Abstract

An iterative algorithm has been developed that takes starting values derived by an envelope method but then minimizes the influence of the envelopes and emphasizes the actual measured data. This combination avoids the difficulties inherent in the accurate drawing of the envelopes and makes it possible to extract the thickness and the optical constants of semiconducting and dielectric films over a wide spectral region, including regions of high absorption.

© 1995 Optical Society of America

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References

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  1. J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E 9, 1002–1004 (1976).
    [CrossRef]
  2. R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E. 16, 1214–1222 (1983).
    [CrossRef]
  3. D. Minkov, R. Swanepoel, “Computerization of the optical characterization of a thin dielectric film,” Opt. Eng. 32, 3333–3337 (1993).
    [CrossRef]
  4. M. McClain, A. Feldman, D. Kahaner, X. Ying, “An algorithm and computer program for the calculation of envelope curves,” Comput. Phys. 5, 45–48 (1991).
    [CrossRef]
  5. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1209 (1965).
    [CrossRef]
  6. J. M. Bennett, E. Pelletier, G. Albrand, J. P. Borgogno, B. Lazarides, C. K. Carniglia, R. A. Schmell, T. H. Allen, T. Tuttle-Hart, K. H. Guenther, A. Saxer, “Comparison of the properties of titanium dioxide films prepared by various techniques,” Appl. Opt. 28, 3303–3317 (1989).
    [CrossRef] [PubMed]
  7. F. K. Urban, “Ellipsometry algorithm for absorbing films,” Appl. Opt. 32, 2339–2344 (1993).
    [CrossRef]

1993 (2)

D. Minkov, R. Swanepoel, “Computerization of the optical characterization of a thin dielectric film,” Opt. Eng. 32, 3333–3337 (1993).
[CrossRef]

F. K. Urban, “Ellipsometry algorithm for absorbing films,” Appl. Opt. 32, 2339–2344 (1993).
[CrossRef]

1991 (1)

M. McClain, A. Feldman, D. Kahaner, X. Ying, “An algorithm and computer program for the calculation of envelope curves,” Comput. Phys. 5, 45–48 (1991).
[CrossRef]

1989 (1)

1983 (1)

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E. 16, 1214–1222 (1983).
[CrossRef]

1976 (1)

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

1965 (1)

Albrand, G.

Allen, T. H.

Bennett, J. M.

Borgogno, J. P.

Carniglia, C. K.

Feldman, A.

M. McClain, A. Feldman, D. Kahaner, X. Ying, “An algorithm and computer program for the calculation of envelope curves,” Comput. Phys. 5, 45–48 (1991).
[CrossRef]

Fillard, J. P.

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

Gasiot, J.

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

Guenther, K. H.

Kahaner, D.

M. McClain, A. Feldman, D. Kahaner, X. Ying, “An algorithm and computer program for the calculation of envelope curves,” Comput. Phys. 5, 45–48 (1991).
[CrossRef]

Lazarides, B.

Malitson, I. H.

Manifacier, J. C.

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

McClain, M.

M. McClain, A. Feldman, D. Kahaner, X. Ying, “An algorithm and computer program for the calculation of envelope curves,” Comput. Phys. 5, 45–48 (1991).
[CrossRef]

Minkov, D.

D. Minkov, R. Swanepoel, “Computerization of the optical characterization of a thin dielectric film,” Opt. Eng. 32, 3333–3337 (1993).
[CrossRef]

Pelletier, E.

Saxer, A.

Schmell, R. A.

Swanepoel, R.

D. Minkov, R. Swanepoel, “Computerization of the optical characterization of a thin dielectric film,” Opt. Eng. 32, 3333–3337 (1993).
[CrossRef]

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E. 16, 1214–1222 (1983).
[CrossRef]

Tuttle-Hart, T.

Urban, F. K.

Ying, X.

M. McClain, A. Feldman, D. Kahaner, X. Ying, “An algorithm and computer program for the calculation of envelope curves,” Comput. Phys. 5, 45–48 (1991).
[CrossRef]

Appl. Opt. (2)

Comput. Phys. (1)

M. McClain, A. Feldman, D. Kahaner, X. Ying, “An algorithm and computer program for the calculation of envelope curves,” Comput. Phys. 5, 45–48 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. E (2)

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E. 16, 1214–1222 (1983).
[CrossRef]

Opt. Eng. (1)

D. Minkov, R. Swanepoel, “Computerization of the optical characterization of a thin dielectric film,” Opt. Eng. 32, 3333–3337 (1993).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Transmission spectrum together with two presumed envelope curves of a TiO2 film system. The measured tangent points and the computed envelope points are marked on curves correspondingly.

Fig. 2
Fig. 2

Merit function of the TiO2 film system. The iteration started at thickness 276.16 nm and ended at thickness 246.16 nm.

Fig. 3
Fig. 3

Measured and calculated transmission spectra of the TiO2 thin-film system. Good agreement is shown at wavelengths larger than 410 nm. At wavelengths shorter than 410 nm, the average discrepancy is ~1.6%.

Fig. 4
Fig. 4

Transmission spectrum together with two true envelopes of a 250-nm film of α-Si:H on a finite transparent substrate. The calculated envelope points depart from the true envelope curves.

Tables (4)

Tables Icon

Table 1 Data Points of Envelopes of a Transmission Curve of TiO2 Film as shown in Fig. 1 a

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Table 2 Order Numbers, Optimized Optical Constants, and Both Measured and Calculated Transmittance of a TiO2 Film System

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Table 3 True and Calculated (in parentheses) Data Points of Envelopes and the Corresponding Indices of Refraction of a 250-nm-thick α-Si:H Film System

Tables Icon

Table 4 Order Numbers, True and Optimized Optical Constants, and Both Measured and Calculated Transmittances of a 250-nm-thick α-Si:H Film System

Equations (25)

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T = A X B - C X + D X 2 ,
A = 16 s ( n 2 + k 2 ) ,
B = [ ( n + 1 ) 2 + k 2 ] [ ( n + 1 ) ( n + s 2 ) + k 2 ] ,
C = [ ( n 2 + k 2 - 1 ) ( n 2 - s 2 + k 2 ) - 2 k 2 ( s 2 + 1 ) ] 2 cos ϕ - k [ 2 ( n 2 - s 2 + k 2 ) + ( s 2 + 1 ) × ( n 2 + k 2 - 1 ) ] 2 sin ϕ ,
D = [ ( n - 1 ) 2 + k 2 ] [ ( n - 1 ) ( n - s 2 ) + k 2 ] ,
ϕ = 4 π n d λ ,             X = exp ( - α d ) ,             α = 4 π k λ .
C = ( c 1 2 + c 2 2 ) 1 / 2 cos ( ϕ - ) ,
c 1 = 2 [ ( n 2 + k 2 - 1 ) ( n 2 + k 2 - s 2 ) - 2 k 2 ( s 2 + 1 ) ] ,
c 2 = - 2 k [ 2 ( n 2 + k 2 - s 2 ) + ( n 2 + k 2 - 1 ) ( s 2 + 1 ) ] ,
= tan - 1 ( c 2 / c 1 ) .
T Top = A X B - ( c 1 2 + c 2 2 ) 1 / 2 X + D X 2 ,
T Bottom = A X B + ( c 1 2 + c 2 2 ) 1 / 2 X + D X 2 ,
X = exp ( - 4 π k d λ ) = E Top - ( E Top 2 - B D ) 1 / 2 D ,
E Top = A 2 T Top + ( c 1 2 + c 2 2 ) 1 / 2 2 ,
X = exp ( - 4 π k d λ ) = E Bottom - ( E Bottom 2 - B D ) 1 / 2 D ,
E Bottom = A 2 T Bottom - ( c 1 2 + c 2 2 ) 1 / 2 2 .
n = [ N + ( N 2 - s 2 ) 1 / 2 ] 1 / 2 ,
N = 2 s T Top - T Bottom T Top T Bottom + s 2 + 1 2 .
ϕ = m π + ,
4 π n d λ = m π .
m 3 = 2 [ ( n 1 λ 3 ) / ( n 3 λ 1 ) - 1 ] ,
Δ = i T cal ( λ i ) - T meas ( λ i )
n ( λ ) = A + B λ 2 + C λ 4 ,
n ( λ ) = 3 × 10 5 λ 2 + 2.6 ,
k ( λ ) = λ 4 π 10 ( 1.5 × 10 6 / λ 2 - 8 ) ,

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