Abstract

We obtain the universal evaluations and expressions of measuring uncertainty for all types of rotating-element spectroscopic ellipsometers. We introduce a general data-reduction process to represent the universal analytic functions of the combined standard uncertainties of the ellipsometric sample parameters. To solve the incompleteness of the analytic expressions, we formulate the estimated covariance for the Fourier coefficient means extracted from the radiant flux waveform using a new Fourier analysis. Our approach can be used for optimization of measurement conditions, instrumentation, simulation, standardization, laboratory accreditation, and metrology.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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2013 (2)

2011 (4)

R. M. A. Azzam, “The intertwined history of polarimetry and ellipsometry,” Thin Solid Films 519(9), 2584–2588 (2011).
[Crossref]

L. Broch, A. E. Naciri, and L. Johann, “Analysis of systematic errors in Mueller matrix ellipsometry as a function of the retardance of the dual rotating compensators,” Thin Solid Films 519(9), 2601–2603 (2011).
[Crossref]

Y. J. Cho, W. Chegal, and H. M. Cho, “Fourier analysis for rotating-element ellipsometers,” Opt. Lett. 36(2), 118–120 (2011).
[Crossref] [PubMed]

J. Li, B. Ramanujam, and R. W. Collins, “Dual rotating compensator ellipsometry: Theory and simulations,” Thin Solid Films 519(9), 2725–2729 (2011).
[Crossref]

2010 (1)

2008 (4)

K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express 16(15), 11589–11603 (2008).
[Crossref] [PubMed]

L. Broch, A. En Naciri, and L. Johann, “Systematic errors for a Mueller matrix dual rotating compensator ellipsometer,” Opt. Express 16(12), 8814–8824 (2008).
[Crossref] [PubMed]

B. Johs and C. M. Herzinger, “Quantifying the accuracy of ellipsometer systems,” Phys. Status Solidi C 5(5), 1031–1035 (2008).
[Crossref]

L. Broch and L. Johann, “Optimizing precision of rotating compensator ellipsometry,” Phys. Status Solidi C 5(5), 1036–1040 (2008).
[Crossref]

2006 (1)

J. Opsal, “How far can one go with optical metrology?” Laser Focus World 42, 101–105 (2006).

2004 (3)

D. E. Aspnes, “Optimizing precision of rotating-analyzer and rotating-compensator ellipsometers,” J. Opt. Soc. Am. A 21(3), 403–410 (2004).
[Crossref] [PubMed]

B. Johs and C. M. Herzinger, “Precision in ellipsometrically determined sample parameters: simulation and experiment,” Thin Solid Films 455–456, 66–71 (2004).
[Crossref]

J. A. Zapien, A. S. Ferlauto, and R. W. Collins, “Application of spectral and temporal weighted error functions for data analysis in real-time spectroscopic ellipsometry,” Thin Solid Films 455–456, 106–111 (2004).
[Crossref]

2003 (1)

C. Chen, I. An, and R. W. Collins, “Multichannel Mueller Matrix Ellipsometry for Simultaneous Real-Time Measurement of Bulk Isotropic and Surface Anisotropic Complex Dielectric Functions of Semiconductors,” Phys. Rev. Lett. 90(21), 217402 (2003).
[Crossref] [PubMed]

2002 (1)

2001 (1)

J. Lee, J. Koh, and R. W. Collins, “Dual rotating-compensator multichannel ellipsometer: Instrument development for high-speed Mueller matrix spectroscopy of surfaces and thin films,” Rev. Sci. Instrum. 72(3), 1742–1754 (2001).
[Crossref]

2000 (1)

1998 (1)

K. Vedam, “Spectroscopic ellipsometry: a historical overview,” Thin Solid Films 313–314, 1–9 (1998).
[Crossref]

1994 (1)

1993 (1)

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: In situ application to the growth of semiconductors,” Prog. Crystal Growth and Charact. 27(1), 1–87 (1993).
[Crossref]

1992 (1)

1991 (2)

N. V. Nguyen, B. S. Pudliner, I. An, and R. W. Collins, “Error correction for calibration and data reduction in rotating-polarizer ellipsometry: applications to a novel multichannel ellipsometer,” J. Opt. Soc. Am. A 8(6), 919–931 (1991).
[Crossref]

I. An and R. W. Collins, “Waveform analysis with optical multichannel detectors: Applications for rapid-scan spectroscopic ellipsometry,” Rev. Sci. Instrum. 62(8), 1904–1911 (1991).
[Crossref]

1990 (2)

D. H. Goldstein and R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 7(4), 693–700 (1990).
[Crossref]

R. W. Collins, “Automatic rotating element ellipsometers: Calibration, operation, and real-time applications,” Rev. Sci. Instrum. 61(8), 2029–2062 (1990).
[Crossref]

1988 (1)

1980 (1)

P. S. Hauge, “Recent development in instrumentation in ellipsometry,” Surf. Sci. 96(1-3), 108–140 (1980).
[Crossref]

1978 (2)

1975 (1)

1970 (1)

An, I.

C. Chen, I. An, and R. W. Collins, “Multichannel Mueller Matrix Ellipsometry for Simultaneous Real-Time Measurement of Bulk Isotropic and Surface Anisotropic Complex Dielectric Functions of Semiconductors,” Phys. Rev. Lett. 90(21), 217402 (2003).
[Crossref] [PubMed]

I. An and R. W. Collins, “Waveform analysis with optical multichannel detectors: Applications for rapid-scan spectroscopic ellipsometry,” Rev. Sci. Instrum. 62(8), 1904–1911 (1991).
[Crossref]

N. V. Nguyen, B. S. Pudliner, I. An, and R. W. Collins, “Error correction for calibration and data reduction in rotating-polarizer ellipsometry: applications to a novel multichannel ellipsometer,” J. Opt. Soc. Am. A 8(6), 919–931 (1991).
[Crossref]

Aspnes, D. E.

Azzam, R. M. A.

R. M. A. Azzam, “The intertwined history of polarimetry and ellipsometry,” Thin Solid Films 519(9), 2584–2588 (2011).
[Crossref]

R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2(6), 148–150 (1978).
[Crossref] [PubMed]

Broch, L.

L. Broch, A. E. Naciri, and L. Johann, “Analysis of systematic errors in Mueller matrix ellipsometry as a function of the retardance of the dual rotating compensators,” Thin Solid Films 519(9), 2601–2603 (2011).
[Crossref]

L. Broch, A. En Naciri, and L. Johann, “Second-order systematic errors in Mueller matrix dual rotating compensator ellipsometry,” Appl. Opt. 49(17), 3250–3258 (2010).
[Crossref] [PubMed]

L. Broch, A. En Naciri, and L. Johann, “Systematic errors for a Mueller matrix dual rotating compensator ellipsometer,” Opt. Express 16(12), 8814–8824 (2008).
[Crossref] [PubMed]

L. Broch and L. Johann, “Optimizing precision of rotating compensator ellipsometry,” Phys. Status Solidi C 5(5), 1036–1040 (2008).
[Crossref]

Chegal, W.

Chen, C.

C. Chen, I. An, and R. W. Collins, “Multichannel Mueller Matrix Ellipsometry for Simultaneous Real-Time Measurement of Bulk Isotropic and Surface Anisotropic Complex Dielectric Functions of Semiconductors,” Phys. Rev. Lett. 90(21), 217402 (2003).
[Crossref] [PubMed]

Chipman, R. A.

Cho, H. M.

Cho, Y. J.

Chu, J.

Collins, R. W.

J. Li, B. Ramanujam, and R. W. Collins, “Dual rotating compensator ellipsometry: Theory and simulations,” Thin Solid Films 519(9), 2725–2729 (2011).
[Crossref]

J. A. Zapien, A. S. Ferlauto, and R. W. Collins, “Application of spectral and temporal weighted error functions for data analysis in real-time spectroscopic ellipsometry,” Thin Solid Films 455–456, 106–111 (2004).
[Crossref]

C. Chen, I. An, and R. W. Collins, “Multichannel Mueller Matrix Ellipsometry for Simultaneous Real-Time Measurement of Bulk Isotropic and Surface Anisotropic Complex Dielectric Functions of Semiconductors,” Phys. Rev. Lett. 90(21), 217402 (2003).
[Crossref] [PubMed]

J. Lee, J. Koh, and R. W. Collins, “Dual rotating-compensator multichannel ellipsometer: Instrument development for high-speed Mueller matrix spectroscopy of surfaces and thin films,” Rev. Sci. Instrum. 72(3), 1742–1754 (2001).
[Crossref]

I. An and R. W. Collins, “Waveform analysis with optical multichannel detectors: Applications for rapid-scan spectroscopic ellipsometry,” Rev. Sci. Instrum. 62(8), 1904–1911 (1991).
[Crossref]

N. V. Nguyen, B. S. Pudliner, I. An, and R. W. Collins, “Error correction for calibration and data reduction in rotating-polarizer ellipsometry: applications to a novel multichannel ellipsometer,” J. Opt. Soc. Am. A 8(6), 919–931 (1991).
[Crossref]

R. W. Collins, “Automatic rotating element ellipsometers: Calibration, operation, and real-time applications,” Rev. Sci. Instrum. 61(8), 2029–2062 (1990).
[Crossref]

de Nijs, J. M. M.

Drévillon, B.

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: In situ application to the growth of semiconductors,” Prog. Crystal Growth and Charact. 27(1), 1–87 (1993).
[Crossref]

El Ghemmaz, A.

En Naciri, A.

Ferlauto, A. S.

J. A. Zapien, A. S. Ferlauto, and R. W. Collins, “Application of spectral and temporal weighted error functions for data analysis in real-time spectroscopic ellipsometry,” Thin Solid Films 455–456, 106–111 (2004).
[Crossref]

Goldstein, D. H.

Han, S. W.

Hauge, P. S.

P. S. Hauge, “Recent development in instrumentation in ellipsometry,” Surf. Sci. 96(1-3), 108–140 (1980).
[Crossref]

P. S. Hauge, “Mueller matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68(11), 1519–1528 (1978).
[Crossref]

Herzinger, C. M.

B. Johs and C. M. Herzinger, “Quantifying the accuracy of ellipsometer systems,” Phys. Status Solidi C 5(5), 1031–1035 (2008).
[Crossref]

B. Johs and C. M. Herzinger, “Precision in ellipsometrically determined sample parameters: simulation and experiment,” Thin Solid Films 455–456, 66–71 (2004).
[Crossref]

Huang, Z.

Johann, L.

L. Broch, A. E. Naciri, and L. Johann, “Analysis of systematic errors in Mueller matrix ellipsometry as a function of the retardance of the dual rotating compensators,” Thin Solid Films 519(9), 2601–2603 (2011).
[Crossref]

L. Broch, A. En Naciri, and L. Johann, “Second-order systematic errors in Mueller matrix dual rotating compensator ellipsometry,” Appl. Opt. 49(17), 3250–3258 (2010).
[Crossref] [PubMed]

L. Broch, A. En Naciri, and L. Johann, “Systematic errors for a Mueller matrix dual rotating compensator ellipsometer,” Opt. Express 16(12), 8814–8824 (2008).
[Crossref] [PubMed]

L. Broch and L. Johann, “Optimizing precision of rotating compensator ellipsometry,” Phys. Status Solidi C 5(5), 1036–1040 (2008).
[Crossref]

Johs, B.

B. Johs and C. M. Herzinger, “Quantifying the accuracy of ellipsometer systems,” Phys. Status Solidi C 5(5), 1031–1035 (2008).
[Crossref]

B. Johs and C. M. Herzinger, “Precision in ellipsometrically determined sample parameters: simulation and experiment,” Thin Solid Films 455–456, 66–71 (2004).
[Crossref]

Kleim, R.

Koh, J.

J. Lee, J. Koh, and R. W. Collins, “Dual rotating-compensator multichannel ellipsometer: Instrument development for high-speed Mueller matrix spectroscopy of surfaces and thin films,” Rev. Sci. Instrum. 72(3), 1742–1754 (2001).
[Crossref]

Kuntzler, L.

Lee, J.

J. Lee, J. Koh, and R. W. Collins, “Dual rotating-compensator multichannel ellipsometer: Instrument development for high-speed Mueller matrix spectroscopy of surfaces and thin films,” Rev. Sci. Instrum. 72(3), 1742–1754 (2001).
[Crossref]

Lee, J. P.

Li, J.

J. Li, B. Ramanujam, and R. W. Collins, “Dual rotating compensator ellipsometry: Theory and simulations,” Thin Solid Films 519(9), 2725–2729 (2011).
[Crossref]

Naciri, A. E.

L. Broch, A. E. Naciri, and L. Johann, “Analysis of systematic errors in Mueller matrix ellipsometry as a function of the retardance of the dual rotating compensators,” Thin Solid Films 519(9), 2601–2603 (2011).
[Crossref]

Nguyen, N. V.

Opsal, J.

J. Opsal, “How far can one go with optical metrology?” Laser Focus World 42, 101–105 (2006).

Pudliner, B. S.

Ramanujam, B.

J. Li, B. Ramanujam, and R. W. Collins, “Dual rotating compensator ellipsometry: Theory and simulations,” Thin Solid Films 519(9), 2725–2729 (2011).
[Crossref]

Schmidt, E.

Silfhout, A. V.

Smith, M. H.

Twietmeyer, K. M.

Vedam, K.

K. Vedam, “Spectroscopic ellipsometry: a historical overview,” Thin Solid Films 313–314, 1–9 (1998).
[Crossref]

Zapien, J. A.

J. A. Zapien, A. S. Ferlauto, and R. W. Collins, “Application of spectral and temporal weighted error functions for data analysis in real-time spectroscopic ellipsometry,” Thin Solid Films 455–456, 106–111 (2004).
[Crossref]

Appl. Opt. (5)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

J. Vac. Sci. Technol. A (1)

D. E. Aspnes, “Spectroscopic ellipsometry – A perspective,” J. Vac. Sci. Technol. A 31(5), 058502 (2013).
[Crossref]

Laser Focus World (1)

J. Opsal, “How far can one go with optical metrology?” Laser Focus World 42, 101–105 (2006).

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

C. Chen, I. An, and R. W. Collins, “Multichannel Mueller Matrix Ellipsometry for Simultaneous Real-Time Measurement of Bulk Isotropic and Surface Anisotropic Complex Dielectric Functions of Semiconductors,” Phys. Rev. Lett. 90(21), 217402 (2003).
[Crossref] [PubMed]

Phys. Status Solidi C (2)

B. Johs and C. M. Herzinger, “Quantifying the accuracy of ellipsometer systems,” Phys. Status Solidi C 5(5), 1031–1035 (2008).
[Crossref]

L. Broch and L. Johann, “Optimizing precision of rotating compensator ellipsometry,” Phys. Status Solidi C 5(5), 1036–1040 (2008).
[Crossref]

Prog. Crystal Growth and Charact. (1)

B. Drévillon, “Phase modulated ellipsometry from the ultraviolet to the infrared: In situ application to the growth of semiconductors,” Prog. Crystal Growth and Charact. 27(1), 1–87 (1993).
[Crossref]

Rev. Sci. Instrum. (3)

R. W. Collins, “Automatic rotating element ellipsometers: Calibration, operation, and real-time applications,” Rev. Sci. Instrum. 61(8), 2029–2062 (1990).
[Crossref]

I. An and R. W. Collins, “Waveform analysis with optical multichannel detectors: Applications for rapid-scan spectroscopic ellipsometry,” Rev. Sci. Instrum. 62(8), 1904–1911 (1991).
[Crossref]

J. Lee, J. Koh, and R. W. Collins, “Dual rotating-compensator multichannel ellipsometer: Instrument development for high-speed Mueller matrix spectroscopy of surfaces and thin films,” Rev. Sci. Instrum. 72(3), 1742–1754 (2001).
[Crossref]

Surf. Sci. (1)

P. S. Hauge, “Recent development in instrumentation in ellipsometry,” Surf. Sci. 96(1-3), 108–140 (1980).
[Crossref]

Thin Solid Films (6)

B. Johs and C. M. Herzinger, “Precision in ellipsometrically determined sample parameters: simulation and experiment,” Thin Solid Films 455–456, 66–71 (2004).
[Crossref]

J. A. Zapien, A. S. Ferlauto, and R. W. Collins, “Application of spectral and temporal weighted error functions for data analysis in real-time spectroscopic ellipsometry,” Thin Solid Films 455–456, 106–111 (2004).
[Crossref]

L. Broch, A. E. Naciri, and L. Johann, “Analysis of systematic errors in Mueller matrix ellipsometry as a function of the retardance of the dual rotating compensators,” Thin Solid Films 519(9), 2601–2603 (2011).
[Crossref]

J. Li, B. Ramanujam, and R. W. Collins, “Dual rotating compensator ellipsometry: Theory and simulations,” Thin Solid Films 519(9), 2725–2729 (2011).
[Crossref]

K. Vedam, “Spectroscopic ellipsometry: a historical overview,” Thin Solid Films 313–314, 1–9 (1998).
[Crossref]

R. M. A. Azzam, “The intertwined history of polarimetry and ellipsometry,” Thin Solid Films 519(9), 2584–2588 (2011).
[Crossref]

Other (15)

G. E. Jellison, Jr. and F. A. Modine, “Polarization Modulation Ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, ed. (Williams Andrew Inc., 2005), Chap. 6.

G. E. Jellison, Jr., “Data Analysis for Spectroscopic Ellipsometry,” in Handbook of Ellipsometry, H. G. Tompkins and E. A. Irene, ed. (Williams Andrew Inc., 2005), Chap. 3.

D. Schmidt, E. Schubert, and M. Schubert, “Generalized Ellipsometry Characterization of Sculptured Thin Films Made by Glancing Angle Deposition,” in Ellipsometry at the Nanoscale, M. Losurdo and K. Hingerl, ed. (Springer, Berlin, 2013), Chap. 10.

BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, and OIML, JCGM 100:2008, “Evaluation of measurement data - Guide to the expression of uncertainty in measurement,” (GUM 1995 with minor corrections) ( www.bipm.org ).

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E. Garcia-Caurel, R. Ossikovski, M. Foldyna, A. Pierangelo, B. Drévillon, and A. D. Martino, “Advanced Mueller Ellipsometry Instrumentation and Data Analysis,” in Ellipsometry at the Nanoscale, M. Losurdo and K. Hingerl, ed. (Springer, Berlin, 2013), Chap. 2.

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D. E. Aspnes, ” Spectroscopic ellipsometry – Past, present, and future,” in Proceeding of the 6th International Conference on Spectroscopic Ellipsometry, Y. Otani, M. Tazawa, and S. Kawabata, ed. Thin Solid Films 571, 334–344 (2014).

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[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Experimental (symbols) and theoretical (line) data of the Fourier coefficients for a bare c-Si wafer obtained at multiple azimuthal angles of the fixed analyzer from 0° to 358° in steps of 2° using a home-made multichannel rotating-polarizer spectroscopic ellipsometer based on a three-polarizer design.
Fig. 2
Fig. 2 Experimental (symbols) and theoretical (line) data of the standard uncertainties for the Fourier coefficients as shown in Fig. 1.
Fig. 3
Fig. 3 Experimental (symbols) and theoretical (line) data of the correlation coefficients for the Fourier coefficients as shown in Fig. 1.
Fig. 4
Fig. 4 Experimental (symbols) and theoretical (line) data of the combined standard uncertainties for the ellipsometric sample parameters N SP and C SP for the wavelengths of 373 nm and 550 nm.

Equations (34)

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S (PDE) = T PSA T PSG M ( DOS ) M ( PSA ) M ( SP ) M ( PSG ) S (LS)
I ( θ r ) = μ QE A PDE S 0 ( PDE ) = I 0 + n = 1 N ho [ A n cos ( n θ r ) + B n sin ( n θ r ) ]
I 0 = γ j = 1 u k = 1 v i 0 , j k M j k = γ i 0 V ( SP ) ,
A n = γ j = 1 u k = 1 v a n , j k M j k = γ a n V ( SP ) ,
B n = γ j = 1 u k = 1 v b n , j k M j k = γ b n V ( SP ) ,
u c ( N SP )
V ( SP ) = C X ,
u c ( v j ) = [ n , m = 1 2 N ho + 1 c j n c j m s ( x ¯ n , x ¯ m ) + k = 1 N u ( v j u k | U ) 2 u 2 ( u k ) ] 1 / 2 ,
u c ( q j ) = [ p , q = 1 u v n , m = 1 2 N ho + 1 ( q j v p | V ( SP ) ) ( q j v q | V ( SP ) ) c p n c q m s ( x ¯ n , x ¯ m ) + k = 1 N u ( q j u k | U ) 2 u 2 ( u k ) ] 1 / 2 .
I ( t ) = I 0 + n = 1 N ho [ A n cos ( n ω t ) + B n sin ( n ω t ) ] + δ ( t ) ,
S j = T i I 0 + n = 1 N ho T i ξ n sin ξ n [ cos ( n θ j ) ( A n cos χ n + B n sin χ n ) sin ( n θ j ) ( A n sin χ n B n cos χ n ) ] + δ S j ,
X ¯ = ( Ξ T Ξ ) 1 Ξ T S ¯ .
H ¯ n c + i H ¯ n s = 2 N J j = 1 N J S j exp ( i n θ j ) ,
I ¯ 0 = H ¯ 0 c / ( 2 T i ) ,
A ¯ n = C n c H ¯ n c C n s H ¯ n s , ( n 1 ) ,
B ¯ n = C n c H ¯ n s + C n s H ¯ n c , ( n 1 ) ,
σ 2 ( S j ) η S j ,
s ( S j , S k ) = σ 2 ( S j ) δ j k ,
s ( I ¯ 0 , I ¯ 0 ) = η I ¯ 0 / ( N J T i ) ,
s ( A ¯ n , A ¯ n ) = q n , n [ 2 I ¯ 0 + sin c ( ξ 2 n ) A ¯ 2 n ] ,
s ( B ¯ n , B ¯ n ) = q n , n [ 2 I ¯ 0 sin c ( ξ 2 n ) A ¯ 2 n ] ,
s ( I ¯ 0 , A ¯ n ) = η A ¯ n / ( N J T i ) ,
s ( I ¯ 0 , B ¯ n ) = η B ¯ n / ( N J T i ) ,
s ( A ¯ n , A ¯ m ) = q n , m [ sin c ( ξ n + m ) A ¯ n + m + sin c ( ξ n m ) A ¯ n m ] , ( n m ) ,
s ( B ¯ n , B ¯ m ) = q n , m [ sin c ( ξ n m ) A ¯ n m sin c ( ξ n + m ) A ¯ n + m ] , ( n m ) ,
s ( A ¯ n , B ¯ m ) = { q n , n sin c ( ξ n ) cos ( ξ n ) B ¯ 2 n , ( n = m ) , q n , m [ sin c ( ξ n + m ) B ¯ n + m sin c ( ξ n m ) B ¯ n m ] , ( n m ) ,
u ( x j ) = s ( x ¯ j , x ¯ j ) ,
r ( x j , x k ) = s ( x ¯ j , x ¯ k ) u ( x j ) u ( x k ) .
I 0 = I 0 ,
A n = A n cos ( n θ r 0 , n ) + B n sin ( n θ r 0 , n ) ,
B n = A n sin ( n θ r 0 , n ) + B n cos ( n θ r 0 , n ) ,
u ( x j ) = [ 1 N ( N 1 ) l = 1 N ( x j , l x ¯ j ) 2 ] 1 / 2 ,
r ( x j , x k ) = l = 1 N ( x j , l x ¯ j ) ( x k , l x ¯ k ) l = 1 N ( x j , l x ¯ j ) 2 l = 1 N ( x k , l x ¯ k ) 2 ,
Ζ ( P , A , ϕ ) = 1 N q N λ j = 1 N q k = 1 N λ u c 2 ( q j ( λ k ) ) ,

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