Abstract

The design of a Fresnel lens with continuous focal length is proposed for use in optical processing. A convex lens is induced in lanthanum-modified lead zirconate titanate through the application of an electric-field profile supplied by the indium tin oxide electrodes that make up the zones of a Fresnel lens. The use of a numerical method based on fast Fourier transform algorithms was required to analyze accurately the induced field inside a Fresnel lens with an initial focal length of 0.4 m (at 470 nm) and 20 indium tin oxide electrodes. The effective focal location obtained by the combined mechanisms is derived. This design is expected to produce continuous variations of ~16% in focal length; the ability of previous designs to achieve focal length switching is maintained.

© 1995 Optical Society of America

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References

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  1. S. H. Lee, S. C. Esener, M. A. Title, T. J. Drabik, “Two-dimensional silicon/PLZT light modulators: design considerations and technology,” Opt. Eng. 25, 250–260 (1986).
  2. E. Van Tomme, P. P. Van Daele, R. G. Baets, P. E. Lagasse, “Integrated optic devices based on nonlinear optical polymers,” IEEE J. Quantum Electron. 27, 778–787 (1991).
    [Crossref]
  3. J. R. Hill, P. L. Dunn, G. J. Davies, S. N. Oliver, P. Pantelis, J. D. Rush, “Efficient frequency-doubling in poled PVDF copolymer guest/host composite,” Electron. Lett. 23, 700–701 (1987).
    [Crossref]
  4. R. T. Chen, “Graded-index polymer-based waveguide lens working at visible wavelengths on GaAs substrate for optoelectronic interconnects,” Appl. Phys. Lett. 62, 2495–2497 (1993).
    [Crossref]
  5. 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]
  6. T. Tatebayashi, T. Yamamoto, H. Sato, “Electro-optic variable focal-length lens using PLZT ceramic,” Appl. Opt. 30, 5049–5055 (1991).
    [Crossref] [PubMed]
  7. K. Rastani, A. Marrakchi, S. F. Habiby, W. M. Hubbard, H. Gilchrist, R. Nahory, “Binary phase Fresnel lenses for generation of two-dimensional beam arrays,” Appl. Opt. 30, 1347–1353 (1991).
    [Crossref] [PubMed]
  8. G. H. Haertling, “Piezoelectric and electrooptic ceramics,” in Ceramic Materials for Electronics: Processing, Properties, and Applications, R. Buchanan, ed. (Dekker, New York, 1984), pp. 166–169.
  9. J. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, San Francisco, 1964), Chap. 2, pp. 77–86.
  10. S. F. Habiby, A. Marrakchi, J. R. Wullert, J. S. Patel, J. T. Meyer, “Programmable coherent source arrays generated by spatial light modulators,” Appl. Opt. 31, 1347–1353 (1992).
    [Crossref]
  11. W. T. Pawlewickz, I. B. Mann, W. H. Lowdermilk, D. Milan, “Laser-damage-resistant transparent conductive indium tin oxide coatings,” Appl. Phys. Lett. 34, 196–198 (1979).
    [Crossref]
  12. R. V. Churchill, Fourier Series and Boundary Value Problems, 1st ed. (McGraw-Hill, New York, 1941), p. 141.
  13. A. D. Poularikas, S. Seely, Signals and Systems, 1st ed. (PWS, Boston, Mass., 1985), pp. 152–155.
  14. B. L. Buzbee, G. H. Golub, C. W. Nielson, “On direct methods for solving Poisson’s equations,” Soc. Ind. Appl. Math. J. Numer. Anal. 7, 627–655 (1970).
  15. P. Swartztrauber, “Fast Poisson solvers,” in Studies in Numerical Analysis, G. Golub, ed. Vol. 24 of MAA Studies in Mathematics (Mathematical Association of America, Washington, D.C., 1984), p. 342.
  16. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipies in fortran(Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 401–407.

1993 (1)

R. T. Chen, “Graded-index polymer-based waveguide lens working at visible wavelengths on GaAs substrate for optoelectronic interconnects,” Appl. Phys. Lett. 62, 2495–2497 (1993).
[Crossref]

1992 (2)

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]

S. F. Habiby, A. Marrakchi, J. R. Wullert, J. S. Patel, J. T. Meyer, “Programmable coherent source arrays generated by spatial light modulators,” Appl. Opt. 31, 1347–1353 (1992).
[Crossref]

1991 (3)

1987 (1)

J. R. Hill, P. L. Dunn, G. J. Davies, S. N. Oliver, P. Pantelis, J. D. Rush, “Efficient frequency-doubling in poled PVDF copolymer guest/host composite,” Electron. Lett. 23, 700–701 (1987).
[Crossref]

1986 (1)

S. H. Lee, S. C. Esener, M. A. Title, T. J. Drabik, “Two-dimensional silicon/PLZT light modulators: design considerations and technology,” Opt. Eng. 25, 250–260 (1986).

1979 (1)

W. T. Pawlewickz, I. B. Mann, W. H. Lowdermilk, D. Milan, “Laser-damage-resistant transparent conductive indium tin oxide coatings,” Appl. Phys. Lett. 34, 196–198 (1979).
[Crossref]

1970 (1)

B. L. Buzbee, G. H. Golub, C. W. Nielson, “On direct methods for solving Poisson’s equations,” Soc. Ind. Appl. Math. J. Numer. Anal. 7, 627–655 (1970).

Baets, R. G.

E. Van Tomme, P. P. Van Daele, R. G. Baets, P. E. Lagasse, “Integrated optic devices based on nonlinear optical polymers,” IEEE J. Quantum Electron. 27, 778–787 (1991).
[Crossref]

Buzbee, B. L.

B. L. Buzbee, G. H. Golub, C. W. Nielson, “On direct methods for solving Poisson’s equations,” Soc. Ind. Appl. Math. J. Numer. Anal. 7, 627–655 (1970).

Chen, R. T.

R. T. Chen, “Graded-index polymer-based waveguide lens working at visible wavelengths on GaAs substrate for optoelectronic interconnects,” Appl. Phys. Lett. 62, 2495–2497 (1993).
[Crossref]

Churchill, R. V.

R. V. Churchill, Fourier Series and Boundary Value Problems, 1st ed. (McGraw-Hill, New York, 1941), p. 141.

Davies, G. J.

J. R. Hill, P. L. Dunn, G. J. Davies, S. N. Oliver, P. Pantelis, J. D. Rush, “Efficient frequency-doubling in poled PVDF copolymer guest/host composite,” Electron. Lett. 23, 700–701 (1987).
[Crossref]

Drabik, T. J.

S. H. Lee, S. C. Esener, M. A. Title, T. J. Drabik, “Two-dimensional silicon/PLZT light modulators: design considerations and technology,” Opt. Eng. 25, 250–260 (1986).

Dunn, P. L.

J. R. Hill, P. L. Dunn, G. J. Davies, S. N. Oliver, P. Pantelis, J. D. Rush, “Efficient frequency-doubling in poled PVDF copolymer guest/host composite,” Electron. Lett. 23, 700–701 (1987).
[Crossref]

Esener, S. C.

S. H. Lee, S. C. Esener, M. A. Title, T. J. Drabik, “Two-dimensional silicon/PLZT light modulators: design considerations and technology,” Opt. Eng. 25, 250–260 (1986).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipies in fortran(Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 401–407.

Gilchrist, H.

Golub, G. H.

B. L. Buzbee, G. H. Golub, C. W. Nielson, “On direct methods for solving Poisson’s equations,” Soc. Ind. Appl. Math. J. Numer. Anal. 7, 627–655 (1970).

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, San Francisco, 1964), Chap. 2, pp. 77–86.

Habiby, S. F.

S. F. Habiby, A. Marrakchi, J. R. Wullert, J. S. Patel, J. T. Meyer, “Programmable coherent source arrays generated by spatial light modulators,” Appl. Opt. 31, 1347–1353 (1992).
[Crossref]

K. Rastani, A. Marrakchi, S. F. Habiby, W. M. Hubbard, H. Gilchrist, R. Nahory, “Binary phase Fresnel lenses for generation of two-dimensional beam arrays,” Appl. Opt. 30, 1347–1353 (1991).
[Crossref] [PubMed]

Haertling, G. H.

G. H. Haertling, “Piezoelectric and electrooptic ceramics,” in Ceramic Materials for Electronics: Processing, Properties, and Applications, R. Buchanan, ed. (Dekker, New York, 1984), pp. 166–169.

Hill, J. R.

J. R. Hill, P. L. Dunn, G. J. Davies, S. N. Oliver, P. Pantelis, J. D. Rush, “Efficient frequency-doubling in poled PVDF copolymer guest/host composite,” Electron. Lett. 23, 700–701 (1987).
[Crossref]

Hubbard, W. M.

Lagasse, P. E.

E. Van Tomme, P. P. Van Daele, R. G. Baets, P. E. Lagasse, “Integrated optic devices based on nonlinear optical polymers,” IEEE J. Quantum Electron. 27, 778–787 (1991).
[Crossref]

Lee, S. H.

S. H. Lee, S. C. Esener, M. A. Title, T. J. Drabik, “Two-dimensional silicon/PLZT light modulators: design considerations and technology,” Opt. Eng. 25, 250–260 (1986).

Lowdermilk, W. H.

W. T. Pawlewickz, I. B. Mann, W. H. Lowdermilk, D. Milan, “Laser-damage-resistant transparent conductive indium tin oxide coatings,” Appl. Phys. Lett. 34, 196–198 (1979).
[Crossref]

Mann, I. B.

W. T. Pawlewickz, I. B. Mann, W. H. Lowdermilk, D. Milan, “Laser-damage-resistant transparent conductive indium tin oxide coatings,” Appl. Phys. Lett. 34, 196–198 (1979).
[Crossref]

Marrakchi, A.

S. F. Habiby, A. Marrakchi, J. R. Wullert, J. S. Patel, J. T. Meyer, “Programmable coherent source arrays generated by spatial light modulators,” Appl. Opt. 31, 1347–1353 (1992).
[Crossref]

K. Rastani, A. Marrakchi, S. F. Habiby, W. M. Hubbard, H. Gilchrist, R. Nahory, “Binary phase Fresnel lenses for generation of two-dimensional beam arrays,” Appl. Opt. 30, 1347–1353 (1991).
[Crossref] [PubMed]

Meyer, J. T.

S. F. Habiby, A. Marrakchi, J. R. Wullert, J. S. Patel, J. T. Meyer, “Programmable coherent source arrays generated by spatial light modulators,” Appl. Opt. 31, 1347–1353 (1992).
[Crossref]

Milan, D.

W. T. Pawlewickz, I. B. Mann, W. H. Lowdermilk, D. Milan, “Laser-damage-resistant transparent conductive indium tin oxide coatings,” Appl. Phys. Lett. 34, 196–198 (1979).
[Crossref]

Nahory, R.

Nielson, C. W.

B. L. Buzbee, G. H. Golub, C. W. Nielson, “On direct methods for solving Poisson’s equations,” Soc. Ind. Appl. Math. J. Numer. Anal. 7, 627–655 (1970).

Oliver, S. N.

J. R. Hill, P. L. Dunn, G. J. Davies, S. N. Oliver, P. Pantelis, J. D. Rush, “Efficient frequency-doubling in poled PVDF copolymer guest/host composite,” Electron. Lett. 23, 700–701 (1987).
[Crossref]

Pantelis, P.

J. R. Hill, P. L. Dunn, G. J. Davies, S. N. Oliver, P. Pantelis, J. D. Rush, “Efficient frequency-doubling in poled PVDF copolymer guest/host composite,” Electron. Lett. 23, 700–701 (1987).
[Crossref]

Patel, J. S.

S. F. Habiby, A. Marrakchi, J. R. Wullert, J. S. Patel, J. T. Meyer, “Programmable coherent source arrays generated by spatial light modulators,” Appl. Opt. 31, 1347–1353 (1992).
[Crossref]

Pawlewickz, W. T.

W. T. Pawlewickz, I. B. Mann, W. H. Lowdermilk, D. Milan, “Laser-damage-resistant transparent conductive indium tin oxide coatings,” Appl. Phys. Lett. 34, 196–198 (1979).
[Crossref]

Poularikas, A. D.

A. D. Poularikas, S. Seely, Signals and Systems, 1st ed. (PWS, Boston, Mass., 1985), pp. 152–155.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipies in fortran(Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 401–407.

Rastani, K.

Rush, J. D.

J. R. Hill, P. L. Dunn, G. J. Davies, S. N. Oliver, P. Pantelis, J. D. Rush, “Efficient frequency-doubling in poled PVDF copolymer guest/host composite,” Electron. Lett. 23, 700–701 (1987).
[Crossref]

Sato, H.

Seely, S.

A. D. Poularikas, S. Seely, Signals and Systems, 1st ed. (PWS, Boston, Mass., 1985), pp. 152–155.

Swartztrauber, P.

P. Swartztrauber, “Fast Poisson solvers,” in Studies in Numerical Analysis, G. Golub, ed. Vol. 24 of MAA Studies in Mathematics (Mathematical Association of America, Washington, D.C., 1984), p. 342.

Tatebayashi, T.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipies in fortran(Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 401–407.

Title, M. A.

S. H. Lee, S. C. Esener, M. A. Title, T. J. Drabik, “Two-dimensional silicon/PLZT light modulators: design considerations and technology,” Opt. Eng. 25, 250–260 (1986).

Van Daele, P. P.

E. Van Tomme, P. P. Van Daele, R. G. Baets, P. E. Lagasse, “Integrated optic devices based on nonlinear optical polymers,” IEEE J. Quantum Electron. 27, 778–787 (1991).
[Crossref]

Van Tomme, E.

E. Van Tomme, P. P. Van Daele, R. G. Baets, P. E. Lagasse, “Integrated optic devices based on nonlinear optical polymers,” IEEE J. Quantum Electron. 27, 778–787 (1991).
[Crossref]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipies in fortran(Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 401–407.

Wullert, J. R.

S. F. Habiby, A. Marrakchi, J. R. Wullert, J. S. Patel, J. T. Meyer, “Programmable coherent source arrays generated by spatial light modulators,” Appl. Opt. 31, 1347–1353 (1992).
[Crossref]

Yamamoto, T.

Appl. Opt. (4)

Appl. Phys. Lett. (2)

W. T. Pawlewickz, I. B. Mann, W. H. Lowdermilk, D. Milan, “Laser-damage-resistant transparent conductive indium tin oxide coatings,” Appl. Phys. Lett. 34, 196–198 (1979).
[Crossref]

R. T. Chen, “Graded-index polymer-based waveguide lens working at visible wavelengths on GaAs substrate for optoelectronic interconnects,” Appl. Phys. Lett. 62, 2495–2497 (1993).
[Crossref]

Electron. Lett. (1)

J. R. Hill, P. L. Dunn, G. J. Davies, S. N. Oliver, P. Pantelis, J. D. Rush, “Efficient frequency-doubling in poled PVDF copolymer guest/host composite,” Electron. Lett. 23, 700–701 (1987).
[Crossref]

IEEE J. Quantum Electron. (1)

E. Van Tomme, P. P. Van Daele, R. G. Baets, P. E. Lagasse, “Integrated optic devices based on nonlinear optical polymers,” IEEE J. Quantum Electron. 27, 778–787 (1991).
[Crossref]

Opt. Eng. (1)

S. H. Lee, S. C. Esener, M. A. Title, T. J. Drabik, “Two-dimensional silicon/PLZT light modulators: design considerations and technology,” Opt. Eng. 25, 250–260 (1986).

Soc. Ind. Appl. Math. J. Numer. Anal. (1)

B. L. Buzbee, G. H. Golub, C. W. Nielson, “On direct methods for solving Poisson’s equations,” Soc. Ind. Appl. Math. J. Numer. Anal. 7, 627–655 (1970).

Other (6)

P. Swartztrauber, “Fast Poisson solvers,” in Studies in Numerical Analysis, G. Golub, ed. Vol. 24 of MAA Studies in Mathematics (Mathematical Association of America, Washington, D.C., 1984), p. 342.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipies in fortran(Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 401–407.

R. V. Churchill, Fourier Series and Boundary Value Problems, 1st ed. (McGraw-Hill, New York, 1941), p. 141.

A. D. Poularikas, S. Seely, Signals and Systems, 1st ed. (PWS, Boston, Mass., 1985), pp. 152–155.

G. H. Haertling, “Piezoelectric and electrooptic ceramics,” in Ceramic Materials for Electronics: Processing, Properties, and Applications, R. Buchanan, ed. (Dekker, New York, 1984), pp. 166–169.

J. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, San Francisco, 1964), Chap. 2, pp. 77–86.

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Figures (7)

Fig. 1
Fig. 1

Schematic of the EO lens electrode configuration and applied potential arrangement.

Fig. 2
Fig. 2

Potential profile computation (n = 150): (a) direct, (b) with padding, (c) with padding and smoothing. Potential profile is in volts; lens position is in micrometers.

Fig. 3
Fig. 3

Cross-sectional view of the EO device with grid-point array for numerical analysis.

Fig. 4
Fig. 4

Potential distribution inside the lens after use of the finite-difference method. Potential is in volts; lens depth and position are in micrometers.

Fig. 5
Fig. 5

Profile of the effective refractive index induced in the substrate by application of surface potential. Lens position is in micrometers.

Fig. 6
Fig. 6

Beam-width variation beyond the plane of the EO lens as a function of applied potential distributions (only center potentials appear).

Fig. 7
Fig. 7

Chromatic aberration reduction in a BP FZP (f F = 0.4 m at 470 nm; wavelength range ±20%) with the addition of a convex lens.

Equations (31)

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X n = [ λ f F 2 ( 2 n - 1 ) ] 1 / 2 ,             n = 1 , 2 , 3 , ,
f F = 2 x 1 2 λ ,
T FZP ( x ) = 1 2 [ 1 + exp ( j θ ) ] + 1 2 [ 1 - exp ( j θ ) ] × n = - sinc [ ( 2 n - 1 ) π 2 ] exp [ j k ( 2 n - 1 ) 4 f F x 2 ] ,
x 2 n x 2 + y 2 n y 2 + z 2 n z 2 = 1 ,
n x = n y n 0 - n 0 3 2 R 12 E z 2 ,
n z n 0 - n 0 3 2 R 11 E z 2 ,
Δ n ( x ) = - n 0 3 2 R 12 E z 2 .
f ind = - W 2 T 4 n 0 3 R 12 V 0 2 ,
T convex ( x ) = exp ( - j Δ n max 4 x 2 W 2 k T ) = exp ( - j x 2 2 f ind k ) ,
u ( x 0 , 0 ) = 2 π 1 ω 0 exp ( - j k x 0 2 2 q ˜ 0 ) ,
u ( x 0 , 0 ) = u ( x 0 , 0 ) T FZP ( x 0 , 0 ) T convex ( x 0 , 0 ) = 2 π 1 ω 0 exp [ - j k 2 ( 1 q ˜ 0 + 1 f ind ) x 0 2 ] × { 1 2 [ 1 + exp ( j θ ) ] + 1 2 [ 1 - exp ( j θ ) ] × m = - sinc [ ( 2 m - 1 ) π 2 ] × exp [ j k 2 ( 2 m - 1 ) f F x 0 2 ] } ,
u ( x , z ) = 2 π 1 ω 0 exp ( j k z ) m = - sinc [ ( 2 m - 1 ) π 2 ] × 1 ω m ( z ) exp { j k 2 z [ 1 ω m 2 ( z ) + 1 ] x 2 } ,
ω m ( z ) = [ 1 + z ( 2 m - 1 f F - 1 f ind + j λ π ω 0 2 ) ] 1 / 2 .
f eff = ( 1 f F 2 + 1 f ind 2 + 2 f ind f F + 1 q ˜ 0 2 ) - 1 / 2 ,
m 1 4 ( x 1 S ) 2 ,
V ( x , z ) = n - 1 P n sin ( a n x ) sinh [ a n ( T - z ) ] sinh ( a n T ) ,
V M ( x , z ) = n = 1 P M , n sin ( a n x ) sinh [ a n ( T - z ) ] sinh ( a n T ) ,
V x x ( x , z ) + V z z ( x , z ) = f ( x , z ) ,
x i = a + i δ x ,
z j = c + j δ z ,
v i + 1 , j - 2 v i , j + v i - 1 , j δ x 2 + v i , j + 1 - 2 v i , j + v i , j - 1 δ z 2 = f i , j = 0 ,
b a ( z ) = v ( a , z ) = 0 ,
b b ( z ) = v ( b , z ) = 0 ,
b c ( x ) = v ( x , c ) ,
b d ( x ) = v ( x , d ) = 0 ,
v i + 1 , j - 2 v i , j + v i - 1 , j δ x 2 + v i , j + 1 - 2 v i , j + v i , j - 1 δ z 2 = f i , j = g i , j ,
v i , j = k = 1 M - 1 V k , j sin ( k i π M ) .
G k , j = 2 M i = 1 M - 1 g i , j sin ( i k π M ) ,
V k , j + 1 - ( 2 + 4 ρ 2 sin 2 k π 2 M ) V k , j + V k , j - 1 = δ z 2 G k , j             for i = 1 , , M - 1 , j = 2 , , N - 2 ;
V k , 2 - ( 2 + 4 ρ 2 sin 2 k π 2 M ) V k , 1 = δ z 2 G k , 1             for i = 1 , M - 1 , j = 1 ;
- ( 2 + 4 ρ 2 sin 2 k π 2 M ) V k , N - 1 + V k , N - 2 = δ z 2 G k , N - 1             for i = 1 , M - 1 , j = N - 1.

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