Abstract

The antireflection coatings consist of layers whose optical thicknesses are quater waves or multiples thereof. Linear programming is used to minimize the reflectance and at the same time restrain the refractive indices to values that are attainable. Examples are given of coatings for visual optics and for higher refractive index substrates such as lithium niobate.

© 1977 Optical Society of America

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References

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  1. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955), Sec. 7.2.
  2. H. A. Macleod, Thin Film Optical Filters (American Elsevier, New York, 1969), Chap. 3.
  3. S. Knittl, Optics of Thin Films (Wiley, New York, 1976), p. 53.
  4. W. F. Geffcken, “Coatings of at least three layers of differing refractive index on nonmetallic objects for the reduction of their surface reflection” (German Patent No. 758767, 9Nov.1944).
  5. A. Thetford, Opt. Acta 16, 37 (1969).
    [Crossref]
  6. A. Mussett and A. Thelen, “Multilayer antireflection coatings,” in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1970), Vol. 8, pp. 201–237.
    [Crossref]
  7. H. Pohlack, “Synthesis of optical interference coatings with prescribed spectral characteristics,” in Jena Jahrbuch (VEB Carl Zeiss, Jena, German Democratic Republic, 1952) (in German).
  8. P. Kard, Opt. Spektrosc. 2, 236 (1957).
  9. R. Pegis, J. Opt. Soc. Am. 51, 1255 (1961).
    [Crossref]
  10. L. Young, IRE Trans. Microwave Theory Tech. MTT-7, 233 (1959).
    [Crossref]
  11. L. Young, J. Opt. Soc. Am. 51, 967 (1961).
    [Crossref]
  12. P. Kard, Loodus ja Matemaatika 1, 67 (1959) (in Estonian);for English translation, see AIP document No. PAPS JOSA-67-1039 for 23 pages of material entilted “Optical Theory of Anti-Reflection Coatings.” Order by PAPS number and journal reference from American Institute of Physics Physics Auxiliary Publication Service, 335 East 45th Street, New York, N. Y. 10017.The price is $1.50 for each microfiche (98 pages), or $5 for photocopies of up to 30 pages with $0.15 for each additional page over 30 pages. Airmail additional. Make checks payable to the American Institute of Physics. This material also appears in Current Physics Microfilm, the monthly microfilm edition of the complete set of journals published by AIP, on the frames immediately following this journal article.
  13. P. Kard, The Analysis and Synthesis of Multilayer Interference Coatings (Valgus, Tallinn, Estonia, 1971), pp. 79–91 (in Russian).
  14. S. Furman, Opt. Spectrosc. (USSR) 22, 344 (1967).
  15. J. Cox and G. Hass, “Antireflection coatings for optical and infrared optical materials,” in Physics of Thin Films, edited by G. Hass and R. Thun, (Academic, New York1964), Vol. 2.
  16. V. Costich, Laser Focus 5, 41, No. 11 (1969).
  17. N. Uchida, S. Miyazawa, and K. Ninomiya, J. Opt. Soc. Am. 60, 1375 (1970).
    [Crossref]
  18. G. Boyd, W. Bond, and H. Carter, J. Appl. Phys. 38, 1941 (1967).
    [Crossref]
  19. G. Matthaei, L. Young, and E. Jones, Microwave Filters Impedance Matching Networks and Coupling Structures (McGraw-Hill, New York, 1964), Chap. 6.
  20. J. Kelley, J. Soc. Ind. Appl. Math. 6, 15 (1958).
    [Crossref]
  21. R. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York, 1962), p. 225.
  22. Mathematical Programing Systems/360 Version 2, Linear and Separable Programing—User’s Manual, Publication No. GH 20-04-76-2 (IBM, 1133 Westchester Ave., White Plains, N. Y., 1975).
  23. R. Jacobsson, “Inhomogeneous and Coevaporated Homogenous Films for Optical Applications,” in Physics of Thin Films, edited by G. Hass, M. Francombe, and R. Hoffman, (Academic, New York, 1975), Vol. 8, p. 51.
  24. S. Fujiwara, J. Opt. Soc. Am. 53, 880 (1963).
    [Crossref]
  25. S. Fujiwara, J. Opt. Soc. Am. 53, 1317 (1963).
    [Crossref]
  26. J. A. Dobrowolski, Appl. Opt. 4, 937 (1965).
    [Crossref]

1970 (1)

1969 (2)

A. Thetford, Opt. Acta 16, 37 (1969).
[Crossref]

V. Costich, Laser Focus 5, 41, No. 11 (1969).

1967 (2)

S. Furman, Opt. Spectrosc. (USSR) 22, 344 (1967).

G. Boyd, W. Bond, and H. Carter, J. Appl. Phys. 38, 1941 (1967).
[Crossref]

1965 (1)

1963 (2)

1961 (2)

1959 (2)

L. Young, IRE Trans. Microwave Theory Tech. MTT-7, 233 (1959).
[Crossref]

P. Kard, Loodus ja Matemaatika 1, 67 (1959) (in Estonian);for English translation, see AIP document No. PAPS JOSA-67-1039 for 23 pages of material entilted “Optical Theory of Anti-Reflection Coatings.” Order by PAPS number and journal reference from American Institute of Physics Physics Auxiliary Publication Service, 335 East 45th Street, New York, N. Y. 10017.The price is $1.50 for each microfiche (98 pages), or $5 for photocopies of up to 30 pages with $0.15 for each additional page over 30 pages. Airmail additional. Make checks payable to the American Institute of Physics. This material also appears in Current Physics Microfilm, the monthly microfilm edition of the complete set of journals published by AIP, on the frames immediately following this journal article.

1958 (1)

J. Kelley, J. Soc. Ind. Appl. Math. 6, 15 (1958).
[Crossref]

1957 (1)

P. Kard, Opt. Spektrosc. 2, 236 (1957).

Bond, W.

G. Boyd, W. Bond, and H. Carter, J. Appl. Phys. 38, 1941 (1967).
[Crossref]

Boyd, G.

G. Boyd, W. Bond, and H. Carter, J. Appl. Phys. 38, 1941 (1967).
[Crossref]

Carter, H.

G. Boyd, W. Bond, and H. Carter, J. Appl. Phys. 38, 1941 (1967).
[Crossref]

Costich, V.

V. Costich, Laser Focus 5, 41, No. 11 (1969).

Cox, J.

J. Cox and G. Hass, “Antireflection coatings for optical and infrared optical materials,” in Physics of Thin Films, edited by G. Hass and R. Thun, (Academic, New York1964), Vol. 2.

Dobrowolski, J. A.

Fujiwara, S.

Furman, S.

S. Furman, Opt. Spectrosc. (USSR) 22, 344 (1967).

Geffcken, W. F.

W. F. Geffcken, “Coatings of at least three layers of differing refractive index on nonmetallic objects for the reduction of their surface reflection” (German Patent No. 758767, 9Nov.1944).

Hamming, R.

R. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York, 1962), p. 225.

Hass, G.

J. Cox and G. Hass, “Antireflection coatings for optical and infrared optical materials,” in Physics of Thin Films, edited by G. Hass and R. Thun, (Academic, New York1964), Vol. 2.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955), Sec. 7.2.

Jacobsson, R.

R. Jacobsson, “Inhomogeneous and Coevaporated Homogenous Films for Optical Applications,” in Physics of Thin Films, edited by G. Hass, M. Francombe, and R. Hoffman, (Academic, New York, 1975), Vol. 8, p. 51.

Jones, E.

G. Matthaei, L. Young, and E. Jones, Microwave Filters Impedance Matching Networks and Coupling Structures (McGraw-Hill, New York, 1964), Chap. 6.

Kard, P.

P. Kard, Loodus ja Matemaatika 1, 67 (1959) (in Estonian);for English translation, see AIP document No. PAPS JOSA-67-1039 for 23 pages of material entilted “Optical Theory of Anti-Reflection Coatings.” Order by PAPS number and journal reference from American Institute of Physics Physics Auxiliary Publication Service, 335 East 45th Street, New York, N. Y. 10017.The price is $1.50 for each microfiche (98 pages), or $5 for photocopies of up to 30 pages with $0.15 for each additional page over 30 pages. Airmail additional. Make checks payable to the American Institute of Physics. This material also appears in Current Physics Microfilm, the monthly microfilm edition of the complete set of journals published by AIP, on the frames immediately following this journal article.

P. Kard, Opt. Spektrosc. 2, 236 (1957).

P. Kard, The Analysis and Synthesis of Multilayer Interference Coatings (Valgus, Tallinn, Estonia, 1971), pp. 79–91 (in Russian).

Kelley, J.

J. Kelley, J. Soc. Ind. Appl. Math. 6, 15 (1958).
[Crossref]

Knittl, S.

S. Knittl, Optics of Thin Films (Wiley, New York, 1976), p. 53.

Macleod, H. A.

H. A. Macleod, Thin Film Optical Filters (American Elsevier, New York, 1969), Chap. 3.

Matthaei, G.

G. Matthaei, L. Young, and E. Jones, Microwave Filters Impedance Matching Networks and Coupling Structures (McGraw-Hill, New York, 1964), Chap. 6.

Miyazawa, S.

Mussett, A.

A. Mussett and A. Thelen, “Multilayer antireflection coatings,” in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1970), Vol. 8, pp. 201–237.
[Crossref]

Ninomiya, K.

Pegis, R.

Pohlack, H.

H. Pohlack, “Synthesis of optical interference coatings with prescribed spectral characteristics,” in Jena Jahrbuch (VEB Carl Zeiss, Jena, German Democratic Republic, 1952) (in German).

Thelen, A.

A. Mussett and A. Thelen, “Multilayer antireflection coatings,” in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1970), Vol. 8, pp. 201–237.
[Crossref]

Thetford, A.

A. Thetford, Opt. Acta 16, 37 (1969).
[Crossref]

Uchida, N.

Young, L.

L. Young, J. Opt. Soc. Am. 51, 967 (1961).
[Crossref]

L. Young, IRE Trans. Microwave Theory Tech. MTT-7, 233 (1959).
[Crossref]

G. Matthaei, L. Young, and E. Jones, Microwave Filters Impedance Matching Networks and Coupling Structures (McGraw-Hill, New York, 1964), Chap. 6.

Appl. Opt. (1)

IRE Trans. Microwave Theory Tech. (1)

L. Young, IRE Trans. Microwave Theory Tech. MTT-7, 233 (1959).
[Crossref]

J. Appl. Phys. (1)

G. Boyd, W. Bond, and H. Carter, J. Appl. Phys. 38, 1941 (1967).
[Crossref]

J. Opt. Soc. Am. (5)

J. Soc. Ind. Appl. Math. (1)

J. Kelley, J. Soc. Ind. Appl. Math. 6, 15 (1958).
[Crossref]

Laser Focus (1)

V. Costich, Laser Focus 5, 41, No. 11 (1969).

Loodus ja Matemaatika (1)

P. Kard, Loodus ja Matemaatika 1, 67 (1959) (in Estonian);for English translation, see AIP document No. PAPS JOSA-67-1039 for 23 pages of material entilted “Optical Theory of Anti-Reflection Coatings.” Order by PAPS number and journal reference from American Institute of Physics Physics Auxiliary Publication Service, 335 East 45th Street, New York, N. Y. 10017.The price is $1.50 for each microfiche (98 pages), or $5 for photocopies of up to 30 pages with $0.15 for each additional page over 30 pages. Airmail additional. Make checks payable to the American Institute of Physics. This material also appears in Current Physics Microfilm, the monthly microfilm edition of the complete set of journals published by AIP, on the frames immediately following this journal article.

Opt. Acta (1)

A. Thetford, Opt. Acta 16, 37 (1969).
[Crossref]

Opt. Spectrosc. (USSR) (1)

S. Furman, Opt. Spectrosc. (USSR) 22, 344 (1967).

Opt. Spektrosc. (1)

P. Kard, Opt. Spektrosc. 2, 236 (1957).

Other (12)

A. Mussett and A. Thelen, “Multilayer antireflection coatings,” in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1970), Vol. 8, pp. 201–237.
[Crossref]

H. Pohlack, “Synthesis of optical interference coatings with prescribed spectral characteristics,” in Jena Jahrbuch (VEB Carl Zeiss, Jena, German Democratic Republic, 1952) (in German).

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955), Sec. 7.2.

H. A. Macleod, Thin Film Optical Filters (American Elsevier, New York, 1969), Chap. 3.

S. Knittl, Optics of Thin Films (Wiley, New York, 1976), p. 53.

W. F. Geffcken, “Coatings of at least three layers of differing refractive index on nonmetallic objects for the reduction of their surface reflection” (German Patent No. 758767, 9Nov.1944).

J. Cox and G. Hass, “Antireflection coatings for optical and infrared optical materials,” in Physics of Thin Films, edited by G. Hass and R. Thun, (Academic, New York1964), Vol. 2.

P. Kard, The Analysis and Synthesis of Multilayer Interference Coatings (Valgus, Tallinn, Estonia, 1971), pp. 79–91 (in Russian).

R. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York, 1962), p. 225.

Mathematical Programing Systems/360 Version 2, Linear and Separable Programing—User’s Manual, Publication No. GH 20-04-76-2 (IBM, 1133 Westchester Ave., White Plains, N. Y., 1975).

R. Jacobsson, “Inhomogeneous and Coevaporated Homogenous Films for Optical Applications,” in Physics of Thin Films, edited by G. Hass, M. Francombe, and R. Hoffman, (Academic, New York, 1975), Vol. 8, p. 51.

G. Matthaei, L. Young, and E. Jones, Microwave Filters Impedance Matching Networks and Coupling Structures (McGraw-Hill, New York, 1964), Chap. 6.

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Figures (9)

FIG. 1
FIG. 1

Computed reflectance vs frequency of a stack whose design is shown in Table IV. The ordinate changes scale at 0.01 and the abscissa at 0.6 and 1.4.

FIG. 2
FIG. 2

Computed reflectance vs frequency of a stack whose design is shown in Table IV. The ordinate changes scale at 0.01 and the abscissa at 0.6 and 1.4.

FIG. 3
FIG. 3

Computed reflectance vs frequency of a stack whose design is shown in Table IV. The ordinate changes scale at 0.01 and the abscissa at 0.6 and 1.4.

FIG. 4
FIG. 4

Computed reflectance vs frequency of a stack whose design is shown in Table IV. The ordinate changes scale at 0.01 and the abscissa at 0.6 and 1.4.

FIG. 5
FIG. 5

Computed reflectance vs frequency of a stack whose design is shown in Table IV. The ordinate changes scale at 0.01.

FIG. 6
FIG. 6

Computed reflectance vs frequency of a stack whose design is shown in Table IV. The ordinate changes scale at 0.01.

FIG. 7
FIG. 7

Computed reflectance vs frequency of a stack whose design is shown in Table IV. The ordinate changes scale at 0.01.

FIG. 8
FIG. 8

Computed reflectance vs frequency of a stack whose design is shown in Table IV. The ordinate changes scale at 0.01.

FIG. 9
FIG. 9

Computed reflectance vs frequency of a stack whose design is shown in Table IV. The ordinate changes scale at 0.01.

Tables (4)

Tables Icon

TABLE I Coefficients that appear in Eqs. (5) and (6).

Tables Icon

TABLE II Coefficients of ui that appear in Eq. (10). The coefficient of u0 in am is given by the binomial coefficient.

Tables Icon

TABLE III The approximate values of the refractive indices for quasimaximally flat designs, as obtained from setting the ai’s in Table II to zero, for i = l, i = l − 1, and so on. The column R = 0 refers to the radiant reflectance when the phase thickness β is 90°. For the QQQQ design α = 15/16 and γ = 11/16.

Tables Icon

TABLE IV Design of ARC’s. The * means that the reflectance curve is dotted in the appropriate figure. The incident medium is air in all cases and ns is the substrate index. L or U means that the refractive index was at its lower or upper bound, respectively. If the design is produced by LP, λ0i and Wi are the frequencies and relative weights at which Eq. (13) is minimized. The bandwidth B is defined in Eq. (14) for reflectance that is less than 0.5%.

Equations (17)

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

n i = n i + 1 .
n i = n i + 1 = n i + 2 .
β = 90 λ 0 / λ
u i = 1 2 ln ( n i / n s ) .
R / T = ( a 0 C 2 a 2 S 2 ) 2 + ( C S a 1 ) 2
R / T = ( a 0 C 3 a 2 C S 2 ) 2 + ( a 1 C 2 S a 3 S 3 ) 2
R / T = ( t 1 ) 2 + ( t 2 ) 2 ,
t 1 = i = 0 l a i j i S i C l i
t 2 = i = 1 l a i j i 1 S i C l i
a m = i = 0 l b m i u i .
R / T = ( R / T ) cos 21 β ,
R / T = a 0 2 C 6 + a 1 2 C 4 ( 1 C 2 ) .
W i | t 1 | min
W i | t 2 | min
B = λ 1 / λ 2 ,
| t 1 | + | t 2 | = 1.2 ,
0.42 R 0.59.