Abstract

A method of glass selection for the design of optical systems with reduced chromatic aberration is presented. This method is based on the unification of two previously published methods adding new contributions and using a multi-objective approach. This new method makes it possible to select sets of compatible glasses suitable for the design of super-apochromatic optical systems. As an example, we present the selection of compatible glasses and the effective designs for all-refractive optical systems corrected in five spectral bands, with central wavelengths going from 485 nm to 1600 nm.

© 2012 OSA

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References

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  1. P. Mouroulis, “Broadband achromatic telecentric lens,” Nasa Tech Briefs, NPO-44059, (2007).
  2. J. L. Rayces and M. Rosete-Aguilar, “Selection of glasses for achromatic doublets with reduced secondary spectrum. I. Tolerance conditions for secondary spectrum, spherochromatism, and fifth-order spherical aberration,” Appl. Opt. 40(31), 5663–5676 (2001).
    [CrossRef] [PubMed]
  3. R. D. Sigler, “Glass selection for airspaced apochromats using the Buchdahl dispersion equation,” Appl. Opt. 25(23), 4311–4320 (1986).
    [CrossRef] [PubMed]
  4. C. Gruescu, I. Nicoara, D. Popov, R. Bodea, and H. Hora, “Optical glass compatibility for the design of apochromatic systems,” Sci. Sin. 40(2), 131–140 (2008).
    [CrossRef]
  5. P. Hariharan, “Superachromatic lens combination,” Opt. Laser Technol. 31(2), 115–118 (1999).
    [CrossRef]
  6. P. Hariharan, “Apochromatic lens combinations, a novel design approach,” Opt. Laser Technol. 29(4), 217–219 (1997).
    [CrossRef]
  7. R. I. Mercado and P. N. Robb, “Color corrected optical systems and method of selecting optical materials therefor,” U.S Patent, 5,210,646, (1993).
  8. P. N. Robb, “Selection of optical glasses. 1: two materials,” Appl. Opt. 24(12), 1864–1877 (1985).
    [CrossRef] [PubMed]
  9. N. V. D. W. Lessing, “Selection of optical glasses in superachromats,” Appl. Opt. 9(7), 1665–1668 (1970).
    [CrossRef] [PubMed]
  10. T. R. Sloan, “Analysis and correction of secondary color in optical systems,” Appl. Opt. 9(4), 853–858 (1970).
    [CrossRef] [PubMed]
  11. M. Herzberger and N. R. McClure, “The design of superachromatic lenses,” Appl. Opt. 2(6), 553–560 (1963).
    [CrossRef]
  12. R. R. Willey., “Machine-aided selection of optical glasses for two-elements, three-color achromats,” Appl. Opt. 1(3), 368–369 (1962).
    [CrossRef]
  13. R. E. Stephens, “Four-color achromats and superchromats,” J. Opt. Soc. Am. 50(10), 1016–1019 (1960).
    [CrossRef]
  14. W. S. Sun, C. H. Chu, and C. L. Tien, “Well-chosen method for an optimal design of doublet lens design,” Opt. Express 17(3), 1414–1428 (2009).
    [CrossRef] [PubMed]
  15. I. Ono, Y. Tatsuzawa, S. Kobayashi, and K. Yoshida, “Designing lens systems taking account of glass selection by real-coded genetic algorithms,” in Proceedings of IEEE International Conference on Systems, Man and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1999), 7803–5731.
  16. Y. C. Fang, C. M. Tsai, J. Macdonald, and Y. C. Pai, “Eliminating chromatic aberration in Gauss-type lens design using a novel genetic algorithm,” Appl. Opt. 46(13), 2401–2410 (2007).
    [CrossRef] [PubMed]
  17. L. Li, Q. H. Wang, X. Q. Xu, and D. H. Li, “Two-step method for lens system design,” Opt. Express 18(12), 13285–13300 (2010).
    [CrossRef] [PubMed]
  18. R. E. Fischer, A. J. Grant, U. Fotheringham, P. Hartmann, and S. Reichel, “Removing the mystique of glass selection,” Proc. SPIE 5524, 134–146 (2004).
    [CrossRef]
  19. W. J. Smith, Modern Optical Engineering (McGraw-Hill, Inc., 1990).
  20. P. N. Robb and R. I. Mercado, “Calculation of refractive indices using Buchdahl’s chromatic coordinate,” Appl. Opt. 22(8), 1198–1215 (1983).
    [CrossRef] [PubMed]
  21. J. Branke, K. Deb, K. Miettinen, and R. Slowinski, Multiobjective Optimization: Interactive and Evolutionary Approaches (Springer-Verlag, Berlin, 2008).
  22. N. Srinivas and K. Deb, “Multi-objective function optimization using non-dominated sorting genetic algorithm,” Evol. Comput. 2(3), 221–248 (1994).
    [CrossRef]
  23. J. Rayces and M. R. Aguilar, “Selection of glasses for achromatic doublets with reduced secondary color,” Proc. SPIE 4093, 36–46 (2000).
    [CrossRef]
  24. N. Lopez, O. Aguirre, J. F. Espiritu, and H. A. Taboada, “Using game theory as a post-Pareto analysis for renewable energy integration problems considering multiple objectives,” in Proceedings of the 41st International Conference on Computers & Industrial Engineering, 678–683 Los Angeles, (2011).
  25. O. Aguirre, H. Taboada, D. Coit, and N. Wattanapongsakorn, “Multiple objective system reliability post-Pareto optimality using self organizing trees,” in Proceedings of IEEE International Conference on Quality and Reliability (Institute of Electrical and Electronics Engineers, New York, 2011), 225–229.
  26. E. Zio and R. Bazzo, “Clustering procedure for reducing the number of representative solutions in the Pareto front of multiobjective optimization problems,” Eur. J. Oper. Res. 210(3), 624–634 (2011).
    [CrossRef]
  27. X. Blasco, J. M. Herrero, J. Sanchis, and M. Martínez, “A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization,” Inf. Sci. 178(20), 3908–3924 (2008).
    [CrossRef]
  28. J. C. Ferreira, C. M. Fonseca, and A. Gaspar-Cunha, “Methodology to select solutions from the Pareto-optimal set: A comparative study,” in Proceedings of the 9th annual conference on Genetic and evolutionary computation, (ACM, New York, NY, 2007), 789–796.
  29. V. Venkat, S. H. Jacobson, and J. A. Stori, “A Post-optimality analysis algorithm for multi-objective optimization,” Comput. Optim. Appl. 28(3), 357–372 (2004).
    [CrossRef]
  30. C. A. Coello Coello, “Handling preferences in evolutionary multiobjective optimization: a survey,” in Proceedings of the 2000 Congress on Evolutionary Computation (Institute of Electrical and Electronics Engineers, New York, 2000), 30–37.
  31. SCHOTT N. America, Inc., “Optical glass catalogue- ZEMAX format, status as of 13th September 2011, http://www.us.schott.com/advanced_optics/english/tools_downloads/download/index.html?PHPSESSID=utt2cbk96nlk3gf7gjpb7ggt54#Optical%20Glass

2011 (1)

E. Zio and R. Bazzo, “Clustering procedure for reducing the number of representative solutions in the Pareto front of multiobjective optimization problems,” Eur. J. Oper. Res. 210(3), 624–634 (2011).
[CrossRef]

2010 (1)

2009 (1)

2008 (2)

C. Gruescu, I. Nicoara, D. Popov, R. Bodea, and H. Hora, “Optical glass compatibility for the design of apochromatic systems,” Sci. Sin. 40(2), 131–140 (2008).
[CrossRef]

X. Blasco, J. M. Herrero, J. Sanchis, and M. Martínez, “A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization,” Inf. Sci. 178(20), 3908–3924 (2008).
[CrossRef]

2007 (1)

2004 (2)

R. E. Fischer, A. J. Grant, U. Fotheringham, P. Hartmann, and S. Reichel, “Removing the mystique of glass selection,” Proc. SPIE 5524, 134–146 (2004).
[CrossRef]

V. Venkat, S. H. Jacobson, and J. A. Stori, “A Post-optimality analysis algorithm for multi-objective optimization,” Comput. Optim. Appl. 28(3), 357–372 (2004).
[CrossRef]

2001 (1)

2000 (1)

J. Rayces and M. R. Aguilar, “Selection of glasses for achromatic doublets with reduced secondary color,” Proc. SPIE 4093, 36–46 (2000).
[CrossRef]

1999 (1)

P. Hariharan, “Superachromatic lens combination,” Opt. Laser Technol. 31(2), 115–118 (1999).
[CrossRef]

1997 (1)

P. Hariharan, “Apochromatic lens combinations, a novel design approach,” Opt. Laser Technol. 29(4), 217–219 (1997).
[CrossRef]

1994 (1)

N. Srinivas and K. Deb, “Multi-objective function optimization using non-dominated sorting genetic algorithm,” Evol. Comput. 2(3), 221–248 (1994).
[CrossRef]

1986 (1)

1985 (1)

1983 (1)

1970 (2)

1963 (1)

1962 (1)

1960 (1)

Aguilar, M. R.

J. Rayces and M. R. Aguilar, “Selection of glasses for achromatic doublets with reduced secondary color,” Proc. SPIE 4093, 36–46 (2000).
[CrossRef]

Bazzo, R.

E. Zio and R. Bazzo, “Clustering procedure for reducing the number of representative solutions in the Pareto front of multiobjective optimization problems,” Eur. J. Oper. Res. 210(3), 624–634 (2011).
[CrossRef]

Blasco, X.

X. Blasco, J. M. Herrero, J. Sanchis, and M. Martínez, “A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization,” Inf. Sci. 178(20), 3908–3924 (2008).
[CrossRef]

Bodea, R.

C. Gruescu, I. Nicoara, D. Popov, R. Bodea, and H. Hora, “Optical glass compatibility for the design of apochromatic systems,” Sci. Sin. 40(2), 131–140 (2008).
[CrossRef]

Chu, C. H.

Deb, K.

N. Srinivas and K. Deb, “Multi-objective function optimization using non-dominated sorting genetic algorithm,” Evol. Comput. 2(3), 221–248 (1994).
[CrossRef]

Fang, Y. C.

Fischer, R. E.

R. E. Fischer, A. J. Grant, U. Fotheringham, P. Hartmann, and S. Reichel, “Removing the mystique of glass selection,” Proc. SPIE 5524, 134–146 (2004).
[CrossRef]

Fotheringham, U.

R. E. Fischer, A. J. Grant, U. Fotheringham, P. Hartmann, and S. Reichel, “Removing the mystique of glass selection,” Proc. SPIE 5524, 134–146 (2004).
[CrossRef]

Grant, A. J.

R. E. Fischer, A. J. Grant, U. Fotheringham, P. Hartmann, and S. Reichel, “Removing the mystique of glass selection,” Proc. SPIE 5524, 134–146 (2004).
[CrossRef]

Gruescu, C.

C. Gruescu, I. Nicoara, D. Popov, R. Bodea, and H. Hora, “Optical glass compatibility for the design of apochromatic systems,” Sci. Sin. 40(2), 131–140 (2008).
[CrossRef]

Hariharan, P.

P. Hariharan, “Superachromatic lens combination,” Opt. Laser Technol. 31(2), 115–118 (1999).
[CrossRef]

P. Hariharan, “Apochromatic lens combinations, a novel design approach,” Opt. Laser Technol. 29(4), 217–219 (1997).
[CrossRef]

Hartmann, P.

R. E. Fischer, A. J. Grant, U. Fotheringham, P. Hartmann, and S. Reichel, “Removing the mystique of glass selection,” Proc. SPIE 5524, 134–146 (2004).
[CrossRef]

Herrero, J. M.

X. Blasco, J. M. Herrero, J. Sanchis, and M. Martínez, “A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization,” Inf. Sci. 178(20), 3908–3924 (2008).
[CrossRef]

Herzberger, M.

Hora, H.

C. Gruescu, I. Nicoara, D. Popov, R. Bodea, and H. Hora, “Optical glass compatibility for the design of apochromatic systems,” Sci. Sin. 40(2), 131–140 (2008).
[CrossRef]

Jacobson, S. H.

V. Venkat, S. H. Jacobson, and J. A. Stori, “A Post-optimality analysis algorithm for multi-objective optimization,” Comput. Optim. Appl. 28(3), 357–372 (2004).
[CrossRef]

Lessing, N. V. D. W.

Li, D. H.

Li, L.

Macdonald, J.

Martínez, M.

X. Blasco, J. M. Herrero, J. Sanchis, and M. Martínez, “A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization,” Inf. Sci. 178(20), 3908–3924 (2008).
[CrossRef]

McClure, N. R.

Mercado, R. I.

Nicoara, I.

C. Gruescu, I. Nicoara, D. Popov, R. Bodea, and H. Hora, “Optical glass compatibility for the design of apochromatic systems,” Sci. Sin. 40(2), 131–140 (2008).
[CrossRef]

Pai, Y. C.

Popov, D.

C. Gruescu, I. Nicoara, D. Popov, R. Bodea, and H. Hora, “Optical glass compatibility for the design of apochromatic systems,” Sci. Sin. 40(2), 131–140 (2008).
[CrossRef]

Rayces, J.

J. Rayces and M. R. Aguilar, “Selection of glasses for achromatic doublets with reduced secondary color,” Proc. SPIE 4093, 36–46 (2000).
[CrossRef]

Rayces, J. L.

Reichel, S.

R. E. Fischer, A. J. Grant, U. Fotheringham, P. Hartmann, and S. Reichel, “Removing the mystique of glass selection,” Proc. SPIE 5524, 134–146 (2004).
[CrossRef]

Robb, P. N.

Rosete-Aguilar, M.

Sanchis, J.

X. Blasco, J. M. Herrero, J. Sanchis, and M. Martínez, “A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization,” Inf. Sci. 178(20), 3908–3924 (2008).
[CrossRef]

Sigler, R. D.

Sloan, T. R.

Srinivas, N.

N. Srinivas and K. Deb, “Multi-objective function optimization using non-dominated sorting genetic algorithm,” Evol. Comput. 2(3), 221–248 (1994).
[CrossRef]

Stephens, R. E.

Stori, J. A.

V. Venkat, S. H. Jacobson, and J. A. Stori, “A Post-optimality analysis algorithm for multi-objective optimization,” Comput. Optim. Appl. 28(3), 357–372 (2004).
[CrossRef]

Sun, W. S.

Tien, C. L.

Tsai, C. M.

Venkat, V.

V. Venkat, S. H. Jacobson, and J. A. Stori, “A Post-optimality analysis algorithm for multi-objective optimization,” Comput. Optim. Appl. 28(3), 357–372 (2004).
[CrossRef]

Wang, Q. H.

Willey, R. R.

Xu, X. Q.

Zio, E.

E. Zio and R. Bazzo, “Clustering procedure for reducing the number of representative solutions in the Pareto front of multiobjective optimization problems,” Eur. J. Oper. Res. 210(3), 624–634 (2011).
[CrossRef]

Appl. Opt. (9)

P. N. Robb, “Selection of optical glasses. 1: two materials,” Appl. Opt. 24(12), 1864–1877 (1985).
[CrossRef] [PubMed]

N. V. D. W. Lessing, “Selection of optical glasses in superachromats,” Appl. Opt. 9(7), 1665–1668 (1970).
[CrossRef] [PubMed]

T. R. Sloan, “Analysis and correction of secondary color in optical systems,” Appl. Opt. 9(4), 853–858 (1970).
[CrossRef] [PubMed]

M. Herzberger and N. R. McClure, “The design of superachromatic lenses,” Appl. Opt. 2(6), 553–560 (1963).
[CrossRef]

R. R. Willey., “Machine-aided selection of optical glasses for two-elements, three-color achromats,” Appl. Opt. 1(3), 368–369 (1962).
[CrossRef]

J. L. Rayces and M. Rosete-Aguilar, “Selection of glasses for achromatic doublets with reduced secondary spectrum. I. Tolerance conditions for secondary spectrum, spherochromatism, and fifth-order spherical aberration,” Appl. Opt. 40(31), 5663–5676 (2001).
[CrossRef] [PubMed]

R. D. Sigler, “Glass selection for airspaced apochromats using the Buchdahl dispersion equation,” Appl. Opt. 25(23), 4311–4320 (1986).
[CrossRef] [PubMed]

Y. C. Fang, C. M. Tsai, J. Macdonald, and Y. C. Pai, “Eliminating chromatic aberration in Gauss-type lens design using a novel genetic algorithm,” Appl. Opt. 46(13), 2401–2410 (2007).
[CrossRef] [PubMed]

P. N. Robb and R. I. Mercado, “Calculation of refractive indices using Buchdahl’s chromatic coordinate,” Appl. Opt. 22(8), 1198–1215 (1983).
[CrossRef] [PubMed]

Comput. Optim. Appl. (1)

V. Venkat, S. H. Jacobson, and J. A. Stori, “A Post-optimality analysis algorithm for multi-objective optimization,” Comput. Optim. Appl. 28(3), 357–372 (2004).
[CrossRef]

Eur. J. Oper. Res. (1)

E. Zio and R. Bazzo, “Clustering procedure for reducing the number of representative solutions in the Pareto front of multiobjective optimization problems,” Eur. J. Oper. Res. 210(3), 624–634 (2011).
[CrossRef]

Evol. Comput. (1)

N. Srinivas and K. Deb, “Multi-objective function optimization using non-dominated sorting genetic algorithm,” Evol. Comput. 2(3), 221–248 (1994).
[CrossRef]

Inf. Sci. (1)

X. Blasco, J. M. Herrero, J. Sanchis, and M. Martínez, “A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization,” Inf. Sci. 178(20), 3908–3924 (2008).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Express (2)

Opt. Laser Technol. (2)

P. Hariharan, “Superachromatic lens combination,” Opt. Laser Technol. 31(2), 115–118 (1999).
[CrossRef]

P. Hariharan, “Apochromatic lens combinations, a novel design approach,” Opt. Laser Technol. 29(4), 217–219 (1997).
[CrossRef]

Proc. SPIE (2)

R. E. Fischer, A. J. Grant, U. Fotheringham, P. Hartmann, and S. Reichel, “Removing the mystique of glass selection,” Proc. SPIE 5524, 134–146 (2004).
[CrossRef]

J. Rayces and M. R. Aguilar, “Selection of glasses for achromatic doublets with reduced secondary color,” Proc. SPIE 4093, 36–46 (2000).
[CrossRef]

Sci. Sin. (1)

C. Gruescu, I. Nicoara, D. Popov, R. Bodea, and H. Hora, “Optical glass compatibility for the design of apochromatic systems,” Sci. Sin. 40(2), 131–140 (2008).
[CrossRef]

Other (10)

R. I. Mercado and P. N. Robb, “Color corrected optical systems and method of selecting optical materials therefor,” U.S Patent, 5,210,646, (1993).

W. J. Smith, Modern Optical Engineering (McGraw-Hill, Inc., 1990).

I. Ono, Y. Tatsuzawa, S. Kobayashi, and K. Yoshida, “Designing lens systems taking account of glass selection by real-coded genetic algorithms,” in Proceedings of IEEE International Conference on Systems, Man and Cybernetics (Institute of Electrical and Electronics Engineers, New York, 1999), 7803–5731.

N. Lopez, O. Aguirre, J. F. Espiritu, and H. A. Taboada, “Using game theory as a post-Pareto analysis for renewable energy integration problems considering multiple objectives,” in Proceedings of the 41st International Conference on Computers & Industrial Engineering, 678–683 Los Angeles, (2011).

O. Aguirre, H. Taboada, D. Coit, and N. Wattanapongsakorn, “Multiple objective system reliability post-Pareto optimality using self organizing trees,” in Proceedings of IEEE International Conference on Quality and Reliability (Institute of Electrical and Electronics Engineers, New York, 2011), 225–229.

J. C. Ferreira, C. M. Fonseca, and A. Gaspar-Cunha, “Methodology to select solutions from the Pareto-optimal set: A comparative study,” in Proceedings of the 9th annual conference on Genetic and evolutionary computation, (ACM, New York, NY, 2007), 789–796.

J. Branke, K. Deb, K. Miettinen, and R. Slowinski, Multiobjective Optimization: Interactive and Evolutionary Approaches (Springer-Verlag, Berlin, 2008).

C. A. Coello Coello, “Handling preferences in evolutionary multiobjective optimization: a survey,” in Proceedings of the 2000 Congress on Evolutionary Computation (Institute of Electrical and Electronics Engineers, New York, 2000), 30–37.

SCHOTT N. America, Inc., “Optical glass catalogue- ZEMAX format, status as of 13th September 2011, http://www.us.schott.com/advanced_optics/english/tools_downloads/download/index.html?PHPSESSID=utt2cbk96nlk3gf7gjpb7ggt54#Optical%20Glass

P. Mouroulis, “Broadband achromatic telecentric lens,” Nasa Tech Briefs, NPO-44059, (2007).

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Figures (7)

Fig. 1
Fig. 1

The graph shows solutions for a generic min-min multi-objective problem, plotted in the objective-functions space F1 and F2. Dominated solutions are represented in blue, while red dots represent non-dominated solutions.

Fig. 2
Fig. 2

Format of the table where the data for each glass arrangement best aplanatic solution is stored.

Fig. 3
Fig. 3

Flowchart of the proposed method of glass combination selection.

Fig. 4
Fig. 4

Typical Pareto front for 2 objective min-min problem, showing the attributes used in the post-Pareto analysis.

Fig. 5
Fig. 5

Chromatic focal Shift for the aplanatic triplets designed with glass combination (a) N-BAF52, N-KZFS11 and N-BAK2, and combination (b) N-KZFS8, P-SF68 and N-SK2.

Fig. 6
Fig. 6

Layout (a) and MTFs (b)(c)(d)(e)(f) for each spectral band and field position for the design made with glass combination N-BAF52, N-KZFS11 and N-BAK2.

Fig. 7
Fig. 7

Layout (a) and MTFs (b)(c)(d)(e)(f) for each spectral band and field position for the design made with glass combination N-KZFS8 P-SF68 and N-SK2.

Tables (5)

Tables Icon

Table 1 Basic Requirements for the optical system used as example.

Tables Icon

Table 2 Output table from the glass selection method for 2 glasses sorted by F2.

Tables Icon

Table 3 Output table from the glass selection method for 3 glasses sorted by | g ¯ i | .

Tables Icon

Table 4 Prescription data for the system shown in Fig. 6.

Tables Icon

Table 5 Prescription data for the system shown in Fig. 7.

Equations (37)

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

N(λ)= N 0 + ν 1 ω(λ)+ ν 2 ω (λ) 2 ++ ν n ω (λ) n
ω= δλ 1+αδλ
D(λ)= i=1 n1 η i ω (λ) i
ϕ= 1 f
ϕ( λ )=[ N( λ )1 ]( C 1 C 2 )
ϕ( λ )=[ N( λ )1 ]K
ϕ( λ 0 )=[ N( λ 0 )1 ]K
ϕ( λ )=ϕ( λ 0 )[ 1+D( λ ) ]
Φ( λ 0 )= j=1 k ϕ j ( λ 0 )
Φ( λ )=Φ( λ 0 )+ j=1 k ϕ j ( λ 0 ) D j ( λ )
Φ( λ 1 )=Φ( λ2 ) Φ( λ 2 )=Φ( λ3 ) Φ( λ n1 )=Φ( λn )
Φ( λ 1 )=Φ( λ 0 )+ ϕ 1 ( λ 0 ) D 1 ( λ 1 )+ϕ ( λ 0 ) k D k ( λ 1 ) Φ( λ 2 )=Φ( λ 0 )+ ϕ 1 ( λ 0 ) D 1 ( λ 2 )+ϕ ( λ 0 ) k D k ( λ 2 ) Φ( λ n )=Φ( λ 0 )+ ϕ 1 ( λ 0 ) D 1 ( λ n )+ϕ ( λ 0 ) k D k ( λ n )
ϕ 1 ( λ 0 )( D 1 ( λ 1 ) D 1 ( λ 2 ) )+... ϕ k ( λ 0 )( D k ( λ 1 ) D k ( λ 2 ) )=0 ϕ 1 ( λ 0 )( D 1 ( λ 2 ) D 1 ( λ 3 ) )+... ϕ k ( λ 0 )( D k ( λ 2 ) D k ( λ 3 ) )=0 ϕ 1 ( λ 0 )( D 1 ( λ n1 ) D 1 ( λ n ) )+... ϕ k ( λ 0 )( D k ( λ n1 ) D k ( λ n ) )=0
D j ( λ 1 , λ 2 )= D j ( λ 1 ) D j ( λ 2 )
D j ( λ 1 , λ 2 )= i=1 n1 η ij [ ω i ( λ 1 ) ω i ( λ 2 ) ]
Δ Ω ¯ η ¯ Φ ¯ = 0 ¯
Δ Ω ¯ =[ ( ω 1 ω 2 ) ( ω 1 2 ω 2 2 ) ( ω 1 n1 ω 2 n1 ) ( ω 2 ω 3 ) ( ω 2 2 ω 3 2 ) ( ω 2 n1 ω 3 n1 ) ( ω n1 ω n ) ( ω n1 2 ω n 2 ) ( ω n1 n1 ω n n1 ) ]
η ¯ =[ η 11 η 12 η 1k η 21 η 22 η 2k η 31 η 32 η 3k η (n1)1 η (n1)2 η (n1)k ]
Φ ¯ =[ ϕ 1 ( λ 0 ) ϕ 2 ( λ 0 ) ϕ K ( λ 0 ) ]
0 ¯ =[ 0 0 0 0 ]
η ¯ Φ ¯ = 0 ¯
j=1 k ϕ j ( λ 0 )=1
S ¯ Φ ¯ =1
[ S ¯ Δ Ω ¯ η ¯ ] Φ ¯ = e ^
e ^ =[ 1 0 0 0 ]
Φ ¯ ^ = ( G ¯ t G ¯ ) 1 G ¯ t e ^
CCP ¯ =Δ Ω ¯ η ¯ Φ ¯ ^
[ f( λ 2 )f( λ 1 ) f( λ 3 )f( λ 2 ) f( λ n )f( λ n1 ) ] CCP ¯ F
F 1 = j=1 k | ϕ j ( λ 0 ) |
Ξ= j=1 k ξ j =0;
Χ= j=1 k χ j =0;
[ N 1 ( λ 0 )1 ]( 1 r 1 1 r 2 )=( ϕ 1 ( λ 0 ) F ) [ N k ( λ 0 )1 ]( 1 r (2k)1 1 r (2k) )=( ϕ k ( λ 0 ) F )
F 3 =( W ¯ 040CL + W ¯ 060 )
W ¯ 060 = 14 W 060 ( λ 0 ) 20 7
W ¯ 040CL = 14 W 040CL ( λ 1 λ n ) 6 5
| g ¯ i |= ob=1 m ( O ob,i O ¯ ob ) 2
ε=±2λ (f#) 2

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