Abstract

Membrane mirrors with transparent electrodes were fabricated for adaptive optics. These devices are capable of generating large, low-spatial-order deformations but exhibit instability for high-order deformations. A variational calculation of the electrostatic and mechanical energy of such membrane devices leads to criteria for stable operation. Simulations based upon this calculation are able to reproduce the observed behavior of fabricated devices and suggest suitable device parameters for improved performance with high-order deformations.

© 2006 Optical Society of America

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References

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  1. R. Aldrich, "Deformable mirrors wavefront correctors," in Adaptive Optics Engineering Handbook, R.Tyson, ed. (Marcel Dekker, 2000), pp. 151-157.
  2. G. Vdovin and P. M. Sarro, "Flexible mirror micromachined in silicon," Appl. Opt. 34, 2968-2972 (1995).
    [CrossRef] [PubMed]
  3. We are preparing a manuscript to be called, "Fabrication of a membrane mirror with transparent electrode."
  4. D. L. Powers, Boundary Value Problems, 3rd ed. (Harcourt Brace Jovanovich, 1987), pp. 264-266.

1995 (1)

Aldrich, R.

R. Aldrich, "Deformable mirrors wavefront correctors," in Adaptive Optics Engineering Handbook, R.Tyson, ed. (Marcel Dekker, 2000), pp. 151-157.

Powers, D. L.

D. L. Powers, Boundary Value Problems, 3rd ed. (Harcourt Brace Jovanovich, 1987), pp. 264-266.

Sarro, P. M.

Vdovin, G.

Appl. Opt. (1)

Other (3)

We are preparing a manuscript to be called, "Fabrication of a membrane mirror with transparent electrode."

D. L. Powers, Boundary Value Problems, 3rd ed. (Harcourt Brace Jovanovich, 1987), pp. 264-266.

R. Aldrich, "Deformable mirrors wavefront correctors," in Adaptive Optics Engineering Handbook, R.Tyson, ed. (Marcel Dekker, 2000), pp. 151-157.

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of a membrane mirror device. The membrane, solid curve, is located between the top transparent electrode located at a distance d T and the bottom actuating electrode located at a distance d A . The membrane experiences deformations relative to the flat position (dotted line) in response to electric fields in regions I and II. Voltages V A and V T refer to the relative voltage between the respective electrode plane and the membrane, which may be at ground or at some nonzero voltage.

Fig. 2
Fig. 2

(Color online) Lowest stability matrix eigenvalue versus peak deformation for several devices with different gap distances. Peak deformations that can be achieved with stable device operation are indicated by positive eigenvalues. Large gap distance devices can achieve a broader range of peak deformations than small gap distance devices. Deformation is the lowest-order membrane function, j = 0. Dashed curves correspond to simulated devices with membrane–transparent electrode distances (equal to membrane–electrode array distances): 75 μm (top), 50 μm (middle), and 25 μm (bottom). The left–right asymmetry of the data is a consequence of differing geometries of the two electrode surfaces. The middle solid curve illustrates the model of an actual device, which was tested in the laboratory, with asymmetrical 27∕36 μm gap distances.

Fig. 3
Fig. 3

(Color online) Lowest stability matrix eigenvalue versus transparent electrode potential for several devices with different gap distances. The voltages for stable device operation are indicated by positive eigenvalues. Large gap distance devices are stable over a wider range of electrode voltages than small gap distance devices. Dashed curves correspond to simulated devices with membrane–transparent electrode distances (equal to membrane–electrode array distances): 75 μm, top curve (dashed), and 50 μm, middle curve (dotted–dashed). Solid lower curve illustrates the model of an actual device, which was tested in the laboratory, with asymmetrical 27∕36 μm gap distances.

Fig. 4
Fig. 4

(Color online) Maximum, stable peak deformation for the lowest eigenmodes of membrane devices for several different membrane–electrode gap distances. Positive deformations are toward the transparent electrode; negative deformations are toward the electrode array. These data quantify the capability of membrane devices to generate deformations at each spatial order.

Fig. 5
Fig. 5

(a) Membrane deformation corresponding to j = 2 eigenmode. Largest magnitude deformation is toward the electrode array (negative value on vertical axis). (b) Voltage distribution on the array necessary to produce the deformation from (a), with a device having 30 um gap distances and 25 V transparent electrode potential.

Fig. 6
Fig. 6

Representative eigenfunctions for a 7.5 mm radius membrane. Each plot graphs the magnitude of the complex eigenfunction ζ j (m−1) versus radial coordinate (mm). (a) First five eigenfunctions, indexed by j value, as defined in the text. (b) The 0th-order eigenfunction and a quadratic fit. The lowest-order eigenfunction is nearly a parabola.

Tables (2)

Tables Icon

Table 1 Maximum Deformation for Lowest-Order ( j = 0) Eigenmode in Several Devices

Tables Icon

Table 2 Maximum Deformation for First-Order ( j = 1) Eigenmode in Several Devices

Equations (42)

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δ U = [ T 2 ξ 1 2 ε 0 V A 2 ( d A ξ ) 2 + 1 2 ε 0 V T 2 ( d T + ξ ) 2 ] δ ξ d S ,
δ 2 U = [ T 2 ( δ ξ ) ε 0 V A 2 ( d A ξ ) 3 δ ξ ε 0 V T 2 ( d T + ξ ) 3 δ ξ ] δ ξ d S .
F ( ξ ) ε 0 V A 2 ( d A ξ ) 3 + ε 0 V T 2 ( d T + ξ ) 3 .
δ 2 U = T 2 ( δ ξ ) δ ξ d S F ( ξ ) ( δ ξ ) 2 d S δ 2 U T + δ 2 U E .
δ ξ = ν , n b ν n ζ ν n .
δ ξ = ν , n b ¯ ν n ζ ¯ ν n .
δ 2 U T = [ T 2 ( ν , n b ν n ζ ν n ) ν , n b ¯ ν n ζ ¯ ν n ] d S .
2 ζ ν n = ( x ν n a ) 2 ζ ν n .
δ 2 U T = [ T ( ν , n b ν n ( x ν n a ) 2 ζ ν n ) ν , n b ¯ ν n ζ ¯ ν n ] d S ,
δ 2 U T = T ν , ν , n , n ( x ν n a ) 2 b ν n b ¯ ν n ζ ν n ζ ¯ ν n d S .
δ 2 U T = T ν , ν , n , n ( x ν n a ) 2 b ν n b ¯ ν n δ νν δ n n ,
δ 2 U T = T ν , n ( x ν n a ) 2 b ν n b ¯ ν n .
δ 2 U E = F ( ξ ) ν , n b ν n ζ ν n ν , n b ¯ ν n ζ ¯ ν n d S .
δ 2 U = b ˜ Ω b ˜ ,
δ 2 U E = F ( ξ ) j b j ζ j j b ¯ j ζ ¯ j d S .
δ 2 U E = j , j b j b ¯ j A j j ,
A j j F ( ξ ) ζ j ζ ¯ j d S .
δ 2 U T = T j ( x j a ) 2 b j b ¯ j .
δ 2 U T = j , j T ( x j a ) 2 b j b ¯ j δ j j ,
δ 2 U = δ 2 U T + δ 2 U E = T j , j ( x j a ) 2 b j b ¯ j δ j j j , j b j b ¯ j A j j ,
δ 2 U = j , j [ T ( x j a ) 2 δ j j A j j ] b j b ¯ j .
Ω [ T ( x 1 a ) 2 A 11 A 1 M A M 1 T ( x M a ) 2 A M M ] .
δ 2 U = j = 1 M λ j b j 2 .
V k     2 = 2 ( d A ξ k ) 2 ε 0 [ ε 0 V T     2 ( d T + ξ k ) 2 T 2 ξ k ] .
F ( r k ) = ε 0 V k     2 ( d A ξ k ) 3 + ε 0 V T     2 ( d T + ξ k ) 3 .
A j j = k F ( r k ) ζ j ( r k ) ζ j ( r k ) Δ s k .
U E = 1 2 ε 0 E I 2 d V I + 1 2 ε 0 E II 2 d V II .
E I = V T M ( d T + ξ ) ,
E II = V A M ( d 0 ξ ) .
U E = ε 0 2 V A 2 d A ξ  d S + ε 0 2 V T 2 d T + ξ  d S .
ε 0 V A 2 2 ( d A ξ )  d S ε 0 2 k V k 2 d A ξ k Δ s k ,
U E = ε 0 2 k V k 2 d A ξ k Δ s k + ε 0 2 V T 2 d T + ξ  d S .
U T = 1 2 T ( ξ ) 2 d S .
U = [ ε 0 2 V A 2 d A ξ + ε 0 2 V T 2 d T + ξ + 1 2 T ( ξ ) 2 ] d S .
δ U = U ( ξ + δ ξ ) U ( ξ ) ,
U ( ξ + δ ξ ) = [ ε 0 2 V A 2 d A ξ + δ ξ + ε 0 2 V T 2 d T + ξ + δ ξ + 1 2 T ( ξ + δ ξ ) ( ξ + δ ξ ) ] d S .
δ U = [ T 2 ξ 1 2 ε 0 V A 2 ( d A ξ ) 2 + 1 2 ε 0 V T 2 ( d T + ξ ) 2 ] δ ξ d S .
T 2 ξ = ε 0 V A 2 2 ( d A ξ ) 2 + ε 0 V T 2 2 ( d T + ξ ) 2 .
2 ζ ν n = λ ν n 2 ζ ν n ,
λ ν n = x ν n a ,
ζ ν n = 1 a π | J ν + 1 ( x ν n ) | J ν ( x ν n ρ a ) e i νϕ .
ζ ν n ζ ¯ ν n d S = δ νν δ n n ,

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