Abstract

Overlay control is of vital importance to good device performances in semiconductor manufacturing. In this work, the differential Mueller matrix calculus is introduced to investigate the Mueller matrices of double-patterned gratings with overlay displacements, which helps to reveal six elementary optical properties hidden in the Mueller matrices. We find and demonstrate that, among these six elementary optical properties, the linear birefringence and dichroism, LB′ and LD′, along the ± 45° axes show a linear response to the overlay displacement and are zero when the overlay displacement is absent at any conical mounting. Although the elements from the two 2 × 2 off-diagonal blocks of the Mueller matrix have a similar property to LB′ and LD′, as reported in the literature, we demonstrate that it is only valid at a special conical mounting with the plane of incidence parallel to grating lines. The better property of LB′ and LD′ than the Mueller matrix elements of the off-diagonal blocks in the presence of overlay displacement verifies them to be a more robust indicator for the diffraction-based overlay metrology.

© 2017 Optical Society of America

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2016 (1)

A. J. den Boef, “Optical wafer metrology sensors for process-robust CD and overlay control in semiconductor device manufacturing,” Surf. Topogr.: Metrol. Prop. 4(2), 023001 (2016).
[Crossref]

2015 (4)

S. Peterhänsel, M. L. Gödecke, V. F. Paz, K. Frenner, and W. Osten, “Detection of overlay error in double patterning gratings using phase-structured illumination,” Opt. Express 23(19), 24246–24256 (2015).
[Crossref] [PubMed]

D. J. Dixit, V. Kamineni, R. Farrell, E. R. Hosler, M. Preil, J. Race, B. Peterson, and A. C. Diebold, “Metrology for block copolymer directed self-assembly structures using Mueller matrix-based scatterometry,” J. Micro/Nanolith. MEMS MOEMS 14(2), 021102 (2015).
[Crossref]

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

S. Liu, X. Chen, and C. Zhang, “Development of a broadband Mueller matrix ellipsometer as a powerful tool for nanostructure metrology,” Thin Solid Films 584, 176–185 (2015).
[Crossref]

2014 (5)

X. Chen, S. Liu, H. Gu, and C. Zhang, “Formulation of error propagation and estimation in grating reconstruction by a dual-rotating compensator Mueller matrix polarimeter,” Thin Solid Films 571, 653–659 (2014).
[Crossref]

O. Arteaga, “Useful Mueller matrix symmetries for ellipsometry,” Thin Solid Films 571, 584–588 (2014).
[Crossref]

R. Ossikovski and V. Devlaminck, “General criterion for the physical realizability of the differential Mueller matrix,” Opt. Lett. 39(5), 1216–1219 (2014).
[Crossref] [PubMed]

X. Chen, C. Zhang, S. Liu, H. Jiang, Z. Ma, and Z. Xu, “Mueller matrix ellipsometric detection of profile asymmetry in nanoimprinted grating structures,” J. Appl. Phys. 116(19), 194305 (2014).
[Crossref]

J. Zhu, S. Liu, X. Chen, C. Zhang, and H. Jiang, “Robust solution to the inverse problem in optical scatterometry,” Opt. Express 22(18), 22031–22042 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (2)

V. F. Paz, S. Peterhänsel, K. Frenner, and W. Osten, “Solving the inverse grating problem by white light interference Fourier scatterometry,” Light Sci. Appl. 1(11), e36 (2012).
[Crossref]

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(12), 12771–12786 (2012).
[Crossref] [PubMed]

2011 (4)

C. Fallet, T. Novikova, M. Foldyna, S. Manhas, B. H. Ibrahim, A. De Martino, C. Vannuffel, and C. Constancias, “Overlay measurements by Mueller polarimetry in back focal plane,” J. Micro/Nanolith. MEMS MOEMS 10(3), 033017 (2011).
[Crossref]

T. Novikova, P. Bulkin, V. Popov, B. H. Ibrahim, and A. De Martino, “Mueller polarimetry as a tool for detecting asymmetry in diffraction grating profiles,” J. Vac. Sci. Technol. B 29(5), 051804 (2011).
[Crossref]

R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. 36(12), 2330–2332 (2011).
[Crossref] [PubMed]

N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition,” Opt. Lett. 36(10), 1942–1944 (2011).
[Crossref] [PubMed]

2010 (4)

J. Li, Y. Liu, P. Dasari, J. Hu, N. Smith, O. Kritsun, and C. Volkman, “Advanced diffraction-based overlay for double patterning,” Proc. SPIE 7638, 76382C (2010).
[Crossref]

J. Li, J. J. Hwu, Y. Liu, S. Rabello, Z. Liu, and J. Hu, “Mueller matrix measurement of asymmetric gratings,” J. Micro/Nanolith. MEMS MOEMS 9(4), 041305 (2010).
[Crossref]

Y. S. Jung, J. B. Chang, E. Verploegen, K. K. Berggren, and C. A. Ross, “A path to ultranarrow patterns using self-assembled lithography,” Nano Lett. 10(3), 1000–1005 (2010).
[Crossref] [PubMed]

S. H. Park, D. O. Shin, B. H. Kim, D. K. Yoon, K. Kim, S. Y. Lee, S. H. Oh, S. W. Choi, S. C. Jeon, and S. O. Kim, “Block copolymer multiple patterning integrated with conventional ArF lithography,” Soft Matter 6(1), 120–125 (2010).
[Crossref]

2009 (2)

J. Finders, M. Dusa, B. Vleeming, B. Hepp, M. Maenhoudt, S. Cheng, and T. Vandeweyer, “Double patterning lithography for 32 nm: critical dimensions uniformity and overlay control considerations,” J. Micro/Nanolith. MEMS MOEMS 8(1), 011002 (2009).
[Crossref]

Y. N. Kim, J. S. Paek, S. Rabello, S. Lee, J. Hu, Z. Liu, Y. Hao, and W. McGahan, “Device based in-chip critical dimension and overlay metrology,” Opt. Express 17(23), 21336–21343 (2009).
[Crossref] [PubMed]

2007 (1)

E. Vogel, “Technology and metrology of new electronic materials and devices,” Nat. Nanotechnol. 2(1), 25–32 (2007).
[Crossref] [PubMed]

2006 (1)

2005 (1)

M. Maenhoudt, J. Versluijs, H. Struyf, J. Van Olmen, and M. Van Hove, “Double patterning scheme for sub-0.25 k1 single damascene structures at NA = 0.75, λ = 193 nm,” Proc. SPIE 5754, 1508–1518 (2005).
[Crossref]

2003 (1)

W. Yang, R. Lowe-Webb, S. Rabello, J. Hu, J. Y. Lin, J. Heaton, M. Dusa, A. den Boef, M. van der Schaar, and A. Hunter, “A novel diffraction based spectroscopic method for overlay metrology,” Proc. SPIE 5038, 200–207 (2003).
[Crossref]

2000 (1)

1996 (1)

1995 (1)

1994 (1)

1990 (1)

S. R. Cloude, “Conditions for the physical realizability of matrix operators in polarimetry,” Proc. SPIE 1166, 177–185 (1990).
[Crossref]

1978 (1)

1966 (1)

Anderson, D. G. M.

Arce-Diego, J. L.

Arteaga, O.

Azzam, R. M. A.

Bär, M.

Barakat, R.

Berggren, K. K.

Y. S. Jung, J. B. Chang, E. Verploegen, K. K. Berggren, and C. A. Ross, “A path to ultranarrow patterns using self-assembled lithography,” Nano Lett. 10(3), 1000–1005 (2010).
[Crossref] [PubMed]

Bulkin, P.

T. Novikova, P. Bulkin, V. Popov, B. H. Ibrahim, and A. De Martino, “Mueller polarimetry as a tool for detecting asymmetry in diffraction grating profiles,” J. Vac. Sci. Technol. B 29(5), 051804 (2011).
[Crossref]

Chang, J. B.

Y. S. Jung, J. B. Chang, E. Verploegen, K. K. Berggren, and C. A. Ross, “A path to ultranarrow patterns using self-assembled lithography,” Nano Lett. 10(3), 1000–1005 (2010).
[Crossref] [PubMed]

Chen, X.

S. Liu, X. Chen, and C. Zhang, “Development of a broadband Mueller matrix ellipsometer as a powerful tool for nanostructure metrology,” Thin Solid Films 584, 176–185 (2015).
[Crossref]

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

X. Chen, S. Liu, H. Gu, and C. Zhang, “Formulation of error propagation and estimation in grating reconstruction by a dual-rotating compensator Mueller matrix polarimeter,” Thin Solid Films 571, 653–659 (2014).
[Crossref]

X. Chen, C. Zhang, S. Liu, H. Jiang, Z. Ma, and Z. Xu, “Mueller matrix ellipsometric detection of profile asymmetry in nanoimprinted grating structures,” J. Appl. Phys. 116(19), 194305 (2014).
[Crossref]

J. Zhu, S. Liu, X. Chen, C. Zhang, and H. Jiang, “Robust solution to the inverse problem in optical scatterometry,” Opt. Express 22(18), 22031–22042 (2014).
[Crossref] [PubMed]

Cheng, S.

J. Finders, M. Dusa, B. Vleeming, B. Hepp, M. Maenhoudt, S. Cheng, and T. Vandeweyer, “Double patterning lithography for 32 nm: critical dimensions uniformity and overlay control considerations,” J. Micro/Nanolith. MEMS MOEMS 8(1), 011002 (2009).
[Crossref]

Choi, S. W.

S. H. Park, D. O. Shin, B. H. Kim, D. K. Yoon, K. Kim, S. Y. Lee, S. H. Oh, S. W. Choi, S. C. Jeon, and S. O. Kim, “Block copolymer multiple patterning integrated with conventional ArF lithography,” Soft Matter 6(1), 120–125 (2010).
[Crossref]

Cloude, S. R.

S. R. Cloude, “Conditions for the physical realizability of matrix operators in polarimetry,” Proc. SPIE 1166, 177–185 (1990).
[Crossref]

Constancias, C.

C. Fallet, T. Novikova, M. Foldyna, S. Manhas, B. H. Ibrahim, A. De Martino, C. Vannuffel, and C. Constancias, “Overlay measurements by Mueller polarimetry in back focal plane,” J. Micro/Nanolith. MEMS MOEMS 10(3), 033017 (2011).
[Crossref]

Dasari, P.

J. Li, Y. Liu, P. Dasari, J. Hu, N. Smith, O. Kritsun, and C. Volkman, “Advanced diffraction-based overlay for double patterning,” Proc. SPIE 7638, 76382C (2010).
[Crossref]

De Martino, A.

C. Fallet, T. Novikova, M. Foldyna, S. Manhas, B. H. Ibrahim, A. De Martino, C. Vannuffel, and C. Constancias, “Overlay measurements by Mueller polarimetry in back focal plane,” J. Micro/Nanolith. MEMS MOEMS 10(3), 033017 (2011).
[Crossref]

T. Novikova, P. Bulkin, V. Popov, B. H. Ibrahim, and A. De Martino, “Mueller polarimetry as a tool for detecting asymmetry in diffraction grating profiles,” J. Vac. Sci. Technol. B 29(5), 051804 (2011).
[Crossref]

den Boef, A.

W. Yang, R. Lowe-Webb, S. Rabello, J. Hu, J. Y. Lin, J. Heaton, M. Dusa, A. den Boef, M. van der Schaar, and A. Hunter, “A novel diffraction based spectroscopic method for overlay metrology,” Proc. SPIE 5038, 200–207 (2003).
[Crossref]

den Boef, A. J.

A. J. den Boef, “Optical wafer metrology sensors for process-robust CD and overlay control in semiconductor device manufacturing,” Surf. Topogr.: Metrol. Prop. 4(2), 023001 (2016).
[Crossref]

Devlaminck, V.

Diebold, A. C.

D. J. Dixit, V. Kamineni, R. Farrell, E. R. Hosler, M. Preil, J. Race, B. Peterson, and A. C. Diebold, “Metrology for block copolymer directed self-assembly structures using Mueller matrix-based scatterometry,” J. Micro/Nanolith. MEMS MOEMS 14(2), 021102 (2015).
[Crossref]

Dixit, D. J.

D. J. Dixit, V. Kamineni, R. Farrell, E. R. Hosler, M. Preil, J. Race, B. Peterson, and A. C. Diebold, “Metrology for block copolymer directed self-assembly structures using Mueller matrix-based scatterometry,” J. Micro/Nanolith. MEMS MOEMS 14(2), 021102 (2015).
[Crossref]

Dusa, M.

J. Finders, M. Dusa, B. Vleeming, B. Hepp, M. Maenhoudt, S. Cheng, and T. Vandeweyer, “Double patterning lithography for 32 nm: critical dimensions uniformity and overlay control considerations,” J. Micro/Nanolith. MEMS MOEMS 8(1), 011002 (2009).
[Crossref]

W. Yang, R. Lowe-Webb, S. Rabello, J. Hu, J. Y. Lin, J. Heaton, M. Dusa, A. den Boef, M. van der Schaar, and A. Hunter, “A novel diffraction based spectroscopic method for overlay metrology,” Proc. SPIE 5038, 200–207 (2003).
[Crossref]

Elster, C.

Fallet, C.

C. Fallet, T. Novikova, M. Foldyna, S. Manhas, B. H. Ibrahim, A. De Martino, C. Vannuffel, and C. Constancias, “Overlay measurements by Mueller polarimetry in back focal plane,” J. Micro/Nanolith. MEMS MOEMS 10(3), 033017 (2011).
[Crossref]

Farrell, R.

D. J. Dixit, V. Kamineni, R. Farrell, E. R. Hosler, M. Preil, J. Race, B. Peterson, and A. C. Diebold, “Metrology for block copolymer directed self-assembly structures using Mueller matrix-based scatterometry,” J. Micro/Nanolith. MEMS MOEMS 14(2), 021102 (2015).
[Crossref]

Finders, J.

J. Finders, M. Dusa, B. Vleeming, B. Hepp, M. Maenhoudt, S. Cheng, and T. Vandeweyer, “Double patterning lithography for 32 nm: critical dimensions uniformity and overlay control considerations,” J. Micro/Nanolith. MEMS MOEMS 8(1), 011002 (2009).
[Crossref]

Foldyna, M.

C. Fallet, T. Novikova, M. Foldyna, S. Manhas, B. H. Ibrahim, A. De Martino, C. Vannuffel, and C. Constancias, “Overlay measurements by Mueller polarimetry in back focal plane,” J. Micro/Nanolith. MEMS MOEMS 10(3), 033017 (2011).
[Crossref]

Frenner, K.

S. Peterhänsel, M. L. Gödecke, V. F. Paz, K. Frenner, and W. Osten, “Detection of overlay error in double patterning gratings using phase-structured illumination,” Opt. Express 23(19), 24246–24256 (2015).
[Crossref] [PubMed]

V. F. Paz, S. Peterhänsel, K. Frenner, and W. Osten, “Solving the inverse grating problem by white light interference Fourier scatterometry,” Light Sci. Appl. 1(11), e36 (2012).
[Crossref]

Gaylord, T. K.

Gödecke, M. L.

Grann, E. B.

Gross, H.

Gu, H.

X. Chen, S. Liu, H. Gu, and C. Zhang, “Formulation of error propagation and estimation in grating reconstruction by a dual-rotating compensator Mueller matrix polarimeter,” Thin Solid Films 571, 653–659 (2014).
[Crossref]

Hao, Y.

Heaton, J.

W. Yang, R. Lowe-Webb, S. Rabello, J. Hu, J. Y. Lin, J. Heaton, M. Dusa, A. den Boef, M. van der Schaar, and A. Hunter, “A novel diffraction based spectroscopic method for overlay metrology,” Proc. SPIE 5038, 200–207 (2003).
[Crossref]

Henn, M. A.

Hepp, B.

J. Finders, M. Dusa, B. Vleeming, B. Hepp, M. Maenhoudt, S. Cheng, and T. Vandeweyer, “Double patterning lithography for 32 nm: critical dimensions uniformity and overlay control considerations,” J. Micro/Nanolith. MEMS MOEMS 8(1), 011002 (2009).
[Crossref]

Hosler, E. R.

D. J. Dixit, V. Kamineni, R. Farrell, E. R. Hosler, M. Preil, J. Race, B. Peterson, and A. C. Diebold, “Metrology for block copolymer directed self-assembly structures using Mueller matrix-based scatterometry,” J. Micro/Nanolith. MEMS MOEMS 14(2), 021102 (2015).
[Crossref]

Hu, J.

J. Li, J. J. Hwu, Y. Liu, S. Rabello, Z. Liu, and J. Hu, “Mueller matrix measurement of asymmetric gratings,” J. Micro/Nanolith. MEMS MOEMS 9(4), 041305 (2010).
[Crossref]

J. Li, Y. Liu, P. Dasari, J. Hu, N. Smith, O. Kritsun, and C. Volkman, “Advanced diffraction-based overlay for double patterning,” Proc. SPIE 7638, 76382C (2010).
[Crossref]

Y. N. Kim, J. S. Paek, S. Rabello, S. Lee, J. Hu, Z. Liu, Y. Hao, and W. McGahan, “Device based in-chip critical dimension and overlay metrology,” Opt. Express 17(23), 21336–21343 (2009).
[Crossref] [PubMed]

W. Yang, R. Lowe-Webb, S. Rabello, J. Hu, J. Y. Lin, J. Heaton, M. Dusa, A. den Boef, M. van der Schaar, and A. Hunter, “A novel diffraction based spectroscopic method for overlay metrology,” Proc. SPIE 5038, 200–207 (2003).
[Crossref]

Hunter, A.

W. Yang, R. Lowe-Webb, S. Rabello, J. Hu, J. Y. Lin, J. Heaton, M. Dusa, A. den Boef, M. van der Schaar, and A. Hunter, “A novel diffraction based spectroscopic method for overlay metrology,” Proc. SPIE 5038, 200–207 (2003).
[Crossref]

Hwu, J. J.

J. Li, J. J. Hwu, Y. Liu, S. Rabello, Z. Liu, and J. Hu, “Mueller matrix measurement of asymmetric gratings,” J. Micro/Nanolith. MEMS MOEMS 9(4), 041305 (2010).
[Crossref]

Ibrahim, B. H.

C. Fallet, T. Novikova, M. Foldyna, S. Manhas, B. H. Ibrahim, A. De Martino, C. Vannuffel, and C. Constancias, “Overlay measurements by Mueller polarimetry in back focal plane,” J. Micro/Nanolith. MEMS MOEMS 10(3), 033017 (2011).
[Crossref]

T. Novikova, P. Bulkin, V. Popov, B. H. Ibrahim, and A. De Martino, “Mueller polarimetry as a tool for detecting asymmetry in diffraction grating profiles,” J. Vac. Sci. Technol. B 29(5), 051804 (2011).
[Crossref]

Jeon, S. C.

S. H. Park, D. O. Shin, B. H. Kim, D. K. Yoon, K. Kim, S. Y. Lee, S. H. Oh, S. W. Choi, S. C. Jeon, and S. O. Kim, “Block copolymer multiple patterning integrated with conventional ArF lithography,” Soft Matter 6(1), 120–125 (2010).
[Crossref]

Jiang, H.

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

X. Chen, C. Zhang, S. Liu, H. Jiang, Z. Ma, and Z. Xu, “Mueller matrix ellipsometric detection of profile asymmetry in nanoimprinted grating structures,” J. Appl. Phys. 116(19), 194305 (2014).
[Crossref]

J. Zhu, S. Liu, X. Chen, C. Zhang, and H. Jiang, “Robust solution to the inverse problem in optical scatterometry,” Opt. Express 22(18), 22031–22042 (2014).
[Crossref] [PubMed]

Jung, Y. S.

Y. S. Jung, J. B. Chang, E. Verploegen, K. K. Berggren, and C. A. Ross, “A path to ultranarrow patterns using self-assembled lithography,” Nano Lett. 10(3), 1000–1005 (2010).
[Crossref] [PubMed]

Kahr, B.

Kamineni, V.

D. J. Dixit, V. Kamineni, R. Farrell, E. R. Hosler, M. Preil, J. Race, B. Peterson, and A. C. Diebold, “Metrology for block copolymer directed self-assembly structures using Mueller matrix-based scatterometry,” J. Micro/Nanolith. MEMS MOEMS 14(2), 021102 (2015).
[Crossref]

Kim, B. H.

S. H. Park, D. O. Shin, B. H. Kim, D. K. Yoon, K. Kim, S. Y. Lee, S. H. Oh, S. W. Choi, S. C. Jeon, and S. O. Kim, “Block copolymer multiple patterning integrated with conventional ArF lithography,” Soft Matter 6(1), 120–125 (2010).
[Crossref]

Kim, K.

S. H. Park, D. O. Shin, B. H. Kim, D. K. Yoon, K. Kim, S. Y. Lee, S. H. Oh, S. W. Choi, S. C. Jeon, and S. O. Kim, “Block copolymer multiple patterning integrated with conventional ArF lithography,” Soft Matter 6(1), 120–125 (2010).
[Crossref]

Kim, S. O.

S. H. Park, D. O. Shin, B. H. Kim, D. K. Yoon, K. Kim, S. Y. Lee, S. H. Oh, S. W. Choi, S. C. Jeon, and S. O. Kim, “Block copolymer multiple patterning integrated with conventional ArF lithography,” Soft Matter 6(1), 120–125 (2010).
[Crossref]

Kim, Y. N.

Ko, C. H.

Kritsun, O.

J. Li, Y. Liu, P. Dasari, J. Hu, N. Smith, O. Kritsun, and C. Volkman, “Advanced diffraction-based overlay for double patterning,” Proc. SPIE 7638, 76382C (2010).
[Crossref]

Ku, Y. S.

Lee, S.

Lee, S. Y.

S. H. Park, D. O. Shin, B. H. Kim, D. K. Yoon, K. Kim, S. Y. Lee, S. H. Oh, S. W. Choi, S. C. Jeon, and S. O. Kim, “Block copolymer multiple patterning integrated with conventional ArF lithography,” Soft Matter 6(1), 120–125 (2010).
[Crossref]

Li, J.

J. Li, Y. Liu, P. Dasari, J. Hu, N. Smith, O. Kritsun, and C. Volkman, “Advanced diffraction-based overlay for double patterning,” Proc. SPIE 7638, 76382C (2010).
[Crossref]

J. Li, J. J. Hwu, Y. Liu, S. Rabello, Z. Liu, and J. Hu, “Mueller matrix measurement of asymmetric gratings,” J. Micro/Nanolith. MEMS MOEMS 9(4), 041305 (2010).
[Crossref]

Li, L.

Lin, J. Y.

W. Yang, R. Lowe-Webb, S. Rabello, J. Hu, J. Y. Lin, J. Heaton, M. Dusa, A. den Boef, M. van der Schaar, and A. Hunter, “A novel diffraction based spectroscopic method for overlay metrology,” Proc. SPIE 5038, 200–207 (2003).
[Crossref]

Liu, S.

S. Liu, X. Chen, and C. Zhang, “Development of a broadband Mueller matrix ellipsometer as a powerful tool for nanostructure metrology,” Thin Solid Films 584, 176–185 (2015).
[Crossref]

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

X. Chen, S. Liu, H. Gu, and C. Zhang, “Formulation of error propagation and estimation in grating reconstruction by a dual-rotating compensator Mueller matrix polarimeter,” Thin Solid Films 571, 653–659 (2014).
[Crossref]

X. Chen, C. Zhang, S. Liu, H. Jiang, Z. Ma, and Z. Xu, “Mueller matrix ellipsometric detection of profile asymmetry in nanoimprinted grating structures,” J. Appl. Phys. 116(19), 194305 (2014).
[Crossref]

J. Zhu, S. Liu, X. Chen, C. Zhang, and H. Jiang, “Robust solution to the inverse problem in optical scatterometry,” Opt. Express 22(18), 22031–22042 (2014).
[Crossref] [PubMed]

Liu, Y.

J. Li, J. J. Hwu, Y. Liu, S. Rabello, Z. Liu, and J. Hu, “Mueller matrix measurement of asymmetric gratings,” J. Micro/Nanolith. MEMS MOEMS 9(4), 041305 (2010).
[Crossref]

J. Li, Y. Liu, P. Dasari, J. Hu, N. Smith, O. Kritsun, and C. Volkman, “Advanced diffraction-based overlay for double patterning,” Proc. SPIE 7638, 76382C (2010).
[Crossref]

Liu, Z.

J. Li, J. J. Hwu, Y. Liu, S. Rabello, Z. Liu, and J. Hu, “Mueller matrix measurement of asymmetric gratings,” J. Micro/Nanolith. MEMS MOEMS 9(4), 041305 (2010).
[Crossref]

Y. N. Kim, J. S. Paek, S. Rabello, S. Lee, J. Hu, Z. Liu, Y. Hao, and W. McGahan, “Device based in-chip critical dimension and overlay metrology,” Opt. Express 17(23), 21336–21343 (2009).
[Crossref] [PubMed]

Lowe-Webb, R.

W. Yang, R. Lowe-Webb, S. Rabello, J. Hu, J. Y. Lin, J. Heaton, M. Dusa, A. den Boef, M. van der Schaar, and A. Hunter, “A novel diffraction based spectroscopic method for overlay metrology,” Proc. SPIE 5038, 200–207 (2003).
[Crossref]

Ma, Z.

X. Chen, C. Zhang, S. Liu, H. Jiang, Z. Ma, and Z. Xu, “Mueller matrix ellipsometric detection of profile asymmetry in nanoimprinted grating structures,” J. Appl. Phys. 116(19), 194305 (2014).
[Crossref]

Maenhoudt, M.

J. Finders, M. Dusa, B. Vleeming, B. Hepp, M. Maenhoudt, S. Cheng, and T. Vandeweyer, “Double patterning lithography for 32 nm: critical dimensions uniformity and overlay control considerations,” J. Micro/Nanolith. MEMS MOEMS 8(1), 011002 (2009).
[Crossref]

M. Maenhoudt, J. Versluijs, H. Struyf, J. Van Olmen, and M. Van Hove, “Double patterning scheme for sub-0.25 k1 single damascene structures at NA = 0.75, λ = 193 nm,” Proc. SPIE 5754, 1508–1518 (2005).
[Crossref]

Manhas, S.

C. Fallet, T. Novikova, M. Foldyna, S. Manhas, B. H. Ibrahim, A. De Martino, C. Vannuffel, and C. Constancias, “Overlay measurements by Mueller polarimetry in back focal plane,” J. Micro/Nanolith. MEMS MOEMS 10(3), 033017 (2011).
[Crossref]

McGahan, W.

Moharam, M. G.

Novikova, T.

C. Fallet, T. Novikova, M. Foldyna, S. Manhas, B. H. Ibrahim, A. De Martino, C. Vannuffel, and C. Constancias, “Overlay measurements by Mueller polarimetry in back focal plane,” J. Micro/Nanolith. MEMS MOEMS 10(3), 033017 (2011).
[Crossref]

T. Novikova, P. Bulkin, V. Popov, B. H. Ibrahim, and A. De Martino, “Mueller polarimetry as a tool for detecting asymmetry in diffraction grating profiles,” J. Vac. Sci. Technol. B 29(5), 051804 (2011).
[Crossref]

Oh, S. H.

S. H. Park, D. O. Shin, B. H. Kim, D. K. Yoon, K. Kim, S. Y. Lee, S. H. Oh, S. W. Choi, S. C. Jeon, and S. O. Kim, “Block copolymer multiple patterning integrated with conventional ArF lithography,” Soft Matter 6(1), 120–125 (2010).
[Crossref]

Ortega-Quijano, N.

Ossikovski, R.

Osten, W.

S. Peterhänsel, M. L. Gödecke, V. F. Paz, K. Frenner, and W. Osten, “Detection of overlay error in double patterning gratings using phase-structured illumination,” Opt. Express 23(19), 24246–24256 (2015).
[Crossref] [PubMed]

V. F. Paz, S. Peterhänsel, K. Frenner, and W. Osten, “Solving the inverse grating problem by white light interference Fourier scatterometry,” Light Sci. Appl. 1(11), e36 (2012).
[Crossref]

Paek, J. S.

Park, S. H.

S. H. Park, D. O. Shin, B. H. Kim, D. K. Yoon, K. Kim, S. Y. Lee, S. H. Oh, S. W. Choi, S. C. Jeon, and S. O. Kim, “Block copolymer multiple patterning integrated with conventional ArF lithography,” Soft Matter 6(1), 120–125 (2010).
[Crossref]

Paz, V. F.

S. Peterhänsel, M. L. Gödecke, V. F. Paz, K. Frenner, and W. Osten, “Detection of overlay error in double patterning gratings using phase-structured illumination,” Opt. Express 23(19), 24246–24256 (2015).
[Crossref] [PubMed]

V. F. Paz, S. Peterhänsel, K. Frenner, and W. Osten, “Solving the inverse grating problem by white light interference Fourier scatterometry,” Light Sci. Appl. 1(11), e36 (2012).
[Crossref]

Peterhänsel, S.

S. Peterhänsel, M. L. Gödecke, V. F. Paz, K. Frenner, and W. Osten, “Detection of overlay error in double patterning gratings using phase-structured illumination,” Opt. Express 23(19), 24246–24256 (2015).
[Crossref] [PubMed]

V. F. Paz, S. Peterhänsel, K. Frenner, and W. Osten, “Solving the inverse grating problem by white light interference Fourier scatterometry,” Light Sci. Appl. 1(11), e36 (2012).
[Crossref]

Peterson, B.

D. J. Dixit, V. Kamineni, R. Farrell, E. R. Hosler, M. Preil, J. Race, B. Peterson, and A. C. Diebold, “Metrology for block copolymer directed self-assembly structures using Mueller matrix-based scatterometry,” J. Micro/Nanolith. MEMS MOEMS 14(2), 021102 (2015).
[Crossref]

Pommet, D. A.

Popov, V.

T. Novikova, P. Bulkin, V. Popov, B. H. Ibrahim, and A. De Martino, “Mueller polarimetry as a tool for detecting asymmetry in diffraction grating profiles,” J. Vac. Sci. Technol. B 29(5), 051804 (2011).
[Crossref]

Preil, M.

D. J. Dixit, V. Kamineni, R. Farrell, E. R. Hosler, M. Preil, J. Race, B. Peterson, and A. C. Diebold, “Metrology for block copolymer directed self-assembly structures using Mueller matrix-based scatterometry,” J. Micro/Nanolith. MEMS MOEMS 14(2), 021102 (2015).
[Crossref]

Rabello, S.

J. Li, J. J. Hwu, Y. Liu, S. Rabello, Z. Liu, and J. Hu, “Mueller matrix measurement of asymmetric gratings,” J. Micro/Nanolith. MEMS MOEMS 9(4), 041305 (2010).
[Crossref]

Y. N. Kim, J. S. Paek, S. Rabello, S. Lee, J. Hu, Z. Liu, Y. Hao, and W. McGahan, “Device based in-chip critical dimension and overlay metrology,” Opt. Express 17(23), 21336–21343 (2009).
[Crossref] [PubMed]

W. Yang, R. Lowe-Webb, S. Rabello, J. Hu, J. Y. Lin, J. Heaton, M. Dusa, A. den Boef, M. van der Schaar, and A. Hunter, “A novel diffraction based spectroscopic method for overlay metrology,” Proc. SPIE 5038, 200–207 (2003).
[Crossref]

Race, J.

D. J. Dixit, V. Kamineni, R. Farrell, E. R. Hosler, M. Preil, J. Race, B. Peterson, and A. C. Diebold, “Metrology for block copolymer directed self-assembly structures using Mueller matrix-based scatterometry,” J. Micro/Nanolith. MEMS MOEMS 14(2), 021102 (2015).
[Crossref]

Ross, C. A.

Y. S. Jung, J. B. Chang, E. Verploegen, K. K. Berggren, and C. A. Ross, “A path to ultranarrow patterns using self-assembled lithography,” Nano Lett. 10(3), 1000–1005 (2010).
[Crossref] [PubMed]

Scholze, F.

Sekera, Z.

Shin, D. O.

S. H. Park, D. O. Shin, B. H. Kim, D. K. Yoon, K. Kim, S. Y. Lee, S. H. Oh, S. W. Choi, S. C. Jeon, and S. O. Kim, “Block copolymer multiple patterning integrated with conventional ArF lithography,” Soft Matter 6(1), 120–125 (2010).
[Crossref]

Smith, N.

J. Li, Y. Liu, P. Dasari, J. Hu, N. Smith, O. Kritsun, and C. Volkman, “Advanced diffraction-based overlay for double patterning,” Proc. SPIE 7638, 76382C (2010).
[Crossref]

Struyf, H.

M. Maenhoudt, J. Versluijs, H. Struyf, J. Van Olmen, and M. Van Hove, “Double patterning scheme for sub-0.25 k1 single damascene structures at NA = 0.75, λ = 193 nm,” Proc. SPIE 5754, 1508–1518 (2005).
[Crossref]

van der Schaar, M.

W. Yang, R. Lowe-Webb, S. Rabello, J. Hu, J. Y. Lin, J. Heaton, M. Dusa, A. den Boef, M. van der Schaar, and A. Hunter, “A novel diffraction based spectroscopic method for overlay metrology,” Proc. SPIE 5038, 200–207 (2003).
[Crossref]

Van Hove, M.

M. Maenhoudt, J. Versluijs, H. Struyf, J. Van Olmen, and M. Van Hove, “Double patterning scheme for sub-0.25 k1 single damascene structures at NA = 0.75, λ = 193 nm,” Proc. SPIE 5754, 1508–1518 (2005).
[Crossref]

Van Olmen, J.

M. Maenhoudt, J. Versluijs, H. Struyf, J. Van Olmen, and M. Van Hove, “Double patterning scheme for sub-0.25 k1 single damascene structures at NA = 0.75, λ = 193 nm,” Proc. SPIE 5754, 1508–1518 (2005).
[Crossref]

Vandeweyer, T.

J. Finders, M. Dusa, B. Vleeming, B. Hepp, M. Maenhoudt, S. Cheng, and T. Vandeweyer, “Double patterning lithography for 32 nm: critical dimensions uniformity and overlay control considerations,” J. Micro/Nanolith. MEMS MOEMS 8(1), 011002 (2009).
[Crossref]

Vannuffel, C.

C. Fallet, T. Novikova, M. Foldyna, S. Manhas, B. H. Ibrahim, A. De Martino, C. Vannuffel, and C. Constancias, “Overlay measurements by Mueller polarimetry in back focal plane,” J. Micro/Nanolith. MEMS MOEMS 10(3), 033017 (2011).
[Crossref]

Verploegen, E.

Y. S. Jung, J. B. Chang, E. Verploegen, K. K. Berggren, and C. A. Ross, “A path to ultranarrow patterns using self-assembled lithography,” Nano Lett. 10(3), 1000–1005 (2010).
[Crossref] [PubMed]

Versluijs, J.

M. Maenhoudt, J. Versluijs, H. Struyf, J. Van Olmen, and M. Van Hove, “Double patterning scheme for sub-0.25 k1 single damascene structures at NA = 0.75, λ = 193 nm,” Proc. SPIE 5754, 1508–1518 (2005).
[Crossref]

Vleeming, B.

J. Finders, M. Dusa, B. Vleeming, B. Hepp, M. Maenhoudt, S. Cheng, and T. Vandeweyer, “Double patterning lithography for 32 nm: critical dimensions uniformity and overlay control considerations,” J. Micro/Nanolith. MEMS MOEMS 8(1), 011002 (2009).
[Crossref]

Vogel, E.

E. Vogel, “Technology and metrology of new electronic materials and devices,” Nat. Nanotechnol. 2(1), 25–32 (2007).
[Crossref] [PubMed]

Volkman, C.

J. Li, Y. Liu, P. Dasari, J. Hu, N. Smith, O. Kritsun, and C. Volkman, “Advanced diffraction-based overlay for double patterning,” Proc. SPIE 7638, 76382C (2010).
[Crossref]

Wurm, M.

Xu, Z.

X. Chen, C. Zhang, S. Liu, H. Jiang, Z. Ma, and Z. Xu, “Mueller matrix ellipsometric detection of profile asymmetry in nanoimprinted grating structures,” J. Appl. Phys. 116(19), 194305 (2014).
[Crossref]

Yang, W.

W. Yang, R. Lowe-Webb, S. Rabello, J. Hu, J. Y. Lin, J. Heaton, M. Dusa, A. den Boef, M. van der Schaar, and A. Hunter, “A novel diffraction based spectroscopic method for overlay metrology,” Proc. SPIE 5038, 200–207 (2003).
[Crossref]

Yoon, D. K.

S. H. Park, D. O. Shin, B. H. Kim, D. K. Yoon, K. Kim, S. Y. Lee, S. H. Oh, S. W. Choi, S. C. Jeon, and S. O. Kim, “Block copolymer multiple patterning integrated with conventional ArF lithography,” Soft Matter 6(1), 120–125 (2010).
[Crossref]

Zhang, C.

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

S. Liu, X. Chen, and C. Zhang, “Development of a broadband Mueller matrix ellipsometer as a powerful tool for nanostructure metrology,” Thin Solid Films 584, 176–185 (2015).
[Crossref]

X. Chen, S. Liu, H. Gu, and C. Zhang, “Formulation of error propagation and estimation in grating reconstruction by a dual-rotating compensator Mueller matrix polarimeter,” Thin Solid Films 571, 653–659 (2014).
[Crossref]

X. Chen, C. Zhang, S. Liu, H. Jiang, Z. Ma, and Z. Xu, “Mueller matrix ellipsometric detection of profile asymmetry in nanoimprinted grating structures,” J. Appl. Phys. 116(19), 194305 (2014).
[Crossref]

J. Zhu, S. Liu, X. Chen, C. Zhang, and H. Jiang, “Robust solution to the inverse problem in optical scatterometry,” Opt. Express 22(18), 22031–22042 (2014).
[Crossref] [PubMed]

Zhu, J.

J. Appl. Phys. (2)

X. Chen, C. Zhang, S. Liu, H. Jiang, Z. Ma, and Z. Xu, “Mueller matrix ellipsometric detection of profile asymmetry in nanoimprinted grating structures,” J. Appl. Phys. 116(19), 194305 (2014).
[Crossref]

X. Chen, H. Jiang, C. Zhang, and S. Liu, “Towards understanding the detection of profile asymmetry from Mueller matrix differential decomposition,” J. Appl. Phys. 118(22), 225308 (2015).
[Crossref]

J. Micro/Nanolith. MEMS MOEMS (4)

J. Finders, M. Dusa, B. Vleeming, B. Hepp, M. Maenhoudt, S. Cheng, and T. Vandeweyer, “Double patterning lithography for 32 nm: critical dimensions uniformity and overlay control considerations,” J. Micro/Nanolith. MEMS MOEMS 8(1), 011002 (2009).
[Crossref]

C. Fallet, T. Novikova, M. Foldyna, S. Manhas, B. H. Ibrahim, A. De Martino, C. Vannuffel, and C. Constancias, “Overlay measurements by Mueller polarimetry in back focal plane,” J. Micro/Nanolith. MEMS MOEMS 10(3), 033017 (2011).
[Crossref]

J. Li, J. J. Hwu, Y. Liu, S. Rabello, Z. Liu, and J. Hu, “Mueller matrix measurement of asymmetric gratings,” J. Micro/Nanolith. MEMS MOEMS 9(4), 041305 (2010).
[Crossref]

D. J. Dixit, V. Kamineni, R. Farrell, E. R. Hosler, M. Preil, J. Race, B. Peterson, and A. C. Diebold, “Metrology for block copolymer directed self-assembly structures using Mueller matrix-based scatterometry,” J. Micro/Nanolith. MEMS MOEMS 14(2), 021102 (2015).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

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

T. Novikova, P. Bulkin, V. Popov, B. H. Ibrahim, and A. De Martino, “Mueller polarimetry as a tool for detecting asymmetry in diffraction grating profiles,” J. Vac. Sci. Technol. B 29(5), 051804 (2011).
[Crossref]

Light Sci. Appl. (1)

V. F. Paz, S. Peterhänsel, K. Frenner, and W. Osten, “Solving the inverse grating problem by white light interference Fourier scatterometry,” Light Sci. Appl. 1(11), e36 (2012).
[Crossref]

Nano Lett. (1)

Y. S. Jung, J. B. Chang, E. Verploegen, K. K. Berggren, and C. A. Ross, “A path to ultranarrow patterns using self-assembled lithography,” Nano Lett. 10(3), 1000–1005 (2010).
[Crossref] [PubMed]

Nat. Nanotechnol. (1)

E. Vogel, “Technology and metrology of new electronic materials and devices,” Nat. Nanotechnol. 2(1), 25–32 (2007).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (4)

Proc. SPIE (4)

S. R. Cloude, “Conditions for the physical realizability of matrix operators in polarimetry,” Proc. SPIE 1166, 177–185 (1990).
[Crossref]

J. Li, Y. Liu, P. Dasari, J. Hu, N. Smith, O. Kritsun, and C. Volkman, “Advanced diffraction-based overlay for double patterning,” Proc. SPIE 7638, 76382C (2010).
[Crossref]

W. Yang, R. Lowe-Webb, S. Rabello, J. Hu, J. Y. Lin, J. Heaton, M. Dusa, A. den Boef, M. van der Schaar, and A. Hunter, “A novel diffraction based spectroscopic method for overlay metrology,” Proc. SPIE 5038, 200–207 (2003).
[Crossref]

M. Maenhoudt, J. Versluijs, H. Struyf, J. Van Olmen, and M. Van Hove, “Double patterning scheme for sub-0.25 k1 single damascene structures at NA = 0.75, λ = 193 nm,” Proc. SPIE 5754, 1508–1518 (2005).
[Crossref]

Soft Matter (1)

S. H. Park, D. O. Shin, B. H. Kim, D. K. Yoon, K. Kim, S. Y. Lee, S. H. Oh, S. W. Choi, S. C. Jeon, and S. O. Kim, “Block copolymer multiple patterning integrated with conventional ArF lithography,” Soft Matter 6(1), 120–125 (2010).
[Crossref]

Surf. Topogr.: Metrol. Prop. (1)

A. J. den Boef, “Optical wafer metrology sensors for process-robust CD and overlay control in semiconductor device manufacturing,” Surf. Topogr.: Metrol. Prop. 4(2), 023001 (2016).
[Crossref]

Thin Solid Films (3)

X. Chen, S. Liu, H. Gu, and C. Zhang, “Formulation of error propagation and estimation in grating reconstruction by a dual-rotating compensator Mueller matrix polarimeter,” Thin Solid Films 571, 653–659 (2014).
[Crossref]

S. Liu, X. Chen, and C. Zhang, “Development of a broadband Mueller matrix ellipsometer as a powerful tool for nanostructure metrology,” Thin Solid Films 584, 176–185 (2015).
[Crossref]

O. Arteaga, “Useful Mueller matrix symmetries for ellipsometry,” Thin Solid Films 571, 584–588 (2014).
[Crossref]

Other (3)

W. Ludwig and C. Falter, Symmetries in Physics: Group Theory Applied to Physical Problems (Springer, 1988).

I. T. Jolliffe, Principal Component Analysis, 2nd ed. (Springer, 2002).

C. J. Raymond, “Scatterometry for Semiconductor Metrology,” in Handbook of Silicon Semiconductor Metrology (Marcel Dekker Inc., 2001).

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Figures (12)

Fig. 1
Fig. 1

Representation of polarized light incidence upon a double-patterned grating structure with a pitch of Λ and an overlay displacement of δ, where E i and E r denote the incident and reflected electric fields, and Es,p refer to the electric field components that perpendicular and parallel to the plane of incidence, respectively.

Fig. 2
Fig. 2

Layout of the DBO target, which consists of two cells per direction used to measure the overlay errors εx and εy along the X and Y directions, respectively. In the DBO target, d is a designed shift, and Λ denotes the pitch of the doubled pattered structure. Without loss of generality, εx and εy are denoted as ε for simplicity.

Fig. 3
Fig. 3

Schematic of the investigated double-patterned grating

Fig. 4
Fig. 4

The spectra of the elementary optical properties LB, LD, LB′, LD′, CB, and CD extracted from the simulated Mueller matrices. The green solid curves with open circles and the red solid curves correspond to the double-patterned grating with an overlay displacement of δ = –10 nm and of δ = 10 nm, respectively. The blue dashed-dotted curves correspond to the grating with δ = 0. The spectra of LB′ and LD′ obtained at ϕ = 30° are multiplied by a factor of 5 for clarity.

Fig. 5
Fig. 5

The difference spectra of Mueller matrix elements m13 and m14 between those calculated for the investigated double-patterned grating with overlay displacements of δ = ± 10 nm and of δ = 0 at different azimuthal angles. The green solid curves with open circles and the red solid curves correspond to the grating with an overlay displacement of δ = –10 nm and of δ = 10 nm, respectively. The difference spectra obtained at ϕ = 30° are multiplied by a factor of 2 for clarity.

Fig. 6
Fig. 6

The norm of the spectra of LB′ for the double-patterned grating with an overlay displacement of δ = 10 nm obtained at different azimuthal angles varied from 0 to 90° with an increment of 15°.

Fig. 7
Fig. 7

Linearity verification results for γ L B . Each subfigure contains two curves, of which the top one shows the linear fitting result while the bottom one presents the corresponding fitting error. (a) and (b) are the results by using the PC weighting approach at ϕ = 60° and ϕ = 90°, respectively. (c) and (d) are the results by using the mean weighting approach at ϕ = 60° and ϕ = 90°, respectively. The R2 inserted in each subfigure represents the coefficient of determination in linear fitting.

Fig. 8
Fig. 8

Linearity verification results for γ m 13 . Each subfigure contains two curves, of which the top one shows the linear fitting result while the bottom one presents the corresponding fitting error. (a) and (b) are the results by the PC weighting approach at ϕ = 60° and ϕ = 90°, respectively. (c) and (d) are the results by the mean weighting approach at ϕ = 60° and ϕ = 90°, respectively.

Fig. 9
Fig. 9

(a) The weights ωi associated with γ L B in the PC-based overlay indicator estimated at ϕ = 60°. The discrete black circles (with a legend title of “all data”) correspond to the ωi estimated by using all the overlay displacements (varied from –50 to 50 nm); the red solid line (with a legend title of “δ = (15, –40)”) corresponds to the ωi estimated by using the overlay displacements of 15 and –40 nm; the blue dash-dot line (with a legend title of “δ = (37, –12)”) corresponds to the ωi estimated by using the overlay displacements of 37 and –12 nm. (b) the red solid line corresponds to the difference between the weights ωi estimated by using all the overlay displacements and those by using the overlay displacements of 15 and –40 nm; the blue dash-dot line corresponds to the difference between the weights ωi estimated by using all the overlay displacements and those by using the overlay displacements of 37 and –12 nm.

Fig. 10
Fig. 10

The measurement errors at different shift values estimated by γ L B at ϕ = 60° (a) and ϕ = 90° (b), as well as by γ m 13 at ϕ = 90° (c). Both γ L B and γ m 13 are calculated using the PC and mean weighting approaches, respectively.

Fig. 11
Fig. 11

The measurement results by using γ L B at ϕ = 60° (a, d) and ϕ = 90° (b, e), as well as by using γ m 13 at ϕ = 90° (c, f). (a), (b) and (c) correspond to the results by γ L B and γ m 13 using the PC weighting approach (PC-based overlay indicator). (d), (e) and (f) correspond to the results by γ L B and γ m 13 using the mean weighting approach (mean-based overlay indicator). The inserted equations associated each subfigure are the fitted linear equation and the coefficient of determination.

Fig. 12
Fig. 12

The measured overlay errors at different azimuthal angles using the original indicator: γ m 13 = i = 1 N ω i m 13 , i and the corrected indicator: γ m 13 = i = 1 N ω i Δ m 13 , i , respectively. The black dash-dot line corresponds to the input overlay error.

Equations (20)

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S = [ I Q U V ] = [ I p + I s I p I s I + 45 ° I 45 ° I R I L ] ,
S o u t = M S i n = [ m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 44 ] S i n ,
d M / d z = m M .
L = ln ( M ) = L m + L u ,
L m = [ 0 L D L D C D L D 0 C B L B L D C B 0 L B C D L B L B 0 ] ,
M = λ 1 M 1 + λ 2 M 2 + λ 3 M 3 + λ 4 M 4 ,
C = 1 4 i , j m i j ( σ i σ j T ) ,
m k , i j = T r [ v k v k ( σ i σ j T ) ] ,
M ^ = M ( δ , ϕ + π ) = O M ( δ , ϕ ) T O 1 ,
M ( δ , ϕ + π ) = M ( δ , ϕ ) = O M ( δ , ϕ ) T O 1 .
L m ( δ , ϕ ) = O L m ( δ , ϕ ) T O 1 ,
L ( δ , ϕ ) = L ( δ , ϕ ) ,
L ( δ , ϕ ) = L ( δ , ϕ ) ,
C ( δ , ϕ ) = C ( δ , ϕ ) ,
m i j ( δ ) = m i j ( δ ) 0 ,
γ L B = i = 1 N ω i L B i ,
ε = γ L B ( d + ε ) + γ L B ( d + ε ) γ L B ( d + ε ) γ L B ( d + ε ) d .
[ E r p E r s ] = J [ E i p E i s ] = [ r p p r p s r s p r s s ] [ E i p E i s ] ,
r p s = C i L T sin T 2 ,
r s p = C + i L T sin T 2 ,

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