Abstract

Proposed is an electro-optic (E-O) variable focal-length lens by combining EO material with numerous transparent fine electrodes on its surface. Coating indium tin oxide transparent electrodes on a lanthanum-modified lead zirconate titanate (9/65/35) ceramic plate and introducing a parabolic index profile with dc voltage, we successfully demonstrate a positive EO variable focal-length lens, obtaining its focal length from ∞ to 1 m with applied voltage up to 300 V to an electrode gap of 50 μm. Theoretical analysis is also given, together with a discussion for improving aberration and resolving power.

© 1991 Optical Society of America

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References

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  1. J. M. Hammer, “Digital electro-optic grating deflector and modulator,” Appl. Phys. Lett. 18, 147–149 (1971).
    [Crossref]
  2. H. Sato, K. Toda, “A new electrically controllable diffraction grating using polarization reflection,” J. Appl. Phys. 47, 4031–4032 (1976).
    [Crossref]
  3. H. Sato, K. Toda, “An application of Pb(Zr,Ti)O3 ceramic to opto-electronic devices,” Appl. Phys. 13, 25–28 (1977).
    [Crossref]
  4. T. Utsunomiya, H. Sato, “Electrically deformable diffraction grating using a piezoelectric material,” Ferroelectrics 27, 27–30 (1980).
    [Crossref]
  5. Y. Nishimoto, “Variable-focal-length lens using an electro-optic effect,” U.S. Patent4, 466,703 (21Aug.1984).
  6. H. Sato, T. Tatebayashi, T. Yamamoto, K. Hayashi, “Electro-optic lens composed of transparent electrodes on PLZT ceramic towards optoelectronic devices,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 493–494 (1990).
  7. For example, see, Crystal Optics, edited by Japanese Society of Applied Physics (Morikita, Tokyo, 1975), p. 217, in Japanese.
  8. T. Ogawa, Crystal Physics (Shōkabo, Tokyo, 1988), p. 47, in Japanese.
  9. W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974), p. 96.

1980 (1)

T. Utsunomiya, H. Sato, “Electrically deformable diffraction grating using a piezoelectric material,” Ferroelectrics 27, 27–30 (1980).
[Crossref]

1977 (1)

H. Sato, K. Toda, “An application of Pb(Zr,Ti)O3 ceramic to opto-electronic devices,” Appl. Phys. 13, 25–28 (1977).
[Crossref]

1976 (1)

H. Sato, K. Toda, “A new electrically controllable diffraction grating using polarization reflection,” J. Appl. Phys. 47, 4031–4032 (1976).
[Crossref]

1971 (1)

J. M. Hammer, “Digital electro-optic grating deflector and modulator,” Appl. Phys. Lett. 18, 147–149 (1971).
[Crossref]

Cathey, W. T.

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974), p. 96.

Hammer, J. M.

J. M. Hammer, “Digital electro-optic grating deflector and modulator,” Appl. Phys. Lett. 18, 147–149 (1971).
[Crossref]

Hayashi, K.

H. Sato, T. Tatebayashi, T. Yamamoto, K. Hayashi, “Electro-optic lens composed of transparent electrodes on PLZT ceramic towards optoelectronic devices,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 493–494 (1990).

Nishimoto, Y.

Y. Nishimoto, “Variable-focal-length lens using an electro-optic effect,” U.S. Patent4, 466,703 (21Aug.1984).

Ogawa, T.

T. Ogawa, Crystal Physics (Shōkabo, Tokyo, 1988), p. 47, in Japanese.

Sato, H.

T. Utsunomiya, H. Sato, “Electrically deformable diffraction grating using a piezoelectric material,” Ferroelectrics 27, 27–30 (1980).
[Crossref]

H. Sato, K. Toda, “An application of Pb(Zr,Ti)O3 ceramic to opto-electronic devices,” Appl. Phys. 13, 25–28 (1977).
[Crossref]

H. Sato, K. Toda, “A new electrically controllable diffraction grating using polarization reflection,” J. Appl. Phys. 47, 4031–4032 (1976).
[Crossref]

H. Sato, T. Tatebayashi, T. Yamamoto, K. Hayashi, “Electro-optic lens composed of transparent electrodes on PLZT ceramic towards optoelectronic devices,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 493–494 (1990).

Tatebayashi, T.

H. Sato, T. Tatebayashi, T. Yamamoto, K. Hayashi, “Electro-optic lens composed of transparent electrodes on PLZT ceramic towards optoelectronic devices,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 493–494 (1990).

Toda, K.

H. Sato, K. Toda, “An application of Pb(Zr,Ti)O3 ceramic to opto-electronic devices,” Appl. Phys. 13, 25–28 (1977).
[Crossref]

H. Sato, K. Toda, “A new electrically controllable diffraction grating using polarization reflection,” J. Appl. Phys. 47, 4031–4032 (1976).
[Crossref]

Utsunomiya, T.

T. Utsunomiya, H. Sato, “Electrically deformable diffraction grating using a piezoelectric material,” Ferroelectrics 27, 27–30 (1980).
[Crossref]

Yamamoto, T.

H. Sato, T. Tatebayashi, T. Yamamoto, K. Hayashi, “Electro-optic lens composed of transparent electrodes on PLZT ceramic towards optoelectronic devices,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 493–494 (1990).

Appl. Phys. (1)

H. Sato, K. Toda, “An application of Pb(Zr,Ti)O3 ceramic to opto-electronic devices,” Appl. Phys. 13, 25–28 (1977).
[Crossref]

Appl. Phys. Lett. (1)

J. M. Hammer, “Digital electro-optic grating deflector and modulator,” Appl. Phys. Lett. 18, 147–149 (1971).
[Crossref]

Ferroelectrics (1)

T. Utsunomiya, H. Sato, “Electrically deformable diffraction grating using a piezoelectric material,” Ferroelectrics 27, 27–30 (1980).
[Crossref]

J. Appl. Phys. (1)

H. Sato, K. Toda, “A new electrically controllable diffraction grating using polarization reflection,” J. Appl. Phys. 47, 4031–4032 (1976).
[Crossref]

Other (5)

Y. Nishimoto, “Variable-focal-length lens using an electro-optic effect,” U.S. Patent4, 466,703 (21Aug.1984).

H. Sato, T. Tatebayashi, T. Yamamoto, K. Hayashi, “Electro-optic lens composed of transparent electrodes on PLZT ceramic towards optoelectronic devices,” in Optics in Complex Systems, F. Lanzl, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 493–494 (1990).

For example, see, Crystal Optics, edited by Japanese Society of Applied Physics (Morikita, Tokyo, 1975), p. 217, in Japanese.

T. Ogawa, Crystal Physics (Shōkabo, Tokyo, 1988), p. 47, in Japanese.

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974), p. 96.

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Figures (8)

Fig. 1
Fig. 1

Model of the EO VFL lens with related coordinates.

Fig. 2
Fig. 2

Experimental setup: (a) schematic diagram and (b) configuration of a positive EO VFL lens.

Fig. 3
Fig. 3

Voltage distribution applied to the EO VFL lens.

Fig. 4
Fig. 4

Beamwidth variation as a function of the observed distance z with different optical wavelengths.

Fig. 5
Fig. 5

Beamwidth variation as a function of z with different electrode sizes.

Fig. 6
Fig. 6

Typical focused pattern observed near the focal point at λ = 515 nm, when the higher-order diffractions take place due to finite thickness of the ITO electrodes.

Fig. 7
Fig. 7

Electric-field distribution within the EO material.

Fig. 8
Fig. 8

Potential model utilized on the EO lens surface.

Tables (2)

Tables Icon

Table I Specifications of the EO Lenses

Tables Icon

Table II Focal Lengths Obtained from EO Lenses

Equations (46)

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E x ( x , z ) = V ( x , z ) / x = m = 1 m n = 1 P m ξ n cos { ξ n [ x + ( h / 2 ) ] } sinh ξ n ( l z ) sinh ( ξ n l ) ,
E z ( x , z ) = V ( x , z ) / z = m = 1 m n = 1 P m ξ n sin { ξ n [ x + ( h / 2 ) ] } cosh ξ n ( l z ) sinh ( ξ n l ) ,
ξ n = n π / h , n = 1 , 2 , 3 , .
Δ ( 1 n ijkl 2 ) = 1 2 k l R ijkl E k E l ( k , l = x , y , z ) ,
R i j = [ R 11 R 12 R 12 0 0 0 R 12 R 11 R 12 0 0 0 R 12 R 12 R 11 0 0 0 0 0 0 R 44 0 0 0 0 0 0 R 44 0 0 0 0 0 0 R 44 ]
x 2 n x 2 + y 2 n y 2 + z 2 n z 2 = 1 ,
n x = n 0 n 0 3 2 ( R 11 E x 2 + R 12 E z 2 ) ,
n y = n 0 n 0 3 2 R 12 ( E x 2 + E z 2 ) ,
n z = n 0 n 0 3 2 ( R 12 E x 2 + R 11 E z 2 ) ,
E x 2 ¯ = 1 l 0 l E x 2 ( x , z ) d z ,
E z 2 ¯ = 1 l 0 l E z 2 ( x , z ) d z .
u ( x 0 , y 0 ) = 2 π 1 w 0 exp ( x 0 2 + y 0 2 w 0 2 ) ,
n ( x 0 ) n 0 n 0 3 2 R 12 E z 2 ( x 0 ) .
n ( x 0 ) = n 0 + Δ n max [ 1 x 0 2 ( h / 2 ) 2 ] ,
Δ n max = n 0 3 2 R 12 E 0 2 = n 0 3 R 12 V 0 2 2 l 2 .
u ( x 0 ) = u ( x 0 ) exp [ i n ( x 0 ) k l ] = u ( x 0 ) exp [ i Δ n max k l x 0 2 ( h / 2 ) 2 ] exp [ i ( n 0 + Δ n max ) k l ] .
u ( x 0 ) = 2 π 1 w 0 exp [ i k 2 ( 1 q 0 1 f ) x 0 2 ] ,
f = h 2 8 Δ n max l .
| u ( x , y , z ) | = [ 2 π w x ( z ) w y ( z ) ] 1 / 2 exp [ x 2 w x 2 ( z ) y 2 w y 2 ( z ) ] ,
w x ( z ) = w 0 [ ( 1 z f ) 2 + ( λ z π w 0 2 ) 2 ] 1 / 2 ,
w y ( z ) = w 0 [ 1 + ( λ z π w 0 2 ) 2 ] 1 / 2
z min = ( 1 f + f q 0 2 ) 1 .
u ( x , z ) = 1 λ z exp [ i ( k z π 4 ) i k x 2 2 z ] × exp [ i k 2 ( 1 z 1 f ) x 0 2 + i k ( x x 0 z ) ] d x 0 .
u ( x , z ) = f f z exp [ ikz i k 2 x 2 ( z f ) ] .
w ( z ) = w 0 { ( 1 z f ) 2 + [ ( 1 z 0 f ) z + z 0 ] 2 ( λ π w 0 2 ) 2 } 1 / 2 ,
p ( x 0 ) = Π ( x 0 / h ) = { 1 h / 2 x 0 h / 2 0 otherwise
Δ x = 2 λ z / h .
Δ x = λ h / 4 Δ n max l ,
u ( x , y , z ) = 1 + exp ( i ϑ 1 ) 2 [ 2 π w x ( z ) w y ( z ) ] 1 / 2 exp { i [ k z ψ ( z ) + ψ 0 ( z ) 2 ] i k 2 [ x 2 R x ( z ) + y 2 R y ( z ) ] [ x 2 w x 2 ( z ) + y 2 w y 2 ( z ) ] } + 1 exp ( i ϑ 1 ) 2 [ 2 π w x ( z ) W y ( z ) ] 1 / 2 exp { i [ k z ψ ( z ) + ψ 0 ( z ) 2 ] } × m = sinc [ ( 2 m 1 ) π 2 ] exp { i k 2 [ x 2 + χ m ( z ) + ω m ( z ) R x ( z ) + y 2 R y ( z ) ] + [ 1 w x 2 ( z ) ( x + λ z 2 d m ) 2 y 2 w y 2 ( z ) ] } ,
θ m = tan 1 [ ( 2 m 1 ) λ 2 d ] .
V m ( x ) = { V L m V R m d x + V L m x R m V R m x L m d , x L m x x R m 0 otherwise
V m ( x , z ) = n = 1 sin ( ξ n x ) [ A n exp ( ξ n z ) + B n exp ( ξ n z ) ] ,
V m ( x , z ) = n = 1 P m sin [ ξ n ( x + h 2 ) ] sinh [ ξ n ( l z ) ] sin ( ξ n l ) ,
P m = 2 h 0 h V m ( x ) sin ( ξ n x ) d x = 2 h ( n π ) 2 V L m V R m d [ sin ( ξ n x R m ) sin ( ξ n x L m ) ] + 2 n π [ V L m cos ( ξ n x L m ) V R m cos ( ξ n x R m ) ] .
V ( x , z ) = m = 1 m V m ( x , z ) = m = 1 m n = 1 P m sin [ ξ n ( x + h 2 ) ] sinh [ ξ n ( l z ) ] sin ( ξ n l ) .
t 1 ( x 0 ) = 1 2 + 1 2 m = sinc [ ( 2 m 1 ) π 2 ] exp ( i k 2 λ d m ) x 0 ,
t 2 ( x 0 ) = 1 2 1 2 m = sinc [ ( 2 m 1 ) π 2 ] exp ( i k 2 λ d m ) x 0 ,
ϑ 1 = ( n 1 1 ) k l 1 .
t total ( x 0 ) = t 1 ( x 0 ) + t 2 ( x 0 ) exp ( i ϑ 1 )
u ( x 0 , y 0 ) = u ( x 0 ) t total ( x 0 ) = 1 + exp ( i ϑ 1 ) 2 2 π 1 w 0 exp { i k 2 [ ( 1 q 0 1 f ) x 0 2 + y 0 2 q 0 ] } + 1 exp ( i ϑ 1 ) 2 2 π 1 w 0 m = sinc [ ( 2 m 1 ) π 2 ] × exp { k 2 [ ( 1 q 0 1 f ) x 0 2 λ d m x 0 + y 0 2 q 0 ] } .
ψ ( z ) = tan 1 [ ( z / z R ) / ( 1 z f ) ] ,
ψ 0 ( z ) = tan 1 ( z / z R ) ,
R x ( z ) = [ f 2 + ( f z ) 2 ( z / z R ) 2 f 2 + ( f z ) z ( z / z R ) 2 ] z ,
R y ( z ) = [ 1 + ( z / z R ) 2 ] z ,
χ m ( z ) = 2 a m f z ( f z ) z R 3 f 2 z R z ( f z ) z R 3 ,
ω m ( z ) = a m f z ( f z ) z R 3 f 2 z R z ( f z ) z R 3 ,

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