Abstract

The authors use a genetic algorithm in the structural design of cemented doublets in accordance with a prespecified set of Gaussian characteristics and primary aberration targets. Actual glasses are taken as variables. Suitable modifications in the algorithm are sought with which to explore the solution space in search of the global as well as useful local optima. Some illustrative examples of structural design are given.

© 2001 Optical Society of America

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References

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  1. M. I. Khan, J. Macdonald, “Cemented doublets, a method for rapid design,” Opt. Acta 29, 807–822 (1982).
    [CrossRef]
  2. P. Turriere, Optique Industrielle (Delagrave, Paris, 1920).
  3. L. N. Hazra, “Structural design of multicomponent lens systems,” Appl. Opt. 23, 4440–4443 (1984).
    [CrossRef] [PubMed]
  4. L. N. Hazra, A. K. Samui, “Design of the individual components of a multicomponent lens system: use of a singlet,” Appl. Opt. 25, 3721–3730 (1986).
    [CrossRef] [PubMed]
  5. S. Banerjee, L. N. Hazra, “Structural design of doublet lenses with prespecified aberration targets,” Opt. Eng. 36, 3111–3118 (1997).
    [CrossRef]
  6. S. Banerjee, L. N. Hazra, “Genetic algorithm for lens optimization at structural design phase,” in Optics and Optoelectronics, Theory, Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K. Singh, eds. (Narossa, New Delhi, 1999), Vol. 1, pp. 499–504.
  7. J. H. Holland, Adaptation in Natural and Artificial Systems (MIT Press, Ann Arbor, Mich., 1975).
  8. I. Ono, S. Kobeyashi, “Global and multi-objective optimization for lens design by real-coded genetic algorithms,” in International Optical Design Conference, L. R. Gardner, K. P. Thomson, eds., Proc. SPIE3482, 110–121 (1998).
  9. X. Chen, K. Yamamoto, “An experiment in genetic optimization in lens design,” J. Mod. Opt. 44, 1693–1702 (1997).
    [CrossRef]
  10. S. Banerjee, “Application of optimization algorithms in structural design of lens systems,” Ph.D. dissertation (University of Calcutta, Calcutta, India, 2000).
  11. D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, New York, 1989).
  12. D. Beasley, D. R. Bull, R. R. Martin, “An overview of genetic algorithms. 1. Research topics,” Univ. Comput. 15, 58–69 (1993).
  13. D. Beasley, D. R. Bull, R. R. Martin, “An overview of genetic algorithms. 2. Research topics,” Univ. Comput. 15, 170–181 (1993).
  14. H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, London, 1950).
  15. W. T. Welford, Aberrations of Optical Systems (Academic, London, 1986).
  16. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).
  17. D. E. Goldberg, K. Deb, “A comparison of selection schemes used in genetic algorithms,” in Foundations of Genetic Algorithms, G. J. E. Rawlins, ed. (Morgan Kaufmann, Los Angeles, Calif., 1991), pp. 69–93.
  18. K. Krishnakumar, “Micro-genetic algorithms for stationary and non-stationary function optimization,” in Intelligent Control and Adaptive Systems, G. Rodriguez, S. Venkataraman, eds., Proc. SPIE1196, 289–296 (1989).
    [CrossRef]
  19. D. E. Goldberg, J. Richardson, “Genetic algorithms with sharing for multimodal function optimization,” in Proceedings of the Second International Conference on Genetic Algorithms, J. J. Grefenstette, ed. (Erlbaum, Hillsdale, N.J., 1987), pp. 41–49.
  20. G. Syswerda, “Uniform crossover in genetic algorithms,” in Proceedings of the Third International Conference on Genetic Algorithms, J. D. Schaffer, ed. (Morgan Kaufmann, New York, 1989), pp. 2–9.
  21. Optical Glass Catalog (Schott Glas, Mainz, Germany, 1999).
  22. D. E. Goldberg, K. Deb, J. H. Clark, “Genetic algorithms: noise, and the sizing of populations,” Complex Syst. 6, 333–362 (1992).
  23. D. Carroll, “Chemical laser modeling with genetic algorithms,” AIAA J. 34, 338–346 (1996).
    [CrossRef]

1997

S. Banerjee, L. N. Hazra, “Structural design of doublet lenses with prespecified aberration targets,” Opt. Eng. 36, 3111–3118 (1997).
[CrossRef]

X. Chen, K. Yamamoto, “An experiment in genetic optimization in lens design,” J. Mod. Opt. 44, 1693–1702 (1997).
[CrossRef]

1996

D. Carroll, “Chemical laser modeling with genetic algorithms,” AIAA J. 34, 338–346 (1996).
[CrossRef]

1993

D. Beasley, D. R. Bull, R. R. Martin, “An overview of genetic algorithms. 1. Research topics,” Univ. Comput. 15, 58–69 (1993).

D. Beasley, D. R. Bull, R. R. Martin, “An overview of genetic algorithms. 2. Research topics,” Univ. Comput. 15, 170–181 (1993).

1992

D. E. Goldberg, K. Deb, J. H. Clark, “Genetic algorithms: noise, and the sizing of populations,” Complex Syst. 6, 333–362 (1992).

1986

1984

1982

M. I. Khan, J. Macdonald, “Cemented doublets, a method for rapid design,” Opt. Acta 29, 807–822 (1982).
[CrossRef]

Banerjee, S.

S. Banerjee, L. N. Hazra, “Structural design of doublet lenses with prespecified aberration targets,” Opt. Eng. 36, 3111–3118 (1997).
[CrossRef]

S. Banerjee, L. N. Hazra, “Genetic algorithm for lens optimization at structural design phase,” in Optics and Optoelectronics, Theory, Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K. Singh, eds. (Narossa, New Delhi, 1999), Vol. 1, pp. 499–504.

S. Banerjee, “Application of optimization algorithms in structural design of lens systems,” Ph.D. dissertation (University of Calcutta, Calcutta, India, 2000).

Beasley, D.

D. Beasley, D. R. Bull, R. R. Martin, “An overview of genetic algorithms. 1. Research topics,” Univ. Comput. 15, 58–69 (1993).

D. Beasley, D. R. Bull, R. R. Martin, “An overview of genetic algorithms. 2. Research topics,” Univ. Comput. 15, 170–181 (1993).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Bull, D. R.

D. Beasley, D. R. Bull, R. R. Martin, “An overview of genetic algorithms. 2. Research topics,” Univ. Comput. 15, 170–181 (1993).

D. Beasley, D. R. Bull, R. R. Martin, “An overview of genetic algorithms. 1. Research topics,” Univ. Comput. 15, 58–69 (1993).

Carroll, D.

D. Carroll, “Chemical laser modeling with genetic algorithms,” AIAA J. 34, 338–346 (1996).
[CrossRef]

Chen, X.

X. Chen, K. Yamamoto, “An experiment in genetic optimization in lens design,” J. Mod. Opt. 44, 1693–1702 (1997).
[CrossRef]

Clark, J. H.

D. E. Goldberg, K. Deb, J. H. Clark, “Genetic algorithms: noise, and the sizing of populations,” Complex Syst. 6, 333–362 (1992).

Deb, K.

D. E. Goldberg, K. Deb, J. H. Clark, “Genetic algorithms: noise, and the sizing of populations,” Complex Syst. 6, 333–362 (1992).

D. E. Goldberg, K. Deb, “A comparison of selection schemes used in genetic algorithms,” in Foundations of Genetic Algorithms, G. J. E. Rawlins, ed. (Morgan Kaufmann, Los Angeles, Calif., 1991), pp. 69–93.

Goldberg, D. E.

D. E. Goldberg, K. Deb, J. H. Clark, “Genetic algorithms: noise, and the sizing of populations,” Complex Syst. 6, 333–362 (1992).

D. E. Goldberg, J. Richardson, “Genetic algorithms with sharing for multimodal function optimization,” in Proceedings of the Second International Conference on Genetic Algorithms, J. J. Grefenstette, ed. (Erlbaum, Hillsdale, N.J., 1987), pp. 41–49.

D. E. Goldberg, K. Deb, “A comparison of selection schemes used in genetic algorithms,” in Foundations of Genetic Algorithms, G. J. E. Rawlins, ed. (Morgan Kaufmann, Los Angeles, Calif., 1991), pp. 69–93.

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, New York, 1989).

Hazra, L. N.

S. Banerjee, L. N. Hazra, “Structural design of doublet lenses with prespecified aberration targets,” Opt. Eng. 36, 3111–3118 (1997).
[CrossRef]

L. N. Hazra, A. K. Samui, “Design of the individual components of a multicomponent lens system: use of a singlet,” Appl. Opt. 25, 3721–3730 (1986).
[CrossRef] [PubMed]

L. N. Hazra, “Structural design of multicomponent lens systems,” Appl. Opt. 23, 4440–4443 (1984).
[CrossRef] [PubMed]

S. Banerjee, L. N. Hazra, “Genetic algorithm for lens optimization at structural design phase,” in Optics and Optoelectronics, Theory, Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K. Singh, eds. (Narossa, New Delhi, 1999), Vol. 1, pp. 499–504.

Holland, J. H.

J. H. Holland, Adaptation in Natural and Artificial Systems (MIT Press, Ann Arbor, Mich., 1975).

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, London, 1950).

Khan, M. I.

M. I. Khan, J. Macdonald, “Cemented doublets, a method for rapid design,” Opt. Acta 29, 807–822 (1982).
[CrossRef]

Kobeyashi, S.

I. Ono, S. Kobeyashi, “Global and multi-objective optimization for lens design by real-coded genetic algorithms,” in International Optical Design Conference, L. R. Gardner, K. P. Thomson, eds., Proc. SPIE3482, 110–121 (1998).

Krishnakumar, K.

K. Krishnakumar, “Micro-genetic algorithms for stationary and non-stationary function optimization,” in Intelligent Control and Adaptive Systems, G. Rodriguez, S. Venkataraman, eds., Proc. SPIE1196, 289–296 (1989).
[CrossRef]

Macdonald, J.

M. I. Khan, J. Macdonald, “Cemented doublets, a method for rapid design,” Opt. Acta 29, 807–822 (1982).
[CrossRef]

Martin, R. R.

D. Beasley, D. R. Bull, R. R. Martin, “An overview of genetic algorithms. 2. Research topics,” Univ. Comput. 15, 170–181 (1993).

D. Beasley, D. R. Bull, R. R. Martin, “An overview of genetic algorithms. 1. Research topics,” Univ. Comput. 15, 58–69 (1993).

Ono, I.

I. Ono, S. Kobeyashi, “Global and multi-objective optimization for lens design by real-coded genetic algorithms,” in International Optical Design Conference, L. R. Gardner, K. P. Thomson, eds., Proc. SPIE3482, 110–121 (1998).

Richardson, J.

D. E. Goldberg, J. Richardson, “Genetic algorithms with sharing for multimodal function optimization,” in Proceedings of the Second International Conference on Genetic Algorithms, J. J. Grefenstette, ed. (Erlbaum, Hillsdale, N.J., 1987), pp. 41–49.

Samui, A. K.

Syswerda, G.

G. Syswerda, “Uniform crossover in genetic algorithms,” in Proceedings of the Third International Conference on Genetic Algorithms, J. D. Schaffer, ed. (Morgan Kaufmann, New York, 1989), pp. 2–9.

Turriere, P.

P. Turriere, Optique Industrielle (Delagrave, Paris, 1920).

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Academic, London, 1986).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Yamamoto, K.

X. Chen, K. Yamamoto, “An experiment in genetic optimization in lens design,” J. Mod. Opt. 44, 1693–1702 (1997).
[CrossRef]

AIAA J.

D. Carroll, “Chemical laser modeling with genetic algorithms,” AIAA J. 34, 338–346 (1996).
[CrossRef]

Appl. Opt.

Complex Syst.

D. E. Goldberg, K. Deb, J. H. Clark, “Genetic algorithms: noise, and the sizing of populations,” Complex Syst. 6, 333–362 (1992).

J. Mod. Opt.

X. Chen, K. Yamamoto, “An experiment in genetic optimization in lens design,” J. Mod. Opt. 44, 1693–1702 (1997).
[CrossRef]

Opt. Acta

M. I. Khan, J. Macdonald, “Cemented doublets, a method for rapid design,” Opt. Acta 29, 807–822 (1982).
[CrossRef]

Opt. Eng.

S. Banerjee, L. N. Hazra, “Structural design of doublet lenses with prespecified aberration targets,” Opt. Eng. 36, 3111–3118 (1997).
[CrossRef]

Univ. Comput.

D. Beasley, D. R. Bull, R. R. Martin, “An overview of genetic algorithms. 1. Research topics,” Univ. Comput. 15, 58–69 (1993).

D. Beasley, D. R. Bull, R. R. Martin, “An overview of genetic algorithms. 2. Research topics,” Univ. Comput. 15, 170–181 (1993).

Other

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, London, 1950).

W. T. Welford, Aberrations of Optical Systems (Academic, London, 1986).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

D. E. Goldberg, K. Deb, “A comparison of selection schemes used in genetic algorithms,” in Foundations of Genetic Algorithms, G. J. E. Rawlins, ed. (Morgan Kaufmann, Los Angeles, Calif., 1991), pp. 69–93.

K. Krishnakumar, “Micro-genetic algorithms for stationary and non-stationary function optimization,” in Intelligent Control and Adaptive Systems, G. Rodriguez, S. Venkataraman, eds., Proc. SPIE1196, 289–296 (1989).
[CrossRef]

D. E. Goldberg, J. Richardson, “Genetic algorithms with sharing for multimodal function optimization,” in Proceedings of the Second International Conference on Genetic Algorithms, J. J. Grefenstette, ed. (Erlbaum, Hillsdale, N.J., 1987), pp. 41–49.

G. Syswerda, “Uniform crossover in genetic algorithms,” in Proceedings of the Third International Conference on Genetic Algorithms, J. D. Schaffer, ed. (Morgan Kaufmann, New York, 1989), pp. 2–9.

Optical Glass Catalog (Schott Glas, Mainz, Germany, 1999).

S. Banerjee, L. N. Hazra, “Genetic algorithm for lens optimization at structural design phase,” in Optics and Optoelectronics, Theory, Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K. Singh, eds. (Narossa, New Delhi, 1999), Vol. 1, pp. 499–504.

J. H. Holland, Adaptation in Natural and Artificial Systems (MIT Press, Ann Arbor, Mich., 1975).

I. Ono, S. Kobeyashi, “Global and multi-objective optimization for lens design by real-coded genetic algorithms,” in International Optical Design Conference, L. R. Gardner, K. P. Thomson, eds., Proc. SPIE3482, 110–121 (1998).

P. Turriere, Optique Industrielle (Delagrave, Paris, 1920).

S. Banerjee, “Application of optimization algorithms in structural design of lens systems,” Ph.D. dissertation (University of Calcutta, Calcutta, India, 2000).

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, New York, 1989).

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Figures (3)

Fig. 1
Fig. 1

Variation of selection pressure (Φmax/Φ) over generations (for proportionate selection operator): Five curves correspond to five runs of the algorithm starting from five different starting populations for an f/4, 400-mm cemented doublet with zero and nonzero [(S 1/8) t = -10λ, (S IIc /2) t = -10λ, (C L /2) t = -10λ)] aberration targets.

Fig. 2
Fig. 2

(n - V) diagram of 64 preferred optical glasses (selected from the Schott Glass catalog21).

Fig. 3
Fig. 3

Evolution of maximum fitness (of a 500-member population) over generations. Curves 1, 2, and 3 correspond to results presented in Tables 1, 2 and 3, respectively.

Tables (3)

Tables Icon

Table 1 Glass Combinations for f/4, 400-mm Focal-Length Aplanatic, Achromatic Cemented Doubletsa

Tables Icon

Table 2 Solutions for f/4, 400-mm Focal-Length Cemented Doublet with Nonzero Aberration Targetsa

Tables Icon

Table 3 Additional Solutions for the Cemented Doublet of Table 1a

Equations (21)

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

X=c1+c2c1-c2,
X2=k¯1n2-1k2n1-1X1-1-1,
k¯2=1-k¯1.
Φ=11+Ψ.
Ψ=i=13 wi2fi2,
fi=Γi1+Γi2-Γit
Γ1j=SI/8j,Γ2j=SIIc/2j,Γ3j=CL/2j,
SIj=14 h4K3k¯j3nj+2njnj-12Xj-2nj2-1nj+2 Yj2+nj2nj-12-njnj+2 Yj2,SIIcj=12 h2K2Hk¯j2nj+1njnj-1 Xj-2nj+1nj Yj,CLj=Kk¯jδnjnj-1,
δn=nF-nC,
V=n-1δn.
Y1=Y+k¯1-1k¯1,Y2=Y+k¯11-k¯1.
Y=u+uu-u,
xi=ximin+ximax-ximin2l-1DVSi,
li=log2ximax-ximinεi.
L=i-1N li.
Crossover mask100101Parent 1101100Parent 2010011Child 1110110
00000  00010.
g=ipopsizedipopsizeL,
shdij=1-dijσshar  dij<σshar=0  otherwise.
Φi=Φimi.
mi=j=1popsizeshdij.

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