Abstract

The design of antireflection coatings at any light incidence is a challenging task in optics. To this aim, a minimax method is presented: it minimizes the maximum deviation of the spectral reflectance from the desired specifications over the wavelength for a given set of incidence angles. Refining is limited to lossless coatings with assigned refractive indices and undetermined thicknesses; the algorithm consists of iterating appropriate linear optimization steps. In the examples some minimax-refined coatings are compared with coatings reported in the literature.

© 1994 Optical Society of America

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References

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  1. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1989).
  2. A. Thelen, R. Langfeld, “Coating design context: antireflecting coating for lenses to be used with normal and infrared photographic films,” presented at the International Symposium on Optical Systems Design, Berlin, 14–18 September 1992.
  3. H. Zycha, “Refining algorithm for the design of multilayer filters,” Appl. Opt. 12, 979–983 (1973).
    [Crossref] [PubMed]
  4. P. Baumeister, R. Moore, K. Walsh, “Application of linear programming to antireflection coating design,” J. Opt. Soc. Am. 67, 1039–1045 (1977).
    [Crossref]
  5. A. L. Bloom, “Refining and optimization in multilayers,” Appl. Opt. 20, 66–73 (1981).
    [Crossref] [PubMed]
  6. J. A. Dobrowolski, R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29, 2876–2893 (1990).
    [Crossref] [PubMed]
  7. A. Premoli, M. L. Rastello, “Minimax refining of optical multilayer systems,” Appl. Opt. 31, 1597–1605 (1992).
    [Crossref] [PubMed]
  8. A. Premoli, “Piecewise-linear programming: the compact (CPLP) algorithm,” Math. Programm. 36, 210–227 (1986).
    [Crossref]
  9. O. S. Heavens, H. M. Liddell, “Least squares method for the automatic design of multilayers,” Opt. Acta 15, 129–138 (1968).
  10. J. F. Tang, Q. Zheng, “Automatic design of optical thin-film systems—merit function and numerical optimization method,” J. Opt. Soc. Am. 72, 1522–1528 (1982).
    [Crossref]
  11. J. A. Dobrowolski, F. C. Ho, A. Belkind, V. A. Koss, “Merit functions for more effective thin-film calculations,” Appl. Opt. 28, 2824–2831 (1989).
    [Crossref] [PubMed]
  12. A. Premoli, M. L. Rastello, “Continuation method for synthesizing antireflection coatings,” Appl. Opt. 31, 6741–6746 (1992).
    [Crossref] [PubMed]

1992 (2)

1990 (1)

1989 (1)

1986 (1)

A. Premoli, “Piecewise-linear programming: the compact (CPLP) algorithm,” Math. Programm. 36, 210–227 (1986).
[Crossref]

1982 (1)

1981 (1)

1977 (1)

1973 (1)

1968 (1)

O. S. Heavens, H. M. Liddell, “Least squares method for the automatic design of multilayers,” Opt. Acta 15, 129–138 (1968).

Baumeister, P.

Belkind, A.

Bloom, A. L.

Dobrowolski, J. A.

Heavens, O. S.

O. S. Heavens, H. M. Liddell, “Least squares method for the automatic design of multilayers,” Opt. Acta 15, 129–138 (1968).

Ho, F. C.

Kemp, R. A.

Koss, V. A.

Langfeld, R.

A. Thelen, R. Langfeld, “Coating design context: antireflecting coating for lenses to be used with normal and infrared photographic films,” presented at the International Symposium on Optical Systems Design, Berlin, 14–18 September 1992.

Liddell, H. M.

O. S. Heavens, H. M. Liddell, “Least squares method for the automatic design of multilayers,” Opt. Acta 15, 129–138 (1968).

Moore, R.

Premoli, A.

Rastello, M. L.

Tang, J. F.

Thelen, A.

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1989).

A. Thelen, R. Langfeld, “Coating design context: antireflecting coating for lenses to be used with normal and infrared photographic films,” presented at the International Symposium on Optical Systems Design, Berlin, 14–18 September 1992.

Walsh, K.

Zheng, Q.

Zycha, H.

Appl. Opt. (6)

J. Opt. Soc. Am. (2)

Math. Programm. (1)

A. Premoli, “Piecewise-linear programming: the compact (CPLP) algorithm,” Math. Programm. 36, 210–227 (1986).
[Crossref]

Opt. Acta (1)

O. S. Heavens, H. M. Liddell, “Least squares method for the automatic design of multilayers,” Opt. Acta 15, 129–138 (1968).

Other (2)

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1989).

A. Thelen, R. Langfeld, “Coating design context: antireflecting coating for lenses to be used with normal and infrared photographic films,” presented at the International Symposium on Optical Systems Design, Berlin, 14–18 September 1992.

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Figures (6)

Fig. 1
Fig. 1

Spectral reflectances for several incidence angles from 0° to 30° of a typical visible–infrared antireflection coating. Solid curves denote the reflectances for 0° and 30°.

Fig. 2
Fig. 2

Spectral reflectances for ϕ = 0° (solid curve) and ϕ = 30° (dashed curve) of (a) the DOCT1 design (M = 9, designer Dick) in Ref. 2, and (b) the design refined according to the method proposed here.

Fig. 3
Fig. 3

Spectral reflectances for ϕ = 0° (solid curve) and ϕ = 30° (dashed curve) of (a) the HACK3 design (M = 8, designer Hacker) in Ref. 2, and (b) the design refined according to the method proposed here.

Fig. 4
Fig. 4

Spectral reflectances for ϕ = 0° (solid curve) and ϕ = 30° (dashed curve) of (a) the HACK4 design (M = 8, designer Hacker) in Ref. 2, and (b) the design refined according to the method proposed here.

Fig. 5
Fig. 5

Spectral reflectances for ϕ = 0° (solid curve) and ϕ = 30° (dashed curve) of (a) the IEN2 design (M = 13, designers Rastello and Premoli) in Ref. 2, and (b) the design refined according to the method proposed here.

Fig. 6
Fig. 6

Spectral reflectances for ϕ = 0° (solid curve), 30° (dashed curve), and 40° (dotted curve) of (a) the IEN2 design (M = 13, designers Rastello and Premoli) in Ref. 2 and (b) the design refined according to the method proposed here, forcing R(λ, ϕ, d) for ϕa = 0° and ϕb = 40°.

Tables (3)

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Table 1 Refractive Indices n of the Coating Materials

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Table 2 Physical Thicknesses and Characteristics of Starting Designs DOCT1, HACK3, HACK4, and Related Minimax-Refined Designs

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Table 3 Physical Thicknesses and Characteristics of the Starting Design IEN2 and the Minimax-Refined Design in Figs. 5 and 6

Equations (16)

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R ( λ , ϕ , d ) = γ R s ( λ , ϕ , d ) + ( 1 γ ) R p ( λ , ϕ , d ) , with 0 γ 1 .
L 2 ( d ) = 1 ( λ b λ a ) ( ϕ b ϕ a ) × ϕ a ϕ b { λ a λ b [ R ( λ , ϕ , d ) R ˆ ( λ , ϕ ) ] 2 d λ } d ϕ .
L ( d ) = max ( λ, ϕ ) [ λ a , λ b ] × [ ϕ a , ϕ b ] [ | R ( λ , ϕ , d ) R ˆ ( λ , ϕ ) | ] .
b U ( λ , ϕ , p ) R ( λ , ϕ , d ) b L ( λ , ϕ , p ) ,
b U ( λ , ϕ , p ) = p b 1 U ( λ , ϕ ) + ( 1 p ) b 0 U ( λ , ϕ ) ,
b L ( λ , ϕ , p ) = p b 1 L ( λ , ϕ ) + ( 1 p ) b 0 L ( λ , ϕ ) .
b 1 U ( λ , ϕ ) > b 0 U ( λ , ϕ ) b 0 L ( λ , ϕ ) > b 1 L ( λ , ϕ ) , ( λ , ϕ ) [ λ a , λ b ] × [ ϕ a , ϕ b ] .
b 0 U ( λ , ϕ ) = b 0 L ( λ , ϕ ) = b 1 L ( λ , ϕ ) = 0 ,
Δ R ˆ p b 1 U ( λ , ϕ ) .
minimize d , Δ R ˆ Δ R ˆ ,
R ( λ , ϕ , d ) Δ R ˆ , ( λ , ϕ ) [ λ a , λ b ] × [ ϕ a , ϕ b ] .
d d m d + ( m = 1 , 2 , , M )
m = 1 M d m d tot ,
m = 1 M d m ζ mat , m d mat tot ,
R max , ϕ = max λ [ λ a , λ b ] [ R ( λ , ϕ , d ) ] 0 . 01 for ϕ = 0 and ϕ = 30 .
F = { 1 2 ( λ b λ a ) × λ a λ b [ ( R ϕ = 0 ) 2 + 1 2 ( R s , ϕ = 30 ) 2 + 1 2 ( R p , ϕ = 30 ) 2 ] Δ λ } 1 / 2 ,

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