Abstract

The optical constants of thin films can be obtained from inversion of spectrophotometric measurements by using minimization gradient methods. The computational approach of these minimization methods requires closed compact formulas for reflectance and∕or transmittance. For normal incidence closed compact formulations for the direct transmittance, both for thin films on transparent or absorbing substrates, and for the reflectance of thin films on transparent substrates, are available in the literature. We report here a closed compact formula to evaluate reflectance spectra of thin films on absorbing substrates, and it is shown that for vanishing substrate absorption this new, to the best of our knowledge, approach gives the same results obtained from the formulation corresponding to thin films supported by transparent substrates.

© 2007 Optical Society of America

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  1. S. D. Ventura, E. G. Birgin, J. M. Martínez, and I. Chambouleyron, "Optimization techniques for the estimation of the thickness and the optical parameters of thin films using reflectance data," J. Appl. Phys. 97, 1-12 (2005).
    [CrossRef]
  2. R. Swanepoel, "Determination of the thickness and optical constants of amorphous silicon," J. Phys. E 16, 1214-1222 (1983).
    [CrossRef]
  3. E. G. Birgin, I. Chambouleyron, and J. M. Martínez, "Estimation of the optical constants and the thickness of thin films using unconstrained optimization," J. Comput. Phys. 151, 862-880 (1999).
    [CrossRef]
  4. W. E. Vargas, D. E. Azofeifa, and N. Clark, "Retrieved optical properties of thin films on absorbing substrates from transmittance measurements by application of a spectral projected gradient method," Thin Solid Films 425, 1-8 (2003).
    [CrossRef]
  5. E. N. Kotlikov and G. V. Tereshchenko "Study of optical constants of films used for the synthesis of broad-band antireflection coatings," Opt. Spectrosc. 82, 603-609 (1997).
  6. B. T. Sullivan, "Optical properties of palladium in the visible and near UV spectral regions," Appl. Opt. 29, 1964-1970 (1990).
    [CrossRef] [PubMed]
  7. A. Ramírez and W. E. Vargas, "Transmission of visible light through oxidized copper films: feasibility of using a spectral projected gradient method," Appl. Opt. 43, 1508-1514 (2004).
    [CrossRef]
  8. R. Swanepoel, "Transmission and reflection of an absorbing thin film on an absorbing substrate," S. Afr. Tydskr. Fis. 12, 148-156 (1989).

2005 (1)

S. D. Ventura, E. G. Birgin, J. M. Martínez, and I. Chambouleyron, "Optimization techniques for the estimation of the thickness and the optical parameters of thin films using reflectance data," J. Appl. Phys. 97, 1-12 (2005).
[CrossRef]

2004 (1)

2003 (1)

W. E. Vargas, D. E. Azofeifa, and N. Clark, "Retrieved optical properties of thin films on absorbing substrates from transmittance measurements by application of a spectral projected gradient method," Thin Solid Films 425, 1-8 (2003).
[CrossRef]

1999 (1)

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, "Estimation of the optical constants and the thickness of thin films using unconstrained optimization," J. Comput. Phys. 151, 862-880 (1999).
[CrossRef]

1997 (1)

E. N. Kotlikov and G. V. Tereshchenko "Study of optical constants of films used for the synthesis of broad-band antireflection coatings," Opt. Spectrosc. 82, 603-609 (1997).

1990 (1)

1989 (1)

R. Swanepoel, "Transmission and reflection of an absorbing thin film on an absorbing substrate," S. Afr. Tydskr. Fis. 12, 148-156 (1989).

1983 (1)

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

Azofeifa, D. E.

W. E. Vargas, D. E. Azofeifa, and N. Clark, "Retrieved optical properties of thin films on absorbing substrates from transmittance measurements by application of a spectral projected gradient method," Thin Solid Films 425, 1-8 (2003).
[CrossRef]

Birgin, E. G.

S. D. Ventura, E. G. Birgin, J. M. Martínez, and I. Chambouleyron, "Optimization techniques for the estimation of the thickness and the optical parameters of thin films using reflectance data," J. Appl. Phys. 97, 1-12 (2005).
[CrossRef]

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, "Estimation of the optical constants and the thickness of thin films using unconstrained optimization," J. Comput. Phys. 151, 862-880 (1999).
[CrossRef]

Chambouleyron, I.

S. D. Ventura, E. G. Birgin, J. M. Martínez, and I. Chambouleyron, "Optimization techniques for the estimation of the thickness and the optical parameters of thin films using reflectance data," J. Appl. Phys. 97, 1-12 (2005).
[CrossRef]

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, "Estimation of the optical constants and the thickness of thin films using unconstrained optimization," J. Comput. Phys. 151, 862-880 (1999).
[CrossRef]

Clark, N.

W. E. Vargas, D. E. Azofeifa, and N. Clark, "Retrieved optical properties of thin films on absorbing substrates from transmittance measurements by application of a spectral projected gradient method," Thin Solid Films 425, 1-8 (2003).
[CrossRef]

Kotlikov, E. N.

E. N. Kotlikov and G. V. Tereshchenko "Study of optical constants of films used for the synthesis of broad-band antireflection coatings," Opt. Spectrosc. 82, 603-609 (1997).

Martínez, J. M.

S. D. Ventura, E. G. Birgin, J. M. Martínez, and I. Chambouleyron, "Optimization techniques for the estimation of the thickness and the optical parameters of thin films using reflectance data," J. Appl. Phys. 97, 1-12 (2005).
[CrossRef]

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, "Estimation of the optical constants and the thickness of thin films using unconstrained optimization," J. Comput. Phys. 151, 862-880 (1999).
[CrossRef]

Ramírez, A.

Sullivan, B. T.

Swanepoel, R.

R. Swanepoel, "Transmission and reflection of an absorbing thin film on an absorbing substrate," S. Afr. Tydskr. Fis. 12, 148-156 (1989).

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

Tereshchenko, G. V.

E. N. Kotlikov and G. V. Tereshchenko "Study of optical constants of films used for the synthesis of broad-band antireflection coatings," Opt. Spectrosc. 82, 603-609 (1997).

Vargas, W. E.

A. Ramírez and W. E. Vargas, "Transmission of visible light through oxidized copper films: feasibility of using a spectral projected gradient method," Appl. Opt. 43, 1508-1514 (2004).
[CrossRef]

W. E. Vargas, D. E. Azofeifa, and N. Clark, "Retrieved optical properties of thin films on absorbing substrates from transmittance measurements by application of a spectral projected gradient method," Thin Solid Films 425, 1-8 (2003).
[CrossRef]

Ventura, S. D.

S. D. Ventura, E. G. Birgin, J. M. Martínez, and I. Chambouleyron, "Optimization techniques for the estimation of the thickness and the optical parameters of thin films using reflectance data," J. Appl. Phys. 97, 1-12 (2005).
[CrossRef]

Appl. Opt. (2)

J. Appl. Phys. (1)

S. D. Ventura, E. G. Birgin, J. M. Martínez, and I. Chambouleyron, "Optimization techniques for the estimation of the thickness and the optical parameters of thin films using reflectance data," J. Appl. Phys. 97, 1-12 (2005).
[CrossRef]

J. Comput. Phys. (1)

E. G. Birgin, I. Chambouleyron, and J. M. Martínez, "Estimation of the optical constants and the thickness of thin films using unconstrained optimization," J. Comput. Phys. 151, 862-880 (1999).
[CrossRef]

J. Phys. E (1)

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

Opt. Spectrosc. (1)

E. N. Kotlikov and G. V. Tereshchenko "Study of optical constants of films used for the synthesis of broad-band antireflection coatings," Opt. Spectrosc. 82, 603-609 (1997).

S. Afr. Tydskr. Fis. (1)

R. Swanepoel, "Transmission and reflection of an absorbing thin film on an absorbing substrate," S. Afr. Tydskr. Fis. 12, 148-156 (1989).

Thin Solid Films (1)

W. E. Vargas, D. E. Azofeifa, and N. Clark, "Retrieved optical properties of thin films on absorbing substrates from transmittance measurements by application of a spectral projected gradient method," Thin Solid Films 425, 1-8 (2003).
[CrossRef]

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Equations (49)

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T = U exp ( β ) 1 R 31 R 34 exp ( 2 β ) ,
R = R 13 + V R 34 exp ( 2 β ) 1 R 31 R 34 exp ( 2 β ) ,
R 34 = [ ( N 1 ) 2 + K 2 ] / z 34 ,
z 34 = [ ( N + 1 ) 2 + K 2 ] ,
U = [ 64 ( n 2 + k 2 ) ( N 2 + K 2 ) X ] / B ¯ z 34 ,
V = [ 256 ( n 2 + k 2 ) 2 ( N 2 + K 2 ) X 2 ] / B ¯ 2 ,
B ¯ = z 12 z 23 + h 12 h 23 X 2 + 2 X [ X m cos φ Y p sin φ ] ,
R 13 = { h 12 z 23 + z 12 h 23 X 2 + 2 X [ X p cos φ Y m sin φ ] } / B ¯ ,
R 31 = { h 23 z 12 + z 23 h 12 X 2 + 2 X [ X p cos φ + Y m sin φ ] } / B ¯ ,
X m = w 12 w 23 v 12 v 23 ,     X p = w 12 w 23 + v 12 v 23 ,
Y p = w 12 v 23 + v 12 w 23 ,     Y m = w 12 v 23 v 12 w 23 ,
v 23 = 2 ( k N K n ) ,     v 12 = 2 k ,
w 23 = n 2 N 2 + k 2 K 2 ,     w 12 = 1 n 2 k 2 ,
h 23 = ( n N ) 2 + ( k K ) 2 ,     h 12 = ( 1 n ) 2 + k 2 ,
z 23 = ( n + N ) 2 + ( k + K ) 2 ,     z 12 = ( 1 + n ) 2 + k 2 .
T = U ¯ Y B ¯ z 34 r 34 R ¯ 31 Y 2 ,     R = R ¯ 13 z 34 + ( V ¯ R ¯ 13 R ¯ 31 ) r 34 Y 2 B ¯ B ¯ z 34 r 34 R ¯ 31 Y 2   ,
Q 0 = [ ( n + 1 ) 2 + k 2 ] [ ( n + N ) 2 + ( k + K ) 2 ] ,
Q 1 = 2 ( Q c cos φ + Q s sin φ ) ,
Q c = 4 k ( k N n K ) ( n 2 + k 2 1 ) ( n 2 N 2 + k 2 K 2 ) ,
Q s = 2 [ ( k N n K ) ( n 2 + k 2 1 ) + k ( n 2 N 2 + k 2 K 2 ) ] ,
Q 2 = [ ( n 1 ) 2 + k 2 ] [ ( n N ) 2 + ( k K ) 2 ] .
B = [ ( n + 1 ) 2 + k 2 ] [ B 1 B 2 ] ,
B 1 = [ ( n + N ) 2 + ( k + K ) 2 ] [ ( N + 1 ) 2 + K 2 ] ,
B 2 = [ ( n N ) 2 + ( k K ) 2 ] [ ( N 1 ) 2 + K 2 ] Y 2 ,
C = 2 { [ C 1 C 2 ] cos φ [ C 3 + C 4 ] sin φ } ,
C 1 = ( n 2 + k 2 1 ) ( n 2 N 2 + k 2 K 2 ) { [ ( N + 1 ) 2 + K 2 ] [ ( N 1 ) 2 + K 2 ] Y 2 } ,
C 2 = 4 k ( k N n K ) { [ ( N + 1 ) 2 + K 2 ] + [ ( N 1 ) 2 + K 2 ] Y 2 } ,
C 3 = 2 ( n 2 + k 2 1 ) ( k N n K ) { [ ( N + 1 ) 2 + K 2 ] + [ ( N 1 ) 2 + K 2 ] Y 2 } ,
C 4 = 2 k ( n 2 N 2 + k 2 K 2 ) { [ ( N + 1 ) 2 + K 2 ] [ ( N 1 ) 2 + K 2 ] Y 2 } ,
D = [ ( n 1 ) 2 + k 2 ] [ D 1 D 2 ] ,
D 1 = [ ( n N ) 2 + ( k K ) 2 ] [ ( N + 1 ) 2 + K 2 ] ,
D 2 = [ ( n + N ) 2 + ( k + K ) 2 ] [ ( N 1 ) 2 + K 2 ] Y 2 .
R = R ¯ 13 z 34 + [ ( V ¯ R ¯ 13 R ¯ 31 ) r 34 Y 2 ] / [ Q 0 + Q 1 X + Q 2 X 2 ] B C X + D X 2 .
R ¯ 13 z 34 + ( V ¯ R ¯ 13 R ¯ 31 ) r 34 Y 2 Q 0 + Q 1 X + Q 2 X 2 = E + G X 2 + X { H + α × ( P 0 + P 1 X + P 2 X 2 Q 0 + Q 1 X + Q 2 X 2 ) } ,
R ¯ 13 z 34 + ( V ¯ R ¯ 13 R ¯ 31 ) r 34 Y 2 Q 0 + Q 1 X + Q 2 X 2 = E + G X 2 + X × [ P 0 + P 1 X + P 2 X 2 Q 0 + Q 1 X + Q 2 X 2 ] .
E = [ ( n 1 ) 2 + k 2 ] { [ ( n + N ) 2 + ( k + K ) 2 ] [ ( N + 1 ) 2 + K 2 ] [ ( n N ) 2 + ( k K ) 2 ] [ ( N 1 ) 2 + K 2 ] Y 2 } ,
G = [ ( n + 1 ) 2 + k 2 ] { [ ( n N ) 2 + ( k K ) 2 ] [ ( N + 1 ) 2 + K 2 ] [ ( n + N ) 2 + ( k + K ) 2 ] [ ( N 1 ) 2 + K 2 ] Y 2 } .
P 0 = 2 ( P 0 ( c ) cos φ + P 0 ( s ) sin φ ) ,
P 1 = P 1 + 4 ( P 1 ( c c ) cos 2 φ + P 1 ( s s ) sin 2 φ + P 1 ( c s ) cos φ sin φ ) ,
P 2 = 2 ( P 2 ( c ) cos φ + P 2 ( s ) sin φ ) ,
P 0 ( c ) = z 34 z 23 z 12 X p r 34 X p Y 2 ( z 12 h 23 + h 12 z 23 ) + h 12 X m h 23 r 34 Y 2 ,
P 0 ( s ) = z 34 z 23 z 12 Y m + r 34 Y m Y 2 ( z 12 h 23 h 12 z 23 ) h 12 Y p h 23 r 34 Y 2 ,
P 1 = r 34 Y 2 ( z 12 2 h 23 2 256 ( k 2 + n 2 ) 2 ( N 2 + K 2 ) + h 12 2 z 23 2 h 12 2 h 23 2 z 12 2 z 23 2 ) ,
P 1 ( c c ) = X p ( z 34 X m r 34 X p Y 2 ) ,
P 1 ( s s ) = Y m ( z 34 Y p + r 34 Y m Y 2 ) ,
P 1 ( c s ) = z 34 ( Y p X p + Y m X m ) ,
P 2 ( c ) = z 34 h 23 h 12 X p r 34 X p Y 2 ( z 12 h 23 + h 12 z 23 ) + z 12 X m z 23 r 34 Y 2 ,
P 2 ( s ) = z 34 h 23 h 12 Y m r 34 Y m Y 2 ( z 12 h 23 h 12 z 23 ) z 12 Y p z 23 r 34 Y 2 .
R = E + G X 2 + X [ P 0 + P 1 X + P 2 X 2 Q 0 + Q 1 X + Q 2 X 2 ] B C X + D X 2 ,

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