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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 24 — Nov. 21, 2011
  • pp: 23989–23996
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Analysis of three-element zoom lens based on refractive variable-focus lenses

Antonin Miks and Jiri Novak  »View Author Affiliations


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

2. Imaging properties of triplet lens

Figure 1
Fig. 1 Scheme of three-element optical system (ξ – object plane, ξ' – image plane, A – axial object point, A' – image of the point A, B – off-axis object point, B' – image of the point B, P – entrance pupil centre, y0- paraxial image height, s – object distance, s- image distance, φ1,φ2,φ3 - individual powers of the lenses, d1,d2- separations of the lenses, s¯- position of the entrance pupil, h1- incidence height of the aperture ray, h¯1 - incidence height of the principal ray.
presents schematically three-element optical system composed of thin lenses. As it is well known, the following relations hold for imaging using such an optical system [5

5. M. Herzberger, Modern geometrical optics (Interscience Publishers, Inc., 1958).

]
φ=1/f=γ,sF=δ/γ,sF=α/γ,s=(δ1/m)/γ,s=(mα)/γ=(βαs)/(γsδ),
(1)
where φ is the power of the optical system, f is the focal length of the optical system, sF is the position of the object focal point, sF is the position of the image focal point, s is the distance of the object plane from the first element of the optical system, s is the distance of the image plane from the last element of the optical system, and m is the transverse magnification of the optical system. Further, we used the following denotation
α=d1d2φ1φ2d2(φ1+φ2)d1φ1+1,β=d1+d2d1d2φ2,γ=[d1d2φ1φ2φ3d2φ3(φ1+φ2)d1φ1(φ2+φ3)+φ1+φ2+φ3],δ=d1d2φ2φ3d2φ3d1(φ2+φ3)+1,
(2)
where φi (i = 1, 2, 3) is the power of the i-th element of the optical system and d1, d2 are the distances between individual elements of the optical system.

3. Chromatic aberration of triplet lens

Let us focus on a problem of achromaticity of a triplet lens. As it is well-known, longitudinal chromatic aberration of a thin lens system is given by relation [2

2. W. Smith, Modern optical engineering, 4th Ed. (McGraw-Hill, 2007).

8

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

,31

31. M. Berek, Grundlagen der praktischen optik, (Walter de Gruyter & Co., 1970).

34

34. M. Laikin, Lens design, (CRC Press, 2006).

]
δsλ=(ms)2i=13hi2φiνi=(ms)2CI,
(3)
where hi is the incidence height of the paraxial aperture ray on i-th lens (we choose h1=1) and νi is the Abbe number of the i-th lens material. Transverse chromatic aberration is then given by relation [2

2. W. Smith, Modern optical engineering, 4th Ed. (McGraw-Hill, 2007).

8

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

,31

31. M. Berek, Grundlagen der praktischen optik, (Walter de Gruyter & Co., 1970).

34

34. M. Laikin, Lens design, (CRC Press, 2006).

]
δyλ=y0i=13hih¯iφiνi=y0CII,
(4)
where y0 is the image height and h¯i is the incidence height of the principal paraxial ray on i-th lens of the optical system. Without loss of generality we can put Lagrange-Helmholtz invariant equal to one. Then, we obtain (h1=1) for h¯1 and for the position of the entrance pupil s¯ the following equations

h¯1=ss¯/(s¯s),s¯=d1/(1φ1d1).
(5)

In case of the triplet where the aperture stop is close behind the middle element we choose h¯2=0. As it can be seen from relations (3) and (4), the conditions for achromaticity of a triplet lens are given by

CI=φ1ν1+h22φ2ν2+h32φ3ν3=0,CII=h¯1φ1ν1+h3h¯3φ3ν3=0.
(6)

Equation (2) for the power φ=γof a triplet can be expressed as

φ=φ1+h2φ2+h3φ3.
(7)

The Petzval sum P of a triplet is then given by [2

2. W. Smith, Modern optical engineering, 4th Ed. (McGraw-Hill, 2007).

8

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

, 31

31. M. Berek, Grundlagen der praktischen optik, (Walter de Gruyter & Co., 1970).

34

34. M. Laikin, Lens design, (CRC Press, 2006).

]

P=φ1/n1+φ2/n2+φ3/n3.

φ1+φ2+φ3nP=p.
(8)

The above mentioned equations can be completed with the formula for an approximate correction of distortion, which has the following form [31

31. M. Berek, Grundlagen der praktischen optik, (Walter de Gruyter & Co., 1970).

]

d1φ1d2φ3=D.
(9)

The values 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 (φ1,φ2,φ3,d1,d2) one can determine the values of these parameters in order to made the triplet achromatic, assuming that we know the Abbe numbers (ν1,ν2,ν3) and refractive indices (n1,n2,n3) of materials of individual elements of the optical system. For simplicity and without loss of generality we set φ=1in the following text. It holds for distances d1,d2 between the lenses, position of the image plane s, and position of the exit pupil s¯ of the optical system (φ=1, h1=1)

d1=1h2φ1,d2=h2h31h3φ3,s=h3,s¯=h2h3h2φ31.
(10)

We obtain using Eqs. (7), (8), and (9)

φ3=D+h21h3D+h2(h31),φ2=pφ3(1h3)11h2,φ1=pφ2φ3.
(11)

Moreover, we can express using conjugate equations

h¯1=h21h2φ1,h¯3=h2h3h2(1h3φ3).
(12)

By substitution of relations (11) and (12) into Eq. (6) we obtain a set of two equations for incidence heights h1 and h2. It holds

a0h23+a1h22+a2h2+a3=0,b0h2+b1=0.
(13)

R=|a0a1a2a3b0b1000b0b1000b0b1|=a3b03+a2b02b1a1b0b12+a0b13=0.
(15)

If we substitute Eq. (14) into Eq. (15), we obtain (after a longer calculation) the following nonlinear equation for the calculation of the incidence height h3. It holds
c4h34+c3h33+c2h32+c1h3+c0=0,
(16)
where

c4=V33(D1)(V1(1p+D)+V2(p1)(1D)),c3=V32p(V1V2(D3D23D+3)V1V3(D1)2V2V3(D1)3+V12(D2+D3))V32(V1V2(D24D+3)+V1V3(D1)V2V3(D1)3+V12(D23)),c2=V1V3p(V1V3(D24D+3)+V1V2(2D23)3V2V3(D1)2V12(D2D3))V1V3(V1V3(D23D+3)+V1V2(2D3)+V2V3(D35D2+7D3)+3V12),c1=V12p((V1V2V12)(1+D)+V1V3(2D3)V2V3(3D3))V12(V1V2V12V1V3(D23D+3)+V2V3(2D25D+3)),c0=V13(V1V2)(D+p1).

By solving Eq. (16) we can determine the incidence height h3. By inserting this value into formulas (13) we can calculate the incidence heighth2. Then, from Eqs. (11) and (10) we can calculate the powers φ1,φ2,φ3 and distances d1,d2 between the individual lenses of the triplet. Our analysis is more complex and exhaustive than the analysis presented in papers [31

31. M. Berek, Grundlagen der praktischen optik, (Walter de Gruyter & Co., 1970).

34

34. M. Laikin, Lens design, (CRC Press, 2006).

].

Assume now that commercially available variable-focus (tunable-focus) lenses from companies Optotune and Varioptic [14,15] will be used for individual elements of the triplet lens. We will assume the lenses to be infinitely thin in the first approximation. If we want to design the triplet lens similar to classical triplets used in photography, i.e. having 6 free radii of curvature, we have to use 4 lenses Optotune to create two outer positive elements and 2 lenses Varioptic for the inner negative lens. A detailed theoretical analysis of refractive variable-focus lenses from the company Varioptic is performed in the papers [28

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

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]

]. For example, the lens ARTIC 416 [14] consists of two immiscible liquid lenses formed by fluids (PC200B, H100), having refractive index nd and Abbe number νd with following values: PC200B - n1 = 1.3999, ν1 = 58.7 and H100 - n2 = 1.489, ν2 = 38.4 for the wavelength λd = 589 nm. The following expression is valid for the equivalent Abbe number νE of this lens
νE=i=12φii=12φi/νi=(n2n1)ν1ν2(n21)ν1(n11)ν2=15.046,
(17)
where φi are the powers of individual liquid lenses. As it is evident from the calculated value the lens suffer from relatively large chromatic aberrations. The equation for longitudinal chromatic aberration of the lens having the focal length f and for the object located at infinity is given by (lims(ms)=f)

δsλ=f/νE=0.0665f.
(18)

Refractive variable-focus lenses OL1024 and OL0901 from the company Optotune [15] have plano-convex shape (positive focal length) and consist of material having refractive index nd and Abbe number νd with following values: OL1024 - nd = 1.30012, νd = 100.177 and OL0901 - nd = 1.55872, νd = 30.276 for the wavelength λd = 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.

Table 1

Table 1. Basic Parameters of Triplets

table-icon
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presents some calculation results of the achromatic (CI = 0, CII = 0) triplets having the unit focal length (φ=1/f=1) and for the object located at infinity (m = 0), where f1,f2,f3 are focal length values of individual lenses of the triplet, d1 and d2 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

φ1=(φsF+d2φ21)/(d1d2φ2d2d1),φ3=(φ+d1φ1φ2φ1φ2)/φsF.
(19)

Table 2

Table 2. Values of Paraxial Incidence Height and Focal Length - Triplet OL1024/N-ZK7/OL1024

table-icon
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shows the results of calculating the three-element zoom lens based on two refractive variable-focus lenses (first and third element) Optotune OL1024. The second lens is formed by a conventional glass lens made of glass N-ZK7 (triplet 2 - Table 1). The triplet 2 in Table 1 was chosen arbitrarily only as an example. One could choose any other triplet in the same way. The ratio of focal distances is fmax/fmin=1.5. Moreover, Table 2 presents values of incidence heights h, h¯ of paraxial aperture and principal rays on individual lenses of the triplet.

Optotune lens OL1024 can change its focal length from fmin=30mm to fmax=100mm [15]. As is evident from Table 2, the zoom lens will change its focal length from fmin=67mm to fmax=100mm. Thus, the focal length of the second (glass) lens will be f2=29.5mm.

6. Conclusion

Acknowledgment

This work has been supported by research projects from the Ministry of Education of Czech Republic, MSM6840770022.

References and links

1.

S. F. Ray, Applied photographic optics, (Focal Press, 2002).

2.

W. Smith, Modern optical engineering, 4th Ed. (McGraw-Hill, 2007).

3.

M. Born and E. Wolf, Principles of optics, (Oxford University Press, 1964).

4.

A. Miks, Applied optics (Czech Technical University Press, 2009).

5.

M. Herzberger, Modern geometrical optics (Interscience Publishers, Inc., 1958).

6.

A. D. Clark, Zoom lenses (Adam Hilger, 1973).

7.

K. Yamaji, Progres in optics, Vol.VI (North-Holland Publishing Co., 1967).

8.

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

10.

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]

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]

12.

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]

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 6034, 603402, 603402-9 (2006). [CrossRef]

14.

http://www.varioptic.com

15.

http://www.optotune.com/

16.

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]

17.

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]

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. 249(1-3), 175–182 (2005). [CrossRef]

19.

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

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

21.

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.

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]

23.

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] [PubMed]

24.

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

25.

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]

26.

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

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]

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]

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]

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]

31.

M. Berek, Grundlagen der praktischen optik, (Walter de Gruyter & Co., 1970).

32.

R. E. Stephens, “The design of triplet anastigmat lenses of the Taylor type,” J. Opt. Soc. Am. 38(12), 1032–1039 (1948). [CrossRef] [PubMed]

33.

W. Wallin, “Design study of air-spaced triplets,” Appl. Opt. 3(3), 421–426 (1964). [CrossRef]

34.

M. Laikin, Lens design, (CRC Press, 2006).

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

  1. S. F. Ray, Applied photographic optics, (Focal Press, 2002).
  2. W. Smith, Modern optical engineering, 4th Ed. (McGraw-Hill, 2007).
  3. M. Born and E. Wolf, Principles of optics, (Oxford University Press, 1964).
  4. A. Miks, Applied optics (Czech Technical University Press, 2009).
  5. M. Herzberger, Modern geometrical optics (Interscience Publishers, Inc., 1958).
  6. A. D. Clark, Zoom lenses (Adam Hilger, 1973).
  7. K. Yamaji, Progres in optics, Vol.VI (North-Holland Publishing Co., 1967).
  8. A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt.47(32), 6088–6098 (2008). [CrossRef] [PubMed]
  9. A. Mikš, “Modification of the formulas for third-order aberration coefficients,” J. Opt. Soc. Am. A19(9), 1867–1871 (2002). [CrossRef] [PubMed]
  10. 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]
  11. 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]
  12. 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]
  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. SPIE6034, 603402, 603402-9 (2006). [CrossRef]
  14. http://www.varioptic.com
  15. http://www.optotune.com/
  16. 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]
  17. 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]
  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.249(1-3), 175–182 (2005). [CrossRef]
  19. H. W. Ren and S. T. Wu, “Variable-focus liquid lens,” Opt. Express15(10), 5931–5936 (2007). [CrossRef] [PubMed]
  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. Express16(16), 11847–11857 (2008). [CrossRef] [PubMed]
  21. 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]
  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.12(3), 255–259 (2005). [CrossRef]
  23. 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]
  24. S. Reichelt and H. Zappe, “Design of spherically corrected, achromatic variable-focus liquid lenses,” Opt. Express15(21), 14146–14154 (2007). [CrossRef] [PubMed]
  25. 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]
  26. Z. Wang, Y. Xu, and Y. Zhao, “Aberration analyses of liquid zooming lenses without moving parts,” Opt. Commun.275(1), 22–26 (2007). [CrossRef]
  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]
  28. A. Miks, J. Novak, and P. Novak, “Generalized refractive tunable-focus lens and its imaging characteristics,” Opt. Express18(9), 9034–9047 (2010). [CrossRef] [PubMed]
  29. A. Miks and J. Novak, “Analysis of two-element zoom systems based on variable power lenses,” Opt. Express18(7), 6797–6810 (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]
  31. M. Berek, Grundlagen der praktischen optik, (Walter de Gruyter & Co., 1970).
  32. R. E. Stephens, “The design of triplet anastigmat lenses of the Taylor type,” J. Opt. Soc. Am.38(12), 1032–1039 (1948). [CrossRef] [PubMed]
  33. W. Wallin, “Design study of air-spaced triplets,” Appl. Opt.3(3), 421–426 (1964). [CrossRef]
  34. M. Laikin, Lens design, (CRC Press, 2006).

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