Abstract

Although the concept of an artificial compound eye has been discussed in the literature, its optical arrangement has never been widely adopted for optical design. A design is presented for a tunable gradient-index microlens array, believed to be new, induced electro-optically inside a cylindrical shell. The transparent electrodes on the both sides of the shell are positioned such that the electrodes on the opposite side compensate the phase delay from the electrodes on the front side for a normally incident plane wave, thus suppressing the intrinsic electrode diffraction for the device without applied voltage. The original technique of the electric field calculation was developed to analyze the induced refractive index inside the shell for two types of electro-optic (EO) ceramics: with linear and with quadratic EO effects. For the linear effect it was shown that for given EO coefficients, electric field strength and intrinsic refractive index, the electrode number should exceed a certain amount to make the focal distance less than the cylinder radius. The quadratic effect provides higher sensitivity to the type of the diffracted wave polarization. It was shown how the quadratic coefficient ratio R 12/R 11 affects the focal-length difference between TE and TM light polarization.

© 2000 Optical Society of America

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References

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  1. M. Kulishov, “Tunable electro-optic microlens array. I. Planar geometry,” Appl. Opt. 39, 2332–2339 (2000).
    [CrossRef]
  2. M. F. Land, “Microlens arrays in the animal kingdom,” Pure Appl. Opt. 6, 599–602 (1997).
    [CrossRef]
  3. J. S. Sanders, C. E. Halford, “Design and analysis of apposition compound eye optical sensors,” Opt. Eng. 34, 222–235 (1995).
    [CrossRef]
  4. S. Ogata, J. Ishida, T. Sasano, “Optical sensor array in an artificial compound eye,” Opt. Eng. 33, 3649–3655 (1994).
    [CrossRef]
  5. G. H. Haertling, “PLZT electrooptic materials and applications—a review,” Ferroelectrics 75, 25–55 (1987).
    [CrossRef]
  6. M. A. Title, S. H. Lee, “Modeling and characterization of embedded electrode performance in transverse electro-optic modulators,” Appl. Opt. 29, 85–98 (1990).
    [CrossRef] [PubMed]
  7. N. Kuleshov (now spelled Kulishov) and G. Beilin, “Optimization of electrode pattern for multichannel spatial light modulators on the basis of PLZT ceramics with quadratic electro-optic effect,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 345–355 (1995).
  8. M. C. Gupta, ed., Handbook of Photonics (CRC, Boca Raton, Fla., 1998).
  9. G. H. Haertling, “Recent developments in bulk and thin film PLZT materials and devices,” Ferroelectrics 131, 1–12 (1992).
    [CrossRef]
  10. M. Kulishov, “Modeling of a converging gradient-index lens with variable focal length in a lanthanum-modified lead zirconate titanate ceramic cylinder with a lateral multielectrode structure,” Appl. Opt. 37, 3506–3514 (1998).
    [CrossRef]
  11. Q. Wang Song, X.-M. Wang, F. Haritatos, “Test and analysis of an electro-optic dynamic diverging lens for three-dimensional optical memories,” Appl. Opt. 36, 1796–1803 (1997).
    [CrossRef]
  12. D. E. Dausch, E. Furman, F. Wang, G. H. Haertling, “PLZT-based multilayer composite thin films. Part II. Modeling of the dielectric and hysteresis properties,” Ferroelectrics 177, 237–253 (1996).
    [CrossRef]
  13. F. A. Lopez, J. M. Cabrera, F. Agullo-Rueda, Electrooptics: Phenomena, Materials and Applications (Academic, London, 1994).

2000 (1)

1998 (1)

1997 (2)

1996 (1)

D. E. Dausch, E. Furman, F. Wang, G. H. Haertling, “PLZT-based multilayer composite thin films. Part II. Modeling of the dielectric and hysteresis properties,” Ferroelectrics 177, 237–253 (1996).
[CrossRef]

1995 (1)

J. S. Sanders, C. E. Halford, “Design and analysis of apposition compound eye optical sensors,” Opt. Eng. 34, 222–235 (1995).
[CrossRef]

1994 (1)

S. Ogata, J. Ishida, T. Sasano, “Optical sensor array in an artificial compound eye,” Opt. Eng. 33, 3649–3655 (1994).
[CrossRef]

1992 (1)

G. H. Haertling, “Recent developments in bulk and thin film PLZT materials and devices,” Ferroelectrics 131, 1–12 (1992).
[CrossRef]

1990 (1)

1987 (1)

G. H. Haertling, “PLZT electrooptic materials and applications—a review,” Ferroelectrics 75, 25–55 (1987).
[CrossRef]

Agullo-Rueda, F.

F. A. Lopez, J. M. Cabrera, F. Agullo-Rueda, Electrooptics: Phenomena, Materials and Applications (Academic, London, 1994).

Beilin, G.

N. Kuleshov (now spelled Kulishov) and G. Beilin, “Optimization of electrode pattern for multichannel spatial light modulators on the basis of PLZT ceramics with quadratic electro-optic effect,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 345–355 (1995).

Cabrera, J. M.

F. A. Lopez, J. M. Cabrera, F. Agullo-Rueda, Electrooptics: Phenomena, Materials and Applications (Academic, London, 1994).

Dausch, D. E.

D. E. Dausch, E. Furman, F. Wang, G. H. Haertling, “PLZT-based multilayer composite thin films. Part II. Modeling of the dielectric and hysteresis properties,” Ferroelectrics 177, 237–253 (1996).
[CrossRef]

Furman, E.

D. E. Dausch, E. Furman, F. Wang, G. H. Haertling, “PLZT-based multilayer composite thin films. Part II. Modeling of the dielectric and hysteresis properties,” Ferroelectrics 177, 237–253 (1996).
[CrossRef]

Haertling, G. H.

D. E. Dausch, E. Furman, F. Wang, G. H. Haertling, “PLZT-based multilayer composite thin films. Part II. Modeling of the dielectric and hysteresis properties,” Ferroelectrics 177, 237–253 (1996).
[CrossRef]

G. H. Haertling, “Recent developments in bulk and thin film PLZT materials and devices,” Ferroelectrics 131, 1–12 (1992).
[CrossRef]

G. H. Haertling, “PLZT electrooptic materials and applications—a review,” Ferroelectrics 75, 25–55 (1987).
[CrossRef]

Halford, C. E.

J. S. Sanders, C. E. Halford, “Design and analysis of apposition compound eye optical sensors,” Opt. Eng. 34, 222–235 (1995).
[CrossRef]

Haritatos, F.

Ishida, J.

S. Ogata, J. Ishida, T. Sasano, “Optical sensor array in an artificial compound eye,” Opt. Eng. 33, 3649–3655 (1994).
[CrossRef]

Kuleshov, N.

N. Kuleshov (now spelled Kulishov) and G. Beilin, “Optimization of electrode pattern for multichannel spatial light modulators on the basis of PLZT ceramics with quadratic electro-optic effect,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 345–355 (1995).

Kulishov, M.

Land, M. F.

M. F. Land, “Microlens arrays in the animal kingdom,” Pure Appl. Opt. 6, 599–602 (1997).
[CrossRef]

Lee, S. H.

Lopez, F. A.

F. A. Lopez, J. M. Cabrera, F. Agullo-Rueda, Electrooptics: Phenomena, Materials and Applications (Academic, London, 1994).

Ogata, S.

S. Ogata, J. Ishida, T. Sasano, “Optical sensor array in an artificial compound eye,” Opt. Eng. 33, 3649–3655 (1994).
[CrossRef]

Sanders, J. S.

J. S. Sanders, C. E. Halford, “Design and analysis of apposition compound eye optical sensors,” Opt. Eng. 34, 222–235 (1995).
[CrossRef]

Sasano, T.

S. Ogata, J. Ishida, T. Sasano, “Optical sensor array in an artificial compound eye,” Opt. Eng. 33, 3649–3655 (1994).
[CrossRef]

Title, M. A.

Wang, F.

D. E. Dausch, E. Furman, F. Wang, G. H. Haertling, “PLZT-based multilayer composite thin films. Part II. Modeling of the dielectric and hysteresis properties,” Ferroelectrics 177, 237–253 (1996).
[CrossRef]

Wang, X.-M.

Wang Song, Q.

Appl. Opt. (4)

Ferroelectrics (3)

D. E. Dausch, E. Furman, F. Wang, G. H. Haertling, “PLZT-based multilayer composite thin films. Part II. Modeling of the dielectric and hysteresis properties,” Ferroelectrics 177, 237–253 (1996).
[CrossRef]

G. H. Haertling, “PLZT electrooptic materials and applications—a review,” Ferroelectrics 75, 25–55 (1987).
[CrossRef]

G. H. Haertling, “Recent developments in bulk and thin film PLZT materials and devices,” Ferroelectrics 131, 1–12 (1992).
[CrossRef]

Opt. Eng. (2)

J. S. Sanders, C. E. Halford, “Design and analysis of apposition compound eye optical sensors,” Opt. Eng. 34, 222–235 (1995).
[CrossRef]

S. Ogata, J. Ishida, T. Sasano, “Optical sensor array in an artificial compound eye,” Opt. Eng. 33, 3649–3655 (1994).
[CrossRef]

Pure Appl. Opt. (1)

M. F. Land, “Microlens arrays in the animal kingdom,” Pure Appl. Opt. 6, 599–602 (1997).
[CrossRef]

Other (3)

N. Kuleshov (now spelled Kulishov) and G. Beilin, “Optimization of electrode pattern for multichannel spatial light modulators on the basis of PLZT ceramics with quadratic electro-optic effect,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 345–355 (1995).

M. C. Gupta, ed., Handbook of Photonics (CRC, Boca Raton, Fla., 1998).

F. A. Lopez, J. M. Cabrera, F. Agullo-Rueda, Electrooptics: Phenomena, Materials and Applications (Academic, London, 1994).

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Figures (7)

Fig. 1
Fig. 1

Cross section of the EO microlens array.

Fig. 2
Fig. 2

Normalized ϕ(r, θ)/V 0 three-dimensional potential distribution (r 0 - Δ ≤ rr 0 + Δ, +2π/N ≤ θ ≤ -2π/N) for N = 9, θ0 = π/18, Δ/r 0 = 0.2, and ε11 = ε33 = 5000.

Fig. 3
Fig. 3

Pseudocapacitance as functions of (a) the electrode duty ratio for N = 36, Δ/r 0 = π/2N (solid curve), and Δ/r 0 = π/N (dashed curve); (b) normalized shell thickness 4NΔ/(πr 0) for N = 36, θ0 = π/N; and (c) electrode number for constant NΔ/r 0 = π and θ0 N = π values.

Fig. 4
Fig. 4

Normalized potential distribution on the internal (dotted curve) and external (dashed curve) surfaces and their difference (solid curve) for -π/N ≤ θ ≤ +π/N, where N = 360, Δ/r 0 = 0.00436, θ0 = π/360. Dotted–dashed curve, polynomial regression of the surface potential difference distribution.

Fig. 5
Fig. 5

β/N parameter versus electrode number N for 2Δ = 2πr 0/N (circles) and 2Δ = πr 0/N (squares).

Fig. 6
Fig. 6

Phase-delay distribution for the z-polarized (solid curve) and the θ-polarized (dashed curve) light on the 9/65/35 PLZT shell with R 11 = 2.42 × 10-16 m2/V2, and R 12 = -1.94 × 10-16 m2/V2, N = 360, 2Δ = πr 0/N, θ0 = π/N, n 0 = 2.5, r 0 = 5 × 10-3 m, V 0 = 80 V.

Fig. 7
Fig. 7

Focal-length-to-radius ratio versus quadratic EO coefficient ratio for V 0 = 40 V, r 0 = 0.5 cm, R 11 = 38.6 × 10-16 m2/V2, N = 360, Δ = 0.0044r 0. Squares, θ-polarized light; circles, z-polarized light.

Equations (25)

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ε331rrr ϕr+ε111r22ϕr, θθ2=0.
ϕr, θ=V0A0+A1 lnrr0+n=1Anrr0δnN+Cnrr0δ-nNcosNnθ,
A0+A1 lnr0-Δr0+n=1Anr0-Δr0δnN+Cnr0δr0-ΔnNcosnNθ=1,  0<θ<θ02, A0+A1 lnr0+Δr0+n=1Anr0+Δr0δnN+Cnr0δr0+ΔnNcosnNθ=0,  πN-θ02<θ<πN.
ε33ϕr=r0-Δr=ε33A1r0r0-Δ+n=1 nNQn×cosnNθ=0,  θ02θπN, ε33ϕr=r0+Δr=ε33A1r0r0+Δ+n=1 nNPn×cosnNθ=0,  0θπN-θ02,
Qn=ε33δAnr0-Δr0δnN-1-Cnr0δr0-ΔnN+1, Pn=ε33δAnr0+Δr0δnN-1-Cnr0δr0+ΔnN+1.
n=1 Qn cosnNθ=r0ε33r0-ΔA0-1+A1 lnr0-Δr0+n=1QnRn1-r0+Δr0-Δ PnRn2×cosnNθ,  0<θ<θ02-n=1 Pn cosnNθ=r0ε33r0+ΔA0+A1 lnr0+Δr0+n=1r0-Δr0+Δ QnRn2-PnRn1×cosnNθ,  πN-θ02<θ<πN,
Rn1=1-r0+Δr0-ΔnN+r0-Δr0+ΔnNr0+Δr0-ΔnN-r0-Δr0+ΔnN, Rn2=2r0+Δr0-ΔnN-r0-Δr0+ΔnN,  limnRni=0.
ε33A1r0r0+Δ+n=1 NnPn cosnNθ=ρ1θπN-θ02θπN00<θ<πN-θ02, ε33A1r0r0-Δ+n=1 NnQn cosnNθ=ρ2θ0<θ<θ020θ02θπN.
ε33A1r0r0-Δ=Nππ/N-θ0/2π/N ρ1ξdξ=Nπ0θ0/2 ρ1π/N-ξdξ=a01, ε33A1r0r0+Δ=Nπ0θ0/2 ρ2ξdξ=a02,  a01=a02,
nPn=2ππ/N-θ0/2π/N ρ1ξcosNnξdξ=-1n2π0θ0/2 ρ1π/N-ξcosNnξdξ, nQn=2π0θ0/2 ρ2ξcosNnξdξ.
ρ1ξξ1dξdξ1=p=0 ap1 cospNξ1, ρ2π/N-ξξ1dξdξ1=p=0 ap2 cospNξ1, cosNnθθ1=i=0n bin cosiNθ1.
x02ε33=A1=1-1Nε33s=1 Gsaslnr0-Δr0+Δ+4NlnsinNθ04+4Ns=1Rn1--1nRn2nb0n2,
A0=1Nε33M=1 Gmxm-A1lnr0+Δr0-2NlnsinNθ04-2Ns=1Rn1--1nRn2nb0n2.
xm=r0-Δr0 am1+r0+Δr0 am2, ym=r0-Δr0 am1-r0+Δr0 am2,
m=1Gms-δmsmxm=2x0Gs,  ys=0,
Gms=n=1Rn1--1nRn2bmnbsnn, Gs=n=1Rn1--1nRn2bsnb0nn.
Qn=2x0b0n+m=1n xmbmnNnδε33r0r0-Δ, Pn=-1n2x0b0n+m=1n xmbmnNnδε33r0r0+Δ, An=Qnr0δr0+ΔnN+1-Pnr0δr0-ΔnN+1r0-Δr0+ΔnN-r0+Δr0-ΔnNr02-Δ2r02δ2, Cn=Qnr0+Δr0δnN-1-Pnr0-Δr0δnN+1r0-Δr0+ΔnN-r0+Δr0-ΔnNr02-Δ2r02δ2.
C=QV0=2r0-ΔV00π/N ρ2ξξ1dξdξ1dξ1=2πN A1ε0ε33.
s=1Rn1--1nRn2nb0n2=0.
nθr, θn01-r13n022 Err, θ-12n04ne2r512Eθ2r, θne2-n02, nzr, θn01-r13n022 Err, θ,
Δzθ=πλ n03r13r0-Δr0+Δ Err, θdr=πλ n03r13r0-Δr0+Δ-ϕrdr=-πλ r13n03ϕ, r0+Δ)-ϕθ, r0-Δ.
f=r02n03V0βr13.
f/r01.
nθr, θn01-n022R11Er2r, θ+R12Eθ2r, θ, nzr, θn01-n022 R12Er2r, θ+Eθ2r, θ,
Δθθ=2πλr0-Δr0+Δ nθr, θdr, Δzθ=2πλr0-Δr0+Δ nzr, θdr.

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