## Four-group stabilized zoom lens design of two focal-length-variable elements |

Optics Express, Vol. 21, Issue 6, pp. 7758-7767 (2013)

http://dx.doi.org/10.1364/OE.21.007758

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### Abstract

We present a theoretical method for analyzing the first-order optics of stabilized zoom lenses with two focal-length-variable elements. The zoom equations are established through the use of the Gaussian brackets method. This is done because the optical power of the focal-length-variable elements varies during the zooming process. The first and second derivatives and the Hessian matrix of the zoom equations with respect to the Gaussian parameters are determined using the equations. These parameters could represent the sensitivity of the zoom ratio of the system to changes in the corresponding system variables. We select the initial values of these system variables, i.e. the magnification of the focal-length-variable element and the structure parameters of the fixed lens group, to be close to the steepest gradient direction. Here the sensitivity of the system focal length is high with respect to variations in the zoom variables. This process leads to an increase in the zoom ratio of the zoom system. The results show successful four-group stabilized zoom lens designs with 2:1 and 5:1 zoom ratios, using two deformable mirrors as focal-length-variable elements. This system, with the inherent characteristics of a steepest gradient, could miniaturize zoom systems.

© 2013 OSA

## 1. Introduction

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

## 2. The zoom equation and its derivatives with respect to magnification

17. M. Herzberger, “Gaussian optics and Gaussian brackets,” J. Opt. Soc. Am. **33**(12), 651–655 (1943). [CrossRef]

## 3. Derivatives of the zoom equation with respect to optical power

## 4. Design examples

### 4.1 Examples for a 2:1 zoom system

12. Y. H. Lin, Y. L. Liu, and G. D. Su, “Optical zoom module based on two deformable mirrors for mobile device applications,” Appl. Opt. **51**(11), 1804–1810 (2012). [CrossRef] [PubMed]

13. B. M. Kaylor, C. R. Wilson, N. J. Greenfield, P. A. Roos, E. M. Seger, M. J. Moghimi, and D. L. Dickensheets, “Miniature non-mechanical zoom camera using deformable MOEMS mirrors,” Proc. SPIE **8252**, 82520N, 82520N-7 (2012). [CrossRef]

### 4.2 Examples for a 5:1 zoom system

## 5. Conclusion and future work

## Acknowledgments

## References and links

1. | K. Yamaji, “Design of zoom lenses,” in |

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

3. | C. Tao, “The varifocal equation for a zoom system,” Kexue Tongbao |

4. | T. ChunKan, “Design of zoom system by the varifocal differential equation. I,” Appl. Opt. |

5. | K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. I: Four-component type,” Appl. Opt. |

6. | K. Tanaka, “Paraxial theory in optical design in terms of gaussian brackets,” in |

7. | E. Betensky, “Forty years of modern zoom lens design,” Proc. SPIE 586506 (2005). [CrossRef] |

8. | S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE |

9. | Q. Hao, X. Cheng, and Y. Song, “Zoom system of MOEMS elements,” PRC Patent 200810119431.4 (18 February 2008). |

10. | P. Valley, M. Reza Dodge, J. Schwiegerling, G. Peyman, and N. Peyghambarian, “Nonmechanical bifocal zoom telescope,” Opt. Lett. |

11. | R. Peng, J. Chen, C. Zhu, and S. Zhuang, “Design of a zoom lens without motorized optical elements,” Opt. Express |

12. | Y. H. Lin, Y. L. Liu, and G. D. Su, “Optical zoom module based on two deformable mirrors for mobile device applications,” Appl. Opt. |

13. | B. M. Kaylor, C. R. Wilson, N. J. Greenfield, P. A. Roos, E. M. Seger, M. J. Moghimi, and D. L. Dickensheets, “Miniature non-mechanical zoom camera using deformable MOEMS mirrors,” Proc. SPIE |

14. | A. Mikš and J. Novák, “Third-order aberrations of the thin refractive tunable-focus lens,” Opt. Lett. |

15. | A. Miks and J. Novak, “Analysis of two-element zoom systems based on variable power lenses,” Opt. Express |

16. | A. Miks, J. Novak, and P. Novak, “Chromatic aberrations of thin refractive variable-focus lens,” Opt. Commun. |

17. | M. Herzberger, “Gaussian optics and Gaussian brackets,” J. Opt. Soc. Am. |

**OCIS Codes**

(080.0080) Geometric optics : Geometric optics

(080.3620) Geometric optics : Lens system design

(080.1753) Geometric optics : Computation methods

**History**

Original Manuscript: January 30, 2013

Manuscript Accepted: March 11, 2013

Published: March 21, 2013

**Citation**

Qun Hao, Xuemin Cheng, and Ke Du, "Four-group stabilized zoom lens design of two focal-length-variable elements," Opt. Express **21**, 7758-7767 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-6-7758

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### References

- K. Yamaji, “Design of zoom lenses,” in Progress in Optics, Vol. 6, E. Wolf, ed. (North-Holland, 1967), pp.105–170.
- S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett.85(7), 1128–1130 (2004). [CrossRef]
- C. Tao, “The varifocal equation for a zoom system,” Kexue Tongbao22(Z1), 207–213 (1977).
- T. ChunKan, “Design of zoom system by the varifocal differential equation. I,” Appl. Opt.31(13), 2265–2273 (1992). [CrossRef] [PubMed]
- K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. I: Four-component type,” Appl. Opt.21(12), 2174–2183 (1982). [CrossRef] [PubMed]
- K. Tanaka, “Paraxial theory in optical design in terms of gaussian brackets,” in Progress in Optics, Vol. 23, E. Wolf, Ed. (North-Holland, Amsterdam, 1986), pp.63–111.
- E. Betensky, “Forty years of modern zoom lens design,” Proc. SPIE586506 (2005). [CrossRef]
- S. Kuiper, B. H. W. Hendriks, J. F. Suijver, S. Deladi, and I. Helwegen, “Zoom camera based on liquid lenses,” Proc. SPIE6466, 64660F, 64660F-7 (2007). [CrossRef]
- Q. Hao, X. Cheng, and Y. Song, “Zoom system of MOEMS elements,” PRC Patent 200810119431.4 (18 February 2008).
- P. Valley, M. Reza Dodge, J. Schwiegerling, G. Peyman, and N. Peyghambarian, “Nonmechanical bifocal zoom telescope,” Opt. Lett.35(15), 2582–2584 (2010). [CrossRef] [PubMed]
- R. Peng, J. Chen, C. Zhu, and S. Zhuang, “Design of a zoom lens without motorized optical elements,” Opt. Express15(11), 6664–6669 (2007). [CrossRef] [PubMed]
- Y. H. Lin, Y. L. Liu, and G. D. Su, “Optical zoom module based on two deformable mirrors for mobile device applications,” Appl. Opt.51(11), 1804–1810 (2012). [CrossRef] [PubMed]
- B. M. Kaylor, C. R. Wilson, N. J. Greenfield, P. A. Roos, E. M. Seger, M. J. Moghimi, and D. L. Dickensheets, “Miniature non-mechanical zoom camera using deformable MOEMS mirrors,” Proc. SPIE8252, 82520N, 82520N-7 (2012). [CrossRef]
- A. Mikš and J. Novák, “Third-order aberrations of the thin refractive tunable-focus lens,” Opt. Lett.35(7), 1031–1033 (2010). [CrossRef] [PubMed]
- A. Miks and J. Novak, “Analysis of two-element zoom systems based on variable power lenses,” Opt. Express18(7), 6797–6810 (2010). [CrossRef] [PubMed]
- A. Miks, J. Novak, and P. Novak, “Chromatic aberrations of thin refractive variable-focus lens,” Opt. Commun.285(10-11), 2506–2509 (2012). [CrossRef]
- M. Herzberger, “Gaussian optics and Gaussian brackets,” J. Opt. Soc. Am.33(12), 651–655 (1943). [CrossRef]

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