Abstract

Ten different optimization methods, representing both local and global minimum seeking algorithms, were applied to the solution of three different optical thin film design problems. Because all methods were incorporated in the same thin film design program, and because a routine was invoked that reads CPU time, the relative efficiencies of the various methods could be compared directly.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. A. Macleod, Thin Film Optical Filters (Macmillan, New York1986).
    [CrossRef]
  2. Z. Knittl, Optics of Thin Films (Wiley, New York1976).
  3. A. Thelen, Design of Optical Interference Filters (McGraw-Hill, New York1988).
  4. H. M. Liddell, Computer-Aided Techniques for the Design of Multilayer Filters (Adam Hilger, Bristol, 1981).
  5. J. A. Dobrowolski, “Computer Design of Optical Coatings,” Thin Solid Films 163, 97–110 (1988).
    [CrossRef]
  6. J. A. Aguilera et al., “Antireflection Coatings for Germanium IR Optics: A Comparison of Numerical Design Methods,” Appl. Opt. 27, 2832–2840 (1988).
    [CrossRef] [PubMed]
  7. J. A. Dobrowolski, F. C. Ho, A. Belkind, V. Koss, “Merit Functions for More Effective Thin Film Calculations,” Appl. Opt. 28, 2824–2831 (1989).
    [CrossRef] [PubMed]
  8. J. A. Dobrowolski, “Completely Automatic Synthesis of Optical Thin Film Systems,” Appl. Opt. 4, 937–946 (1965).
    [CrossRef]
  9. J. A. Dobrowolski, “Versatile Computer Program for Absorbing Optical Thin Film Systems,” Appl. Opt. 20, 74–81 (1981).
    [CrossRef] [PubMed]
  10. J. A. Dobrowolski, F. C. Ho, A. Waldorf, “Determination of Optical Constants of Thin Film Coating Materials Based on Inverse Synthesis,” Appl. Opt. 22, 3191–3200 (1983).
    [CrossRef] [PubMed]
  11. J. A. Dobrowolski, “On the Determination of Optical Constants of Films in Multilayers,” Proc. Soc. Photo-Opt. Instrum. Eng. 652, 206–208 (1986).
  12. J. Bartela et al., “Multiple Analysis of an Unknown Optical Multilayer Coating,” Appl. Opt. 24, 2625–2646 (1985).
    [CrossRef]
  13. C. Van der Laan, H. J. Frankena, “Fast Computation Method for Derivatives of Multilayer Stack Reflectance,” Appl. Opt. 17, 538–541 (1978).
    [CrossRef]
  14. K.-O. Peng, M. R. de la Fonteijne, “Derivatives of Transmittance and Reflectance for an Absorbing Multilayer Stack,” Appl. Opt. 24, 501–503 (1985).
    [CrossRef] [PubMed]
  15. S. F. Masri, G. A. Bekey, “A Global Optimization Algorithm Using Adaptive Random Search,” Appl. Math. Comput. 7, 353–375 (1980).
    [CrossRef]
  16. P. W. Baumeister, “Design of Multilayer Filters by Successive Approximations,” J. Opt. Soc. Am. 48, 955–958 (1958).
    [CrossRef]
  17. A. L. Bloom, “Refining and Optimization in Multilayers,” Appl. Opt. 20, 66–73 (1981).
    [CrossRef] [PubMed]
  18. J. A. Dobrowolski, “Automatic Refinement of Optical Multilayer Assemblies,” J. Opt. Soc. Am. 51, 1475 (1961).
  19. L. E. Scales, Introduction to Non-Linear Optimization (Springer-Verlag, New York, 1985), pp. 110–113.
  20. Ref. 19, pp. 115–118.
  21. P. D. Crout, “A Short Method for Evaluating Determinants and Solving Systems of Linear Equations with Real or Complex Coefficients,” Trans. Am. Inst. Electr. Eng. 60, 1235–1241 (1941).
    [CrossRef]
  22. Ref. 20, pp. 56–61.
  23. C. HoIm, “Optical Thin Film Production with Continuous Reoptimization of Layer Thicknesses,” Appl. Opt. 18, 1978–1982 (1979).
    [CrossRef]
  24. T. E. Shoup, F. Mistree, Optimization Methods with Applications to Personal Computers (Springer-Verlag, New York, 1985), pp. 36–41.
  25. Ref. 24, pp. 110–118.
  26. M. J. D. Powell, “An Efficient Method for Finding the Minimum of a Function of Several Variables without Calculating Derivatives,” Computer J. 7, 155–162 (1964).
    [CrossRef]
  27. H. H. Rosenbrock, “An Automatic Method for Finding the Greatest or Least Value of a Function,” Computer J. 7, 308–313 (1964).
  28. J. Stoer, Introduction to Numerical Analysis (Springer-Verlag, New York, 1980), pp. 195–197.
  29. S. Kirkpatrick, C. D. Gellatt, M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220, 671–680 (1984).
    [CrossRef]
  30. I. O. Bohachevsky, M. E. Johnson, M. L. Stein, “Generalized Simulated Annealing for Function Optimization,” Technometrics 28, 209–217 (1986).
    [CrossRef]
  31. R. Morf, R. E. Kuns, “Dielectric Filter Optimization by Simulated Thermal Annealing,” Proc. Soc. Photo-Opt. Instrum. Eng. 1019, 211–217 (1988).
  32. D. Vanderbilt, S. Louie, “A Monte Carlo Simulated Annealing Approach to Optimization over Continuous Variables,” J. Comput. Phys. 56, 155–162 (1984).
    [CrossRef]
  33. E. Pelletier, “Calcul et Réalisation de Revétments Multi-dielectriques Présentant des Charactéristiques Spectrales Imposées,” Thesis, Faculté des Sciences d’Orsay, 1970.
  34. J. A. Nelder, R. Mead, “A Simplex Method for Function Minimization,” Computer J. 7, 308–313 (1964).
    [CrossRef]
  35. E. D. Palik, Ed., Handbook of Optical Constants of Solids (Academic, New York, 1985), pp. 313–323.

1989 (1)

1988 (3)

J. A. Aguilera et al., “Antireflection Coatings for Germanium IR Optics: A Comparison of Numerical Design Methods,” Appl. Opt. 27, 2832–2840 (1988).
[CrossRef] [PubMed]

J. A. Dobrowolski, “Computer Design of Optical Coatings,” Thin Solid Films 163, 97–110 (1988).
[CrossRef]

R. Morf, R. E. Kuns, “Dielectric Filter Optimization by Simulated Thermal Annealing,” Proc. Soc. Photo-Opt. Instrum. Eng. 1019, 211–217 (1988).

1986 (2)

I. O. Bohachevsky, M. E. Johnson, M. L. Stein, “Generalized Simulated Annealing for Function Optimization,” Technometrics 28, 209–217 (1986).
[CrossRef]

J. A. Dobrowolski, “On the Determination of Optical Constants of Films in Multilayers,” Proc. Soc. Photo-Opt. Instrum. Eng. 652, 206–208 (1986).

1985 (2)

1984 (2)

D. Vanderbilt, S. Louie, “A Monte Carlo Simulated Annealing Approach to Optimization over Continuous Variables,” J. Comput. Phys. 56, 155–162 (1984).
[CrossRef]

S. Kirkpatrick, C. D. Gellatt, M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220, 671–680 (1984).
[CrossRef]

1983 (1)

1981 (2)

A. L. Bloom, “Refining and Optimization in Multilayers,” Appl. Opt. 20, 66–73 (1981).
[CrossRef] [PubMed]

J. A. Dobrowolski, “Versatile Computer Program for Absorbing Optical Thin Film Systems,” Appl. Opt. 20, 74–81 (1981).
[CrossRef] [PubMed]

1980 (1)

S. F. Masri, G. A. Bekey, “A Global Optimization Algorithm Using Adaptive Random Search,” Appl. Math. Comput. 7, 353–375 (1980).
[CrossRef]

1979 (1)

1978 (1)

C. Van der Laan, H. J. Frankena, “Fast Computation Method for Derivatives of Multilayer Stack Reflectance,” Appl. Opt. 17, 538–541 (1978).
[CrossRef]

1965 (1)

J. A. Dobrowolski, “Completely Automatic Synthesis of Optical Thin Film Systems,” Appl. Opt. 4, 937–946 (1965).
[CrossRef]

1964 (3)

J. A. Nelder, R. Mead, “A Simplex Method for Function Minimization,” Computer J. 7, 308–313 (1964).
[CrossRef]

M. J. D. Powell, “An Efficient Method for Finding the Minimum of a Function of Several Variables without Calculating Derivatives,” Computer J. 7, 155–162 (1964).
[CrossRef]

H. H. Rosenbrock, “An Automatic Method for Finding the Greatest or Least Value of a Function,” Computer J. 7, 308–313 (1964).

1961 (1)

J. A. Dobrowolski, “Automatic Refinement of Optical Multilayer Assemblies,” J. Opt. Soc. Am. 51, 1475 (1961).

1958 (1)

P. W. Baumeister, “Design of Multilayer Filters by Successive Approximations,” J. Opt. Soc. Am. 48, 955–958 (1958).
[CrossRef]

1941 (1)

P. D. Crout, “A Short Method for Evaluating Determinants and Solving Systems of Linear Equations with Real or Complex Coefficients,” Trans. Am. Inst. Electr. Eng. 60, 1235–1241 (1941).
[CrossRef]

Aguilera, J. A.

Bartela, J.

Baumeister, P. W.

P. W. Baumeister, “Design of Multilayer Filters by Successive Approximations,” J. Opt. Soc. Am. 48, 955–958 (1958).
[CrossRef]

Bekey, G. A.

S. F. Masri, G. A. Bekey, “A Global Optimization Algorithm Using Adaptive Random Search,” Appl. Math. Comput. 7, 353–375 (1980).
[CrossRef]

Belkind, A.

Bloom, A. L.

Bohachevsky, I. O.

I. O. Bohachevsky, M. E. Johnson, M. L. Stein, “Generalized Simulated Annealing for Function Optimization,” Technometrics 28, 209–217 (1986).
[CrossRef]

Crout, P. D.

P. D. Crout, “A Short Method for Evaluating Determinants and Solving Systems of Linear Equations with Real or Complex Coefficients,” Trans. Am. Inst. Electr. Eng. 60, 1235–1241 (1941).
[CrossRef]

de la Fonteijne, M. R.

Dobrowolski, J. A.

J. A. Dobrowolski, F. C. Ho, A. Belkind, V. Koss, “Merit Functions for More Effective Thin Film Calculations,” Appl. Opt. 28, 2824–2831 (1989).
[CrossRef] [PubMed]

J. A. Dobrowolski, “Computer Design of Optical Coatings,” Thin Solid Films 163, 97–110 (1988).
[CrossRef]

J. A. Dobrowolski, “On the Determination of Optical Constants of Films in Multilayers,” Proc. Soc. Photo-Opt. Instrum. Eng. 652, 206–208 (1986).

J. A. Dobrowolski, F. C. Ho, A. Waldorf, “Determination of Optical Constants of Thin Film Coating Materials Based on Inverse Synthesis,” Appl. Opt. 22, 3191–3200 (1983).
[CrossRef] [PubMed]

J. A. Dobrowolski, “Versatile Computer Program for Absorbing Optical Thin Film Systems,” Appl. Opt. 20, 74–81 (1981).
[CrossRef] [PubMed]

J. A. Dobrowolski, “Completely Automatic Synthesis of Optical Thin Film Systems,” Appl. Opt. 4, 937–946 (1965).
[CrossRef]

J. A. Dobrowolski, “Automatic Refinement of Optical Multilayer Assemblies,” J. Opt. Soc. Am. 51, 1475 (1961).

Frankena, H. J.

C. Van der Laan, H. J. Frankena, “Fast Computation Method for Derivatives of Multilayer Stack Reflectance,” Appl. Opt. 17, 538–541 (1978).
[CrossRef]

Gellatt, C. D.

S. Kirkpatrick, C. D. Gellatt, M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220, 671–680 (1984).
[CrossRef]

Ho, F. C.

HoIm, C.

Johnson, M. E.

I. O. Bohachevsky, M. E. Johnson, M. L. Stein, “Generalized Simulated Annealing for Function Optimization,” Technometrics 28, 209–217 (1986).
[CrossRef]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gellatt, M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220, 671–680 (1984).
[CrossRef]

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, New York1976).

Koss, V.

Kuns, R. E.

R. Morf, R. E. Kuns, “Dielectric Filter Optimization by Simulated Thermal Annealing,” Proc. Soc. Photo-Opt. Instrum. Eng. 1019, 211–217 (1988).

Liddell, H. M.

H. M. Liddell, Computer-Aided Techniques for the Design of Multilayer Filters (Adam Hilger, Bristol, 1981).

Louie, S.

D. Vanderbilt, S. Louie, “A Monte Carlo Simulated Annealing Approach to Optimization over Continuous Variables,” J. Comput. Phys. 56, 155–162 (1984).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin Film Optical Filters (Macmillan, New York1986).
[CrossRef]

Masri, S. F.

S. F. Masri, G. A. Bekey, “A Global Optimization Algorithm Using Adaptive Random Search,” Appl. Math. Comput. 7, 353–375 (1980).
[CrossRef]

Mead, R.

J. A. Nelder, R. Mead, “A Simplex Method for Function Minimization,” Computer J. 7, 308–313 (1964).
[CrossRef]

Mistree, F.

T. E. Shoup, F. Mistree, Optimization Methods with Applications to Personal Computers (Springer-Verlag, New York, 1985), pp. 36–41.

Morf, R.

R. Morf, R. E. Kuns, “Dielectric Filter Optimization by Simulated Thermal Annealing,” Proc. Soc. Photo-Opt. Instrum. Eng. 1019, 211–217 (1988).

Nelder, J. A.

J. A. Nelder, R. Mead, “A Simplex Method for Function Minimization,” Computer J. 7, 308–313 (1964).
[CrossRef]

Pelletier, E.

E. Pelletier, “Calcul et Réalisation de Revétments Multi-dielectriques Présentant des Charactéristiques Spectrales Imposées,” Thesis, Faculté des Sciences d’Orsay, 1970.

Peng, K.-O.

Powell, M. J. D.

M. J. D. Powell, “An Efficient Method for Finding the Minimum of a Function of Several Variables without Calculating Derivatives,” Computer J. 7, 155–162 (1964).
[CrossRef]

Rosenbrock, H. H.

H. H. Rosenbrock, “An Automatic Method for Finding the Greatest or Least Value of a Function,” Computer J. 7, 308–313 (1964).

Scales, L. E.

L. E. Scales, Introduction to Non-Linear Optimization (Springer-Verlag, New York, 1985), pp. 110–113.

Shoup, T. E.

T. E. Shoup, F. Mistree, Optimization Methods with Applications to Personal Computers (Springer-Verlag, New York, 1985), pp. 36–41.

Stein, M. L.

I. O. Bohachevsky, M. E. Johnson, M. L. Stein, “Generalized Simulated Annealing for Function Optimization,” Technometrics 28, 209–217 (1986).
[CrossRef]

Stoer, J.

J. Stoer, Introduction to Numerical Analysis (Springer-Verlag, New York, 1980), pp. 195–197.

Thelen, A.

A. Thelen, Design of Optical Interference Filters (McGraw-Hill, New York1988).

Van der Laan, C.

C. Van der Laan, H. J. Frankena, “Fast Computation Method for Derivatives of Multilayer Stack Reflectance,” Appl. Opt. 17, 538–541 (1978).
[CrossRef]

Vanderbilt, D.

D. Vanderbilt, S. Louie, “A Monte Carlo Simulated Annealing Approach to Optimization over Continuous Variables,” J. Comput. Phys. 56, 155–162 (1984).
[CrossRef]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gellatt, M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220, 671–680 (1984).
[CrossRef]

Waldorf, A.

Appl. Opt. (3)

C. Van der Laan, H. J. Frankena, “Fast Computation Method for Derivatives of Multilayer Stack Reflectance,” Appl. Opt. 17, 538–541 (1978).
[CrossRef]

J. A. Dobrowolski, “Completely Automatic Synthesis of Optical Thin Film Systems,” Appl. Opt. 4, 937–946 (1965).
[CrossRef]

J. A. Dobrowolski, “Versatile Computer Program for Absorbing Optical Thin Film Systems,” Appl. Opt. 20, 74–81 (1981).
[CrossRef] [PubMed]

Appl. Math. Comput. (1)

S. F. Masri, G. A. Bekey, “A Global Optimization Algorithm Using Adaptive Random Search,” Appl. Math. Comput. 7, 353–375 (1980).
[CrossRef]

Appl. Opt. (7)

Computer J. (3)

J. A. Nelder, R. Mead, “A Simplex Method for Function Minimization,” Computer J. 7, 308–313 (1964).
[CrossRef]

M. J. D. Powell, “An Efficient Method for Finding the Minimum of a Function of Several Variables without Calculating Derivatives,” Computer J. 7, 155–162 (1964).
[CrossRef]

H. H. Rosenbrock, “An Automatic Method for Finding the Greatest or Least Value of a Function,” Computer J. 7, 308–313 (1964).

J. Comput. Phys. (1)

D. Vanderbilt, S. Louie, “A Monte Carlo Simulated Annealing Approach to Optimization over Continuous Variables,” J. Comput. Phys. 56, 155–162 (1984).
[CrossRef]

J. Opt. Soc. Am. (2)

P. W. Baumeister, “Design of Multilayer Filters by Successive Approximations,” J. Opt. Soc. Am. 48, 955–958 (1958).
[CrossRef]

J. A. Dobrowolski, “Automatic Refinement of Optical Multilayer Assemblies,” J. Opt. Soc. Am. 51, 1475 (1961).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

J. A. Dobrowolski, “On the Determination of Optical Constants of Films in Multilayers,” Proc. Soc. Photo-Opt. Instrum. Eng. 652, 206–208 (1986).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

R. Morf, R. E. Kuns, “Dielectric Filter Optimization by Simulated Thermal Annealing,” Proc. Soc. Photo-Opt. Instrum. Eng. 1019, 211–217 (1988).

Science (1)

S. Kirkpatrick, C. D. Gellatt, M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220, 671–680 (1984).
[CrossRef]

Technometrics (1)

I. O. Bohachevsky, M. E. Johnson, M. L. Stein, “Generalized Simulated Annealing for Function Optimization,” Technometrics 28, 209–217 (1986).
[CrossRef]

Thin Solid Films (1)

J. A. Dobrowolski, “Computer Design of Optical Coatings,” Thin Solid Films 163, 97–110 (1988).
[CrossRef]

Trans. Am. Inst. Electr. Eng. (1)

P. D. Crout, “A Short Method for Evaluating Determinants and Solving Systems of Linear Equations with Real or Complex Coefficients,” Trans. Am. Inst. Electr. Eng. 60, 1235–1241 (1941).
[CrossRef]

Other (12)

Ref. 20, pp. 56–61.

T. E. Shoup, F. Mistree, Optimization Methods with Applications to Personal Computers (Springer-Verlag, New York, 1985), pp. 36–41.

Ref. 24, pp. 110–118.

J. Stoer, Introduction to Numerical Analysis (Springer-Verlag, New York, 1980), pp. 195–197.

L. E. Scales, Introduction to Non-Linear Optimization (Springer-Verlag, New York, 1985), pp. 110–113.

Ref. 19, pp. 115–118.

H. A. Macleod, Thin Film Optical Filters (Macmillan, New York1986).
[CrossRef]

Z. Knittl, Optics of Thin Films (Wiley, New York1976).

A. Thelen, Design of Optical Interference Filters (McGraw-Hill, New York1988).

H. M. Liddell, Computer-Aided Techniques for the Design of Multilayer Filters (Adam Hilger, Bristol, 1981).

E. Pelletier, “Calcul et Réalisation de Revétments Multi-dielectriques Présentant des Charactéristiques Spectrales Imposées,” Thesis, Faculté des Sciences d’Orsay, 1970.

E. D. Palik, Ed., Handbook of Optical Constants of Solids (Academic, New York, 1985), pp. 313–323.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Block diagram illustrating the flow of calculations.

Fig. 2
Fig. 2

Example A: calculations on a broadband infrared antireflection coating.

Fig. 3
Fig. 3

Example B: calculations on a 50% all-dielectric reflector for the 0.4–1.0-μm spectral region.

Fig. 4
Fig. 4

Example C: calculations on a wide-angle absorber for the 3.0–5.0-μm spectral region.

Fig. 5
Fig. 5

Best merit function vs time curves obtained with the various optimization methods for each example.

Fig. 6
Fig. 6

Additional calculations on examples A and B.

Fig. 7
Fig. 7

Average nonnormal incidence spectral reflectance of the multilayer of Fig. 4(D).

Tables (5)

Tables Icon

Table I Names and Properties of Optimization Methods Investigated

Tables Icon

Table II Construction Parameters of Solutions of Problem A Depicted In Figure 2

Tables Icon

Table III Construction Parameters of Solutions of Problem B Depicted In Figure 3

Tables Icon

Table IV Construction Parameters of Solutions of Problem C Depicted in Figure 4

Tables Icon

Table V Construction Parameters of Solutions Depicted In Figure 6

Equations (20)

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

M = [ 1 / m i = 1 m ( ( Q i T - Q i ) / δ Q i ) 2 ] 1 / 2
δ Q = D δ p ,
δ Q = [ δ Q 1 δ Q 2 . . . . δ Q m ] , D = [ d Q 1 / d p 1 d Q 1 / d p 2 d Q 1 / d p l d Q 2 / d p 1 . . . . d Q m / d p 1 ....... d Q m / d p l ] , δ p = [ δ p 1 δ p 2 . . δ p l ]
[ δ Q - D δ p ] 2 .
0 = B δ p .
B = [ B 1 , 1 0 . . . . 0 0 B 2 , 2 . . . . 0 . . . . . . . . . . . . . . 0 0 . . . . B l , l ] .
B t = [ i = 1 m ( δ Q i ) 2 ] 1 / 2 / ( 4 / Δ t ) ;             B n = [ i = 1 m ( δ Q i ) 2 ] 1 / 2 / ( 4 / Δ n ) .
[ δ Q - D δ p ] 2 + [ B δ p ] 2 .
p = p 0 - s d M / d p ,
δ p j = - d M / d p j / d 2 M / d p j 2
M = M ( p + k Δ p )
p 1 = p min + 0.382 ( p max - p min )
p 2 = p min + 0.618 ( p max - p min ) .
P = 1
P = exp [ β ( M 1 - M 2 ) / M 1 g ]
a i = 1 / K k = 1 K p i ( k ) ,
b i j = 1 / K k = 1 K ( p i ( k ) - a i ) ( p j ( k ) - a j ) .
s n = ( α γ b ) / ( K β ) + ( 1 - α ) s .
P F 1 = 1 / 3 i = 1 3 ( M B , i / M S , i )
P F 2 = 1 / 3 i = 1 3 [ ( M A , i + σ A , i ) / M S , i ] ,

Metrics