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

Optical systems with variable optical parameters (zoom lens systems) find a large number of applications in many areas and a high importance is paid to their development [127]. Individual optical elements of these classical optical systems mainly consist of glass lenses and the change of their optical parameters is done by translation of several elements of such optical systems [68]. Recently, optical elements with a continuously variable focal length are available, which makes possible to design special optical systems with variable focus and no mechanical movement of optical elements inside the optical system [1130]. We performed a detailed theoretical analysis [29] of a two-element zoom lens with the variable focal length. The aim of this work is to perform a detailed theoretical analysis of a three element optical system (triplet) with the variable focal length and apply the derived relations to the optical design of such system composed of commercially available refractive variable-focus lenses [14,15] and classical lenses. The derived formulas, described in the following sections of our paper, enable to calculate parameters of the triplet with a variable focal length, which may serve as input parameters (starting point) for further optimization of the triplet using appropriate optimization procedures.

2. Imaging properties of triplet lens

Figure 1 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]

φ=1/f=γ,sF=δ/γ,sF=α/γ,s=(δ1/m)/γ,s=(mα)/γ=(βαs)/(γsδ),
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,
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.

 

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.

Download Full Size | PPT Slide | PDF

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 [28,3134]

δsλ=(ms)2i=13hi2φiνi=(ms)2CI,
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 [28,3134]
δyλ=y0i=13hih¯iφiνi=y0CII,
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).

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.

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

φ=φ1+h2φ2+h3φ3.

The Petzval sum P of a triplet is then given by [28, 3134]

P=φ1/n1+φ2/n2+φ3/n3.

If we assume that it holds approximately n1=n2=n3=n, we can simplify previous relation to

φ1+φ2+φ3nP=p.

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

d1φ1d2φ3=D.

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.

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

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

Moreover, we can express using conjugate equations

h¯1=h21h2φ1,h¯3=h2h3h2(1h3φ3).

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.

The coefficients in Eq. (13) are given by the following formulas

a0=V2p(1h3),a1=V3h32+(V2+V1pDV2p)h3+V2(D1)V1(p1),a2=V3(D2)h32+V1(Dph3D+2),a3=V3(1D)h32V1h3,b0=V1V3h3,b1=(1D)V3h3V,
where we denoted Vi=1/νi (i = 1, 2, 3). By solving previous equations one can calculate the incidence heights h1 and h2. 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

R=|a0a1a2a3b0b1000b0b1000b0b1|=a3b03+a2b02b1a1b0b12+a0b13=0.

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,
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 [3134].

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 [28,30]. 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,
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.

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 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.

Tables Icon

Table 1. Basic Parameters of Triplets

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

In the previous section, we derived formulas for calculating powers and distances of individual lenses of the triplet. We require now that this triplet can change its power (focal length f=1/φ), without moving its individual elements. Another requirement is that the triplet lens will not change the position of the image plane with respect to the last member of the triplet. The continuous change of focal length is achieved by changing the focal length of its two elements. Because the magnitude of powers of commercially available refractive variable-focus lenses (Optotune, Varioptics) is limited, we choose the first and third lenses as the lenses with a continuously variable power. We obtain for the powers φ1 a φ3 of the first and the third element using Eqs. (1) and (2) the following formulas

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

Table 2 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.

Tables Icon

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

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.

Table 3 presents chromatic aberration coefficients (CI, CII) and Seidel aberration coefficients (SI,SII,SIII,SVI,SV) for different values of the focal length, where SIis the coefficient of spherical aberration, SII is the coefficient of coma, SIIIis the coefficient of astigmatism, SIVis the Petzval sum, and SVis the coefficient of distortion. Moreover, as it is evident from Table 4 , the longitudinal (CI) and transversal (CII) 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 δs is the longitudinal spherical aberration, Kt is the tangential coma, δsts is astigmatism, δss is the sagittal field curvature, δst is the tangential field curvature, and 100δy/y0 is the relative distortion in percents. Linear dimensions in Table 3 and 4 are given in millimeters and the calculation is performed for the triplet consisting of thin lenses. Aberrations in Table 4 are calculated for the object at infinity, the angle of field of view 10°, and the incidence height of the aperture ray at the first lens 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 [8,9,31] and therefore this issue is not addressed in this work. Table 3 also shows the resulting shape parameters (bending factors) [9] X1, X2, and X3 of each lens. The lens Optotune OL1024 has diameter 10 mm and thus the F-number of the zoom lens will have value F10. Using the previous analysis initial design parameters of the triplet can be calculated, which may serve as the starting point for further optimization using appropriate software such as, for example, ZEMAX, OSLO, etc. The thickness of individual lenses can be calculated by the approach mentioned in Ref [8].

Tables Icon

Table 3. Aberration Coefficients of Triplet - OL1024/N-ZK7/OL1024

Tables Icon

Table 4. Residual Aberrations of Triplet - OL1024/N-ZK7/OL1024

6. Conclusion

General formulas for calculating basic parameters of three-element zoom lens based on refractive variable-focus lenses were derived in this work. These relations can be used for the calculation of powers of the individual elements of the zoom lens and their distances, so that the zoom lens has corrected longitudinal and transverse chromatic aberration. None of the three elements of the zoom lens changes its position when the focal length of the zoom lens is changing. Thus, there is no need to use a complicated mechanical system, such as in the conventional zoom lenses. The analysis of the three-element optical system is more complex and exhaustive than the analysis presented in papers [3134] because it also makes possible to affect distortion of the triplet by the parameter 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 fmin=67mm to fmax=100mm. It is evident from the presented example that the change of the focal length of the optical system does not vary its chromatic aberrations significantly. The position of the image plane is constant as well. Using the previous analysis one can obtain initial design parameters of the triplet, which can be used as the starting point for further optimization using appropriate optical design software such as, for example, ZEMAX, OSLO, etc.

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).

References

  • View by:
  • |
  • |
  • |

  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).

2011

2010

2009

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]

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]

2008

2007

2006

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]

2005

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]

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]

2004

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]

2002

2000

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]

1964

1948

As, M. A. J.

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]

Baer, E.

Beadie, G.

Berge, B.

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]

Bräuer, A.

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]

Chen, J.

Cho, S. H.

Craen, P.

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]

Fan, Y. H.

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]

Fang, Y.-Ch.

Gauza, S.

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]

Hazra, L.

Hendriks, B. H. W.

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]

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]

Hiltner, A.

Hsueh, B.-R.

Hu, C. C.

Justis, N.

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]

Kazmierczak, T.

Kuiper, S.

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]

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]

Lepkowicz, R. S.

Lo, Y. H.

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]

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]

MacDonald, J.

Miks, A.

Mikš, A.

Noguchi, M.

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]

Novak, J.

Novak, P.

Novák, J.

Novák, P.

Pal, S.

Peng, R.

Peseux, J.

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]

Ponting, M.

Reichelt, S.

Ren, H. W.

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

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]

Renders, C. A.

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]

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]

Sandrock, M. L.

Sato, S.

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]

Schreiber, P.

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]

Shirk, J. S.

Stephens, R. E.

Sun, J.-H.

Tsai, F. S.

Tukker, T. W.

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]

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]

van As, M. A. J.

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]

Vasko, B.

Vasko, J.

Wallin, W.

Wang, B.

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]

Wang, Z.

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

Wiggins, M. J.

Wippermann, F. C.

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]

Wu, S. T.

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

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]

Xu, Y.

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

Yang, Y.

Ye, M.

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]

Zappe, H.

Zhang, D. Y.

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]

Zhao, Y.

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

Zhu, Ch.

Zhuang, S.

Appl. Opt.

Appl. Phys. Lett.

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]

Electron. Lett.

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]

Eur. Phys. J. E

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]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

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]

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

Opt. Express

Opt. Lett.

Opt. Rev.

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]

Proc. SPIE

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]

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]

Other

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).

http://www.varioptic.com

http://www.optotune.com/

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

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 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, y 0 - paraxial image height, s – object distance, s - image distance, φ 1 , φ 2 , φ 3 - individual powers of the lenses, d 1 , d 2 - separations of the lenses, s ¯ - position of the entrance pupil, h 1 - incidence height of the aperture ray, h ¯ 1 - incidence height of the principal ray.

Tables (4)

Tables Icon

Table 1 Basic Parameters of Triplets

Tables Icon

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

Tables Icon

Table 3 Aberration Coefficients of Triplet - OL1024/N-ZK7/OL1024

Tables Icon

Table 4 Residual Aberrations of Triplet - OL1024/N-ZK7/OL1024

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

φ=1/ f =γ, s F =δ/γ, s F =α/γ, s=(δ1/m)/γ, s =(mα)/γ=(βαs)/(γsδ),
α= d 1 d 2 φ 1 φ 2 d 2 ( φ 1 + φ 2 ) d 1 φ 1 +1,β= d 1 + d 2 d 1 d 2 φ 2 , γ=[ d 1 d 2 φ 1 φ 2 φ 3 d 2 φ 3 ( φ 1 + φ 2 ) d 1 φ 1 ( φ 2 + φ 3 )+ φ 1 + φ 2 + φ 3 ], δ= d 1 d 2 φ 2 φ 3 d 2 φ 3 d 1 ( φ 2 + φ 3 )+1,
δ s λ = (ms) 2 i=1 3 h i 2 φ i ν i = (ms) 2 C I ,
δ y λ = y 0 i=1 3 h i h ¯ i φ i ν i = y 0 C II ,
h ¯ 1 =s s ¯ /( s ¯ s), s ¯ = d 1 /(1 φ 1 d 1 ).
C I = φ 1 ν 1 + h 2 2 φ 2 ν 2 + h 3 2 φ 3 ν 3 =0, C II = h ¯ 1 φ 1 ν 1 + h 3 h ¯ 3 φ 3 ν 3 =0.
φ= φ 1 + h 2 φ 2 + h 3 φ 3 .
P= φ 1 / n 1 + φ 2 / n 2 + φ 3 / n 3 .
φ 1 + φ 2 + φ 3 nP=p.
d 1 φ 1 d 2 φ 3 =D.
d 1 = 1 h 2 φ 1 , d 2 = h 2 h 3 1 h 3 φ 3 , s = h 3 , s ¯ = h 2 h 3 h 2 φ 3 1 .
φ 3 = D+ h 2 1 h 3 D+ h 2 ( h 3 1) , φ 2 = p φ 3 (1 h 3 )1 1 h 2 , φ 1 =p φ 2 φ 3 .
h ¯ 1 = h 2 1 h 2 φ 1 , h ¯ 3 = h 2 h 3 h 2 (1 h 3 φ 3 ) .
a 0 h 2 3 + a 1 h 2 2 + a 2 h 2 + a 3 =0, b 0 h 2 + b 1 =0.
a 0 = V 2 p(1 h 3 ), a 1 = V 3 h 3 2 +( V 2 + V 1 pD V 2 p ) h 3 + V 2 (D1) V 1 (p1), a 2 = V 3 ( D2 ) h 3 2 + V 1 (Dp h 3 D+2), a 3 = V 3 ( 1D ) h 3 2 V 1 h 3 , b 0 = V 1 V 3 h 3 , b 1 =( 1D ) V 3 h 3 V,
R=| a 0 a 1 a 2 a 3 b 0 b 1 0 0 0 b 0 b 1 0 0 0 b 0 b 1 |= a 3 b 0 3 + a 2 b 0 2 b 1 a 1 b 0 b 1 2 + a 0 b 1 3 =0.
c 4 h 3 4 + c 3 h 3 3 + c 2 h 3 2 + c 1 h 3 + c 0 =0,
c 4 = V 3 3 ( D1 )( V 1 ( 1p+D )+ V 2 ( p1 )( 1D ) ), c 3 = V 3 2 p( V 1 V 2 ( D 3 D 2 3D+3 ) V 1 V 3 ( D1 ) 2 V 2 V 3 ( D1 ) 3 + V 1 2 ( D 2 +D3 ) ) V 3 2 ( V 1 V 2 ( D 2 4D+3 )+ V 1 V 3 ( D1 ) V 2 V 3 ( D1 ) 3 + V 1 2 ( D 2 3 ) ), c 2 = V 1 V 3 p( V 1 V 3 ( D 2 4D+3 )+ V 1 V 2 ( 2 D 2 3 )3 V 2 V 3 ( D1 ) 2 V 1 2 ( D 2 D3 ) ) V 1 V 3 ( V 1 V 3 ( D 2 3D+3 )+ V 1 V 2 ( 2D3 )+ V 2 V 3 ( D 3 5 D 2 +7D3 )+3 V 1 2 ), c 1 = V 1 2 p( ( V 1 V 2 V 1 2 )( 1+D )+ V 1 V 3 ( 2D3 ) V 2 V 3 ( 3D3 ) ) V 1 2 ( V 1 V 2 V 1 2 V 1 V 3 ( D 2 3D+3 )+ V 2 V 3 ( 2 D 2 5D+3 ) ), c 0 = V 1 3 ( V 1 V 2 )( D+p1 ).
ν E = i=1 2 φ i i=1 2 φ i / ν i = ( n 2 n 1 ) ν 1 ν 2 ( n 2 1) ν 1 ( n 1 1) ν 2 =15.046,
δ s λ = f / ν E =0.0665 f .
φ 1 =(φ s F + d 2 φ 2 1)/( d 1 d 2 φ 2 d 2 d 1 ), φ 3 =(φ+ d 1 φ 1 φ 2 φ 1 φ 2 )/φ s F .

Metrics