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June  2015, 8(3): 497-519. doi: 10.3934/dcdss.2015.8.497

Modeling aspects to improve the solution of the inverse problem in scatterometry

1. 

Physikalisch-Techn. Bundesanstalt, Berlin, Abbestr. 2-12, Germany, Germany, Germany, Germany

2. 

Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Mohrenstr. 39, Germany

Received  October 2013 Revised  January 2014 Published  October 2014

The precise and accurate determination of critical dimensions (CDs) of photo masks and their uncertainties is relevant to the lithographic process. Scatterometry is a fast, non-destructive optical method for the indirect determination of geometry parameters of periodic surface structures from scattered light intensities. Shorter wavelengths like extreme ultraviolet (EUV) at 13.5 nm ensure that the measured light diffraction pattern has many higher diffraction orders and is sensitive to the structure details. We present a fast non-rigorous method for the analysis of stochastic line edge roughness with amplitudes in the range of a few nanometers, based on a 2D Fourier transform method. The mean scattering light efficiencies of rough line edges reveal an exponential decrease in terms of the diffraction orders and the standard deviation of the roughness amplitude. Former results obtained by rigorous finite element methods (FEM) are confirmed. The implicated extension of the mathematical model of scatterometry by an exponential damping factor is demonstrated by a maximum likelihood method used to reconstruct the geometrical parameters. Approximate uncertainties are determined by employing the Fisher information matrix and additionally a Monte Carlo method with a limited amount of samplings. It turns out that using incomplete mathematical models may lead to underestimated uncertainties calculated by the Fisher matrix approach and to substantially larger uncertainties for the Monte Carlo method.
Citation: Hermann Gross, Sebastian Heidenreich, Mark-Alexander Henn, Markus Bär, Andreas Rathsfeld. Modeling aspects to improve the solution of the inverse problem in scatterometry. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 497-519. doi: 10.3934/dcdss.2015.8.497
References:
[1]

R. Al-Assaad and D. Byrne, Error analysis in inverse scatterometry. I. Modeling,, J. Opt. Soc. Am. A, 24 (2007), 326.  doi: 10.1364/JOSAA.24.000326.  Google Scholar

[2]

G. Bao and D. C. Dobson, Modeling and optimal design of diffractive optical structures,, Surveys on Mathematics for Industry, 8 (1998), 37.   Google Scholar

[3]

G. Bao, Finite element approximation of time harmonic waves in periodic structures,, SIAM J.Numer.Anal., 32 (1995), 1155.  doi: 10.1137/0732053.  Google Scholar

[4]

B. Bergner, T. Germer and T. Suleski, Effective medium approximations for modeling optical reflectance from gratings with rough edges,, J. Opt. Soc. Am. A, 27 (2010), 1083.  doi: 10.1364/JOSAA.27.001083.  Google Scholar

[5]

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML, Evaluation of measurement data - Guide to the expression of uncertainty in measurement,, Joint Committee for Guides in Metrology, (2008).   Google Scholar

[6]

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML, Evaluation of measurement data - Supplement 1 to the Guide to the expression of uncertainty in measurement - Propagation of distributions using a Monte Carlo method,, Joint Committee for Guides in Metrology, (2008).   Google Scholar

[7]

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 (1998), 255.  doi: 10.1137/S0036142995285873.  Google Scholar

[8]

D. Champeney, Fourier Transforms and Their Physical Applications,, Academic Press, (1973).   Google Scholar

[9]

J. Chandezon, G. Raoult and D. Maystre, A new theoretical method for diffraction gratings and its numerical application,, J. Opt., 11 (1980).   Google Scholar

[10]

A. Chen and X. Friedmann, Maxwell's equation in a periodic structure,, Trans. Amer. Math. Soc., 323 (1991), 811.   Google Scholar

[11]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Applied Mathematical Sciences 93, (1998).  doi: 10.1007/978-3-662-03537-5.  Google Scholar

[12]

J. Elschner, R. Hinder, A. Rathsfeld and G. Schmidt, =, ().   Google Scholar

[13]

J. Elschner, R. Hinder and G. Schmidt, Finite element solution of conical diffraction problems,, Adv. Comput. Math., 16 (2002), 139.  doi: 10.1023/A:1014456026778.  Google Scholar

[14]

O. Ersoy, Diffraction, Fourier Optics, and Imaging,, Wiley-Interscience, (2006).  doi: 10.1002/0470085002.  Google Scholar

[15]

T. Germer, Effect of line and trench profile variation on specular and diffusive reflectance from periodic structure,, J. Opt. Soc. Am., A24 (2007), 696.  doi: 10.1364/JOSAA.24.000696.  Google Scholar

[16]

J. Goodman, Introduction to Fourier Optics,, Roberts & Company, (2005).   Google Scholar

[17]

H. Gross, S. Heidenreich, M.-A. Henn, G. Dai, F. Scholze and M. Bär, Modelling line edge roughness in periodic line-space structures by Fourier optics to improve scatterometry,, J. Europ. Opt. Soc. Rap. Public., 9 (2014).  doi: 10.2971/jeos.2014.14003.  Google Scholar

[18]

H. Gross, M.-A. Henn, S. Heidenreich, A. Rathsfeld and M. Bär, Modeling of line roughness and its impact on the diffraction intensities and the reconstructed critical dimensions in scatterometry,, Appl. Optics, 51 (2012), 7384.  doi: 10.1364/AO.51.007384.  Google Scholar

[19]

H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann and A. Rathsfeld, Mathematical modelling of indirect measurements in scatterometry,, Measurement, 39 (2006), 782.  doi: 10.1016/j.measurement.2006.04.009.  Google Scholar

[20]

H. Gross, A. Rathsfeld, F. Scholze and M. Bär, Profile reconstruction in extreme ultraviolet (EUV) scatterometry: modeling and uncertainty estimates,, Meas. Sci. Technol., 20 (2009).  doi: 10.1088/0957-0233/20/10/105102.  Google Scholar

[21]

H. Gross, A. Rathsfeld, F. Scholze, R. Model and M. Bär, Computational methods estimating uncertainties for profile reconstruction in scatterometry,, Proc. SPIE, 6995 (2008).  doi: 10.1117/12.781006.  Google Scholar

[22]

M.-A. Henn, H. Gross, S. Heidenreich, F. Scholze, C. Elster and M. Bär, Improved reconstruction of Critical Dimensions in Extreme Ultraviolet Scatterometry by Modeling Systematic Errors,, Meas. Sci. Technol., 25 (2014).  doi: 10.1088/0957-0233/25/4/044003.  Google Scholar

[23]

M.-A. Henn, H. Gross, F. Scholze, M. Wurm, C. Elster and M. Bär, A maximum likelihood approach to the inverse problem of scatterometry,, Optics Express, 20 (2012), 12771.  doi: 10.1364/OE.20.012771.  Google Scholar

[24]

M.-A. Henn, S. Heidenreich, H. Gross, A. Rathsfeld, F. Scholze and M. Bär, Improved grating reconstruction by determination of line roughness in extreme ultraviolet scatterometry,, Optics Letters, 37 (2012), 5229.  doi: 10.1364/OL.37.005229.  Google Scholar

[25]

A. Hesford and W. Chew, A frequency-domain formulation of the Frechet derivative to exploit the inherent parallelism of the distorted Born iterative method,, Waves in Random and Complex Media, 16 (2006), 495.  doi: 10.1080/17455030600675830.  Google Scholar

[26]

H. Huang and F. Terry Jr, Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,, Thin Solid Films, 455 (2004), 828.  doi: 10.1016/j.tsf.2004.04.010.  Google Scholar

[27]

F. Ihlenburg, Finite Element Analysis of Acoustic Scattering,, Springer-Verlag, (1998).  doi: 10.1007/b98828.  Google Scholar

[28]

A. Kato and F. Scholze, Effect of line roughness on the diffraction intensities in angular resolved scatterometry,, Appl. Opt., 49 (2010), 6102.  doi: 10.1364/AO.49.006102.  Google Scholar

[29]

B. Kleemann, Elektromagnetische Analyse von Oberflächengittern von IR bis XUV Mittels Einer Parametrisierten Randintegralmethode: Theorie, Vergleich und Anwendungen,, PhD thesis, (2002).   Google Scholar

[30]

G. Lalanne P. and Morris, Highly improved convergence of the coupled-wave method for TM-polarization,, JOSA A, 13 (1996), 779.   Google Scholar

[31]

L. Li, New formulation of the Fourier modal method for crossed surface-relief gratings,, JOSA A, 14 (1997), 2758.  doi: 10.1364/JOSAA.14.002758.  Google Scholar

[32]

C. Mack, Analytic form for the power spectral density in one, two, and three dimensions,, J. Micro/Nanolith. MEMS MOEMS, 10 (2011).  doi: 10.1117/1.3663567.  Google Scholar

[33]

C. Mack, Generating random rough edges, surfaces, and volumes,, Appl. Optics, 52 (2013), 1472.  doi: 10.1364/AO.52.001472.  Google Scholar

[34]

J. Melenk and I. Babuška, The partition of unity finite element method: Basic theory and applications,, Comput. Meth. Appl. Mech. Eng., 139 (1996), 289.  doi: 10.1016/S0045-7825(96)01087-0.  Google Scholar

[35]

R. Millar, Maximum Likelihood Estimation and Inference,, Wiley, (2011).  doi: 10.1002/9780470094846.  Google Scholar

[36]

B. Minhas, S. Coulombe, S. Sohail, H. Naqvi and J. McNeil, Ellipsometric scatterometry for metrology of sub-0.1$\mu$m-linewidth structures,, Appl. Optics, 37 (1998), 5112.   Google Scholar

[37]

M. Moharam, E. Grann, D. Pommet and T. Gaylord, Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,, JOSA A, 12 (1995), 1077.  doi: 10.1364/JOSAA.12.001077.  Google Scholar

[38]

T. Moharam and M. G. Gaylord, Rigorous coupled wave analysis,, J. Opt. Soc. Amer., 71 (1981), 811.   Google Scholar

[39]

M. Wurm, Über die Dimensionelle Charakterisierung von Gitterstrukturen auf Fotomasken mit Einem Neuartigen DUV-Scatterometer,, PhD thesis, (2008).   Google Scholar

[40]

H. Patrick, T. Germer, R. Silver and B. Bunday, Developing an uncertainty analysis for optical scatterometry,, Proc. SPIE, 7272 (2009).   Google Scholar

[41]

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. SPIE, 7271 (2009).  doi: 10.1117/12.815360.  Google Scholar

[42]

R. Petit and L. Botten, Electromagnetic Theory of Gratings,, Springer-Verlag, (1980).   Google Scholar

[43]

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., 15 (1997), 361.  doi: 10.1116/1.589320.  Google Scholar

[44]

C. Raymond, M. Murnane, S. Sohail, H. Naqvi and J. McNeil, Metrology of subwavelength photoresist gratings using optical scatterometry,, J. Vac. Sci. Technol., 13 (1995), 1484.  doi: 10.1116/1.588176.  Google Scholar

[45]

A. Schaedle, L. Zschiedrich, S. Burger, R. Klose and F. Schmidt, Domain Decomposition Method for Maxwell's Equations: Scattering of Periodic Structures,, ZIB-Report 06-04, (2006), 06.   Google Scholar

[46]

F. Scholze, C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner and G. Ulm, Status of EUV reflectometry at PTB,, Proc. SPIE, 5751 (2005), 749.   Google Scholar

[47]

F. Scholze, V. Soltwisch, G. Dai, M.-A. Henn and H. Gross, Comparison of CD measurements of an EUV photomask by EUV scatterometry and CD-AFM,, Proc. SPIE, 8880 (2013), 1.  doi: 10.1117/12.2025827.  Google Scholar

[48]

F. Scholze, J. Tümmler and G. Ulm, High-accuracy radiometry in the EUV range at the PTB soft X-ray radiometry beam line,, Metrologia, 40 (2003).   Google Scholar

[49]

T. Schuster, S. Rafler, V. Paz, F. Frenner and W. Osten, Fieldstitching with Kirchhoff-boundaries as a model based description for line edge roughness (LER) in scatterometry,, Microelectronic Engineering, 86 (2009), 1029.  doi: 10.1016/j.mee.2008.11.019.  Google Scholar

[50]

A. Tarantola, Inverse Problem Theory,, Elsevier Amsterdam etc., (1987).   Google Scholar

[51]

A. Tavrov, M. Totzeck, N. Kerwien and H. Tiziani, Rigorous coupled-wave analysis calculus of submicrometer interference pattern and resolving,, Opt.Eng., 41 (2002), 1886.   Google Scholar

[52]

F. Torcal-Milla, L. Sanchez-Brea and E. Bernabeu, Diffraction of gratings with rough edges,, Optics Express, 16 (2008), 19757.  doi: 10.1364/OE.16.019757.  Google Scholar

[53]

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

[54]

H. Urbach, Convergence of the Galerkin method for two-dimensional electromagnetic problems,, SIAM J. Numer. Anal., 28 (1991), 697.  doi: 10.1137/0728037.  Google Scholar

[55]

D. Voelz, Computational Fourier Optics,, TT89, (2011).   Google Scholar

[56]

M. Wurm, B. Bodermann and W. Mirandé, Evaluation of scatterometry tools for critical dimension metrology,, DGaO Proceedings, 106 ().   Google Scholar

show all references

References:
[1]

R. Al-Assaad and D. Byrne, Error analysis in inverse scatterometry. I. Modeling,, J. Opt. Soc. Am. A, 24 (2007), 326.  doi: 10.1364/JOSAA.24.000326.  Google Scholar

[2]

G. Bao and D. C. Dobson, Modeling and optimal design of diffractive optical structures,, Surveys on Mathematics for Industry, 8 (1998), 37.   Google Scholar

[3]

G. Bao, Finite element approximation of time harmonic waves in periodic structures,, SIAM J.Numer.Anal., 32 (1995), 1155.  doi: 10.1137/0732053.  Google Scholar

[4]

B. Bergner, T. Germer and T. Suleski, Effective medium approximations for modeling optical reflectance from gratings with rough edges,, J. Opt. Soc. Am. A, 27 (2010), 1083.  doi: 10.1364/JOSAA.27.001083.  Google Scholar

[5]

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML, Evaluation of measurement data - Guide to the expression of uncertainty in measurement,, Joint Committee for Guides in Metrology, (2008).   Google Scholar

[6]

BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML, Evaluation of measurement data - Supplement 1 to the Guide to the expression of uncertainty in measurement - Propagation of distributions using a Monte Carlo method,, Joint Committee for Guides in Metrology, (2008).   Google Scholar

[7]

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 (1998), 255.  doi: 10.1137/S0036142995285873.  Google Scholar

[8]

D. Champeney, Fourier Transforms and Their Physical Applications,, Academic Press, (1973).   Google Scholar

[9]

J. Chandezon, G. Raoult and D. Maystre, A new theoretical method for diffraction gratings and its numerical application,, J. Opt., 11 (1980).   Google Scholar

[10]

A. Chen and X. Friedmann, Maxwell's equation in a periodic structure,, Trans. Amer. Math. Soc., 323 (1991), 811.   Google Scholar

[11]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Applied Mathematical Sciences 93, (1998).  doi: 10.1007/978-3-662-03537-5.  Google Scholar

[12]

J. Elschner, R. Hinder, A. Rathsfeld and G. Schmidt, =, ().   Google Scholar

[13]

J. Elschner, R. Hinder and G. Schmidt, Finite element solution of conical diffraction problems,, Adv. Comput. Math., 16 (2002), 139.  doi: 10.1023/A:1014456026778.  Google Scholar

[14]

O. Ersoy, Diffraction, Fourier Optics, and Imaging,, Wiley-Interscience, (2006).  doi: 10.1002/0470085002.  Google Scholar

[15]

T. Germer, Effect of line and trench profile variation on specular and diffusive reflectance from periodic structure,, J. Opt. Soc. Am., A24 (2007), 696.  doi: 10.1364/JOSAA.24.000696.  Google Scholar

[16]

J. Goodman, Introduction to Fourier Optics,, Roberts & Company, (2005).   Google Scholar

[17]

H. Gross, S. Heidenreich, M.-A. Henn, G. Dai, F. Scholze and M. Bär, Modelling line edge roughness in periodic line-space structures by Fourier optics to improve scatterometry,, J. Europ. Opt. Soc. Rap. Public., 9 (2014).  doi: 10.2971/jeos.2014.14003.  Google Scholar

[18]

H. Gross, M.-A. Henn, S. Heidenreich, A. Rathsfeld and M. Bär, Modeling of line roughness and its impact on the diffraction intensities and the reconstructed critical dimensions in scatterometry,, Appl. Optics, 51 (2012), 7384.  doi: 10.1364/AO.51.007384.  Google Scholar

[19]

H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann and A. Rathsfeld, Mathematical modelling of indirect measurements in scatterometry,, Measurement, 39 (2006), 782.  doi: 10.1016/j.measurement.2006.04.009.  Google Scholar

[20]

H. Gross, A. Rathsfeld, F. Scholze and M. Bär, Profile reconstruction in extreme ultraviolet (EUV) scatterometry: modeling and uncertainty estimates,, Meas. Sci. Technol., 20 (2009).  doi: 10.1088/0957-0233/20/10/105102.  Google Scholar

[21]

H. Gross, A. Rathsfeld, F. Scholze, R. Model and M. Bär, Computational methods estimating uncertainties for profile reconstruction in scatterometry,, Proc. SPIE, 6995 (2008).  doi: 10.1117/12.781006.  Google Scholar

[22]

M.-A. Henn, H. Gross, S. Heidenreich, F. Scholze, C. Elster and M. Bär, Improved reconstruction of Critical Dimensions in Extreme Ultraviolet Scatterometry by Modeling Systematic Errors,, Meas. Sci. Technol., 25 (2014).  doi: 10.1088/0957-0233/25/4/044003.  Google Scholar

[23]

M.-A. Henn, H. Gross, F. Scholze, M. Wurm, C. Elster and M. Bär, A maximum likelihood approach to the inverse problem of scatterometry,, Optics Express, 20 (2012), 12771.  doi: 10.1364/OE.20.012771.  Google Scholar

[24]

M.-A. Henn, S. Heidenreich, H. Gross, A. Rathsfeld, F. Scholze and M. Bär, Improved grating reconstruction by determination of line roughness in extreme ultraviolet scatterometry,, Optics Letters, 37 (2012), 5229.  doi: 10.1364/OL.37.005229.  Google Scholar

[25]

A. Hesford and W. Chew, A frequency-domain formulation of the Frechet derivative to exploit the inherent parallelism of the distorted Born iterative method,, Waves in Random and Complex Media, 16 (2006), 495.  doi: 10.1080/17455030600675830.  Google Scholar

[26]

H. Huang and F. Terry Jr, Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) for critical dimension measurement and in situ, real-time process monitoring,, Thin Solid Films, 455 (2004), 828.  doi: 10.1016/j.tsf.2004.04.010.  Google Scholar

[27]

F. Ihlenburg, Finite Element Analysis of Acoustic Scattering,, Springer-Verlag, (1998).  doi: 10.1007/b98828.  Google Scholar

[28]

A. Kato and F. Scholze, Effect of line roughness on the diffraction intensities in angular resolved scatterometry,, Appl. Opt., 49 (2010), 6102.  doi: 10.1364/AO.49.006102.  Google Scholar

[29]

B. Kleemann, Elektromagnetische Analyse von Oberflächengittern von IR bis XUV Mittels Einer Parametrisierten Randintegralmethode: Theorie, Vergleich und Anwendungen,, PhD thesis, (2002).   Google Scholar

[30]

G. Lalanne P. and Morris, Highly improved convergence of the coupled-wave method for TM-polarization,, JOSA A, 13 (1996), 779.   Google Scholar

[31]

L. Li, New formulation of the Fourier modal method for crossed surface-relief gratings,, JOSA A, 14 (1997), 2758.  doi: 10.1364/JOSAA.14.002758.  Google Scholar

[32]

C. Mack, Analytic form for the power spectral density in one, two, and three dimensions,, J. Micro/Nanolith. MEMS MOEMS, 10 (2011).  doi: 10.1117/1.3663567.  Google Scholar

[33]

C. Mack, Generating random rough edges, surfaces, and volumes,, Appl. Optics, 52 (2013), 1472.  doi: 10.1364/AO.52.001472.  Google Scholar

[34]

J. Melenk and I. Babuška, The partition of unity finite element method: Basic theory and applications,, Comput. Meth. Appl. Mech. Eng., 139 (1996), 289.  doi: 10.1016/S0045-7825(96)01087-0.  Google Scholar

[35]

R. Millar, Maximum Likelihood Estimation and Inference,, Wiley, (2011).  doi: 10.1002/9780470094846.  Google Scholar

[36]

B. Minhas, S. Coulombe, S. Sohail, H. Naqvi and J. McNeil, Ellipsometric scatterometry for metrology of sub-0.1$\mu$m-linewidth structures,, Appl. Optics, 37 (1998), 5112.   Google Scholar

[37]

M. Moharam, E. Grann, D. Pommet and T. Gaylord, Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,, JOSA A, 12 (1995), 1077.  doi: 10.1364/JOSAA.12.001077.  Google Scholar

[38]

T. Moharam and M. G. Gaylord, Rigorous coupled wave analysis,, J. Opt. Soc. Amer., 71 (1981), 811.   Google Scholar

[39]

M. Wurm, Über die Dimensionelle Charakterisierung von Gitterstrukturen auf Fotomasken mit Einem Neuartigen DUV-Scatterometer,, PhD thesis, (2008).   Google Scholar

[40]

H. Patrick, T. Germer, R. Silver and B. Bunday, Developing an uncertainty analysis for optical scatterometry,, Proc. SPIE, 7272 (2009).   Google Scholar

[41]

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. SPIE, 7271 (2009).  doi: 10.1117/12.815360.  Google Scholar

[42]

R. Petit and L. Botten, Electromagnetic Theory of Gratings,, Springer-Verlag, (1980).   Google Scholar

[43]

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., 15 (1997), 361.  doi: 10.1116/1.589320.  Google Scholar

[44]

C. Raymond, M. Murnane, S. Sohail, H. Naqvi and J. McNeil, Metrology of subwavelength photoresist gratings using optical scatterometry,, J. Vac. Sci. Technol., 13 (1995), 1484.  doi: 10.1116/1.588176.  Google Scholar

[45]

A. Schaedle, L. Zschiedrich, S. Burger, R. Klose and F. Schmidt, Domain Decomposition Method for Maxwell's Equations: Scattering of Periodic Structures,, ZIB-Report 06-04, (2006), 06.   Google Scholar

[46]

F. Scholze, C. Laubis, C. Buchholz, A. Fischer, S. Plöger, F. Scholz, H. Wagner and G. Ulm, Status of EUV reflectometry at PTB,, Proc. SPIE, 5751 (2005), 749.   Google Scholar

[47]

F. Scholze, V. Soltwisch, G. Dai, M.-A. Henn and H. Gross, Comparison of CD measurements of an EUV photomask by EUV scatterometry and CD-AFM,, Proc. SPIE, 8880 (2013), 1.  doi: 10.1117/12.2025827.  Google Scholar

[48]

F. Scholze, J. Tümmler and G. Ulm, High-accuracy radiometry in the EUV range at the PTB soft X-ray radiometry beam line,, Metrologia, 40 (2003).   Google Scholar

[49]

T. Schuster, S. Rafler, V. Paz, F. Frenner and W. Osten, Fieldstitching with Kirchhoff-boundaries as a model based description for line edge roughness (LER) in scatterometry,, Microelectronic Engineering, 86 (2009), 1029.  doi: 10.1016/j.mee.2008.11.019.  Google Scholar

[50]

A. Tarantola, Inverse Problem Theory,, Elsevier Amsterdam etc., (1987).   Google Scholar

[51]

A. Tavrov, M. Totzeck, N. Kerwien and H. Tiziani, Rigorous coupled-wave analysis calculus of submicrometer interference pattern and resolving,, Opt.Eng., 41 (2002), 1886.   Google Scholar

[52]

F. Torcal-Milla, L. Sanchez-Brea and E. Bernabeu, Diffraction of gratings with rough edges,, Optics Express, 16 (2008), 19757.  doi: 10.1364/OE.16.019757.  Google Scholar

[53]

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

[54]

H. Urbach, Convergence of the Galerkin method for two-dimensional electromagnetic problems,, SIAM J. Numer. Anal., 28 (1991), 697.  doi: 10.1137/0728037.  Google Scholar

[55]

D. Voelz, Computational Fourier Optics,, TT89, (2011).   Google Scholar

[56]

M. Wurm, B. Bodermann and W. Mirandé, Evaluation of scatterometry tools for critical dimension metrology,, DGaO Proceedings, 106 ().   Google Scholar

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