Abstract

Electrostatic membrane mirrors are potentially useful as aberration generators. To realize this capability, it is necessary to know the influence functions (each giving mirror response to a single electrode) and then to use a fitting procedure to determine the optimal voltage settings for reproducing a desired surface shape. We show that an approximate analytical solution of Poisson’s equation exists that can generate the influence functions for a circular electrostatic mirror. We also show, by demonstration, that it is computationally feasible to calculate these influence functions and to use them with a fitting procedure to fit the surface of a 109-electrode mirror to a desired shape. Our methods allow one to test the theoretical performance of the 109-electrode mirror; we find that good fits are obtainable for Zernike polynomials of up to degree 6.

© 1986 Optical Society of America

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References

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  1. J. E. Harvey, G. M. Callahan, “Wave front error compensation capabilities of multi-actuator deformable mirrors,” Proc. Soc. Photo-Opt. Instrum. Eng. 141, 50–57 (1978).
  2. T. Sato, H. Ishikawa, O. Ikeda, S. Nomura, K. Uchino, “Deformable 2-D mirror using multilayered electrostrictors,” Appl. Opt. 21, 3669–3672 (1982).
    [CrossRef] [PubMed]
  3. F. Merkle, K. Freischlad, J. Bille, “Development of an active optical mirror for astronomical applications,” presented at the ESO Conference on Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, Garching, Federal Republic of Germany, March 1981.
  4. R. P. Grosso, M. J. Yellin, “The membrane mirror as an adaptive optical element,” J. Opt. Soc. Am. 67, 399–406 (1977).
    [CrossRef]
  5. T. Myint-U, Partial Differential Equations of Mathematical Physics, 2nd ed. (North-Holland, New York, 1980), pp. 13–15.
  6. P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), pp. 793–795.
  7. S. L. Meyer, Data Analysis for Scientists and Engineers (Wiley, New York, 1975), p. 389.
  8. P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, New York, 1953), p. 1191.
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 464–466.
  10. J. Y. Wang, D. E. Silva, “Wave front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  11. S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), pp. 461–462.
  12. F. Otto, ed., Tensile Structures (MIT Press, Cambridge, Mass., 1973), p. 178.

1982 (1)

1980 (1)

1978 (1)

J. E. Harvey, G. M. Callahan, “Wave front error compensation capabilities of multi-actuator deformable mirrors,” Proc. Soc. Photo-Opt. Instrum. Eng. 141, 50–57 (1978).

1977 (1)

Bille, J.

F. Merkle, K. Freischlad, J. Bille, “Development of an active optical mirror for astronomical applications,” presented at the ESO Conference on Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, Garching, Federal Republic of Germany, March 1981.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 464–466.

Callahan, G. M.

J. E. Harvey, G. M. Callahan, “Wave front error compensation capabilities of multi-actuator deformable mirrors,” Proc. Soc. Photo-Opt. Instrum. Eng. 141, 50–57 (1978).

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, New York, 1953), p. 1191.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), pp. 793–795.

Freischlad, K.

F. Merkle, K. Freischlad, J. Bille, “Development of an active optical mirror for astronomical applications,” presented at the ESO Conference on Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, Garching, Federal Republic of Germany, March 1981.

Grosso, R. P.

Harvey, J. E.

J. E. Harvey, G. M. Callahan, “Wave front error compensation capabilities of multi-actuator deformable mirrors,” Proc. Soc. Photo-Opt. Instrum. Eng. 141, 50–57 (1978).

Ikeda, O.

Ishikawa, H.

Merkle, F.

F. Merkle, K. Freischlad, J. Bille, “Development of an active optical mirror for astronomical applications,” presented at the ESO Conference on Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, Garching, Federal Republic of Germany, March 1981.

Meyer, S. L.

S. L. Meyer, Data Analysis for Scientists and Engineers (Wiley, New York, 1975), p. 389.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), pp. 793–795.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, New York, 1953), p. 1191.

Myint-U, T.

T. Myint-U, Partial Differential Equations of Mathematical Physics, 2nd ed. (North-Holland, New York, 1980), pp. 13–15.

Nomura, S.

Sato, T.

Silva, D. E.

Timoshenko, S.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), pp. 461–462.

Uchino, K.

Wang, J. Y.

Woinowsky-Krieger, S.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), pp. 461–462.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 464–466.

Yellin, M. J.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

J. E. Harvey, G. M. Callahan, “Wave front error compensation capabilities of multi-actuator deformable mirrors,” Proc. Soc. Photo-Opt. Instrum. Eng. 141, 50–57 (1978).

Other (8)

F. Merkle, K. Freischlad, J. Bille, “Development of an active optical mirror for astronomical applications,” presented at the ESO Conference on Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, Garching, Federal Republic of Germany, March 1981.

T. Myint-U, Partial Differential Equations of Mathematical Physics, 2nd ed. (North-Holland, New York, 1980), pp. 13–15.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), pp. 793–795.

S. L. Meyer, Data Analysis for Scientists and Engineers (Wiley, New York, 1975), p. 389.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, New York, 1953), p. 1191.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 464–466.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), pp. 461–462.

F. Otto, ed., Tensile Structures (MIT Press, Cambridge, Mass., 1973), p. 178.

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Figures (12)

Fig. 1
Fig. 1

Cross section of deformable membrane mirror.3

Fig. 2
Fig. 2

Electrode configuration of deformable mirror. The actual mirror has narrow insulating gaps between electrodes. The gaps have been ignored in this simulation.

Fig. 3
Fig. 3

Approximated electrode shapes chosen to permit analytical integration of matrix elements.

Fig. 4
Fig. 4

Points used for least-squares fitting of surface.

Fig. 5
Fig. 5

Displacement of the membrane surface along a radial line at ϕ = 0 for Zernike polynomial U 3 3, with an amplitude of 1 μm.

Fig. 6
Fig. 6

Displacement of the membrane surface along a radial line at ϕ = 0 for Zernike polynomial U 6 0, with an amplitude of 0.109 μm.

Fig. 7
Fig. 7

Membrane pressure distribution for the Zernike polynomial U 3 3 (1-μm amplitude).

Fig. 8
Fig. 8

Membrane pressure distribution for the Zernike polynomial U 6 0 (0.109-μm amplitude).

Fig. 9
Fig. 9

Electrode voltage distribution required to generate the displacements of Fig. 5 and the pressure distribution of Fig. 7. Units are volts. Values are rounded to the nearest volt.

Fig. 10
Fig. 10

Electrode voltage distribution required to generate the displacements of Fig. 6 and the pressure distribution of Fig. 8. Units are volts. Values are rounded to the nearest volt.

Fig. 11
Fig. 11

Surface response to actuation of a single electrode with a pressure of 0.24 N/m2. Curves show surface displacement from each of six electrodes along a radial line at ϕ = 0.

Fig. 12
Fig. 12

Effect of approximated electrode shapes on surface shape. Curves show surface displacement along a radial line at ϕ = 0 due to a constant pressure of 0.24 N/m2 applied by all electrodes.

Equations (20)

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z i = j = 1 N e A i j P j ,
S = i = 1 N p ( z i z i ) 2 .
z = A P ,
P = ( A t A ) 1 A t z ,
2 z = P / T ,
z ( r , ϕ ) = a 2 2 π T 0 2 π d ϕ × ( 0 r r d r { ln ( 1 / r ) n = 1 1 n [ ( r r ) n ( r / r ) n ] × cos n ( ϕ ϕ ) { P ( r , ϕ ) + r 1 r d r { ln ( 1 / r ) n = 1 1 n [ ( r r ) n ( r / r ) n ] × cos n ( ϕ ϕ ) } P ( r , ϕ ) ) ,
z i = a 2 T j = 1 N e A i j P j ,
A i j = 1 2 π { Δ ϕ j [ ln ( 1 / r i ) 0 r i r d r + r i 1 r d r ln ( 1 / r ) ] 0 r i r d r n = 1 1 n 2 [ ( r i r ) n ( r / r i ) n ] × [ sin n ( ϕ 2 j ϕ i ) sin n ( ϕ 1 j ϕ i ) ] r i 1 r d r n = 1 1 n 2 [ ( r i r ) n ( r i / r ) n ] × [ sin n ( ϕ 2 j ϕ i ) sin n ( ϕ 1 j ϕ i ) ] } ,
A i j = 1 2 π Δ ϕ j r 2 j 2 ( 1 2 ln r 2 j ) / 2 .
A i j = 1 2 π Δ ϕ j [ r 2 j 2 ( 1 2 ln r 2 j ) r 1 j 2 ( 1 2 ln r 1 j ) ] / 2 .
A i j = 1 2 π { Δ ϕ j [ ln r i ] [ r 2 j 2 r 1 j 2 ] / 2 n = 1 r i 2 n 2 ( n + 2 ) [ r i 2 n 1 ] × [ ( r 2 j r i ) n + 2 ( r 1 j r i ) n + 2 ] × [ sin n ( ϕ 2 j ϕ i ) sin n ( ϕ 1 j ϕ i ) ] } .
A i j = 1 2 π { Δ ϕ j [ r 2 j 2 ( 1 2 ln r 2 j ) r 1 j 2 ( 1 2 ln r 1 j ) ] / 2 n = 1 1 n 2 [ r i n ( r 2 j n + 2 r 1 j n + 2 ) / ( n + 2 ) + α ] × [ sin n ( ϕ 2 j ϕ i ) sin n ( ϕ 1 j ϕ i ) ] } ,
α = r i ( r 2 j r 1 j ) when n = 1 , α = r i 2 ( ln r 2 j ln r 1 j ) when n = 2 , α = r i 2 [ ( r i r 2 j ) n 2 ( r i r 1 j ) n 2 ] / ( n 2 ) when n 3.
P = 0 2 ( V 2 2 d 2 2 V 1 2 d 1 2 ) ,
Δ V = d 1 ( V 2 2 d 2 2 2 0 P ) 1 / 2 V 1 B .
U n m = R n m ( r ) cos m ϕ , m 0 ,
U n m = R n m ( r ) sin m ϕ , m > 0 ,
R n m ( r ) = s = 0 ( n m ) / 2 ( 1 ) s ( n s ) ! s ! [ ( n + m ) / 2 s ] ! [ ( n m ) / 2 s ] ! r n 2 s .
U 3 3 = r 3 cos 3 ϕ ,
U 6 0 = 20 r 6 30 r 4 + 12 r 2 1.

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