Abstract

The synthesis (as opposed to analysis) of multilayer dielectric films as antireflection coatings over any specified frequency band has not been attempted before. The similarity with transmission lines, which can be synthesized as multisection quarter-wave transformers to minimize reflection over any prescribed frequency band, is utilized by applying recently developed synthesis procedures for quarter-wave transformers to the synthesis of multiple antireflection films. [ See O. S. Heavens, Optical Properties of Thin Solid Films ( Academic Press, Inc., New York, 1955); and P. J. Leurgans, J. Opt. Soc. Am. 41, 714 ( 1951).] A brief review of network synthesis is given, leading up to the synthesis of quarter-wave transformers and multilayer films. Numerical tables are presented in this paper from which antireflection coatings of up to four layers can be designed by interpolation. The design procedure (synthesis) described herein leads to the best possible antireflection films, but demands precisely controlled materials with certain refractive indices. It is hoped that this paper will help to promote the development of such materials, and will encourage opticists to try experimentally for the optimum performances possible in theory.

© 1961 Optical Society of America

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References

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  1. O. S. Heavens, Optical Properties of Thin Solid Films (Academic Press, Inc., New York, 1955).
  2. P. J. Leurgans, J. Opt. Soc. Am. 41, 714 (1951).
    [Crossref]
  3. A. F. Turner, J. phys. radium 11, 444 (1950).
    [Crossref]
  4. P. Baumeister, J. Opt. Soc. Am. 48, 955 (1958).
    [Crossref]
  5. W. W. Weinstein (now T. Welford), Vacuum 4, 3 (1954).
    [Crossref]
  6. H. J. Riblet, I. R. E. Trans. on Microwave Theory Tech. MTT-8, 169 (1960).
    [Crossref]
  7. J. E. Eaton, I. R. E. Trans. on Microwave Theory Tech. MTT-8, 171 (1960).
    [Crossref]
  8. J. T. Cox, G. Hass, and R. F. Rowntree, Vacuum 4, 445 (1954).
    [Crossref]
  9. P. H. Berning, J. Opt. Soc. Am. 47, 121 (A) (1957).
    [Crossref]
  10. W. F. Koehler, J. Opt. Soc. Am. 46, 370 (A) (1956).
  11. Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-7, 99 (1959).
    [Crossref]
  12. M. Iwata, Sci. Light (Tokyo) 2, 116 (1953).
  13. M. Iwata, S. Katsube, and T. Fukuda, Sci. Light (Tokyo) 7, 33 (1958).
  14. P. H. Lissberger, J. Opt. Soc. Am. 49, 121 (1959).
    [Crossref]
  15. P. H. Lissberger and W. L. Wilcock, J. Opt. Soc. Am. 49, 126 (1959).
    [Crossref]
  16. O. Brune, J. Math. and Phys. 10, 191 (1931).
  17. Leo Young, I. R. E. Trans. on Circuit Theory CT-7, 247 (1960).
    [Crossref]
  18. S. Darlington, J. Math. and Phys. 18, 257 (1939).
  19. M. E. Van Valkenburg, Introduction to Modern Network Synthesis, (John Wiley & Sons, Inc., New York, 1960).
  20. E. A. Guillemin, Synthesis of Passive Networks (John Wiley & Sons, Inc., New York, 1957).
  21. D. F. Tuttle, Network Synthesis (John Wiley & Sons, Inc., New York, 1958), Vol. I.
  22. P. I. Richards, Proc. I. R. E. 36, 217 (1948).
    [Crossref]
  23. H. J. Riblet, I. R. E. Trans. on Microwave Theory Tech. MTT-5, 36 (1957).
    [Crossref]
  24. H. Ozaki, I. R. E. Trans. on Microwave Theory Tech. MTT-6, 331 (1958).
    [Crossref]
  25. H. J. Riblet, I. R. E. Trans. on Microwave Theory Tech. MTT-7, 297 (1959).
    [Crossref]
  26. Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-7, 477 (1959).
    [Crossref]
  27. Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-8, 645 (1960).
    [Crossref]
  28. Leo Young, Quart. Appl. Math. 19, 25 (1961).
  29. P. I. Richards, Duke Math. J. 14, 777 (1947).
    [Crossref]
  30. Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-7, 233 (1959), and I. R. E. Trans. on Microwave Theory Tech. MTT-8, 243 (1960).
    [Crossref]
  31. Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-8, 483 (1960).
    [Crossref]

1961 (1)

Leo Young, Quart. Appl. Math. 19, 25 (1961).

1960 (5)

Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-8, 645 (1960).
[Crossref]

Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-8, 483 (1960).
[Crossref]

H. J. Riblet, I. R. E. Trans. on Microwave Theory Tech. MTT-8, 169 (1960).
[Crossref]

J. E. Eaton, I. R. E. Trans. on Microwave Theory Tech. MTT-8, 171 (1960).
[Crossref]

Leo Young, I. R. E. Trans. on Circuit Theory CT-7, 247 (1960).
[Crossref]

1959 (6)

Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-7, 99 (1959).
[Crossref]

P. H. Lissberger, J. Opt. Soc. Am. 49, 121 (1959).
[Crossref]

P. H. Lissberger and W. L. Wilcock, J. Opt. Soc. Am. 49, 126 (1959).
[Crossref]

Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-7, 233 (1959), and I. R. E. Trans. on Microwave Theory Tech. MTT-8, 243 (1960).
[Crossref]

H. J. Riblet, I. R. E. Trans. on Microwave Theory Tech. MTT-7, 297 (1959).
[Crossref]

Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-7, 477 (1959).
[Crossref]

1958 (3)

H. Ozaki, I. R. E. Trans. on Microwave Theory Tech. MTT-6, 331 (1958).
[Crossref]

M. Iwata, S. Katsube, and T. Fukuda, Sci. Light (Tokyo) 7, 33 (1958).

P. Baumeister, J. Opt. Soc. Am. 48, 955 (1958).
[Crossref]

1957 (2)

P. H. Berning, J. Opt. Soc. Am. 47, 121 (A) (1957).
[Crossref]

H. J. Riblet, I. R. E. Trans. on Microwave Theory Tech. MTT-5, 36 (1957).
[Crossref]

1956 (1)

W. F. Koehler, J. Opt. Soc. Am. 46, 370 (A) (1956).

1954 (2)

J. T. Cox, G. Hass, and R. F. Rowntree, Vacuum 4, 445 (1954).
[Crossref]

W. W. Weinstein (now T. Welford), Vacuum 4, 3 (1954).
[Crossref]

W. W. Weinstein (now T. Welford), Vacuum 4, 3 (1954).
[Crossref]

1953 (1)

M. Iwata, Sci. Light (Tokyo) 2, 116 (1953).

1951 (1)

1950 (1)

A. F. Turner, J. phys. radium 11, 444 (1950).
[Crossref]

1948 (1)

P. I. Richards, Proc. I. R. E. 36, 217 (1948).
[Crossref]

1947 (1)

P. I. Richards, Duke Math. J. 14, 777 (1947).
[Crossref]

1939 (1)

S. Darlington, J. Math. and Phys. 18, 257 (1939).

1931 (1)

O. Brune, J. Math. and Phys. 10, 191 (1931).

Baumeister, P.

Berning, P. H.

P. H. Berning, J. Opt. Soc. Am. 47, 121 (A) (1957).
[Crossref]

Brune, O.

O. Brune, J. Math. and Phys. 10, 191 (1931).

Cox, J. T.

J. T. Cox, G. Hass, and R. F. Rowntree, Vacuum 4, 445 (1954).
[Crossref]

Darlington, S.

S. Darlington, J. Math. and Phys. 18, 257 (1939).

Eaton, J. E.

J. E. Eaton, I. R. E. Trans. on Microwave Theory Tech. MTT-8, 171 (1960).
[Crossref]

Fukuda, T.

M. Iwata, S. Katsube, and T. Fukuda, Sci. Light (Tokyo) 7, 33 (1958).

Guillemin, E. A.

E. A. Guillemin, Synthesis of Passive Networks (John Wiley & Sons, Inc., New York, 1957).

Hass, G.

J. T. Cox, G. Hass, and R. F. Rowntree, Vacuum 4, 445 (1954).
[Crossref]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Academic Press, Inc., New York, 1955).

Iwata, M.

M. Iwata, S. Katsube, and T. Fukuda, Sci. Light (Tokyo) 7, 33 (1958).

M. Iwata, Sci. Light (Tokyo) 2, 116 (1953).

Katsube, S.

M. Iwata, S. Katsube, and T. Fukuda, Sci. Light (Tokyo) 7, 33 (1958).

Koehler, W. F.

W. F. Koehler, J. Opt. Soc. Am. 46, 370 (A) (1956).

Leurgans, P. J.

Lissberger, P. H.

Ozaki, H.

H. Ozaki, I. R. E. Trans. on Microwave Theory Tech. MTT-6, 331 (1958).
[Crossref]

Riblet, H. J.

H. J. Riblet, I. R. E. Trans. on Microwave Theory Tech. MTT-8, 169 (1960).
[Crossref]

H. J. Riblet, I. R. E. Trans. on Microwave Theory Tech. MTT-7, 297 (1959).
[Crossref]

H. J. Riblet, I. R. E. Trans. on Microwave Theory Tech. MTT-5, 36 (1957).
[Crossref]

Richards, P. I.

P. I. Richards, Proc. I. R. E. 36, 217 (1948).
[Crossref]

P. I. Richards, Duke Math. J. 14, 777 (1947).
[Crossref]

Rowntree, R. F.

J. T. Cox, G. Hass, and R. F. Rowntree, Vacuum 4, 445 (1954).
[Crossref]

Turner, A. F.

A. F. Turner, J. phys. radium 11, 444 (1950).
[Crossref]

Tuttle, D. F.

D. F. Tuttle, Network Synthesis (John Wiley & Sons, Inc., New York, 1958), Vol. I.

Van Valkenburg, M. E.

M. E. Van Valkenburg, Introduction to Modern Network Synthesis, (John Wiley & Sons, Inc., New York, 1960).

Weinstein, W. W.

W. W. Weinstein (now T. Welford), Vacuum 4, 3 (1954).
[Crossref]

Welford, T.

W. W. Weinstein (now T. Welford), Vacuum 4, 3 (1954).
[Crossref]

Wilcock, W. L.

Young, Leo

Leo Young, Quart. Appl. Math. 19, 25 (1961).

Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-8, 645 (1960).
[Crossref]

Leo Young, I. R. E. Trans. on Circuit Theory CT-7, 247 (1960).
[Crossref]

Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-8, 483 (1960).
[Crossref]

Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-7, 99 (1959).
[Crossref]

Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-7, 477 (1959).
[Crossref]

Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-7, 233 (1959), and I. R. E. Trans. on Microwave Theory Tech. MTT-8, 243 (1960).
[Crossref]

Duke Math. J. (1)

P. I. Richards, Duke Math. J. 14, 777 (1947).
[Crossref]

I. R. E. Trans. on Circuit Theory (1)

Leo Young, I. R. E. Trans. on Circuit Theory CT-7, 247 (1960).
[Crossref]

I. R. E. Trans. on Microwave Theory Tech. (10)

H. J. Riblet, I. R. E. Trans. on Microwave Theory Tech. MTT-5, 36 (1957).
[Crossref]

H. Ozaki, I. R. E. Trans. on Microwave Theory Tech. MTT-6, 331 (1958).
[Crossref]

H. J. Riblet, I. R. E. Trans. on Microwave Theory Tech. MTT-7, 297 (1959).
[Crossref]

Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-7, 477 (1959).
[Crossref]

Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-8, 645 (1960).
[Crossref]

Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-7, 233 (1959), and I. R. E. Trans. on Microwave Theory Tech. MTT-8, 243 (1960).
[Crossref]

Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-8, 483 (1960).
[Crossref]

H. J. Riblet, I. R. E. Trans. on Microwave Theory Tech. MTT-8, 169 (1960).
[Crossref]

J. E. Eaton, I. R. E. Trans. on Microwave Theory Tech. MTT-8, 171 (1960).
[Crossref]

Leo Young, I. R. E. Trans. on Microwave Theory Tech. MTT-7, 99 (1959).
[Crossref]

J. Math. and Phys. (2)

O. Brune, J. Math. and Phys. 10, 191 (1931).

S. Darlington, J. Math. and Phys. 18, 257 (1939).

J. Opt. Soc. Am. (6)

J. phys. radium (1)

A. F. Turner, J. phys. radium 11, 444 (1950).
[Crossref]

Proc. I. R. E. (1)

P. I. Richards, Proc. I. R. E. 36, 217 (1948).
[Crossref]

Quart. Appl. Math. (1)

Leo Young, Quart. Appl. Math. 19, 25 (1961).

Sci. Light (Tokyo) (2)

M. Iwata, Sci. Light (Tokyo) 2, 116 (1953).

M. Iwata, S. Katsube, and T. Fukuda, Sci. Light (Tokyo) 7, 33 (1958).

Vacuum (2)

J. T. Cox, G. Hass, and R. F. Rowntree, Vacuum 4, 445 (1954).
[Crossref]

W. W. Weinstein (now T. Welford), Vacuum 4, 3 (1954).
[Crossref]

Other (4)

M. E. Van Valkenburg, Introduction to Modern Network Synthesis, (John Wiley & Sons, Inc., New York, 1960).

E. A. Guillemin, Synthesis of Passive Networks (John Wiley & Sons, Inc., New York, 1957).

D. F. Tuttle, Network Synthesis (John Wiley & Sons, Inc., New York, 1958), Vol. I.

O. S. Heavens, Optical Properties of Thin Solid Films (Academic Press, Inc., New York, 1955).

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Figures (3)

Fig. 1
Fig. 1

Quarter-wave transformer of k sections.

Fig. 2
Fig. 2

Typical perfromance of transformer or multilayer film.

Fig. 3
Fig. 3

Three-layer antireflection film with numerical solution from text.

Tables (2)

Tables Icon

Table I Maximum (amplitude) reflection coefficients.

Tables Icon

Table II Optimum values of the refractive indices.

Equations (34)

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

Z = ( μ μ 0 / 0 ) 1 2
Z 1 / Z 2 = ( 2 / 1 ) 1 2 = n 2 / n 1 ,
Y 1 / Y 2 = Z 2 / Z 1 = n 1 / n 2 .
Γ = Z 2 - Z 1 Z 2 + Z 1 = Y 1 - Y 2 Y 1 + Y 2 = n 1 - n 2 n 1 + n 2 .
s = i tan θ ,
θ = 2 π l / λ g radians ,
P L = 1 + ( n - 1 ) 2 4 n T k 2 ( cos θ / μ 0 ) T k 2 ( 1 / μ 0 ) ,
T k ( x ) = cos ( k arc cos x )
μ 0 = sin ( 1 2 W 90° ) ,
W = 2 [ ( λ 2 - λ 1 ) / ( λ 2 + λ 1 ) ] ,
P L = 1 + [ ( n - 1 ) 2 / 4 n ] cos 2 n θ .
W = 2 ( 7672 - 4080 7672 + 4080 ) = 0.611 μ 0 = 0.464 and ( n - 1 ) 2 4 n · 1 T 3 2 ( 1 / μ 0 ) = 1.578 × 10 - 4 } .
Γ 2 = P L - 1 P L
= 0.253 cos 6 θ - 0.0817 cos 4 θ + 0.00659 cos 2 θ 0.253 cos 6 θ - 0.0817 cos 4 θ + 0.00659 cos 2 θ + 1
= 0.1779 + 0.0685 s 2 + 0.00659 s 4 1.1779 - 2.9315 s 2 + 3.00659 s 4 - s 6 ,
D ( s ) = s 3 + 3.030 s 2 + 3.085 s + 1.086 ,
Γ = const · T 3 ( cos θ / μ 0 ) ( s 2 - 1 ) 3 2 D ( s )
= const · ( 39.96 cos 3 θ - 6.47 cos θ ) ( 1 - s 2 ) 3 2 D ( s ) = const · 6.47 s 2 + 33.49 D ( s ) = - 0.081 s 2 + 0.422 s 3 + 3.030 s 2 + 3.085 s + 1.086 ,
Γ = ( 1 - n ) / ( 1 + n ) when s = 0.
Y ( s ) = 1 - Γ 1 + Γ = s 3 + 3.111 s 2 + 3.085 s + 1.508 s 3 + 2.949 s 2 + 3.085 s + 0.664 .
n 1 = Y ( 1 ) = - Y ( - 1 ) = 1.13.
Y ( s ) = n 1 ( n 1 s - Y ( s ) s Y ( s ) - n 1 ) .
n 2 = Y ( 1 ) = - Y ( - 1 ) = 1.50.
Y ( s ) = n 2 ( n 2 s - Y ( s ) s Y ( s ) - n 2 ) .
n 3 = Y ( 1 ) = - Y ( - 1 ) = 2.20 ,
Y ( s ) = n 3 ( n 3 s - Y ( s ) s Y ( s ) - n 3 ) 2.27 ,
l 0 = λ 1 λ 2 / 2 ( λ 1 + λ 2 ) ,
( 4080 × 7672 ) 2 ( 4080 + 7672 ) = 1332 A ,
Single - layer film ( k = 1 ) n 1 = n
Two - layer film ( k = 2 ) n 2 = n / n 1
There - layer film ( k = 3 ) n 2 = n n 3 = n / n 1 }
Four - layer film ( k = 4 ) n 3 = n / n 2 n 4 = n / n 1 }
n 1 ,             n 1 2 n 2 ,             n 1 2 n 3 n 2 2 ,             n 1 2 n 3 2 n 2 2 n 4 ,            
1 n 1 ,             n 2 n 1 2 ,             n 2 2 n 1 2 n 3 ,             n 2 2 n 4 n 1 2 n 3 2 ,             .