Abstract

A description is given of a microlens array design, believed to be new, with tunable focal length. Converging or diverging periodic refractive-index distribution is induced in a linear electro-optic wafer through the application of electric field profiles on both sides of the wafer. The transparent electrodes on both sides of the wafer are positioned such that the electrodes on the opposite side compensate the phase delay from the electrodes on the front side of the wafer for a normally incident plane wave, suppressing the intrinsic electrode diffraction for the device without applied voltage. The original technique of the electric field calculation was developed to analyze accurately the induced refractive index inside the wafer. Its focal length changes from 7 mm to infinity at a 1-µm wavelength with an external voltage range of 0–100 V.

© 2000 Optical Society of America

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References

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  1. N. F. Borrelli, Microoptics Technology (Marcel Dekker, New York, 1999).
  2. H. P. Herzig, ed., Micro-Optics: Elements, Systems and Applications (Taylor & Francis, London, 1997).
  3. M. Oikawa, H. Nemoto, K. Hamanaka, E. Okuda, “High-numerical-aperture planar microlens with swelled structure,” Appl. Opt. 29, 4077–4080 (1990).
    [CrossRef] [PubMed]
  4. T. Tatebayashi, T. Yamamoto, H. Sato, “Dual focal point electro-optic lens with a Fresnel-zone plate on a PLZT ceramic,” Appl. Opt. 31, 2770–2775 (1992).
    [CrossRef] [PubMed]
  5. Q. Wang Song and X.-M. Wang, “Lanthanum-modified lead zirconate titanate ceramic wafer-based electro-optic dynamic diverging lens,” Opt. Lett. 21, 243–245 (1996).
  6. 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] [PubMed]
  7. M. Kulishov, “Adjustable electro-optic microlens with two concentric ring electrode,” Opt. Lett. 23, 1936–1938 (1998).
    [CrossRef]
  8. 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]
  9. S. Masuda, S. Fujioka, M. Honma, T. Nose, S. Sato, “Dependence of optical properties on the device and material parameters in liquid crystal microlenses,” Jpn. J. Appl. Phys. Part 1 35, 4668–4672 (1996).
    [CrossRef]
  10. 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).
  11. I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980), Secs. 1.441.1 and 1.441.4.
  12. 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]
  13. H. P. Herzig, D. Prongue, R. Dandliker, “Design and fabrication of highly efficient fan out elements,” Jpn. J. Appl. Phys. 29, L1307–L1309 (1990).
    [CrossRef]

1998

1997

1996

Q. Wang Song and X.-M. Wang, “Lanthanum-modified lead zirconate titanate ceramic wafer-based electro-optic dynamic diverging lens,” Opt. Lett. 21, 243–245 (1996).

S. Masuda, S. Fujioka, M. Honma, T. Nose, S. Sato, “Dependence of optical properties on the device and material parameters in liquid crystal microlenses,” Jpn. J. Appl. Phys. Part 1 35, 4668–4672 (1996).
[CrossRef]

1992

1990

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).

Borrelli, N. F.

N. F. Borrelli, Microoptics Technology (Marcel Dekker, New York, 1999).

Dandliker, R.

H. P. Herzig, D. Prongue, R. Dandliker, “Design and fabrication of highly efficient fan out elements,” Jpn. J. Appl. Phys. 29, L1307–L1309 (1990).
[CrossRef]

Fujioka, S.

S. Masuda, S. Fujioka, M. Honma, T. Nose, S. Sato, “Dependence of optical properties on the device and material parameters in liquid crystal microlenses,” Jpn. J. Appl. Phys. Part 1 35, 4668–4672 (1996).
[CrossRef]

Gradshtein, I. S.

I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980), Secs. 1.441.1 and 1.441.4.

Hamanaka, K.

Haritatos, F.

Herzig, H. P.

H. P. Herzig, D. Prongue, R. Dandliker, “Design and fabrication of highly efficient fan out elements,” Jpn. J. Appl. Phys. 29, L1307–L1309 (1990).
[CrossRef]

Honma, M.

S. Masuda, S. Fujioka, M. Honma, T. Nose, S. Sato, “Dependence of optical properties on the device and material parameters in liquid crystal microlenses,” Jpn. J. Appl. Phys. Part 1 35, 4668–4672 (1996).
[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.

Lee, S. H.

Masuda, S.

S. Masuda, S. Fujioka, M. Honma, T. Nose, S. Sato, “Dependence of optical properties on the device and material parameters in liquid crystal microlenses,” Jpn. J. Appl. Phys. Part 1 35, 4668–4672 (1996).
[CrossRef]

Nemoto, H.

Nose, T.

S. Masuda, S. Fujioka, M. Honma, T. Nose, S. Sato, “Dependence of optical properties on the device and material parameters in liquid crystal microlenses,” Jpn. J. Appl. Phys. Part 1 35, 4668–4672 (1996).
[CrossRef]

Oikawa, M.

Okuda, E.

Prongue, D.

H. P. Herzig, D. Prongue, R. Dandliker, “Design and fabrication of highly efficient fan out elements,” Jpn. J. Appl. Phys. 29, L1307–L1309 (1990).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980), Secs. 1.441.1 and 1.441.4.

Sato, H.

Sato, S.

S. Masuda, S. Fujioka, M. Honma, T. Nose, S. Sato, “Dependence of optical properties on the device and material parameters in liquid crystal microlenses,” Jpn. J. Appl. Phys. Part 1 35, 4668–4672 (1996).
[CrossRef]

Song, Q. Wang

Song and X.-M. Wang, Q. Wang

Q. Wang Song and X.-M. Wang, “Lanthanum-modified lead zirconate titanate ceramic wafer-based electro-optic dynamic diverging lens,” Opt. Lett. 21, 243–245 (1996).

Tatebayashi, T.

Title, M. A.

Wang, X.-M.

Yamamoto, T.

Appl. Opt.

Jpn. J. Appl. Phys.

H. P. Herzig, D. Prongue, R. Dandliker, “Design and fabrication of highly efficient fan out elements,” Jpn. J. Appl. Phys. 29, L1307–L1309 (1990).
[CrossRef]

Jpn. J. Appl. Phys. Part 1

S. Masuda, S. Fujioka, M. Honma, T. Nose, S. Sato, “Dependence of optical properties on the device and material parameters in liquid crystal microlenses,” Jpn. J. Appl. Phys. Part 1 35, 4668–4672 (1996).
[CrossRef]

Opt. Lett.

Q. Wang Song and X.-M. Wang, “Lanthanum-modified lead zirconate titanate ceramic wafer-based electro-optic dynamic diverging lens,” Opt. Lett. 21, 243–245 (1996).

M. Kulishov, “Adjustable electro-optic microlens with two concentric ring electrode,” Opt. Lett. 23, 1936–1938 (1998).
[CrossRef]

Other

N. F. Borrelli, Microoptics Technology (Marcel Dekker, New York, 1999).

H. P. Herzig, ed., Micro-Optics: Elements, Systems and Applications (Taylor & Francis, London, 1997).

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).

I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980), Secs. 1.441.1 and 1.441.4.

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Figures (7)

Fig. 1
Fig. 1

Cross section of EO microlens array.

Fig. 2
Fig. 2

Normalized ϕ(x, h)/V 0 and ϕ(x, -h)/V 0 surface potential distributions for a/ l = 0.6, 2h = l/4 and LiNbO3 wafer (ε11 = 85, ε33 = 29), ε = 1 (solid curve, bottom side; dashed curve, top side).

Fig. 3
Fig. 3

Pseudocapacitance as a function of (a) normalized spatial frequency at a/ l = 0.5 and (b) electrode duty ratio at 2h = l/4 for EO wafer with the double-sided electrode structure (solid curve) and one-sided traditional interdigitated electrodes (ε11 = 85, ε33 = 29, ε = 1).

Fig. 4
Fig. 4

(a) Surface potential difference for 2h = l/2 (solid curve) and 2h = l/4 (dashed curve) and (b) polynomial regression of the surface potential difference for 2h = l/4, a/ l = 0.5, ε11 = 85, ε33 = 29, ε = 1.

Fig. 5
Fig. 5

Contour plots of refractive-index distributions for x-polarized light wave: 2h = l/2, a/ l = 0.5, ε11 = 500, ε33= 3000, ε = l, n o = 2.32, n e = 2.273, r 13 = -266 × 10-12 m/V, r 15 = 120 × 10-12 m/V, V 0 = 100 V.

Fig. 6
Fig. 6

Three-dimensional distribution of the phase delay for the wafer with an orthogonal arrays of ITO stripe electrodes.

Fig. 7
Fig. 7

Normalized surface potential distribution on the top wafer surface (dashed curve), on the bottom wafer surface (dotted curve), and potential difference (solid curve) for the following structure parameters: a/ l = 0.25, 2h = l/4, ε11 = 500, ε33 = 3000, ε = 1.

Equations (37)

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ϕ1x, z=V0A1+n=1 En1 exp-nkzcosnkx,ϕ2x, z=V0A2+B2z+n=1En2coshnkδzsinhnkδh+Dn2sinhnkδzcoshnkδhcosnkx,ϕ3x, z=V0A3+n=1 En3 expnkzcosnkx,
ε ϕ1x, z=+hz=ε33ϕ2x, z=+hz, a2xl2,ε ϕ3x, z=-hz=ε33ϕ2x, z=-hz, 0xl-a2.
ε33B2h+n=1 nkhPn cosnkx=0, a2xl2,ε33B2h+n=1 nkhQn cosnkx=0, 0xl-a2,
Qn=En2ε33δ+ε cothnkδh+Dn2ε33δ+ε tanhnkδh,Pn=-En2ε33δ+ε cothnkδh+Dn2ε33δ+ε tanhnkδh.
ϕ1x, z=+h=ϕ2x, z=+h=0, 0x<a2,ϕ2x, z=-h=ϕ3x, z=-h=V0, l-a2x<l2,
A2-B2h+n=1PnFn2+QnFn1cosnkx=0,0xa2,A2+B2h+n=1PnFn1+QnFn2cosnkx=1,l-a2xl2,
Fn1=ε33δtanhnkδh-cothnkδh2ε33δ+ε cothhkδhε33δ+ε cothhkδh,Fn2=ε33δtanhnkδh+cothnkδh+2ε2ε33δ+ε cothhkδhε33δ+ε cothhkδh.
n=1 Qn cosnkx=A2-B2hε33δ+ε+n=1PnRn2-QnRn1cosnkx,  0xa2,n=1 Pn cosnkx=1-A2-B2hε33δ+ε+n=1PnRn1-QnRn2cosnkx,  l-a2xl2,
Rn1=1-Fn1ε33δ+ε,  Rn2=Fn2ε33δ+ε.
ε33B2h+n=1 nkhQn cosnkx=00xl-a2ρ1xl-a2<x<l2,ε33B2h+n=1 nkhPn cosnkx=ρ2x0xa20a2<x<l2.
ε33B2h=2ll-a/2l/2 ρ1ξdξ=2l0a/2 ρ1l/2-ξdξ=a01,ε33B2h=2l0a/2 ρ2ξdξ=a02,  a01=a02,
nkhQn=4ll-a/2l/2 ρ1ξcosnkξdξ=-1n4l0a/2 ρ1l/2-ξcosnkξdξ,nkhPn=4l0a/2 ρ2ξcosnkξdξ.
-1πh0a/2 ρ1ξln2|coskx+coskξ|dξ=1-A2-B2hε33δ+ε+2πhn=1Rn2n0a/2 ρ2ξcosnkξdξ--1nRn1n0a/2 ρ1l/2-ξcosnkξdξ×cosnkx,  l-a2xl2,
-1πh0a/2 ρ2ξln2|coskx-coskξ|dξ=A2-B2hε33δ+ε+2πhn=1Rn1n0a/2 ρ2ξcosnkξdξ--1nRn2n0a/2 ρ1l/2-ξcosnkξdξ×cosnkx;  0xa2,
n=1cosnkxcosnkξn=-12ln2×|coskx-coskξ|,n=1-1n cosnkxcosnkξn=-12ln2×|coskx+coskξ|.
coskx=-cos2ka/4-sin2ka/4coskηcoskξ=cos2ka/4+sin2ka/4coskζ for Eq. 13,
coskx=cos2ka/4+sin2ka/4coskηcoskξ=cos2ka/4+sin2ka/4coskζ for Eq. 14.
ρ1l/2-ξζdξdζ=s=0 as1 cosskζ,ρ2ξζdξdζ=s=0 as2 cosskζ.
ln2|coskx-coskξ|=lnsin2ka4-2 m=1cosmkζcosmkηm,cosnkx=s=1n bsn cosmkη,
-a02 lnsin2ka/4+m=1am2mcosmkη=A2-B2hε33δ+εkh+m=0ns=0as2n=1Rn1n bmnbsn-as1×n=1-1nRn2n bmnbsncosmkη,
-a01 lnsin2ka/4+m=1am1mcosmkη=1-A2-B2hε33δ+εkh+m=0ns=0as2n=1-1nRn2n bmnbsn-as1×n=1Rn1n bmnbsncosmkη.
am1=am2=am,  A2=12,m=1n=1Rn1+-1nRn2bmnbinn-δmimam=2a0n=1Rn1+-1nRn2bmnb0nn,
a0=ε33δ+εkh2+s=1 asn=1Rn1+-1n+Rn2n bsnb0nε33δ+εε33 kh-lnsin2ka4-2 s=1Rn1+-1n+Rn2nb0n2.
D2n2=2a0b02n+m=12n ambm2n2nkhε33δ+ε tanh2nkδh, D2n-12=0,E2n-12=2a0b02n-1+m=12n-1 ambm2n-12n-1khε33δ+ε coth2n-1kδh,E2n2=0.
C=QV0=2V00a/2 ρ2xηdxdηdη=2πa0ε0kh.
s=1Rn1+-1n+Rn2nb0n2=0.
rij=00r1300r1300r330r150r1500000.
1n2ij=1n02+r13Ez0r51Ex01n02+r13Ez0r51Ex01ne2+r33Ez,
1n2i,jΓj-νiΓi=0,
nxx, zn01-r13n022 Ezx, z-12n04ne2r512Ex2x, zne2-n02,nyx, zn01-r13n022 Ezx, z.
Δyx=πλ n03r13-h+h Ezx, zdz=πλ n03r13-h+h-ϕzdz=-πλ r13n03ϕx, +h-ϕx, -h,
exp-j 2πλ n03r13V0q2πxl2.
f=l24π2n03r13V0q,  NA=2π2n03r13V0ql.
U3x, y, -h=14π2--++ G2ωx, ωy, +h×expi2hκ2-ωx2-ωy21/2×expiωxx+iωyydωxdωy,
2hκ2-ωx2-ωy21/22κh-h ωx2+ωy2κ+.
U3x, y, -h14π2--++ G2ωx, ωy, +h×expiκzexpiωxx+iωyydωxdωy=expi2κhU2x, y, +h.
U4x, y, -hU0 expi2κhexpiκαx+iκβyt1x, yt2x, y.

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