Abstract

Scatterometry is frequently used as a non-imaging indirect optical method to reconstruct the critical dimensions (CD) of periodic nanostructures. A particular promising direction is EUV scatterometry with wavelengths in the range of 13 – 14 nm. The conventional approach to determine CDs is the minimization of a least squares function (LSQ). In this paper, we introduce an alternative method based on the maximum likelihood estimation (MLE) that determines the statistical error model parameters directly from measurement data. By using simulation data, we show that the MLE method is able to correct the systematic errors present in LSQ results and improves the accuracy of scatterometry. In a second step, the MLE approach is applied to measurement data from both extreme ultraviolet (EUV) and deep ultraviolet (DUV) scatterometry. Using MLE removes the systematic disagreement of EUV with other methods such as scanning electron microscopy and gives consistent results for DUV.

© 2012 OSA

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2011

M. Wurm, S. Bonifer, B. Bodermann, and J. Richter, “Deep ultraviolet scatterometer for dimensional characterization of nanostructures: system improvements and test measurements,” Meas. Sci. Technol.22, 094024 (2011).
[CrossRef]

M. Wurm, S. Bonifer, B. Bodermann, and M. Gerhard, “Comparison of far field characterisation of DOEs with a goniometric DUV-scatterometer and a CCD-based system,” J. Eur. Opt. Soc. Rapid Publ.6, 11015s (2011).
[CrossRef]

2010

2009

H. Patrick, T. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE7271, 727128 (2009).
[CrossRef]

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE7390, 73900Q (2009).
[CrossRef]

H. Gross, A. Rathsfeld, F. Scholze, and M. Bär, “Profile reconstruction in extreme ultraviolet (EUV) scatterometry: modeling and uncertainty estimates,” Meas. Sci. Technol.20, 105102 (2009).
[CrossRef]

H. Patrick, T. Germer, R. Silver, and B. Bunday, “Developing an uncertainty analysis for optical scatterometry,” Proc. SPIE7272, 72720T (2009).

2008

F. Scholze and C. Laubis, “Use of EUV scatterometry for the characterization of line profiles and line roughness on photomasks,” Proc. SPIE6792, 6792OU (2008).

J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV Masks by EUV-Scatterometry and Finite Element Analysis,” Proc. SPIE7028, 70280P (2008).
[CrossRef]

H. Gross, R. Model, F. Scholze, M. Wurm, B. Bodermann, M. Bär, and A. Rathsfeld, “Modellbildung, Bestimmung der Messunsicherheit und Validierung für diskrete inverse Probleme am Beispiel der Scatterometrie,” VDI-Berichte2011, 337–346 (2008).

J. Richter, J. Rudolf, B. Bodermann, and J. C. Lam, “Comparative scatterometric CD measurements on a MoSi photo mask using different metrology tools,” Proc. SPIE7122, 71222U (2008).
[CrossRef]

2007

2006

C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE6151, 61510I (2006).
[CrossRef]

H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, and A. Rathsfeld, “Mathematical modelling of indirect measurements in scatterometry,” Measurement39, 782–794 (2006).
[CrossRef]

2004

J. Perlich, F. Kamm, J. Rau, F. Scholze, and G. Ulm, “Characterization of extreme ultraviolet masks by extreme ultraviolet scatterometry,” J. Vac. Sci. Technol. B22, 3059 (2004).
[CrossRef]

H. Huang and F. Terry, “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films455, 828–836 (2004).
[CrossRef]

2002

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math.16, 139–156 (2002).
[CrossRef]

1999

X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE3677, 159–168 (1999).
[CrossRef]

1998

O. Cessenat and B. Despres, “Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem,” SIAM J. Numer. Anal.35, 255–299 (1998).
[CrossRef]

S. Coulombe, B. Minhas, C. Raymond, S. Naqvi, and J. McNeil, “Scatterometry measurement of sub-0.1 μm linewidth gratings,” J. Vac. Sci. Technol. B16, 80 (1998).
[CrossRef]

1997

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B15, 361–368 (1997).
[CrossRef]

1996

J. Melenk and I. Babuška, “The partition of unity finite element method: basic theory and applications,” Comput. Meth. Appl. Mech. Eng.139, 289–314 (1996).
[CrossRef]

P. Lalanne and G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A13, 779–784 (1996).
[CrossRef]

1995

1981

M. Moharam and T. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A71, 811–818 (1981).
[CrossRef]

1886

A. Tavrov, M. Totzeck, N. Kerwien, and H. Tiziani, “Rigorous coupled-wave analysis calculus of submicrometer interference pattern and resolving edge position versus signal-to-noise ratio,” Opt. Eng.41, 1886 (2002).

Al-Assaad, R.

Babuška, I.

J. Melenk and I. Babuška, “The partition of unity finite element method: basic theory and applications,” Comput. Meth. Appl. Mech. Eng.139, 289–314 (1996).
[CrossRef]

Bao, J.

X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE3677, 159–168 (1999).
[CrossRef]

Bär, M.

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE7390, 73900Q (2009).
[CrossRef]

H. Gross, A. Rathsfeld, F. Scholze, and M. Bär, “Profile reconstruction in extreme ultraviolet (EUV) scatterometry: modeling and uncertainty estimates,” Meas. Sci. Technol.20, 105102 (2009).
[CrossRef]

H. Gross, R. Model, F. Scholze, M. Wurm, B. Bodermann, M. Bär, and A. Rathsfeld, “Modellbildung, Bestimmung der Messunsicherheit und Validierung für diskrete inverse Probleme am Beispiel der Scatterometrie,” VDI-Berichte2011, 337–346 (2008).

H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, and A. Rathsfeld, “Mathematical modelling of indirect measurements in scatterometry,” Measurement39, 782–794 (2006).
[CrossRef]

H. Gross, A. Rathsfeld, and M. Bär, “Modelling and uncertainty estimates for numerically reconstructed profiles in scatterometry,” in “Advanced mathematical and computational tools in metrology and testing: AMCTM VIII,” F. Pavese, M. Bär, A. B. Forbes, J.-M. Linares, C. Perruchet, and N.-F. Zhang, eds. (World Scientific Pub. Co. Inc., 2009), 142–147.

H. Gross, M.-A. Henn, A. Rathsfeld, and M. Bär, “Stochastic modelling aspects for an improved solution of the inverse problem in scatterometry,” in “Advanced mathematical and computational tools in metrology and testing: AMCTM IX,” F. Pavese, M. Bär, J.-R. Filtz, A. B. Forbes, L. Pendrill, and H. Shirono, eds. (World Scientific Pub. Co. Inc., 2012), 202–209.

Berger, J.

J. Berger, Statistical decision theory and Bayesian analysis (Springer, 1985).

Bodermann, B.

M. Wurm, S. Bonifer, B. Bodermann, and M. Gerhard, “Comparison of far field characterisation of DOEs with a goniometric DUV-scatterometer and a CCD-based system,” J. Eur. Opt. Soc. Rapid Publ.6, 11015s (2011).
[CrossRef]

M. Wurm, S. Bonifer, B. Bodermann, and J. Richter, “Deep ultraviolet scatterometer for dimensional characterization of nanostructures: system improvements and test measurements,” Meas. Sci. Technol.22, 094024 (2011).
[CrossRef]

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE7390, 73900Q (2009).
[CrossRef]

J. Richter, J. Rudolf, B. Bodermann, and J. C. Lam, “Comparative scatterometric CD measurements on a MoSi photo mask using different metrology tools,” Proc. SPIE7122, 71222U (2008).
[CrossRef]

H. Gross, R. Model, F. Scholze, M. Wurm, B. Bodermann, M. Bär, and A. Rathsfeld, “Modellbildung, Bestimmung der Messunsicherheit und Validierung für diskrete inverse Probleme am Beispiel der Scatterometrie,” VDI-Berichte2011, 337–346 (2008).

H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, and A. Rathsfeld, “Mathematical modelling of indirect measurements in scatterometry,” Measurement39, 782–794 (2006).
[CrossRef]

Bonifer, S.

M. Wurm, S. Bonifer, B. Bodermann, and J. Richter, “Deep ultraviolet scatterometer for dimensional characterization of nanostructures: system improvements and test measurements,” Meas. Sci. Technol.22, 094024 (2011).
[CrossRef]

M. Wurm, S. Bonifer, B. Bodermann, and M. Gerhard, “Comparison of far field characterisation of DOEs with a goniometric DUV-scatterometer and a CCD-based system,” J. Eur. Opt. Soc. Rapid Publ.6, 11015s (2011).
[CrossRef]

Botten, L.

R. Petit and L. Botten, Electromagnetic theory of gratings (Springer, 1980).
[CrossRef]

Buchholz, C.

C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE6151, 61510I (2006).
[CrossRef]

Bunday, B.

H. Patrick, T. Germer, R. Silver, and B. Bunday, “Developing an uncertainty analysis for optical scatterometry,” Proc. SPIE7272, 72720T (2009).

Burger, S.

J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV Masks by EUV-Scatterometry and Finite Element Analysis,” Proc. SPIE7028, 70280P (2008).
[CrossRef]

Byrne, D.

Cessenat, O.

O. Cessenat and B. Despres, “Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem,” SIAM J. Numer. Anal.35, 255–299 (1998).
[CrossRef]

Ciarlet, P.

P. Ciarlet, The finite element method for elliptic problems (North-Holland, 1978).

Coulombe, S.

S. Coulombe, B. Minhas, C. Raymond, S. Naqvi, and J. McNeil, “Scatterometry measurement of sub-0.1 μm linewidth gratings,” J. Vac. Sci. Technol. B16, 80 (1998).
[CrossRef]

Dersch, U.

J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV Masks by EUV-Scatterometry and Finite Element Analysis,” Proc. SPIE7028, 70280P (2008).
[CrossRef]

Despres, B.

O. Cessenat and B. Despres, “Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem,” SIAM J. Numer. Anal.35, 255–299 (1998).
[CrossRef]

Ding, Y.

H. Patrick, T. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE7271, 727128 (2009).
[CrossRef]

Elschner, J.

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math.16, 139–156 (2002).
[CrossRef]

Enkisch, H.

C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE6151, 61510I (2006).
[CrossRef]

Fischer, A.

C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE6151, 61510I (2006).
[CrossRef]

Fitzgerald, W.

J. Ruanaidh and W. Fitzgerald, Numerical Bayesian Methods Applied to Signal Processing (Springer, 1996).
[CrossRef]

Gaylord, T.

Gerhard, M.

M. Wurm, S. Bonifer, B. Bodermann, and M. Gerhard, “Comparison of far field characterisation of DOEs with a goniometric DUV-scatterometer and a CCD-based system,” J. Eur. Opt. Soc. Rapid Publ.6, 11015s (2011).
[CrossRef]

Germer, T.

H. Patrick, T. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE7271, 727128 (2009).
[CrossRef]

H. Patrick, T. Germer, R. Silver, and B. Bunday, “Developing an uncertainty analysis for optical scatterometry,” Proc. SPIE7272, 72720T (2009).

Grann, E.

Gross, H.

H. Gross, A. Rathsfeld, F. Scholze, and M. Bär, “Profile reconstruction in extreme ultraviolet (EUV) scatterometry: modeling and uncertainty estimates,” Meas. Sci. Technol.20, 105102 (2009).
[CrossRef]

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE7390, 73900Q (2009).
[CrossRef]

H. Gross, R. Model, F. Scholze, M. Wurm, B. Bodermann, M. Bär, and A. Rathsfeld, “Modellbildung, Bestimmung der Messunsicherheit und Validierung für diskrete inverse Probleme am Beispiel der Scatterometrie,” VDI-Berichte2011, 337–346 (2008).

H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, and A. Rathsfeld, “Mathematical modelling of indirect measurements in scatterometry,” Measurement39, 782–794 (2006).
[CrossRef]

H. Gross, A. Rathsfeld, and M. Bär, “Modelling and uncertainty estimates for numerically reconstructed profiles in scatterometry,” in “Advanced mathematical and computational tools in metrology and testing: AMCTM VIII,” F. Pavese, M. Bär, A. B. Forbes, J.-M. Linares, C. Perruchet, and N.-F. Zhang, eds. (World Scientific Pub. Co. Inc., 2009), 142–147.

H. Gross, M.-A. Henn, A. Rathsfeld, and M. Bär, “Stochastic modelling aspects for an improved solution of the inverse problem in scatterometry,” in “Advanced mathematical and computational tools in metrology and testing: AMCTM IX,” F. Pavese, M. Bär, J.-R. Filtz, A. B. Forbes, L. Pendrill, and H. Shirono, eds. (World Scientific Pub. Co. Inc., 2012), 202–209.

Henn, M.-A.

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE7390, 73900Q (2009).
[CrossRef]

H. Gross, M.-A. Henn, A. Rathsfeld, and M. Bär, “Stochastic modelling aspects for an improved solution of the inverse problem in scatterometry,” in “Advanced mathematical and computational tools in metrology and testing: AMCTM IX,” F. Pavese, M. Bär, J.-R. Filtz, A. B. Forbes, L. Pendrill, and H. Shirono, eds. (World Scientific Pub. Co. Inc., 2012), 202–209.

Hinder, R.

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math.16, 139–156 (2002).
[CrossRef]

Hosch, J.

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B15, 361–368 (1997).
[CrossRef]

Huang, H.

H. Huang and F. Terry, “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films455, 828–836 (2004).
[CrossRef]

Ihlenburg, F.

F. Ihlenburg, Finite element analysis of acoustic scattering (Springer, 1998).
[CrossRef]

Jakatdar, N.

X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE3677, 159–168 (1999).
[CrossRef]

Kaipio, J.

J. Kaipio and E. Somersalo, Statistical and computational inverse problems (Springer, 2005).

Kamm, F.

J. Perlich, F. Kamm, J. Rau, F. Scholze, and G. Ulm, “Characterization of extreme ultraviolet masks by extreme ultraviolet scatterometry,” J. Vac. Sci. Technol. B22, 3059 (2004).
[CrossRef]

Kato, A.

Kerwien, N.

A. Tavrov, M. Totzeck, N. Kerwien, and H. Tiziani, “Rigorous coupled-wave analysis calculus of submicrometer interference pattern and resolving edge position versus signal-to-noise ratio,” Opt. Eng.41, 1886 (2002).

Lalanne, P.

Lam, J. C.

J. Richter, J. Rudolf, B. Bodermann, and J. C. Lam, “Comparative scatterometric CD measurements on a MoSi photo mask using different metrology tools,” Proc. SPIE7122, 71222U (2008).
[CrossRef]

Landau, L.

L. Landau and J. Lifschitz, Lehrbuch der theoretischen Physik: 2, Klassische Feldtheorie (Akademie Verlag, 1977).

Laubis, C.

J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV Masks by EUV-Scatterometry and Finite Element Analysis,” Proc. SPIE7028, 70280P (2008).
[CrossRef]

F. Scholze and C. Laubis, “Use of EUV scatterometry for the characterization of line profiles and line roughness on photomasks,” Proc. SPIE6792, 6792OU (2008).

C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE6151, 61510I (2006).
[CrossRef]

Lifschitz, J.

L. Landau and J. Lifschitz, Lehrbuch der theoretischen Physik: 2, Klassische Feldtheorie (Akademie Verlag, 1977).

Louis, E.

C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE6151, 61510I (2006).
[CrossRef]

McNeil, J.

S. Coulombe, B. Minhas, C. Raymond, S. Naqvi, and J. McNeil, “Scatterometry measurement of sub-0.1 μm linewidth gratings,” J. Vac. Sci. Technol. B16, 80 (1998).
[CrossRef]

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B15, 361–368 (1997).
[CrossRef]

Melenk, J.

J. Melenk and I. Babuška, “The partition of unity finite element method: basic theory and applications,” Comput. Meth. Appl. Mech. Eng.139, 289–314 (1996).
[CrossRef]

Millar, R.

R. Millar, Maximum Likelihood Estimation and Inference (Wiley, 2011).
[CrossRef]

Minhas, B.

S. Coulombe, B. Minhas, C. Raymond, S. Naqvi, and J. McNeil, “Scatterometry measurement of sub-0.1 μm linewidth gratings,” J. Vac. Sci. Technol. B16, 80 (1998).
[CrossRef]

Model, R.

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE7390, 73900Q (2009).
[CrossRef]

H. Gross, R. Model, F. Scholze, M. Wurm, B. Bodermann, M. Bär, and A. Rathsfeld, “Modellbildung, Bestimmung der Messunsicherheit und Validierung für diskrete inverse Probleme am Beispiel der Scatterometrie,” VDI-Berichte2011, 337–346 (2008).

H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, and A. Rathsfeld, “Mathematical modelling of indirect measurements in scatterometry,” Measurement39, 782–794 (2006).
[CrossRef]

Moharam, M.

Morris, G.

Müllender, S.

C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE6151, 61510I (2006).
[CrossRef]

Murnane, M.

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B15, 361–368 (1997).
[CrossRef]

Naqvi, H.

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B15, 361–368 (1997).
[CrossRef]

Naqvi, S.

S. Coulombe, B. Minhas, C. Raymond, S. Naqvi, and J. McNeil, “Scatterometry measurement of sub-0.1 μm linewidth gratings,” J. Vac. Sci. Technol. B16, 80 (1998).
[CrossRef]

Niu, X.

X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE3677, 159–168 (1999).
[CrossRef]

Patrick, H.

H. Patrick, T. Germer, R. Silver, and B. Bunday, “Developing an uncertainty analysis for optical scatterometry,” Proc. SPIE7272, 72720T (2009).

H. Patrick, T. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE7271, 727128 (2009).
[CrossRef]

Perlich, J.

J. Perlich, F. Kamm, J. Rau, F. Scholze, and G. Ulm, “Characterization of extreme ultraviolet masks by extreme ultraviolet scatterometry,” J. Vac. Sci. Technol. B22, 3059 (2004).
[CrossRef]

Petit, R.

R. Petit and L. Botten, Electromagnetic theory of gratings (Springer, 1980).
[CrossRef]

Plöger, S.

C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE6151, 61510I (2006).
[CrossRef]

Pommet, D.

Pomplun, J.

J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV Masks by EUV-Scatterometry and Finite Element Analysis,” Proc. SPIE7028, 70280P (2008).
[CrossRef]

Prins, S.

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B15, 361–368 (1997).
[CrossRef]

Rathsfeld, A.

H. Gross, A. Rathsfeld, F. Scholze, and M. Bär, “Profile reconstruction in extreme ultraviolet (EUV) scatterometry: modeling and uncertainty estimates,” Meas. Sci. Technol.20, 105102 (2009).
[CrossRef]

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE7390, 73900Q (2009).
[CrossRef]

H. Gross, R. Model, F. Scholze, M. Wurm, B. Bodermann, M. Bär, and A. Rathsfeld, “Modellbildung, Bestimmung der Messunsicherheit und Validierung für diskrete inverse Probleme am Beispiel der Scatterometrie,” VDI-Berichte2011, 337–346 (2008).

H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, and A. Rathsfeld, “Mathematical modelling of indirect measurements in scatterometry,” Measurement39, 782–794 (2006).
[CrossRef]

H. Gross, A. Rathsfeld, and M. Bär, “Modelling and uncertainty estimates for numerically reconstructed profiles in scatterometry,” in “Advanced mathematical and computational tools in metrology and testing: AMCTM VIII,” F. Pavese, M. Bär, A. B. Forbes, J.-M. Linares, C. Perruchet, and N.-F. Zhang, eds. (World Scientific Pub. Co. Inc., 2009), 142–147.

H. Gross, M.-A. Henn, A. Rathsfeld, and M. Bär, “Stochastic modelling aspects for an improved solution of the inverse problem in scatterometry,” in “Advanced mathematical and computational tools in metrology and testing: AMCTM IX,” F. Pavese, M. Bär, J.-R. Filtz, A. B. Forbes, L. Pendrill, and H. Shirono, eds. (World Scientific Pub. Co. Inc., 2012), 202–209.

Rau, J.

J. Perlich, F. Kamm, J. Rau, F. Scholze, and G. Ulm, “Characterization of extreme ultraviolet masks by extreme ultraviolet scatterometry,” J. Vac. Sci. Technol. B22, 3059 (2004).
[CrossRef]

Raymond, C.

S. Coulombe, B. Minhas, C. Raymond, S. Naqvi, and J. McNeil, “Scatterometry measurement of sub-0.1 μm linewidth gratings,” J. Vac. Sci. Technol. B16, 80 (1998).
[CrossRef]

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B15, 361–368 (1997).
[CrossRef]

Richter, J.

M. Wurm, S. Bonifer, B. Bodermann, and J. Richter, “Deep ultraviolet scatterometer for dimensional characterization of nanostructures: system improvements and test measurements,” Meas. Sci. Technol.22, 094024 (2011).
[CrossRef]

J. Richter, J. Rudolf, B. Bodermann, and J. C. Lam, “Comparative scatterometric CD measurements on a MoSi photo mask using different metrology tools,” Proc. SPIE7122, 71222U (2008).
[CrossRef]

Richter, L.

H. Patrick, T. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE7271, 727128 (2009).
[CrossRef]

Ro, H.

H. Patrick, T. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE7271, 727128 (2009).
[CrossRef]

Ruanaidh, J.

J. Ruanaidh and W. Fitzgerald, Numerical Bayesian Methods Applied to Signal Processing (Springer, 1996).
[CrossRef]

Rudolf, J.

J. Richter, J. Rudolf, B. Bodermann, and J. C. Lam, “Comparative scatterometric CD measurements on a MoSi photo mask using different metrology tools,” Proc. SPIE7122, 71222U (2008).
[CrossRef]

Schmidt, F.

J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV Masks by EUV-Scatterometry and Finite Element Analysis,” Proc. SPIE7028, 70280P (2008).
[CrossRef]

Schmidt, G.

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math.16, 139–156 (2002).
[CrossRef]

Scholz, F.

C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE6151, 61510I (2006).
[CrossRef]

Scholze, F.

A. Kato and F. Scholze, “Effect of line roughness on the diffraction intensities in angular resolved scatterometry,” Appl. Opt.49, 6102–6110 (2010).
[CrossRef]

H. Gross, A. Rathsfeld, F. Scholze, and M. Bär, “Profile reconstruction in extreme ultraviolet (EUV) scatterometry: modeling and uncertainty estimates,” Meas. Sci. Technol.20, 105102 (2009).
[CrossRef]

J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV Masks by EUV-Scatterometry and Finite Element Analysis,” Proc. SPIE7028, 70280P (2008).
[CrossRef]

F. Scholze and C. Laubis, “Use of EUV scatterometry for the characterization of line profiles and line roughness on photomasks,” Proc. SPIE6792, 6792OU (2008).

H. Gross, R. Model, F. Scholze, M. Wurm, B. Bodermann, M. Bär, and A. Rathsfeld, “Modellbildung, Bestimmung der Messunsicherheit und Validierung für diskrete inverse Probleme am Beispiel der Scatterometrie,” VDI-Berichte2011, 337–346 (2008).

C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE6151, 61510I (2006).
[CrossRef]

J. Perlich, F. Kamm, J. Rau, F. Scholze, and G. Ulm, “Characterization of extreme ultraviolet masks by extreme ultraviolet scatterometry,” J. Vac. Sci. Technol. B22, 3059 (2004).
[CrossRef]

Silver, R.

H. Patrick, T. Germer, R. Silver, and B. Bunday, “Developing an uncertainty analysis for optical scatterometry,” Proc. SPIE7272, 72720T (2009).

Sohail, S.

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B15, 361–368 (1997).
[CrossRef]

Soles, C.

H. Patrick, T. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE7271, 727128 (2009).
[CrossRef]

Somersalo, E.

J. Kaipio and E. Somersalo, Statistical and computational inverse problems (Springer, 2005).

Spanos, C.

X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE3677, 159–168 (1999).
[CrossRef]

Tarantola, A.

A. Tarantola, Inverse problem theory (Elsevier, 1987).

Tavrov, A.

A. Tavrov, M. Totzeck, N. Kerwien, and H. Tiziani, “Rigorous coupled-wave analysis calculus of submicrometer interference pattern and resolving edge position versus signal-to-noise ratio,” Opt. Eng.41, 1886 (2002).

Terry, F.

H. Huang and F. Terry, “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films455, 828–836 (2004).
[CrossRef]

Tiziani, H.

A. Tavrov, M. Totzeck, N. Kerwien, and H. Tiziani, “Rigorous coupled-wave analysis calculus of submicrometer interference pattern and resolving edge position versus signal-to-noise ratio,” Opt. Eng.41, 1886 (2002).

Totzeck, M.

A. Tavrov, M. Totzeck, N. Kerwien, and H. Tiziani, “Rigorous coupled-wave analysis calculus of submicrometer interference pattern and resolving edge position versus signal-to-noise ratio,” Opt. Eng.41, 1886 (2002).

Ulm, G.

C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE6151, 61510I (2006).
[CrossRef]

J. Perlich, F. Kamm, J. Rau, F. Scholze, and G. Ulm, “Characterization of extreme ultraviolet masks by extreme ultraviolet scatterometry,” J. Vac. Sci. Technol. B22, 3059 (2004).
[CrossRef]

Wagner, H.

C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE6151, 61510I (2006).
[CrossRef]

Wedowski, M.

C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE6151, 61510I (2006).
[CrossRef]

Wurm, M.

M. Wurm, S. Bonifer, B. Bodermann, and M. Gerhard, “Comparison of far field characterisation of DOEs with a goniometric DUV-scatterometer and a CCD-based system,” J. Eur. Opt. Soc. Rapid Publ.6, 11015s (2011).
[CrossRef]

M. Wurm, S. Bonifer, B. Bodermann, and J. Richter, “Deep ultraviolet scatterometer for dimensional characterization of nanostructures: system improvements and test measurements,” Meas. Sci. Technol.22, 094024 (2011).
[CrossRef]

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE7390, 73900Q (2009).
[CrossRef]

H. Gross, R. Model, F. Scholze, M. Wurm, B. Bodermann, M. Bär, and A. Rathsfeld, “Modellbildung, Bestimmung der Messunsicherheit und Validierung für diskrete inverse Probleme am Beispiel der Scatterometrie,” VDI-Berichte2011, 337–346 (2008).

H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, and A. Rathsfeld, “Mathematical modelling of indirect measurements in scatterometry,” Measurement39, 782–794 (2006).
[CrossRef]

M. Wurm, “Über die dimensionelle Charakterisierung von Gitterstrukturen auf Fotomasken mit einem neuartigen DUV-Scatterometer,” Ph.D. thesis, Friedrich-Schiller-Universität Jena (2008).

Yedur, S.

X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE3677, 159–168 (1999).
[CrossRef]

Zoethout, E.

C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE6151, 61510I (2006).
[CrossRef]

Adv. Comput. Math.

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math.16, 139–156 (2002).
[CrossRef]

Appl. Opt.

Comput. Meth. Appl. Mech. Eng.

J. Melenk and I. Babuška, “The partition of unity finite element method: basic theory and applications,” Comput. Meth. Appl. Mech. Eng.139, 289–314 (1996).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ.

M. Wurm, S. Bonifer, B. Bodermann, and M. Gerhard, “Comparison of far field characterisation of DOEs with a goniometric DUV-scatterometer and a CCD-based system,” J. Eur. Opt. Soc. Rapid Publ.6, 11015s (2011).
[CrossRef]

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. B

C. Raymond, M. Murnane, S. Prins, S. Sohail, H. Naqvi, J. McNeil, and J. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B15, 361–368 (1997).
[CrossRef]

J. Perlich, F. Kamm, J. Rau, F. Scholze, and G. Ulm, “Characterization of extreme ultraviolet masks by extreme ultraviolet scatterometry,” J. Vac. Sci. Technol. B22, 3059 (2004).
[CrossRef]

S. Coulombe, B. Minhas, C. Raymond, S. Naqvi, and J. McNeil, “Scatterometry measurement of sub-0.1 μm linewidth gratings,” J. Vac. Sci. Technol. B16, 80 (1998).
[CrossRef]

Meas. Sci. Technol.

H. Gross, A. Rathsfeld, F. Scholze, and M. Bär, “Profile reconstruction in extreme ultraviolet (EUV) scatterometry: modeling and uncertainty estimates,” Meas. Sci. Technol.20, 105102 (2009).
[CrossRef]

M. Wurm, S. Bonifer, B. Bodermann, and J. Richter, “Deep ultraviolet scatterometer for dimensional characterization of nanostructures: system improvements and test measurements,” Meas. Sci. Technol.22, 094024 (2011).
[CrossRef]

Measurement

H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, and A. Rathsfeld, “Mathematical modelling of indirect measurements in scatterometry,” Measurement39, 782–794 (2006).
[CrossRef]

Opt. Eng.

A. Tavrov, M. Totzeck, N. Kerwien, and H. Tiziani, “Rigorous coupled-wave analysis calculus of submicrometer interference pattern and resolving edge position versus signal-to-noise ratio,” Opt. Eng.41, 1886 (2002).

Proc. SPIE

J. Richter, J. Rudolf, B. Bodermann, and J. C. Lam, “Comparative scatterometric CD measurements on a MoSi photo mask using different metrology tools,” Proc. SPIE7122, 71222U (2008).
[CrossRef]

H. Patrick, T. Germer, R. Silver, and B. Bunday, “Developing an uncertainty analysis for optical scatterometry,” Proc. SPIE7272, 72720T (2009).

C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner, F. Scholze, G. Ulm, H. Enkisch, S. Müllender, M. Wedowski, E. Louis, and E. Zoethout, “Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB,” Proc. SPIE6151, 61510I (2006).
[CrossRef]

J. Pomplun, S. Burger, F. Schmidt, F. Scholze, C. Laubis, and U. Dersch, “Metrology of EUV Masks by EUV-Scatterometry and Finite Element Analysis,” Proc. SPIE7028, 70280P (2008).
[CrossRef]

X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE3677, 159–168 (1999).
[CrossRef]

H. Patrick, T. Germer, Y. Ding, H. Ro, L. Richter, and C. Soles, “In situ measurement of annealing-induced line shape evolution in nanoimprinted polymers using scatterometry,” Proc. SPIE7271, 727128 (2009).
[CrossRef]

F. Scholze and C. Laubis, “Use of EUV scatterometry for the characterization of line profiles and line roughness on photomasks,” Proc. SPIE6792, 6792OU (2008).

M.-A. Henn, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld, and H. Gross, “On numerical reconstructions of lithographic masks in DUV scatterometry,” Proc. SPIE7390, 73900Q (2009).
[CrossRef]

SIAM J. Numer. Anal.

O. Cessenat and B. Despres, “Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem,” SIAM J. Numer. Anal.35, 255–299 (1998).
[CrossRef]

Thin Solid Films

H. Huang and F. Terry, “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films455, 828–836 (2004).
[CrossRef]

VDI-Berichte

H. Gross, R. Model, F. Scholze, M. Wurm, B. Bodermann, M. Bär, and A. Rathsfeld, “Modellbildung, Bestimmung der Messunsicherheit und Validierung für diskrete inverse Probleme am Beispiel der Scatterometrie,” VDI-Berichte2011, 337–346 (2008).

Other

J. Turunen and F. Wyrowski, eds., Diffractive optics for industrial and commercial applications (Wiley-VCH, 1997).

A. Tarantola, Inverse problem theory (Elsevier, 1987).

P. Ciarlet, The finite element method for elliptic problems (North-Holland, 1978).

J. Elschner, R. Hinder, A. Rathsfeld, and G. Schmidt, http://www.wias-berlin.de/software/DIPOG .

R. Millar, Maximum Likelihood Estimation and Inference (Wiley, 2011).
[CrossRef]

H. Gross, A. Rathsfeld, and M. Bär, “Modelling and uncertainty estimates for numerically reconstructed profiles in scatterometry,” in “Advanced mathematical and computational tools in metrology and testing: AMCTM VIII,” F. Pavese, M. Bär, A. B. Forbes, J.-M. Linares, C. Perruchet, and N.-F. Zhang, eds. (World Scientific Pub. Co. Inc., 2009), 142–147.

M. Wurm, “Über die dimensionelle Charakterisierung von Gitterstrukturen auf Fotomasken mit einem neuartigen DUV-Scatterometer,” Ph.D. thesis, Friedrich-Schiller-Universität Jena (2008).

J. Kaipio and E. Somersalo, Statistical and computational inverse problems (Springer, 2005).

J. Berger, Statistical decision theory and Bayesian analysis (Springer, 1985).

F. Ihlenburg, Finite element analysis of acoustic scattering (Springer, 1998).
[CrossRef]

H. Gross, M.-A. Henn, A. Rathsfeld, and M. Bär, “Stochastic modelling aspects for an improved solution of the inverse problem in scatterometry,” in “Advanced mathematical and computational tools in metrology and testing: AMCTM IX,” F. Pavese, M. Bär, J.-R. Filtz, A. B. Forbes, L. Pendrill, and H. Shirono, eds. (World Scientific Pub. Co. Inc., 2012), 202–209.

J. Ruanaidh and W. Fitzgerald, Numerical Bayesian Methods Applied to Signal Processing (Springer, 1996).
[CrossRef]

R. Petit and L. Botten, Electromagnetic theory of gratings (Springer, 1980).
[CrossRef]

L. Landau and J. Lifschitz, Lehrbuch der theoretischen Physik: 2, Klassische Feldtheorie (Akademie Verlag, 1977).

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Figures (12)

Fig. 1
Fig. 1

a) scheme of the the spectroscopic reflectometer and b) cross section of the investigated EUV-mask.

Fig. 2
Fig. 2

a) scheme of the goniometric reflectometer and b) cross section of the investigated MoSi photomask.

Fig. 3
Fig. 3

χ2 in dependence on bottom CD and top CD for different ratios b/a.

Fig. 4
Fig. 4

a) reconstructed CDs and b) reconstructed SWA in dependence on ratio b/a for a simulated data set (SWA values are shown with approximate 95 % confidence intervalls based on LSQ variance estimation).

Fig. 5
Fig. 5

a) reconstructed noise parameter a and b) reconstructed ratio b/a with approximate 95% confidence intervals for MLE, the green dotted lines represent the actual values, the red dotted lines represent the mean values.

Fig. 6
Fig. 6

Comparison of the reconstructed sidewall angles with approximate 95% confidence intervals for the solutions a) for LSQ and b) for MLE, the green dotted lines represent the actual values, the red dotted lines represent the mean values.

Fig. 7
Fig. 7

Comparison of the RMSD and mean estimated standard deviations in % of the actual value a) for LSQ and b) for MLE.

Fig. 8
Fig. 8

a) reconstructed CDs and b) reconstructed SWA in dependence on ratio b/a for measurement data set D4.

Fig. 9
Fig. 9

a) reconstructed noise parameter a and b) reconstructed ratio b/a with approximate 95% confidence intervals for measured data from the EUV scatterometer, the dotted lines represent the mean values of the reconstructed values.

Fig. 10
Fig. 10

Reconstructed sidewall angles with approximate 95% confidence intervals for measured data from EUV scatterometry, the dotted lines represent the mean values for the two methods.

Fig. 11
Fig. 11

a) reconstructed noise parameter a and b) reconstructed ratio b/a with approximate 95% confidence intervals for measured data from DUV scatterometry, the dotted lines represent the mean values of the reconstructed values.

Fig. 12
Fig. 12

Reconstructed sidewall angles with approximate 95% confidence intervals for measured data from the DUV scatterometer a) for LSQ and b) for MLE, the dotted lines represent the mean values.

Tables (3)

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Table 1 Geometrical parameters and optical constants at a wavelength of 13.0 nm of the EUV mask used for simulations, period d=720 nm.

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Table 2 Geometrical parameters and optical constants at a wavelength of 193 nm of the MoSi mask, period d=560 nm.

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Table 3 Design values of the EUV mask.

Equations (15)

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Δ u ( x , y ) + k 2 u ( x , y ) = 0.
k ( x , y ) = ω ( μ 0 ε ( x , y ) ) 1 / 2 ,
y j = f j ( p ) + ε j .
ε j 𝒩 ( 0 , σ j 2 ) ,
σ j 2 = ( a f j ( p ) ) 2 + b 2 .
f ( p ^ ) y 2 = min p .
χ 2 ( p ) = f ( p ) y 2 = j = 1 m ω j [ f j ( p ) y j ] 2 .
𝒧 ( a , b , p ) = j = 1 m ( 2 π ( ( a f j ( p ) ) 2 + b 2 ) ) 1 / 2 exp [ ( f j ( p ) y j ) 2 2 ( ( a f j ( p ) ) 2 + b 2 ]
θ ^ ML = ( a ^ , b ^ , p ^ ) = arg max a , b , p 𝒧 ( a , b , p ) .
Σ [ J T J ] 1 , J = ( f j p i 1 σ j ) i , j ,
u ( p i ) = ( Σ i , i ) 1 / 2
χ min 2 = j = 1 m σ j 2 [ f j ( p ^ ) y j ] 2 [ χ ν , α / 2 2 , χ ν , 1 α / 2 2 ]
σ ^ j 2 = σ j 2 κ ,
I = ( 2 log 𝒧 θ i θ j ) i , j .
u ( θ ^ i ) = ( Σ i , i ) 1 / 2 , with Σ = I 1 .

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