Abstract

Scatterometry is an important nonimaging and noncontact method for optical metrology. In scatterometry certain parameters of interest are determined by solving an inverse problem. This is done by minimizing a cost functional that quantifies the discrepancy among measured data and model evaluation. Solving the inverse problem is mathematically challenging owing to the instability of the inversion and to the presence of several local minima that are caused by correlation among parameters. This is a relevant issue, particularly when the inverse problem to be solved requires the retrieval of a high number of parameters. In such cases, methods to reduce the complexity of the problem are to be sought. In this work, we propose an algorithm suitable to automatically determine which subset of the parameters is mostly relevant in the model, and we apply it to the reconstruction of 2D and 3D scatterers. We compare the results with local sensitivity analysis and with the screening method proposed by Morris.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2016 (1)

2014 (3)

2013 (1)

2012 (1)

2011 (1)

H. J. Wonsuk Lee and S. H. Han, “Measurement of critical dimension in scanning electron microscope mask images,” J. Micro/Nanolithogr., MEMS, MOEMS 10, 1–8 (2011).
[Crossref]

2009 (2)

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. Friedman, T. Hastie, and R. Tibshirani, “Regularization paths for generalized linear models via coordinate descent,” J. Stat. Softw. 33, 1–22 (2009).
[Crossref]

2008 (1)

H. Gross and A. Rathsfeld, “Sensitivity analysis for indirect measurement in scatterometry and the reconstruction of periodic grating structures,” Waves Random Complex Media 18, 129–149 (2008).
[Crossref]

2007 (1)

F. Campolongo, J. Cariboni, and A. Saltelli, “An effective screening design for sensitivity analysis of large models,” Environ. Model. Software 22, 1509–1518 (2007).
[Crossref]

2006 (1)

G. Dahlen, M. Osborn, H.-C. Liu, R. Jain, W. Foreman, and J. R. Osborne, “Critical dimension AFM tip characterization and image reconstruction applied to the 45-nm node,” Proc. SPIE 6152, 61522R (2006).
[Crossref]

2005 (1)

H. Zou and T. Hastie, “Regularization and variable selection via the elastic net,” J. R. Statist. Soc. B 67, 301–320 (2005).
[Crossref]

2004 (1)

H.-T. Huang and F. Terry, “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films 455-456, 828–836 (2004).
[Crossref]

2002 (1)

1997 (2)

P. Lalanne, “Convergence performance of the coupled-wave and the differential methods for thin gratings,” J. Opt. Soc. Am. A 14, 1583–1591 (1997).
[Crossref]

C. J. Raymond, M. R. Murnane, S. L. Prins, S. Sohail, H. Naqvi, J. R. McNeil, and J. W. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (1997).
[Crossref]

1996 (1)

R. Tibshirani, “Regression shrinkage and selection via the lasso,” J. R. Stat. Soc. Ser. B 58, 267–288 (1996).

1995 (1)

G. Bao, “Finite element approximation of time harmonic waves in periodic structures,” SIAM J. Numer. Anal. 32, 1155–1169 (1995).
[Crossref]

1994 (2)

P. C. Hansen, “Regularization tools—a Matlab package for analysis and solution of discrete ill-posed problems,” Numer. Algorithms 6, 1–35 (1994).
[Crossref]

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[Crossref]

1991 (1)

M. D. Morris, “Factorial sampling plans for preliminary computational experiments,” Technometrics 33, 161–174 (1991).
[Crossref]

1980 (1)

J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[Crossref]

1944 (1)

K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Quart. Appl. Math. 2, 164–168 (1944).

Allard, A.

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

Bao, G.

G. Bao, “Finite element approximation of time harmonic waves in periodic structures,” SIAM J. Numer. Anal. 32, 1155–1169 (1995).
[Crossref]

Bär, M.

M.-A. Henn, H. Gross, F. Scholze, M. Wurm, C. Elster, and M. Bär, “A maximum likelihood approach to the inverse problem of scatterometry,” Opt. Express 20, 12771–12786 (2012).
[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]

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

Barton, E.

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[Crossref]

Bodermann, B.

B. Bodermann, M. Wurm, A. Diener, F. Scholze, and H. Gross, “EUV and DUV scatterometry for CD and edge profile metrology on EUV masks,” in 25th European Mask and Lithography Conference (2009), pp. 1–12.

Campolongo, F.

F. Campolongo, J. Cariboni, and A. Saltelli, “An effective screening design for sensitivity analysis of large models,” Environ. Model. Software 22, 1509–1518 (2007).
[Crossref]

Cariboni, J.

F. Campolongo, J. Cariboni, and A. Saltelli, “An effective screening design for sensitivity analysis of large models,” Environ. Model. Software 22, 1509–1518 (2007).
[Crossref]

Chandezon, J.

J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[Crossref]

Chen, M.-C.

Chen, X.

Chen, Y.-C.

Coene, W. M. J.

Dahlen, G.

G. Dahlen, M. Osborn, H.-C. Liu, R. Jain, W. Foreman, and J. R. Osborne, “Critical dimension AFM tip characterization and image reconstruction applied to the 45-nm node,” Proc. SPIE 6152, 61522R (2006).
[Crossref]

Demeyer, S.

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

Diener, A.

B. Bodermann, M. Wurm, A. Diener, F. Scholze, and H. Gross, “EUV and DUV scatterometry for CD and edge profile metrology on EUV masks,” in 25th European Mask and Lithography Conference (2009), pp. 1–12.

Doicu, A.

A. Doicu, T. Trautmann, and F. Schreier, Numerical Regularization for Atmospheric Inverse Problems (Springer, 2010).

Dong, Z.

Z. Dong, S. Liu, X. Chen, and C. Zhang, “Determination of an optimal measurement configuration in optical scatterometry using global sensitivity analysis,” Thin Solid Films 562, 16–23 (2014).
[Crossref]

Elster, C.

Eriksson, J.

J. Eriksson, “Optimization and regularization of nonlinear least squares problems,” Ph.D. thesis, (Dept. of Computing Science, Umea University, Umea, Sweden, 1996).

Fiebach, H. G. A.

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

Fischer, N.

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

Foreman, W.

G. Dahlen, M. Osborn, H.-C. Liu, R. Jain, W. Foreman, and J. R. Osborne, “Critical dimension AFM tip characterization and image reconstruction applied to the 45-nm node,” Proc. SPIE 6152, 61522R (2006).
[Crossref]

Friedman, J.

J. Friedman, T. Hastie, and R. Tibshirani, “Regularization paths for generalized linear models via coordinate descent,” J. Stat. Softw. 33, 1–22 (2009).
[Crossref]

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer Series in Statistics (Springer, 2009).

Gawhary, O. E.

Gross, H.

M.-A. Henn, H. Gross, F. Scholze, M. Wurm, C. Elster, and M. Bär, “A maximum likelihood approach to the inverse problem of scatterometry,” Opt. Express 20, 12771–12786 (2012).
[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 and A. Rathsfeld, “Sensitivity analysis for indirect measurement in scatterometry and the reconstruction of periodic grating structures,” Waves Random Complex Media 18, 129–149 (2008).
[Crossref]

B. Bodermann, M. Wurm, A. Diener, F. Scholze, and H. Gross, “EUV and DUV scatterometry for CD and edge profile metrology on EUV masks,” in 25th European Mask and Lithography Conference (2009), pp. 1–12.

Han, S. H.

H. J. Wonsuk Lee and S. H. Han, “Measurement of critical dimension in scanning electron microscope mask images,” J. Micro/Nanolithogr., MEMS, MOEMS 10, 1–8 (2011).
[Crossref]

Hansen, P. C.

P. C. Hansen, “Regularization tools—a Matlab package for analysis and solution of discrete ill-posed problems,” Numer. Algorithms 6, 1–35 (1994).
[Crossref]

Hastie, T.

J. Friedman, T. Hastie, and R. Tibshirani, “Regularization paths for generalized linear models via coordinate descent,” J. Stat. Softw. 33, 1–22 (2009).
[Crossref]

H. Zou and T. Hastie, “Regularization and variable selection via the elastic net,” J. R. Statist. Soc. B 67, 301–320 (2005).
[Crossref]

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer Series in Statistics (Springer, 2009).

Heidenreich, S.

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

Henn, M.-A.

M.-A. Henn, H. Gross, F. Scholze, M. Wurm, C. Elster, and M. Bär, “A maximum likelihood approach to the inverse problem of scatterometry,” Opt. Express 20, 12771–12786 (2012).
[Crossref]

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

Hosch, J. W.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. Sohail, H. Naqvi, J. R. McNeil, and J. W. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (1997).
[Crossref]

Huang, H.-T.

H.-T. Huang and F. Terry, “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films 455-456, 828–836 (2004).
[Crossref]

Iooss, B.

B. Iooss and P. Lemaître, “A review on global sensitivity analysis methods,” in Uncertainty Management in Simulation–Optimization of Complex Systems: Algorithms and Applications, G. Dellino and C. Meloni, eds. (Springer, 2015), Chap. 5, pp. 101–122.

Jain, R.

G. Dahlen, M. Osborn, H.-C. Liu, R. Jain, W. Foreman, and J. R. Osborne, “Critical dimension AFM tip characterization and image reconstruction applied to the 45-nm node,” Proc. SPIE 6152, 61522R (2006).
[Crossref]

Jiang, H.

Kok, G.

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

Kondrup, J. B.

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

Ku, Y.-S.

Kumar, N.

Lalanne, P.

Lemaître, P.

B. Iooss and P. Lemaître, “A review on global sensitivity analysis methods,” in Uncertainty Management in Simulation–Optimization of Complex Systems: Algorithms and Applications, G. Dellino and C. Meloni, eds. (Springer, 2015), Chap. 5, pp. 101–122.

Levenberg, K.

K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Quart. Appl. Math. 2, 164–168 (1944).

Liu, H.-C.

G. Dahlen, M. Osborn, H.-C. Liu, R. Jain, W. Foreman, and J. R. Osborne, “Critical dimension AFM tip characterization and image reconstruction applied to the 45-nm node,” Proc. SPIE 6152, 61522R (2006).
[Crossref]

Liu, S.

Lo, C.-W.

Logofatu, P. C.

Madsen, K.

K. Madsen, H. B. Nielsen, and O. Tingleff, Methods for Non-Linear Least Squares Problems, 2nd ed. (2004).

Maystre, D.

J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[Crossref]

McNeil, J. R.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. Sohail, H. Naqvi, J. R. McNeil, and J. W. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (1997).
[Crossref]

Model, R.

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

Morris, M. D.

M. D. Morris, “Factorial sampling plans for preliminary computational experiments,” Technometrics 33, 161–174 (1991).
[Crossref]

Murnane, M. R.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. Sohail, H. Naqvi, J. R. McNeil, and J. W. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (1997).
[Crossref]

Naqvi, H.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. Sohail, H. Naqvi, J. R. McNeil, and J. W. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (1997).
[Crossref]

Nielsen, H. B.

K. Madsen, H. B. Nielsen, and O. Tingleff, Methods for Non-Linear Least Squares Problems, 2nd ed. (2004).

Osborn, M.

G. Dahlen, M. Osborn, H.-C. Liu, R. Jain, W. Foreman, and J. R. Osborne, “Critical dimension AFM tip characterization and image reconstruction applied to the 45-nm node,” Proc. SPIE 6152, 61522R (2006).
[Crossref]

Osborne, J. R.

G. Dahlen, M. Osborn, H.-C. Liu, R. Jain, W. Foreman, and J. R. Osborne, “Critical dimension AFM tip characterization and image reconstruction applied to the 45-nm node,” Proc. SPIE 6152, 61522R (2006).
[Crossref]

Partridge, D.

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

Pelevic, N.

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

Pereira, S. F.

Petrik, P.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

Prins, S. L.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. Sohail, H. Naqvi, J. R. McNeil, and J. W. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (1997).
[Crossref]

Ramanandan, G. K. P.

Raoult, G.

J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[Crossref]

Rasmussen, K.

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

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]

H. Gross and A. Rathsfeld, “Sensitivity analysis for indirect measurement in scatterometry and the reconstruction of periodic grating structures,” Waves Random Complex Media 18, 129–149 (2008).
[Crossref]

Raymond, C. J.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. Sohail, H. Naqvi, J. R. McNeil, and J. W. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (1997).
[Crossref]

Roy, S.

Saltelli, A.

F. Campolongo, J. Cariboni, and A. Saltelli, “An effective screening design for sensitivity analysis of large models,” Environ. Model. Software 22, 1509–1518 (2007).
[Crossref]

Schmelter, S.

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

Scholze, F.

M.-A. Henn, H. Gross, F. Scholze, M. Wurm, C. Elster, and M. Bär, “A maximum likelihood approach to the inverse problem of scatterometry,” Opt. Express 20, 12771–12786 (2012).
[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]

B. Bodermann, M. Wurm, A. Diener, F. Scholze, and H. Gross, “EUV and DUV scatterometry for CD and edge profile metrology on EUV masks,” in 25th European Mask and Lithography Conference (2009), pp. 1–12.

Schreier, F.

A. Doicu, T. Trautmann, and F. Schreier, Numerical Regularization for Atmospheric Inverse Problems (Springer, 2010).

Sohail, S.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. Sohail, H. Naqvi, J. R. McNeil, and J. W. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (1997).
[Crossref]

Terry, F.

H.-T. Huang and F. Terry, “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films 455-456, 828–836 (2004).
[Crossref]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

Tibshirani, R.

J. Friedman, T. Hastie, and R. Tibshirani, “Regularization paths for generalized linear models via coordinate descent,” J. Stat. Softw. 33, 1–22 (2009).
[Crossref]

R. Tibshirani, “Regression shrinkage and selection via the lasso,” J. R. Stat. Soc. Ser. B 58, 267–288 (1996).

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer Series in Statistics (Springer, 2009).

Tingleff, O.

K. Madsen, H. B. Nielsen, and O. Tingleff, Methods for Non-Linear Least Squares Problems, 2nd ed. (2004).

Trautmann, T.

A. Doicu, T. Trautmann, and F. Schreier, Numerical Regularization for Atmospheric Inverse Problems (Springer, 2010).

Urbach, H. P.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

Vogel, C.

C. Vogel, Computational Methods for Inverse Problems (Society for Industrial and Applied Mathematics, 2002).

Wang, W.-T.

Wonsuk Lee, H. J.

H. J. Wonsuk Lee and S. H. Han, “Measurement of critical dimension in scanning electron microscope mask images,” J. Micro/Nanolithogr., MEMS, MOEMS 10, 1–8 (2011).
[Crossref]

Wright, L.

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

Wurm, M.

M.-A. Henn, H. Gross, F. Scholze, M. Wurm, C. Elster, and M. Bär, “A maximum likelihood approach to the inverse problem of scatterometry,” Opt. Express 20, 12771–12786 (2012).
[Crossref]

B. Bodermann, M. Wurm, A. Diener, F. Scholze, and H. Gross, “EUV and DUV scatterometry for CD and edge profile metrology on EUV masks,” in 25th European Mask and Lithography Conference (2009), pp. 1–12.

Yeh, C.-L.

Zhang, C.

Zhu, J.

Zou, H.

H. Zou and T. Hastie, “Regularization and variable selection via the elastic net,” J. R. Statist. Soc. B 67, 301–320 (2005).
[Crossref]

Appl. Opt. (2)

Environ. Model. Software (1)

F. Campolongo, J. Cariboni, and A. Saltelli, “An effective screening design for sensitivity analysis of large models,” Environ. Model. Software 22, 1509–1518 (2007).
[Crossref]

J. Comput. Phys. (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[Crossref]

J. Micro/Nanolithogr., MEMS, MOEMS (1)

H. J. Wonsuk Lee and S. H. Han, “Measurement of critical dimension in scanning electron microscope mask images,” J. Micro/Nanolithogr., MEMS, MOEMS 10, 1–8 (2011).
[Crossref]

J. Opt. (1)

J. Chandezon, G. Raoult, and D. Maystre, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
[Crossref]

J. Opt. Soc. Am. A (1)

J. R. Stat. Soc. Ser. B (1)

R. Tibshirani, “Regression shrinkage and selection via the lasso,” J. R. Stat. Soc. Ser. B 58, 267–288 (1996).

J. R. Statist. Soc. B (1)

H. Zou and T. Hastie, “Regularization and variable selection via the elastic net,” J. R. Statist. Soc. B 67, 301–320 (2005).
[Crossref]

J. Stat. Softw. (1)

J. Friedman, T. Hastie, and R. Tibshirani, “Regularization paths for generalized linear models via coordinate descent,” J. Stat. Softw. 33, 1–22 (2009).
[Crossref]

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

C. J. Raymond, M. R. Murnane, S. L. Prins, S. Sohail, H. Naqvi, J. R. McNeil, and J. W. Hosch, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (1997).
[Crossref]

Meas. Sci. Technol. (1)

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]

Numer. Algorithms (1)

P. C. Hansen, “Regularization tools—a Matlab package for analysis and solution of discrete ill-posed problems,” Numer. Algorithms 6, 1–35 (1994).
[Crossref]

Opt. Express (4)

Proc. SPIE (1)

G. Dahlen, M. Osborn, H.-C. Liu, R. Jain, W. Foreman, and J. R. Osborne, “Critical dimension AFM tip characterization and image reconstruction applied to the 45-nm node,” Proc. SPIE 6152, 61522R (2006).
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K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Quart. Appl. Math. 2, 164–168 (1944).

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G. Bao, “Finite element approximation of time harmonic waves in periodic structures,” SIAM J. Numer. Anal. 32, 1155–1169 (1995).
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[Crossref]

Thin Solid Films (2)

Z. Dong, S. Liu, X. Chen, and C. Zhang, “Determination of an optimal measurement configuration in optical scatterometry using global sensitivity analysis,” Thin Solid Films 562, 16–23 (2014).
[Crossref]

H.-T. Huang and F. Terry, “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,” Thin Solid Films 455-456, 828–836 (2004).
[Crossref]

Waves Random Complex Media (1)

H. Gross and A. Rathsfeld, “Sensitivity analysis for indirect measurement in scatterometry and the reconstruction of periodic grating structures,” Waves Random Complex Media 18, 129–149 (2008).
[Crossref]

Other (11)

B. Iooss and P. Lemaître, “A review on global sensitivity analysis methods,” in Uncertainty Management in Simulation–Optimization of Complex Systems: Algorithms and Applications, G. Dellino and C. Meloni, eds. (Springer, 2015), Chap. 5, pp. 101–122.

A. Doicu, T. Trautmann, and F. Schreier, Numerical Regularization for Atmospheric Inverse Problems (Springer, 2010).

K. Rasmussen, J. B. Kondrup, A. Allard, S. Demeyer, N. Fischer, E. Barton, D. Partridge, L. Wright, M. Bär, H. G. A. Fiebach, S. Heidenreich, M.-A. Henn, R. Model, S. Schmelter, G. Kok, and N. Pelevic, “Novel mathematical and statistical approaches to uncertainty evaluation: best practice guide to uncertainty evaluation for computationally expensive models,” (Euramet, 2015).

J. Eriksson, “Optimization and regularization of nonlinear least squares problems,” Ph.D. thesis, (Dept. of Computing Science, Umea University, Umea, Sweden, 1996).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

K. Madsen, H. B. Nielsen, and O. Tingleff, Methods for Non-Linear Least Squares Problems, 2nd ed. (2004).

Y. Khare and R. Muñoz-Carpena, “Global sensitivity analysis: elementary effects method of Morris using sampling for uniformity (SU) Matlab code manual,” 2014 https://abe.ufl.edu/faculty/carpena/software/SUMorris.shtml .

https://jcmwave.com .

B. Bodermann, M. Wurm, A. Diener, F. Scholze, and H. Gross, “EUV and DUV scatterometry for CD and edge profile metrology on EUV masks,” in 25th European Mask and Lithography Conference (2009), pp. 1–12.

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer Series in Statistics (Springer, 2009).

C. Vogel, Computational Methods for Inverse Problems (Society for Industrial and Applied Mathematics, 2002).

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Figures (7)

Fig. 1.
Fig. 1. (a) Grating with parameterized profile. The independent degrees of freedom are the X and Y coordinates of the yellow points. The materials are given in Table 1. (b) Diffracted efficiencies in percentage. For the given geometry and wavelengths, only a subset of the diffraction orders can be detected.
Fig. 2.
Fig. 2. Plots at the last iteration of the automatic variable selection algorithm. 5% Gaussian noise is added to the synthetic data. p 0 is [ X BL , Y BL , X AL , Y AL , X ARC , Y ARC ] = [66.9, 22.47, 73.41, 81.65, 60.21, 93.7] nm. (a) Elastic net coefficients as a function of regularization parameter strength. γ 0 is the regularization strength selected according to the criteria of Eq. (6). (b) Normalized local sensitivities in percentage.
Fig. 3.
Fig. 3. Plots at the last iteration of the automatic variable selection algorithm. 10% Gaussian noise is added to the synthetic data. p 0 is [ X BL , Y BL , X AL , Y AL , X ARC , Y ARC ] = [74.4, 26.05, 61.78, 81.13, 66.85, 84.97] nm. (a) Elastic net coefficients as a function of regularization parameter strength. γ 0 is the regularization strength selected according to the criteria in Eq. (6). (b) Normalized local sensitivities in percentage.
Fig. 4.
Fig. 4. Plots at the last iteration of the automatic variable selection algorithm. 15% Gaussian noise is added to the synthetic data. p 0 is [ X BL , Y AL , X AL , Y AL , X ARC , Y ARC ] = [68.15, 23.25, 70.92, 72.81, 71, 91.7] nm. (a) Elastic net coefficients as a function of regularization parameter strength. γ 0 is the regularization strength selected according to the criteria in Eq. (6). (b) Normalized local sensitivities in percentage.
Fig. 5.
Fig. 5. Morris plots for four different diffracted orders at λ = 13.398 nm : (a) order 6 , (b) order 1 , (c) order 4, and (d) order 9.
Fig. 6.
Fig. 6. (a) Scatterer with parameterized profile. The parameter p7, not indicated in the figure, is the thickness of the anti-reflective layer. (b) Diffuse scattering given by the structure in (a).
Fig. 7.
Fig. 7. (a) Elastic net coefficients as a function of the regularization parameter strength. (b) Normalized local sensitivities in percentage.

Tables (3)

Tables Icon

Table 1. Layer Thicknesses and Material Properties at λ = 13.5 nm

Tables Icon

Table 2. Reconstruction Results

Tables Icon

Table 3. Layer Thicknesses and Material Properties at λ = 13.5 nm

Equations (10)

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

χ 2 ( p ) = | | y δ F ( p ) | | 2 2 σ 2 .
σ ( λ ) 2 = [ a · E ± ( λ ) ] 2 + b g 2 ,
Δ p n = arg min Δ p n y δ F ( p n ) Δ p n T J n 2 p n + 1 = p n + Δ p n ,
p n + 1 = p n + arg min Δ p n Δ F Δ p n T J + γ P α ( Δ p n ) ,
P α ( Δ p n ) = ( 1 α ) 1 2 Δ p n 2 2 + α Δ p n 1 .
γ n = { ϵ γ n 1 + ( 1 ϵ ) γ if γ < γ n 1 γ n 1 otherwise .
E i ( j ) = F ( p 1 ( j ) , p 2 ( j ) , , p i ( j ) + Δ p i ( j ) , , p n ( j ) ) F ( p ( j ) ) Δ p i ( j ) ,
μ i * = 1 R j = 1 R | E i ( j ) |
σ i = 1 2 j = 1 R ( E i ( j ) 1 R j = 1 R E i ( j ) ) 2 .
Δ p j = { i = 1 N r i j + α γ if i = 1 N r i j < α γ 0 if α γ < i = 1 N r i j < α γ i = 1 N r i j α γ if i = 1 N r i j > α γ .

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