Abstract

Sensitivity calculations are given for a novel ellipsometric technique that involves a rotating polarizer or analyzer and the measurement of signal phase angles. It is superior to optical null techniques under signal-limited conditions, and avoids the difficulties that attend precise signal-amplitude measurements. Noise coefficients as functions of optical-component settings and surface properties are presented for the technique and compared with those for the conventional ellipsometer.

© 1975 Optical Society of America

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References

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  1. Optical Studies of Adsorbed Layers at Interfaces, edited by F. C. Tompkins (Faraday Society, London, 1970).
  2. R. J. Archer, Manual on Ellipsometry (Gaertner, Chicago, 1963).
  3. J. L. Ord and B. L. Wills, Appl. Opt. 6, 1673 (1967).
    [Crossref] [PubMed]
  4. J. R. Beatties, Philos. Mag. 46, 235 (1955).
  5. S. Roberts, Phys. Rev. 114, 104 (1959).
    [Crossref]
  6. C. V. Kent and J. Lawson, J. Opt. Soc. Am. 27, 117 (1937).
    [Crossref]
  7. Y. J. van der Meulen and N. C. Hien, J. Opt. Soc. Am. 64, 804 (1974).
    [Crossref]
  8. J. Archard, P. L. Clegg, and A. M. Taylor, Proc. Phys. Soc. Lond. B 65, 758 (1952).
    [Crossref]
  9. T. E. Faber and N. V. Smith, J. Opt. Soc. Am. 58, 102 (1968).
    [Crossref]
  10. R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941).
    [Crossref]
  11. R. C. Jones, J. Opt. Soc. Am. 37, 110 (1947).
    [Crossref]
  12. D. E. Aspnes, Opt. Commun. 8, 222 (1973).
    [Crossref]
  13. M. J. Dignam, B. Rao, M. Moskovits, and R. W. Stobie, Can. J. Chem. 49, 1115 (1971).
    [Crossref]
  14. M. J. D. Powell, Computer J. 7, 155 (1964).
    [Crossref]
  15. D. E. Aspnes, J. Opt. Soc. Am. 64, 639 (1974).
    [Crossref]
  16. M. J. Dignam, B. Rao, and R. W. Stobie, Surf. Sci. 46, 308 (1974).
    [Crossref]

1974 (3)

1973 (1)

D. E. Aspnes, Opt. Commun. 8, 222 (1973).
[Crossref]

1971 (1)

M. J. Dignam, B. Rao, M. Moskovits, and R. W. Stobie, Can. J. Chem. 49, 1115 (1971).
[Crossref]

1968 (1)

1967 (1)

1964 (1)

M. J. D. Powell, Computer J. 7, 155 (1964).
[Crossref]

1959 (1)

S. Roberts, Phys. Rev. 114, 104 (1959).
[Crossref]

1955 (1)

J. R. Beatties, Philos. Mag. 46, 235 (1955).

1952 (1)

J. Archard, P. L. Clegg, and A. M. Taylor, Proc. Phys. Soc. Lond. B 65, 758 (1952).
[Crossref]

1947 (1)

1941 (1)

1937 (1)

Archard, J.

J. Archard, P. L. Clegg, and A. M. Taylor, Proc. Phys. Soc. Lond. B 65, 758 (1952).
[Crossref]

Archer, R. J.

R. J. Archer, Manual on Ellipsometry (Gaertner, Chicago, 1963).

Aspnes, D. E.

Beatties, J. R.

J. R. Beatties, Philos. Mag. 46, 235 (1955).

Clegg, P. L.

J. Archard, P. L. Clegg, and A. M. Taylor, Proc. Phys. Soc. Lond. B 65, 758 (1952).
[Crossref]

Dignam, M. J.

M. J. Dignam, B. Rao, and R. W. Stobie, Surf. Sci. 46, 308 (1974).
[Crossref]

M. J. Dignam, B. Rao, M. Moskovits, and R. W. Stobie, Can. J. Chem. 49, 1115 (1971).
[Crossref]

Faber, T. E.

Hien, N. C.

Jones, R. C.

Kent, C. V.

Lawson, J.

Moskovits, M.

M. J. Dignam, B. Rao, M. Moskovits, and R. W. Stobie, Can. J. Chem. 49, 1115 (1971).
[Crossref]

Ord, J. L.

Powell, M. J. D.

M. J. D. Powell, Computer J. 7, 155 (1964).
[Crossref]

Rao, B.

M. J. Dignam, B. Rao, and R. W. Stobie, Surf. Sci. 46, 308 (1974).
[Crossref]

M. J. Dignam, B. Rao, M. Moskovits, and R. W. Stobie, Can. J. Chem. 49, 1115 (1971).
[Crossref]

Roberts, S.

S. Roberts, Phys. Rev. 114, 104 (1959).
[Crossref]

Smith, N. V.

Stobie, R. W.

M. J. Dignam, B. Rao, and R. W. Stobie, Surf. Sci. 46, 308 (1974).
[Crossref]

M. J. Dignam, B. Rao, M. Moskovits, and R. W. Stobie, Can. J. Chem. 49, 1115 (1971).
[Crossref]

Taylor, A. M.

J. Archard, P. L. Clegg, and A. M. Taylor, Proc. Phys. Soc. Lond. B 65, 758 (1952).
[Crossref]

van der Meulen, Y. J.

Wills, B. L.

Appl. Opt. (1)

Can. J. Chem. (1)

M. J. Dignam, B. Rao, M. Moskovits, and R. W. Stobie, Can. J. Chem. 49, 1115 (1971).
[Crossref]

Computer J. (1)

M. J. D. Powell, Computer J. 7, 155 (1964).
[Crossref]

J. Opt. Soc. Am. (6)

Opt. Commun. (1)

D. E. Aspnes, Opt. Commun. 8, 222 (1973).
[Crossref]

Philos. Mag. (1)

J. R. Beatties, Philos. Mag. 46, 235 (1955).

Phys. Rev. (1)

S. Roberts, Phys. Rev. 114, 104 (1959).
[Crossref]

Proc. Phys. Soc. Lond. B (1)

J. Archard, P. L. Clegg, and A. M. Taylor, Proc. Phys. Soc. Lond. B 65, 758 (1952).
[Crossref]

Surf. Sci. (1)

M. J. Dignam, B. Rao, and R. W. Stobie, Surf. Sci. 46, 308 (1974).
[Crossref]

Other (2)

Optical Studies of Adsorbed Layers at Interfaces, edited by F. C. Tompkins (Faraday Society, London, 1970).

R. J. Archer, Manual on Ellipsometry (Gaertner, Chicago, 1963).

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Figures (5)

FIG. 1
FIG. 1

Contour plots of the noise coefficients for the real and imaginary parts of the optical density, Re(D) and Im(D), for a surface with optical properties defined by tanψ = 0.95 and Δ = π/4.

FIG. 2
FIG. 2

Optimum azimuthal positions for the first polarizer ϕ1, for different values of Δ and tanψ (solid line, tanψ = 0.35; dashed line, tanψ = 0.55; dot-dashed line, tanψ = 0.95); both the real part and the imaginary part of the optical density are optimized.

FIG. 3
FIG. 3

Optimum azimuthal positions for the third polarizer ϕ3, for different values of Δ and tanψ; both the real part and the imaginary part of the optical density are optimized.

FIG. 4
FIG. 4

Noise coefficients for the real part of the optical density for the rotating-polarizer method (dashed line, optimizing Re(D) only; solid line, optimizing Re(D) and Im(D)) and for an idealized null-balancing ellipsometer (dot-dashed line), as a function of tanψ.

FIG. 5
FIG. 5

Noise coefficient for Im(D), the imaginary part of the optical density for the rotating polarizer method as a function of Δ. Corresponding values for the conventional ellipsometer are indicated by *.

Equations (17)

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I = A 0 ( A 1 + A 2 sin 2 ω t + A 3 cos 2 ω t + A 4 sin 4 ω t + A 5 cos 4 ω t ) ,
A 2 = 1 4 [ sin ( 2 ϕ 1 ) ( cos 2 ϕ 3 ρ p 2 + sin 2 ϕ 3 ρ s 2 ) + sin ( 2 ϕ 3 ) ρ p ρ s cos Δ ] ,
A 3 = 1 2 ( cos 2 ϕ 1 cos 2 ϕ 3 ρ p 2 sin 2 ϕ 1 sin 2 ϕ 3 ρ s 2 ) ,
A 4 = 1 8 [ sin ( 2 ϕ 1 ) ( cos 2 ϕ 3 ρ p 2 sin 2 ϕ 3 ρ s 2 ) + cos ( 2 ϕ 1 ) sin ( 2 ϕ 3 ) ρ p ρ s cos Δ ] ,
A 5 = 1 8 [ cos ( 2 ϕ 1 ) ( cos 2 ϕ 3 ρ p 2 sin 2 ϕ 3 ρ s 2 ) sin ( 2 ϕ 1 ) sin ( 2 ϕ 3 ) ρ p ρ s cos Δ ] ,
I = A 0 [ A 1 + B 2 cos ( 2 ω t γ 2 ω ) + B 4 cos ( 4 ω t γ 4 ω ) ] ,
B 2 = ( A 2 2 + A 3 2 ) 1 / 2 ,
B 4 = ( A 4 2 + A 5 2 ) 1 / 2
tan γ 2 ω = A 2 / A 3 ,
tan γ 4 ω = A 4 / A 5 .
( ρ p / ρ s ) 2 tan 2 ϕ 3 = 2 sin ( 2 ϕ 1 ) cos 2 ϕ 1 + 2 cos ( 2 ϕ 1 ) sin 2 ϕ 1 tan γ 2 ω + tan γ 4 ω [ cos 2 ϕ 1 + sin 2 2 ϕ 1 + 2 sin ( 2 ϕ 1 ) sin 2 ϕ 1 tan γ 2 ω ] 2 sin ( 2 ϕ 1 ) sin 2 ϕ 1 + 2 cos ( 2 ϕ 1 ) cos 2 ϕ 1 tan γ 2 ω + tan 4 ω [ cos 2 ϕ 1 + 2 sin ( 2 ϕ 1 ) cos 2 ϕ 1 tan γ 2 ω sin 2 ( 2 ϕ 1 ) ] ,
cos Δ = 2 tan γ 2 ω [ cos 2 ϕ 1 cos 2 ϕ 3 ( ρ p / ρ s ) 2 sin 2 ϕ 1 sin 2 ϕ 3 ] sin 2 ϕ 3 [ cos 2 ϕ 3 ( ρ p / ρ s ) 2 + sin 2 ϕ 3 ] sin ( 2 ϕ 3 ) ρ p / ρ s
Noise Re ( D ) = 2 ( Re ( D ) γ 2 ω ) γ 4 ω 2 ( σ γ 2 ω ) 2 + 2 ( Re ( D ) γ 4 ω ) γ 2 ω 2 ( σ γ 4 ω ) 2 ,
D = ln tan ψ + i Δ ,
σ j = σ sig / B j .
Re ( D ) γ n ω = 1 tan ψ tan ψ γ n ω
Im ( D ) γ n ω = ( 1 cos 2 Δ ) 1 / 2 d cos Δ γ n ω ,