Abstract

We reconsider the problem of locating the globally optimal solution of a multilayer-optical-coating design problem, within some predetermined space of parameters, with the aim of obtaining a robust technique that requires a minimum of user intervention. The approach we adopt centers on exploring the space of the parameters of interest by using a Markov-chain Monte Carlo sampling algorithm. This technique enables one to locate the global optimum automatically with high confidence and without the need for a good starting design. It also allows the trivial inclusion of prior constraints on the variables and provides a natural means for investigating the robustness of the optimal solution.

© 2004 Optical Society of America

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References

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  1. P. Baumeister, “Design of multilayer filters by successive approximations,” J. Opt. Soc. Am. 48, 955–958 (1958).
    [CrossRef]
  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. A. Premoli, M. L. Rastello, “Minimax refining of wideband antireflection coatings for wide angular indicence,” Appl. Opt. 33, 2018–2024 (1994).
    [CrossRef] [PubMed]
  4. J. A. Dobrowolski, “Completely automatic synthesis of optical thin film systems,” Appl. Opt. 4, 937–946 (1965).
    [CrossRef]
  5. C. G. Snedaker, “New numerical thin-film synthesis technique,” J. Opt. Soc. Am. 72, 1732A (1982).
  6. W. H. Southwell, “Coating design using very thin high- and low-index layers,” Appl. Opt. 24, 457–460 (1985).
    [CrossRef] [PubMed]
  7. A. V. Tikhonravov, M. K. Trubetskov, G. W. DeBell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35, 5493–5508 (1996).
    [CrossRef] [PubMed]
  8. P. G. Verly, “Fourier transform technique with refinement in the frequency domain for the synthesis of optical thin films,” Appl. Opt. 35, 5148–5154 (1996).
    [CrossRef] [PubMed]
  9. T. Boudet, P. Chaton, “Thin film design using simulated annealing and the study of filter robustness,” in Developments in Optical Component Coatings, I. Reid, ed., Proc. SPIE2776, 27–38 (1996).
    [CrossRef]
  10. S. Martin, J. Rivory, M. Schoenauer, “Synthesis of optical multilayer systems using genetic algorithms,” Appl. Opt. 34, 2247–2254 (1995).
    [CrossRef] [PubMed]
  11. D. G. Li, A. C. Watson, “Global optimization for optical thin film design using Latin squares,” in Optical Thin Films V: New Developments, R. L. Hall, ed., Proc. SPIE3133, 8–15 (1997).
    [CrossRef]
  12. P. G. Verly, A. V. Tikhonravov, M. K. Trubetskov, “Efficient refinement algorithm for the synthesis of inhomogeneous optical coatings,” Appl. Opt. 36, 1487–1495 (1997).
    [CrossRef] [PubMed]
  13. P. G. Verly, “Optical coating synthesis by simultaneous refractive-index and thickness refinement of inhomogeneous films,” Appl. Opt. 37, 7327–7333 (1998).
    [CrossRef]
  14. Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992), pp. 1–149.
  15. W. R. Gilks, S. Richardson, D. J. Spiegelhalter, Markov-Chain Monte Carlo in Practice (Chapman Hall, London, 1995).
  16. M. Kalos, P. Whitlock, Monte Carlo Methods (Wiley, New York, 1986).
    [CrossRef]
  17. W. H. Press, S. A. Teukolski, W. T. Vetterlin, B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge University, Cambridge, England, 1992), pp. 409–413.

1998

1997

1996

1995

1994

1990

1985

1982

C. G. Snedaker, “New numerical thin-film synthesis technique,” J. Opt. Soc. Am. 72, 1732A (1982).

1965

1958

Baumeister, P.

Boudet, T.

T. Boudet, P. Chaton, “Thin film design using simulated annealing and the study of filter robustness,” in Developments in Optical Component Coatings, I. Reid, ed., Proc. SPIE2776, 27–38 (1996).
[CrossRef]

Chaton, P.

T. Boudet, P. Chaton, “Thin film design using simulated annealing and the study of filter robustness,” in Developments in Optical Component Coatings, I. Reid, ed., Proc. SPIE2776, 27–38 (1996).
[CrossRef]

DeBell, G. W.

Dobrowolski, J. A.

Flannery, B. P.

W. H. Press, S. A. Teukolski, W. T. Vetterlin, B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge University, Cambridge, England, 1992), pp. 409–413.

Furman, Sh. A.

Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992), pp. 1–149.

Gilks, W. R.

W. R. Gilks, S. Richardson, D. J. Spiegelhalter, Markov-Chain Monte Carlo in Practice (Chapman Hall, London, 1995).

Kalos, M.

M. Kalos, P. Whitlock, Monte Carlo Methods (Wiley, New York, 1986).
[CrossRef]

Kemp, R. A.

Li, D. G.

D. G. Li, A. C. Watson, “Global optimization for optical thin film design using Latin squares,” in Optical Thin Films V: New Developments, R. L. Hall, ed., Proc. SPIE3133, 8–15 (1997).
[CrossRef]

Martin, S.

Premoli, A.

Press, W. H.

W. H. Press, S. A. Teukolski, W. T. Vetterlin, B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge University, Cambridge, England, 1992), pp. 409–413.

Rastello, M. L.

Richardson, S.

W. R. Gilks, S. Richardson, D. J. Spiegelhalter, Markov-Chain Monte Carlo in Practice (Chapman Hall, London, 1995).

Rivory, J.

Schoenauer, M.

Snedaker, C. G.

C. G. Snedaker, “New numerical thin-film synthesis technique,” J. Opt. Soc. Am. 72, 1732A (1982).

Southwell, W. H.

Spiegelhalter, D. J.

W. R. Gilks, S. Richardson, D. J. Spiegelhalter, Markov-Chain Monte Carlo in Practice (Chapman Hall, London, 1995).

Teukolski, S. A.

W. H. Press, S. A. Teukolski, W. T. Vetterlin, B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge University, Cambridge, England, 1992), pp. 409–413.

Tikhonravov, A. V.

Trubetskov, M. K.

Verly, P. G.

Vetterlin, W. T.

W. H. Press, S. A. Teukolski, W. T. Vetterlin, B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge University, Cambridge, England, 1992), pp. 409–413.

Watson, A. C.

D. G. Li, A. C. Watson, “Global optimization for optical thin film design using Latin squares,” in Optical Thin Films V: New Developments, R. L. Hall, ed., Proc. SPIE3133, 8–15 (1997).
[CrossRef]

Whitlock, P.

M. Kalos, P. Whitlock, Monte Carlo Methods (Wiley, New York, 1986).
[CrossRef]

Appl. Opt.

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

A. Premoli, M. L. Rastello, “Minimax refining of wideband antireflection coatings for wide angular indicence,” Appl. Opt. 33, 2018–2024 (1994).
[CrossRef] [PubMed]

J. A. Dobrowolski, “Completely automatic synthesis of optical thin film systems,” Appl. Opt. 4, 937–946 (1965).
[CrossRef]

W. H. Southwell, “Coating design using very thin high- and low-index layers,” Appl. Opt. 24, 457–460 (1985).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, G. W. DeBell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35, 5493–5508 (1996).
[CrossRef] [PubMed]

P. G. Verly, “Fourier transform technique with refinement in the frequency domain for the synthesis of optical thin films,” Appl. Opt. 35, 5148–5154 (1996).
[CrossRef] [PubMed]

P. G. Verly, A. V. Tikhonravov, M. K. Trubetskov, “Efficient refinement algorithm for the synthesis of inhomogeneous optical coatings,” Appl. Opt. 36, 1487–1495 (1997).
[CrossRef] [PubMed]

P. G. Verly, “Optical coating synthesis by simultaneous refractive-index and thickness refinement of inhomogeneous films,” Appl. Opt. 37, 7327–7333 (1998).
[CrossRef]

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

J. Opt. Soc. Am.

P. Baumeister, “Design of multilayer filters by successive approximations,” J. Opt. Soc. Am. 48, 955–958 (1958).
[CrossRef]

C. G. Snedaker, “New numerical thin-film synthesis technique,” J. Opt. Soc. Am. 72, 1732A (1982).

Other

T. Boudet, P. Chaton, “Thin film design using simulated annealing and the study of filter robustness,” in Developments in Optical Component Coatings, I. Reid, ed., Proc. SPIE2776, 27–38 (1996).
[CrossRef]

D. G. Li, A. C. Watson, “Global optimization for optical thin film design using Latin squares,” in Optical Thin Films V: New Developments, R. L. Hall, ed., Proc. SPIE3133, 8–15 (1997).
[CrossRef]

Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992), pp. 1–149.

W. R. Gilks, S. Richardson, D. J. Spiegelhalter, Markov-Chain Monte Carlo in Practice (Chapman Hall, London, 1995).

M. Kalos, P. Whitlock, Monte Carlo Methods (Wiley, New York, 1986).
[CrossRef]

W. H. Press, S. A. Teukolski, W. T. Vetterlin, B. P. Flannery, Numerical Recipes, 2nd ed. (Cambridge University, Cambridge, England, 1992), pp. 409–413.

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Figures (4)

Fig. 1
Fig. 1

System of complex field amplitudes and internal angles of refraction in a multilayer optical coating subject to incoming radiation incident at angle θ0.

Fig. 2
Fig. 2

Transmission line.

Fig. 3
Fig. 3

Reflectivity R(λ) of the optimal wideband antireflection coating (solid curve) as in Table 1: solid circles, target wavelengths; error bars, 68% confidence intervals in reflectivity corresponding to an rms manufacturing error of 1 nm in the layer thicknesses.

Fig. 4
Fig. 4

Reflectivity R(λ) of the optimal wideband dichroic coating (solid curve) as in Table 1: solid circles, target wavelengths; error bars, 68% confidence intervals in reflectivity corresponding to an rms manufacturing error of 1 nm in the layer thicknesses.

Tables (2)

Tables Icon

Table 1 Optimal Wideband Antireflection Coating

Tables Icon

Table 2 Optimal Wideband Dichroic Coating

Equations (14)

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

aj+bj expiϕj=aj-1 expiϕj-1+bj-1,
aj-bj expiϕjnj cos θj=aj-1 expiϕj-1-bj-1nj-1 cos θj-1,
nj sin θj=nj-1 sin θj-1=sin θ0,
aj+bj expiϕjcos θj=aj-1 expiϕj-1+bj-1cos θj-1,
aj-bj expiϕjnj=aj-1 expiϕj-1-bj-1nj-1,
Mx=c,
Δλλ=Δll=λ2l.
Δλλ=2λMλ0.
δλλλ2Mλ0.
fp=k=1K wkRλk; p-R˜λk2,
χ2p=k=1KRkp-R˜kδRk2,
Prp|R˜=PrR˜|pPrpPrR˜,
Cijpi-pˆipj-pˆj,
αp, pn=min1, πpqpn|pπpnqp|pn.

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