Abstract

A finite-element model (FEM) is employed to study the pressure response of deformable elastic membranes used as tunable optical elements. The model is capable of determining in situ both the modulus and the prestrain from a measurement of peak deflection versus pressure. Given accurate values for modulus and prestrain, it is shown that the two parameters of a standard optical shape function (radius of curvature and conic constant) can be accurately predicted. The effects of prestrain in polydimethylsiloxane (PDMS) membranes are investigated in detail. It was found that prestrain reduces the sensitivity of the membrane shape to the details of the edge clamping. It also reduces the variation of the conic constant with changes in curvature. Thus the ability to control the prestrain as well as thickness and modulus is important to developing robust optical designs based on fluid-driven polymer lenses.

© 2008 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. D. Y. Zhang, N. Justis, V. Lien, Y. Berdichevstky, and Y. H. Lo, “High-performance fluidic adaptive lenses,” Appl. Opt. 43, 783-787 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2006 (2)

P. S. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
[CrossRef]

H. Ren, D. Fox, P. A. Anderson, B. Wu, and S. T. Wu, “Tunale-focus liquid lens controlled using a servo motor,” Opt. Express 14, 8031-8036 (2006).
[CrossRef] [PubMed]

2005 (2)

A. Werber and H. Zappe, “Tunable microfluidic microlenses,” Appl. Opt. 44, 3238-3244 (2005).
[CrossRef] [PubMed]

R. A. Gunasekaran, M. Agarwal, A. Singh, P. Dubasi, P. Coane, and K. Varahramyan, “Design and fabrication of fluid controlled dynamic optical lens system,” Opt. Lasers Eng. 43, 686-703 (2005).
[CrossRef]

2004 (5)

D. Y. Zhang, N. Justis, V. Lien, Y. Berdichevstky, and Y. H. Lo, “High-performance fluidic adaptive lenses,” Appl. Opt. 43, 783-787 (2004).
[CrossRef] [PubMed]

M. Agarwal, R. A. Gunasekaran, P. Coane, and K. Varahramyan, “Polymer-based variable focal length microlens system,” J. Micromech. Microeng. 14, 1665-1673 (2004).
[CrossRef]

J. Chen, W. Wang, J. Fang, and K. Varahramyan, “Variable-focusing microlens with microfluidic chip,” J. Micromech. Microeng. 14, 675-680 (2004).
[CrossRef]

D. Y. Zhang, N. Justis, and Y. H. Lo, “Fluidic adaptive lens of transformable lens type,” Appl. Phys. Lett. 84, 4194-4196(2004).
[CrossRef]

H. Oku, K. Hashimoto, and M. Ishikawa, “Variable-focus lens with 1 kHz bandwidth,” Opt. Express 12, 2138-2149 (2004).
[CrossRef] [PubMed]

2003 (2)

T. S. Krupenkin, T. S. Yang, and P. Mach, “Tunable liquid micro-lens,” Appl. Phys. Lett. 82, 316-318 (2003).
[CrossRef]

D. Y. Zhang, V. Lien, Y. Berdichevstky, J. Choi, and Y. H. Lo, “Fluidic adaptive lens with high focal length tunability,” Appl. Phys. Lett. 82, 3171-3172 (2003).
[CrossRef]

2000 (2)

G. Commander, S. E. Day, and D. R. Selviah, “Variable focal length microlenses,” Opt. Commun. 177, 157-170 (2000).
[CrossRef]

B. Berge and J. Perseux, “Variable focal length controlled by an external voltage: an application of electrowetting,” Eur. Phys. J. E 3, 159-163 (2000).
[CrossRef]

1998 (1)

M. Sheplak and J. Dugundji, “Large deflections of clamped circular plates under initial tension and transitions to membrane behavior,” ASME J. Appl. Mech. 65, 107-115 (1998).
[CrossRef]

1993 (1)

Agarwal, M.

R. A. Gunasekaran, M. Agarwal, A. Singh, P. Dubasi, P. Coane, and K. Varahramyan, “Design and fabrication of fluid controlled dynamic optical lens system,” Opt. Lasers Eng. 43, 686-703 (2005).
[CrossRef]

M. Agarwal, R. A. Gunasekaran, P. Coane, and K. Varahramyan, “Polymer-based variable focal length microlens system,” J. Micromech. Microeng. 14, 1665-1673 (2004).
[CrossRef]

Anderson, P. A.

Berdichevstky, Y.

D. Y. Zhang, N. Justis, V. Lien, Y. Berdichevstky, and Y. H. Lo, “High-performance fluidic adaptive lenses,” Appl. Opt. 43, 783-787 (2004).
[CrossRef] [PubMed]

D. Y. Zhang, V. Lien, Y. Berdichevstky, J. Choi, and Y. H. Lo, “Fluidic adaptive lens with high focal length tunability,” Appl. Phys. Lett. 82, 3171-3172 (2003).
[CrossRef]

Berge, B.

B. Berge and J. Perseux, “Variable focal length controlled by an external voltage: an application of electrowetting,” Eur. Phys. J. E 3, 159-163 (2000).
[CrossRef]

Chan, M. L.

P. S. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
[CrossRef]

Chen, J.

J. Chen, W. Wang, J. Fang, and K. Varahramyan, “Variable-focusing microlens with microfluidic chip,” J. Micromech. Microeng. 14, 675-680 (2004).
[CrossRef]

Choi, J.

D. Y. Zhang, V. Lien, Y. Berdichevstky, J. Choi, and Y. H. Lo, “Fluidic adaptive lens with high focal length tunability,” Appl. Phys. Lett. 82, 3171-3172 (2003).
[CrossRef]

Coane, P.

R. A. Gunasekaran, M. Agarwal, A. Singh, P. Dubasi, P. Coane, and K. Varahramyan, “Design and fabrication of fluid controlled dynamic optical lens system,” Opt. Lasers Eng. 43, 686-703 (2005).
[CrossRef]

M. Agarwal, R. A. Gunasekaran, P. Coane, and K. Varahramyan, “Polymer-based variable focal length microlens system,” J. Micromech. Microeng. 14, 1665-1673 (2004).
[CrossRef]

Commander, G.

G. Commander, S. E. Day, and D. R. Selviah, “Variable focal length microlenses,” Opt. Commun. 177, 157-170 (2000).
[CrossRef]

Day, S. E.

G. Commander, S. E. Day, and D. R. Selviah, “Variable focal length microlenses,” Opt. Commun. 177, 157-170 (2000).
[CrossRef]

Dharmatilleke, S.

P. S. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
[CrossRef]

Dubasi, P.

R. A. Gunasekaran, M. Agarwal, A. Singh, P. Dubasi, P. Coane, and K. Varahramyan, “Design and fabrication of fluid controlled dynamic optical lens system,” Opt. Lasers Eng. 43, 686-703 (2005).
[CrossRef]

Dugundji, J.

M. Sheplak and J. Dugundji, “Large deflections of clamped circular plates under initial tension and transitions to membrane behavior,” ASME J. Appl. Mech. 65, 107-115 (1998).
[CrossRef]

Fang, J.

J. Chen, W. Wang, J. Fang, and K. Varahramyan, “Variable-focusing microlens with microfluidic chip,” J. Micromech. Microeng. 14, 675-680 (2004).
[CrossRef]

Fox, D.

Gunasekaran, R. A.

R. A. Gunasekaran, M. Agarwal, A. Singh, P. Dubasi, P. Coane, and K. Varahramyan, “Design and fabrication of fluid controlled dynamic optical lens system,” Opt. Lasers Eng. 43, 686-703 (2005).
[CrossRef]

M. Agarwal, R. A. Gunasekaran, P. Coane, and K. Varahramyan, “Polymer-based variable focal length microlens system,” J. Micromech. Microeng. 14, 1665-1673 (2004).
[CrossRef]

Hashimoto, K.

Ishikawa, M.

Justis, N.

D. Y. Zhang, N. Justis, V. Lien, Y. Berdichevstky, and Y. H. Lo, “High-performance fluidic adaptive lenses,” Appl. Opt. 43, 783-787 (2004).
[CrossRef] [PubMed]

D. Y. Zhang, N. Justis, and Y. H. Lo, “Fluidic adaptive lens of transformable lens type,” Appl. Phys. Lett. 84, 4194-4196(2004).
[CrossRef]

Khaw, A. H.

P. S. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
[CrossRef]

Krupenkin, T. S.

T. S. Krupenkin, T. S. Yang, and P. Mach, “Tunable liquid micro-lens,” Appl. Phys. Lett. 82, 316-318 (2003).
[CrossRef]

Lien, V.

D. Y. Zhang, N. Justis, V. Lien, Y. Berdichevstky, and Y. H. Lo, “High-performance fluidic adaptive lenses,” Appl. Opt. 43, 783-787 (2004).
[CrossRef] [PubMed]

D. Y. Zhang, V. Lien, Y. Berdichevstky, J. Choi, and Y. H. Lo, “Fluidic adaptive lens with high focal length tunability,” Appl. Phys. Lett. 82, 3171-3172 (2003).
[CrossRef]

Lo, Y. H.

D. Y. Zhang, N. Justis, and Y. H. Lo, “Fluidic adaptive lens of transformable lens type,” Appl. Phys. Lett. 84, 4194-4196(2004).
[CrossRef]

D. Y. Zhang, N. Justis, V. Lien, Y. Berdichevstky, and Y. H. Lo, “High-performance fluidic adaptive lenses,” Appl. Opt. 43, 783-787 (2004).
[CrossRef] [PubMed]

D. Y. Zhang, V. Lien, Y. Berdichevstky, J. Choi, and Y. H. Lo, “Fluidic adaptive lens with high focal length tunability,” Appl. Phys. Lett. 82, 3171-3172 (2003).
[CrossRef]

Mach, P.

T. S. Krupenkin, T. S. Yang, and P. Mach, “Tunable liquid micro-lens,” Appl. Phys. Lett. 82, 316-318 (2003).
[CrossRef]

Moran, P. S.

P. S. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
[CrossRef]

Morita, S.

Oku, H.

Perseux, J.

B. Berge and J. Perseux, “Variable focal length controlled by an external voltage: an application of electrowetting,” Eur. Phys. J. E 3, 159-163 (2000).
[CrossRef]

Ren, H.

Rodriguez, I.

P. S. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
[CrossRef]

Selviah, D. R.

G. Commander, S. E. Day, and D. R. Selviah, “Variable focal length microlenses,” Opt. Commun. 177, 157-170 (2000).
[CrossRef]

Sheplak, M.

M. Sheplak and J. Dugundji, “Large deflections of clamped circular plates under initial tension and transitions to membrane behavior,” ASME J. Appl. Mech. 65, 107-115 (1998).
[CrossRef]

Singh, A.

R. A. Gunasekaran, M. Agarwal, A. Singh, P. Dubasi, P. Coane, and K. Varahramyan, “Design and fabrication of fluid controlled dynamic optical lens system,” Opt. Lasers Eng. 43, 686-703 (2005).
[CrossRef]

Sugiura, N.

Tan, K. W.

P. S. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
[CrossRef]

Varahramyan, K.

R. A. Gunasekaran, M. Agarwal, A. Singh, P. Dubasi, P. Coane, and K. Varahramyan, “Design and fabrication of fluid controlled dynamic optical lens system,” Opt. Lasers Eng. 43, 686-703 (2005).
[CrossRef]

M. Agarwal, R. A. Gunasekaran, P. Coane, and K. Varahramyan, “Polymer-based variable focal length microlens system,” J. Micromech. Microeng. 14, 1665-1673 (2004).
[CrossRef]

J. Chen, W. Wang, J. Fang, and K. Varahramyan, “Variable-focusing microlens with microfluidic chip,” J. Micromech. Microeng. 14, 675-680 (2004).
[CrossRef]

Walish, J.

J. Walish, personal communication (MIT, 2007).

Wang, W.

J. Chen, W. Wang, J. Fang, and K. Varahramyan, “Variable-focusing microlens with microfluidic chip,” J. Micromech. Microeng. 14, 675-680 (2004).
[CrossRef]

Werber, A.

Wu, B.

Wu, S. T.

Yang, T. S.

T. S. Krupenkin, T. S. Yang, and P. Mach, “Tunable liquid micro-lens,” Appl. Phys. Lett. 82, 316-318 (2003).
[CrossRef]

Zappe, H.

Zhang, D. Y.

D. Y. Zhang, N. Justis, V. Lien, Y. Berdichevstky, and Y. H. Lo, “High-performance fluidic adaptive lenses,” Appl. Opt. 43, 783-787 (2004).
[CrossRef] [PubMed]

D. Y. Zhang, N. Justis, and Y. H. Lo, “Fluidic adaptive lens of transformable lens type,” Appl. Phys. Lett. 84, 4194-4196(2004).
[CrossRef]

D. Y. Zhang, V. Lien, Y. Berdichevstky, J. Choi, and Y. H. Lo, “Fluidic adaptive lens with high focal length tunability,” Appl. Phys. Lett. 82, 3171-3172 (2003).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (4)

D. Y. Zhang, V. Lien, Y. Berdichevstky, J. Choi, and Y. H. Lo, “Fluidic adaptive lens with high focal length tunability,” Appl. Phys. Lett. 82, 3171-3172 (2003).
[CrossRef]

T. S. Krupenkin, T. S. Yang, and P. Mach, “Tunable liquid micro-lens,” Appl. Phys. Lett. 82, 316-318 (2003).
[CrossRef]

P. S. Moran, S. Dharmatilleke, A. H. Khaw, K. W. Tan, M. L. Chan, and I. Rodriguez, “Fluidic lenses with variable focal length,” Appl. Phys. Lett. 88, 041120 (2006).
[CrossRef]

D. Y. Zhang, N. Justis, and Y. H. Lo, “Fluidic adaptive lens of transformable lens type,” Appl. Phys. Lett. 84, 4194-4196(2004).
[CrossRef]

ASME J. Appl. Mech. (1)

M. Sheplak and J. Dugundji, “Large deflections of clamped circular plates under initial tension and transitions to membrane behavior,” ASME J. Appl. Mech. 65, 107-115 (1998).
[CrossRef]

Eur. Phys. J. E (1)

B. Berge and J. Perseux, “Variable focal length controlled by an external voltage: an application of electrowetting,” Eur. Phys. J. E 3, 159-163 (2000).
[CrossRef]

J. Micromech. Microeng. (2)

M. Agarwal, R. A. Gunasekaran, P. Coane, and K. Varahramyan, “Polymer-based variable focal length microlens system,” J. Micromech. Microeng. 14, 1665-1673 (2004).
[CrossRef]

J. Chen, W. Wang, J. Fang, and K. Varahramyan, “Variable-focusing microlens with microfluidic chip,” J. Micromech. Microeng. 14, 675-680 (2004).
[CrossRef]

Opt. Commun. (1)

G. Commander, S. E. Day, and D. R. Selviah, “Variable focal length microlenses,” Opt. Commun. 177, 157-170 (2000).
[CrossRef]

Opt. Express (2)

Opt. Lasers Eng. (1)

R. A. Gunasekaran, M. Agarwal, A. Singh, P. Dubasi, P. Coane, and K. Varahramyan, “Design and fabrication of fluid controlled dynamic optical lens system,” Opt. Lasers Eng. 43, 686-703 (2005).
[CrossRef]

Other (1)

J. Walish, personal communication (MIT, 2007).

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Figures (9)

Fig. 1
Fig. 1

Illustration of a thin-film polymer membrane (of radius a and thickness h) bonded to a rigid washer.

Fig. 2
Fig. 2

(a) Effects of prestrain ( ε 0 ) on the relationship between normalized peak deflection ( δ 0 / a ) and applied pressure ( p / E ) for various membrane geometries ( h / a ), and (b) magnified view of δ 0 / a p / E at typical operational regime.

Fig. 3
Fig. 3

Test configuration for the pressure-dependent shape measurements.

Fig. 4
Fig. 4

3D axial-symmetric FEM of the bonded membrane system used for membrane deformation calculations

Fig. 5
Fig. 5

(a) FEM predicted normalized deformation shape ( δ ( r ) / a ) as a function of normalized position ( ( r ) / a ) at various deformation levels for ε 0 = 0 % (red dashed curves with circles), ε 0 = 2 % (black single-dotted–dashed curves), and ε 0 = 4 % (pink solid curves). Also superimposed are spherical shapes at the corresponding peak deformation levels (blue double-dotted–dashed curves) for comparison. (b) Microsoft Excel fitted conic constant (K) and normalized radius of curvature ( R / a ) from the deformation data of (a) plotted as functions of applied pressure ( p / E ). The solid curves refer to the left Y axis and dashed curves to the right Y axis

Fig. 6
Fig. 6

(a) Best fit of FEM result of peak deflection ( δ 0 ) as a function of applied pressure (p) (red solid curve) as compared with the experimentally measured data (squares). Two other FEM curves of δ 0 p using different moduli and prestrains are also plotted to show the influence of these two parameters on the δ 0 p data. (b) FEM predicted deformation profiles (black solid curves) under various high pressure levels, compared with the experimentally measured deformation (curves with high-frequency undulations). (c) Comparison of FEM predicted deformations (black solid curves) with experimental measurements (curves with undulations) at lower pressure levels.

Fig. 7
Fig. 7

(a) FEM calculated δ 0 p using best-fit E and ε 0 as indicated for three membranes (solid curves) compared with experimentally measured data (symbols). (b) Normalized δ 0 / a p / E curves showing the different initial slopes that are inversely proportional to the level of prestrains in the three membranes.

Fig. 8
Fig. 8

(a) Comparison of FEM predicted (solid curves) and experimentally measured (symbols) normalized radius of curvature ( R / a ) as functions of normalized pressure ( p / E ) for the three membranes with different mechanical properties. (b) Conic constant K predicted (solid) and extracted from experimental measurements (symbols).

Fig. 9
Fig. 9

ZEMAX calculation for the focal spot size of a single double convex fluidic lens with diameter 8 mm , aperture 6 mm , material n = 1.67 , and λ = 550 nm . Triangles are for a lens with radius of curvature R = 25 mm , and the squares are for R = 8 mm . Above the plot are schematic representations of lenses with negative (left) and positive (right) conic constants.

Tables (2)

Tables Icon

Table 1 Process Parameters for Fabrication of the PDMS Films Used in this Study

Tables Icon

Table 2 Membrane Geometries and Calculated Modulus and Prestrain

Equations (8)

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W ( r ) = δ 0 [ 1 ( r / a ) 2 ] 2 ,
p E = 1 21 ( 1 ν 2 ) [ 112 ( h a ) 2 + ( 46 + 28 ν 18 ν 2 ) ( δ 0 a ) 2 + 84 ε 0 ( 1 + v ) ] ( δ 0 a ) ( h a ) .
p E = 16 3 ( 1 ν 2 ) ( h a ) 3 δ 0 a
p E = 4 ε 0 ( 1 ν ) h a δ 0 a ( δ 0 a 1 , h a 1 ) .
p E = 46 + 28 ν 18 ν 2 21 ( 1 ν 2 ) h a ( δ 0 a ) 3 ( large   δ 0 a ) .
δ ( r ) a = ( r / a ) 2 ( a / R ) 1 + 1 ( 1 + K ) ( r / a ) 2 ( a / R ) 2 ,
R = ± b 1 2 / b 2 , K = ( b 1 2 b 2 2 ) / b 1 2 .
ε 0 ( 1 ν ) 4 a h a k E ,

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