Abstract

A 1-D deformable mirror which uses multilayered PVDF piezoelectric films as the actuators is proposed. First, a set of desired wave fronts is considered and the electrode shape of each film is designed so that the film generates one component of the set. In actual operation of the mirror a combination of the supply voltages is used to control the surface profile. The principle of construction and several basic experimental results are presented.

© 1982 Optical Society of America

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References

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  1. J. W. Hardy, Proc. IEEE 66, 651 (1978).
    [CrossRef]
  2. T. R. O’Meara, J. Opt. Soc. Am. 67, 306 (1977).
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  3. T. R. O’Meara, J. Opt. Soc. Am. 67, 318 (1977).
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  4. J. E. Pearson, S. Hansen, J. Opt. Soc. Am. 67, 325 (1977).
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  5. E. Steinhaus, S. G. Lipson, J. Opt. Soc. Am. 69, 478 (1979).
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  6. N. T. Adelman, Appl. Opt. 16, 3075 (1977).
    [CrossRef] [PubMed]
  7. S. A. Kokorowski, J. Opt. Soc. Am. 69, 181 (1979).
    [CrossRef]
  8. T. Sato, H. Ishida, O. Ikeda, Appl. Opt. 19, 1430 (1980).
    [CrossRef] [PubMed]
  9. K. Uchino, Y. Tsuchiya, S. Nomura, T. Sato, H. Ishikawa, O. Ikeda, Appl. Opt. 20, 3077 (1981).
    [CrossRef] [PubMed]
  10. “Technical Data of Kureha KF Piezo Film,” Kureha Chemical Industry Co., Ltd., Tokyo, 1975.

1981 (1)

1980 (1)

1979 (2)

1978 (1)

J. W. Hardy, Proc. IEEE 66, 651 (1978).
[CrossRef]

1977 (4)

Adelman, N. T.

Hansen, S.

Hardy, J. W.

J. W. Hardy, Proc. IEEE 66, 651 (1978).
[CrossRef]

Ikeda, O.

Ishida, H.

Ishikawa, H.

Kokorowski, S. A.

Lipson, S. G.

Nomura, S.

O’Meara, T. R.

Pearson, J. E.

Sato, T.

Steinhaus, E.

Tsuchiya, Y.

Uchino, K.

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Figures (9)

Fig. 1
Fig. 1

Schematic construction of MDM consisting of several PVDF piezoelectric films that are bonded to the mirror glass plate: tg is the thickness of the glass plate; ti, i = 1,2, …,n are the thicknesses of the PVDF films; Wg is the width of MDM, Wi(x), i = 1,2, …,n are the integrated widths of the electrodes along the y axis. The electrodes are distributed to achieve 1-D deformation.

Fig. 2
Fig. 2

Cross sections of (a) bimorphous and (b) multilayered deformable mirrors, where tp is the thickness of the PVDF film, a is the distance of the neutral plane from the mirror surface, lx is the width of the MDM in the x direction, and si, i = 1,2, …,n are the thicknesses of the adhesive layers.

Fig. 3
Fig. 3

MDM used in experiments, which consists of double PVDF film actuators with square and triangular electrodes.

Fig. 4
Fig. 4

Experimental results when only the PVDF film with the square electrode of the MDM shown in Fig. 3 was actuated. The upper part shows the interferometric fringe patterns representing the deformations and the lower part shows the magnitudes of the deformations in terms of the coefficient C1(V1) calculated by C1(V1) = f1(x;V1)/x2. ● and — show the experimentally obtained and the theoretically expected values, respectively.

Fig. 5
Fig. 5

Experimental results when only the PVDF film with the triangular electrode of the MDM shown in Fig. 3 was actuated. C2(V2) is obtained from the relation C2(V2) = f2(x;V2)/x3. ● and — show the experimentally obtained and the theoretically expected values, respectively.

Fig. 6
Fig. 6

Experimental results showing the superposition of the deformations of the MDM shown in Fig. 3, where λ is the wavelength (0.6328 μm).

Fig. 7
Fig. 7

Efficiency of deformation for the MDM consisting of triple PVDF film actuators with square electrodes: ρ is the radius of curvature of the mirror surface; ●, ○, × show the experimentally obtained values; and — shows the theoretically expected values.

Fig. 8
Fig. 8

Superposition of deformations for the same MDM as in Fig. 7: ● shows the experimental values; and — shows the quadratic curves in x obtained by the least-mean-square error law.

Fig. 9
Fig. 9

Effects of the shape of the electrode on the deformation where W(x) is unchanged from (a) to (d).

Tables (1)

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Table I Specifications of the Materials Used to Build PVDF MDMs

Equations (24)

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U 1 = ( ½ ) - a t g - a i j k l C i j k l ɛ i j ɛ k l g d z + ( ½ ) t g - a t g - a + t p i j k l C i j k l p ɛ i j ɛ k l d z ,
ɛ 11 = z ( 2 f / x 2 ) ,             ɛ 22 = z ( 2 f / y 2 ) ,             ɛ 12 = ɛ 21 = z ( 2 f / x y ) ,             C i i i i = C 1 ,             C i i j j = C 2 ,             C i j i j = C 1 - C 2 ,
C 1 p / C 1 g C 2 p / C 2 g ,
U 1 ( K 1 / 2 ) { C 1 g [ ( 2 f / x 2 ) 2 + ( 2 f / y 2 ) 2 ] + 2 C 2 g ( 2 f / x 2 ) ( 2 f / y 2 ) + ( C 1 g - C 2 g ) ( 2 f / y 2 ) 2 } ,
K 1 = { a 3 - ( a - t g ) 3 + ( C 1 p / C 1 g ) [ ( a - t g ) 3 - ( a - t g - t p ) 3 ] } / 3 ,
U 2 = t g - a t g - a + t p i j k d i j k σ j k E i d z = K 2 C 1 p V ( x , y ) [ d 31 ( 2 f / x 2 ) + d 32 ( 2 f / y 2 ) ] ,
K 2 = t g - a + t p / 2 ,
U = U 1 + U 2 .
A O [ 4 f / x 4 + 2 ( 4 f / x 2 y 2 ) + 4 f / y 4 ] + B O [ d 31 ( 2 V / x 2 ) + d 32 ( 2 V / y 2 ) ] = 0 ,
A O = { C 1 g [ a 3 - ( a - t g ) 3 ] + C 1 p [ ( t g - a + t p ) 3 - ( t g - a ) 3 ] } / 3 ,
B O = C 1 p ( t g - a + t p / 2 ) .
A [ 4 f i / x 4 + 2 ( 4 f i / x 2 y 2 ) + 4 f i / y 4 ] + B i [ d 31 ( 2 V i / x 2 ) + d 32 ( 2 V i / y 2 ) ] = 0 ,             i = 1 , 2 , , n ,
A = ( ) { C 1 g [ a 3 - ( a - t g ) 3 ] + C 1 p i = 1 n [ ( t g - a + s 1 + t 1 + + s i + t i ) 3 - ( t g - a + s 1 + t 1 + + s i ) 3 ] + C 1 a i = 1 n [ ( t g - a + s 1 + t 1 + + t i - 1 + s i ) 3 - ( t g - a + s 1 + t 1 + + t i - 1 ) 3 ] } ,
B i = C 1 p ( t g - a + s 1 + t 1 + + s i + t i / 2 ) ,
a = { E g t g 2 + E p i = 1 n [ ( t g + s 1 + t 1 + + s i + t i ) 2 - ( t g + s 1 + t 1 + + s i ) 2 ] + E a i = 1 n [ ( t g + s 1 + t 1 + + t i - 1 + s i ) 2 - ( t g + s 1 + t 1 + + t i - 1 ) 2 ] } / 2 [ E g t g + i = 1 n ( E p t i + E a s i ) ] ,
f ( x , y ) = i = 1 n f i ( x , y ) .
A ( d 4 f i / d x 4 ) + d 31 B i ( d 2 V i / d x 2 ) = 0 ,             i = 1 , 2 , , n .
f i ( x ) = 0 for V i ( x ) = 0 ,             i = 1 , 2 , , n ,
A [ d 2 f i ( x ) / d x 2 ] + B i d 31 V i ( x ) = 0 ,             i = 1 , 2 , , n .
V i ( x ) = V O i W i ( x ) / W g ,             i = 1 , 2 , , n .
U 2 = W g U 2 = C 1 p K 2 V i ( x ) W g d 31 [ d 2 f i ( x ) / d x 2 ] ,             i = 1 , 2 , , n .
A [ d 2 f i ( x ) / d x 2 ] + ( d 31 / W g ) B i V O i W i ( x ) = 0 ,             i = 1 , 2 , , n .
f i ( x ) = V O i u i ( x ) ,             i = 1 , 2 , , n ,
f ( x ) = i = 1 n V O i u i ( x ) .

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