Abstract

The deformation properties of an adaptive optical element made from a piezoelectric bimorph plate are analyzed. The fundamental relationship between the deformation of the optical surface and the voltage distribution applied across the thickness of the plate is derived and the general solution for an infinite plate is presented. A particular solution for a finite rectangular plate is also presented.

© 1979 Optical Society of America

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References

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  1. J. Opt. Soc. Am. 67, 269–422 (1977).
  2. J. E. Pearson, W. B. Bridges, S. Hansen, T. A. Nussmeir, and M. E. Pedinoff, “Coherent Optical Adaptive Techniques: Design and Performance of an 18-element Visible Multidither COAT System,” Appl. Opt. 15, 611 (1976).
    [Crossref] [PubMed]
  3. J. E. Pearson and S. Hansen, “Experimental studies of a deformablemirror adaptive optical system,” J. Opt. Soc. Am. 67, 325 (1977).
    [Crossref]
  4. J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic Piezoelectric Mirror for Wavefront Correction,” Appl. Phys. Lett. 25, 311 (1974).
    [Crossref]
  5. Richard Hudgin, “Wave-front compensation error due to finite corrector-element size,” J. Opt. Soc. Am. 67, 393 (1977).
    [Crossref]
  6. E. Bin-Nun and F. Dothan-Deutsch, “Mirror with Adjustable Radius of Curvature,” Rev. Sci. Instrum. 44, 512 (1973).
    [Crossref]
  7. S. Mikoshiba and B. Ahlborn, “Laser Mirror with Variable Focal Length,” Rev. of Sci. Instrum. 44, 508 (1973).
    [Crossref]
  8. N. T. Adelman, “Spherical Mirror with Piezoelectrically Controlled Curvature,” Appl. Opt. 16, 3075 (1977).
    [Crossref] [PubMed]
  9. Ronald P. Grosso and Martin Yellin, “The membrane mirror as an adaptive optical element,” J. Opt. Soc. Am. 67, 399 (1977).
    [Crossref]
  10. V. Wang and T. R. O’Meara, personal communication.
  11. Physical Acoustics Vol. I, Part A, edited by W. P. Mason, (Academic, New York, 1964).
  12. B. A. Auld, Acoustic Fields and Waves in Solids, (Wiley, New York, 1973).
  13. W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, (Van Nostrand, Princeton, 1950).
  14. Most present manufacturing techniques employ uniaxial stretching prior to poling PVF2 in order to enhance the piezoelectric properties. This produces a biased piezoelectric response in the stretched direction (i.e., h31 ≠ h32) as well as possible mechanical inhomogeneity in the x,y plane. Biaxial or perhaps even radial stretching would be necessary to achieve the assumed properties, (i) and (iii). Both of these techniques are technically feasible, but there has not been a need to produce such a material for the present applications involving PVF2. Another possibility would be not to stretch the material at all, but this would probably reduce the magnitude of the piezoelectric constants.
  15. J. H. Shames, Mechanics of Deformable Solids, (Prentice Hall, Englewood Cliffs, N.J., 1964).
  16. L. D. Landau and E. M. Litshitz, Theory of Elasticity, (Addison-Wesley, Reading, Mass.1959).
  17. G. Arfken, Mathematical Methods for Physicists, (Academic, New York, 1971).
  18. H. Ohigashi, R. Shigenari, and M. Yokota, “Light Modulation by Ultrasonic Waves from Piezoelectric Polyvinylidene Fluoride Films,” Jpn. J. Appl. Phys.,  14, No. 7 (1975).
    [Crossref]
  19. P. M. Morse and H. Feshback, Methods of Theoretical Physics, (McGraw-Hill, New York, 1953).

1977 (5)

1976 (1)

1975 (1)

H. Ohigashi, R. Shigenari, and M. Yokota, “Light Modulation by Ultrasonic Waves from Piezoelectric Polyvinylidene Fluoride Films,” Jpn. J. Appl. Phys.,  14, No. 7 (1975).
[Crossref]

1974 (1)

J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic Piezoelectric Mirror for Wavefront Correction,” Appl. Phys. Lett. 25, 311 (1974).
[Crossref]

1973 (2)

E. Bin-Nun and F. Dothan-Deutsch, “Mirror with Adjustable Radius of Curvature,” Rev. Sci. Instrum. 44, 512 (1973).
[Crossref]

S. Mikoshiba and B. Ahlborn, “Laser Mirror with Variable Focal Length,” Rev. of Sci. Instrum. 44, 508 (1973).
[Crossref]

Adelman, N. T.

Ahlborn, B.

S. Mikoshiba and B. Ahlborn, “Laser Mirror with Variable Focal Length,” Rev. of Sci. Instrum. 44, 508 (1973).
[Crossref]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, (Academic, New York, 1971).

Auld, B. A.

B. A. Auld, Acoustic Fields and Waves in Solids, (Wiley, New York, 1973).

Bin-Nun, E.

E. Bin-Nun and F. Dothan-Deutsch, “Mirror with Adjustable Radius of Curvature,” Rev. Sci. Instrum. 44, 512 (1973).
[Crossref]

Bridges, W. B.

Cone, P. F.

J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic Piezoelectric Mirror for Wavefront Correction,” Appl. Phys. Lett. 25, 311 (1974).
[Crossref]

Dothan-Deutsch, F.

E. Bin-Nun and F. Dothan-Deutsch, “Mirror with Adjustable Radius of Curvature,” Rev. Sci. Instrum. 44, 512 (1973).
[Crossref]

Feinleib, J.

J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic Piezoelectric Mirror for Wavefront Correction,” Appl. Phys. Lett. 25, 311 (1974).
[Crossref]

Feshback, H.

P. M. Morse and H. Feshback, Methods of Theoretical Physics, (McGraw-Hill, New York, 1953).

Grosso, Ronald P.

Hansen, S.

Hudgin, Richard

Landau, L. D.

L. D. Landau and E. M. Litshitz, Theory of Elasticity, (Addison-Wesley, Reading, Mass.1959).

Lipson, S. G.

J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic Piezoelectric Mirror for Wavefront Correction,” Appl. Phys. Lett. 25, 311 (1974).
[Crossref]

Litshitz, E. M.

L. D. Landau and E. M. Litshitz, Theory of Elasticity, (Addison-Wesley, Reading, Mass.1959).

Mason, W. P.

W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, (Van Nostrand, Princeton, 1950).

Mikoshiba, S.

S. Mikoshiba and B. Ahlborn, “Laser Mirror with Variable Focal Length,” Rev. of Sci. Instrum. 44, 508 (1973).
[Crossref]

Morse, P. M.

P. M. Morse and H. Feshback, Methods of Theoretical Physics, (McGraw-Hill, New York, 1953).

Nussmeir, T. A.

O’Meara, T. R.

V. Wang and T. R. O’Meara, personal communication.

Ohigashi, H.

H. Ohigashi, R. Shigenari, and M. Yokota, “Light Modulation by Ultrasonic Waves from Piezoelectric Polyvinylidene Fluoride Films,” Jpn. J. Appl. Phys.,  14, No. 7 (1975).
[Crossref]

Pearson, J. E.

Pedinoff, M. E.

Shames, J. H.

J. H. Shames, Mechanics of Deformable Solids, (Prentice Hall, Englewood Cliffs, N.J., 1964).

Shigenari, R.

H. Ohigashi, R. Shigenari, and M. Yokota, “Light Modulation by Ultrasonic Waves from Piezoelectric Polyvinylidene Fluoride Films,” Jpn. J. Appl. Phys.,  14, No. 7 (1975).
[Crossref]

Wang, V.

V. Wang and T. R. O’Meara, personal communication.

Yellin, Martin

Yokota, M.

H. Ohigashi, R. Shigenari, and M. Yokota, “Light Modulation by Ultrasonic Waves from Piezoelectric Polyvinylidene Fluoride Films,” Jpn. J. Appl. Phys.,  14, No. 7 (1975).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic Piezoelectric Mirror for Wavefront Correction,” Appl. Phys. Lett. 25, 311 (1974).
[Crossref]

J. Opt. Soc. Am. (4)

Jpn. J. Appl. Phys. (1)

H. Ohigashi, R. Shigenari, and M. Yokota, “Light Modulation by Ultrasonic Waves from Piezoelectric Polyvinylidene Fluoride Films,” Jpn. J. Appl. Phys.,  14, No. 7 (1975).
[Crossref]

Rev. of Sci. Instrum. (1)

S. Mikoshiba and B. Ahlborn, “Laser Mirror with Variable Focal Length,” Rev. of Sci. Instrum. 44, 508 (1973).
[Crossref]

Rev. Sci. Instrum. (1)

E. Bin-Nun and F. Dothan-Deutsch, “Mirror with Adjustable Radius of Curvature,” Rev. Sci. Instrum. 44, 512 (1973).
[Crossref]

Other (9)

P. M. Morse and H. Feshback, Methods of Theoretical Physics, (McGraw-Hill, New York, 1953).

V. Wang and T. R. O’Meara, personal communication.

Physical Acoustics Vol. I, Part A, edited by W. P. Mason, (Academic, New York, 1964).

B. A. Auld, Acoustic Fields and Waves in Solids, (Wiley, New York, 1973).

W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, (Van Nostrand, Princeton, 1950).

Most present manufacturing techniques employ uniaxial stretching prior to poling PVF2 in order to enhance the piezoelectric properties. This produces a biased piezoelectric response in the stretched direction (i.e., h31 ≠ h32) as well as possible mechanical inhomogeneity in the x,y plane. Biaxial or perhaps even radial stretching would be necessary to achieve the assumed properties, (i) and (iii). Both of these techniques are technically feasible, but there has not been a need to produce such a material for the present applications involving PVF2. Another possibility would be not to stretch the material at all, but this would probably reduce the magnitude of the piezoelectric constants.

J. H. Shames, Mechanics of Deformable Solids, (Prentice Hall, Englewood Cliffs, N.J., 1964).

L. D. Landau and E. M. Litshitz, Theory of Elasticity, (Addison-Wesley, Reading, Mass.1959).

G. Arfken, Mathematical Methods for Physicists, (Academic, New York, 1971).

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Figures (5)

FIG. 1
FIG. 1

Illustration of elementary piezoelectric effect. Large arrows indicate poling direction.

FIG. 2
FIG. 2

Illustration of bending thickness displacements due to piezoelectric effect in a bimorph configuration. Large arrows indicate poling directions.

FIG. 3
FIG. 3

Definition of coordinates and geometry of an infinite piezoelectric bimorph.

FIG. 4
FIG. 4

Displacement sensitivity of PVF2.

FIG. 5
FIG. 5

Geometry of finite rectangular plate.

Tables (1)

Tables Icon

TABLE I Values of material properties of PVF2

Equations (66)

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T i = C i j D S j h n i D n
E m = h m j S j + B m n S D n
( τ x x τ x y τ x z τ y x τ y y τ y z τ z x τ z y τ z z ) = ( T 1 T 4 T 6 T 4 T 2 T 5 T 6 T 5 T 3 )
( u x x u x y u x z u y x u y y u y z u z x u z y u z z ) = ( S 1 S 4 2 S 6 2 S 4 2 S 2 S 5 2 S 6 2 S 5 2 S 3 )
T 1 = ( 2 G + λ ) S 1 + λ S 2 + λ S 3 h 31 D 3 ,
T 2 = λ S 1 + ( 2 G + λ ) S 2 + λ S 3 h 31 D 3 ,
T 3 = λ ( S 1 + S 2 ) + κ S 3 h 33 D 3 ,
T 4 = G S 4 ,
T 5 = G S 5 ,
T 6 = G S 6 ,
S 3 = ( h 33 / κ ) D 3 ( λ / κ ) ( S 1 + S 2 ) .
T 1 = ( 2 G + λ ( λ ) 2 κ ) S 1 + ( λ ( λ ) 2 κ ) S 2 + ( h 33 λ κ h 31 ) D 3 ,
T 2 = ( 2 G + λ ( λ ) 2 κ ) S 2 + ( λ ( λ ) 2 κ ) S 1 + ( h 33 λ κ h 31 ) D 3 .
u z ( x , y , z ) = ψ 0 ( x , y ) + ψ 1 ( x , y , z ) ,
u x ( x , y , z ) = z ψ 0 x ,
u y ( x , y , z ) = z ψ 0 y .
S 1 u x x = z 2 ψ 0 x 2 ,
S 2 u y y = z 2 ψ 0 y 2 ,
S 3 u z z = ψ 1 z ,
S 4 u x y + u y x = 2 z 2 ψ 0 x y ,
S 5 u y z + u z y = ψ 1 y ,
S 6 u x z + u z x = ψ 1 x .
T 1 x + T 4 y + T 6 z = 0 ,
T 4 x + T 2 y + T 5 z = 0 ,
T 6 x + T 5 y + T 3 z = 0 .
x [ z ( 2 G + λ ( λ ) 2 κ G λ κ ) ¯ 2 ψ 0 + ( h 33 κ ( λ + G ) h 31 ) D 3 ] = 0 ,
y [ z ( 2 G + λ ( λ ) 2 κ G λ κ ) ¯ 2 ψ 0 + ( h 33 κ ( λ + G ) h 31 ) D 3 ] = 0 ,
¯ 2 ψ 1 = 0 .
¯ 2 2 x 2 + 2 y 2 .
f ( z ) z ( 2 G + λ ( λ ) 2 κ G λ κ ) ¯ 2 ψ 0 + ( h 33 κ ( λ + G ) h 31 ) D 3 .
E 3 = h 31 ( S 1 + S 2 ) h 33 S 3 + β 33 S D 3 .
D 3 = ( β 33 S h 33 2 κ ) 1 ( E 3 z ( h 31 h 33 λ κ ) ¯ 2 ψ 0 ) .
E 3 = z α 0 ¯ 2 ψ 0 ( x , y ) ,
α 0 ( h 31 h 33 λ κ ) + ( h 33 κ ( λ + G ) h 31 ) 1 × ( 2 G + λ ( λ ) 2 κ G λ κ ) ( β 33 S h 33 2 κ ) .
ϕ ( x , y ) = h / 2 h / 2 d z E 3 ,
¯ 2 ψ 0 ( x , y ) = ( 4 / α 0 h 2 ) ϕ ( x , y ) .
ψ 1 z = z α 1 2 ψ 0 ( x , y ) ,
α 1 1 κ ( λ + h 33 [ κ ( 2 G + λ ) ( λ ) 2 G λ ] h 33 ( G + λ ) h 31 κ ) .
ψ 1 ( x , y , z ) = 0 z d z ψ z + ψ 1 ( x , y , 0 ) = z 2 2 α 1 ¯ 2 ψ 0 .
ψ 1 ( x , y , z ) = ( 2 z 2 α 1 / α 0 h 2 ) ϕ ( x , y ) .
u z ( x , y , h / 2 ) = ψ 0 ( x , y ) ( α 1 / 2 α 0 ) ϕ ( x , y ) ,
¯ 2 G ( x , x ) = δ ( x x ) .
ψ 0 ( x ) = 4 α 0 h 2 d 2 x G ( x , x ) ϕ ( x ) ,
u 2 ( x , h 2 ) = α 1 2 α 0 ϕ ( x ) + 4 α 0 h 2 d 2 x G ( x , x ) ϕ ( x ) .
G ( x , x ) = 1 2 π ln | x x | .
1 2 π d 2 k U z ( k ) e i k x = α 1 4 π α 0 d 2 k Φ ( k ) e i k x + 4 ( 2 π ) 2 α 0 h 2 d 2 x d 2 k G ( k ) e i k ( x x ) × d 2 k Φ ( k ) e i k x ,
k k x î + k y ĵ .
δ ( k k ) 1 ( 2 π ) 2 d 2 x e i ( k k ) x .
1 2 π d 2 k U z ( k ) e i k x = α 1 4 π α 0 d 2 k Φ ( k ) e i k x + 4 α 0 h 2 d 2 k G ( k ) Φ ( k ) e i k x .
U z ( k ) = Φ ( k ) [ ( 8 π / α 0 h 2 ) G ( k ) α 1 / 2 α 0 ]
G ( k ) = 1 / 2 π k 2 ,
U z ( k ) = Φ ( k ) ( 4 / α 0 h 2 k 2 α 1 / 2 α 0 ) .
ψ 0 ( x , y ) 0 if { x = 0 , L x or y = 0 , L y .
G 1 ( x , x ) n = 1 m = 1 S n m ( x ) S n m ( x ) k n m 2 ,
k n m 2 ( π ) 2 ( n 2 / L x 2 + m 2 / L y 2 ) ,
S n m ( x ) ( 4 / L x L y ) 1 / 2 sin [ ( π n / L x ) x ] sin [ ( π m / L y ) y ] .
ψ 0 ( x ) = 4 α 0 h 2 0 L x d x 0 L y d y G 1 ( x , x ) ϕ ( x ) .
ψ 0 ( x ) = n = 1 m = 1 Ψ n m S n m ( x ) ,
Ψ n m 4 α 0 h 2 0 L x d x 0 L y d y S n m ( x ) ϕ ( x ) k n m 2 .
2 ψ 0 = n = 1 m = 1 k n m 2 Ψ n m S n m ( x ) .
ϕ ( x ) = α 0 h 2 4 n = 1 m = 1 k n m 2 Ψ n m S n m ( x ) .
ϕ ( x ) = n = 1 m = 1 Φ n m S n m ( x ) ,
Φ n m = ( α 0 h 2 k n m 2 / 4 ) Ψ n m .
U z ( x , h 2 ) = n = 1 m = 1 U n m S n m ( x ) .
U n m = Φ n m ( 4 / α 0 h 2 k n m 2 α 1 / 2 α 0 ) .
β 11 S = β 22 S = β 33 S