## Generalized refractive tunable-focus lens and its imaging characteristics

Optics Express, Vol. 18, Issue 9, pp. 9034-9047 (2010)

http://dx.doi.org/10.1364/OE.18.009034

Acrobat PDF (1097 KB)

### Abstract

Conventional lenses made from optical glass or plastics have fixed properties (e.g. focal length) that depend on the index of refraction and geometrical parameters of the lens. We present an approach to the problem of calculation of basic paraxial parameters and the third order aberration coefficients of compound optical elements analogical to classical lenses which are based on refractive tunable-focus lenses. A detailed theoretical analysis is performed for a simple tunable-focus lens, a generalized tunable-focus lens, a generalized tunable-focus lens with minimum spherical aberration, and three-element tunable-focus lens (a tunable-focus doublet).

© 2010 OSA

## 1. Introduction

18. R. Marks, D. L. Mathine, G. Peyman, J. Schwiegerling, and N. Peyghambarian, “Adjustable fluidic lenses for ophthalmic corrections,” Opt. Lett. **34**(4), 515–517 (2009). [CrossRef] [PubMed]

3. B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E **3**(2), 159–163 (2000). [CrossRef]

8. R. Peng, J. Chen, and S. Zhuang, “Electrowetting-actuated zoom lens with spherical-interface liquid lenses,” J. Opt. Soc. Am. A **25**(11), 2644–2650 (2008). [CrossRef]

9. 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]

11. H. W. Ren and S. T. Wu, “Variable-focus liquid lens,” Opt. Express **15**(10), 5931–5936 (2007). [CrossRef] [PubMed]

12. 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 **16**(16), 11847–11857 (2008). [CrossRef] [PubMed]

13. A. F. Naumov, G. D. Love, M. Y. Loktev, and F. L. Vladimirov, “Control optimization of spherical modal liquid crystal lenses,” Opt. Express **4**(9), 344–352 (1999). [CrossRef] [PubMed]

17. P. Valley, D. L. Mathine, M. R. Dodge, J. Schwiegerling, G. Peyman, and N. Peyghambarian, “Tunable-focus flat liquid-crystal diffractive lens,” Opt. Lett. **35**(3), 336–338 (2010). [CrossRef] [PubMed]

18. R. Marks, D. L. Mathine, G. Peyman, J. Schwiegerling, and N. Peyghambarian, “Adjustable fluidic lenses for ophthalmic corrections,” Opt. Lett. **34**(4), 515–517 (2009). [CrossRef] [PubMed]

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

21. 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]

3. B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E **3**(2), 159–163 (2000). [CrossRef]

22. A. Miks, *Applied Optics* (Czech Technical University Press, Prague 2009). [PubMed]

27. S. Reichelt and H. Zappe, “Design of spherically corrected, achromatic variable-focus liquid lenses,” Opt. Express **15**(21), 14146–14154 (2007). [CrossRef] [PubMed]

29. Z. Wang, Y. Xu, and Y. Zhao, “Aberration analyses of liquid zooming lenses without moving parts,” Opt. Commun. **275**(1), 22–26 (2007). [CrossRef]

## 2. Basic formulas for calculation of parameters of refractive tunable-focus lenses

3. B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E **3**(2), 159–163 (2000). [CrossRef]

16. 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]

30. M. Herzberger, “Replacing a thin lens by a thick lens,” J. Opt. Soc. Am. **34**(2), 114–115 (1944). [CrossRef]

*K*surfaces [22

22. A. Miks, *Applied Optics* (Czech Technical University Press, Prague 2009). [PubMed]

*i*-th surface of the optical system,

*i*-th surface,

*i*-th surface,

*i*-th surface,

*i*-th surface,

*i*-th surface,

*i*-th surface and the vertex of (

*i*+1)-st surface,

*i*-th surface, from

*i*-th surface,

*i*surfaces, from the

*i*-th surface of the optical system. The transverse magnification

*m*is given by the formulaNow, consider imaging of the object at infinity (

*m*is the transverse magnification. Equations (5) enable to calculate

*s*and

*s*' for a given value of the transverse magnification

*m*. Further, it holds the following image equation [22

22. A. Miks, *Applied Optics* (Czech Technical University Press, Prague 2009). [PubMed]

*q*is the distance of point A from the object focal point F, and

*q*' is the distance of point A' from the image focal point F'.

## 3. Third order aberrations of tunable-focus lenses

*K*spherical surfaces. In case we know radii of curvature of lenses, their thicknesses, indices of refraction and distances between individual lenses we can simply calculate aberration coefficients of the third order [22

22. A. Miks, *Applied Optics* (Czech Technical University Press, Prague 2009). [PubMed]

*K*surfaces

*i*-th surface,

*i*-th surface,

*i*-th surface, from

*i*-th surface,

*i*surfaces, from

*i*-th surface of the optical system. The meaning of other symbols is the same as in the case of the paraxial aperture ray. The angular magnification in pupils of the optical system can be expressed as

_{1}as the distance of the object plane from the first surface of the optical system,

22. A. Miks, *Applied Optics* (Czech Technical University Press, Prague 2009). [PubMed]

22. A. Miks, *Applied Optics* (Czech Technical University Press, Prague 2009). [PubMed]

*S*

_{I}is the coefficient of spherical aberration,

*S*

_{II}is the coefficient of coma,

*S*

_{III}is the coefficient of astigmatism,

*S*

_{IV}is Petzval coefficient, and

*S*

_{V}is the coefficient of distortion. The quantity

*H*is the Lagrange-Helmholtz invariant. It is evident from previous equations that one can use an arbitrary choice of input parameters

*d*

_{1}= 0,

*d*

_{2}= 0) in air (

*n*

_{1}=1,

*n*

_{4}=1) the following formulaswhere Functions

*A*(

*P,Q*) and

*B*(

*P,Q*) on parameters

*P*and

*Q*(refractive indices) is shown in Fig. 3 for the value

*P*= 1.38.

*x*and

*y*are the coordinates of an arbitrary point of the lens surface meridian,

*r*is the radius of curvature on the optical axis, and

*b*is the aspheric coefficient that characterizes the shape of the aspheric surface. We can determine the type of the curve by the value of the coefficient

*b*. The curve represents hyperbola, if

*b*≠ 0 or circle, if

*b*= 0. If the inner surface tunable-focus lens is aspheric then we must replace the variable

*M*in Eq. (11) by the following expressionWe can determine aberration coefficients of the third order of a tunable-focus lens using Eqs. (10-13) for an arbitrary value of optical power φ and position of the object plane

*s*. We can write

*p*= I, II, III, IV, V) denotes the aberration coefficient of the

*j*-th element of the optical system. We can provide calculations of the third-order aberration coefficients of an arbitrary optical system of thin tunable-focus lenses using Eq. (8) and Eqs. (10-14). The provided analysis may serve for the initial design of optical systems, and the calculated parameters can be used for its further optimization using optical design software. Chromatic aberrations can be simply calculated by substitution of refractive indices for corresponding wavelengths into Eqs. (8) – (14). The error due to neglecting the finite thickness of lenses is relatively small because the change of aberration coefficients with respect to the lens thickness is approximately few percents.

## 4. Imaging properties of generalized tunable-focus lens

**)**to obtain the element similar to a classical simple spherical lens. As we can see form Fig. 4 it holds that

*d*of the object principal plane of the second lens from the image principal plane of the first lensWe can derive for the optical power φ, the position of the image focal point

22. A. Miks, *Applied Optics* (Czech Technical University Press, Prague 2009). [PubMed]

*K*thin simple tunable-focus lenses in contact in air (

## 6. Imaging properties of three thin tunable-focus lenses

*R*of Eqs. (26) and (27) can be expressed asWe can calculate the resultant (28) from the following formula

*k*

_{10}and

*p*

_{6}are changed in the following wayIt can be noted that we can proceed in a similar way even in the case of more complicated optical systems which consist of a larger number of thin tunable-focus lenses. For example, an equivalent of a traditional non-cemented doublet must be composed of four tunable-focus lenses, a triplet must be composed of six tunable-focus lenses (i.e. three generalized tunable-focus lenses), the Petzval lens must be composed of six tunable-focus lenses (i.e. two tunable-focus doublets), etc. It is possible to combine tunable-focus lenses, traditional lenses and optical systems and design hybrid optical systems with variable optical characteristics (e.g. focal length, magnification). The fundamental advantage of optical systems with tunable-focus lenses over classical optical systems is the possibility to continuously change properties of these systems without the need for changing position of individual elements of the optical system.

## 7. Examples of tunable-focus lenses

### Example 1

*S*is given in Table 1 for both cases of refractive indices.We obtain for the transverse spherical aberration

_{I}*n*and the minimum spherical aberration for the object at infinity. We have for the longitudinal spherical aberration

*n*= 1.516 (λ = 589 nm) we can calculate for the focal length

*n*= 1.904 (λ = 589 nm) we obtain

*n*

_{2}= 1.38 and

*n*

_{3}= 1.55). In the second case (

*n*

_{2}= 1.38 and

*n*

_{3}= 1.99) the tunable-focus lens has almost the same residual spherical aberration as the classical lens from the glass BK7. It is clear from the presented example that the difference (

*n*

_{3}

*– n*

_{2}) between indices of refraction must be relatively large for achieving small residual aberration of the simple tunable-focus lens. It is also evident from Fig. 3.

### Example 2

*S*)

_{I}_{min}are presented in Table 2 . As we can see from Table 2 the first solution gives always lower value

*S*of spherical aberration.

_{I}### Example 3

*n*= 1.99 in the previous examples was chosen intentionally in order to show from the theoretical point of view that we need the difference of refractive indices as large as possible for obtaining small values of Seidel coefficients.

## 8. Summary

*A*,

*B*and

*C*that depend only on refractive indices of fluids forming the tunable-focus lens and do not depend on the position and size of the object and the position of the entrance pupil. These functions are constant for a given type of the tunable-focus lens. A detailed theoretical analysis was performed for a simple tunable-focus lens, a generalized tunable-focus lens, a generalized tunable-focus lens with minimum spherical aberration, and three-element tunable-focus lens (a tunable-focus doublet), which is the equivalent of the classical cemented doublet. The derived equations enable to carry out calculations of all parameters of above-mentioned optical systems and are also fundamental for solving of more complex optical systems using tunable-focus lenses. For example, an analogy of a traditional non-cemented doublet is composed of four tunable-focus lenses, a triplet must is composed of six tunable-focus lenses (i.e. three generalized tunable-focus lenses), the Petzval lens is composed of six tunable-focus lenses (i.e. two tunable-focus doublets), etc. The calculation of parameters of the optical systems with tunable-focus lenses was presented on several examples. The provided analysis may serve for better understanding aberration and imaging properties of the refractive tunable-focus lenses and for the initial design of optical systems using such non-conventional lens systems. Tunable-focus lenses start to be used in various practical applications and in near future these lenses will impact considerably the design of modern non-conventional optical systems, e.g. zoom lenses.

## Acknowledgements

## References and links

1. | |

2. | |

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

4. | C. Gabay, B. Berge, G. Dovillaire, and S. Bucourt, “Dynamic study of a Varioptic variable focal lens,” SPIE Proc. |

5. | B. Berge, “Liquid lens technology |

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

7. | S. Kuiper and B. H. W. Hendriks, “Variable |

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

9. | 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. |

10. | H. Ren, D. Fox, P. A. Anderson, B. Wu, and S. T. Wu, “Tunable-focus liquid lens controlled using a servo motor,” Opt. Express |

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

12. | 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 |

13. | A. F. Naumov, G. D. Love, M. Y. Loktev, and F. L. Vladimirov, “Control optimization of spherical modal liquid crystal lenses,” Opt. Express |

14. | M. Ye and S. Sato, “Optical properties of liquid crystal lens of any size,” Jpn. J. Appl. Phys. |

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

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

17. | P. Valley, D. L. Mathine, M. R. Dodge, J. Schwiegerling, G. Peyman, and N. Peyghambarian, “Tunable-focus flat liquid-crystal diffractive lens,” Opt. Lett. |

18. | R. Marks, D. L. Mathine, G. Peyman, J. Schwiegerling, and N. Peyghambarian, “Adjustable fluidic lenses for ophthalmic corrections,” Opt. Lett. |

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

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

21. | 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 |

22. | A. Miks, |

23. | W. Smith, |

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

25. | P. Mouroulis, and J. Macdonald, |

26. | M. Herzberger, |

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

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

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

30. | M. Herzberger, “Replacing a thin lens by a thick lens,” J. Opt. Soc. Am. |

31. | K. Rektorys, |

**OCIS Codes**

(080.3620) Geometric optics : Lens system design

(110.0110) Imaging systems : Imaging systems

(220.3630) Optical design and fabrication : Lenses

(110.1080) Imaging systems : Active or adaptive optics

**History**

Original Manuscript: March 1, 2010

Revised Manuscript: March 31, 2010

Manuscript Accepted: March 31, 2010

Published: April 14, 2010

**Citation**

Antonin Miks, Jiri Novak, and Pavel Novak, "Generalized refractive tunable-focus lens and its imaging characteristics," Opt. Express **18**, 9034-9047 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-9-9034

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### References

- http://www.varioptic.com
- http://www.optotune.com/
- B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” Eur. Phys. J. E 3(2), 159–163 (2000). [CrossRef]
- C. Gabay, B. Berge, G. Dovillaire, and S. Bucourt, “Dynamic study of a Varioptic variable focal lens,” SPIE Proc. 4767, 159–165 (2002). [CrossRef]
- B. Berge, “Liquid lens technology: Principle of electrowetting based lenses and applications to imaging”, Proc. IEEE MEMS, 227–230 (2004).
- B. H. W. Hendriks, S. Kuiper, M. A. J. Van As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005). [CrossRef]
- S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. 85(7), 1128–1130 (2004). [CrossRef]
- R. Peng, J. Chen, and S. Zhuang, “Electrowetting-actuated zoom lens with spherical-interface liquid lenses,” J. Opt. Soc. Am. A 25(11), 2644–2650 (2008). [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. Ren, D. Fox, P. A. Anderson, B. Wu, and S. T. Wu, “Tunable-focus liquid lens controlled using a servo motor,” Opt. Express 14(18), 8031–8036 (2006). [CrossRef] [PubMed]
- H. W. Ren and S. T. Wu, “Variable-focus liquid lens,” Opt. Express 15(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. Express 16(16), 11847–11857 (2008). [CrossRef] [PubMed]
- A. F. Naumov, G. D. Love, M. Y. Loktev, and F. L. Vladimirov, “Control optimization of spherical modal liquid crystal lenses,” Opt. Express 4(9), 344–352 (1999). [CrossRef] [PubMed]
- M. Ye and S. Sato, “Optical properties of liquid crystal lens of any size,” Jpn. J. Appl. Phys. 41(Part 2, No. 5B), L571–L573 (2002). [CrossRef]
- 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]
- P. Valley, D. L. Mathine, M. R. Dodge, J. Schwiegerling, G. Peyman, and N. Peyghambarian, “Tunable-focus flat liquid-crystal diffractive lens,” Opt. Lett. 35(3), 336–338 (2010). [CrossRef] [PubMed]
- R. Marks, D. L. Mathine, G. Peyman, J. Schwiegerling, and N. Peyghambarian, “Adjustable fluidic lenses for ophthalmic corrections,” Opt. Lett. 34(4), 515–517 (2009). [CrossRef] [PubMed]
- F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” SPIE Proc. 6501, 650109 (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. SPIE 6034, 603402 (2006). [CrossRef]
- A. Miks, Applied Optics (Czech Technical University Press, Prague 2009). [PubMed]
- W. Smith, Modern Optical Engineering, 4th Ed. (McGraw-Hill, New York 2007).
- M. Born, and E. Wolf, Principles of Optics, (Oxford University Press, New York 1964).
- P. Mouroulis, and J. Macdonald, Geometrical Optics and Optical Design (Oxford University Press, New York 1997).
- M. Herzberger, Modern Geometrical Optics (Interscience Publishers, Inc., New York 1958).
- S. Reichelt and H. Zappe, “Design of spherically corrected, achromatic variable-focus liquid lenses,” Opt. Express 15(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. Express 15(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]
- M. Herzberger, “Replacing a thin lens by a thick lens,” J. Opt. Soc. Am. 34(2), 114–115 (1944). [CrossRef]
- K. Rektorys, Survey of Applicable Mathematics. (Kluwer Academic Publisher, Dodrecht 1994)

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