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 third order aberration analysis of possible optical designs of two-element zoom lens systems based on variable power lenses with a variable focal length. First order chromatic aberrations of the variable power 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.

© 2010 OSA

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References

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  1. S. F. Ray, Applied photographic optics, (Focal Press, New York 2002).
  2. W. Smith, Modern optical engineering, 4th Ed. (McGraw-Hill, New York 2007).
  3. M. Born, and E. Wolf, Principles of optics, (Oxford University Press, New York 1964).
  4. P. Mouroulis, and J. Macdonald, Geometrical optics and optical design (Oxford University Press, New York 1997).
  5. A. Miks, Applied optics (Czech Technical University Press, Prague 2009).
    [PubMed]
  6. M. Herzberger, Modern geometrical optics (Interscience Publishers, Inc., New York 1958).
  7. A. D. Clark, Zoom lenses (Adam Hilger, London, 1973).
  8. K. Yamaji, Progres in optics, Vol.VI (North-Holland Publishing Co., Amsterdam 1967).
  9. A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt. 47(32), 6088–6098 (2008).
    [CrossRef] [PubMed]
  10. A. Mikš, “Modification of the formulas for third-order aberration coefficients,” J. Opt. Soc. Am. A 19(9), 1867–1871 (2002).
    [CrossRef]
  11. A. Walther, “Angle eikonals for a perfect zoom system,” J. Opt. Soc. Am. A 18(8), 1968–1971 (2001).
    [CrossRef]
  12. G. Wooters and E. W. Silvertooth, “Optically compensated zoom lens,” J. Opt. Soc. Am. 55(4), 347–351 (1965).
    [CrossRef]
  13. D. F. Kienholz, “The design of a zoom lens with a large computer,” Appl. Opt. 9(6), 1443–1452 (1970).
    [CrossRef] [PubMed]
  14. A. V. Grinkevich, “Version of an objective with variable focal length,” J. Opt. Technol. 73, 343–345 (2006).
    [CrossRef]
  15. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 1: Four-component Type,” Appl. Opt. 21(12), 2174–2183 (1982).
    [CrossRef] [PubMed]
  16. G. H. Matter and E. T. Luszcz, “A family of optically compensated zoom lenses,” Appl. Opt. 9(4), 844–848 (1970).
    [CrossRef] [PubMed]
  17. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses 1: Four-component type Errata,” Appl. Opt. 21(21), 3805 (1982).
    [CrossRef] [PubMed]
  18. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 2: Generalization of Yamaji type V,” Appl. Opt. 21(22), 4045–4053 (1982).
    [CrossRef] [PubMed]
  19. K. Tanaka, “Paraxial analysis of mechanically compensated zoom lenses. 3: Five-component type,” Appl. Opt. 22(4), 541–553 (1983).
    [CrossRef] [PubMed]
  20. D. F. Horne, Lens mechanism technology (Adam Hilger, Bristol 1975)
  21. 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]
  22. F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” Proc. SPIE 6501, 650109 (2007).
    [CrossRef]
  23. 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]
  24. 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]
  25. http://www.varioptic.com
  26. http://www.optotune.com/
  27. 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]
  28. 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]
  29. 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]
  30. H. W. Ren and S. T. Wu, “Variable-focus liquid lens,” Opt. Express 15(10), 5931–5936 (2007).
    [CrossRef] [PubMed]
  31. 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]
  32. 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]
  33. 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]
  34. 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]
  35. S. Reichelt and H. Zappe, “Design of spherically corrected, achromatic variable-focus liquid lenses,” Opt. Express 15(21), 14146–14154 (2007).
    [CrossRef] [PubMed]
  36. 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]
  37. Z. Wang, Y. Xu, and Y. Zhao, “Aberration analyses of liquid zooming lenses without moving parts,” Opt. Commun. 275(1), 22–26 (2007).
    [CrossRef]
  38. 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]

2009

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 (2006).
[CrossRef]

A. V. Grinkevich, “Version of an objective with variable focal length,” J. Opt. Technol. 73, 343–345 (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

2001

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]

1983

1982

1970

1965

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 (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 (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]

Grinkevich, A. V.

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 (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.

Kienholz, D. F.

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 (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]

Luszcz, E. T.

MacDonald, J.

Marks, R.

Mathine, D. L.

Matter, G. H.

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]

Novák, J.

Novák, P.

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]

Peyghambarian, N.

Peyman, G.

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 (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 (2007).
[CrossRef]

Schwiegerling, J.

Shirk, J. S.

Silvertooth, E. W.

Sun, J.-H.

Tanaka, K.

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 (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 (2006).
[CrossRef]

Vasko, B.

Vasko, J.

Walther, A.

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 (2007).
[CrossRef]

Wooters, G.

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

J. Opt. Technol.

Opt. Commun.

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

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

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

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]

Other

http://www.varioptic.com

http://www.optotune.com/

D. F. Horne, Lens mechanism technology (Adam Hilger, Bristol 1975)

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

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).

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

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

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

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

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Figures (6)

Fig. 1
Fig. 1

Two-element optical system

Fig. 2
Fig. 2

Principal and aperture paraxial rays

Fig. 3
Fig. 3

Simple variable power lens

Fig. 4
Fig. 4

Dependence of power on transverse magnification (example 1)

Fig. 5
Fig. 5

Dependence of power values on transverse magnification (example 2)

Fig. 6
Fig. 6

Spot diagrams for imaging of axial point in dependence on transverse magnification m (thin lens system). Transverse ray aberrations dx, dy are given in mm, numerical aperture is N.A. = - 0.02.

Tables (3)

Tables Icon

Table 1 Parameters of zoom lens system

Tables Icon

Table 3 Aberration coefficients of optical system

Tables Icon

Table 2 Parameters of optical system

Equations (33)

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n i + 1 σ i + 1 = n i σ i + h i ϕ i , h i + 1 = h i d i σ i + 1 ,     i   =  1,2 , N ,
ϕ = ϕ 1 + ϕ 2 d ϕ 1 ϕ 2 = 1 f 1 + 1 f 2 d f 1 f 2 , f = f 1 f 2 f 1 + f 2 d , m = σ 1 / σ 3 , a F = f ( 1 d / f 1 ) , a F = f ( 1 d / f 2 ) , a H = a F f , a H = a F + f ,
a 1 = f ( 1 m 1 + d f 2 ) ,     a 2 = f ( 1 m d f 1 )     .
L = a 1 + Δ H 1 + d + Δ H 2 + a 2 ,
d 2 L d + ( f 1 + f 2 ) L + f 1 f 2 ( m 1 ) 2 / m = 0     .
d = 1 2 [ L ± L 2 4 ( f 1 + f 2 ) L 4 f 1 f 2 ( m 1 ) 2 / m ]     .
a 1 = d L + ( 1 m ) f 2 f 1 + f 2 m f 1 , a 2 = m ( L d ) + ( 1 m ) f 1 f 1 + f 2 m f .
α 2 ϕ 2 2 + α 1 ϕ 2 + α 0 = 0 ,
α 2 = a 2 d 2 , α 1 = d [ a 2 ( 1 / m 2 ) + a 1 m d ]     , α 0 = a 1 ( 1 m ) ( a 2 + d ) ( 1 / m 1 ) .
Δ = α 1 2 4 α 2 α 0 = d 2 ( a 2 / m a 1 m + d ) 2 0.
ϕ 2 = a 2 + d a 1 m a 2 d = A + B m .
ϕ 1 = ( 1 / m 1 ) + ϕ 2 ( d a 1 ) a 1 ( 1 ϕ 2 d ) = ( 1 d 1 a 1 ) ( a 2 a 1 d ) 1 m = C + D / m     .
ϕ 1 = 1 a 2 ϕ d = 1 a 2 / f d , ϕ 2 = ϕ ( a 2 + d ) 1 a 2 ϕ d = a 2 + d f a 2 d .
n i σ i = n i σ i + h i ( n i n i ) / r i , h i + 1 = h i d i σ i , σ i + 1 = σ , n i + 1 = n i , i = 1 , 2 , ... , K , n i / s i n i / s i = ( n i n i ) / r i , s i + 1 = s i d i , n i + 1 = n i , i = 1 , 2 , ... , K ,
m = y 0 y 0 = n 1 σ 1 n K σ K .
n i σ ¯ i = n i σ ¯ i + h ¯ i ( n i n i ) / r i , h ¯ i + 1 = h ¯ i d i σ ¯ i , σ ¯ i + 1 = σ ¯ i , n i / s ¯ i n i / s ¯ i = ( n i n i ) / r i , s ¯ i + 1 = s ¯ i d i , n i + 1 = n i , i = 1 , 2 , ... , K ,
δ x = x P ( y P 2 + x P 2 ) 2 n K σ K ( s 1 s ¯ 1 ) 3 σ 1 3 S I + 2 y 0 y P x P 2 n K σ K ( s 1 s ¯ 1 ) 3 σ 1 2 σ ¯ 1 S I I y 0 2 y P ( S I I I + H 2 S I V ) 2 n K σ K ( s 1 s ¯ 1 ) 3 σ 1 σ ¯ 1 2 ,
δ y = y P ( y P 2 + x P 2 ) 2 n K σ K ( s 1 s ¯ 1 ) 3 σ 1 3 S I + y 0 ( 3 y P 2 + x P 2 ) 2 n K σ K ( s 1 s ¯ 1 ) 3 σ 1 2 σ ¯ 1 S I I y 0 2 y P 2 n K σ K ( s 1 s ¯ 1 ) 3 σ 1 σ ¯ 1 2 ( 3 S I I I + H 2 S I V ) + y 0 3 2 n K σ K ( s 1 s ¯ 1 ) 3 σ ¯ 1 3 S V ,
S I = i = 1 i = K h i U i , S I I = i = 1 i = K h i U i ( Δ σ ¯ i Δ σ i ) , S I I I = i = 1 i = K h i U i ( Δ σ ¯ i Δ σ i ) 2 , S I V = i = 1 i = K 1 h i Δ ( n i σ i ) n i n i , S V = i = 1 i = K [ h i U i ( Δ σ ¯ i Δ σ i ) 2 + H 2 h i Δ ( n i σ i ) n i n i ] ( Δ σ ¯ i Δ σ i ) ,
U i = ( Δ σ i Δ ( 1 / n i ) ) 2 Δ ( σ i n i ) , H = n 1 σ 1 y 0 = n K σ K y 0 = n 1 ( h ¯ 1 σ 1 h 1 σ ¯ 1 ) = n K ( h ¯ K σ K h K σ ¯ K ) = konst .
S I = h M , S I I = h ¯ M H N , S I I I = h ¯ 2 h M 2 H h ¯ h N + H 2 ϕ ,
S I V = ϕ n 2 n 3 , S V = h ¯ 3 h 2 M 3 H h ¯ 2 h 2 N + H 2 h ¯ h ϕ ( 3 + 1 n 2 n 3 ) ,
M = ( h ϕ ) 3 A + σ ( h ϕ ) 2 ( 4 B 1 ) + σ 2 ( h ϕ ) ( 3 + 2 n 2 n 3 ) ,
N = ( h ϕ ) 2 B + σ ( h ϕ ) ( 2 + 1 n 2 n 3 ) ,
A = 2 n 3 + 1 ( n 3 1 ) 2 [ ( 1 n 2 n 3 n 2 ) 2 1 ] + n 3 + 2 n 3 ( n 3 1 ) ( n 3 n 2 ) + + [ ( n 2 n 2 1 ) 2 ( n 3 n 3 1 ) 2 ] ( 1 n 2 n 3 n 2 ) 3 + ( n 3 n 3 1 ) 2 , B = ( n 2 n 2 1 n 3 n 3 1 ) ( 1 n 2 n 3 n 2 ) 2 + n 3 n 3 1 n 3 + 1 n 3 ( n 3 n 2 ) .
S I = j = 1 K ( S I ) j , S I I = j = 1 K ( S I I ) j , S I I I = j = 1 K ( S I I I ) j , S I V = j = 1 K ( S I V ) j , S V = j = 1 K ( S V ) j ,
x = y 2 2 r 2 + ( 1 + b ) y 4 8 r 2 3 ,
M a s f = M + b ( h r 2 ) 3 ( n 3 n 2 )     .
d s = s n 2 d n 2 + s n 3 d n 3 .
s = r 2 s r 2 + s ( n 3 n 2 ) .
d s = s 2 r 2 ( n 1 ) ( 1 ν 2 1 ν 3 ) ,
d y y = d s s ¯ s d σ σ ,
d y y = s ( n 1 ) ( 1 ν 2 1 ν 3 ) ( s ( s ¯ s ) 2 + 1 r 2 )     .

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