Abstract

Theoretical studies were conducted for thickness measurements using transparent substrates on the external and internal reflection configurations. For three-phase systems consisting of ambient, film, and substrate, the refractive index of the substrate could be optimized to obtain the high sensitivity of an ellipsometric quantity Δ to the film thickness and the small susceptibility of Δ to errors in the incident angle. It was shown that the combination of an ordinary glass substrate and an additional dielectric layer with an appropriate layer thickness works as a synthetic high-index single substrate (SHIS). The optical effect of the combination was approximately described by use of the effective refractive index of SHIS. A method to select the refractive index of the additional layer was also given.

© 2005 Optical Society of America

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References

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  1. R. M. A. Azzam, M. Elshazly-Zaghloul, N. M. Bashara, “Combined reflection and transmission thin-film ellipsometry: A unified linear analysis,” Appl. Opt. 14, 1652–1663 (1975).
    [Crossref] [PubMed]
  2. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  3. S. Otsuki, K. Ohta, K. Tamada, S. Wakida, “Thickness measurements on transparent substrates based on reflection ellipsometry. II. Estimation of the theoretical sensitivity and precision,” submitted for publication in Appl. Opt.
  4. S. Otsuki, K. Tamada, S. Wakida, “Two-dimensional thickness measurements based on internal reflection ellipsometry,” Appl. Opt. 44, 1410–1415 (2005).
    [Crossref] [PubMed]
  5. D. Den Engelsen, “Ellipsometry of anisotropic films,” J. Opt. Soc. Am. 61, 1460–1466 (1971).
    [Crossref]
  6. L. A. Catalán, “p-Polarized reflectance for transparent thin films on transparent substrates,” J. Opt. Soc. Am. 55, 857–859 (1965).
    [Crossref]
  7. F. M. W. Kroesen, G. S. Oehrlein, E. de Frésart, M. Haverlag, “Depth profiling of Ge concentration in SiGe alloys using in situ ellipsometry during reactive-ion etching,” J. Appl. Phys. 73, 8017–8026 (1993).
    [Crossref]
  8. D. E. Aspnes, “Minimal-data approaches for determining outer-layer dielectric responses of films from kinetic reflectometric and ellipsometric measurements,” J. Opt. Soc. Am. A 10, 974–983 (1993).
    [Crossref]

2005 (1)

1993 (2)

F. M. W. Kroesen, G. S. Oehrlein, E. de Frésart, M. Haverlag, “Depth profiling of Ge concentration in SiGe alloys using in situ ellipsometry during reactive-ion etching,” J. Appl. Phys. 73, 8017–8026 (1993).
[Crossref]

D. E. Aspnes, “Minimal-data approaches for determining outer-layer dielectric responses of films from kinetic reflectometric and ellipsometric measurements,” J. Opt. Soc. Am. A 10, 974–983 (1993).
[Crossref]

1975 (1)

1971 (1)

1965 (1)

Aspnes, D. E.

Azzam, R. M. A.

Bashara, N. M.

Catalán, L. A.

de Frésart, E.

F. M. W. Kroesen, G. S. Oehrlein, E. de Frésart, M. Haverlag, “Depth profiling of Ge concentration in SiGe alloys using in situ ellipsometry during reactive-ion etching,” J. Appl. Phys. 73, 8017–8026 (1993).
[Crossref]

Den Engelsen, D.

Elshazly-Zaghloul, M.

Haverlag, M.

F. M. W. Kroesen, G. S. Oehrlein, E. de Frésart, M. Haverlag, “Depth profiling of Ge concentration in SiGe alloys using in situ ellipsometry during reactive-ion etching,” J. Appl. Phys. 73, 8017–8026 (1993).
[Crossref]

Kroesen, F. M. W.

F. M. W. Kroesen, G. S. Oehrlein, E. de Frésart, M. Haverlag, “Depth profiling of Ge concentration in SiGe alloys using in situ ellipsometry during reactive-ion etching,” J. Appl. Phys. 73, 8017–8026 (1993).
[Crossref]

Oehrlein, G. S.

F. M. W. Kroesen, G. S. Oehrlein, E. de Frésart, M. Haverlag, “Depth profiling of Ge concentration in SiGe alloys using in situ ellipsometry during reactive-ion etching,” J. Appl. Phys. 73, 8017–8026 (1993).
[Crossref]

Ohta, K.

S. Otsuki, K. Ohta, K. Tamada, S. Wakida, “Thickness measurements on transparent substrates based on reflection ellipsometry. II. Estimation of the theoretical sensitivity and precision,” submitted for publication in Appl. Opt.

Otsuki, S.

S. Otsuki, K. Tamada, S. Wakida, “Two-dimensional thickness measurements based on internal reflection ellipsometry,” Appl. Opt. 44, 1410–1415 (2005).
[Crossref] [PubMed]

S. Otsuki, K. Ohta, K. Tamada, S. Wakida, “Thickness measurements on transparent substrates based on reflection ellipsometry. II. Estimation of the theoretical sensitivity and precision,” submitted for publication in Appl. Opt.

Tamada, K.

S. Otsuki, K. Tamada, S. Wakida, “Two-dimensional thickness measurements based on internal reflection ellipsometry,” Appl. Opt. 44, 1410–1415 (2005).
[Crossref] [PubMed]

S. Otsuki, K. Ohta, K. Tamada, S. Wakida, “Thickness measurements on transparent substrates based on reflection ellipsometry. II. Estimation of the theoretical sensitivity and precision,” submitted for publication in Appl. Opt.

Wakida, S.

S. Otsuki, K. Tamada, S. Wakida, “Two-dimensional thickness measurements based on internal reflection ellipsometry,” Appl. Opt. 44, 1410–1415 (2005).
[Crossref] [PubMed]

S. Otsuki, K. Ohta, K. Tamada, S. Wakida, “Thickness measurements on transparent substrates based on reflection ellipsometry. II. Estimation of the theoretical sensitivity and precision,” submitted for publication in Appl. Opt.

Appl. Opt. (2)

J. Appl. Phys. (1)

F. M. W. Kroesen, G. S. Oehrlein, E. de Frésart, M. Haverlag, “Depth profiling of Ge concentration in SiGe alloys using in situ ellipsometry during reactive-ion etching,” J. Appl. Phys. 73, 8017–8026 (1993).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Other (2)

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

S. Otsuki, K. Ohta, K. Tamada, S. Wakida, “Thickness measurements on transparent substrates based on reflection ellipsometry. II. Estimation of the theoretical sensitivity and precision,” submitted for publication in Appl. Opt.

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Figures (10)

Fig. 1
Fig. 1

(a) ER configuration and (b) IR configuration for ellipsometry using transparent substrates.

Fig. 2
Fig. 2

Optical system composed of three phases; light-incident medium (0)–film (1)–light-transmitted medium (2).

Fig. 3
Fig. 3

Δ–incident-angle relationship at 600 nm with varying thicknesses of the sample for three-phase systems composed of glass–sample–air (IR): (a) BK7 (n = 1.52) and (b) SF18 (n = 1.722).

Fig. 4
Fig. 4

Values of the factors G and G′ as a function of the refractive index of the substrate on the IR and ER configurations; refractive indices: 1.0 (ambient, air), 1.33 (ambient, water), and 1.45 (sample).

Fig. 5
Fig. 5

Structures of the composed substrate (a) without and (b) with a sample in comparison with the ordinary single substrate (c) without and (d) with a sample.

Fig. 6
Fig. 6

Δ–incident-angle relationship at a wavelength of 600 nm with varying thicknesses of the sample; (a) air–sample (n = 1.45)–additional layer (n = 2.43, d = 67.2 nm)–substrate (n = 1.458) (ER) and (b) substrate (n = 1.458)–additional layer (n = 2.43, d = 67.2 nm)–sample (n = 1.45)–air (IR).

Fig. 7
Fig. 7

Relationship of the imaginary part of ρ and the thickness of the additional layer at 600 nm with varying incident angle; substrate (n = 1.458)–additional layer (n = 2.43, d = 67.2 nm)–air (IR).

Fig. 8
Fig. 8

Relationship of the relative variation of (n12n02 sin2 ϕ)−1/2 and the incident angle; ambient–additional layer–substrate (ER) and substrate–additional layer–ambient (IR); refractive indices: 1.0 (ambient, air), 1.33 (ambient, water), 2.43 (additional layer), and 1.458 (substrate); all phases are assumed to be transparent.

Fig. 9
Fig. 9

Optical system composed of four phases; ambient (0)–sample (1)–additional layer (2)–substrate (3) (ER); substrate (0)–additional layer (1)–sample (2)–ambient (3) (IR).

Fig. 10
Fig. 10

Δ–incident-angle relationship at 600 nm with varying thicknesses of the sample; (a) air–sample (n = 1.45)–additional layer (n = 2.43, d = 67.2 nm)–substrate (n = 1.458) and air–sample (n = 1.45)–SHIS (n′ = 3.469) and (b) substrate (n = 1.458)–additional layer (n = 2.43, d = 67.2 nm)–sample (n = 1.45)–air and SHIS (n′ = 3.469)–sample (n = 1.45)–air.

Equations (34)

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ρ = R p R s = tan Ψ exp ( j Δ ) ,
R ν = r 01 ν + r 12 ν X 1 1 + r 01 ν r 12 ν X 1 ,
X 1 = exp [ - j 4 π ( d 1 / λ ) ( n 1 2 - n 0 2 sin 2 ϕ ) 1 / 2 ] ,
r 01 ν r 21 ν ,             r 12 ν r 10 ν .
r 21 ν = - r 12 ν ,             r 10 ν = - r 01 ν .
R ν = - r 12 ν + r 01 ν X 1 1 + r 01 ν r 12 ν X 1 ,
δ Δ = Δ - Δ ¯ = - 4 π d 1 λ G sin ϕ tan ϕ 1 - ( n 0 / n 2 ) 2 tan 2 ϕ .
G = n 0 n 1 2 ( n 1 2 - n 0 2 ) n 2 2 - n 1 2 n 2 2 - n 0 2 ,
d ( δ Δ ) d ϕ = - 4 π d 1 λ G { ( cos ϕ + sec ϕ ) tan ϕ 1 - ( n 0 / n 2 ) 2 tan 2 ϕ + 2 ( n 0 / n 2 ) 2 tan ϕ sec 2 ϕ [ 1 - ( n 0 / n 2 ) 2 tan 2 ϕ ] 2 } .
d ( δ Δ ) d ϕ = - 8 π d 1 λ G tan ϕ sec 2 ϕ [ 1 - ( n 0 / n 2 ) 2 tan 2 ϕ ] 2 ,
G = n 0 3 n 1 2 n 2 2 ( n 1 2 - n 0 2 ) n 2 2 - n 1 2 n 2 2 - n 0 2 .
ρ = r 01 p + r 12 p X 1 1 + r 01 p r 12 p X 1 1 + r 01 s r 12 s X 1 r 01 s + r 12 s X 1 .
X 1 = exp [ - j 2 π ( d 1 / D ϕ ) ] ,
D ϕ = ( 1 / 2 ) λ ( n 1 2 - n 0 2 sin 2 ϕ ) - 1 / 2 .
d 1 = [ ( 2 m + 1 ) / 2 ] D ϕ             ( m = 0 , 1 , 2 , ) .
r 01 p = r 12 p .
n 1 = ( n 0 n 2 cos 2 ϕ 1 / cos ϕ cos ϕ 2 ) 1 / 2 ,
R ν = r 01 ν - r 12 ν 1 - r 01 ν r 12 ν .
g j p = n j cos ϕ j ,             g j s = n j cos ϕ j ,
r j ( j + 1 ) ν = g j + 1 ν - g j ν g j + 1 ν + g j ν
R ν = g 1 ν 2 - g 0 ν g 2 ν g 1 ν 2 + g 0 ν g 2 ν .
R ν = g 1 ν 2 / g 2 ν - g 0 ν g 1 ν 2 / g 2 ν + g 0 ν = g 2 ν - g 0 ν g 2 ν + g 0 ν ,
g 2 ν = g 1 ν 2 / g 2 ν .
R ν = - g 2 ν - g 1 ν 2 / g 0 ν g 2 ν + g 1 ν 2 / g 0 ν = - g 2 ν - g 0 ν g 2 ν + g 0 ν ,
g 0 ν = g 1 ν 2 / g 0 ν .
r 02 ν = - r 02 ν
R ν = r 01 ν + r ¯ 13 ν X 1 1 + r 01 ν r ¯ 13 ν X 1 ,
r ¯ 13 ν = r 12 ν + r 23 ν X 2 1 + r 12 ν r 23 ν X 2 ,
R ν = r ¯ 02 ν + α r 23 ν X 2 1 + r ¯ 02 ν r 23 ν X 2 ,
r ¯ 02 ν = r 01 ν + r 12 ν X 1 1 + r 01 r 12 ν X 1 ,
r ¯ 02 ν = r 01 ν X 1 + r 12 ν 1 + r 01 ν r 12 ν X 1 ,
α = r 01 ν r 12 ν + X 1 1 + r 01 ν r 12 ν X 1 ,
R ν = r 02 ν + r 23 ν X 2 1 + r 02 ν r 23 ν X 2 .
n = n 0 tan ϕ B .

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