Abstract

A closed-form inversion expression for obtaining the optical constant (complex refractive index) of the substrate of an absorbing-film–absorbing-substrate system from one reflection ellipsometry measurement is derived. If, in addition, the film thickness is to be determined, a second measurement at another angle of incidence may or may not be used. The derived formula does not introduce errors itself, and tolerates errors in input variables very well. Random and systematic errors in the measured ellipsometric parameters do not affect the value obtained for the optical constant of the substrate: it is always the true value to two decimal places. Two examples in ellipsometry and in the design of reflection-type optical devices, one each, are presented and discussed. In addition, experimental results for a commercially available wafer are also presented. Two other closed-form inversion expressions for obtaining the optical constant of the substrate from two and three measurements are also presented and briefly discussed.

© 2005 Optical Society of America

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References

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  1. See for example, D. Goldstein, Polarized Light (Marcel Dekker, 2003).
  2. D. Kouznetsov, Al. Hofrichter, B. Drevillon, “Direct numerical inversion method for kinetic ellipsometric data. I. Presentation of the method and numerical evaluation,” Appl. Opt. 41, 4510–4518 (2002).
    [CrossRef] [PubMed]
  3. A. Hofrichter, D. Kouznetsov, P. Bulkin, B. Drevillon, “Direct numerical inversion method for kinetic ellipsometric data. II. Implementation and experimental verification,” Appl. Opt. 41, 4519–4525 (2002).
    [CrossRef] [PubMed]
  4. R. M.A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  5. M. Kildemo, “Real-time monitoring and growth control of Si-gradient-index structures by multiwavelength ellipsometry,” Appl. Opt. 37, 113–124 (1998).
    [CrossRef]
  6. R. M.A. Azzam, A. R.M. Zaghloul, N. M. Bashara, “Polarizer-surface-analyzer null ellipsometry for film–substrate systems,” J. Opt. Soc. Am. 65, 1464–1471 (1975).
    [CrossRef]
  7. H. Zhu, L. Liu, Y. Wen, Z. Lu, B. Zhang, “High-precision system for automatic null ellipsometric measurement,” Appl. Opt. 41, 4536–4540 (2002).
    [CrossRef] [PubMed]
  8. S. C. Russev, T. Vl. Arguiro, “Rotating analyzer-fixed analyzer ellipsometer based on null type ellipsometer,” Rev. Sci. Instrum. 70, 3077–3082 (1999).
    [CrossRef]
  9. R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation, and real-time applications,” Rev. Sci. Instrum. 61, 2029–2062 (1990).
    [CrossRef]
  10. A. R.M. Zaghloul, R. M.A. Azzam, N. M. Bashara, “Inversion of the nonlinear equations of reflection ellipsometry on film–substrate systems,” Surf. Sci. 5, 87–96 (1976).
    [CrossRef]
  11. Y. Yoriume, “Method of numerical inversion of the ellipsometry equation for transparent film,” J. Opt. Soc. Am. 73, 888–891 (1983).
    [CrossRef]
  12. D. A. Tonova, “Inverse profiling by ellipsometry: a Newton–Kantorovitch algorithm,” Opt. Commun. 105, 104–112 (1994).
    [CrossRef]
  13. T. Easwarakhanthan, S. Ravelet, P. Renard, “An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates,” Appl. Surf. Sci. 90, 251–259 (1995).
    [CrossRef]
  14. S. Bosch, J. Perez, A. Canillas, “Numerical algorithm for spectroscopic ellipsometry of thick transparent films,” Appl. Opt. 37, 1177–1179 (1998).
    [CrossRef]
  15. J. Lekner, “Inversion of reflection ellipsometric data,” Appl. Opt. 33, 5159–5165 (1994).
    [CrossRef] [PubMed]
  16. J. Lekner, “Analytic inversion of ellipsometric data for an unsupported nonabsorbing uniform layer,” J. Opt. Soc. Am. A 7, 1875–1877 (1990).
    [CrossRef]
  17. D. Charlot, A. Maruani, “Ellipsometric data processing: an efficient method and an analysis of the relative errors,” Appl. Opt. 24, 3368–3373 (1985).
    [CrossRef] [PubMed]
  18. S. C. Russev, M. I. Boyanov, J.-P. Drolet, R. M. Leblanc, “Analytical determination of the optical constants of a substrate in the presence of a covering layer by use of ellipsometric data,” J. Opt. Soc. Am. A 16, 1496–1500 (1999).
    [CrossRef]
  19. S. C. Russev, D. D. Georgieva, “Analytical solution of another ellipsometric inverse problem,” J. Mod. Opt. 38, 1217–1222 (1991).
    [CrossRef]
  20. R. M.A. Azzam, A. R.M. Zaghloul, N. M. Bashara, “Ellipsometric function of a film–substrate system: applications to the design of reflection-type optical devices and to ellipsometry,” J. Opt. Soc. Am. 65, 252–260 (1975).
    [CrossRef]
  21. For a transparent film on an absorbing substrate, Dϕ is real. For an absorbing film on an absorbing substrate it is not. For a discussion of the complex film-thickness period, see Refs. [20] and [22].
  22. A. R.M. Zaghloul, M. S.A. Yousef, “Unified analysis and mathematical representation of film-thickness behavior of film–substrate systems,” Appl. Opt. (to be published).

2002 (3)

1999 (2)

1998 (2)

1995 (1)

T. Easwarakhanthan, S. Ravelet, P. Renard, “An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates,” Appl. Surf. Sci. 90, 251–259 (1995).
[CrossRef]

1994 (2)

J. Lekner, “Inversion of reflection ellipsometric data,” Appl. Opt. 33, 5159–5165 (1994).
[CrossRef] [PubMed]

D. A. Tonova, “Inverse profiling by ellipsometry: a Newton–Kantorovitch algorithm,” Opt. Commun. 105, 104–112 (1994).
[CrossRef]

1991 (1)

S. C. Russev, D. D. Georgieva, “Analytical solution of another ellipsometric inverse problem,” J. Mod. Opt. 38, 1217–1222 (1991).
[CrossRef]

1990 (2)

J. Lekner, “Analytic inversion of ellipsometric data for an unsupported nonabsorbing uniform layer,” J. Opt. Soc. Am. A 7, 1875–1877 (1990).
[CrossRef]

R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation, and real-time applications,” Rev. Sci. Instrum. 61, 2029–2062 (1990).
[CrossRef]

1985 (1)

1983 (1)

1976 (1)

A. R.M. Zaghloul, R. M.A. Azzam, N. M. Bashara, “Inversion of the nonlinear equations of reflection ellipsometry on film–substrate systems,” Surf. Sci. 5, 87–96 (1976).
[CrossRef]

1975 (2)

Arguiro, T. Vl.

S. C. Russev, T. Vl. Arguiro, “Rotating analyzer-fixed analyzer ellipsometer based on null type ellipsometer,” Rev. Sci. Instrum. 70, 3077–3082 (1999).
[CrossRef]

Azzam, R. M.A.

Bashara, N. M.

Bosch, S.

Boyanov, M. I.

Bulkin, P.

Canillas, A.

Charlot, D.

Collins, R. W.

R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation, and real-time applications,” Rev. Sci. Instrum. 61, 2029–2062 (1990).
[CrossRef]

Drevillon, B.

Drolet, J.-P.

Easwarakhanthan, T.

T. Easwarakhanthan, S. Ravelet, P. Renard, “An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates,” Appl. Surf. Sci. 90, 251–259 (1995).
[CrossRef]

Georgieva, D. D.

S. C. Russev, D. D. Georgieva, “Analytical solution of another ellipsometric inverse problem,” J. Mod. Opt. 38, 1217–1222 (1991).
[CrossRef]

Goldstein, D.

See for example, D. Goldstein, Polarized Light (Marcel Dekker, 2003).

Hofrichter, A.

Hofrichter, Al.

Kildemo, M.

Kouznetsov, D.

Leblanc, R. M.

Lekner, J.

Liu, L.

Lu, Z.

Maruani, A.

Perez, J.

Ravelet, S.

T. Easwarakhanthan, S. Ravelet, P. Renard, “An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates,” Appl. Surf. Sci. 90, 251–259 (1995).
[CrossRef]

Renard, P.

T. Easwarakhanthan, S. Ravelet, P. Renard, “An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates,” Appl. Surf. Sci. 90, 251–259 (1995).
[CrossRef]

Russev, S. C.

S. C. Russev, T. Vl. Arguiro, “Rotating analyzer-fixed analyzer ellipsometer based on null type ellipsometer,” Rev. Sci. Instrum. 70, 3077–3082 (1999).
[CrossRef]

S. C. Russev, M. I. Boyanov, J.-P. Drolet, R. M. Leblanc, “Analytical determination of the optical constants of a substrate in the presence of a covering layer by use of ellipsometric data,” J. Opt. Soc. Am. A 16, 1496–1500 (1999).
[CrossRef]

S. C. Russev, D. D. Georgieva, “Analytical solution of another ellipsometric inverse problem,” J. Mod. Opt. 38, 1217–1222 (1991).
[CrossRef]

Tonova, D. A.

D. A. Tonova, “Inverse profiling by ellipsometry: a Newton–Kantorovitch algorithm,” Opt. Commun. 105, 104–112 (1994).
[CrossRef]

Wen, Y.

Yoriume, Y.

Yousef, M. S.A.

A. R.M. Zaghloul, M. S.A. Yousef, “Unified analysis and mathematical representation of film-thickness behavior of film–substrate systems,” Appl. Opt. (to be published).

Zaghloul, A. R.M.

A. R.M. Zaghloul, R. M.A. Azzam, N. M. Bashara, “Inversion of the nonlinear equations of reflection ellipsometry on film–substrate systems,” Surf. Sci. 5, 87–96 (1976).
[CrossRef]

R. M.A. Azzam, A. R.M. Zaghloul, N. M. Bashara, “Polarizer-surface-analyzer null ellipsometry for film–substrate systems,” J. Opt. Soc. Am. 65, 1464–1471 (1975).
[CrossRef]

R. M.A. Azzam, A. R.M. Zaghloul, N. M. Bashara, “Ellipsometric function of a film–substrate system: applications to the design of reflection-type optical devices and to ellipsometry,” J. Opt. Soc. Am. 65, 252–260 (1975).
[CrossRef]

A. R.M. Zaghloul, M. S.A. Yousef, “Unified analysis and mathematical representation of film-thickness behavior of film–substrate systems,” Appl. Opt. (to be published).

Zhang, B.

Zhu, H.

Appl. Opt. (7)

Appl. Surf. Sci. (1)

T. Easwarakhanthan, S. Ravelet, P. Renard, “An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates,” Appl. Surf. Sci. 90, 251–259 (1995).
[CrossRef]

J. Mod. Opt. (1)

S. C. Russev, D. D. Georgieva, “Analytical solution of another ellipsometric inverse problem,” J. Mod. Opt. 38, 1217–1222 (1991).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

D. A. Tonova, “Inverse profiling by ellipsometry: a Newton–Kantorovitch algorithm,” Opt. Commun. 105, 104–112 (1994).
[CrossRef]

Rev. Sci. Instrum. (2)

S. C. Russev, T. Vl. Arguiro, “Rotating analyzer-fixed analyzer ellipsometer based on null type ellipsometer,” Rev. Sci. Instrum. 70, 3077–3082 (1999).
[CrossRef]

R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation, and real-time applications,” Rev. Sci. Instrum. 61, 2029–2062 (1990).
[CrossRef]

Surf. Sci. (1)

A. R.M. Zaghloul, R. M.A. Azzam, N. M. Bashara, “Inversion of the nonlinear equations of reflection ellipsometry on film–substrate systems,” Surf. Sci. 5, 87–96 (1976).
[CrossRef]

Other (4)

See for example, D. Goldstein, Polarized Light (Marcel Dekker, 2003).

R. M.A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

For a transparent film on an absorbing substrate, Dϕ is real. For an absorbing film on an absorbing substrate it is not. For a discussion of the complex film-thickness period, see Refs. [20] and [22].

A. R.M. Zaghloul, M. S.A. Yousef, “Unified analysis and mathematical representation of film-thickness behavior of film–substrate systems,” Appl. Opt. (to be published).

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Equations (35)

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ρ = R p R s ,
R p = r 01 p + r 12 p X 1 + r 01 p r 12 p X ,
R S = r 01 S + r 12 S X 1 + r 01 S r 12 S X ,
r 01 p = N 1 cos ϕ 0 N 0 cos ϕ 1 N 1 cos ϕ 0 + N 0 cos ϕ 1 ,
r 01 s = N 0 cos ϕ 0 N 1 cos ϕ 1 N 0 cos ϕ 0 + N 1 cos ϕ 1 ,
r 12 p = N 2 cos ϕ 1 N 1 cos ϕ 2 N 2 cos ϕ 1 + N 1 cos ϕ 2 ,
r 12 s = N 1 cos ϕ 1 N 2 cos ϕ 2 N 1 cos ϕ 1 + N 2 cos ϕ 2 ,
N 0 sin ϕ 0 = N 1 sin ϕ 1 = N 2 sin ϕ 2 .
X = exp ( j 4 π d N 1 cos ( ϕ 1 ) λ ) .
X = exp ( j 2 π d D ϕ ) ,
D ϕ = λ 2 N 1 cos ϕ 1 ,
ρ = A + B X + C X 2 D + E X + F X 2 ,
( A , B ) = ( r 01 p , r 12 p + r 01 p r 01 s r 12 s ) ,
( C , D ) = ( r 12 p r 01 s r 12 s , r 01 s ) ,
( E , F ) = ( r 12 s + r 01 p r 12 p r 01 s , r 01 p r 12 p r 12 s ) .
ρ = ( tan ψ ) exp ( j Δ ) .
ρ = ( A 1 N 2 2 + A 2 C 0 ) ( B 3 C 0 + B 4 ) ( A 3 N 2 2 + A 4 C 0 ) ( B 1 C 0 + B 2 ) ,
( A 0 , B 0 , C 0 ) = ( N 0 sin ϕ 0 , N 1 cos ϕ 1 , N 2 cos ϕ 2 ) ,
( A 1 , B 1 ) = ( B 0 ( r 01 p + X ) , r 01 s X ) ,
( A 2 , B 2 ) = ( N 1 2 ( r 01 p X ) , B 0 ( r 01 s + X ) ) ,
( A 3 , B 3 ) = ( B 0 ( 1 + r 01 p X ) , 1 r 01 s X ) ,
( A 4 , B 4 ) = ( N 1 2 ( 1 r 01 p X ) , B 0 ( 1 + r 01 s X ) ) .
C 0 2 = B 5 2 N 2 4 + 2 B 5 B 6 N 2 2 + B 6 2 A 5 2 N 2 4 + 2 A 5 A 6 N 2 2 + A 6 2 ,
( A 5 , B 5 ) = ( A 1 B 3 ρ A 3 B 1 , ρ ( A 3 B 2 + A 4 B 1 ) ( A 1 B 4 + A 2 B 3 ) ) ,
( A 6 , B 6 ) = ( A 2 B 4 ρ A 4 B 2 , A 0 2 ( A 2 B 3 ρ A 4 B 1 ) ) .
N 2 6 + A 7 N 2 4 + A 8 N 2 2 A 9 = 0 ,
( A 7 , A 8 , A 9 ) = ( 2 A 5 A 6 A 0 2 A 5 2 B 5 2 A 5 2 , A 6 2 2 A 0 2 A 5 A 6 2 B 5 B 6 A 5 2 , A 0 2 A 6 2 + B 6 2 A 5 2 ) .
( N 21 , N 22 , N 23 ) = ( C 4 C 1 3 C 4 A 7 3 , C 5 C 1 3 C 5 A 7 3 , C 6 C 1 3 C 6 A 7 3 ) ,
( C 1 , C 2 , C 3 , C 4 , C 5 , C 6 ) = ( A 8 A 7 2 3 , A 9 A 7 A 8 3 + 2 A 7 3 27 , C 2 2 + C 2 2 4 + C 1 3 27 , C 3 3 , exp ( j π 1.5 ) C 4 , exp ( j π 1.5 ) C 5 ) .
N 2
= { ( A 88 A 8 ) ± [ ( A 88 A 8 ) 2 + 4 ( A 77 A 7 ) ( A 99 A 9 ) ] 1 2 2 ( A 7 A 77 ) } 1 2 .
N 2 = [ ( A 9 A 8 ) ( F 2 F 1 ) ] 1 2 ,
F 2 = [ ( 1 A 99 A 9 ) ( 1 A 77 A 7 ) ] [ ( 1 A 999 A 9 ) ( 1 A 777 A 7 ) ] ,
F 1 = [ ( 1 A 88 A 8 ) ( 1 A 77 A 7 ) ] [ ( 1 A 888 A 8 ) ( 1 A 777 A 7 ) ] .
N 2 = N 0 6 tan 2 ϕ 0 ( ρ 2 + 1 + 2 ρ cos 2 ϕ 0 ) ( ρ + 1 ) 2 .

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