Abstract

The optical constants nf and kf of an absorbing thin solid film, deposited on a transparent or a slightly absorbing substrate, depend in a complex way on measurable quantities such as the transmittance T and the air-incident and substrate-incident reflectances R and R′. Simple analytical formulas, which minimize propagation of experimental errors, are derived for determining ks of the substrate and kf and nf of the film. The formulas are applied to Te alloy films of varying thickness deposited on glass substrates.

© 1983 Optical Society of America

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References

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  1. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965).
  2. H. Wolter, Z. Physik, 105, 269 (1937).
  3. H. E. Bennett, J. M. Bennett, Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, New York, 1967), p. 1.
  4. B. J. Mulder, Philips Res. Rep.27, 315 (1972).
  5. D. Y. Lou, Appl. Opt. 21, 1602 (1982).
    [CrossRef] [PubMed]
  6. S. G. Tomlin, J. Phys. D 1, 1667 (1968).
    [CrossRef]
  7. S. G. Tomlin, J. Phys. D 5, 847 (1972).
    [CrossRef]
  8. R. E. Denton, R. D. Campbell, S. G. Tomlin, J. Phys. D 5, 852 (1972).
    [CrossRef]
  9. K. Truszkowska, T. Borowicz, C. Wesolowska, Appl. Opt. 17, 1579 (1978).
    [CrossRef] [PubMed]
  10. P. O. Nilsson, Appl. Opt. 7, 435 (1968).
    [CrossRef] [PubMed]
  11. A. Hjortsberg, Appl. Opt. 20, 1254 (1981).
    [CrossRef] [PubMed]
  12. C. J. Powell, J. Opt. Soc. Am. 60, 78 (1970);see also references herein.
    [CrossRef]
  13. H. E. Bennett, M. Silver, E. J. Ashley, J. Opt. Soc. Am. 53, 1089 (1963).
    [CrossRef]
  14. J. Stuke, H. Keller, Phys. Status Solidi 7, 189 (1964).
    [CrossRef]
  15. H. Keller, J. Stuke, Phys. Status Solidi 8, 831 (1965).
    [CrossRef]

1982 (1)

1981 (1)

1978 (1)

1972 (2)

S. G. Tomlin, J. Phys. D 5, 847 (1972).
[CrossRef]

R. E. Denton, R. D. Campbell, S. G. Tomlin, J. Phys. D 5, 852 (1972).
[CrossRef]

1970 (1)

1968 (2)

1965 (1)

H. Keller, J. Stuke, Phys. Status Solidi 8, 831 (1965).
[CrossRef]

1964 (1)

J. Stuke, H. Keller, Phys. Status Solidi 7, 189 (1964).
[CrossRef]

1963 (1)

Ashley, E. J.

Bennett, H. E.

H. E. Bennett, M. Silver, E. J. Ashley, J. Opt. Soc. Am. 53, 1089 (1963).
[CrossRef]

H. E. Bennett, J. M. Bennett, Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, New York, 1967), p. 1.

Bennett, J. M.

H. E. Bennett, J. M. Bennett, Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, New York, 1967), p. 1.

Borowicz, T.

Campbell, R. D.

R. E. Denton, R. D. Campbell, S. G. Tomlin, J. Phys. D 5, 852 (1972).
[CrossRef]

Denton, R. E.

R. E. Denton, R. D. Campbell, S. G. Tomlin, J. Phys. D 5, 852 (1972).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965).

Hjortsberg, A.

Keller, H.

H. Keller, J. Stuke, Phys. Status Solidi 8, 831 (1965).
[CrossRef]

J. Stuke, H. Keller, Phys. Status Solidi 7, 189 (1964).
[CrossRef]

Lou, D. Y.

Mulder, B. J.

B. J. Mulder, Philips Res. Rep.27, 315 (1972).

Nilsson, P. O.

Physik, Z.

H. Wolter, Z. Physik, 105, 269 (1937).

Powell, C. J.

Silver, M.

Stuke, J.

H. Keller, J. Stuke, Phys. Status Solidi 8, 831 (1965).
[CrossRef]

J. Stuke, H. Keller, Phys. Status Solidi 7, 189 (1964).
[CrossRef]

Tomlin, S. G.

R. E. Denton, R. D. Campbell, S. G. Tomlin, J. Phys. D 5, 852 (1972).
[CrossRef]

S. G. Tomlin, J. Phys. D 5, 847 (1972).
[CrossRef]

S. G. Tomlin, J. Phys. D 1, 1667 (1968).
[CrossRef]

Truszkowska, K.

Wesolowska, C.

Wolter, H.

H. Wolter, Z. Physik, 105, 269 (1937).

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

J. Phys. D (3)

S. G. Tomlin, J. Phys. D 1, 1667 (1968).
[CrossRef]

S. G. Tomlin, J. Phys. D 5, 847 (1972).
[CrossRef]

R. E. Denton, R. D. Campbell, S. G. Tomlin, J. Phys. D 5, 852 (1972).
[CrossRef]

Phys. Status Solidi (2)

J. Stuke, H. Keller, Phys. Status Solidi 7, 189 (1964).
[CrossRef]

H. Keller, J. Stuke, Phys. Status Solidi 8, 831 (1965).
[CrossRef]

Other (4)

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965).

H. Wolter, Z. Physik, 105, 269 (1937).

H. E. Bennett, J. M. Bennett, Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, New York, 1967), p. 1.

B. J. Mulder, Philips Res. Rep.27, 315 (1972).

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram indicating the transmittance Ts and reflectance Rs of the substrate, the transmittance Tf and reflectance Rf of the film, and the transmittance Tt and the air-incident and substrate-incident reflectances Rt and R t of the thin-film substrate combination.

Fig. 2
Fig. 2

Absorption coefficient αs = 4πkshs0 of the glass substrate calculated with Eq. (8) (open circles and solid curve) and with Eq. (9) (triangles and dashed curve) using experimental values of Ts and Rs and ns = 1.5. The margins show the propagation of errors due to variation of ns from 1.45 to 1.55.

Fig. 3
Fig. 3

Absorption index kf for the eight Te alloy films in the crystalline state calculated using Eq. (41), the first-order approximation which disregards the phase factor q. The film thicknesses are: 20 nm (—), 32 nm (▴ - - - ▴), 39.5 nm (x- -x), 53.8 nm (▫ ‥ … ▫), 69.5 nm (▪ — ▪), 77.7 nm (● - - - ●), 97 nm (○ ‥ … ○), and 120 nm (△ - - △).

Fig. 4
Fig. 4

Absorption index kf for the eight Te alloy films in the amorphous state, calculated using Eq. (41). The film thicknesses as measured in the crystalline state are indicated in Fig. 3. The measured thickness contraction due to crystallization was ∼4%.

Fig. 5
Fig. 5

Absorption index kf for the crystalline and amorphous Te alloy films as calculated with Eq. (42); kf is found to be independent of film thickness hf in the range from 20 to 500 nm. The kf values are accurate within 2% at λ0 = 500 nm, 6–11% at 800 nm, and 17–35% at 1200 nm.

Fig. 6
Fig. 6

Optical constant nf for the crystalline and amorphous Tealloy films as calculated with Eqs. (21) and (42); nf is independent of film thickness from 20 to 500 nm. The nf values are accurate within 3% at λ0 = 500 nm, 1% at 800 nm, and 1% at 1200 nm.

Tables (1)

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Table I Standard Deviations

Equations (42)

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n ̂ s = n s + i k s ,
T a , s = T s , a = 4 n s / ( n s + 1 ) 2 ,
R a , s = R s , a = ( n s 1 ) 2 / ( n s + 1 ) 2 .
T s = T a , s 2 exp ( α s ) 1 R a , s 2 exp ( 2 α s ) ,
R s = R a , s + T a , s 2 R a , s exp ( 2 α s ) 1 R a , s 2 exp ( 2 α s ) ,
α s = 4 π k s h s / λ 0 ,
1 R s T s = 1 R a , s exp ( 2 α s ) T a , s exp ( α s ) ,
exp ( α s ) = T a , s 2 R a , s ( 1 R s T s ) { [ 1 + 4 R a , s T a , s 2 ( T s 1 R s ) 2 ] 1 / 2 1 } .
exp ( α s ) = T s T a , s R a , s + R a , s R s .
n ̂ f = n f + i k f .
t ̂ j , f = 2 n ̂ j / ( n ̂ j + n ̂ f ) ,
r ̂ j , f = ( n ̂ j n ̂ f ) / ( n ̂ j + n ̂ f ) .
T f = ( n 1 / n 0 ) | t ̂ f | 2 ,
R f = | r ̂ f | 2 ,
t ̂ f = t ̂ 0 , f t ̂ f , 1 exp ( i δ ̂ f ) 1 + r ̂ 0 , f r ̂ f , 1 exp ( 2 i δ ̂ f ) ,
r ̂ f = r ̂ 0 , f + r ̂ f , 1 exp ( 2 i δ ̂ f ) 1 + r ̂ 0 , f r ̂ f , 1 exp ( 2 i δ ̂ f ) .
δ ̂ f = 2 π n ̂ f h f / λ 0 ,
1 ± R f T f = ( 1 ± R 0 , f ) [ 1 ± R f , 1 exp ( 2 α f ) ] T 0 , f T f , 1 exp ( α f ) + ( r ̂ 0 , f ± r ̂ 0 , f * ) [ r ̂ f , 1 exp ( 2 i δ ̂ f ) ± r ̂ f , 1 * exp ( 2 i δ ̂ f * ) ] T 0 , f T f , 1 exp ( α f ) ,
α f = 4 π k f h f / λ 0 ,
T 0 , f = T f , 0 = 4 n 0 ( n f 2 + k f 2 ) 1 / 2 ( n 0 + n f ) 2 + k f 2 ,
R 0 , f = R f , 0 = ( n 0 n f ) 2 + k f 2 ( n 0 + n f ) 2 + k f 2 ,
R f , 1 exp ( 2 α f ) + ( x q ) exp ( α f ) 1 = 0 ,
R f , 1 exp ( 2 α f ) + ( p y ) exp ( α f ) + 1 = 0 ,
x = ( 1 R f , 1 ) ( 1 R f T f ) ,
y = ( T 0 , f T f , 1 1 + R 0 , f ) ( 1 + R f T f ) ,
q = [ ( 1 n 1 n f ) + R f , 1 ( 1 + n 1 n f ) ] k f n f × sin ϕ f + k f 2 n f 2 ( 1 R f , 1 ) cos ϕ f ,
p = [ n 0 n f ( 1 R 0 , f 1 + R 0 , f ) 1 ] [ { ( 1 n 1 n f ) + R f , 1 ( 1 + n 1 n f ) } cos ϕ f k f n f ( 1 R f , 1 ) sin ϕ f ] ,
ϕ f = 4 π n f h f / λ 0 .
exp ( α f ) = x q 2 R f , 1 [ ( 1 + 4 R f , 1 ( x q ) 2 ) 1 / 2 1 ] .
exp ( α f ) = 2 / ( x + y q p )
exp ( α f ) = ( y + q x p ) / 2 R f , 1 .
T t = T s , a T f exp ( α s ) 1 R s , a R f exp ( 2 α s ) ,
R t = R f + T f 2 R s , a exp ( 2 α s ) 1 R s , a R f exp ( 2 α s ) ,
R t = R s , a + T s , a 2 R f exp ( 2 α s ) 1 R s , a R f exp ( 2 α s ) ,
T f = T t T s , a exp ( α s ) R s , a R t + T s , a R s , a ,
R f = R t T t 2 R s , a R s , a R t + T s , a R s , a ,
R f = ( R t R s , a ) exp ( 2 α s ) R s , a R t + T s , a R s , a .
1 R t T t = 1 R f exp ( 2 α s ) T f exp ( α s ) .
1 R f T f = [ 1 + ( n s 1 ) 2 2 n s α s ] ( 1 R t T t ) α s ( 1 + R t T t ) + 0 ( α s 2 ) .
1 R f T f = ( 1 R t T t ) α s ( 1 + R t T t ) ,
k f = λ 0 4 π h f ln [ x 2 R f , a { ( 1 + 4 R f , a x 2 ) 1 / 2 1 } ] .
k f = λ 0 4 π h f ln [ ( x q ) 2 R f , a { ( 1 + 4 R f , a ( x q ) 2 ) 1 / 2 1 } ] ,

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