## Design of a zoom lens without motorized optical elements

Optics Express, Vol. 15, Issue 11, pp. 6664-6669 (2007)

http://dx.doi.org/10.1364/OE.15.006664

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### Abstract

A novel design of a zoom lens system without motorized movements is proposed. The lens system consists of a fixed lens and two double-liquid variable-focus lenses. The liquid lenses, made out of two immiscible liquids, are based on the principle of electrowetting: an effect controlling the wetting properties of a liquid on a solid by modifying the applied voltage at the solid-liquid interface. The structure and principle of the lens system are introduced in this paper. Detailed calculations and simulation examples are presented to show that this zoom lens system appears viable as the next-generation zoom lens.

© 2007 Optical Society of America

## 1. Introduction

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]

5. 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**, 603402-1–603402-9 (2006). [CrossRef]

## 2. Principle of the zoom lens

*n*

_{1}and the other insulating (Liquid 2) with refractive index

*n*

_{2}. These two liquids have the same density, thus the interface of liquid 1 and liquid 2 is perfectly spherical [6

6. B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage,” Eur. Phys. J. E. **3**, 159–163 (2000). [CrossRef]

*d*and relative permittivity

*ε*are subsequently coated inside the cylindrical chamber. A voltage

_{r}*U*applied between the conducting liquid and the electrode results in a variation of the interface curvature. If

*n*

_{1}≠

*n*

_{2}, the double-liquid system can be considered as a variable focus lens. Equation (1) [7

7. S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. **85**, 1128–1130 (2004). [CrossRef]

*r*.

*a*is the inner radius of the cylindrical chamber,

*θ*

_{0}is the initial contact angle when there is no external voltage,

*ε*

_{0}is the permittivity of the free space and

*γ*

_{12}is the tension coefficient of the interface between liquid 1 and liquid 2.

## 3. Theoretical analysis

*ĺ*, the distance from the focus

*P*to the back surface of the second liquid lens, can be given by

*n*

_{0}is the refractive index of air. For a given zoom lens system,

*ĺ*is a constant. We mark the thickness of the first and second liquid lens as

*d*

_{01}and

*d*

_{02}respectively, and the volume percentage of liquid 2 for the first and second liquid lens is denoted by

*k*

_{1}and

*k*

_{2}times. That is

*d*

_{2}=

*k*

_{1}

*d*

_{01}and

*d*

_{5}=

*k*

_{2}

*d*

_{02}, and Eq. (2) can be rewritten as

*r*

_{2}and

*r*

_{5}should have opposite signs. Figure 4 shows such a zoom lens with a concave interface (i.e. negative

*r*

_{2}) for the first double-liquid lens under some external voltage. The focus position of the system in this case can be derived and given by

*a*

_{1}and

*a*

_{2}are the inner radius of the cylindrical chamber for the first and second liquid lens respectively. Equations (6) and (7) also shows that the absolute value for both

*r*

_{2}and

*r*

_{5}have a lower limit, that is |

*r*

_{2}| ≥

*a*

_{1}and |

*r*

_{5}| ≥

*a*

_{2}.

*α*is a function of

*r*

_{2}and

*r*

_{5}is implicated in

*d́*

_{5}. When

*r*

_{2}, which is determined by the external voltage, is given, Eq. (9) can be solved analytically. However there will appear a high-order equation in terms of

*r*

_{5}and extraneous roots will occur. The relation between

*r*

_{5}and

*r*

_{2}is then achieved by a numerically iterative method.

*U*

_{1}and

*U*

_{2}respectively. According to Eq. (1), we have

*r*

_{5}and

*r*

_{2}can be readily converted into that of

*U*

_{2}and

*U*

_{1}.

*f*, can be expressed in

*r*

_{2}and

*r*

_{5}by

## 4. Results and discussion

*n*

_{0}= 1,

*n*

_{1}= 1.38,

*n*

_{2}= 1.55,

*d*

_{01}=

*d*

_{02}= 2

*mm*,

*d*

_{1}= 0.05

*mm*,

*d*

_{4}= 2.95

*mm*,

*k*

_{1}= 1/3,

*k*

_{2}= 2/3 and

*f*

_{0}= 10

*mm*. Using Eq. (9), we can plot

*r*

_{5}as a function of

*r*

_{2}in Fig. 5. In agreement with earlier analysis, the signs of

*r*

_{5}and

*r*

_{2}remain opposite. Due to the limit we mentioned earlier for

*r*

_{2}and

*r*

_{5}, there exists a gap on the curve in Fig. 5 corresponding to the limited range.

*a*

_{1}=

*a*

_{2}= 1

*mm*) and choose two liquids with an interface tension coefficient

*γ*

_{12}=38.1×10

^{-3}

*N/m*for two double-liquid variable-focus lenses. According to Eqs. (9), (10) and (11), the relation between voltage

*U*

_{2}and voltage

*U*

_{1}can be simulated for different dielectric thicknesses, and the results are shown in Fig. 6(a). Figure 6(b) shows the focal length of the system as a function of the applied voltage

*U*

_{1}for two different thicknesses of the dielectric layer.

## 5. Acknowledgments

## References and links

1. | L. Saurei, G. Mathieu, and B. Berge, “Design of an autofocus lens for VGA 1/4-in. CCD and CMOS sensors,” Proc. SPIE |

2. | G. I. Swanson and W. B. Veldkamp, “Infrared applications of diffractive optical elements,” Proc. SPIE |

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. Opt. |

4. | A. H. Robert and B. J. Feenstra, “Video-speed electronic paper based on electrowetting,” Nature |

5. | 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 |

6. | B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage,” Eur. Phys. J. E. |

7. | S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. |

**OCIS Codes**

(220.2560) Optical design and fabrication : Propagating methods

(220.2740) Optical design and fabrication : Geometric optical design

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: April 30, 2007

Revised Manuscript: May 2, 2007

Manuscript Accepted: May 3, 2007

Published: May 15, 2007

**Citation**

Runling Peng, Jiabi Chen, Cheng Zhu, and Songlin Zhuang, "Design of a zoom lens without motorized optical
elements," Opt. Express **15**, 6664-6669 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-11-6664

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### References

- L. Saurei, G. Mathieu, and B. Berge, "Design of an autofocus lens for VGA ¼-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).
- 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. Opt. 43, 6278-6284 (2004).
- A. H. Robert and B. J. Feenstra, "Video-speed electronic paper based on electrowetting," Nature 25, 383-385 (2003).
- 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, 603402-1-603402-9 (2006). [CrossRef]
- B. Berge and J. Peseux, "Variable focal lens controlled by an external voltage," Eur. Phys. J. E. 3, 159-163 (2000). [CrossRef]
- S. Kuiper and B. H. W. Hendriks, "Variable-focus liquid lens for miniature cameras," Appl. Phys. Lett. 85, 1128-1130 (2004). [CrossRef]

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