Abstract

The nonlinear response and strong coupling of control channels in micromachined membrane deformable mirror (MMDM) devices make it difficult for one to control the MMDM to obtain the desired mirror surface shapes. A closed-loop adaptive control algorithm is developed for a continuous-surface MMDM used for aberration compensation. The algorithm iteratively adjusts the control voltages of all electrodes to reduce the variance of the optical wave front measured with a Hartmann–Shack wave-front sensor. Zernike polynomials are used to represent the mirror surface shape as well as the optical wave front. An adaptive experimental system to compensate for the wave-front aberrations of a model eye has been built in which the developed adaptive mirror-control algorithm is used to control a deformable mirror with 19 active channels. The experimental results show that the algorithm can adaptively update control voltages to generate an optimum continuous mirror surface profile, compensating for the aberrations within the operating range of the deformable mirror.

© 1999 Optical Society of America

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References

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  1. R. K. Tyson, Principles of Adaptive Optics. 2nd ed. (Academic, San Diego, Calif., 1998).
  2. G. Vdovin, P. M. Sarro, “Flexible mirror micromachined in silicon,” Appl. Opt. 34, 2968–2972 (1995).
    [CrossRef] [PubMed]
  3. T. G. Bifano, R. Krishnamoorthy, J. K. Dorton, J. Perreault, N. Vandelli, M. N. Horenstein, D. A. Castanon, “Continuous-membrane surface-micromachined silicon deformable mirror,” Opt. Eng. 36, 1354–1360 (1997).
    [CrossRef]
  4. G. Vdovin, “Model of an adaptive optical system controlled by a neural network,” Opt. Eng. 34, 3249–3253 (1995).
    [CrossRef]
  5. P. K. C. Wang, F. Y. Hadaegh, “Computation of static shapes and voltages for micromachined deformable mirrors with nonlinear electrostatic actuators,” J. Microelectromech. Syst. 5, 205–220 (1996).
    [CrossRef]
  6. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  7. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  8. D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 461–472.
  9. S. S. Haykin, Adaptive Filter Theory, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1996).
  10. N. J. O’Connor, D.-U. Bartsch, W. R. Freeman, A. J. Mueller, T. J. Holmes, “Fluorescent infrared scanning-laser ophthalmoscope for three-dimensional visualization: automatic random-eye-motion correction and deconvolution,” Appl. Opt. 37, 2021–2033 (1998).
    [CrossRef]

1998 (1)

1997 (1)

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

1996 (1)

P. K. C. Wang, F. Y. Hadaegh, “Computation of static shapes and voltages for micromachined deformable mirrors with nonlinear electrostatic actuators,” J. Microelectromech. Syst. 5, 205–220 (1996).
[CrossRef]

1995 (2)

G. Vdovin, “Model of an adaptive optical system controlled by a neural network,” Opt. Eng. 34, 3249–3253 (1995).
[CrossRef]

G. Vdovin, P. M. Sarro, “Flexible mirror micromachined in silicon,” Appl. Opt. 34, 2968–2972 (1995).
[CrossRef] [PubMed]

1980 (2)

Bartsch, D.-U.

Bifano, T. G.

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

Castanon, D. A.

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

DeVore, S. L.

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 461–472.

Dorton, J. K.

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

Freeman, W. R.

Hadaegh, F. Y.

P. K. C. Wang, F. Y. Hadaegh, “Computation of static shapes and voltages for micromachined deformable mirrors with nonlinear electrostatic actuators,” J. Microelectromech. Syst. 5, 205–220 (1996).
[CrossRef]

Haykin, S. S.

S. S. Haykin, Adaptive Filter Theory, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1996).

Holmes, T. J.

Horenstein, M. N.

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

Krishnamoorthy, R.

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

Malacara, D.

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 461–472.

Mueller, A. J.

O’Connor, N. J.

Perreault, J.

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

Sarro, P. M.

Silva, D. E.

Southwell, W. H.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics. 2nd ed. (Academic, San Diego, Calif., 1998).

Vandelli, N.

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

Vdovin, G.

G. Vdovin, P. M. Sarro, “Flexible mirror micromachined in silicon,” Appl. Opt. 34, 2968–2972 (1995).
[CrossRef] [PubMed]

G. Vdovin, “Model of an adaptive optical system controlled by a neural network,” Opt. Eng. 34, 3249–3253 (1995).
[CrossRef]

Wang, J. Y.

Wang, P. K. C.

P. K. C. Wang, F. Y. Hadaegh, “Computation of static shapes and voltages for micromachined deformable mirrors with nonlinear electrostatic actuators,” J. Microelectromech. Syst. 5, 205–220 (1996).
[CrossRef]

Appl. Opt. (3)

J. Microelectromech. Syst. (1)

P. K. C. Wang, F. Y. Hadaegh, “Computation of static shapes and voltages for micromachined deformable mirrors with nonlinear electrostatic actuators,” J. Microelectromech. Syst. 5, 205–220 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (2)

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

G. Vdovin, “Model of an adaptive optical system controlled by a neural network,” Opt. Eng. 34, 3249–3253 (1995).
[CrossRef]

Other (3)

R. K. Tyson, Principles of Adaptive Optics. 2nd ed. (Academic, San Diego, Calif., 1998).

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 461–472.

S. S. Haykin, Adaptive Filter Theory, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1996).

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the micromachined continuous-membrane deformable mirror.

Fig. 2
Fig. 2

Schematic diagram of wave-front propagation in an adaptive optical system with a deformable mirror.

Fig. 3
Fig. 3

Principle of the Hartmann–Shack wave-front sensor.

Fig. 4
Fig. 4

Flow chart describing the adaptive control algorithm used for aberration compensation in a typical adaptive optical system (e.g., Fig. 2).

Fig. 5
Fig. 5

Schematic diagram of the adaptive optical system with a MMDM applied to aberration compensation for a model eye.

Fig. 6
Fig. 6

Example of an HSWS image with slope measurement windows and beam aperture.

Fig. 7
Fig. 7

Electrode layout of the MMDM used in the system. The 19 active electrode channels are labeled. The inner circle indicates the active aperture of the system.

Fig. 8
Fig. 8

Measured coefficient a 4 as a function of (a) the control voltage and (b) the control voltage square for the actuator in channel 10 of the MMDM in the system (Fig. 5).

Fig. 9
Fig. 9

Convergence of the Zernike coefficients a k (k = 1, 2, … , 20) as a function of the iteration index.

Fig. 10
Fig. 10

Example of aberration-compensation results: (a), (b) Wave front before and after 50 iterations for mirror shape self-compensation; (c), (d) wave front before and after 50 iterations for defocus and astigmatism compensation. The vertical bar indicates the maximum wave-front variation over the beam size.

Equations (22)

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ϕx, y-ϕ0x, y=k=1M akzkx, y,
αij=2πλf Δxij,  βij=2πλf Δyij,
a=ATA-1ATs,
δd  Vl2.
cl=Vl2,
ΔSx, y=l=1P clφlx, y,
φlx, y=k=1M bklzkx, y,
ΔSx, y=l=1P clk=1M bklzkx, y=k=1Ml=1P clbklzkx, yk=1M akzkx, y,
ak=l=1P clbkl for k=1, 2,, M.
ac=Bc,
S0x, y=k=1M a0kzkx, y
a=ac+a0+ain=Bc+a0+ain=Bc+aaber,
aaber=a0+ain,
σϕ2=1Aapertureϕx, y-ϕ0x, y2dxdy=1πunit circleϕx, y-ϕ0x, y2dxdy,
E=σϕ2=k=1M σk2,
σk2=1πunit circleakzkx, y2dxdy=ak21πunit circlezkx, y2dxdy=ak2wk2
wk2=1πunit circlezkx, y2dxdy.
E=k=1M ak2wk2=a*w2,
E=a*w2=Bc+aaber*w2.
Ec=2BTa*w2,
cnew=cold-μ Ec=cold-2μBTa*w2,
s=α1,3 α1,4α1,11 α2,2α2,12α13,11 β1,3β13,11T,

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