Abstract

With the increasing power of computers, new methods in synthesis of optical multilayer systems have appeared. Among these, the simulated-annealing algorithm has proved its efficiency in several fields of physics. We propose to show its performances in the field of optical multilayer systems through different filter designs.

© 1996 Optical Society of America

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  1. R. Morf, R. E. Kunz, “Dielectric filter optimization by simulated thermal annealing,” in Thin Film Technologies III, K. H. Guenther, H. K. Pulker, eds., Proc. SPIE1019, 211–217 (1989).
  2. J. A. Dobrowolski, R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29, 2876–2893 (1990).
    [CrossRef] [PubMed]
  3. S. Martin, J. Rivory, M. Schoenauer, “Synthesis of optical multilayer systems using genetic algorithms,” Appl. Opt. 34, 2247–2254 (1995).
    [CrossRef] [PubMed]
  4. P. J. M. Van Laarhoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications (Kluwer, Dordrecht, The Netherlands, 1989), pp. 17–71.
  5. N. Metropolis, A. W. Rosenbluth, A. H. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
    [CrossRef]
  6. L. Herault, R. Horaud, “Figure-ground discrimination: a combinatorial optimization approach,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 899–916 (1993).
    [CrossRef]
  7. P. Chaton, P. Pinston, J. P. Gailliard, “Synthesis of optical coatings using a simulated annealing algorithm,” Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 73–80 (1994).
  8. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965).
  9. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef] [PubMed]
  10. R. Morf, R. E. Kunz, “Dielectric filter optimization by simulated thermal annealing: a simulated zone-melting approach,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. SPIE1270, 11–17 (1990).
  11. M. P. Vecchi, S. Kirkpatrick, “Global wiring by simulated annealing,” IEEE Trans. Comput. Aided Des. CAD-2, 215–222 (1983).
    [CrossRef]
  12. F. Abeles, “Recherches sur la propagation des ondes électromagénetiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. 5, 596–640, 706–784 (1950).
  13. 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]

1995 (1)

1993 (1)

L. Herault, R. Horaud, “Figure-ground discrimination: a combinatorial optimization approach,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 899–916 (1993).
[CrossRef]

1990 (1)

1988 (1)

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]

1983 (2)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

M. P. Vecchi, S. Kirkpatrick, “Global wiring by simulated annealing,” IEEE Trans. Comput. Aided Des. CAD-2, 215–222 (1983).
[CrossRef]

1953 (1)

N. Metropolis, A. W. Rosenbluth, A. H. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

1950 (1)

F. Abeles, “Recherches sur la propagation des ondes électromagénetiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. 5, 596–640, 706–784 (1950).

Aarts, E. H. L.

P. J. M. Van Laarhoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications (Kluwer, Dordrecht, The Netherlands, 1989), pp. 17–71.

Abeles, F.

F. Abeles, “Recherches sur la propagation des ondes électromagénetiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. 5, 596–640, 706–784 (1950).

Aguilera, J.

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]

Aguilera, J. A.

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]

Baumeister, P.

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]

Bloom, A.

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]

Chaton, P.

P. Chaton, P. Pinston, J. P. Gailliard, “Synthesis of optical coatings using a simulated annealing algorithm,” Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 73–80 (1994).

Coursen, D.

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]

Dobrowolski, J. A.

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

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]

Gailliard, J. P.

P. Chaton, P. Pinston, J. P. Gailliard, “Synthesis of optical coatings using a simulated annealing algorithm,” Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 73–80 (1994).

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Goldstein, F. T.

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]

Gustafson, D. E.

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]

Herault, L.

L. Herault, R. Horaud, “Figure-ground discrimination: a combinatorial optimization approach,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 899–916 (1993).
[CrossRef]

Horaud, R.

L. Herault, R. Horaud, “Figure-ground discrimination: a combinatorial optimization approach,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 899–916 (1993).
[CrossRef]

Kemp, R. A.

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

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]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

M. P. Vecchi, S. Kirkpatrick, “Global wiring by simulated annealing,” IEEE Trans. Comput. Aided Des. CAD-2, 215–222 (1983).
[CrossRef]

Kunz, R. E.

R. Morf, R. E. Kunz, “Dielectric filter optimization by simulated thermal annealing,” in Thin Film Technologies III, K. H. Guenther, H. K. Pulker, eds., Proc. SPIE1019, 211–217 (1989).

R. Morf, R. E. Kunz, “Dielectric filter optimization by simulated thermal annealing: a simulated zone-melting approach,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. SPIE1270, 11–17 (1990).

Martin, S.

Metropolis, N.

N. Metropolis, A. W. Rosenbluth, A. H. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Morf, R.

R. Morf, R. E. Kunz, “Dielectric filter optimization by simulated thermal annealing,” in Thin Film Technologies III, K. H. Guenther, H. K. Pulker, eds., Proc. SPIE1019, 211–217 (1989).

R. Morf, R. E. Kunz, “Dielectric filter optimization by simulated thermal annealing: a simulated zone-melting approach,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. SPIE1270, 11–17 (1990).

Pinston, P.

P. Chaton, P. Pinston, J. P. Gailliard, “Synthesis of optical coatings using a simulated annealing algorithm,” Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 73–80 (1994).

Reif, F.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965).

Rivory, J.

Rosenbluth, A. W.

N. Metropolis, A. W. Rosenbluth, A. H. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Schoenauer, M.

Teller, A. H.

N. Metropolis, A. W. Rosenbluth, A. H. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Van Laarhoven, P. J. M.

P. J. M. Van Laarhoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications (Kluwer, Dordrecht, The Netherlands, 1989), pp. 17–71.

Vecchi, M. P.

M. P. Vecchi, S. Kirkpatrick, “Global wiring by simulated annealing,” IEEE Trans. Comput. Aided Des. CAD-2, 215–222 (1983).
[CrossRef]

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Ann. Phys. (1)

F. Abeles, “Recherches sur la propagation des ondes électromagénetiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. 5, 596–640, 706–784 (1950).

Appl. Opt. (1)

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]

Appl. Opt. (2)

IEEE Trans. Comput. Aided Des. (1)

M. P. Vecchi, S. Kirkpatrick, “Global wiring by simulated annealing,” IEEE Trans. Comput. Aided Des. CAD-2, 215–222 (1983).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

L. Herault, R. Horaud, “Figure-ground discrimination: a combinatorial optimization approach,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 899–916 (1993).
[CrossRef]

J. Chem. Phys. (1)

N. Metropolis, A. W. Rosenbluth, A. H. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Science (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Other (5)

R. Morf, R. E. Kunz, “Dielectric filter optimization by simulated thermal annealing: a simulated zone-melting approach,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. SPIE1270, 11–17 (1990).

R. Morf, R. E. Kunz, “Dielectric filter optimization by simulated thermal annealing,” in Thin Film Technologies III, K. H. Guenther, H. K. Pulker, eds., Proc. SPIE1019, 211–217 (1989).

P. J. M. Van Laarhoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications (Kluwer, Dordrecht, The Netherlands, 1989), pp. 17–71.

P. Chaton, P. Pinston, J. P. Gailliard, “Synthesis of optical coatings using a simulated annealing algorithm,” Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 73–80 (1994).

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965).

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Figures (10)

Fig. 1
Fig. 1

Landscape of a merit function used for the synthesis of the bandpass filter with high rejection bands discussed in Subsection 4.B. We have reported the variation of merit function M3 (see Subsection 4.B) versus the thicknesses of two layers of the bandpass filter presented in Subsection 4.B, Figs. 9 and 10. This plot outlines the great number of local minima.

Fig. 2
Fig. 2

Variations of free energy [see Eqs. (2.2)] versus normalized temperature in the case of the antireflection coating B 2 (see Table 1).

Fig. 3
Fig. 3

Variations of entropy [see Eqs. (2.2)] versus normalized temperature in the case of the antireflection coating B 2 (see Table 1).

Fig. 4
Fig. 4

Variations of specific heat [see Eqs. (2.2)] versus normalized temperature in the case of the antireflection coating B 2 (see Table 1).

Fig. 5
Fig. 5

Results of the antireflection problem over the region 7.7 ≤ λ ≤ 12.3 μm found by Martin et al. 3 (see Table 2). Systems A 1, B 1, and C 1 correspond to the solutions found by the genetic algorithm with 31-, 27-, and 32-μm, respectively, total optical thickness and 17, 18, and 20 layers, respectively. Optimizations were made with a 0.1-nm step on the thicknesses.

Fig. 6
Fig. 6

Antireflection problem over the region 7.7 ≤ λ ≤ 12.3 μm (see Table 1). Systems A 2, B 2, and C 2 correspond to the solutions found by the simulated algorithm with a 30-, 31-, and 29-μm, respectively, total optical thickness and 16, 17, and 20 layers, respectively. Optimizations were made with a 1-nm step on the thicknesses, and the maximum thickness for each layer was 1.4 μm.

Fig. 7
Fig. 7

Antireflection problem over the region 7.7 ≤ λ ≤ 12.3 μm (see Table 3). Systems A 3 and B 3 correspond to the solutions found by the simulated algorithm with an 18- and 24-μm, respectively, total optical thickness and 20 and 24 layers, respectively. Optimizations were made with a 1-nm step on the thicknesses, and the maximum thickness for each layer was 0.6 μm to keep in mind a possible realization of the filters.

Fig. 8
Fig. 8

Bandpass filter with a high rejection band problem. This system was synthesized with merit function M2. It was made up of 50 layers of SiO2 and TiO2, and the total optical thickness was 10 μm.

Fig. 9
Fig. 9

Bandpass filter with a high rejection band problem. This system was synthesized with merit function M3. It was made up of 50 layers of SiO2 and TiO2, and the total optical thickness was approximately 9 μm. This plot shows the importance of the merit function.

Fig. 10
Fig. 10

Same plot as in Fig. 9 in a semilogarithm scale. The aim is to show the quality of the rejection bands of the filter.

Tables (3)

Tables Icon

Table 1 Antireflection Coatings a

Tables Icon

Table 2 Antireflection Coatings

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Table 3 Antireflection Coatings a

Equations (8)

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

Pr [ E ( σ ) = E n ] = q n ( T ) = exp ( E n T ) Z ( T ) ,
Z ( T ) = n exp ( E n T )
F ( T ) = E T S = T ln Z ( T ) ; S ( T ) = n q n ( T ) ln [ q n ( T ) ] ; C υ ( T ) = d E ( T ) d T = E ( T ) 2 E ( T ) 2 T 2
E ( T ) = n q n E n .
Pr = 1 if Δ E < 0 , Pr = exp ( Δ E T ) otherwise .
if Δ E < 0 , accept the new filter , if Δ E 0 , draw a random number δ between 0 and 1 ; if exp ( Δ E / T ) > δ , accept the filter , otherwise reject it .
M 1 ( X ) = { 1 n j = 1 n [ R ( λ j ) R target ( λ j ) δ R j ] 2 } 1 / 2
M 2 ( X ) = k = s , p j = 1 n W j { [ R k ( λ j ) R k target ( λ j ) ] 2 + [ T k ( λ j ) T k target ( λ j ) ] 2 } ,

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