Abstract

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

© 2010 OSA

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References

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  1. http://www.varioptic.com
  2. http://www.optotune.com/
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    [CrossRef]
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    [CrossRef]
  5. B. Berge, “Liquid lens technology: Principle of electrowetting based lenses and applications to imaging”, Proc. IEEE MEMS, 227–230 (2004).
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    [CrossRef]
  7. S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. 85(7), 1128–1130 (2004).
    [CrossRef]
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    [CrossRef]
  9. D. Y. Zhang, N. Justis, and Y. H. Lo, “Fluidic adaptive zoom lens with high zoom ratio and widely tunable field of view,” Opt. Commun. 249(1-3), 175–182 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  15. 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]
  16. M. Ye, M. Noguchi, B. Wang, and S. Sato, “Zoom lens system without moving elements realised using liquid crystal lenses,” Electron. Lett. 45(12), 646–648 (2009).
    [CrossRef]
  17. P. Valley, D. L. Mathine, M. R. Dodge, J. Schwiegerling, G. Peyman, and N. Peyghambarian, “Tunable-focus flat liquid-crystal diffractive lens,” Opt. Lett. 35(3), 336–338 (2010).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  19. F. C. Wippermann, P. Schreiber, A. Bräuer, and P. Craen, “Bifocal liquid lens zoom objective for mobile phone applications,” SPIE Proc. 6501, 650109 (2007).
    [CrossRef]
  20. 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]
  21. B. H. W. Hendriks, S. Kuiper, M. A. J. van As, C. A. Renders, and T. W. Tukker, “Variable liquid lenses for electronic products,” Proc. SPIE 6034, 603402 (2006).
    [CrossRef]
  22. A. Miks, Applied Optics (Czech Technical University Press, Prague 2009).
    [PubMed]
  23. W. Smith, Modern Optical Engineering, 4th Ed. (McGraw-Hill, New York 2007).
  24. M. Born, and E. Wolf, Principles of Optics, (Oxford University Press, New York 1964).
  25. P. Mouroulis, and J. Macdonald, Geometrical Optics and Optical Design (Oxford University Press, New York 1997).
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    [CrossRef] [PubMed]
  28. 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]
  29. Z. Wang, Y. Xu, and Y. Zhao, “Aberration analyses of liquid zooming lenses without moving parts,” Opt. Commun. 275(1), 22–26 (2007).
    [CrossRef]
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    [CrossRef]
  31. K. Rektorys, Survey of Applicable Mathematics. (Kluwer Academic Publisher, Dodrecht 1994)

2010 (1)

2009 (2)

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]

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 (3)

2007 (5)

2006 (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 (2006).
[CrossRef]

H. Ren, D. Fox, P. A. Anderson, B. Wu, and S. T. Wu, “Tunable-focus liquid lens controlled using a servo motor,” Opt. Express 14(18), 8031–8036 (2006).
[CrossRef] [PubMed]

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. Van As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005).
[CrossRef]

2004 (2)

S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. 85(7), 1128–1130 (2004).
[CrossRef]

H. W. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. 84(23), 4789–4791 (2004).
[CrossRef]

2002 (2)

M. Ye and S. Sato, “Optical properties of liquid crystal lens of any size,” Jpn. J. Appl. Phys. 41(Part 2, No. 5B), L571–L573 (2002).
[CrossRef]

C. Gabay, B. Berge, G. Dovillaire, and S. Bucourt, “Dynamic study of a Varioptic variable focal lens,” SPIE Proc. 4767, 159–165 (2002).
[CrossRef]

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]

1999 (1)

1944 (1)

Anderson, P. A.

Baer, E.

Beadie, G.

Berge, B.

C. Gabay, B. Berge, G. Dovillaire, and S. Bucourt, “Dynamic study of a Varioptic variable focal lens,” SPIE Proc. 4767, 159–165 (2002).
[CrossRef]

B. Berge 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,” SPIE Proc. 6501, 650109 (2007).
[CrossRef]

Bucourt, S.

C. Gabay, B. Berge, G. Dovillaire, and S. Bucourt, “Dynamic study of a Varioptic variable focal lens,” SPIE Proc. 4767, 159–165 (2002).
[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,” SPIE Proc. 6501, 650109 (2007).
[CrossRef]

Dodge, M. R.

Dovillaire, G.

C. Gabay, B. Berge, G. Dovillaire, and S. Bucourt, “Dynamic study of a Varioptic variable focal lens,” SPIE Proc. 4767, 159–165 (2002).
[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]

Fox, D.

Gabay, C.

C. Gabay, B. Berge, G. Dovillaire, and S. Bucourt, “Dynamic study of a Varioptic variable focal lens,” SPIE Proc. 4767, 159–165 (2002).
[CrossRef]

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]

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. Van As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005).
[CrossRef]

S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. 85(7), 1128–1130 (2004).
[CrossRef]

Herzberger, M.

Hiltner, A.

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 (2006).
[CrossRef]

B. H. W. Hendriks, S. Kuiper, M. A. J. Van As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005).
[CrossRef]

S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. 85(7), 1128–1130 (2004).
[CrossRef]

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]

Loktev, M. Y.

Love, G. D.

Marks, R.

Mathine, D. L.

Naumov, A. F.

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]

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.

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. Van As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005).
[CrossRef]

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]

M. Ye and S. Sato, “Optical properties of liquid crystal lens of any size,” Jpn. J. Appl. Phys. 41(Part 2, No. 5B), L571–L573 (2002).
[CrossRef]

Schreiber, P.

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

Schwiegerling, J.

Shirk, J. S.

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. Van As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005).
[CrossRef]

Valley, P.

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]

B. H. W. Hendriks, S. Kuiper, M. A. J. Van As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005).
[CrossRef]

Vasko, B.

Vasko, J.

Vladimirov, F. L.

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,” SPIE Proc. 6501, 650109 (2007).
[CrossRef]

Wu, B.

Wu, S. T.

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]

M. Ye and S. Sato, “Optical properties of liquid crystal lens of any size,” Jpn. J. Appl. Phys. 41(Part 2, No. 5B), L571–L573 (2002).
[CrossRef]

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. Phys. Lett. (2)

S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. 85(7), 1128–1130 (2004).
[CrossRef]

H. W. Ren, Y. H. Fan, S. Gauza, and S. T. Wu, “Tunable-focus flat liquid crystal spherical lens,” Appl. Phys. Lett. 84(23), 4789–4791 (2004).
[CrossRef]

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 (1)

Jpn. J. Appl. Phys. (1)

M. Ye and S. Sato, “Optical properties of liquid crystal lens of any size,” Jpn. J. Appl. Phys. 41(Part 2, No. 5B), L571–L573 (2002).
[CrossRef]

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. (3)

Opt. Rev. (1)

B. H. W. Hendriks, S. Kuiper, M. A. J. Van As, C. A. Renders, and T. W. Tukker, “Electrowetting-based variable-focus lens for miniature systems,” Opt. Rev. 12(3), 255–259 (2005).
[CrossRef]

Proc. SPIE (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 (2006).
[CrossRef]

SPIE Proc. (2)

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

C. Gabay, B. Berge, G. Dovillaire, and S. Bucourt, “Dynamic study of a Varioptic variable focal lens,” SPIE Proc. 4767, 159–165 (2002).
[CrossRef]

Other (9)

B. Berge, “Liquid lens technology: Principle of electrowetting based lenses and applications to imaging”, Proc. IEEE MEMS, 227–230 (2004).

http://www.varioptic.com

http://www.optotune.com/

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

W. Smith, Modern Optical Engineering, 4th Ed. (McGraw-Hill, New York 2007).

M. Born, and E. Wolf, Principles of Optics, (Oxford University Press, New York 1964).

P. Mouroulis, and J. Macdonald, Geometrical Optics and Optical Design (Oxford University Press, New York 1997).

M. Herzberger, Modern Geometrical Optics (Interscience Publishers, Inc., New York 1958).

K. Rektorys, Survey of Applicable Mathematics. (Kluwer Academic Publisher, Dodrecht 1994)

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Figures (5)

Fig. 1
Fig. 1

Simple refractive tunable-focus lens

Fig. 2
Fig. 2

General optical system

Fig. 3
Fig. 3

Graph of functions A(P,Q) and B(P,Q) for value P = 1.38

Fig. 4
Fig. 4

Generalized tunable-focus lens

Fig. 5
Fig. 5

Thin generalized tunable-focus lens

Tables (4)

Tables Icon

Table 1 Simple thin tunable-focus lens

Tables Icon

Table 2 Generalized thin tunable-focus lens

Tables Icon

Table 3 Three tunable-focus lenses (n 2=n 5=n 8, n 3=n 6=n 9, A 1=A 2=A 3, B 1=B 2=B 3)

Tables Icon

Table 4 Three tunable-focus lenses (n 2=n 6=n 8, n 3=n 5=n 9, A 1=A 3, B 1=B 3)

Equations (48)

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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 + 1 = n 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 ,
m = y 0 y 0 = n 1 σ 1 n K σ K .
φ = ( n 3 n 2 ) / r 2 , σ 4 = h 1 φ , h 3 = h 1 ( 1 d 2 n 3 φ ) ,
f = 1 φ = h 1 σ 4 = r 2 ( n 3 n 2 ) , s F = h 3 σ 4 = f d 2 n 3 , s F = f + d 1 n 2 .
s f = 1 m 1 + d 1 n 2 f , s f = 1 m d 2 n 3 f ,
q q = f 2 , m = q / f = f / q ,
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 , 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 2 n K σ K ( s 1 s ¯ 1 ) 3 σ 1 σ ¯ 1 2 ( S I I I + H 2 S I V ) , 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 ) , and 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 .
σ ¯ 1 = h ¯ 1 s ¯ 1 = y 0 s ¯ 1 s 1 .
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     C , S V = h ¯ 3 h 2 M 3 H h ¯ 2 h 2 N + H 2 h ¯ h φ ( 3 + C ) ,
M = ( h φ ) 3 A + σ ( h φ ) 2 ( 4 B 1 ) + σ 2 ( h φ ) ( 3 + 2 C ) ,
N = ( h φ ) 2 B + σ ( h φ ) ( 2 + C ) .
A = 1 + 2 P Q Q ( P Q ) 2 , B = 1 + 1 Q ( P Q ) , C = 1 P Q ,
x = y 2 2 r + ( 1 + b ) y 4 8 r 3 ,
M a s f = M + b ( h r 2 ) 3 ( Q P ) .
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 ,
φ 1 = 1 / f 1 = ( n 3 n 2 ) / r 2 , φ 2 = 1 / f 2 = ( n 6 n 5 ) / r 5 , d = d 2 / n 3 + d 3 + d 4 / n 5 .
φ = 1 / f = φ 1 + φ 2 + d φ 1 φ 2 , s F = f ( 1 d / f 1 ) , s F = f ( 1 d / f 2 ) .
σ 1 σ 1 = h 1 φ 1 , σ 2 = σ , σ 2 σ 2 = h 2 φ 2 , σ 2 σ 1 = h 1 φ 1 + h 2 φ 2 = h φ , φ = φ 1 + φ 2 ,
σ K σ 1 = i = 1 K h i φ i = h φ .
A 1 = A ( P = n 2 , Q = n 3 ) , B 1 = B ( P = n 2 , Q = n 3 ) , C 1 = C ( P = n 2 , Q = n 3 ) .
A 2 = A ( P = n 5 , Q = n 6 ) , B 2 = B ( P = n 5 , Q = n 6 ) , C 2 = C ( P = n 5 , Q = n 6 ) .
1 ) n 2 = n 5 , n 3 = n 6 , A 1 = A 2 , B 1 = B 2 , C 1 = C 2 , 2 ) n 2 = n 6 , n 3 = n 5 , A 1 A 2 , B 1 B 2 , C 1 = C 2 .
M 12 = M 1 + M 2 = h 3 [ φ 1 3 A 1 + ( φ φ 1 ) 3 A 2 + φ 1 2 ( φ φ 1 ) ( 3 + 2 C 2 ) + φ 1 ( φ φ 1 ) 2 ( 4 B 2 1 ) ] , N 12 = N 1 + N 2 = h 2 [ φ 1 2 B 1 + ( φ φ 1 ) 2 B 2 + φ 1 ( φ φ 1 ) ( 2 + C 2 ) ] .
S I = h M 12 , S I I = H N 12 , S I I I = H 2 φ , S I V = φ     C , S V = 0.
3 ( A 1 A 2 + 4 B 2 2 C 2 4 ) φ 1 2 + 2 φ ( 3 A 2 + 2 C 2 8 B 2 + 4 ) φ 1 + φ 2 ( 4 B 2 3 A 2 1 ) = 0.
σ 1 = 0 , σ 2 = h φ 1 , σ 3 = h ( φ 1 + φ 2 ) , φ 3 = φ ( φ 1 + φ 2 ) .
M 123 = M 1 + M 2 + M 3 , N 123 = N 1 + N 2 + N 3 ,
M 123 = h 3 ( k 1 φ 1 3 + k 2 φ 2 3 + k 3 φ 1 2 φ 2 + k 4 φ 1 φ 2 2 + k 5 φ 1 2 + k 6 φ 2 2 + k 7 φ 1 φ 2 + k 8 φ 1 + k 9 φ 2 + k 10 ) ,
N 123 = h 2 ( p 1 φ 1 2 + p 2 φ 2 2 + p 3 φ 1 φ 2 + p 4 φ 1 + p 5 φ 2 + p 6 ) .
b 2 = 4 B 2 1 , b 3 = 4 B 3 1 , c 2 = 3 + 2 C 2 , c 3 = 3 + 2 C 3 ,
k 1 = A 1 A 3 + b 3 c 3 , k 2 = A 2 A 3 + b 3 c 3 , k 3 = 3 A 3 + 3 b 3 3 c 3 + c 2 ,
k 4 = 3 A 3 + 3 b 3 3 c 3 + b 2 , k 5 = φ ( 3 A 3 3 b 3 + c 3 ) , k 6 = k 5 , k 7 = 2 k 5 ,
k 8 = φ 2 ( 3 A 3 + b 3 ) , k 9 = k 8 , k 10 = φ 3 A 3 ,
p 1 = B 1 + B 3 ( 2 + C 3 ) , p 2 = B 2 + B 3 ( 2 + C 3 ) , p 3 = 2 B 3 + ( 2 + C 2 ) 2 ( 2 + C 3 ) ,
p 4 = ( 2 + C 3 ) 2 B 3 , p 5 = p 4 , p 6 = φ 2 B 3 .
S I = h M 123 , S I I = H N 123 , S I I I = H 2 φ , S I V = φ     C , S V = 0.
α 0 φ 1 3 + α 1 φ 1 2 + α 2 φ 1 + α 3 = 0 ,
β 0 φ 1 2 + β 1 φ 1 + β 2 = 0 ,
α 0 = k 1 , α 1 = k 3 φ 2 + k 5 , α 2 = k 4 φ 2 2 + k 7 φ 2 + k 8 , α 3 = k 2 φ 2 3 + k 6 φ 2 2 + k 9 φ 2 + k 10 ,
β 0 = p 1 , β 1 = p 3 φ 2 + p 4 , β 2 = p 2 φ 2 2 + p 5 φ 2 + p 6 .
R = | α 0 α 1 α 2 α 3 0 0 α 0 α 1 α 2 α 3 β 0 β 1 β 2 0 0 0 β 0 β 1 β 2 0 0 0 β 0 β 1 β 2 | = 0.
R = α 0 α 3 ( 2 β 0 β 1 β 2 β 1 3 ) + α 0 β 2 ( α 0 β 2 2 + α 2 β 1 2 α 2 β 0 β 2 α 1 β 1 β 2 ) + + α 3 β 0 ( α 1 β 1 2 + α 3 β 0 2 α 2 β 0 β 1 α 1 β 0 β 2 ) + + β 0 β 2 ( α 1 2 β 2 + α 2 2 β 0 + α 0 α 3 β 1 α 1 α 3 β 0 α 0 α 2 β 2 α 1 α 2 β 1 ) = 0.
k 10 ( k 10 S I / h 4 ) , p 6 ( p 6 + S I I / H h 2 ) .
δ y = y P 3 2 f 3 S I = 1 2 sin 3 U S I , δ s = 1 2 sin 2 U S I ,
δ s min = 1 2 sin 2 U ( S I ) min ,
( S I ) min = n ( 4 n 1 ) 4 ( n + 2 ) ( n 1 ) 2 f .

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