Abstract

A method of obtaining accurate values for the optical parameters of thin film substrates is described. Calculations for reflected and transmitted fluxes in substrate sets are developed allowing for double reflection. A simple procedure to fit measurements is also illustrated. Values for substrate parameters are extracted and used to correct for substrate influence in thin film spectrophotometric measurements. Corrected thin film measurements and data calculated with obtained optical constants are compared.

© 1983 Optical Society of America

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References

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  1. J. P. Marton, M. Schlesinger, J. Appl. Phys. 40, 4529 (1969).
    [CrossRef]
  2. G. Baldini, L. Rigaldi, J. Opt. Soc. Am. 60, 495 (1970).
    [CrossRef]
  3. A. Panatta, M. Matzeu, unpublished.
  4. F. Abelès, in Advanced Optical Techniques, A. C. S. van Heel, Ed. (North-Holland, Amsterdam, 1967).
  5. H. L. Alder, E. R. Roessler, Introduction to Probability and Statistics (Freeman, San Francisco, 1964).

1970 (1)

1969 (1)

J. P. Marton, M. Schlesinger, J. Appl. Phys. 40, 4529 (1969).
[CrossRef]

Abelès, F.

F. Abelès, in Advanced Optical Techniques, A. C. S. van Heel, Ed. (North-Holland, Amsterdam, 1967).

Alder, H. L.

H. L. Alder, E. R. Roessler, Introduction to Probability and Statistics (Freeman, San Francisco, 1964).

Baldini, G.

Marton, J. P.

J. P. Marton, M. Schlesinger, J. Appl. Phys. 40, 4529 (1969).
[CrossRef]

Matzeu, M.

A. Panatta, M. Matzeu, unpublished.

Panatta, A.

A. Panatta, M. Matzeu, unpublished.

Rigaldi, L.

Roessler, E. R.

H. L. Alder, E. R. Roessler, Introduction to Probability and Statistics (Freeman, San Francisco, 1964).

Schlesinger, M.

J. P. Marton, M. Schlesinger, J. Appl. Phys. 40, 4529 (1969).
[CrossRef]

J. Appl. Phys. (1)

J. P. Marton, M. Schlesinger, J. Appl. Phys. 40, 4529 (1969).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (3)

A. Panatta, M. Matzeu, unpublished.

F. Abelès, in Advanced Optical Techniques, A. C. S. van Heel, Ed. (North-Holland, Amsterdam, 1967).

H. L. Alder, E. R. Roessler, Introduction to Probability and Statistics (Freeman, San Francisco, 1964).

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Figures (3)

Fig. 1
Fig. 1

(a) Parameters used to characterize optical behavior of one substrate: surface reflectance (ρ), surface transmittance (τ), and internal transmittance (θ). (b) Schematic representation of a set of substrates as used.

Fig. 2
Fig. 2

Comparison between measured and calculated reflectance R, transmittance T, and −lnT for a set of substrates vs number n of set elements: (●) measured, (▲) calculated by parameter values given by fitting and (full curves) behavior according to refined values. The straight line in (b), through the experimental values, is the regression line through the origin, with a = 0.0702, r = 0.999 (see the Appendix for details).

Fig. 3
Fig. 3

Behavior of final obtained values of substrate parameters vs wavelength (see also Table I).

Tables (2)

Tables Icon

Table I Values for Substrate Parameters as Determined by Best Fit and After Refining

Tables Icon

Table II Reflectance and Transmittance Values Measured and Recalculated for a 10-nm Gold Film Deposited on Quartz Glass Substrate a

Equations (39)

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R m = R f + T f 2 ρ τ 2 / B , R m = ρ + ( 1 ρ ) 2 τ 2 R f / B , T m = T f τ ( 1 ρ ) / B ,
B = 1 R f ρ τ 2
R f = R m ρ T m / D , R f = ( R m ρ ) / τ 2 D , T f = T m ( 1 ρ ) / τ D ,
D = 1 + R m ρ 2 ρ .
R s = ρ [ 1 + ( 1 ρ ) 2 τ 2 / ( 1 ρ 2 τ 2 ) ] , T s = τ ( 1 ρ ) 2 / ( 1 ρ 2 τ 2 ) .
R 1 = ρ ( 1 + θ 2 τ 2 ) , T 1 = τ θ 2 .
R = k = 1 n ( T 1 k 1 R 1 ) T k 1 = R 1 k = 0 n 1 T 1 2 k .
θ = 1 ρ , θ 2 1 2 ρ ,
R ρ k = 0 n 1 τ 2 k { 1 + τ 2 2 ρ [ τ 2 + 2 k ( 1 + τ 2 ] } .
T 1 k 1 θ 2 τ 3 ρ 2 = T 1 k ( τ ρ ) 2 .
I = T 1 k 1 R 1 T k j 1 .
I = R 1 T 1 k j = R 2 T 1 k 2 · T 1 2 ( k j ) .
T 1 k 2 R 1 2 j = 1 k 1 T 1 2 ( k 1 ) = T 1 k R 1 2 j = 0 k 2 T 2 j .
T 1 n [ n ( τ ρ ) 2 + R 1 2 k = 2 n j = 0 k 2 T 1 2 j ] .
k = 0 n 2 ( n k 1 ) T 1 2 k .
T = T 1 n [ 1 + n ( τ ρ ) 2 + R 1 2 k = 0 n 2 ( n k 1 ) T 2 k ] .
T = τ n { 1 2 n ρ + ρ 2 [ n ( 2 n 1 ) + n τ 2 + ( 1 + τ 2 ) k = 0 n 2 ( n k 1 ) τ 2 k ] } .
R = 1 τ 2 n 1 τ 2 ( 1 + τ 2 2 ρ τ 2 { 1 + 2 ( 1 + τ 2 ) ( 1 τ 2 ) ( 1 τ 2 n ) [ 1 n τ 2 ( n 1 ) + ( n 1 ) τ 2 n ] } ) ,
T = τ n { 1 2 n ρ + ρ 2 [ n ( 2 n 1 ) + n τ 2 + ( 1 + τ 2 1 τ 2 ) ( n 1 n τ 2 + τ 2 n ) ] } .
ln T = ( 2 ρ ln τ ) · n .
R = 1 τ 2 n 1 τ 2 ( 1 + τ 2 ) .
R = 2 ρ n [ 1 n ( 1 τ ) ] ;
R 2 ρ τ · n .
2 ρ ln τ = a ,
2 ρ τ = b ,
ln ( 1 τ ) = a b / τ .
δ b = δ R = R ( δ ρ ρ + δ τ τ ) , δ a = 2 δ ρ δ τ / τ
δ a 7 × 10 5 ,
R f = R ( n f , k f , d / λ , n o , n s ) , T f = T ( n f , k f , d / λ , n o , n s ) ,
k = 0 n 1 τ k = 1 τ 2 n 1 T 2 ,
k = 0 n 1 k τ 2 k = τ 2 ( 1 τ 2 ) 2 [ 1 n τ 2 ( n 1 ) + ( n 1 ) τ 2 n ] ,
k = 0 n 2 ( n k 1 ) T 2 k = n 1 n τ 2 + τ 2 n ( 1 τ 2 ) 2 .
R = 2 ρ n [ 1 ( 2 n 1 ) ρ ] , T = 1 2 ρ n + 2 ρ 2 n ( 2 n 1 ) .
for n = 1 : R = ρ ( 1 + τ 2 2 ρ τ 2 ) , T = τ [ 1 2 ρ + ρ 2 ( 1 + τ 2 ) ] ,
for n = 2 : R = ρ ( 1 + τ 2 ) ( 1 + τ 2 6 ρ τ 2 ) , T = τ 2 [ 1 4 ρ + ρ 2 ( 7 + 4 τ 2 + τ 4 ) ] ,
for n = 3 : R = ρ [ ( 1 + τ 2 ) ( 1 + τ 2 + τ 4 ) 2 ρ τ 2 ( 3 + 7 τ 2 + 5 τ 4 ) ] T = τ 3 [ 1 6 ρ + ρ 2 ( 17 + 8 τ 2 + 4 τ 4 + τ 6 ) ] .
y = a x + q ( regression of y on x ) , x = ( 1 / a ) y + q ( regression of x on y ) .
a ¯ δ a = a , a ¯ + δ a = a , r = a / a
δ a = 1 r 2 2 r p ,

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