Abstract

A new method of analysis for the simultaneous determination of the optical constants and the thickness of thin films is proposed. It requires measurements under normal incidence of the transmission and of the reflections from both sides of a thin film deposited on a nonabsorbing substrate. An algebraic inversion technique is developed involving a numerical interpolation procedure in the last step. There are no missing solutions. The physical solution can be isolated by a comparison with some film thickness estimates or by measurements in a wavelength range.

© 1991 Optical Society of America

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References

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  1. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955), Chap. 5.
  2. H. M. Liddell, Computer Aided Techniques for the Design of Multilayer Filters (Hilger, Bristol, 1981), Chap. 6.
  3. D. P. Arndt et al., “Multiple Determination of the Optical Constants of Thin-Film Coating Materials,” Appl. Opt. 24, 3571–3596 (1984).
    [CrossRef]
  4. F. Abeles, M. L. Theye, “Méthode De Calcul Des Constantes Optiques Des Couches Minces Absorbantes À Partir De Mesures De Réflexion Et De Transmission,” Surf. Sci. 5, 325–331 (1966).
    [CrossRef]
  5. J. M. Bennett, M. J. Booty, “Computational Method for Determining n and k for a Thin Film from the Measured Reflectance, Transmittance, and Film Thickness,” Appl. Opt. 5, 41–43 (1966).
    [CrossRef] [PubMed]
  6. W. N. Hansen, “Optical Characterization of Thin Films: Theory,” J. Opt. Soc. Am. 63, 793–802 (1973).
    [CrossRef]
  7. J. E. Nestell, R. W. Christy, “Derivation of Optical Constants of Metals from Thin-Film Measurements at Oblique Incidence,” Appl. Opt. 11, 643–651 (1972).
    [CrossRef] [PubMed]
  8. W. E. Case, “Algebraic Method for Extracting Thin-Film Optical Parameters from Spectrophotometer Measurements,” Appl. Opt. 22, 1832–1836 (1983).
    [CrossRef] [PubMed]
  9. D. M. Spink, C. B. Thomas, “Optical Constant Determination of Thin Films: an Analytical Solution,” Appl. Opt. 27, 4362–4362 (1988).
    [CrossRef] [PubMed]
  10. C. L. Nagendra, G. K. M. Thutupalli, “Optical constants of absorbing films,” Vacuum 31, 141–145 (1981).
    [CrossRef]
  11. A. Hjortsberg, “Determination of Optical Constants of Absorbing Materials Using Transmission and Reflection of Thin Films on Partially Metallized Substrates: Analysis of the new (T,Rm) Technique,” Appl. Opt. 20, 1254–1263 (1981).
    [CrossRef] [PubMed]
  12. Ref. 1, pp. 55–58.
  13. Ref. 2, p. 7.
  14. R. C. McPhedran, L. C. Botten, D. R. McKenzie, R. P. Netterfield, “Unambiguous Determination of Optical Constants of Absorbing Films by Reflectance and Transmittance Measurements,” Appl. Opt. 23, 1197–1205 (1984).
    [CrossRef] [PubMed]
  15. Ref. 2, p. 19.

1988

1984

1983

1981

1973

1972

1966

F. Abeles, M. L. Theye, “Méthode De Calcul Des Constantes Optiques Des Couches Minces Absorbantes À Partir De Mesures De Réflexion Et De Transmission,” Surf. Sci. 5, 325–331 (1966).
[CrossRef]

J. M. Bennett, M. J. Booty, “Computational Method for Determining n and k for a Thin Film from the Measured Reflectance, Transmittance, and Film Thickness,” Appl. Opt. 5, 41–43 (1966).
[CrossRef] [PubMed]

Abeles, F.

F. Abeles, M. L. Theye, “Méthode De Calcul Des Constantes Optiques Des Couches Minces Absorbantes À Partir De Mesures De Réflexion Et De Transmission,” Surf. Sci. 5, 325–331 (1966).
[CrossRef]

Arndt, D. P.

D. P. Arndt et al., “Multiple Determination of the Optical Constants of Thin-Film Coating Materials,” Appl. Opt. 24, 3571–3596 (1984).
[CrossRef]

Bennett, J. M.

Booty, M. J.

Botten, L. C.

Case, W. E.

Christy, R. W.

Hansen, W. N.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955), Chap. 5.

Hjortsberg, A.

Liddell, H. M.

H. M. Liddell, Computer Aided Techniques for the Design of Multilayer Filters (Hilger, Bristol, 1981), Chap. 6.

McKenzie, D. R.

McPhedran, R. C.

Nagendra, C. L.

C. L. Nagendra, G. K. M. Thutupalli, “Optical constants of absorbing films,” Vacuum 31, 141–145 (1981).
[CrossRef]

Nestell, J. E.

Netterfield, R. P.

Spink, D. M.

Theye, M. L.

F. Abeles, M. L. Theye, “Méthode De Calcul Des Constantes Optiques Des Couches Minces Absorbantes À Partir De Mesures De Réflexion Et De Transmission,” Surf. Sci. 5, 325–331 (1966).
[CrossRef]

Thomas, C. B.

Thutupalli, G. K. M.

C. L. Nagendra, G. K. M. Thutupalli, “Optical constants of absorbing films,” Vacuum 31, 141–145 (1981).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

Surf. Sci.

F. Abeles, M. L. Theye, “Méthode De Calcul Des Constantes Optiques Des Couches Minces Absorbantes À Partir De Mesures De Réflexion Et De Transmission,” Surf. Sci. 5, 325–331 (1966).
[CrossRef]

Vacuum

C. L. Nagendra, G. K. M. Thutupalli, “Optical constants of absorbing films,” Vacuum 31, 141–145 (1981).
[CrossRef]

Other

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955), Chap. 5.

H. M. Liddell, Computer Aided Techniques for the Design of Multilayer Filters (Hilger, Bristol, 1981), Chap. 6.

Ref. 1, pp. 55–58.

Ref. 2, p. 7.

Ref. 2, p. 19.

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Figures (6)

Fig. 1
Fig. 1

Basic configuration for the computations.

Fig. 2
Fig. 2

Dependence of n and k on ϕb according to Eqs. (6) and (7), respectively. The valid region for ϕb is restricted by k ≥ 0 (λ = 600 nm; ns = 1.5, T = 0.1763, Rf = 0.1956, and Rb = 0.0671).

Fig. 3
Fig. 3

Dependence of nd on ϕb according to Eq. (16). The individual ranges of nd are marked according to the parameter m in Eq. (16). The region with negative nd is meaningless (λ, ns, T, Rf, and Rb, the same as in Fig. 2).

Fig. 4
Fig. 4

Dependence of kd on ϕb according to Eq. (17) (λ, ns, T, Rf, and Rb, the same as in Fig. 2).

Fig. 5
Fig. 5

Dependence of F on ϕb according to Eq. (18). Solutions (see Table I) exist for those ϕb where F is zero (λ, ns, T, Rf, and Rb, the same as in Fig. 2).

Fig. 6
Fig. 6

Dependence of the refractive index n on the thickness d = nd/n for a nonabsorbing thin film (k = 0, λ = 700 nm, ns = 1.5, T = 0.8). For d = 100 nm (the dashed line) the first six solutions are presented on the figure.

Tables (2)

Tables Icon

Table I Numerical Test Example

Tables Icon

Table II Solutions for the Example Problem of Case8

Equations (29)

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r ¯ f = r ¯ 1 + r ¯ 2 β ¯ 2 1 + r ¯ 1 r ¯ 2 β ¯ 2 = R f × exp ( i ϕ f ) ,
t ¯ f = ( 1 + r ¯ 1 ) ( 1 + r ¯ 2 ) β ¯ 1 + r ¯ 1 r ¯ 2 β 2 = T n s × exp ( i ϕ t ) ,
r ¯ b = - r ¯ 2 + r ¯ 1 β ¯ 2 1 + r ¯ 1 r ¯ 2 β ¯ 2 = R b × exp ( i ϕ b ) ,
t ¯ b = ( 1 - r ¯ 1 ) ( 1 - r ¯ 2 ) β ¯ 1 + r ¯ 1 r ¯ 2 β ¯ 2 = n s T × exp ( i ϕ t ) = n s t ¯ f ,
r ¯ 1 = ( 1 - n ¯ ) / ( 1 + n ¯ ) ,
r ¯ 2 = ( n ¯ - n s ) / ( n ¯ + n s ) ,
β ¯ = exp ( i 2 π n ¯ d λ ) ,
n ¯ = n + i k .
β ¯ 2 = r ¯ f - r ¯ 1 r ¯ 2 ( 1 - r ¯ 1 r ¯ f ) ,
β ¯ 2 = - r ¯ b + r ¯ 2 r ¯ 1 ( 1 + r ¯ 2 r ¯ b ) ,
n ¯ 2 = n s 2 ( r ¯ b - 1 ) - n s ( r ¯ f - 1 ) ( r ¯ b + 1 ) - n s ( r ¯ f + 1 ) ,
n = N x + N x 2 + N y 2 2 ,
k = N y 2 n ,
n ¯ 2 = N x + i N y .
r s t ¯ f t ¯ b = r s r ¯ f r ¯ b + r ¯ f - r ¯ b - r s ,
r s = ( 1 - n s ) / ( 1 + n s ) .
A × cos ϕ f + B × cos ϕ b + C × cos ϕ f cos ϕ b + D × sin ϕ f sin ϕ b = E ,
A = ( 1 + R b ) / R b , B = - ( 1 + R f ) / R f , C = - ( 1 + R s ) / R s , D = - ( 1 - R s ) / R s , E = - ( R s R f R b + R s + R f + R b - R s T 2 ) / 2 / R s R f R b , R s = r s 2 .
tan ϕ f 2 = y ± y 2 + x 2 - z 2 x + z ,
x = A + C × cos ϕ b , y = D × sin ϕ b , z = E - B × cos ϕ b .
a × cos 2 ϕ b + 2 b × cos ϕ b + c 0 ,
a = C 2 - B 2 - D 2 , b = A C + B E , c = A 2 + D 2 - E 2 .
- b + b 2 - a c a cos ϕ b - b - b 2 - a c a .
β ¯ 2 = B x + i B y ,
n d = λ 4 π [ arctg ( B y B x + 2 π m ) ] , m = 0 , 1 , 2
k d = - λ 4 π ln B x 2 + B y 2 .
F = n × k d - k × n d
n 2 = R ( 1 - r s ) cos ϕ f - n s ( R + r s ) R ( 1 - r s ) cos ϕ f + ( R - r s ) .
F = n × d - n d

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