Abstract

Analysis of the response function for various possible bimorphtype deformable mirrors is presented by numerical calculation. Using this response function, sensitivity comparison of a five-layer bimorph with two- and three-layer structures is carried out. It is shown that the displacement of a bimorph deformable mirror surface reduces when the number of layers increases, and the displacement is closely related to the distance of the control layer from the median plane. The farther electrode contributes the larger deformation on the surface. Furthermore, for a fixed position of the control layer, the displacement is directly proportional to the loading voltage and inversely proportional to the square of the thickness of the PZT layer, and it increases about 1.4 times when the area of the electrode doubles.

© 2007 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  12. S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (McGraw- Hill, 1959).

2007

A. V. Kudryashov and A. Sobolev, "Novel development of tiny bimorph mirrors," Proc. SPIE 6467, 1-8 (2007).

2005

G. Vdovin, M. Loktev, and A. Simonov, "Low-cost deformable mirrors: technologies and goals," Proc. SPIE 5894, 1-10 (2005).

Z. Jin and J. Zhang, "Control of filamentation induced by femtosecond laser pulses propagating in air," Opt. Express 13, 10424-10430 (2005).
[CrossRef] [PubMed]

P. Wnuk, C. Radzewicz, and J. Krasiñski, "Bimorph piezo deformable mirror for femtosecond pulse shaping," Opt. Express 13, 4154- 4159 (2005).
[CrossRef] [PubMed]

1998

1996

A. V. Kudryashov and V. I. Shmalhausen, "Semipassive bimorph flexible mirrors for atmospheric adaptive optics applications," Opt. Eng. 35, 3064-3073 (1996).
[CrossRef]

1994

1983

1979

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

A. V. Kudryashov and V. I. Shmalhausen, "Semipassive bimorph flexible mirrors for atmospheric adaptive optics applications," Opt. Eng. 35, 3064-3073 (1996).
[CrossRef]

Opt. Express

Proc. SPIE

G. Vdovin, M. Loktev, and A. Simonov, "Low-cost deformable mirrors: technologies and goals," Proc. SPIE 5894, 1-10 (2005).

A. V. Kudryashov and A. Sobolev, "Novel development of tiny bimorph mirrors," Proc. SPIE 6467, 1-8 (2007).

Other

E. M. Ellis, "Low-cost bimorph mirrors in adaptive optics," Ph.D. dissertation (University of London, 1999).

S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (McGraw- Hill, 1959).

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Figures (7)

Fig. 1.
Fig. 1.

Various-layer bimorphs in cross section. (a) two-layer, (b) three-layer, (c) multilayer (n≥4). Light-colored layer indicates optical polished passive layer (e.g., glass); deep-colored layer indicates PZT.

Fig. 2.
Fig. 2.

(a) Illustration of point voltage loading; (b) region of sectorial electrode; (c) electrodes distribution of a bimorph with 17 elements.

Fig. 3.
Fig. 3.

Cross section of three kinds of bimorphs with indication of control electrodes.

Fig. 4.
Fig. 4.

Variation of f(k) as a function of thickness ratio k for three kinds of bimorphs.

Fig. 5.
Fig. 5.

Cross profile of the response function of electrode number 1 in Fig. 2(c) at different control layer for various-layer bimorphs.

Fig. 6.
Fig. 6.

Schematic plan of the position of the median plane for various bimorphs.

Fig. 7.
Fig. 7.

Cross profile of the response function of electrode number 1 in Fig. 2(c), calculated with adjusted thickness of the passive layer.

Equations (26)

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4 W ( r , θ ) = Γ 2 V ( r , θ ) ,
W ( r , θ ) = n = 0 w n cos ( n ( θ ϕ ) ) ,
{ w 0 = A 0 + B 0 r 2 + C 0 log r + D 0 r 2 log r w 1 = A 1 r + B 1 r 3 + C 1 r 1 + D 1 r log r w n = A n r n + B n r n + 2 + C n r n + D n r ( 2 n ) , ( n 2 ) .
w n r = ρ = w n r = ρ , d w n dr r = ρ = d w n dr r = ρ , d 2 w n d r 2 r = ρ = d 2 w n d r 2 r = ρ ( n 0 ) .
( Q r Q r ) r = ρ = ( 1 2 + n = 1 cos ( n ( θ ϕ ) ) ) P π ρ a ,
A 0 = 1 16 D π ( a 2 P + a 2 P ρ 2 2 a 2 P ρ 2 Log ρ ) ;
B 0 = 1 16 D π ( a 2 P + a 2 P ρ 2 2 a 2 P Log ρ ) ;
A 1 = 1 8 D π ( a 2 P ρ a 2 P ρ 3 + 2 a 2 P ρ Log ρ ) ; B 1 = a 2 P 2 a 2 P ρ 2 + a 2 P ρ 4 16 D π ρ ;
A n = a 2 P ρ n ( ρ 2 n ρ 2 n n ρ 2 + 2 n + n ρ 2 + 2 n ) 8 D ( n 1 ) n π ; B n = a 2 P ρ n ( 1 ρ 2 n n ρ 2 n + n ρ 2 + 2 n ) 8 D n ( n + 1 ) π ;
C 0 = D 0 = C 1 = D 1 = C n = D n = 0 ;
A 0 = a 2 P + a 2 P ρ 2 16 D π ; B 0 = a 2 P + a 2 P ρ 2 16 D π ; C 0 = a 2 P ρ 2 8 D π ; D 0 = a 2 P 8 D π ;
A 1 = a 2 P ρ a 2 P ρ 3 8 D π ; B 1 = 2 a 2 P ρ + a 2 P ρ 2 16 D π ; C 1 = a 2 P ρ 3 16 D π ; D 1 = a 2 P ρ 4 D π ;
A n = a 2 P ρ n ( n + ρ 2 n ρ 2 ) 8 D ( n 1 ) n π ; B n = a 2 P ρ n ( n + 1 n ρ 2 ) 8 D ( n + 1 ) n π ;
C n = a 2 P ρ n + 2 8 D ( n + 1 ) n π ; D n = a 2 P ρ n 8 D ( n 1 ) n π ; ( n 2 ) .
W ( r , θ ; ρ , ϕ ) = { n = 0 w n cos ( n ( θ ϕ ) ) , 0 r ρ n = 0 w ' n cos ( n ( θ ϕ ) ) , ρ r 1 .
n = 0 : W 0 ( r ρ 1 ) = ( ρ w 0 ρ ρ = ρ 2 ρ w 0 ρ ρ = ρ 1 ) ( ϕ 2 ϕ 1 ) ,
W 0 ( ρ 1 r ρ 2 ) = ( ρ w 0 ρ ρ = ρ 2 ρ w ' 0 ρ ρ = ρ 1 ) ( ϕ 2 ϕ 1 ) ,
W 0 ( ρ 2 r ) = ( ρ w ' 0 ρ ρ = ρ 2 ρ w ' 0 ρ ρ = ρ 1 ) ( ϕ 2 ϕ 1 ) ,
n 1 : W n ( r ρ 1 ) = ( ρ w n ρ ρ = ρ 2 ρ w n ρ = ρ 1 ) ( 1 n ( sin ( n ( θ ϕ 1 ) ) sin ( n ( θ ϕ 2 ) ) ) )
n sin ( n ( θ ϕ 1 ) ) ρ 1 ρ 2 w n ρ d ρ + n sin ( n ( θ ϕ 2 ) ) ρ 1 ρ 2 w n ρ d ρ .
W n ( ρ 1 r ρ 2 ) = ( ρ w n ρ ρ = ρ 2 ρ w n ρ ρ = ρ 1 ) ( 1 n ( sin ( n ( θ ϕ 1 ) ) sin ( n ( θ ϕ 2 ) ) ) )
( ρ 1 r w n ρ d ρ + r ρ 2 w n ρ d ρ ) n sin ( n ( θ ϕ 1 ) ) + ( ρ 1 r w n ρ d ρ + r ρ 2 w n ρ d ρ ) n sin ( n ( θ ϕ 2 ) ) ,
W n ( ρ 2 r ) = ( ρ w n ρ ρ = ρ 2 ρ w n ρ ρ = ρ 1 ) ( 1 n ( sin ( n ( θ ϕ 1 ) ) sin ( n ( θ ϕ 2 ) ) ) )
n sin ( n ( θ ϕ 1 ) ) ρ 1 ρ 2 w n ρ d ρ + n sin ( n ( θ ϕ 2 ) ) ρ 1 ρ 2 w n ρ d ρ .
W = { W 0 ( r ) + W 1 ( r ) + n = 2 n max W n ( r ) ( r ρ 1 ) W 0 ( r ) + W 1 ( r ) + n = 2 n max W n ( r ) ( ρ 1 r ρ 2 ) W 0 ( r ) + W 1 ( r ) + n = 2 n max W n ( r ) ( ρ 2 r ) ,
Δ = ( j = 1 n E j ( ( i = 1 j t i ) 2 ( i = 0 j 1 t i ) 2 ) ) ( 2 j = 1 n E j t j ) , t 0 = 0 ,

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