Abstract

The normal-incidence transmissivity and reflectivities (front and back) for the incoherent case are calculated for an absorbing-plane sample with linear variation of the thickness. Closed-form expressions for the direct determination of the energy (intensity) coefficients of a free-standing sample and of a film on a transparent substrate are given. The results are illustrated with the simulated infrared spectra of semiconductor InSb films of different thicknesses.

© 1996 Optical Society of America

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References

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  1. D. B. Kushev, “Influence of finite spectral width on the interference spectra of a thin absorbing film on a transparent substrate,” Infrared Phys. Technol. (to be published).
  2. D. B. Kushev, N. N. Zheleva, “Transmittivity, reflectivities and absorptivities of semiconductor film with linear variation in thickness,” J. Phys. D 28, 1239–1243 (1995).
    [CrossRef]
  3. D. Siapkas, D. B. Kushev, N. N. Zheleva, J. Siapkas, I. Lelidis, “Optical constants of tin telluride determined from infrared interference spectra,” Infrared Phys. 31, 425–433 (1991).
    [CrossRef]
  4. E. E. Bell, “Optical constants and their measurements,” in Encyclopedia of Physics, S. Flügge, ed. (Springer-Verlag, Berlin, 1967), Vol. 25, Part 2a, pp. 1–58.

1995

D. B. Kushev, N. N. Zheleva, “Transmittivity, reflectivities and absorptivities of semiconductor film with linear variation in thickness,” J. Phys. D 28, 1239–1243 (1995).
[CrossRef]

1991

D. Siapkas, D. B. Kushev, N. N. Zheleva, J. Siapkas, I. Lelidis, “Optical constants of tin telluride determined from infrared interference spectra,” Infrared Phys. 31, 425–433 (1991).
[CrossRef]

Bell, E. E.

E. E. Bell, “Optical constants and their measurements,” in Encyclopedia of Physics, S. Flügge, ed. (Springer-Verlag, Berlin, 1967), Vol. 25, Part 2a, pp. 1–58.

Kushev, D. B.

D. B. Kushev, N. N. Zheleva, “Transmittivity, reflectivities and absorptivities of semiconductor film with linear variation in thickness,” J. Phys. D 28, 1239–1243 (1995).
[CrossRef]

D. Siapkas, D. B. Kushev, N. N. Zheleva, J. Siapkas, I. Lelidis, “Optical constants of tin telluride determined from infrared interference spectra,” Infrared Phys. 31, 425–433 (1991).
[CrossRef]

D. B. Kushev, “Influence of finite spectral width on the interference spectra of a thin absorbing film on a transparent substrate,” Infrared Phys. Technol. (to be published).

Lelidis, I.

D. Siapkas, D. B. Kushev, N. N. Zheleva, J. Siapkas, I. Lelidis, “Optical constants of tin telluride determined from infrared interference spectra,” Infrared Phys. 31, 425–433 (1991).
[CrossRef]

Siapkas, D.

D. Siapkas, D. B. Kushev, N. N. Zheleva, J. Siapkas, I. Lelidis, “Optical constants of tin telluride determined from infrared interference spectra,” Infrared Phys. 31, 425–433 (1991).
[CrossRef]

Siapkas, J.

D. Siapkas, D. B. Kushev, N. N. Zheleva, J. Siapkas, I. Lelidis, “Optical constants of tin telluride determined from infrared interference spectra,” Infrared Phys. 31, 425–433 (1991).
[CrossRef]

Zheleva, N. N.

D. B. Kushev, N. N. Zheleva, “Transmittivity, reflectivities and absorptivities of semiconductor film with linear variation in thickness,” J. Phys. D 28, 1239–1243 (1995).
[CrossRef]

D. Siapkas, D. B. Kushev, N. N. Zheleva, J. Siapkas, I. Lelidis, “Optical constants of tin telluride determined from infrared interference spectra,” Infrared Phys. 31, 425–433 (1991).
[CrossRef]

Infrared Phys.

D. Siapkas, D. B. Kushev, N. N. Zheleva, J. Siapkas, I. Lelidis, “Optical constants of tin telluride determined from infrared interference spectra,” Infrared Phys. 31, 425–433 (1991).
[CrossRef]

J. Phys. D

D. B. Kushev, N. N. Zheleva, “Transmittivity, reflectivities and absorptivities of semiconductor film with linear variation in thickness,” J. Phys. D 28, 1239–1243 (1995).
[CrossRef]

Other

D. B. Kushev, “Influence of finite spectral width on the interference spectra of a thin absorbing film on a transparent substrate,” Infrared Phys. Technol. (to be published).

E. E. Bell, “Optical constants and their measurements,” in Encyclopedia of Physics, S. Flügge, ed. (Springer-Verlag, Berlin, 1967), Vol. 25, Part 2a, pp. 1–58.

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Figures (3)

Fig. 1
Fig. 1

Thin absorbing film on a transparent substrate. The thickness of the film varies from d ¯ − Δd to d ¯ + Δd.

Fig. 2
Fig. 2

Infrared transmissivity of an InSb film, with linear variation in its thickness, on a transparent substrate. The spectra of a film with a uniform (mean d ¯ ) thickness are also shown: (a) d ¯ = 2 μm, Δd = 2 μm, and nS = 1.5; (b) d ¯ = 20 μm, Δd = 10 μm, and nS = 1.5; and (c) d ¯ = 0.5 mm, Δd = 0.5 μm, and nS = 1.

Fig. 3
Fig. 3

Differences between the infrared spectra of a film with linear variation in its thickness and of a plane-parallel film with a uniform thickness d ¯ : The parameters for (a), (b), and (c) are the same as for (a), (b), and (c), respectively, in Fig. 2. The thickest curves represent the transmissivity, the medium-weight curves, the back reflectivity, and the thinnest curves, the front reflectivity.

Equations (16)

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T = T T A 1 R 2 A 2 = T T R m = 1 R 2 m 1 A 2 m 1 ,
R = ( n 1 ) 2 + k 2 ( n + 1 ) 2 + k 2 , T = 4 n ( n + 1 ) 2 + k 2 , T = 4 n ( 1 + n ) 2 + k 2 ,
1 β 2 β 1 β 1 β 2 exp ( m β ) d β = exp ( m β ) sinh ( m Δ β ) m Δ β , β 2 , 1 = α ( d ¯ ± Δ d ) = β ± Δ β , β = α d ¯ , Δ β = α Δ d ,
T = T T R m = 1 R 2 m 1 A 2 m 1 sinh [ ( 2 m 1 ) Δ β ] ( 2 m 1 ) Δ β = T T R { m = 1 R m A m sinh ( m Δ β ) m Δ β m = 1 R 2 m A 2 m sinh ( m 2 Δ β ) m 2 Δ β } = T T R { 1 Δ β arctanh [ R A sinh ( Δ β ) 1 R A cosh ( Δ β ) ] 1 2 Δ β arctanh [ R 2 A 2 sinh ( 2 Δ β ) 1 R 2 A 2 cosh ( 2 Δ β ) ] } .
R = R + T T R A 2 1 R 2 A 2 = R + T T R m = 1 R 2 m A 2 m .
R = R + T T R m = 1 R 2 m A 2 m sinh ( m 2 Δ β ) m 2 Δ β = R + T T R 1 2 Δ β arctanh [ R 2 A 2 sinh ( 2 Δ β ) 1 R 2 A 2 cosh ( 2 Δ β ) ] .
A = 1 R T .
T = T 12 T 23 A 1 R 21 R 23 A 2 = T 12 T 23 ( R 21 R 23 ) 1 / 2 m = 1 R 21 m 1 / 2 R 23 m 1 / 2 A 2 m 1 ,
R i t = ( n i n t ) 2 + ( k i k t ) 2 ( n i + n t ) 2 + ( k i + k t ) 2 , T i t = 4 ( n i n t + k i k t ) ( n i + n t ) 2 + ( k i + k t ) 2 , i , t = 1 , 2 , 3.
T = T 12 T 23 R 21 R 23 { m = 1 R 21 m / 2 R 23 m / 2 A m sinh ( m Δ β ) m Δ β m = 1 R 21 m R 23 m A 2 m sinh ( m 2 Δ β ) m 2 Δ β } = T 12 T 23 ( R 21 R 23 ) 1 / 2 × { 1 Δ β arctanh [ ( R 21 R 23 ) 1 / 2 A sinh ( Δ β ) 1 ( R 21 R 23 ) 1 / 2 A cosh ( Δ β ) ] 1 2 Δ β arctanh [ R 21 R 23 A 2 sinh ( 2 Δ β ) 1 R 21 R 23 A 2 cosh ( 2 Δ β ) ] } .
R = R 12 + T 12 T 21 R 23 A 2 1 R 21 R 23 A 2 = R 12 + T 12 T 21 R 21 m = 1 R 21 m R 23 m A 2 m ,
R = R 12 + T 12 T 21 R 21 m = 1 R 21 m R 23 m A 2 m sinh ( m 2 Δ β ) m 2 Δ β = R 12 + T 12 T 21 R 21 1 2 Δ β × arctanh [ R 21 R 23 A 2 sinh ( 2 Δ β ) 1 R 21 R 23 A 2 cosh ( 2 Δ β ) ] .
R = R 32 + T 32 T 23 R 21 A 2 1 R 21 R 23 A 2 = R 32 + T 32 T 23 R 23 m = 1 R 21 m R 23 m A 2 m ,
R = R 32 + T 32 T 23 R 23 m = 1 R 21 m R 23 m A 2 m sinh ( m 2 Δ β ) m 2 Δ β = R 32 + T 32 T 23 R 23 1 2 Δ β × arctanh [ R 21 R 23 A 2 sinh ( 2 Δ β ) 1 R 21 R 23 A 2 cosh ( 2 Δ β ) ] .
A = 1 R T , A = 1 R T .
ɛ 1 ( ω ) = ɛ + ( ɛ S ɛ ) ( ω T 2 ω 2 ) ω T 2 ( ω T 2 ω 2 ) 2 + ω 2 γ 2 ɛ ω p 2 ω 2 + γ 0 2 , ɛ 2 ( ω ) = ( ɛ S ɛ ) ω T 2 ω γ ( ω T 2 ω 2 ) 2 + ω 2 γ 2 + ɛ ω p 2 γ 0 ω ( ω 2 + γ 0 2 ) .

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