Abstract

The optical performance of interference filters depends on systematic and statistical variations of the thicknesses and indices of refraction of the layers that occur during production and use. Assuming that their distributions are known, the expected performance can be optimized as a function of the nominal layer thicknesses with the help of strategies that mimic biological evolution. This results in filter designs that are easier to manufacture and more robust to use. The method is illustrated for color shifts that are rather sensitive to layer thickness variations. Its scope is entirely general, and it could be applied to other tolerancing problems that arise in optical design.

© 1996 Optical Society of America

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References

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  1. W. P. Thoeni, “Deposition of optical coatings process control and automation,” Thin Solid Films 88, 385–397 (1982).
    [CrossRef]
  2. J. D. Rancourt, Optical Thin Films User’s Handbook (McGraw-Hill, New York, 1987), pp. 33–38, 71–75.
  3. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1988), Chap. 12, pp. 235–246.
  4. Y.-F. Zheng, J. F. Tang, “New automatic design technique for optical coatings,” Appl. Opt. 26, 1546–1549 (1987).
    [CrossRef] [PubMed]
  5. A. Antamoshkin, H. P. Schwefel, A. Törn, G. Yin, A. Zilinskas, System Analysis, Design and Optimization: An Introduction (Russian Academy of Engineering, Siberian Department, Krasnoyarsk, Russia, 1993), pp. 1–173.
  6. Y. Ermoliev, R. Wets, eds., Numerical Techniques for Stochastic Optimization (Springer, Berlin, 1988), pp. 1–33.
    [CrossRef]
  7. A. Törn, A. Zilinskas, Global Optimization, Vol. 350 of Springer Lecture Notes in Computer Science (Springer, Berlin, 1987), pp. 1–24.
    [CrossRef]
  8. D. B. Fogel, Evolutionary Computation (IEEE, Piscataway, N.J., 1995), pp. 67–187.
  9. H. P. Schwefel, Evolution and Optimum Seeking (Wiley, New York, 1995), pp. 105–249.
  10. M. S. Phadke, Quality Engineering Using Robust Design (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 1–27.
  11. G. S. Peace, Taguchi Methods (Addison-Wesley, Reading, Mass., 1992), pp. 1–13.
  12. W. Klug, R. Herrmann, G. Saur, “Design and manufacturing of ophthalmic antireflection coatings with low angular color shift,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. SPIE1270, 278–286 (1990).
  13. H. Greiner, “Robust filter design by stochastic optimization,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 150–161 (1994).
  14. W. J. Wild, H. Buhay, “Thin-film multilayer design optimization using a Monte-Carlo approach,” Opt. Lett. 11, 745–747 (1986).
    [CrossRef] [PubMed]
  15. S. Martin, A. Brunet-Bruneau, J. Rivory, “Simulated Darwinian evolution of homogeneous multilayer systems: a new method for optical coating design,” Opt. Commun. 110, 503–506 (1994).
    [CrossRef]
  16. S. Martin, J. Rivory, M. Schoenauer, “Synthesis of optical multilayer systems using genetic algorithms,” Appl. Opt. 34, 2247–2254 (1995).
    [CrossRef] [PubMed]
  17. 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]
  18. U. Hammel, T. Bäck, “Evolution strategies on noisy functions: how to improve convergence,” in Proceedings of the International Conference on Evolutionary Computation, Y. Davidor, H. P. Schwefel, R. Männer, eds. (Springer-Verlag, Berlin, 1994), pp. 159–168.
  19. D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).

1995 (1)

1994 (2)

D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).

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

1993 (1)

1987 (1)

1986 (1)

1982 (1)

W. P. Thoeni, “Deposition of optical coatings process control and automation,” Thin Solid Films 88, 385–397 (1982).
[CrossRef]

Antamoshkin, A.

A. Antamoshkin, H. P. Schwefel, A. Törn, G. Yin, A. Zilinskas, System Analysis, Design and Optimization: An Introduction (Russian Academy of Engineering, Siberian Department, Krasnoyarsk, Russia, 1993), pp. 1–173.

Bäck, T.

U. Hammel, T. Bäck, “Evolution strategies on noisy functions: how to improve convergence,” in Proceedings of the International Conference on Evolutionary Computation, Y. Davidor, H. P. Schwefel, R. Männer, eds. (Springer-Verlag, Berlin, 1994), pp. 159–168.

Brunet-Bruneau, A.

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

Buhay, H.

Eisenhammer, T.

Fogel, D. B.

D. B. Fogel, Evolutionary Computation (IEEE, Piscataway, N.J., 1995), pp. 67–187.

Greiner, H.

H. Greiner, “Robust filter design by stochastic optimization,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 150–161 (1994).

Hammel, U.

U. Hammel, T. Bäck, “Evolution strategies on noisy functions: how to improve convergence,” in Proceedings of the International Conference on Evolutionary Computation, Y. Davidor, H. P. Schwefel, R. Männer, eds. (Springer-Verlag, Berlin, 1994), pp. 159–168.

Herrmann, R.

W. Klug, R. Herrmann, G. Saur, “Design and manufacturing of ophthalmic antireflection coatings with low angular color shift,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. SPIE1270, 278–286 (1990).

Klug, W.

W. Klug, R. Herrmann, G. Saur, “Design and manufacturing of ophthalmic antireflection coatings with low angular color shift,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. SPIE1270, 278–286 (1990).

Lazarov, M.

Leutbecher, M.

Martin, S.

S. Martin, J. Rivory, M. Schoenauer, “Synthesis of optical multilayer systems using genetic algorithms,” Appl. Opt. 34, 2247–2254 (1995).
[CrossRef] [PubMed]

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

Peace, G. S.

G. S. Peace, Taguchi Methods (Addison-Wesley, Reading, Mass., 1992), pp. 1–13.

Phadke, M. S.

M. S. Phadke, Quality Engineering Using Robust Design (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 1–27.

Rancourt, J. D.

J. D. Rancourt, Optical Thin Films User’s Handbook (McGraw-Hill, New York, 1987), pp. 33–38, 71–75.

Rivory, J.

S. Martin, J. Rivory, M. Schoenauer, “Synthesis of optical multilayer systems using genetic algorithms,” Appl. Opt. 34, 2247–2254 (1995).
[CrossRef] [PubMed]

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

Saur, G.

W. Klug, R. Herrmann, G. Saur, “Design and manufacturing of ophthalmic antireflection coatings with low angular color shift,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. SPIE1270, 278–286 (1990).

Schoenauer, M.

Schöffel, U.

Schwefel, H. P.

H. P. Schwefel, Evolution and Optimum Seeking (Wiley, New York, 1995), pp. 105–249.

A. Antamoshkin, H. P. Schwefel, A. Törn, G. Yin, A. Zilinskas, System Analysis, Design and Optimization: An Introduction (Russian Academy of Engineering, Siberian Department, Krasnoyarsk, Russia, 1993), pp. 1–173.

Shafer, D.

D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).

Sizmann, R.

Tang, J. F.

Thelen, A.

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1988), Chap. 12, pp. 235–246.

Thoeni, W. P.

W. P. Thoeni, “Deposition of optical coatings process control and automation,” Thin Solid Films 88, 385–397 (1982).
[CrossRef]

Törn, A.

A. Antamoshkin, H. P. Schwefel, A. Törn, G. Yin, A. Zilinskas, System Analysis, Design and Optimization: An Introduction (Russian Academy of Engineering, Siberian Department, Krasnoyarsk, Russia, 1993), pp. 1–173.

A. Törn, A. Zilinskas, Global Optimization, Vol. 350 of Springer Lecture Notes in Computer Science (Springer, Berlin, 1987), pp. 1–24.
[CrossRef]

Wild, W. J.

Yin, G.

A. Antamoshkin, H. P. Schwefel, A. Törn, G. Yin, A. Zilinskas, System Analysis, Design and Optimization: An Introduction (Russian Academy of Engineering, Siberian Department, Krasnoyarsk, Russia, 1993), pp. 1–173.

Zheng, Y.-F.

Zilinskas, A.

A. Antamoshkin, H. P. Schwefel, A. Törn, G. Yin, A. Zilinskas, System Analysis, Design and Optimization: An Introduction (Russian Academy of Engineering, Siberian Department, Krasnoyarsk, Russia, 1993), pp. 1–173.

A. Törn, A. Zilinskas, Global Optimization, Vol. 350 of Springer Lecture Notes in Computer Science (Springer, Berlin, 1987), pp. 1–24.
[CrossRef]

Appl. Opt. (3)

Comput. Phys. (1)

D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).

Opt. Commun. (1)

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

Opt. Lett. (1)

Thin Solid Films (1)

W. P. Thoeni, “Deposition of optical coatings process control and automation,” Thin Solid Films 88, 385–397 (1982).
[CrossRef]

Other (12)

J. D. Rancourt, Optical Thin Films User’s Handbook (McGraw-Hill, New York, 1987), pp. 33–38, 71–75.

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1988), Chap. 12, pp. 235–246.

U. Hammel, T. Bäck, “Evolution strategies on noisy functions: how to improve convergence,” in Proceedings of the International Conference on Evolutionary Computation, Y. Davidor, H. P. Schwefel, R. Männer, eds. (Springer-Verlag, Berlin, 1994), pp. 159–168.

A. Antamoshkin, H. P. Schwefel, A. Törn, G. Yin, A. Zilinskas, System Analysis, Design and Optimization: An Introduction (Russian Academy of Engineering, Siberian Department, Krasnoyarsk, Russia, 1993), pp. 1–173.

Y. Ermoliev, R. Wets, eds., Numerical Techniques for Stochastic Optimization (Springer, Berlin, 1988), pp. 1–33.
[CrossRef]

A. Törn, A. Zilinskas, Global Optimization, Vol. 350 of Springer Lecture Notes in Computer Science (Springer, Berlin, 1987), pp. 1–24.
[CrossRef]

D. B. Fogel, Evolutionary Computation (IEEE, Piscataway, N.J., 1995), pp. 67–187.

H. P. Schwefel, Evolution and Optimum Seeking (Wiley, New York, 1995), pp. 105–249.

M. S. Phadke, Quality Engineering Using Robust Design (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 1–27.

G. S. Peace, Taguchi Methods (Addison-Wesley, Reading, Mass., 1992), pp. 1–13.

W. Klug, R. Herrmann, G. Saur, “Design and manufacturing of ophthalmic antireflection coatings with low angular color shift,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. SPIE1270, 278–286 (1990).

H. Greiner, “Robust filter design by stochastic optimization,” in Optical Interference Coatings, F. Abelès, ed., Proc. SPIE2253, 150–161 (1994).

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Figures (10)

Fig. 1
Fig. 1

Simulation of the transmittance variations of a hot mirror induced by statistical variations of the layer thicknesses and indices of refraction. Independent normal distributions with standard deviations of 2% were assumed.

Fig. 2
Fig. 2

Nonrobust and robust minima of the merit function f(x, ω). x denotes the design, and ω denotes noise variables. The aim of robust design is to find settings for x, that for an acceptable value of the figure of merit, minimize the effect of the noise factors.

Fig. 3
Fig. 3

Emphasis on meeting the specification limits leads to a uniform distribution of the performance (broken curve) within the specification range, whereas concentrating on meeting the target leads to a distribution (solid curve) peaked around the target, even at the expense of a few samples outside the allowed range (from Phadke10).

Fig. 4
Fig. 4

Ellipses indicate the distribution of the directions of the mutation steps relative to the function landscape defined by the contour lines of the function of merit. Their adaptation and alignment (e.g., along narrow valleys) lead to a more effective search. Left, no adaptation; right, with adaptation (picture courtesy of F. Hofmeister).

Fig. 5
Fig. 5

Robust evolution: expected figure of merit (based on all evaluation since birth) versus number of generations for a (15 + 60) strategy with only one random sample per evaluation (taken from Ref. 13).

Fig. 6
Fig. 6

Reflectance (in percentage) versus wavelength (in nanometers) for design A given in Table 1. Solid curve, nominal reflectance; dotted curves, perturbed reflectances.

Fig. 7
Fig. 7

Color spread of designs A, B, C, and D defined in Table 1 around the target value (0.281, 0.351) in the (x, y) chromaticity diagram. The spread is caused by independent random perturbations of the layer thicknesses and refractive indices with a standard deviation of 1%.

Fig. 8
Fig. 8

Reflectance (in percentage) versus wavelength (in nanometers) for design D given in Table 1. Solid curve, nominal reflectance; dotted curve, perturbed reflectances.

Fig. 9
Fig. 9

Reflectance (in percentage) versus wavelength (in nanometers) for designs A, B, C, and D given in Table 1.

Fig. 10
Fig. 10

Cumulative distribution of the figure of merit for designs A, B, C, and D given in Table 1. The figure of merit is defined in the text.

Tables (1)

Tables Icon

Table 1 Design Parameters

Equations (10)

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P d ( Δ d ) = P d ( Δ d 1 , , Δ d k ) , P n ( Δ n ) = P n ( Δ n 1 , , Δ n k ) .
E f ( d ) = f ( d + Δ d , n + Δ n ) d P d ( Δ d ) d P n ( Δ n ) ,
f ( x ) = min ! ,             x = ( x 1 , , x n ) S R n ,
P 0 = { x 1 , , x μ } ,
x = ( x 1 , , x n ) ,             y = ( y 1 , , y n ) z = ( z 1 , , z n )
σ = ( σ 1 , , σ n ) z z + ζ .
p ( ζ ) = [ det ( C ) ( 2 π ) n ] 1 / 2 exp ( - 0.5 ζ t C ζ ) ,
E f ( x ) = Ω f ( x , ω ) d P ( ω ) = min ! ,
E f ( x ) 1 k i = 1 k f ( x , ω i ) ,
F ( d , n ) = 100 { [ x ( d , n ) - 0.281 ] 2 + [ y ( d , n ) - 0.351 ] 2 } 1 / 2 ,

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