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

Full Article  |  PDF Article

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 (1)

2010 (3)

2009 (2)

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]

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]

2008 (4)

2007 (5)

2006 (1)

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 (2)

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 (1)

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 (1)

2000 (1)

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 (1)

1948 (1)

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. (4)

Appl. Phys. Lett. (1)

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. (1)

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 (1)

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. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

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

Opt. Express (6)

Opt. Lett. (2)

Opt. Rev. (1)

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 (2)

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 (11)

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