Abstract

We describe a new method providing the complex index of an absorbing material from interferometric measurements of the transmittance and the corresponding phase shift for a series of unbacked films, of medium to large thickness. In fact, the usual method, giving the optical constants independently from the slope of the line obtained by a least-squares fit to the experimental points, may be used only in the thick-film limit, although it has the advantage of being insensitive to the surface conditions. The new method suggested here retains this advantage and allows the thickness range to be extended to more transparent specimens. Furthermore, it provides information on the surface quality and the thin-film model validity. Application to unbacked gold films is shown in the visible and near ultraviolet.

© 1979 Optical Society of America

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References

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  1. P. O. Nilsson, “Determination of optical constants from intensity measurements at normal incidence,” Appl. Opt. 7, 435–442 (1968).
    [Crossref] [PubMed]
  2. E. Kretschmann, “The determination of the optical constants of thin films by means of constant transmission,” Z. Physik 221, 346–356 (1969).
    [Crossref]
  3. K. Ishiguro and G. Kuwamara, “Determination of the optical constants of Ag films from measurements of intensity and phase change,” J. Phys. Soc. Jpn. 6, 71–75 (1951).
    [Crossref]
  4. J. Casset, “New method for simultaneous determination of the thickness and the complex refractive index of thin unbacked absorbing films,” Thin Solid Films 41, 315–328 (1977).
    [Crossref]
  5. J. Casset and P. Bousquet, “Measurements of phase shifts by transmission produced by thin unsupported films,” Rev. d’Opt. 42, 13–21 (1963).
  6. J. Casset, “On a new method for determining the optical constants and the thickness of thin unbacked films,” Rev. d’Opt. 43, 161–168 (1964).
  7. L. G. Schulz, “An experimental confirmation of the Drude free electron theory of the optical properties of metals for silver, gold and cooper in the near infrared,” J. Opt. Soc. Am. 44, 540–545 (1954).
    [Crossref]
  8. W. F. Koehler, F. K. Odercrantz, and W. C. White, “Optical constants of evaporated selenium films by successive approximations,” J. Opt. Soc. Am. 49, 109–115 (1959).
    [Crossref]
  9. W. R. Hunter, “On the optical constants of metals at wavelengths shorter than their critical wavelengths,” J. Phys. (Paris) 25, 154–160 (1964).
    [Crossref]
  10. W. R. Hunter, “Observation of absorption edges in the extreme ultraviolet by transmittance measurements through thin unbacked metal films,” in Optical Properties and Electronic Structure of Metals and Alloys, edited by F. Abelès, (North-Holland, Amsterdam, 1966) pp 136–145.
  11. P. Joensen, J. C. Irwin, J. F. Cochran, and A. E. Curzon, “Transmission method for determining the optical constants of metals,” J. Opt. Soc. Am. 63, 1556–1562 (1973).
    [Crossref]
  12. T. A. Mc Math, R. A. D. Hewko, O. Singh, A. E. Curzon, and J. C. Irwin, “Optical constants of gold by transmission interferometry,” J. Opt. Soc. Am.67, 630–634 (1977).
    [Crossref]
  13. E. Tallet, Thèse de spécialité, Marseille (1969) (unpublished).
  14. J. Casset, “Optical transmission in the visible and near ultraviolet of thin unbacked noble metal films,” Thin Solid Films 18, 99–103 (1973).
    [Crossref]
  15. G. P. Pells and M. Shiga, “The optical properties of cooper and gold as a function of temperature,” J. Phys. C 2, 1835–1846 (1969).
    [Crossref]
  16. J. Casset, “Determination of optical constants of a triple, symmetrical, absorbent film,” C. R. Acad. Sc. (Paris) 263, 299–302 (1966).

1977 (1)

J. Casset, “New method for simultaneous determination of the thickness and the complex refractive index of thin unbacked absorbing films,” Thin Solid Films 41, 315–328 (1977).
[Crossref]

1973 (2)

P. Joensen, J. C. Irwin, J. F. Cochran, and A. E. Curzon, “Transmission method for determining the optical constants of metals,” J. Opt. Soc. Am. 63, 1556–1562 (1973).
[Crossref]

J. Casset, “Optical transmission in the visible and near ultraviolet of thin unbacked noble metal films,” Thin Solid Films 18, 99–103 (1973).
[Crossref]

1969 (2)

G. P. Pells and M. Shiga, “The optical properties of cooper and gold as a function of temperature,” J. Phys. C 2, 1835–1846 (1969).
[Crossref]

E. Kretschmann, “The determination of the optical constants of thin films by means of constant transmission,” Z. Physik 221, 346–356 (1969).
[Crossref]

1968 (1)

1966 (1)

J. Casset, “Determination of optical constants of a triple, symmetrical, absorbent film,” C. R. Acad. Sc. (Paris) 263, 299–302 (1966).

1964 (2)

W. R. Hunter, “On the optical constants of metals at wavelengths shorter than their critical wavelengths,” J. Phys. (Paris) 25, 154–160 (1964).
[Crossref]

J. Casset, “On a new method for determining the optical constants and the thickness of thin unbacked films,” Rev. d’Opt. 43, 161–168 (1964).

1963 (1)

J. Casset and P. Bousquet, “Measurements of phase shifts by transmission produced by thin unsupported films,” Rev. d’Opt. 42, 13–21 (1963).

1959 (1)

1954 (1)

1951 (1)

K. Ishiguro and G. Kuwamara, “Determination of the optical constants of Ag films from measurements of intensity and phase change,” J. Phys. Soc. Jpn. 6, 71–75 (1951).
[Crossref]

Bousquet, P.

J. Casset and P. Bousquet, “Measurements of phase shifts by transmission produced by thin unsupported films,” Rev. d’Opt. 42, 13–21 (1963).

Casset, J.

J. Casset, “New method for simultaneous determination of the thickness and the complex refractive index of thin unbacked absorbing films,” Thin Solid Films 41, 315–328 (1977).
[Crossref]

J. Casset, “Optical transmission in the visible and near ultraviolet of thin unbacked noble metal films,” Thin Solid Films 18, 99–103 (1973).
[Crossref]

J. Casset, “Determination of optical constants of a triple, symmetrical, absorbent film,” C. R. Acad. Sc. (Paris) 263, 299–302 (1966).

J. Casset, “On a new method for determining the optical constants and the thickness of thin unbacked films,” Rev. d’Opt. 43, 161–168 (1964).

J. Casset and P. Bousquet, “Measurements of phase shifts by transmission produced by thin unsupported films,” Rev. d’Opt. 42, 13–21 (1963).

Cochran, J. F.

Curzon, A. E.

P. Joensen, J. C. Irwin, J. F. Cochran, and A. E. Curzon, “Transmission method for determining the optical constants of metals,” J. Opt. Soc. Am. 63, 1556–1562 (1973).
[Crossref]

T. A. Mc Math, R. A. D. Hewko, O. Singh, A. E. Curzon, and J. C. Irwin, “Optical constants of gold by transmission interferometry,” J. Opt. Soc. Am.67, 630–634 (1977).
[Crossref]

Hewko, R. A. D.

T. A. Mc Math, R. A. D. Hewko, O. Singh, A. E. Curzon, and J. C. Irwin, “Optical constants of gold by transmission interferometry,” J. Opt. Soc. Am.67, 630–634 (1977).
[Crossref]

Hunter, W. R.

W. R. Hunter, “On the optical constants of metals at wavelengths shorter than their critical wavelengths,” J. Phys. (Paris) 25, 154–160 (1964).
[Crossref]

W. R. Hunter, “Observation of absorption edges in the extreme ultraviolet by transmittance measurements through thin unbacked metal films,” in Optical Properties and Electronic Structure of Metals and Alloys, edited by F. Abelès, (North-Holland, Amsterdam, 1966) pp 136–145.

Irwin, J. C.

P. Joensen, J. C. Irwin, J. F. Cochran, and A. E. Curzon, “Transmission method for determining the optical constants of metals,” J. Opt. Soc. Am. 63, 1556–1562 (1973).
[Crossref]

T. A. Mc Math, R. A. D. Hewko, O. Singh, A. E. Curzon, and J. C. Irwin, “Optical constants of gold by transmission interferometry,” J. Opt. Soc. Am.67, 630–634 (1977).
[Crossref]

Ishiguro, K.

K. Ishiguro and G. Kuwamara, “Determination of the optical constants of Ag films from measurements of intensity and phase change,” J. Phys. Soc. Jpn. 6, 71–75 (1951).
[Crossref]

Joensen, P.

Koehler, W. F.

Kretschmann, E.

E. Kretschmann, “The determination of the optical constants of thin films by means of constant transmission,” Z. Physik 221, 346–356 (1969).
[Crossref]

Kuwamara, G.

K. Ishiguro and G. Kuwamara, “Determination of the optical constants of Ag films from measurements of intensity and phase change,” J. Phys. Soc. Jpn. 6, 71–75 (1951).
[Crossref]

Mc Math, T. A.

T. A. Mc Math, R. A. D. Hewko, O. Singh, A. E. Curzon, and J. C. Irwin, “Optical constants of gold by transmission interferometry,” J. Opt. Soc. Am.67, 630–634 (1977).
[Crossref]

Nilsson, P. O.

Odercrantz, F. K.

Pells, G. P.

G. P. Pells and M. Shiga, “The optical properties of cooper and gold as a function of temperature,” J. Phys. C 2, 1835–1846 (1969).
[Crossref]

Schulz, L. G.

Shiga, M.

G. P. Pells and M. Shiga, “The optical properties of cooper and gold as a function of temperature,” J. Phys. C 2, 1835–1846 (1969).
[Crossref]

Singh, O.

T. A. Mc Math, R. A. D. Hewko, O. Singh, A. E. Curzon, and J. C. Irwin, “Optical constants of gold by transmission interferometry,” J. Opt. Soc. Am.67, 630–634 (1977).
[Crossref]

Tallet, E.

E. Tallet, Thèse de spécialité, Marseille (1969) (unpublished).

White, W. C.

Appl. Opt. (1)

C. R. Acad. Sc. (Paris) (1)

J. Casset, “Determination of optical constants of a triple, symmetrical, absorbent film,” C. R. Acad. Sc. (Paris) 263, 299–302 (1966).

J. Opt. Soc. Am. (3)

J. Phys. (Paris) (1)

W. R. Hunter, “On the optical constants of metals at wavelengths shorter than their critical wavelengths,” J. Phys. (Paris) 25, 154–160 (1964).
[Crossref]

J. Phys. C (1)

G. P. Pells and M. Shiga, “The optical properties of cooper and gold as a function of temperature,” J. Phys. C 2, 1835–1846 (1969).
[Crossref]

J. Phys. Soc. Jpn. (1)

K. Ishiguro and G. Kuwamara, “Determination of the optical constants of Ag films from measurements of intensity and phase change,” J. Phys. Soc. Jpn. 6, 71–75 (1951).
[Crossref]

Rev. d’Opt. (2)

J. Casset and P. Bousquet, “Measurements of phase shifts by transmission produced by thin unsupported films,” Rev. d’Opt. 42, 13–21 (1963).

J. Casset, “On a new method for determining the optical constants and the thickness of thin unbacked films,” Rev. d’Opt. 43, 161–168 (1964).

Thin Solid Films (2)

J. Casset, “Optical transmission in the visible and near ultraviolet of thin unbacked noble metal films,” Thin Solid Films 18, 99–103 (1973).
[Crossref]

J. Casset, “New method for simultaneous determination of the thickness and the complex refractive index of thin unbacked absorbing films,” Thin Solid Films 41, 315–328 (1977).
[Crossref]

Z. Physik (1)

E. Kretschmann, “The determination of the optical constants of thin films by means of constant transmission,” Z. Physik 221, 346–356 (1969).
[Crossref]

Other (3)

W. R. Hunter, “Observation of absorption edges in the extreme ultraviolet by transmittance measurements through thin unbacked metal films,” in Optical Properties and Electronic Structure of Metals and Alloys, edited by F. Abelès, (North-Holland, Amsterdam, 1966) pp 136–145.

T. A. Mc Math, R. A. D. Hewko, O. Singh, A. E. Curzon, and J. C. Irwin, “Optical constants of gold by transmission interferometry,” J. Opt. Soc. Am.67, 630–634 (1977).
[Crossref]

E. Tallet, Thèse de spécialité, Marseille (1969) (unpublished).

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Figures (7)

FIG. 1
FIG. 1

Transmission and reflection geometry.

FIG. 2
FIG. 2

Curve A is the theoretical asymptote, g(x) = ax + b; curve C is the theoretical curve, f(x) = ax + bh(x); M is an experimental point (xj, yj), close to the theoretical curve C; M′ is the corresponding “corrected” point [xj, yj + h(Xj)], close to the theoretical asymptote A.

FIG. 3
FIG. 3

Basic flowchart.

FIG. 4
FIG. 4

Determination of n and k for λ = 577.0 nm. It is noted that (i) the first value obtained for n is negative and (ii) the relative variation of k, from one iteration to the next, are much smaller than those of n.

FIG. 5
FIG. 5

Gold self-supporting films, phase shift versus thickness. λ = 480.0 nm; n = 1.240; k = 1.794. (a) dots, corrected points; line, asymptote found by least squares fit to corrected points; dashed line, asymptote calculated from indices n and k. (b) dots, experimental points; line, experimental curve; dashed line, calculated curve.

FIG. 6
FIG. 6

Optical constants of gold self-supporting films; (a) n vs λ, (b) k vs λ: ●, volume; ○, surface; plain line, bulk (Pells and Shiga, 1969).15

FIG. 7
FIG. 7

Dielectric function of gold self-supporting films: ●, volume; ○, surface; plain line, bulk (Pells and Shiga, 1969).15

Equations (26)

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exp ( α d / λ ) 1 + α d / λ ,
exp ( α d / λ ) 1 + α d / λ + 1 2 ( α d / λ ) 2
t = E 0 t E 0 + = 4 n 0 ñ exp [ i 2 π ( ñ n 0 ) d / λ ] ( n 0 + ñ ) 2 ( n 0 ñ ) 2 exp ( i 4 π ñ d / λ )
t = T exp ( i ϕ ) .
ϕ = 2 π ( n n 0 ) d / λ ϕ 0 + ϕ r
T = 16 n 0 2 ( n 2 + k 2 ) exp ( 4 π k d / λ ) m 2 + 2 ( υ cos u + w sin u ) exp ( 4 π k d / λ ) + l 2 exp ( 8 π k d / λ ) ,
l m = ( n n 0 ) 2 + k 2 ,
u = 4 π n d / λ ,
υ = 4 n 0 2 k 2 ( n 2 + k 2 n 0 2 ) 2 ,
w = 4 n 0 k ( n 2 + k 2 n 0 2 ) ,
tan ϕ 0 = w / 4 n 0 ( m n + 2 n 0 k 2 ) ,
tan ϕ r = ( w cos u υ sin u ) exp ( 4 π k d / λ ) m 2 + ( υ cos u + w sin u ) exp ( 4 π k d / λ ) .
ln T = 4 π k d / λ + ln T 0 ln T r ,
T 0 = 16 n 0 2 ( n 2 + k 2 ) m 2 = 16 n 0 2 ( n 2 + k 2 ) [ ( n 0 + n ) 2 + k 2 ] 2 ,
T r = 1 + 1 m 2 [ 2 ( υ cos u + w sin u ) exp ( 4 π k d / λ ) + l 2 exp ( 8 π k d / λ ) ] .
t a = 4 n 0 ñ exp [ i 2 π ( ñ n 0 ) d / λ ] ( n 0 + n ) 2 ,
ϕ a = 2 π ( n n 0 ) d / λ ϕ 0 ,
T a = 16 n 0 2 ( n 2 + k 2 ) exp ( 4 π k d / λ ) [ ( n 0 + n ) 2 + k 2 ] 2 ,
ln T a = 4 π k d / λ + ln T 0 ;
p = 2 π ( n n 0 ) / λ
q = 4 π k / λ
n = n 0 + λ p / 2 λ
k = λ q / 4 π .
f ( x ) = a x + b h ( x ) ,
g ( x ) = a x + b ,
g ( x ) = f ( x ) + h ( x ) .