Abstract

The optical properties of aggregated gold films have been studied in the visible and near infrared. From the measurements of the normal transmittance and reflectances at two different angles, the effective optical constants and the effective optical thickness could be derived. The films were produced by evaporation on a hot substrate to form a highly aggregated structure; the particles that form the films were relatively large; i.e., the thickest films had particles attaining modal sizes up to 1500 Å. By considering these particles as rotational ellipsoids and by assuming a plausible distribution function of the depolarization factor, we were able to describe the optical properties of the aggregated films and to obtain satisfactory agreement with experiments. The proposed analysis permitted determination of several parameters that characterize the films, in particular the filling factor and the mean effective depolarization factor.

© 1976 Optical Society of America

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  1. J. C. Maxwell-Garnett, Philos. Trans. R. Soc. London Ser. A 203, 385 (1904); Philos. Trans. R. Soc. London Ser. A 205, 237 (1906).
    [CrossRef]
  2. R. S. Sennett and G. D. Scott, J. Opt. Soc. Am. 40, 203 (1950).
    [CrossRef]
  3. D. Malé, C. R. Acad. Sci. (Paris) B 230, 286 (1950).
  4. I. N. Shklyarevskii and T. I. Korneeva, Opt. Spektrosk. 31, 270 (1971) [Opt. Spectrosc. 31, 144 (1971)].
  5. J. P. Marton and J. R. Lemon, Phys. Rev, B 4, 271 (1971).
    [CrossRef]
  6. E. David, Z. Phys. 114, 389 (1939).
    [CrossRef]
  7. S. Yamaguchi, J. Phys. Soc. Jpn. 17, 184 (1962).
    [CrossRef]
  8. S. Yoshida, T. Yamaguchi, and A. Kinbara, J. Opt. Soc. Am. 61, 62 (1971).
    [CrossRef]
  9. A. Meessen, J. Phys. (Paris) 33, 371 (1972).
    [CrossRef]
  10. R. W. Tokarsky and J. P. Marton, J. Appl. Phys. 45, 3051 (1974).
    [CrossRef]
  11. R. H. Doremus, J. Chem. Phys. 40, 2389 (1964); J. Appl. Phys. 37, 2775 (1966).
    [CrossRef]
  12. S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Clarendon, Oxford, 1948).
  13. A. Kawabata and R. Kubo, J. Phys. Soc. Jpn. 21, 1765 (1966).
    [CrossRef]
  14. H. E. Bennett and J. M. Bennett, in Physics of Thin Films, Vol. 4, edited by G. Hass and R. E. Thun (Academic, New York, 1967), p. 1.
  15. See, for example, F. Abelès, in Physics of Thin Films, Vol. 6, edited by G. Hass and R. E. Thun (Academic, New York, 1971), p. 151.
  16. J. E. Nestell and R. W. Christy, Appl. Opt. 11, 643 (1972).
    [CrossRef] [PubMed]
  17. D. Bedeaux and J. Vlieger, Physica (Utr.) 73, 287 (1974).
    [CrossRef]
  18. T. Yamaguchi, S. Yoshida, and A. Kinbara, J. Opt. Soc. Am. 64, 1563 (1974).
    [CrossRef]
  19. P. B. Johnson and R. W. Christy, Phys, Rev. B 6, 4370 (1972).
    [CrossRef]
  20. S. Yoshida, T. Yamaguchi, and A. Kinbara, J. Opt. Soc. Am. 62, 1415 (1972).
    [CrossRef]
  21. B. R. Cooper, H. Ehrenreich, and H. R. Philipp, Phys. Rev. 138, A494 (1965).
    [CrossRef]
  22. I. N. Shklyarenskii and P. L. Pakhomov, Opt. Spektrosk,  34, 163 (1973) [Opt. Spectrosc. 34, 90 (1973)].
  23. M. Schlüter, Z. Phys. 250, 87 (1972).
    [CrossRef]
  24. J. P. Marton and J. R. Lemon, J. Appl. Phys. 44, 3955 (1973).
    [CrossRef]

1974 (3)

R. W. Tokarsky and J. P. Marton, J. Appl. Phys. 45, 3051 (1974).
[CrossRef]

D. Bedeaux and J. Vlieger, Physica (Utr.) 73, 287 (1974).
[CrossRef]

T. Yamaguchi, S. Yoshida, and A. Kinbara, J. Opt. Soc. Am. 64, 1563 (1974).
[CrossRef]

1973 (2)

I. N. Shklyarenskii and P. L. Pakhomov, Opt. Spektrosk,  34, 163 (1973) [Opt. Spectrosc. 34, 90 (1973)].

J. P. Marton and J. R. Lemon, J. Appl. Phys. 44, 3955 (1973).
[CrossRef]

1972 (5)

P. B. Johnson and R. W. Christy, Phys, Rev. B 6, 4370 (1972).
[CrossRef]

J. E. Nestell and R. W. Christy, Appl. Opt. 11, 643 (1972).
[CrossRef] [PubMed]

M. Schlüter, Z. Phys. 250, 87 (1972).
[CrossRef]

A. Meessen, J. Phys. (Paris) 33, 371 (1972).
[CrossRef]

S. Yoshida, T. Yamaguchi, and A. Kinbara, J. Opt. Soc. Am. 62, 1415 (1972).
[CrossRef]

1971 (3)

S. Yoshida, T. Yamaguchi, and A. Kinbara, J. Opt. Soc. Am. 61, 62 (1971).
[CrossRef]

I. N. Shklyarevskii and T. I. Korneeva, Opt. Spektrosk. 31, 270 (1971) [Opt. Spectrosc. 31, 144 (1971)].

J. P. Marton and J. R. Lemon, Phys. Rev, B 4, 271 (1971).
[CrossRef]

1966 (1)

A. Kawabata and R. Kubo, J. Phys. Soc. Jpn. 21, 1765 (1966).
[CrossRef]

1965 (1)

B. R. Cooper, H. Ehrenreich, and H. R. Philipp, Phys. Rev. 138, A494 (1965).
[CrossRef]

1964 (1)

R. H. Doremus, J. Chem. Phys. 40, 2389 (1964); J. Appl. Phys. 37, 2775 (1966).
[CrossRef]

1962 (1)

S. Yamaguchi, J. Phys. Soc. Jpn. 17, 184 (1962).
[CrossRef]

1950 (2)

D. Malé, C. R. Acad. Sci. (Paris) B 230, 286 (1950).

R. S. Sennett and G. D. Scott, J. Opt. Soc. Am. 40, 203 (1950).
[CrossRef]

1939 (1)

E. David, Z. Phys. 114, 389 (1939).
[CrossRef]

1904 (1)

J. C. Maxwell-Garnett, Philos. Trans. R. Soc. London Ser. A 203, 385 (1904); Philos. Trans. R. Soc. London Ser. A 205, 237 (1906).
[CrossRef]

Abelès, F.

See, for example, F. Abelès, in Physics of Thin Films, Vol. 6, edited by G. Hass and R. E. Thun (Academic, New York, 1971), p. 151.

Bedeaux, D.

D. Bedeaux and J. Vlieger, Physica (Utr.) 73, 287 (1974).
[CrossRef]

Bennett, H. E.

H. E. Bennett and J. M. Bennett, in Physics of Thin Films, Vol. 4, edited by G. Hass and R. E. Thun (Academic, New York, 1967), p. 1.

Bennett, J. M.

H. E. Bennett and J. M. Bennett, in Physics of Thin Films, Vol. 4, edited by G. Hass and R. E. Thun (Academic, New York, 1967), p. 1.

Christy, R. W.

P. B. Johnson and R. W. Christy, Phys, Rev. B 6, 4370 (1972).
[CrossRef]

J. E. Nestell and R. W. Christy, Appl. Opt. 11, 643 (1972).
[CrossRef] [PubMed]

Cooper, B. R.

B. R. Cooper, H. Ehrenreich, and H. R. Philipp, Phys. Rev. 138, A494 (1965).
[CrossRef]

David, E.

E. David, Z. Phys. 114, 389 (1939).
[CrossRef]

Doremus, R. H.

R. H. Doremus, J. Chem. Phys. 40, 2389 (1964); J. Appl. Phys. 37, 2775 (1966).
[CrossRef]

Ehrenreich, H.

B. R. Cooper, H. Ehrenreich, and H. R. Philipp, Phys. Rev. 138, A494 (1965).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, Phys, Rev. B 6, 4370 (1972).
[CrossRef]

Kawabata, A.

A. Kawabata and R. Kubo, J. Phys. Soc. Jpn. 21, 1765 (1966).
[CrossRef]

Kinbara, A.

Korneeva, T. I.

I. N. Shklyarevskii and T. I. Korneeva, Opt. Spektrosk. 31, 270 (1971) [Opt. Spectrosc. 31, 144 (1971)].

Kubo, R.

A. Kawabata and R. Kubo, J. Phys. Soc. Jpn. 21, 1765 (1966).
[CrossRef]

Lemon, J. R.

J. P. Marton and J. R. Lemon, J. Appl. Phys. 44, 3955 (1973).
[CrossRef]

J. P. Marton and J. R. Lemon, Phys. Rev, B 4, 271 (1971).
[CrossRef]

Malé, D.

D. Malé, C. R. Acad. Sci. (Paris) B 230, 286 (1950).

Marton, J. P.

R. W. Tokarsky and J. P. Marton, J. Appl. Phys. 45, 3051 (1974).
[CrossRef]

J. P. Marton and J. R. Lemon, J. Appl. Phys. 44, 3955 (1973).
[CrossRef]

J. P. Marton and J. R. Lemon, Phys. Rev, B 4, 271 (1971).
[CrossRef]

Maxwell-Garnett, J. C.

J. C. Maxwell-Garnett, Philos. Trans. R. Soc. London Ser. A 203, 385 (1904); Philos. Trans. R. Soc. London Ser. A 205, 237 (1906).
[CrossRef]

Meessen, A.

A. Meessen, J. Phys. (Paris) 33, 371 (1972).
[CrossRef]

Nestell, J. E.

Pakhomov, P. L.

I. N. Shklyarenskii and P. L. Pakhomov, Opt. Spektrosk,  34, 163 (1973) [Opt. Spectrosc. 34, 90 (1973)].

Philipp, H. R.

B. R. Cooper, H. Ehrenreich, and H. R. Philipp, Phys. Rev. 138, A494 (1965).
[CrossRef]

Schlüter, M.

M. Schlüter, Z. Phys. 250, 87 (1972).
[CrossRef]

Scott, G. D.

Sennett, R. S.

Shklyarenskii, I. N.

I. N. Shklyarenskii and P. L. Pakhomov, Opt. Spektrosk,  34, 163 (1973) [Opt. Spectrosc. 34, 90 (1973)].

Shklyarevskii, I. N.

I. N. Shklyarevskii and T. I. Korneeva, Opt. Spektrosk. 31, 270 (1971) [Opt. Spectrosc. 31, 144 (1971)].

Tokarsky, R. W.

R. W. Tokarsky and J. P. Marton, J. Appl. Phys. 45, 3051 (1974).
[CrossRef]

Tolansky, S.

S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Clarendon, Oxford, 1948).

Vlieger, J.

D. Bedeaux and J. Vlieger, Physica (Utr.) 73, 287 (1974).
[CrossRef]

Yamaguchi, S.

S. Yamaguchi, J. Phys. Soc. Jpn. 17, 184 (1962).
[CrossRef]

Yamaguchi, T.

Yoshida, S.

Appl. Opt. (1)

C. R. Acad. Sci. (Paris) B (1)

D. Malé, C. R. Acad. Sci. (Paris) B 230, 286 (1950).

J. Appl. Phys. (2)

R. W. Tokarsky and J. P. Marton, J. Appl. Phys. 45, 3051 (1974).
[CrossRef]

J. P. Marton and J. R. Lemon, J. Appl. Phys. 44, 3955 (1973).
[CrossRef]

J. Chem. Phys. (1)

R. H. Doremus, J. Chem. Phys. 40, 2389 (1964); J. Appl. Phys. 37, 2775 (1966).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Phys. (Paris) (1)

A. Meessen, J. Phys. (Paris) 33, 371 (1972).
[CrossRef]

J. Phys. Soc. Jpn. (2)

S. Yamaguchi, J. Phys. Soc. Jpn. 17, 184 (1962).
[CrossRef]

A. Kawabata and R. Kubo, J. Phys. Soc. Jpn. 21, 1765 (1966).
[CrossRef]

Opt. Spektrosk (1)

I. N. Shklyarenskii and P. L. Pakhomov, Opt. Spektrosk,  34, 163 (1973) [Opt. Spectrosc. 34, 90 (1973)].

Opt. Spektrosk. (1)

I. N. Shklyarevskii and T. I. Korneeva, Opt. Spektrosk. 31, 270 (1971) [Opt. Spectrosc. 31, 144 (1971)].

Philos. Trans. R. Soc. London Ser. A (1)

J. C. Maxwell-Garnett, Philos. Trans. R. Soc. London Ser. A 203, 385 (1904); Philos. Trans. R. Soc. London Ser. A 205, 237 (1906).
[CrossRef]

Phys, Rev. B (1)

P. B. Johnson and R. W. Christy, Phys, Rev. B 6, 4370 (1972).
[CrossRef]

Phys. Rev, B (1)

J. P. Marton and J. R. Lemon, Phys. Rev, B 4, 271 (1971).
[CrossRef]

Phys. Rev. (1)

B. R. Cooper, H. Ehrenreich, and H. R. Philipp, Phys. Rev. 138, A494 (1965).
[CrossRef]

Physica (Utr.) (1)

D. Bedeaux and J. Vlieger, Physica (Utr.) 73, 287 (1974).
[CrossRef]

Z. Phys. (2)

M. Schlüter, Z. Phys. 250, 87 (1972).
[CrossRef]

E. David, Z. Phys. 114, 389 (1939).
[CrossRef]

Other (3)

H. E. Bennett and J. M. Bennett, in Physics of Thin Films, Vol. 4, edited by G. Hass and R. E. Thun (Academic, New York, 1967), p. 1.

See, for example, F. Abelès, in Physics of Thin Films, Vol. 6, edited by G. Hass and R. E. Thun (Academic, New York, 1971), p. 151.

S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Clarendon, Oxford, 1948).

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Figures (11)

FIG. 1
FIG. 1

Electron micrographs of some aggregated gold films. The micrographs are labeled with the masses of evaporant used in the preparation of the films.

FIG. 2
FIG. 2

Typical distribution function for the effective depolarization factor f *. The analytical form of the function is g ( f * ) = K ( f * - A ) 3 exp ( - [ ( f * - A ) / f ¯ ] 2 }. In this case, K = 10 000, f = 0. 10, and A = 0.

FIG. 3
FIG. 3

Normal transmittance for seven aggregated gold films. The figures shown on the curves are the masses of gold used in the preparation of the films.

FIG. 4
FIG. 4

Typical behavior of the normal transmittance and the reflectance at 7.5°. These data are for the 60 mg film.

FIG. 5
FIG. 5

Experimental values for the effective optical constants of the 15, 60, and 80 mg films. These values were calculated from the measurements of the transmittance and the reflectance. The data for nbulk and kbulk are from Ref. 16. —neff; --keff; I 15 mg; II 60 mg; III 80 mg; ⋯ nbulk and kbulk.

FIG. 6
FIG. 6

Experimental values for the imaginary part of the effective dielectric constant of the 15, 60, 80, and 140 mg films.

FIG. 7
FIG. 7

Experimental values for the real part of the effective dielectric constant of the 15, 60, and 80 mg films.

FIG. 8
FIG. 8

Loss function Im(−1/∊eff) for the 15, 60, and 80 mg films.

FIG. 9
FIG. 9

Theoretical real part of the effective dielectric constant, as given by a rotational-ellipsoid model.

FIG. 10
FIG. 10

Theoretical imaginary part of the effective dielectric constant, as given by a rotational-ellipsoid model.

FIG. 11
FIG. 11

Variations of the resonance wavelength with the film thickness.

Equations (10)

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R exp = R f + R s T f 2 1 - R f R s , T exp = T f ( 1 - R s ) 1 - R f R s ,
1 i = a + q [ ( 1 - a ) + ( 1 - a ) 2 f i * / a + 2 2 f i * / a ] [ 1 + ( 1 - a ) f i * / a ] 2 + [ 2 f i * / a ] 2 ,
2 i = q 2 [ 1 + ( 1 - a ) f i * / a ] 2 + [ 2 f i * / a ] 2 ,
f i * = f i - β i q d a ,
β i = { + 1 2 for i = - 1 for i = ,
1 = a + 0 1 g ( f * ) [ ( 1 - a ) + ( 1 - a ) 2 f * / a + 2 2 f * / a ] [ 1 + ( 1 - a ) f * / a ] 2 + [ 2 f * / a ] 2 d f * ,
2 = 0 1 g ( f * ) 2 [ 1 + ( 1 - a ) f * / a ] 2 + [ 2 f * / a ] 2 d f * .
π a g ( f * 1 ) = 2 ( λ 1 ) ,
f * 1 = - ( 1 - a ) a ( 1 - a ) 2 + 2 2 ;
g ( f * ) = K ( f * - A ) 3 exp { - [ ( f * - A ) / f ¯ ] 2 } ,