Abstract

General equations describing the theoretical uncertainty in measurements of the complex reflectance ratio, ρ, are given for fixed-polarizer, rotating-polarizer, sample, and fixed-analyzer (PPrSA) ellipsometers operating under shot-noise-limited (SNL) and detector-noise-limited (DNL) conditions. Optimum values of the fixed-polarizer and fixed-analyzer azimuths are calculated as a function of ellipsometric parameters (ψ and Δ) for SNL and DNL systems. The uncertainty of complex reflectance ratio, calculated from the optimum values of the fixed polarizer and analyzer azimuths, decreases linearly in ψ as ψ → 0 under the SNL condition but approaches arbitrary large under the DNL condition. This suggests that a compensator is not needed under the SNL condition to achieve high precision on dielectric surfaces but that measurements at small ψ values should be avoided under the DNL condition. A discrepancy of the fixed polarizer or analyzer azimuth from its optimum value for various Δ is discussed with ψ = 5° and 45° under the SNL condition.

© 2000 Optical Society of America

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References

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  1. B. D. Cahan, R. F. Spanier, “A high-speed automatic ellipsometer,” Surf. Sci. 16, 166–176 (1969).
    [CrossRef]
  2. P. S. Hauge, F. H. Dill, “Design and operation of ETA, an automated ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
    [CrossRef]
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    [CrossRef]
  4. L. Y. Chen, D. W. Lynch, “Scanning ellipsometer by rotating polarizer and analyzer,” Appl. Opt. 26, 5221–5228 (1987).
    [CrossRef] [PubMed]
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  6. R. M. A. Azzam, “A simple Fourier photopolarimeter with rotating polarizer and analyzer for measuring Jones and Mueller matrices,” Opt. Commun. 25, 137–140 (1978).
    [CrossRef]
  7. P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
    [CrossRef]
  8. S. H. Russev, “Correction for nonlinearity and polarization-dependent sensitivity in the detection system of analyzer ellipsometers,” Appl. Opt. 28, 1504–1507 (1989).
    [CrossRef] [PubMed]
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    [CrossRef]
  14. R. M. A. Azzam, N. M. Bashara, “Choice of compensator azimuth and position in ellipsometry,” J. Opt. Soc. Am. 62, 700–701 (1972).
    [CrossRef]
  15. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  16. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955).

1998

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, R. Kleim, “Systemic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).

1994

1990

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

1989

1987

1980

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

1978

R. M. A. Azzam, “A simple Fourier photopolarimeter with rotating polarizer and analyzer for measuring Jones and Mueller matrices,” Opt. Commun. 25, 137–140 (1978).
[CrossRef]

1975

1974

1973

P. S. Hauge, F. H. Dill, “Design and operation of ETA, an automated ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
[CrossRef]

1972

1970

1969

B. D. Cahan, R. F. Spanier, “A high-speed automatic ellipsometer,” Surf. Sci. 16, 166–176 (1969).
[CrossRef]

Archer, R. J.

R. J. Archer, Manual on Ellipsometry (Gaertner, Chicago, 1968).

Aspnes, D. E.

Azzam, R. M. A.

R. M. A. Azzam, “A simple Fourier photopolarimeter with rotating polarizer and analyzer for measuring Jones and Mueller matrices,” Opt. Commun. 25, 137–140 (1978).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, “Choice of compensator azimuth and position in ellipsometry,” J. Opt. Soc. Am. 62, 700–701 (1972).
[CrossRef]

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

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, “Choice of compensator azimuth and position in ellipsometry,” J. Opt. Soc. Am. 62, 700–701 (1972).
[CrossRef]

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

Bertucci, S.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, R. Kleim, “Systemic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).

Cahan, B. D.

B. D. Cahan, R. F. Spanier, “A high-speed automatic ellipsometer,” Surf. Sci. 16, 166–176 (1969).
[CrossRef]

Chen, L. Y.

Collins, R. W.

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

Dignam, M. J.

Dill, F. H.

P. S. Hauge, F. H. Dill, “Design and operation of ETA, an automated ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
[CrossRef]

El Ghemmaz, A.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, R. Kleim, “Systemic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).

Feng, X. W.

Hauge, P. S.

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

P. S. Hauge, F. H. Dill, “Design and operation of ETA, an automated ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955).

Johann, L.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, R. Kleim, “Systemic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).

Kleim, R.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, R. Kleim, “Systemic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).

Lynch, D. W.

Ma, H. Z.

Nicolas, N.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, R. Kleim, “Systemic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).

Pawlowski, A.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, R. Kleim, “Systemic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).

Qian, Y. H.

Rao, B.

Russev, S. H.

Schmidt, E.

Spanier, R. F.

B. D. Cahan, R. F. Spanier, “A high-speed automatic ellipsometer,” Surf. Sci. 16, 166–176 (1969).
[CrossRef]

Stein, N.

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, R. Kleim, “Systemic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).

Stobie, R. W.

Su, Y.

Appl. Opt.

IBM J. Res. Dev.

P. S. Hauge, F. H. Dill, “Design and operation of ETA, an automated ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

R. M. A. Azzam, “A simple Fourier photopolarimeter with rotating polarizer and analyzer for measuring Jones and Mueller matrices,” Opt. Commun. 25, 137–140 (1978).
[CrossRef]

Rev. Sci. Instrum.

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

Surf. Sci.

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

B. D. Cahan, R. F. Spanier, “A high-speed automatic ellipsometer,” Surf. Sci. 16, 166–176 (1969).
[CrossRef]

Thin Solid Films

S. Bertucci, A. Pawlowski, N. Nicolas, L. Johann, A. El Ghemmaz, N. Stein, R. Kleim, “Systemic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).

Other

R. J. Archer, Manual on Ellipsometry (Gaertner, Chicago, 1968).

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

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955).

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Figures (8)

Fig. 1
Fig. 1

Optimum values (a) P opt, of fixed-polarizer azimuth, and (b) A opt, of fixed-analyzer azimuth for a PPrSA ellipsometer system, as a function of ψ under the SNL condition, for Δ = 10°, 90°, 110°, 130°, and 170°. Note that the curves obtained for specific values of Δ are identical to those obtained for Δ with the same values of cos Δ. This remark also applies to the results shown in Figs. 28 below.

Fig. 2
Fig. 2

Optimum values (a) P opt, of fixed-polarizer azimuth, and (b) A opt, of fixed-analyzer azimuth for a PPrSA ellipsometer system, as a function of ψ for DNL condition, under Δ = 10°, 90°, 110°, 130°, and 170°.

Fig. 3
Fig. 3

Uncertainty, U SNL, calculated for P = P opt and A = A opt from Fig. 1 as a function of ψ and Δ for a SNL system.

Fig. 4
Fig. 4

Uncertainty, U DNL, calculated for P = P opt and A = A opt from Fig. 2 as a function of ψ and Δ for a DNL system.

Fig. 5
Fig. 5

Effect of fixed-polarizer azimuth P on uncertainty U SNL for various Δ with A = A opt and ψ = 5°.

Fig. 6
Fig. 6

Effect of fixed-polarizer azimuth P on uncertainty U SNL for various Δ with A = A opt and ψ = 45°.

Fig. 7
Fig. 7

Effect of fixed-analyzer azimuth A on uncertainty U SNL for various Δ with P = P opt and ψ = 5°.

Fig. 8
Fig. 8

Effect of fixed-analyzer azimuth A on uncertainty U SNL for various Δ with P = P opt and ψ = 45°.

Equations (24)

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I=Id+I2c cos 2Pr+I2s sin 2Pr+I4c cos 4Pr+I4s sin 4Pr,
Id=I0|rs|216 cos2 ψcos 2Acos 2P-2 cos 2ψ+sin 2A sin 2P sin 2ψ cos Δ-cos 2P cos 2ψ+2,
I2c=I0|rs|28 cos2 ψcos 2A1-cos 2P cos 2ψ+cos 2P-cos 2ψ,
I2s=I0|rs|28 cos2 ψ-cos 2A sin 2P cos 2ψ+sin 2A sin 2ψ cos Δ+sin 2P,
I4c=I0|rs|216 cos2 ψcos 2A cos 2P-sin 2A sin 2P sin 2ψ cos Δ-cos 2P cos 2ψ,
I4s=I0|rs|216 cos2 ψcos 2A sin 2P+sin 2A cos 2P sin 2ψ cos Δ-sin 2P cos 2ψ.
tan ψ=tan A2I4c+I2c cos 2P+I2s sin 2P+21-2 cos 2PI4c cos 2P+I4s sin 2P1/2/2I4c+I2c cos 2P+I2s sin 2P-21+2 cos 2PI4c cos 2P+I4s sin 2P1/2,
cos Δ=2I4s cos 2P-I4c sin 2P/2I4c+I2c cos 2P+I2s sin 2P-21+2 cos 2PI4c cos 2P+I4s sin 2P2I4c+I2c cos 2P+I2s sin 2P+21-2 cos 2PI4c cos 2P+I4s sin 2P1/2.
=asin2 ϕ+sin2 ϕ tan2 ϕ1-ρ1+ρ2,
Id+δId=N-1i=1N Ii,
I2c+δI2c=2N-1i=1N Ii cos 2Pri,
I2s+δI2s=2N-1i=1N Ii sin 2Pri,
I4c+δI4c=2N-1i=1N Ii cos 4Pri,
I4s+δI4s=2N-1i=1N Ii sin 4Pri,
Ii=It+δIi,
δIi,rmsηωIt+Inep/ΔT1/2,
δId,rms=ηωId+Inep/Ttot1/2,
δI2c,rms=δI2s,rms=δI4c,rms=δI4s,rms=2δId,rms,
|δρ|rms=δtan ψrms2+tan2 ψδΔrms21/2.
|δρ|rms=2ηωI0Ttot|rs|21/2USNLψ, Δ, P, A,
USNLψ, Δ, P, A=1/2cos ψcos 2A cos 2P-2 cos 2A cos 2ψ+sin 2A sin 2P×sin 2ψ cos Δ - cos 2P cos 2ψ+2Vψ, Δ, P, A,
Vψ, Δ, P, A=-48 cos2 A cos4 ψ+4 cos4 A cos4 ψ cos2 Δ-4 cos2 A cos4 ψ cos2 Δ-16 cos2 P cos4 ψ+32 cos4 A cos4 ψ+17 cos4 ψ+32 cos2 A cos2 P cos4 ψ+4 cos2 ψ cos Δ sin 2P sin 2A sin 2ψ-8 cos2 ψ sin 2P sin 2A sin 2ψ cos Δ×cos2 A-16 cos4 A cos2 ψ-4 cos4 A cos2 ψ cos2 Δ-32 cos4 A cos2 P cos2 ψ+14 cos2 A cos2 ψ+4 cos2 A cos2 ψ cos2 Δ+4 cos2 A cos Δ sin 2ψ sin 2P sin 2A+cos4 A+16 cos4 A cos2 P1/2/|sin Δ|sin2 2A sin ψ cos3 ψ,
|δρ|rms=4ηωInepI02Ttot|rs|41/2UDNLψ, Δ, P, A,
UDNLψ, Δ, P, A=4/2cos2 ψVψ, Δ, P, A.

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