. Microelectromech. Syst. 9, 409–417(2000).
[CrossRef]

Laut, S. P.

D. A. Horsley, H. Park, S. P. Laut, and J. S. Werner, “Characterization of a bimorph deformable mirror using stroboscopic phase-shifting interferometry,” Sens. Actuators A 134, 221–230 (2007).
[CrossRef]

Lavergne, E.

E. Odlund, M. Navarro, E. Lavergne, F. Martins, X. Levecq, and A. Dubra, “Optimization of the temporal performance of a deformable mirror for use in ophthalmic applications,” Proc. SPIE 7139, 713912 (2008).
[CrossRef]

Levecq, X.

E. Odlund, M. Navarro, E. Lavergne, F. Martins, X. Levecq, and A. Dubra, “Optimization of the temporal performance of a deformable mirror for use in ophthalmic applications,” Proc. SPIE 7139, 713912 (2008).
[CrossRef]

Ljung, L.

F. Gustafsson, L. Ljung, and M. Millnert, Signalbehandling (Studentlitteratur, 2001).

L. Ljung and T. Glad, Modeling of Dynamic Systems (Prentice-Hall, 1994).

Looze, D. P.

Mackey, D.

N. Devaney, D. Coburn, C. Coleman, C. Dainty, E. Dalimier, T. Farrell, D. Lara, D. Mackey, and R. Mackey, “Characterisation of MEMs mirrors for use in atmospheric and ocular wavefront correction,” Proc. SPIE 6888, 688802 (2008).
[CrossRef]

Mackey, R.

N. Devaney, D. Coburn, C. Coleman, C. Dainty, E. Dalimier, T. Farrell, D. Lara, D. Mackey, and R. Mackey, “Characterisation of MEMs mirrors for use in atmospheric and ocular wavefront correction,” Proc. SPIE 6888, 688802 (2008).
[CrossRef]

Martins, F.

E. Odlund, M. Navarro, E. Lavergne, F. Martins, X. Levecq, and A. Dubra, “Optimization of the temporal performance of a deformable mirror for use in ophthalmic applications,” Proc. SPIE 7139, 713912 (2008).
[CrossRef]

Millnert, M.

F. Gustafsson, L. Ljung, and M. Millnert, Signalbehandling (Studentlitteratur, 2001).

Muller, R. S.

M. R. Hart, R. A. Conant, K. Y. Lau, and R. S. Muller, “Stroboscopic interferometer system for dynamic MEMS characterization,” IEEE J. Microelectromech. Syst. 9, 409–417(2000).
[CrossRef]

Navarro, M.

E. Odlund, M. Navarro, E. Lavergne, F. Martins, X. Levecq, and A. Dubra, “Optimization of the temporal performance of a deformable mirror for use in ophthalmic applications,” Proc. SPIE 7139, 713912 (2008).
[CrossRef]

Odlund, E.

E. Odlund, M. Navarro, E. Lavergne, F. Martins, X. Levecq, and A. Dubra, “Optimization of the temporal performance of a deformable mirror for use in ophthalmic applications,” Proc. SPIE 7139, 713912 (2008).
[CrossRef]

Park, H.

D. A. Horsley, H. Park, S. P. Laut, and J. S. Werner, “Characterization of a bimorph deformable mirror using stroboscopic phase-shifting interferometry,” Sens. Actuators A 134, 221–230 (2007).
[CrossRef]

Povazay, B.

Pronzato, L.

E. Walter and L. Pronzato, Identification of Parametric Models from Experimental Data (Springer-Verlag, 1997).

Raynaud, H.-F.

Sun, H.-B.

Taisuke, O.

Unterhuber, A.

Vabre, L.

Velinsky, S. A.

S. E. Winters, J. H. Chung, and S. A. Velinsky, “Dynamic modeling and control of a deformable mirror,” Mech. Des. Struct. Mach. 32, 195–213 (2004).
[CrossRef]

Walter, E.

E. Walter and L. Pronzato, Identification of Parametric Models from Experimental Data (Springer-Verlag, 1997).

Werner, J. S.

D. A. Horsley, H. Park, S. P. Laut, and J. S. Werner, “Characterization of a bimorph deformable mirror using stroboscopic phase-shifting interferometry,” Sens. Actuators A 134, 221–230 (2007).
[CrossRef]

Wilson, T.

Winters, S. E.

S. E. Winters, J. H. Chung, and S. A. Velinsky, “Dynamic modeling and control of a deformable mirror,” Mech. Des. Struct. Mach. 32, 195–213 (2004).
[CrossRef]

Appl. Opt. (1)

IEEE J. Microelectromech. Syst. (1)

M. R. Hart, R. A. Conant, K. Y. Lau, and R. S. Muller, “Stroboscopic interferometer system for dynamic MEMS characterization,” IEEE J. Microelectromech. Syst. 9, 409–417(2000).
[CrossRef]

J. Opt. Soc. Am. A (2)

Mech. Des. Struct. Mach. (1)

S. E. Winters, J. H. Chung, and S. A. Velinsky, “Dynamic modeling and control of a deformable mirror,” Mech. Des. Struct. Mach. 32, 195–213 (2004).
[CrossRef]

Opt. Express (2)

Proc. SPIE (2)

N. Devaney, D. Coburn, C. Coleman, C. Dainty, E. Dalimier, T. Farrell, D. Lara, D. Mackey, and R. Mackey, “Characterisation of MEMs mirrors for use in atmospheric and ocular wavefront correction,” Proc. SPIE 6888, 688802 (2008).
[CrossRef]

E. Odlund, M. Navarro, E. Lavergne, F. Martins, X. Levecq, and A. Dubra, “Optimization of the temporal performance of a deformable mirror for use in ophthalmic applications,” Proc. SPIE 7139, 713912 (2008).
[CrossRef]

Sens. Actuators A (1)

D. A. Horsley, H. Park, S. P. Laut, and J. S. Werner, “Characterization of a bimorph deformable mirror using stroboscopic phase-shifting interferometry,” Sens. Actuators A 134, 221–230 (2007).
[CrossRef]

Smart Mater. Struct. (1)

C. L. Hom, “Simulating electrostrictive deformable mirror: II. Nonlinear dynamic analysis,” Smart Mater. Struct. 8, 700–708(1999).
[CrossRef]

Other (4)

International Electrotechnical Commission standards on “Safety requirements for electrical equipment for measurement, control, and laboratory use,” http://www.iec.ch/.

L. Ljung and T. Glad, Modeling of Dynamic Systems (Prentice-Hall, 1994).

F. Gustafsson, L. Ljung, and M. Millnert, Signalbehandling (Studentlitteratur, 2001).

E. Walter and L. Pronzato, Identification of Parametric Models from Experimental Data (Springer-Verlag, 1997).

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Figures (11)

Fig. 1
Fig. 1

DM mirao 52-e with (i) the actuator distribution over a 15 mm effective pupil, (ii) a cross section showing the coils and the magnet array under the reflective membrane, (iii) picture and dimensions of the device. Courtesy: Adrian Bradu, Internal report, University of Kent.

Fig. 2
Fig. 2

Schematic description of the AO test bench used in the study.

Fig. 3
Fig. 3

Principle of the sampling algorithm. For every DM signal application, the moment of acquisition is delayed by a time interval t s (upper), which gives the sampling frequency ω = 1 / t s . Then all samples are merged together into one response pulse (lower). For clarity, the actuator signals are deliberately illustrated with exaggerated inclination of the leading and trailing edges.

Fig. 4
Fig. 4

Schematic diagram describing the timing between events for the different devices: DM triggering actuator, DM central actuator, pulse generator, and the WFS. When the pulse generator receives the pulse IN, its program execution starts and trigger signals, OUT, are sent to the WFS after time delays i t s .

Fig. 5
Fig. 5

Sampling the step response of the DM at 1 kHz (solid line) and at 10 kHz (dash–dots).

Fig. 6
Fig. 6

Periodogram of the step response of the central actuator. There are clear peaks at frequencies ω 1 200 Hz , ω 2 400 Hz , ω 3 690 Hz , ω 4 1070 Hz , ω 5 1470 Hz , and ω 6 1514 Hz .

Fig. 7
Fig. 7

(a) Identified model of the mirao 52-e step response (solid line) versus the measured result (dots). (b) Decomposition of the identified model into three second-order systems. The frequencies are ω 1 = 1406 Hz (dots), ω 2 = 692 Hz (dashes), and ω 3 = 193 Hz (dash–dots). (c) Spectrum for the signal (solid line) and the model (dots), respectively.

Fig. 8
Fig. 8

Residuals (estimation errors) for the model relative to a data series sampled at 10 kHz . The residuals were calculated for five data series. This leads to graphs displaying a similar behavior and, therefore, only one is presented. The estimated errors are more important in the beginning of the series for all the data sets, with a quick decay below the noise threshold by the end of the series.

Fig. 9
Fig. 9

Inverse filtering is made using Simulink. The system response is denoted by F ( s ) and the desired response by G ( s ) . An input step filtered through G ( s ) / F ( s ) gives the trajectory of the model reference control signal. This trajectory is retrieved as a numerical array and uploaded in the DM control software.

Fig. 10
Fig. 10

Simulated output (dots) versus the real measured data (solid lines) for the two controls obtained from filters A (upper) and B (lower).

Fig. 11
Fig. 11

Trajectories of the model reference control inputs for filters A (left) and B (right). The applied voltage (dashes) was slightly attenuated to ensure that it never exceeds the maximum absolute voltage value of 1 V that can be applied to a DM actuator.

Tables (1)

Tables Icon

Table 1 Rise Times, Settling Times, and Overshoot for an Unfiltered Control Signal Versus Two Filtered Control Signals a

Equations (18)

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Y ( s ) = F ( s ) U ( s ) .
Y ref ( s ) = G ( s ) Y r ( s ) .
U ( s ) = G ( s ) F ( s ) Y r ( s )
Y ( s ) = F ( s ) U ( s ) = F ( s ) G ( s ) F ( s ) Y r ( s ) = Y ref ( s ) .
U ( s ) = G ( s ) F m ( s ) Y r ( s )
Y ( s ) = F ( s ) U ( s ) = G ( s ) F ( s ) F m ( s ) Y r ( s ) .
y ¯ [ k ] = 1 n i = 1 n y i [ k ] ,
σ [ k ] = 1 n 1 i = 1 n ( y i [ k ] y ¯ [ k ] ) 2 for     k = [ 1 , , m ] ,
σ r = 1 m k = 1 m σ [ k ] 2 y ss .
y ¨ + 2 ζ ω n y ˙ + ω n 2 y = C ω n 2 u ,
y step ( t ) = C 1 ζ 2 ( 1 e ζ ω n t sin ( ω n 1 ζ 2 t + arccos ( ζ ) ) ) .
F ( s ) = C s 2 ω n 2 + 2 ζ ω n s + 1 .
F m ( s ) = F 1 ( s ) + F 2 ( s ) + F 3 ( s ) ,
F 1 ( s ) = C 1 s 2 ω n 1 2 + 2 ζ 1 ω n 1 s + 1 , F 2 ( s ) = C 2 s 2 ω n 2 2 + 2 ζ 2 ω n 2 s + 1 , F 3 ( s ) = C 3 s 2 ω n 3 2 + 2 ζ 3 ω n 3 s + 1 .
y ( t , θ ) = j = 1 3 y j ( t , θ i ) , where     θ j = ( ω n j , ζ j , C j ) T .
l ( θ ) = y ( t , θ ) y ( t ) y ss .
ω n 1 = 1406 Hz , ζ 1 = 0.38 , C 1 = 0.57 μm ω n 2 = 692 Hz , ζ 2 = 0.017 , C 2 = 0.66 μm ω n 3 = 193 Hz , ζ 3 = 0.11 , C 3 = 0.33 μm .
ϵ ( t ) = ϵ ( t , θ ^ ) = y ( t ) y ( t , θ ^ ) ,

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