Abstract

The interface shape of two immiscible liquids in a conical chamber is discussed. The analytical solution of the differential equation describing the interface shape shows that the interface shape is completely spherical when the density difference of two liquids is zero. On the basis of the spherical-interface shape and an energy-minimization method, explicit calculations and detailed analyses of an extended Young-type equation for the conical double-liquid lens are given. Finally, a novel design of a zoom lens system without motorized movements is proposed. The lens system consists of a fixed lens and two conical double-liquid variable-focus lenses. The structure and principle of the lens system are introduced in this paper. Taking finite objects as example, detailed calculations and simulation examples are presented to predict how two liquid lenses are related to meet the basic requirements of zoom lenses.

© 2008 Optical Society of America

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References

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  1. L. Saurei, G. Mathieu, and B. Berge, “Design of an autofocus lens for VGA 1/4-in. CCD and CMOS sensors,” Proc. SPIE 5249, 288-296 (2004).
    [CrossRef]
  2. G. I. Swanson and W. B. Veldkamp, “Infrared applications of diffractive optical elements,” Proc. SPIE 883, 155-158 (1988).
  3. M. Ignacio, L. Claudio, M. Andres, C. Juan, and J. Y. Maria, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Phys. Lett. 43, 6728-6284 (2004).
  4. R. A. Hayes and B. J. Feenstra, “Video-speed electronic paper based on electrowetting,” Nature 425, 383-385 (2003).
    [CrossRef] [PubMed]
  5. B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: an application of electrowetting,” Eur. Phys. J. E 3, 159-163 (2000).
    [CrossRef]
  6. J. Lee, H. Moon, J. Fowler, T. Schoellhammer, and C.-J. Kim, “Electrowetting and electrowetting-on-dielectric for microscale liquid handling,” Sens. Actuators A 95, 259-268 (2002).
    [CrossRef]
  7. B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
    [CrossRef]
  8. R. Peng, J. Chen, and S. Zhuang, “Design of a zoom lens without motorized optical elements,” Opt. Express 15, 6664-6669 (2007).
    [CrossRef] [PubMed]
  9. E. M. Blokhuis, “Liquid drops at surfaces,” in Surface and Interfacial Tension: Measurement, Theory, and ApplicationS.Hartland, ed. (Academic, 2004), pp. 149-193.
  10. F. Mugele and J.-C. Baret, “Electrowetting: from basics to applications,” J. Phys.: Condens. Matter 17, R705-R774 (2005).
    [CrossRef]
  11. W. Ramsay and F. G. Donnan, A System of Physical Chemistry (Longmans, Green; Academic, 1925), Chap. IV.
  12. T. Young, “An essay on the cohesion of fluids,” Philos. Trans. R. Soc. London 95, 65-87 (1805).
    [CrossRef]

2007 (1)

2006 (1)

B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
[CrossRef]

2005 (1)

F. Mugele and J.-C. Baret, “Electrowetting: from basics to applications,” J. Phys.: Condens. Matter 17, R705-R774 (2005).
[CrossRef]

2004 (2)

L. Saurei, G. Mathieu, and B. Berge, “Design of an autofocus lens for VGA 1/4-in. CCD and CMOS sensors,” Proc. SPIE 5249, 288-296 (2004).
[CrossRef]

M. Ignacio, L. Claudio, M. Andres, C. Juan, and J. Y. Maria, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Phys. Lett. 43, 6728-6284 (2004).

2003 (1)

R. A. Hayes and B. J. Feenstra, “Video-speed electronic paper based on electrowetting,” Nature 425, 383-385 (2003).
[CrossRef] [PubMed]

2002 (1)

J. Lee, H. Moon, J. Fowler, T. Schoellhammer, and C.-J. Kim, “Electrowetting and electrowetting-on-dielectric for microscale liquid handling,” Sens. Actuators A 95, 259-268 (2002).
[CrossRef]

2000 (1)

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

1988 (1)

G. I. Swanson and W. B. Veldkamp, “Infrared applications of diffractive optical elements,” Proc. SPIE 883, 155-158 (1988).

1805 (1)

T. Young, “An essay on the cohesion of fluids,” Philos. Trans. R. Soc. London 95, 65-87 (1805).
[CrossRef]

Andres, M.

M. Ignacio, L. Claudio, M. Andres, C. Juan, and J. Y. Maria, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Phys. Lett. 43, 6728-6284 (2004).

Baret, J.-C.

F. Mugele and J.-C. Baret, “Electrowetting: from basics to applications,” J. Phys.: Condens. Matter 17, R705-R774 (2005).
[CrossRef]

Berge, B.

L. Saurei, G. Mathieu, and B. Berge, “Design of an autofocus lens for VGA 1/4-in. CCD and CMOS sensors,” Proc. SPIE 5249, 288-296 (2004).
[CrossRef]

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

Blokhuis, E. M.

E. M. Blokhuis, “Liquid drops at surfaces,” in Surface and Interfacial Tension: Measurement, Theory, and ApplicationS.Hartland, ed. (Academic, 2004), pp. 149-193.

Chen, J.

Claudio, L.

M. Ignacio, L. Claudio, M. Andres, C. Juan, and J. Y. Maria, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Phys. Lett. 43, 6728-6284 (2004).

Donnan, F. G.

W. Ramsay and F. G. Donnan, A System of Physical Chemistry (Longmans, Green; Academic, 1925), Chap. IV.

Feenstra, B. J.

R. A. Hayes and B. J. Feenstra, “Video-speed electronic paper based on electrowetting,” Nature 425, 383-385 (2003).
[CrossRef] [PubMed]

Fowler, J.

J. Lee, H. Moon, J. Fowler, T. Schoellhammer, and C.-J. Kim, “Electrowetting and electrowetting-on-dielectric for microscale liquid handling,” Sens. Actuators A 95, 259-268 (2002).
[CrossRef]

Hayes, R. A.

R. A. Hayes and B. J. Feenstra, “Video-speed electronic paper based on electrowetting,” Nature 425, 383-385 (2003).
[CrossRef] [PubMed]

Hendriks, B. H. W.

B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
[CrossRef]

Ignacio, M.

M. Ignacio, L. Claudio, M. Andres, C. Juan, and J. Y. Maria, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Phys. Lett. 43, 6728-6284 (2004).

Juan, C.

M. Ignacio, L. Claudio, M. Andres, C. Juan, and J. Y. Maria, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Phys. Lett. 43, 6728-6284 (2004).

Kim, C.-J.

J. Lee, H. Moon, J. Fowler, T. Schoellhammer, and C.-J. Kim, “Electrowetting and electrowetting-on-dielectric for microscale liquid handling,” Sens. Actuators A 95, 259-268 (2002).
[CrossRef]

Kuiper, S.

B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
[CrossRef]

Lee, J.

J. Lee, H. Moon, J. Fowler, T. Schoellhammer, and C.-J. Kim, “Electrowetting and electrowetting-on-dielectric for microscale liquid handling,” Sens. Actuators A 95, 259-268 (2002).
[CrossRef]

Maria, J. Y.

M. Ignacio, L. Claudio, M. Andres, C. Juan, and J. Y. Maria, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Phys. Lett. 43, 6728-6284 (2004).

Mathieu, G.

L. Saurei, G. Mathieu, and B. Berge, “Design of an autofocus lens for VGA 1/4-in. CCD and CMOS sensors,” Proc. SPIE 5249, 288-296 (2004).
[CrossRef]

Moon, H.

J. Lee, H. Moon, J. Fowler, T. Schoellhammer, and C.-J. Kim, “Electrowetting and electrowetting-on-dielectric for microscale liquid handling,” Sens. Actuators A 95, 259-268 (2002).
[CrossRef]

Mugele, F.

F. Mugele and J.-C. Baret, “Electrowetting: from basics to applications,” J. Phys.: Condens. Matter 17, R705-R774 (2005).
[CrossRef]

Peng, R.

Peseux, J.

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

Ramsay, W.

W. Ramsay and F. G. Donnan, A System of Physical Chemistry (Longmans, Green; Academic, 1925), Chap. IV.

Renders, C. A.

B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
[CrossRef]

Saurei, L.

L. Saurei, G. Mathieu, and B. Berge, “Design of an autofocus lens for VGA 1/4-in. CCD and CMOS sensors,” Proc. SPIE 5249, 288-296 (2004).
[CrossRef]

Schoellhammer, T.

J. Lee, H. Moon, J. Fowler, T. Schoellhammer, and C.-J. Kim, “Electrowetting and electrowetting-on-dielectric for microscale liquid handling,” Sens. Actuators A 95, 259-268 (2002).
[CrossRef]

Swanson, G. I.

G. I. Swanson and W. B. Veldkamp, “Infrared applications of diffractive optical elements,” Proc. SPIE 883, 155-158 (1988).

Tukker, T. W.

B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
[CrossRef]

van As, M. A. J.

B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
[CrossRef]

Veldkamp, W. B.

G. I. Swanson and W. B. Veldkamp, “Infrared applications of diffractive optical elements,” Proc. SPIE 883, 155-158 (1988).

Young, T.

T. Young, “An essay on the cohesion of fluids,” Philos. Trans. R. Soc. London 95, 65-87 (1805).
[CrossRef]

Zhuang, S.

Appl. Phys. Lett. (1)

M. Ignacio, L. Claudio, M. Andres, C. Juan, and J. Y. Maria, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Phys. Lett. 43, 6728-6284 (2004).

Eur. Phys. J. E (1)

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

J. Phys.: Condens. Matter (1)

F. Mugele and J.-C. Baret, “Electrowetting: from basics to applications,” J. Phys.: Condens. Matter 17, R705-R774 (2005).
[CrossRef]

Nature (1)

R. A. Hayes and B. J. Feenstra, “Video-speed electronic paper based on electrowetting,” Nature 425, 383-385 (2003).
[CrossRef] [PubMed]

Opt. Express (1)

Philos. Trans. R. Soc. London (1)

T. Young, “An essay on the cohesion of fluids,” Philos. Trans. R. Soc. London 95, 65-87 (1805).
[CrossRef]

Proc. SPIE (3)

B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
[CrossRef]

L. Saurei, G. Mathieu, and B. Berge, “Design of an autofocus lens for VGA 1/4-in. CCD and CMOS sensors,” Proc. SPIE 5249, 288-296 (2004).
[CrossRef]

G. I. Swanson and W. B. Veldkamp, “Infrared applications of diffractive optical elements,” Proc. SPIE 883, 155-158 (1988).

Sens. Actuators A (1)

J. Lee, H. Moon, J. Fowler, T. Schoellhammer, and C.-J. Kim, “Electrowetting and electrowetting-on-dielectric for microscale liquid handling,” Sens. Actuators A 95, 259-268 (2002).
[CrossRef]

Other (2)

E. M. Blokhuis, “Liquid drops at surfaces,” in Surface and Interfacial Tension: Measurement, Theory, and ApplicationS.Hartland, ed. (Academic, 2004), pp. 149-193.

W. Ramsay and F. G. Donnan, A System of Physical Chemistry (Longmans, Green; Academic, 1925), Chap. IV.

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Figures (7)

Fig. 1
Fig. 1

Conical structure of a double-liquid lens and shape of the liquid interface in terms of y ( x ) .

Fig. 2
Fig. 2

Conical double-liquid lens labeled with different parameters.

Fig. 3
Fig. 3

Schematic of the proposed zoom lens. 1: fixed lens. 2: exit pupil of the fixed lens. 3, 4: double-liquid lens.

Fig. 4
Fig. 4

Schematic of the system with zero focal power for both liquid lenses. P is the object point and P is the image point.

Fig. 5
Fig. 5

Schematic of the system with nonzero focal power for both liquid lenses. P is the object point and P is the image point.

Fig. 6
Fig. 6

Voltage U 2 as a function of voltage U 1 for two different thicknesses of the dielectric layer.

Fig. 7
Fig. 7

Zoom factor of the whole system as a function of the voltage U 1 for two different thicknesses of the dielectric layer.

Equations (53)

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Δ p = γ 12 ( 1 R 1 + 1 R 2 ) ,
1 R 1 = y ( x ) x [ 1 + y ( x ) 2 ] 1 2 ,
1 R 2 = y ( x ) [ 1 + y ( x ) 2 ] 3 2 .
p 10 p 20 = 2 γ 12 R 0 .
p 1 = p 10 ρ 1 g ( y 0 y ) ,
p 2 = p 20 ρ 2 g ( y 0 y ) ,
Δ p = p 1 p 2 = p 10 p 20 ( ρ 2 ρ 1 ) g ( y 0 y ) .
Δ p = p 10 p 20 = 2 γ 12 R 0 .
1 x y ( x ) [ 1 + y ( x ) 2 ] 1 2 + y ( x ) [ 1 + y ( x ) 2 ] 3 2 = 2 R 0 .
y ( x ) = tan φ ( x ) .
y ( x ) = sec 2 φ ( x ) φ ( x ) .
φ ( x ) x cos φ ( x ) + sin φ ( x ) = 2 R 0 x .
φ ( x ) x cos φ ( x ) d x + sin φ ( x ) d x = 2 R 0 x d x + C .
x d sin φ ( x ) + sin φ ( x ) d x = 1 R 0 x 2 + C .
x sin φ ( x ) = 1 R 0 x 2 + C ,
x = R 0 sin φ ( x ) .
y ( x ) = ( d y d φ ) ( d x d φ ) = tan φ ( x ) ,
d x d φ = R 0 cos φ ( x ) .
d y d φ = R 0 sin φ ( x ) ,
y = R 0 cos φ ( x ) + C 1 .
C 1 = y 0 R 0 .
x 2 + ( y ( y 0 R 0 ) ) 2 = R 0 2 .
x 2 + ( y ( y 0 R 0 ) ) 2 + z 2 = R 0 2 .
V = π [ r 3 cos 3 ( θ α ) a 1 3 ] 3 tan α + 2 3 π r 3 π r 3 sin ( θ α ) + 1 3 π r 3 sin 3 ( θ α ) .
d V = V ( r , θ ) r d r + V ( r , θ ) θ d θ = 0 ,
d r = r tan α cos 2 ( θ α ) sin ( θ α ) + r cos 3 ( θ α ) 1 tan α cos 3 ( θ α ) + 2 3 sin ( θ α ) + sin 3 ( θ α ) d θ .
d F = d W ,
A s 1 = π ( a 2 a 1 tan α h 1 ) ( a 2 + a 1 + π h 1 tan α ) + π a 2 2 ,
A s 2 = 2 π a 1 h 1 + π h 1 2 tan α + π a 1 2 ,
A 12 = 2 π r 2 ( 1 sin ( θ α ) ) .
d F = γ s 1 d A s 1 + γ s 2 d A s 2 + γ 12 d A 12 = ( γ s 2 γ s 1 ) 2 π ( a 1 + h 1 tan α ) d h 1 + γ 12 [ 4 π r ( 1 sin ( θ α ) ) d r 2 π r 2 cos ( θ α ) d θ ] .
d F = 0 .
d h 1 = γ 12 [ 2 π r 2 cos ( θ α ) d θ 4 π r ( 1 sin ( θ α ) ) d r ] ( γ s 1 γ s 2 ) 2 π ( a 1 + h 1 tan α ) .
a 1 + h 1 tan α = r cos ( θ α ) .
d h 1 = cos ( θ α ) d r r sin ( θ α ) d θ tan α .
cos ( θ α ) tan α sin ( θ α ) = γ s 2 γ s 1 γ 12 .
cos ( θ α ) tan α sin ( θ α ) = cos ( θ 0 α ) tan α sin ( θ 0 α ) + ϵ 0 ϵ r 2 γ 12 d U 2 .
cos θ = γ s 2 γ s 1 γ 12 .
l = l 0 f 0 f 0 + l 0 d 1 1 n 1 d 2 1 n 2 d 3 d 4 1 n 1 d 5 1 n 2 d 6 ,
l f = f 0 2 + 2 f 0 l min 2 ( f 0 + l min ) d 1 d 4 n 2 n 1 n 1 n 2 ( d 2 + d 5 ) d 01 + d 02 n 2 .
l = r 5 β n 1 r 5 + ( n 2 n 1 ) β 1 n 2 ( d 02 d 5 ) ,
β = ( n 1 ( f 0 l 0 f 0 + l 0 d 1 ) d 2 ) r 2 r 2 + ( n 2 n 1 ) ( f 0 l 0 f 0 + l 0 d 1 d 2 n 1 ) n 1 n 2 ( d 01 d 2 ) n 1 d 4 d 5 , d i ( i = 2 , 3 , 5 , 6 )
r i = ( a i d i tan α ) 3 tan α cos 3 ( θ i α ) tan α + ( 1 sin ( θ i α ) ) 2 ( 2 + sin ( θ i α ) ) 3 ,
d i = a i r i tan α ( 1 sin ( θ i α ) ) r i cos ( θ i α ) tan α ,
l = l f .
r 5 = ( d 02 + n 2 l f d 5 ) ( n 2 n 1 ) β n 2 β n 1 ( d 02 d 5 + n 2 l f ) ,
k = r 2 r 5 ( n 2 n 1 ) 2 ( d 4 + d 01 d 2 n 2 + d 5 n 1 r 2 + r 5 n 2 n 1 ) ( d 1 + d 2 n 1 f 0 ) ( n 2 n 1 ) r 2 ( d 4 + d 01 d 2 n 2 + d 5 n 1 ) + r 2 r 5 .
θ 0 = 2 π 3 , α = π 12 , ε r = 1.993 ,
γ 12 = 38.1 × 10 3 N m ,
n 1 = 1.38 , n 2 = 1.55 , d 1 = 1 mm ,
d 4 = 2.95 mm , l min = 250 mm ,
f 0 = 10 mm , d 2 = 1 mm , d 5 = 1 mm ,
d 01 = d 02 = 2 mm ; a 2 = a 5 = 2 mm .

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