## Analysis of three-element zoom lens based on refractive variable-focus lenses |

Optics Express, Vol. 19, Issue 24, pp. 23989-23996 (2011)

http://dx.doi.org/10.1364/OE.19.023989

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### Abstract

Traditional optical systems with variable optical characteristics are composed of several optical elements that can be shifted with respect to each other mechanically. A motorized change of position of individual elements (or group of elements) then makes possible to achieve desired optical properties of such zoom lens systems. A disadvantage of such systems is the fact that individual elements of these optical systems have to move very precisely, which results in high requirements on mechanical construction of such optical systems. Our work is focused on a paraxial and aberration analysis of possible optical designs of three-element zoom lens systems based on variable-focus (tunable-focus) lenses with a variable focal length. First order chromatic aberrations of the variable-focus lenses are also described. Computer simulation examples are presented to show that such zoom lens systems without motorized movements of lenses appear to be promising for the next-generation of zoom lens design.

© 2011 OSA

## 1. Introduction

27. J.-H. Sun, B.-R. Hsueh, Y.-Ch. Fang, J. MacDonald, and C. C. Hu, “Optical design and multiobjective optimization of miniature zoom optics with liquid lens element,” Appl. Opt. **48**(9), 1741–1757 (2009). [CrossRef] [PubMed]

8. A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt. **47**(32), 6088–6098 (2008). [CrossRef] [PubMed]

11. F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE **6501**, 650109, 650109-9 (2007). [CrossRef]

30. A. Mikš and J. Novák, “Third-order aberrations of the thin refractive tunable-focus lens,” Opt. Lett. **35**(7), 1031–1033 (2010). [CrossRef] [PubMed]

29. A. Miks and J. Novak, “Analysis of two-element zoom systems based on variable power lenses,” Opt. Express **18**(7), 6797–6810 (2010). [CrossRef] [PubMed]

## 2. Imaging properties of triplet lens

*s*is the distance of the object plane from the first element of the optical system,

*m*is the transverse magnification of the optical system. Further, we used the following denotation

*i*= 1, 2, 3) is the power of the

*i*-th element of the optical system and

## 3. Chromatic aberration of triplet lens

8. A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt. **47**(32), 6088–6098 (2008). [CrossRef] [PubMed]

*i*-th lens (we choose

*i*-th lens material. Transverse chromatic aberration is then given by relation [2–8

8. A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt. **47**(32), 6088–6098 (2008). [CrossRef] [PubMed]

*i*-th lens of the optical system. Without loss of generality we can put Lagrange-Helmholtz invariant equal to one. Then, we obtain (

*P*of a triplet is then given by [2–8

**47**(32), 6088–6098 (2008). [CrossRef] [PubMed]

*p*and

*D*are parameters, which are can be chosen appropriately. Using these parameters one can affect distortion and Petzval sum of the triplet. These parameters can be used for optimization of the triplet. Using Eqs. (6), (7), (8) and (9) with five unknown parameters (

*h*

_{1}and

*h*

_{2}. It holds

*i*= 1, 2, 3). By solving previous equations one can calculate the incidence heights

*h*

_{1}and

*h*

_{2}. In order to obtain the common solution of a set of Eqs. (13), the resultant of these equations must equal to zero. The resultant

*R*is given by

28. A. Miks, J. Novak, and P. Novak, “Generalized refractive tunable-focus lens and its imaging characteristics,” Opt. Express **18**(9), 9034–9047 (2010). [CrossRef] [PubMed]

30. A. Mikš and J. Novák, “Third-order aberrations of the thin refractive tunable-focus lens,” Opt. Lett. **35**(7), 1031–1033 (2010). [CrossRef] [PubMed]

*n*and Abbe number ν

_{d}*with following values: PC200B -*

_{d}*n*

_{1}= 1.3999, ν

_{1}= 58.7 and H100 -

*n*

_{2}= 1.489, ν

_{2}= 38.4 for the wavelength λ

*= 589 nm. The following expression is valid for the equivalent Abbe number ν*

_{d}*of this lenswhere*

_{E}*n*and Abbe number ν

_{d}*with following values: OL1024 -*

_{d}*n*= 1.30012, ν

_{d}*= 100.177 and OL0901 -*

_{d}*n*= 1.55872, ν

_{d}*= 30.276 for the wavelength λ*

_{d}*= 589 nm. Chromatic aberration of the Optotune lenses is lower compared with the lens ARTIC 416 at the same focal length. The disadvantage of the Optotune lenses compared to the lens ARTIC 416 is that they only can form a positive power, while ARTIC 416 can be both positive and negative lenses depending on the applied voltage.*

_{d}*C*= 0,

_{I}*C*= 0) triplets having the unit focal length (

_{II}*m*= 0), where

*d*

_{1}and

*d*

_{2}are separations between individual lenses. The triplets 1 and 2 have the lens OL1024 as the first and third lens, the second lens is made of Schott glass N-BK7 and N-ZK7. The triplet 3 has the lens Optotune OL0901 as the first and third element, the second lens is made of Schott glass SF67-P. The triplet 4 has lens OL1024 as the first elements, the second elements is the lens ARTIC 416, and the third elements is the lens OL0901.

## 4. Example of three-element zoom lens based on refractive variable-focus lenses

*h*,

*C*,

_{I}*C*) and Seidel aberration coefficients (

_{II}*C*) and transversal (

_{I}*C*) chromatic aberrations coefficients remain well-adjusted when changing the focal length, and chromatic aberrations of the triplet do not almost change. Table 4 also describes calculated ray aberrations for different values of the focal length, where

_{II}*H*= 0.05 mm. The first and the third lens have a plano-convex shape [15]. The shape of the second lens can be calculated using methods and relationships described in detail in papers [8

**47**(32), 6088–6098 (2008). [CrossRef] [PubMed]

9. A. Mikš, “Modification of the formulas for third-order aberration coefficients,” J. Opt. Soc. Am. A **19**(9), 1867–1871 (2002). [CrossRef] [PubMed]

9. A. Mikš, “Modification of the formulas for third-order aberration coefficients,” J. Opt. Soc. Am. A **19**(9), 1867–1871 (2002). [CrossRef] [PubMed]

*X*

_{1},

*X*

_{2}, and

*X*

_{3}of each lens. The lens Optotune OL1024 has diameter 10 mm and thus the F-number of the zoom lens will have value

**47**(32), 6088–6098 (2008). [CrossRef] [PubMed]

## 6. Conclusion

*D*. It also is not necessary to use an iterative process as suggested, for example, in Ref [2]. Using derived formulas we obtain all solutions of a given problem, which is not possible with the iterative method. The disadvantage of the iterative method is the fact that it provides only one solution near the starting point if the method converges. Derived formulas in our work can be used very easily, e.g. with the MATLAB system, the calculation is much faster and more complex than iterative methods, because all solutions are find. Furthermore, the proposed method makes possible to find out if a real solution exists (Eq. (16) must have real roots) for given input parameters, which is not possible using the iterative method. The example presented the calculation method for three-element zoom lens, where the first and the last lenses are commercially available lenses with a variable focal length from the company Optotune. Given the parameters of the Optotune lenses the focal length of the zoom lens can vary from

## Acknowledgment

## References and links

1. | S. F. Ray, |

2. | W. Smith, |

3. | M. Born and E. Wolf, |

4. | A. Miks, |

5. | M. Herzberger, |

6. | A. D. Clark, |

7. | K. Yamaji, |

8. | A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt. |

9. | A. Mikš, “Modification of the formulas for third-order aberration coefficients,” J. Opt. Soc. Am. A |

10. | S. Pal and L. Hazra, “ |

11. | F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE |

12. | F. S. Tsai, S. H. Cho, Y. H. Lo, B. Vasko, and J. Vasko, “Miniaturized universal imaging device using fluidic lens,” Opt. Lett. |

13. | 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 |

14. | |

15. | |

16. | H. W. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. |

17. | M. Ye, M. Noguchi, B. Wang, and S. Sato, “Zoom lens system without moving elements realised using liquid crystal lenses,” Electron. Lett. |

18. | D. Y. Zhang, N. Justis, and Y. H. Lo, “Fluidic adaptive zoom lens with high zoom ratio and widely tunable field of view,” Opt. Commun. |

19. | H. W. Ren and S. T. Wu, “Variable-focus liquid lens,” Opt. Express |

20. | G. Beadie, M. L. Sandrock, M. J. Wiggins, R. S. Lepkowicz, J. S. Shirk, M. Ponting, Y. Yang, T. Kazmierczak, A. Hiltner, and E. Baer, “Tunable polymer lens,” Opt. Express |

21. | B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E |

22. | B. H. W. Hendriks, S. Kuiper, M. A. J. As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. |

23. | R. Peng, J. Chen, and S. Zhuang, “Electrowetting-actuated zoom lens with spherical-interface liquid lenses,” J. Opt. Soc. Am. A |

24. | S. Reichelt and H. Zappe, “Design of spherically corrected, achromatic variable-focus liquid lenses,” Opt. Express |

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

26. | Z. Wang, Y. Xu, and Y. Zhao, “Aberration analyses of liquid zooming lenses without moving parts,” Opt. Commun. |

27. | J.-H. Sun, B.-R. Hsueh, Y.-Ch. Fang, J. MacDonald, and C. C. Hu, “Optical design and multiobjective optimization of miniature zoom optics with liquid lens element,” Appl. Opt. |

28. | A. Miks, J. Novak, and P. Novak, “Generalized refractive tunable-focus lens and its imaging characteristics,” Opt. Express |

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

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

31. | M. Berek, |

32. | R. E. Stephens, “The design of triplet anastigmat lenses of the Taylor type,” J. Opt. Soc. Am. |

33. | W. Wallin, “Design study of air-spaced triplets,” Appl. Opt. |

34. | M. Laikin, |

**OCIS Codes**

(080.3620) Geometric optics : Lens system design

(110.0110) Imaging systems : Imaging systems

(220.3620) Optical design and fabrication : Lens system design

(110.1080) Imaging systems : Active or adaptive optics

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: August 16, 2011

Revised Manuscript: September 30, 2011

Manuscript Accepted: October 23, 2011

Published: November 10, 2011

**Citation**

Antonin Miks and Jiri Novak, "Analysis of three-element zoom lens based on refractive variable-focus lenses," Opt. Express **19**, 23989-23996 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-24-23989

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### References

- S. F. Ray, Applied photographic optics, (Focal Press, 2002).
- W. Smith, Modern optical engineering, 4th Ed. (McGraw-Hill, 2007).
- M. Born and E. Wolf, Principles of optics, (Oxford University Press, 1964).
- A. Miks, Applied optics (Czech Technical University Press, 2009).
- M. Herzberger, Modern geometrical optics (Interscience Publishers, Inc., 1958).
- A. D. Clark, Zoom lenses (Adam Hilger, 1973).
- K. Yamaji, Progres in optics, Vol.VI (North-Holland Publishing Co., 1967).
- A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt.47(32), 6088–6098 (2008). [CrossRef] [PubMed]
- A. Mikš, “Modification of the formulas for third-order aberration coefficients,” J. Opt. Soc. Am. A19(9), 1867–1871 (2002). [CrossRef] [PubMed]
- S. Pal and L. Hazra, “Ab initio synthesis of linearly compensated zoom lenses by evolutionary programming,” Appl. Opt.50(10), 1434–1441 (2011). [CrossRef] [PubMed]
- F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE6501, 650109, 650109-9 (2007). [CrossRef]
- F. S. Tsai, S. H. Cho, Y. H. Lo, B. Vasko, and J. Vasko, “Miniaturized universal imaging device using fluidic lens,” Opt. Lett.33(3), 291–293 (2008). [CrossRef] [PubMed]
- 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. SPIE6034, 603402, 603402-9 (2006). [CrossRef]
- http://www.varioptic.com
- http://www.optotune.com/
- H. W. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett.84(23), 4789–4791 (2004). [CrossRef]
- M. Ye, M. Noguchi, B. Wang, and S. Sato, “Zoom lens system without moving elements realised using liquid crystal lenses,” Electron. Lett.45(12), 646–648 (2009). [CrossRef]
- D. Y. Zhang, N. Justis, and Y. H. Lo, “Fluidic adaptive zoom lens with high zoom ratio and widely tunable field of view,” Opt. Commun.249(1-3), 175–182 (2005). [CrossRef]
- H. W. Ren and S. T. Wu, “Variable-focus liquid lens,” Opt. Express15(10), 5931–5936 (2007). [CrossRef] [PubMed]
- G. Beadie, M. L. Sandrock, M. J. Wiggins, R. S. Lepkowicz, J. S. Shirk, M. Ponting, Y. Yang, T. Kazmierczak, A. Hiltner, and E. Baer, “Tunable polymer lens,” Opt. Express16(16), 11847–11857 (2008). [CrossRef] [PubMed]
- B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E3(2), 159–163 (2000). [CrossRef]
- B. H. W. Hendriks, S. Kuiper, M. A. J. As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev.12(3), 255–259 (2005). [CrossRef]
- R. Peng, J. Chen, and S. Zhuang, “Electrowetting-actuated zoom lens with spherical-interface liquid lenses,” J. Opt. Soc. Am. A25(11), 2644–2650 (2008). [CrossRef] [PubMed]
- S. Reichelt and H. Zappe, “Design of spherically corrected, achromatic variable-focus liquid lenses,” Opt. Express15(21), 14146–14154 (2007). [CrossRef] [PubMed]
- R. Peng, J. Chen, Ch. Zhu, and S. Zhuang, “Design of a zoom lens without motorized optical elements,” Opt. Express15(11), 6664–6669 (2007). [CrossRef] [PubMed]
- Z. Wang, Y. Xu, and Y. Zhao, “Aberration analyses of liquid zooming lenses without moving parts,” Opt. Commun.275(1), 22–26 (2007). [CrossRef]
- J.-H. Sun, B.-R. Hsueh, Y.-Ch. Fang, J. MacDonald, and C. C. Hu, “Optical design and multiobjective optimization of miniature zoom optics with liquid lens element,” Appl. Opt.48(9), 1741–1757 (2009). [CrossRef] [PubMed]
- A. Miks, J. Novak, and P. Novak, “Generalized refractive tunable-focus lens and its imaging characteristics,” Opt. Express18(9), 9034–9047 (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. Mikš and J. Novák, “Third-order aberrations of the thin refractive tunable-focus lens,” Opt. Lett.35(7), 1031–1033 (2010). [CrossRef] [PubMed]
- M. Berek, Grundlagen der praktischen optik, (Walter de Gruyter & Co., 1970).
- R. E. Stephens, “The design of triplet anastigmat lenses of the Taylor type,” J. Opt. Soc. Am.38(12), 1032–1039 (1948). [CrossRef] [PubMed]
- W. Wallin, “Design study of air-spaced triplets,” Appl. Opt.3(3), 421–426 (1964). [CrossRef]
- M. Laikin, Lens design, (CRC Press, 2006).

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