Abstract

Optical scatterometry has been given much credit during the past few years in the semiconductor industry. The geometry of an optical diffracted structure is deduced from the scattered intensity by solving an inverse problem. This step always requires a previously defined geometrical model. We develop an artificial neural network classifier whose purpose is to identify the structural geometry of a diffraction grating from its measured ellipsometric signature. This will take place before the characterization stage. Two types of geometry will be treated: sinusoidal and symmetric trapezoidal. Experimental results are performed on two manufactured samples: a sinusoidal photoresist grating deposited on a glass substrate and a trapezoidal grating etched on a SiO2 substrate with periods of 2μm and 0.565μm, respectively.

© 2008 Optical Society of America

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2007

I. Gereige, S. Robert, G. Granet, D. Jamon, and J. J. Rousseau, “Rapid control of submicrometer periodic structures by a neural inversion of ellipsometric measurement,” Opt. Commun. 278, 270-273 (2007).
[CrossRef]

2006

2004

H. T. Huang and F. L. Terry, “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension control and in situ, real-time process monitoring,” Thin Solid Films 468, 339-346 (2004).
[CrossRef]

S. Robert, A. Mure-Ravaud, S. Thyria, M. Yacoub, and F. Badran, “Neural selection of the optimal signature for a rapid characterization of a submicrometer period grating,” Opt. Commun. 238, 215-228 (2004).
[CrossRef]

2003

C. J. Raymond, M. Littau, B. Youn, C. J. Sohn, J. A. Kim, and Y. S. Kang, “Applications of optical scatterometry for the measurement of multiple periodic features,” Proc. SPIE 5038, 577-584 (2003).
[CrossRef]

2002

J. M. Holden, T. Gubiotti, W. A. McGaham, M. Dusa, and T. Kiers, “Normal-incidence spectroscopic ellipsometry and polarized reflectometry for measurement and control of photoresist critical dimension,” Proc. SPIE 4689, 1110-1121 (2002).
[CrossRef]

E. Drége, J. Reed, and D. Byrne, “Linearized inversion of scatterometric data to obtain surface profile information,” Opt. Eng. (Bellingham) 41, 225-236 (2002).
[CrossRef]

S. Robert, A. M. Ravaud, S. Reynaud, S. Fourment, F. Carcenac, and P. Arguel, “Experimental characterization of subwavelength diffraction gratings by an inverse scattering neural method,” J. Opt. Soc. Am. A 19, 2394-2402 (2002).
[CrossRef]

P. Logofatu, “Phase-modulation scatterometry,” Appl. Opt. 41, 7187-7192 (2002).
[CrossRef] [PubMed]

2000

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification (Wiley-Interscience, 2000).

1999

X. Ni, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE 3677, 159-168 (1999).
[CrossRef]

1998

S. A. Coulombe, P. Logofatu, B. K. Minhas, S. S. H. Naqvi, and J. R. McNeil, “Ellipsometric scatterometry for sub-0.1μm CD measurements,” Proc. SPIE 3332, 282-293 (1998).
[CrossRef]

I. Kallioniemi, J. Saarinen, and E. Oja, “Optical scatterometry of subwavelength diffraction gratings: neural network approach,” Appl. Opt. 37, 5830-5834 (1998).
[CrossRef]

1996

1994

1993

R. Krukar, A. Cornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Rietman, and R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698-3706 (1993).
[CrossRef]

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, and S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration,” Proc. SPIE 1926, 60-71 (1993).
[CrossRef]

1991

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, E. A. Rietman, and D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664-668 (1991).
[CrossRef]

1990

K. Fukunaga, Introduction to Statistical Pattern Recognition, 2nd ed. (Academic, 1990).

1989

G. Cybenko, “Approximation by superpositions of sigmoidal functions,” Math. Control, Signals, Syst. 2, 303-314 (1989).
[CrossRef]

1986

T. Y. Young and K. S. Fu, Handbook of Pattern Recognition and Image Processing (Academic, 1986).

Arguel, P.

Badran, F.

S. Robert, A. Mure-Ravaud, S. Thyria, M. Yacoub, and F. Badran, “Neural selection of the optimal signature for a rapid characterization of a submicrometer period grating,” Opt. Commun. 238, 215-228 (2004).
[CrossRef]

Bao, J.

X. Ni, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE 3677, 159-168 (1999).
[CrossRef]

Ben Hatit, S.

Byrne, D.

E. Drége, J. Reed, and D. Byrne, “Linearized inversion of scatterometric data to obtain surface profile information,” Opt. Eng. (Bellingham) 41, 225-236 (2002).
[CrossRef]

Carcenac, F.

Clark, L. A.

R. Krukar, A. Cornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Rietman, and R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698-3706 (1993).
[CrossRef]

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, E. A. Rietman, and D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664-668 (1991).
[CrossRef]

Cornblit, A.

R. Krukar, A. Cornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Rietman, and R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698-3706 (1993).
[CrossRef]

Coulombe, S. A.

S. A. Coulombe, P. Logofatu, B. K. Minhas, S. S. H. Naqvi, and J. R. McNeil, “Ellipsometric scatterometry for sub-0.1μm CD measurements,” Proc. SPIE 3332, 282-293 (1998).
[CrossRef]

Cybenko, G.

G. Cybenko, “Approximation by superpositions of sigmoidal functions,” Math. Control, Signals, Syst. 2, 303-314 (1989).
[CrossRef]

De Martino, A.

Drége, E.

E. Drége, J. Reed, and D. Byrne, “Linearized inversion of scatterometric data to obtain surface profile information,” Opt. Eng. (Bellingham) 41, 225-236 (2002).
[CrossRef]

Drévillon, B.

Duda, R. O.

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification (Wiley-Interscience, 2000).

Dusa, M.

J. M. Holden, T. Gubiotti, W. A. McGaham, M. Dusa, and T. Kiers, “Normal-incidence spectroscopic ellipsometry and polarized reflectometry for measurement and control of photoresist critical dimension,” Proc. SPIE 4689, 1110-1121 (2002).
[CrossRef]

Fourment, S.

Franke, J. E.

Fu, K. S.

T. Y. Young and K. S. Fu, Handbook of Pattern Recognition and Image Processing (Academic, 1986).

Fukunaga, K.

K. Fukunaga, Introduction to Statistical Pattern Recognition, 2nd ed. (Academic, 1990).

Gaspar, S. M.

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, and S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration,” Proc. SPIE 1926, 60-71 (1993).
[CrossRef]

Gereige, I.

I. Gereige, S. Robert, G. Granet, D. Jamon, and J. J. Rousseau, “Rapid control of submicrometer periodic structures by a neural inversion of ellipsometric measurement,” Opt. Commun. 278, 270-273 (2007).
[CrossRef]

Giapas, K. P.

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, E. A. Rietman, and D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664-668 (1991).
[CrossRef]

Gottscho, R. A.

R. Krukar, A. Cornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Rietman, and R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698-3706 (1993).
[CrossRef]

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, E. A. Rietman, and D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664-668 (1991).
[CrossRef]

Gottsho, R. A.

Granet, G.

I. Gereige, S. Robert, G. Granet, D. Jamon, and J. J. Rousseau, “Rapid control of submicrometer periodic structures by a neural inversion of ellipsometric measurement,” Opt. Commun. 278, 270-273 (2007).
[CrossRef]

Gubiotti, T.

J. M. Holden, T. Gubiotti, W. A. McGaham, M. Dusa, and T. Kiers, “Normal-incidence spectroscopic ellipsometry and polarized reflectometry for measurement and control of photoresist critical dimension,” Proc. SPIE 4689, 1110-1121 (2002).
[CrossRef]

Haaland, D. M.

Hagan, M. T.

M. T. Hagan and M. Menhaj, “Training feedforward networks with the Marquardt algorithm,” IEEE Trans. Neural Netw. 5, 989-993 (1994).
[CrossRef] [PubMed]

Hart, P. E.

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification (Wiley-Interscience, 2000).

Holden, J. M.

J. M. Holden, T. Gubiotti, W. A. McGaham, M. Dusa, and T. Kiers, “Normal-incidence spectroscopic ellipsometry and polarized reflectometry for measurement and control of photoresist critical dimension,” Proc. SPIE 4689, 1110-1121 (2002).
[CrossRef]

Huang, H. T.

H. T. Huang and F. L. Terry, “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension control and in situ, real-time process monitoring,” Thin Solid Films 468, 339-346 (2004).
[CrossRef]

Jakatdar, N.

X. Ni, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE 3677, 159-168 (1999).
[CrossRef]

Jamon, D.

I. Gereige, S. Robert, G. Granet, D. Jamon, and J. J. Rousseau, “Rapid control of submicrometer periodic structures by a neural inversion of ellipsometric measurement,” Opt. Commun. 278, 270-273 (2007).
[CrossRef]

Kallioniemi, I.

Kang, Y. S.

C. J. Raymond, M. Littau, B. Youn, C. J. Sohn, J. A. Kim, and Y. S. Kang, “Applications of optical scatterometry for the measurement of multiple periodic features,” Proc. SPIE 5038, 577-584 (2003).
[CrossRef]

Kiers, T.

J. M. Holden, T. Gubiotti, W. A. McGaham, M. Dusa, and T. Kiers, “Normal-incidence spectroscopic ellipsometry and polarized reflectometry for measurement and control of photoresist critical dimension,” Proc. SPIE 4689, 1110-1121 (2002).
[CrossRef]

Kim, J. A.

C. J. Raymond, M. Littau, B. Youn, C. J. Sohn, J. A. Kim, and Y. S. Kang, “Applications of optical scatterometry for the measurement of multiple periodic features,” Proc. SPIE 5038, 577-584 (2003).
[CrossRef]

Ko, C.

Kornblit, A.

Krukar, D. M.

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, and S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration,” Proc. SPIE 1926, 60-71 (1993).
[CrossRef]

Krukar, R.

R. Krukar, A. Cornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Rietman, and R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698-3706 (1993).
[CrossRef]

Krukar, R. H.

S. S. H. Naqvi, R. H. Krukar, J. R. McNeil, J. E. Franke, T. M. Niemczyk, D. M. Haaland, R. A. Gottsho, and A. Kornblit, “Etch depth estimation of large-period silicon gratings with multivariate calibration of rigorously simulated diffraction profiles,” J. Opt. Soc. Am. A 11, 2485-2493 (1994).
[CrossRef]

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, and S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration,” Proc. SPIE 1926, 60-71 (1993).
[CrossRef]

Kruskal, J.

R. Krukar, A. Cornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Rietman, and R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698-3706 (1993).
[CrossRef]

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, E. A. Rietman, and D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664-668 (1991).
[CrossRef]

Ku, Y.

Ku, Y. S.

Lambert, D.

R. Krukar, A. Cornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Rietman, and R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698-3706 (1993).
[CrossRef]

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, E. A. Rietman, and D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664-668 (1991).
[CrossRef]

Li, L.

Littau, M.

C. J. Raymond, M. Littau, B. Youn, C. J. Sohn, J. A. Kim, and Y. S. Kang, “Applications of optical scatterometry for the measurement of multiple periodic features,” Proc. SPIE 5038, 577-584 (2003).
[CrossRef]

Logofatu, P.

P. Logofatu, “Phase-modulation scatterometry,” Appl. Opt. 41, 7187-7192 (2002).
[CrossRef] [PubMed]

S. A. Coulombe, P. Logofatu, B. K. Minhas, S. S. H. Naqvi, and J. R. McNeil, “Ellipsometric scatterometry for sub-0.1μm CD measurements,” Proc. SPIE 3332, 282-293 (1998).
[CrossRef]

McGaham, W. A.

J. M. Holden, T. Gubiotti, W. A. McGaham, M. Dusa, and T. Kiers, “Normal-incidence spectroscopic ellipsometry and polarized reflectometry for measurement and control of photoresist critical dimension,” Proc. SPIE 4689, 1110-1121 (2002).
[CrossRef]

McNeil, J. R.

S. A. Coulombe, P. Logofatu, B. K. Minhas, S. S. H. Naqvi, and J. R. McNeil, “Ellipsometric scatterometry for sub-0.1μm CD measurements,” Proc. SPIE 3332, 282-293 (1998).
[CrossRef]

S. S. H. Naqvi, R. H. Krukar, J. R. McNeil, J. E. Franke, T. M. Niemczyk, D. M. Haaland, R. A. Gottsho, and A. Kornblit, “Etch depth estimation of large-period silicon gratings with multivariate calibration of rigorously simulated diffraction profiles,” J. Opt. Soc. Am. A 11, 2485-2493 (1994).
[CrossRef]

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, and S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration,” Proc. SPIE 1926, 60-71 (1993).
[CrossRef]

Menhaj, M.

M. T. Hagan and M. Menhaj, “Training feedforward networks with the Marquardt algorithm,” IEEE Trans. Neural Netw. 5, 989-993 (1994).
[CrossRef] [PubMed]

Minhas, B. K.

S. A. Coulombe, P. Logofatu, B. K. Minhas, S. S. H. Naqvi, and J. R. McNeil, “Ellipsometric scatterometry for sub-0.1μm CD measurements,” Proc. SPIE 3332, 282-293 (1998).
[CrossRef]

Mure-Ravaud, A.

S. Robert, A. Mure-Ravaud, S. Thyria, M. Yacoub, and F. Badran, “Neural selection of the optimal signature for a rapid characterization of a submicrometer period grating,” Opt. Commun. 238, 215-228 (2004).
[CrossRef]

Naqvi, S. S. H.

S. A. Coulombe, P. Logofatu, B. K. Minhas, S. S. H. Naqvi, and J. R. McNeil, “Ellipsometric scatterometry for sub-0.1μm CD measurements,” Proc. SPIE 3332, 282-293 (1998).
[CrossRef]

S. S. H. Naqvi, R. H. Krukar, J. R. McNeil, J. E. Franke, T. M. Niemczyk, D. M. Haaland, R. A. Gottsho, and A. Kornblit, “Etch depth estimation of large-period silicon gratings with multivariate calibration of rigorously simulated diffraction profiles,” J. Opt. Soc. Am. A 11, 2485-2493 (1994).
[CrossRef]

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, and S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration,” Proc. SPIE 1926, 60-71 (1993).
[CrossRef]

Ni, X.

X. Ni, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE 3677, 159-168 (1999).
[CrossRef]

Niemczyk, T. M.

Novikova, T.

Oja, E.

Peterson, G. A.

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, and S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration,” Proc. SPIE 1926, 60-71 (1993).
[CrossRef]

Prins, S. L.

R. H. Krukar, S. L. Prins, D. M. Krukar, G. A. Peterson, S. M. Gaspar, J. R. McNeil, and S. S. H. Naqvi, “Using scattered light modeling for semiconductor critical dimension metrology and calibration,” Proc. SPIE 1926, 60-71 (1993).
[CrossRef]

Ravaud, A. M.

Raymond, C. J.

C. J. Raymond, M. Littau, B. Youn, C. J. Sohn, J. A. Kim, and Y. S. Kang, “Applications of optical scatterometry for the measurement of multiple periodic features,” Proc. SPIE 5038, 577-584 (2003).
[CrossRef]

Reed, J.

E. Drége, J. Reed, and D. Byrne, “Linearized inversion of scatterometric data to obtain surface profile information,” Opt. Eng. (Bellingham) 41, 225-236 (2002).
[CrossRef]

Reynaud, S.

Rietman, E. A.

R. Krukar, A. Cornblit, L. A. Clark, J. Kruskal, D. Lambert, E. A. Rietman, and R. A. Gottscho, “Reactive ion etching profile and depth characterization using statistical and neural analysis of light scattering data,” J. Appl. Phys. 74, 3698-3706 (1993).
[CrossRef]

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, E. A. Rietman, and D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664-668 (1991).
[CrossRef]

Robert, S.

I. Gereige, S. Robert, G. Granet, D. Jamon, and J. J. Rousseau, “Rapid control of submicrometer periodic structures by a neural inversion of ellipsometric measurement,” Opt. Commun. 278, 270-273 (2007).
[CrossRef]

S. Robert, A. Mure-Ravaud, S. Thyria, M. Yacoub, and F. Badran, “Neural selection of the optimal signature for a rapid characterization of a submicrometer period grating,” Opt. Commun. 238, 215-228 (2004).
[CrossRef]

S. Robert, A. M. Ravaud, S. Reynaud, S. Fourment, F. Carcenac, and P. Arguel, “Experimental characterization of subwavelength diffraction gratings by an inverse scattering neural method,” J. Opt. Soc. Am. A 19, 2394-2402 (2002).
[CrossRef]

Rousseau, J. J.

I. Gereige, S. Robert, G. Granet, D. Jamon, and J. J. Rousseau, “Rapid control of submicrometer periodic structures by a neural inversion of ellipsometric measurement,” Opt. Commun. 278, 270-273 (2007).
[CrossRef]

Saarinen, J.

Shyu, D. M.

Sinatore, D.

K. P. Giapas, R. A. Gottscho, L. A. Clark, J. Kruskal, D. Lambert, E. A. Rietman, and D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664-668 (1991).
[CrossRef]

Smith, N.

Sohn, C. J.

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Figures (7)

Fig. 1
Fig. 1

Architecture of the multilayer perceptron used for classification with one hidden layer. The input layer contains the intensities composing the ellipsometric signature, and the output vector is binary coded.

Fig. 2
Fig. 2

Geometrical profiles of two different gratings: a sinusoidal model (left) defined by two parameters b (half-period of the sinus function) and h (height); and a symmetrical trapezoidal model (right) defined by three parameters b 1 (sidewall projection), b 2 (linewidth), and h (height).

Fig. 3
Fig. 3

Representation of the two simulated geometrical profiles B 1 and B 2 for one period distance 0.565 μ m : B 1 is a trapezoidal profile with b 1 = 10 nm , b 2 = 100 nm , and h = 170 nm ; B 2 is a sinusoidal profile with b = 170 nm and h = 200 nm .

Fig. 4
Fig. 4

Variation of the ellipsometric signature (intensities I s and I c versus wavelength at a 60 ° incident angle in the classical mounting) calculated for simulated gratings B 1 and B 2 .

Fig. 5
Fig. 5

Cross-sectional scanning electron microscopy view of the 0.565 μ m period SiO 2 relief grating G 1 .

Fig. 6
Fig. 6

Geometrical profile of the 2 μ m period photoresist grating G 2 characterized by atomic force microscopy.

Fig. 7
Fig. 7

Performances of the ANN classifier according to the different ellipsometric configurations C 1 , C 2 , C 3 , and C 4 described in Table 5. (a) Percentage of correct classifications, (b) percentage of incorrect classifications, (c) percentage of nonclassifications.

Tables (5)

Tables Icon

Table 1 ANN Performance Test (with Two Hidden Neurons) Accuracy for Different Numbers of Training Pairs in the Case of Grating Models A and B

Tables Icon

Table 2 ANN Performance Test (with 2000 Training Pairs) Accuracy for Different Numbers of Hidden Neurons in the Case of Grating Models A and B

Tables Icon

Table 3 Estimation of the Class Probability According to Several ANN Architectures for the Simulated Trapezoidal Grating B 1 and for the Simulated Sinusoidal Grating B 2

Tables Icon

Table 4 Experimental Classification of Ellipsometric Signatures for the Manufactured Gratings G 1 and G 2

Tables Icon

Table 5 Ellipsometric Signatures C 1 , C 2 , C 3 , and C 4 Defined from a Set of Different Wavelengths λ, Incident Angles θ, and Azimuth Angles φ

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

ρ = r p r s = tan ψ e j Δ ,
I s = sin 2 ψ sin Δ , I c = sin 2 ψ cos Δ .
O j = f ( i = 1 N w j , i x i ) ,
f ( a ) = 1 1 + e a .
S m = e E m m = 1 M e E m ,
E m = j = 1 H z m , j O j .

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