Abstract

The role of the refractive index in the design of optical multilayer coatings is discussed. The theory of the Herpin equivalent-index concept is reviewed. New results are presented for an extension of this concept first proposed by Ikeda. A computer program is described which automatically transforms a multilayer design utilizing many refractive indices into a system based on the use of two materials only. Experimental results are given to illustrate this. It is shown how, by the use of Herpin equivalent indices, computation time during thin film synthesis can be drastically reduced.

© 1982 Optical Society of America

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References

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  1. S. Fujiwara, J. Opt. Soc. Am. 53, 880 (1963).
    [CrossRef]
  2. W. Geffcken, U.S. Patent3,147,132, 1Sept.1964.
  3. F. Stetter, R. Esselborn, N. Harder, M. Friz, P. Tolles, Appl. Opt. 15, 2315 (1976).
    [CrossRef] [PubMed]
  4. S. Wei-ren, Opt. Spectra 14, 57, (July1980).
  5. P. Bourgeois, P. Moch, Le Vide 20, No. 119, 376 (1965).
  6. J. Wales, P. W. Black, Thin Solid Films 7, 325 (1971).
    [CrossRef]
  7. M. Peršin, A. Peršin, B. Čelustka, B. Etlinger, Thin Solid Films 11, 153 (1972).
    [CrossRef]
  8. R. Jacobsson, J.-O. Martensson, Jpn. J. Appl. Phys. Suppl. I 4, 333 (1965).
  9. P. W. Baumeister, G. Borak, L. Stensland, in Optical Instruments and Techniques 1969, J. H. Dickson, Ed. (Oriel, Newcastle-upon-Tyne, England, 1970), p. 147.
  10. J. A. Dobrowolski, A. J. Waldorf, J. Opt. Soc. Am. 60, 725 (1970);Proc. Soc. Photo-Opt. Instrumen. Eng. 140, 102 (1978).
  11. V. N. Yadava, S. K. Sharma, K. L. Chopra, Thin Solid Films 17, 243 (1973).
    [CrossRef]
  12. A. Herpin, C.R. Acad. Sci. 225, 182 (1947).
  13. L. I. Epstein, J. Opt. Soc. Am. 42, 806 (1952).
    [CrossRef]
  14. A. Thelen, in Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, New York, 1969), p. 47.
  15. H. A. Macleod, Thin-Film Optical Filters (American Elsevier, New York, 1969).
  16. P. H. Berning, J. Opt. Soc. Am. 52, 431 (1962).
    [CrossRef]
  17. C. Ufford, P. W. Baumeister, J. Opt. Soc. Am. 64, 329 (1974).
    [CrossRef]
  18. Z. Knittl, Optics of Thin Films (Wiley, London, 1976).
  19. H. Ikeda, H. Akasaka, Z. Wakimoto, in Space Optics, B. J. Thompson, R. R. Shannon, Eds. (National Academy of Sciences, Washington, D.C., 1974), p. 554.
  20. H. Ikeda, U.S. Patent3,799,653, 26Mar.1974.
  21. H. Ikeda, H. Akasaka, U.S. Patent3,936,136, 3Feb.1976.
  22. H. Zycha, Appl. Opt. 12, 979 (1973).
    [CrossRef] [PubMed]
  23. J. A. Dobrowolski, Appl. Opt. 4, 937, (1965);Appl. Opt.20, 74 (1981).
    [CrossRef] [PubMed]
  24. J. A. Dobrowolski, Appl. Opt. 12, 1885 (1973).
    [CrossRef] [PubMed]
  25. M. C. Ohmer, J. Opt. Soc. Am. 68, 137 (1978).
    [CrossRef]
  26. J. A. Dobrowolski, Appl. Opt. 9, 1396 (1970).
    [CrossRef] [PubMed]
  27. P. Berning, in Physics of Thin Films, G. Hass, Ed. (Academic, New York, 1963), p. 69.
  28. P. H. Berning, A. F. Turner, J. Opt. Soc. Am. 47, 230 (1957).
    [CrossRef]
  29. J. A. Dobrowolski, Thin Solid Films 34, 313 (1976).
    [CrossRef]

1980 (1)

S. Wei-ren, Opt. Spectra 14, 57, (July1980).

1978 (1)

1976 (2)

1974 (1)

1973 (3)

1972 (1)

M. Peršin, A. Peršin, B. Čelustka, B. Etlinger, Thin Solid Films 11, 153 (1972).
[CrossRef]

1971 (1)

J. Wales, P. W. Black, Thin Solid Films 7, 325 (1971).
[CrossRef]

1970 (2)

J. A. Dobrowolski, A. J. Waldorf, J. Opt. Soc. Am. 60, 725 (1970);Proc. Soc. Photo-Opt. Instrumen. Eng. 140, 102 (1978).

J. A. Dobrowolski, Appl. Opt. 9, 1396 (1970).
[CrossRef] [PubMed]

1965 (3)

J. A. Dobrowolski, Appl. Opt. 4, 937, (1965);Appl. Opt.20, 74 (1981).
[CrossRef] [PubMed]

R. Jacobsson, J.-O. Martensson, Jpn. J. Appl. Phys. Suppl. I 4, 333 (1965).

P. Bourgeois, P. Moch, Le Vide 20, No. 119, 376 (1965).

1963 (1)

1962 (1)

1957 (1)

1952 (1)

1947 (1)

A. Herpin, C.R. Acad. Sci. 225, 182 (1947).

Akasaka, H.

H. Ikeda, H. Akasaka, U.S. Patent3,936,136, 3Feb.1976.

H. Ikeda, H. Akasaka, Z. Wakimoto, in Space Optics, B. J. Thompson, R. R. Shannon, Eds. (National Academy of Sciences, Washington, D.C., 1974), p. 554.

Baumeister, P. W.

C. Ufford, P. W. Baumeister, J. Opt. Soc. Am. 64, 329 (1974).
[CrossRef]

P. W. Baumeister, G. Borak, L. Stensland, in Optical Instruments and Techniques 1969, J. H. Dickson, Ed. (Oriel, Newcastle-upon-Tyne, England, 1970), p. 147.

Berning, P.

P. Berning, in Physics of Thin Films, G. Hass, Ed. (Academic, New York, 1963), p. 69.

Berning, P. H.

Black, P. W.

J. Wales, P. W. Black, Thin Solid Films 7, 325 (1971).
[CrossRef]

Borak, G.

P. W. Baumeister, G. Borak, L. Stensland, in Optical Instruments and Techniques 1969, J. H. Dickson, Ed. (Oriel, Newcastle-upon-Tyne, England, 1970), p. 147.

Bourgeois, P.

P. Bourgeois, P. Moch, Le Vide 20, No. 119, 376 (1965).

Celustka, B.

M. Peršin, A. Peršin, B. Čelustka, B. Etlinger, Thin Solid Films 11, 153 (1972).
[CrossRef]

Chopra, K. L.

V. N. Yadava, S. K. Sharma, K. L. Chopra, Thin Solid Films 17, 243 (1973).
[CrossRef]

Dobrowolski, J. A.

J. A. Dobrowolski, Thin Solid Films 34, 313 (1976).
[CrossRef]

J. A. Dobrowolski, Appl. Opt. 12, 1885 (1973).
[CrossRef] [PubMed]

J. A. Dobrowolski, A. J. Waldorf, J. Opt. Soc. Am. 60, 725 (1970);Proc. Soc. Photo-Opt. Instrumen. Eng. 140, 102 (1978).

J. A. Dobrowolski, Appl. Opt. 9, 1396 (1970).
[CrossRef] [PubMed]

J. A. Dobrowolski, Appl. Opt. 4, 937, (1965);Appl. Opt.20, 74 (1981).
[CrossRef] [PubMed]

Epstein, L. I.

Esselborn, R.

Etlinger, B.

M. Peršin, A. Peršin, B. Čelustka, B. Etlinger, Thin Solid Films 11, 153 (1972).
[CrossRef]

Friz, M.

Fujiwara, S.

Geffcken, W.

W. Geffcken, U.S. Patent3,147,132, 1Sept.1964.

Harder, N.

Herpin, A.

A. Herpin, C.R. Acad. Sci. 225, 182 (1947).

Ikeda, H.

H. Ikeda, U.S. Patent3,799,653, 26Mar.1974.

H. Ikeda, H. Akasaka, U.S. Patent3,936,136, 3Feb.1976.

H. Ikeda, H. Akasaka, Z. Wakimoto, in Space Optics, B. J. Thompson, R. R. Shannon, Eds. (National Academy of Sciences, Washington, D.C., 1974), p. 554.

Jacobsson, R.

R. Jacobsson, J.-O. Martensson, Jpn. J. Appl. Phys. Suppl. I 4, 333 (1965).

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, London, 1976).

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (American Elsevier, New York, 1969).

Martensson, J.-O.

R. Jacobsson, J.-O. Martensson, Jpn. J. Appl. Phys. Suppl. I 4, 333 (1965).

Moch, P.

P. Bourgeois, P. Moch, Le Vide 20, No. 119, 376 (1965).

Ohmer, M. C.

Peršin, A.

M. Peršin, A. Peršin, B. Čelustka, B. Etlinger, Thin Solid Films 11, 153 (1972).
[CrossRef]

Peršin, M.

M. Peršin, A. Peršin, B. Čelustka, B. Etlinger, Thin Solid Films 11, 153 (1972).
[CrossRef]

Sharma, S. K.

V. N. Yadava, S. K. Sharma, K. L. Chopra, Thin Solid Films 17, 243 (1973).
[CrossRef]

Stensland, L.

P. W. Baumeister, G. Borak, L. Stensland, in Optical Instruments and Techniques 1969, J. H. Dickson, Ed. (Oriel, Newcastle-upon-Tyne, England, 1970), p. 147.

Stetter, F.

Thelen, A.

A. Thelen, in Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, New York, 1969), p. 47.

Tolles, P.

Turner, A. F.

Ufford, C.

Wakimoto, Z.

H. Ikeda, H. Akasaka, Z. Wakimoto, in Space Optics, B. J. Thompson, R. R. Shannon, Eds. (National Academy of Sciences, Washington, D.C., 1974), p. 554.

Waldorf, A. J.

J. A. Dobrowolski, A. J. Waldorf, J. Opt. Soc. Am. 60, 725 (1970);Proc. Soc. Photo-Opt. Instrumen. Eng. 140, 102 (1978).

Wales, J.

J. Wales, P. W. Black, Thin Solid Films 7, 325 (1971).
[CrossRef]

Wei-ren, S.

S. Wei-ren, Opt. Spectra 14, 57, (July1980).

Yadava, V. N.

V. N. Yadava, S. K. Sharma, K. L. Chopra, Thin Solid Films 17, 243 (1973).
[CrossRef]

Zycha, H.

Appl. Opt. (5)

C.R. Acad. Sci. (1)

A. Herpin, C.R. Acad. Sci. 225, 182 (1947).

J. Opt. Soc. Am. (7)

Jpn. J. Appl. Phys. Suppl. I (1)

R. Jacobsson, J.-O. Martensson, Jpn. J. Appl. Phys. Suppl. I 4, 333 (1965).

Le Vide (1)

P. Bourgeois, P. Moch, Le Vide 20, No. 119, 376 (1965).

Opt. Spectra (1)

S. Wei-ren, Opt. Spectra 14, 57, (July1980).

Thin Solid Films (4)

J. A. Dobrowolski, Thin Solid Films 34, 313 (1976).
[CrossRef]

V. N. Yadava, S. K. Sharma, K. L. Chopra, Thin Solid Films 17, 243 (1973).
[CrossRef]

J. Wales, P. W. Black, Thin Solid Films 7, 325 (1971).
[CrossRef]

M. Peršin, A. Peršin, B. Čelustka, B. Etlinger, Thin Solid Films 11, 153 (1972).
[CrossRef]

Other (9)

P. W. Baumeister, G. Borak, L. Stensland, in Optical Instruments and Techniques 1969, J. H. Dickson, Ed. (Oriel, Newcastle-upon-Tyne, England, 1970), p. 147.

W. Geffcken, U.S. Patent3,147,132, 1Sept.1964.

A. Thelen, in Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, New York, 1969), p. 47.

H. A. Macleod, Thin-Film Optical Filters (American Elsevier, New York, 1969).

Z. Knittl, Optics of Thin Films (Wiley, London, 1976).

H. Ikeda, H. Akasaka, Z. Wakimoto, in Space Optics, B. J. Thompson, R. R. Shannon, Eds. (National Academy of Sciences, Washington, D.C., 1974), p. 554.

H. Ikeda, U.S. Patent3,799,653, 26Mar.1974.

H. Ikeda, H. Akasaka, U.S. Patent3,936,136, 3Feb.1976.

P. Berning, in Physics of Thin Films, G. Hass, Ed. (Academic, New York, 1963), p. 69.

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Figures (12)

Fig. 1
Fig. 1

Maximum and minimum values of refractive indices and their ratios available for the manufacture of multilayer coatings in different wavelength regions.

Fig. 2
Fig. 2

Schematic construction of three-layer symmetric and quasi-symmetric Herpin equivalent-index systems.

Fig. 3
Fig. 3

Variation with relative wave number of the equivalent phase thicknesses and of the ratios of the equivalent refractive indices to the index of the outside layers of symmetric (full curves) and quasi-symmetric systems of the first (broken curves) and second kind (dotted curves) for a system for which np/nq = 0.6 and 2ϕp/ϕq = 1.0.

Fig. 4
Fig. 4

Change in Herpin equivalent index of a quasi-symmetric system of the first kind with 2ϕp/ϕq for different values of λ0/λ (see text).

Fig. 5
Fig. 5

Block diagram of the flow of calculations in program herpin.

Fig. 6
Fig. 6

Spectral transmittance curves for multilayers of Table I. Crosses represent experimental results.

Fig. 7
Fig. 7

Spectral transmittance curves for multilayers of Table II. Crosses represent experimental results.

Fig. 8
Fig. 8

Calculated spectral transmittance curves for multilayers of Table III.

Fig. 9
Fig. 9

Calculated spectral transmittance curves for multilayers of Table IV.

Fig. 10
Fig. 10

Design of a Fabry-Perot interference filter using Herpin equivalent-index films.

Fig. 11
Fig. 11

Design of an induced transmission filter using Herpin equivalent-index films.

Fig. 12
Fig. 12

Design of a wide-angle antireflection coating for λ = 0.6328 μm using Herpin equivalent-index films.

Tables (4)

Tables Icon

Table I Optical Thicknesses and Refractive Indices for Multilayers AC

Tables Icon

Table II Optical Thicknesses and Refractive Indices for Multilayers DE

Tables Icon

Table III Optical Thicknesses and Refractive Indices for Multilayers FH

Tables Icon

Table IV Optical Thicknesses and Refractive Indices for Multilayers IJ

Equations (12)

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( M 11 i M 12 i M 21 M 22 ) = j = 1 3 ( cos ϕ j i n j sin ϕ j i n j sin ϕ j cos ϕ j ) .
M 11 = a 1 + a 2 ; M 12 = b 1 + b 2 , M 21 = c 1 + c 2 ; M 22 = d 1 + d 2 ,
a 1 = cos ϕ 1 ( cos ϕ 2 cos ϕ 3 n 3 n 2 sin ϕ 2 sin ϕ 3 ) , a 2 = sin ϕ 1 n 1 ( n 2 sin ϕ 2 cos ϕ 3 + n 3 cos ϕ 2 sin ϕ 3 ) , b 1 = sin ϕ 1 n 1 ( cos ϕ 2 cos ϕ 3 n 2 n 3 sin ϕ 2 sin ϕ 3 ) , b 2 = cos ϕ 1 ( 1 n 3 cos ϕ 2 sin ϕ 3 + 1 n 2 sin ϕ 2 cos 3 ) , c 1 = n 1 sin ϕ 1 ( cos ϕ 2 cos ϕ 3 n 3 n 2 sin ϕ 2 sin ϕ 3 ) , c 2 = cos ϕ 1 ( n 2 sin ϕ 2 cos ϕ 3 + n 3 cos ϕ 2 sin ϕ 3 ) , d 1 = cos ϕ 1 ( cos ϕ 2 cos ϕ 3 n 2 n 3 sin ϕ 2 sin ϕ 3 ) , d 2 = n 1 sin ϕ 1 ( 1 n 3 cos ϕ 2 sin ϕ 3 + 1 n 2 sin ϕ 2 cos ϕ 3 ) .
N = ( M 21 M 12 ) 1 / 2 , cos Γ = ( M 11 × M 22 ) 1 / 2 .
Γ 2 ϕ p + ϕ q
N n p ( 2 ϕ p ϕ q + n q n p 2 ϕ p ϕ q + n p n q ) 1 / 2 , Γ ( 2 ϕ p + ϕ q ) [ 1 + ( n p n q ) 2 × 2 ϕ p ϕ q n p n q ( 1 + 2 ϕ p ϕ q ) 2 ] 1 / 2 .
N 1 * = [ M 21 + c 1 δ n n p M 12 b 1 δ n n p ] 1 / 2 N [ 1 + ½ c 1 M 21 + b 1 M 12 δ n n p ] , cos Γ 1 * = [ ( M 11 + d 2 δ n n p ) ( M 22 + a 2 δ n n p ) ] 1 / 2 cos Γ [ 1 + ½ ( d 2 M 11 + a 2 M 22 ) δ n n p ] .
N 2 * = N ( sin Γ cos δ ϕ + n p N cos Γ sin δ ϕ sin Γ cos δ ϕ + N n p cos Γ sin δ ϕ ) 1 / 2 , cos Γ 2 * = [ ( cos Γ cos δ ϕ n p N sin Γ sin δ ϕ ) × ( cos Γ cos δ ϕ N n p sin Γ sin δ ϕ ) ] 1 / 2 .
sin ϕ q = ( n p / N N / n p n p / n q n q / n p ) sin Γ , sin ( 2 ϕ p + α ) = c r for a = r cos α , b = r sin α ,
a = ½ ( n q / n p + n p / n q ) sin ϕ q , b = cos ϕ q , c = cos Γ , tan α = b a .
M . F . = 1 m i = 1 m ( T i 0 T i δ T ) 2 .
N = n A ( n A n B ) k .

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