Abstract

Stochastic algorithms are a promising method for the synthesis of optical multilayer systems. A method based on the use of genetic algorithms is described and applied to the design of three very different optical filters. Solutions found by genetic algorithms are refined, and results are compared with those of previous publications.

© 1995 Optical Society of America

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References

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  1. J. A. Dobrowolski, R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29, 2876–2893 (1990).
    [CrossRef] [PubMed]
  2. A. V. Tikhonravov, J. A. Dobrowolski, “Quasi-optimal synthesis for antireflection coatings: a new method,” Appl. Opt. 32, 4265–42751993).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. J. A. Dobrowolski, D. Lowe, “Optical thin film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039–3050 (1978).
    [CrossRef] [PubMed]
  5. P. G. Verly, J. A. Dobrowolski, W. J. Wild, R. L. Burton, “Synthesis of high rejection filters with the Fourier transform method,” Appl. Opt. 28, 2864–2875 (1989).
    [CrossRef] [PubMed]
  6. S. Martin, J. Rivory, M. Schoenauer, “Simulated Darwinian evolution of homogeneous multilayer systems: a new method for optical coatings design,” Opt. Commun. 110, 503–506 (1994).
    [CrossRef]
  7. D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley, Reading, Mass., 1989).
  8. J. Holland, Adaptation in Natural and Artificial Systems (U. Michigan Press, Ann Arbor, Mich., 1975).
  9. R. Cerf, “Une théorie assymptotique des algorithmes génétiques,” Ph.D. dissertation (Université de Montpellier, Montpellier, France, 1994).
    [PubMed]
  10. H. A. Macleod, Thin-Film Optical Filters (Hilger, Bristol, 1986), Chap. 2, pp. 32–40.
  11. J. A. Aguilera, J. Aguilera, P. Baumeister, A. Bloom, D. Coursen, J. A. Dobrowolski, F. T. Goldstein, D. E. Gustafson, R. A. Kemp, “Antireflection coatings for germanium IR optics: a comparison of numerical design methods,” Appl. Opt. 27, 2832–2840 (1988).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  14. M. Schoenauer, Z. Wu, “Conception optimale discrète de structures par algorithmes génétiques,” presented at the National Symposium on Structure Calculus, Giens, France, May 1993.
  15. T. Eisenhammer, M. Lazarov, M. Leutbecher, U. Schöffel, R. Sizmann, “Optimization of interference filters with genetic algorithms applied to silver-based heat mirrors,” Appl. Opt. 32, 6310–6315 (1993).
    [CrossRef] [PubMed]
  16. D. Hillis, “Co-evolving parasites improved simulated evolution as an optimisation procedure,” Physica (D) 42, 228–234 (1990).

1994

S. Martin, J. Rivory, M. Schoenauer, “Simulated Darwinian evolution of homogeneous multilayer systems: a new method for optical coatings design,” Opt. Commun. 110, 503–506 (1994).
[CrossRef]

1993

1990

J. A. Dobrowolski, R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29, 2876–2893 (1990).
[CrossRef] [PubMed]

D. Hillis, “Co-evolving parasites improved simulated evolution as an optimisation procedure,” Physica (D) 42, 228–234 (1990).

1989

1988

1985

1978

Aguilera, J.

Aguilera, J. A.

Baumeister, P.

Bloom, A.

Bovard, B. G.

Burton, R. L.

Cerf, R.

R. Cerf, “Une théorie assymptotique des algorithmes génétiques,” Ph.D. dissertation (Université de Montpellier, Montpellier, France, 1994).
[PubMed]

Coursen, D.

Dobrowolski, J. A.

Druessel, J.

Eisenhammer, T.

Goldberg, D. E.

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

Goldstein, F. T.

Grantham, J.

Gustafson, D. E.

Haaland, P.

Hillis, D.

D. Hillis, “Co-evolving parasites improved simulated evolution as an optimisation procedure,” Physica (D) 42, 228–234 (1990).

Holland, J.

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

Kemp, R. A.

Lazarov, M.

Leutbecher, M.

Lowe, D.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Hilger, Bristol, 1986), Chap. 2, pp. 32–40.

Martin, S.

S. Martin, J. Rivory, M. Schoenauer, “Simulated Darwinian evolution of homogeneous multilayer systems: a new method for optical coatings design,” Opt. Commun. 110, 503–506 (1994).
[CrossRef]

Rivory, J.

S. Martin, J. Rivory, M. Schoenauer, “Simulated Darwinian evolution of homogeneous multilayer systems: a new method for optical coatings design,” Opt. Commun. 110, 503–506 (1994).
[CrossRef]

Schoenauer, M.

S. Martin, J. Rivory, M. Schoenauer, “Simulated Darwinian evolution of homogeneous multilayer systems: a new method for optical coatings design,” Opt. Commun. 110, 503–506 (1994).
[CrossRef]

M. Schoenauer, Z. Wu, “Conception optimale discrète de structures par algorithmes génétiques,” presented at the National Symposium on Structure Calculus, Giens, France, May 1993.

Schöffel, U.

Sizmann, R.

Southwell, W. H.

Tikhonravov, A. V.

Verly, P. G.

Wild, W. J.

Wu, Z.

M. Schoenauer, Z. Wu, “Conception optimale discrète de structures par algorithmes génétiques,” presented at the National Symposium on Structure Calculus, Giens, France, May 1993.

Appl. Opt.

Opt. Commun.

S. Martin, J. Rivory, M. Schoenauer, “Simulated Darwinian evolution of homogeneous multilayer systems: a new method for optical coatings design,” Opt. Commun. 110, 503–506 (1994).
[CrossRef]

Opt. Lett.

Physica (D)

D. Hillis, “Co-evolving parasites improved simulated evolution as an optimisation procedure,” Physica (D) 42, 228–234 (1990).

Other

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

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

R. Cerf, “Une théorie assymptotique des algorithmes génétiques,” Ph.D. dissertation (Université de Montpellier, Montpellier, France, 1994).
[PubMed]

H. A. Macleod, Thin-Film Optical Filters (Hilger, Bristol, 1986), Chap. 2, pp. 32–40.

M. Schoenauer, Z. Wu, “Conception optimale discrète de structures par algorithmes génétiques,” presented at the National Symposium on Structure Calculus, Giens, France, May 1993.

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Figures (7)

Fig. 1
Fig. 1

Antireflection problem over the region 7.7 ≤ λ ≤ 12.3 μm for a substrate of index 4, using coating materials with indices between 2.2 and 4.2. Systems A1 (A2) correspond to the solutions found by the GA with a 32-μm (25-μm) total optical thickness and 20 (25) layers. System A3 is a solution synthesized by the GA, with 31 layers, using only two coating materials with indices of 2.2 and 4.2.

Fig. 2
Fig. 2

Antireflection problem over the region 7.7 ≤ λ ≤ 12.3 μm for a substrate of index 4, using coating materials with indices between 2.2 and 4.2. Systems B1 (B2) are the two-material equivalent systems of the solutions found by the GA with a 32-μm (25-μm) total optical thickness and 20 (25) layers.

Fig. 3
Fig. 3

Antireflection problem over the region 7.7 ≤ λ ≤ 12.3 μm for a substrate of index 4, using coating materials with indices of 2.2 and 4.2. Systems C1 (C2) are the refined solutions of the systems found by GA with a 32-μm (25-μm) total optical thickness and 20 (25) layers. System C3 is a solution found by GA, using only two coating materials with indices of 2.2 and 4.2 after refinement by the gradient method.

Fig. 4
Fig. 4

Beam-splitter problem over the region 0.4 ≤ λ ≤ 1.0 μm for a substrate of index 1.52, using coating materials with indices between 1.35 and 2.35. System D1 is a 20-layer solution with a 2-μm total optical thickness synthesized by the GA. System D2 corresponds to system D1 after refinement with the gradient method.

Fig. 5
Fig. 5

Rejection filter of 90% over the region 0.5 ≤ λ ≤ 0.7 μm, with a 40-nm bandwidth centered at 0.6 μm, using coating materials with indices between 1.35 and 2.20 (substrate is glass, n = 1.52). System E1 is the solution found by GA (20-μm total optical thickness, 40 layers). System E2 is obtained after refinement of system E1 by the gradient method.

Fig. 6
Fig. 6

Rejection filter of 90% over the region 0.5 ≤ λ ≤ 0.7 μm, with a 40-nm bandwidth centered at 0.6 μm, using coating materials with indices between 1.35 and 2.20 (substrate and incident medium are identical, n = 1.52). The refractive index profile of system F was synthesized by the inverse Fourier transform method, using optimal phase modulation.

Fig. 7
Fig. 7

Rejection filter of 90% over the region 0.5 ≤ λ ≤ 0.7 μm, with a 40-nm bandwidth centered at 0.6 μm, using coating materials with indices between 1.35 and 2.20. System E2 is the solution found by GA after refinement. System F is synthesized by the inverse Fourier transform method and has the refractive index profile shown in Fig. 6. The reflectance of system F is calculated with glass as the incident medium.

Tables (5)

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Table 1 Construction Parameters of Solutions Found by GA for Antireflection Coatings

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Table 2 Two-Material Equivalents of the Systems Found by GA for Antireflection Coatings Presented in Table 1

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Table 3 Construction Parameters of the Systems Found by GA (Table 1) Transformed into Two-Material Systems (Table 2), after Refinement by the Gradient Method, for Antireflection Coatings

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Table 4 Solution Found by GA before (D1) and after (D2) Refinement for Beam Splitters

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Table 5 Construction Parameters of the System Synthesized by GA before (E1) and after (E2) Refinement for Rejection Filters

Equations (2)

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n i 2 = α ( n i , 1 ) 2 + ( 1 - α ) ( n i , 2 ) 2 , n i t i = α ( n i , 1 t i , 1 ) + ( 1 - α ) ( n i , 2 t i , 2 ) .
F ( X ) = ( 1 p j = 1 p { [ R ( λ j ) - R opt ( λ j ) ] / δ R j } 2 ) - 1 ,

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