Abstract

Scatterometry is now proven to be a very powerful technique for measurement of subwavelength periodic structures. However it requires heavy numerical calculations of the scattered optical waves from the structure. For periodic nanoarrays with feature size less than 100  nm, it is possible to simplify this using the Rytov near-quasi-static approximation valid for feature periods only few time less than the wavelength. The validity is investigated by way of comparison with exact numerical results obtained with the eigenfunctions approach. It is shown to be adequate for the determination of the structure parameters from the specularly reflected or transmitted waves and their polarization or ellipsometric properties. The validity of this approach is applied to lamellar nanoscale grating photoresist lines on Si substrate. The high sensitivity of the signals to the structure parameters is demonstrated using wavelengths of only few times the period.

© 2007 Optical Society of America

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2004 (1)

H.-T. Huang and F. L. Terry, Jr., "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).

2003 (2)

H. Chu, "Finite difference approach to optical scattering of gratings," in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, A. Duparre and B. Singh, eds., Proc. SPIE 5188, 358-370 (2003).

J. Opsal, H. Chu, Y. Wen, G. Li, and Y. C. Chang, "Contact hole inspection by real-time optical CD metrology," in Metrology, Inspection, and Process Control for Lithography XVII,, D. J. Herr, ed., Proc. SPIE 5038, 597-607 (2003).

2002 (2)

J. Opsal, H. Chu, Y. Wen, Y. C. Chang, and G. Li, "Fundamental solutions for real-time optical CD metrology," in Metrology, Inspection, and Process Control for Lithography XVI, D. J. Herr, ed., Proc. SPIE 4689, 163-176 (2002).

J. Allgair, R. R. Hershey, L. C. Litt, D. C. Benoit, P. Herrera, A. Levy, Y. Xu, U. K. Whitney, J. C. Robinson, B. Braymer, I. Abdulhalim, and M. Faeyrman, "Spectroscopic CD offers higher precision metrology for sub-0.18 μm linewidth control," KLA-Tencor Magazine on Yield Management Solutions, 8-13 (2002), www.kla-tencor.com/company/magazine/fall01.

2001 (2)

H.-T. Huang, W. Kong, and F. L. Terry, Jr., "Normal incidence spectroscopic ellipsometry for critical dimension monitoring," Appl. Phys. Lett. 78, 3983-3985 (2001).

S. Hava and M. Auslender, "Optical scatterometry evaluation of groove depth in lamellar silicon grating structures," Opt. Eng. 40, 1244-1248 (2001).

2000 (5)

I. J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Seligson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. SPIE 3998, 125-134 (2000).

B. S. Stutzman, H.-T. Huang, and F. L. Terry, Jr., "Two-channel spectroscopic reflectometry for in situ monitoring of blanket and patterned structures during reactive ion etching," J. Vac. Sci. Technol. B 18, 2785-2793 (2000).

E. Popov and M. Neviere, "Grating theory: new equations in Fourier space leading to fast converging results for TM polarization," J. Opt. Soc. Am. A 17, 1773-1784 (2000).

S. Hava and M. Auslender, "Groove depth dependence of IR transmission spectra through silicon gratings: experiment versus theory," Infrared Phys. Technol. 41, 149-154 (2000).

C. Zhang, B. Yang, X. Wu, T. Lu, Y. Zheng, and W. Su, "Calculation of the effective dielectric function of composites with periodic geometry," Physica B 293, 16-32 (2000).

1999 (3)

1998 (3)

1997 (4)

F. García-Vidal, J. M. Pitarke, and J. B. Pendry, "Effective medium theory of the optical properties of aligned carbon nanotubes," Phys. Rev. B 78, 4289-4292 (1997).

C.-Y. You, S.-C. Shin, and S.-Y. Kim, "Modified effective-medium theory for magneto-optical spectra of magnetic materials," Phys. Rev. B 55, 5953-5958 (1997).

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. W. Hosch, and J. R. McNeil, "Multiparameter grating metrology using optical scatterometry," J. Vacuum Sci. Technol. B 15, 361-368 (1997).

P. Lalanne, "Improved formulation of the coupled-wave method for two-dimensional gratings," J. Opt. Soc. Am. A 14, 1592-1598 (1997).

1996 (6)

1995 (3)

C. J. Raymond, M. R. Murnane, S. S. H. Naqvi, and J. R. McNeil, "Metrology of subwavelength photoresist gratings using optical scatterometry," J. Vacuum Sci. Technol. B 13, 1484-1495 (1995).

M. G. Moharam, E. B. Grann, and D. A. Pommet, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995).

H. Kikuta, H. Yoshida, and K. Iwata, "Ability and limitation of effective medium theory for subwavelength gratings," Opt. Rev. 2, 92-99 (1995).

1994 (4)

1993 (5)

L. Li, "A modal analysis of lamellar diffraction gratings in conical mountings," J. Mod. Opt. 40, 553-573 (1993).

L. Li, "Multilayer modal method for diffraction gratings of arbitary profile, depth, and permittivity," J. Opt. Soc. Am. A 10, 2583-2591 (1993).

L. Li and C. Haggans, "Convergence of the coupled-wave method for metallic lamellar diffraction gratings," J. Opt. Soc. Am. A 10, 1184-1189 (1993).

S. S. H. Naqvi, J. R. McNeil, R. H. Krukar, and K. P. Bishop, "Scatterometry and the simulation of diffraction-based metrology," Microlithogr. World 2, 5-16 (1993).

D. H. Raguin and G. M. Morris, "Antireflection structured surfaces for the infrared spectral region," J. Opt. Soc. Am. A 32, 1154-1167 (1993).

1985 (1)

G. F. Mendes, L. Cescato, J. Frejlich, E. S. Braga, and A. P. Mammana, "Continuous optical measurement of the dry etching of silicon using the diffraction of a lamellar grating," J. Electrochem. Soc. 132, 190-193 (1985).

1984 (1)

1978 (1)

H. P. Kleinknecht and H. Meier, "Optical monitoring of the etching of SiO2 and Si3N4 by the use of grating test pattern," J. Electrochem. Soc. 125, 798-803 (1978).

1956 (1)

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Abdulhalim, I.

J. Allgair, R. R. Hershey, L. C. Litt, D. C. Benoit, P. Herrera, A. Levy, Y. Xu, U. K. Whitney, J. C. Robinson, B. Braymer, I. Abdulhalim, and M. Faeyrman, "Spectroscopic CD offers higher precision metrology for sub-0.18 μm linewidth control," KLA-Tencor Magazine on Yield Management Solutions, 8-13 (2002), www.kla-tencor.com/company/magazine/fall01.

I. J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Seligson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. SPIE 3998, 125-134 (2000).

I. Abdulhalim, "Analytic propagation matrix method for linear optics of arbitrary biaxial layered media," J. Opt. A 1, 646-653 (1999).

Y. Xu and I. Abdulhalim, "Spectroscopic scatterometer system," U.S. patent 6,483,080, 19 November 2002.

I. Abdulhalim, M. Adel, M. Friedmann, and M. Faeyrman, "Periodic patterns and techniques to control misalignment," U.S. Patents Application Publication #2003/0002043 A1 (2 January 2003).

S. Lakkaparagada, K. A. Brown, M. Hankinson, A. Levy, and I. Abdulhalim, "Methods and systems for lithography process control," U.S. patent application no. 2004/0005507 (8 January 2004).

A. Levy, K. A. Brown, R. Smedt, G. Bultman, M. Nikoonahad, D. Wack, J. Fielden, and I. Abdulhalim, "Methods and systems for determining a critical dimension and overlay of a specimen," U.S. patent application no. 2004/0235205 (25 November 2004).

Adel, M.

I. Abdulhalim, M. Adel, M. Friedmann, and M. Faeyrman, "Periodic patterns and techniques to control misalignment," U.S. Patents Application Publication #2003/0002043 A1 (2 January 2003).

Allgair, I. J.

I. J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Seligson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. SPIE 3998, 125-134 (2000).

Allgair, J.

J. Allgair, R. R. Hershey, L. C. Litt, D. C. Benoit, P. Herrera, A. Levy, Y. Xu, U. K. Whitney, J. C. Robinson, B. Braymer, I. Abdulhalim, and M. Faeyrman, "Spectroscopic CD offers higher precision metrology for sub-0.18 μm linewidth control," KLA-Tencor Magazine on Yield Management Solutions, 8-13 (2002), www.kla-tencor.com/company/magazine/fall01.

Auslender, M.

S. Hava and M. Auslender, "Optical scatterometry evaluation of groove depth in lamellar silicon grating structures," Opt. Eng. 40, 1244-1248 (2001).

S. Hava and M. Auslender, "Groove depth dependence of IR transmission spectra through silicon gratings: experiment versus theory," Infrared Phys. Technol. 41, 149-154 (2000).

M. Auslender and S. Hava, "Scattering-matrix propagation algorithm in full-vectorial optics of multilayer grating structures," Opt. Lett. 21, 1765-1767 (1996).
[PubMed]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1986), Chap. 4.

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1986), Chap. 4.

Benoit, D.

I. J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Seligson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. SPIE 3998, 125-134 (2000).

Benoit, D. C.

J. Allgair, R. R. Hershey, L. C. Litt, D. C. Benoit, P. Herrera, A. Levy, Y. Xu, U. K. Whitney, J. C. Robinson, B. Braymer, I. Abdulhalim, and M. Faeyrman, "Spectroscopic CD offers higher precision metrology for sub-0.18 μm linewidth control," KLA-Tencor Magazine on Yield Management Solutions, 8-13 (2002), www.kla-tencor.com/company/magazine/fall01.

Bishop, K. P.

S. S. H. Naqvi, J. R. McNeil, R. H. Krukar, and K. P. Bishop, "Scatterometry and the simulation of diffraction-based metrology," Microlithogr. World 2, 5-16 (1993).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Braga, E. S.

G. F. Mendes, L. Cescato, J. Frejlich, E. S. Braga, and A. P. Mammana, "Continuous optical measurement of the dry etching of silicon using the diffraction of a lamellar grating," J. Electrochem. Soc. 132, 190-193 (1985).

Braymer, B.

J. Allgair, R. R. Hershey, L. C. Litt, D. C. Benoit, P. Herrera, A. Levy, Y. Xu, U. K. Whitney, J. C. Robinson, B. Braymer, I. Abdulhalim, and M. Faeyrman, "Spectroscopic CD offers higher precision metrology for sub-0.18 μm linewidth control," KLA-Tencor Magazine on Yield Management Solutions, 8-13 (2002), www.kla-tencor.com/company/magazine/fall01.

I. J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Seligson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. SPIE 3998, 125-134 (2000).

Brown, K. A.

S. Lakkaparagada, K. A. Brown, M. Hankinson, A. Levy, and I. Abdulhalim, "Methods and systems for lithography process control," U.S. patent application no. 2004/0005507 (8 January 2004).

A. Levy, K. A. Brown, R. Smedt, G. Bultman, M. Nikoonahad, D. Wack, J. Fielden, and I. Abdulhalim, "Methods and systems for determining a critical dimension and overlay of a specimen," U.S. patent application no. 2004/0235205 (25 November 2004).

Brueck, S. R.

S. S. H. Naqvi, S. H. Zaidi, S. R. Brueck, and J. R. McNeil, "Diffractive techniques for lithographic process monitoring and control," J. Vac. Sci. Technol. B 12, 3600-3606 (1994).

Bultman, G.

A. Levy, K. A. Brown, R. Smedt, G. Bultman, M. Nikoonahad, D. Wack, J. Fielden, and I. Abdulhalim, "Methods and systems for determining a critical dimension and overlay of a specimen," U.S. patent application no. 2004/0235205 (25 November 2004).

Cescato, L.

G. F. Mendes, L. Cescato, J. Frejlich, E. S. Braga, and A. P. Mammana, "Continuous optical measurement of the dry etching of silicon using the diffraction of a lamellar grating," J. Electrochem. Soc. 132, 190-193 (1985).

G. F. Mendes, L. Cescato, and J. Frejlich, "Gratings for metrology and process control. 2: Thin film thickness measurement," Appl. Opt. 23, 576-583 (1984).
[PubMed]

Chandezon, J.

Chang, Y. C.

J. Opsal, H. Chu, Y. Wen, G. Li, and Y. C. Chang, "Contact hole inspection by real-time optical CD metrology," in Metrology, Inspection, and Process Control for Lithography XVII,, D. J. Herr, ed., Proc. SPIE 5038, 597-607 (2003).

J. Opsal, H. Chu, Y. Wen, Y. C. Chang, and G. Li, "Fundamental solutions for real-time optical CD metrology," in Metrology, Inspection, and Process Control for Lithography XVI, D. J. Herr, ed., Proc. SPIE 4689, 163-176 (2002).

Chu, H.

J. Opsal, H. Chu, Y. Wen, G. Li, and Y. C. Chang, "Contact hole inspection by real-time optical CD metrology," in Metrology, Inspection, and Process Control for Lithography XVII,, D. J. Herr, ed., Proc. SPIE 5038, 597-607 (2003).

H. Chu, "Finite difference approach to optical scattering of gratings," in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies, A. Duparre and B. Singh, eds., Proc. SPIE 5188, 358-370 (2003).

J. Opsal, H. Chu, Y. Wen, Y. C. Chang, and G. Li, "Fundamental solutions for real-time optical CD metrology," in Metrology, Inspection, and Process Control for Lithography XVI, D. J. Herr, ed., Proc. SPIE 4689, 163-176 (2002).

Coulombe, S. A.

Faeyrman, M.

J. Allgair, R. R. Hershey, L. C. Litt, D. C. Benoit, P. Herrera, A. Levy, Y. Xu, U. K. Whitney, J. C. Robinson, B. Braymer, I. Abdulhalim, and M. Faeyrman, "Spectroscopic CD offers higher precision metrology for sub-0.18 μm linewidth control," KLA-Tencor Magazine on Yield Management Solutions, 8-13 (2002), www.kla-tencor.com/company/magazine/fall01.

I. J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Seligson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. SPIE 3998, 125-134 (2000).

I. Abdulhalim, M. Adel, M. Friedmann, and M. Faeyrman, "Periodic patterns and techniques to control misalignment," U.S. Patents Application Publication #2003/0002043 A1 (2 January 2003).

Fielden, J.

A. Levy, K. A. Brown, R. Smedt, G. Bultman, M. Nikoonahad, D. Wack, J. Fielden, and I. Abdulhalim, "Methods and systems for determining a critical dimension and overlay of a specimen," U.S. patent application no. 2004/0235205 (25 November 2004).

Franke, J. E.

Frejlich, J.

G. F. Mendes, L. Cescato, J. Frejlich, E. S. Braga, and A. P. Mammana, "Continuous optical measurement of the dry etching of silicon using the diffraction of a lamellar grating," J. Electrochem. Soc. 132, 190-193 (1985).

G. F. Mendes, L. Cescato, and J. Frejlich, "Gratings for metrology and process control. 2: Thin film thickness measurement," Appl. Opt. 23, 576-583 (1984).
[PubMed]

Friedmann, M.

I. Abdulhalim, M. Adel, M. Friedmann, and M. Faeyrman, "Periodic patterns and techniques to control misalignment," U.S. Patents Application Publication #2003/0002043 A1 (2 January 2003).

García-Vidal, F.

F. García-Vidal, J. M. Pitarke, and J. B. Pendry, "Effective medium theory of the optical properties of aligned carbon nanotubes," Phys. Rev. B 78, 4289-4292 (1997).

Gasvik, Kjell J.

Kjell J. Gasvik, Optical Metrology, 3rd ed. (Wiley, 1996).

Gottscho, R. A.

Granet, G.

Grann, E. B.

Guizal, B.

Haaland, D. M.

Haggans, C.

Hankinson, M.

S. Lakkaparagada, K. A. Brown, M. Hankinson, A. Levy, and I. Abdulhalim, "Methods and systems for lithography process control," U.S. patent application no. 2004/0005507 (8 January 2004).

Hava, S.

S. Hava and M. Auslender, "Optical scatterometry evaluation of groove depth in lamellar silicon grating structures," Opt. Eng. 40, 1244-1248 (2001).

S. Hava and M. Auslender, "Groove depth dependence of IR transmission spectra through silicon gratings: experiment versus theory," Infrared Phys. Technol. 41, 149-154 (2000).

M. Auslender and S. Hava, "Scattering-matrix propagation algorithm in full-vectorial optics of multilayer grating structures," Opt. Lett. 21, 1765-1767 (1996).
[PubMed]

Herrera, P.

J. Allgair, R. R. Hershey, L. C. Litt, D. C. Benoit, P. Herrera, A. Levy, Y. Xu, U. K. Whitney, J. C. Robinson, B. Braymer, I. Abdulhalim, and M. Faeyrman, "Spectroscopic CD offers higher precision metrology for sub-0.18 μm linewidth control," KLA-Tencor Magazine on Yield Management Solutions, 8-13 (2002), www.kla-tencor.com/company/magazine/fall01.

Hershey, R.

I. J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Seligson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. SPIE 3998, 125-134 (2000).

Hershey, R. R.

J. Allgair, R. R. Hershey, L. C. Litt, D. C. Benoit, P. Herrera, A. Levy, Y. Xu, U. K. Whitney, J. C. Robinson, B. Braymer, I. Abdulhalim, and M. Faeyrman, "Spectroscopic CD offers higher precision metrology for sub-0.18 μm linewidth control," KLA-Tencor Magazine on Yield Management Solutions, 8-13 (2002), www.kla-tencor.com/company/magazine/fall01.

Hosch, J. W.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. W. Hosch, and J. R. McNeil, "Multiparameter grating metrology using optical scatterometry," J. Vacuum Sci. Technol. B 15, 361-368 (1997).

Huang, H.-T.

H.-T. Huang and F. L. Terry, Jr., "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).

H.-T. Huang, W. Kong, and F. L. Terry, Jr., "Normal incidence spectroscopic ellipsometry for critical dimension monitoring," Appl. Phys. Lett. 78, 3983-3985 (2001).

B. S. Stutzman, H.-T. Huang, and F. L. Terry, Jr., "Two-channel spectroscopic reflectometry for in situ monitoring of blanket and patterned structures during reactive ion etching," J. Vac. Sci. Technol. B 18, 2785-2793 (2000).

Iwata, K.

H. Kikuta, H. Yoshida, and K. Iwata, "Ability and limitation of effective medium theory for subwavelength gratings," Opt. Rev. 2, 92-99 (1995).

Kallioniemi, I.

Kikuta, H.

H. Kikuta, H. Yoshida, and K. Iwata, "Ability and limitation of effective medium theory for subwavelength gratings," Opt. Rev. 2, 92-99 (1995).

Kim, S.-Y.

C.-Y. You, S.-C. Shin, and S.-Y. Kim, "Modified effective-medium theory for magneto-optical spectra of magnetic materials," Phys. Rev. B 55, 5953-5958 (1997).

Kleinknecht, H. P.

H. P. Kleinknecht and H. Meier, "Optical monitoring of the etching of SiO2 and Si3N4 by the use of grating test pattern," J. Electrochem. Soc. 125, 798-803 (1978).

Kong, W.

H.-T. Huang, W. Kong, and F. L. Terry, Jr., "Normal incidence spectroscopic ellipsometry for critical dimension monitoring," Appl. Phys. Lett. 78, 3983-3985 (2001).

Kornblit, A.

Krukar, R. H.

Lakkaparagada, S.

S. Lakkaparagada, K. A. Brown, M. Hankinson, A. Levy, and I. Abdulhalim, "Methods and systems for lithography process control," U.S. patent application no. 2004/0005507 (8 January 2004).

Lalanne, P.

Levy, A.

J. Allgair, R. R. Hershey, L. C. Litt, D. C. Benoit, P. Herrera, A. Levy, Y. Xu, U. K. Whitney, J. C. Robinson, B. Braymer, I. Abdulhalim, and M. Faeyrman, "Spectroscopic CD offers higher precision metrology for sub-0.18 μm linewidth control," KLA-Tencor Magazine on Yield Management Solutions, 8-13 (2002), www.kla-tencor.com/company/magazine/fall01.

S. Lakkaparagada, K. A. Brown, M. Hankinson, A. Levy, and I. Abdulhalim, "Methods and systems for lithography process control," U.S. patent application no. 2004/0005507 (8 January 2004).

A. Levy, K. A. Brown, R. Smedt, G. Bultman, M. Nikoonahad, D. Wack, J. Fielden, and I. Abdulhalim, "Methods and systems for determining a critical dimension and overlay of a specimen," U.S. patent application no. 2004/0235205 (25 November 2004).

Li, G.

J. Opsal, H. Chu, Y. Wen, G. Li, and Y. C. Chang, "Contact hole inspection by real-time optical CD metrology," in Metrology, Inspection, and Process Control for Lithography XVII,, D. J. Herr, ed., Proc. SPIE 5038, 597-607 (2003).

J. Opsal, H. Chu, Y. Wen, Y. C. Chang, and G. Li, "Fundamental solutions for real-time optical CD metrology," in Metrology, Inspection, and Process Control for Lithography XVI, D. J. Herr, ed., Proc. SPIE 4689, 163-176 (2002).

Li, L.

Litt, L. C.

J. Allgair, R. R. Hershey, L. C. Litt, D. C. Benoit, P. Herrera, A. Levy, Y. Xu, U. K. Whitney, J. C. Robinson, B. Braymer, I. Abdulhalim, and M. Faeyrman, "Spectroscopic CD offers higher precision metrology for sub-0.18 μm linewidth control," KLA-Tencor Magazine on Yield Management Solutions, 8-13 (2002), www.kla-tencor.com/company/magazine/fall01.

I. J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Seligson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. SPIE 3998, 125-134 (2000).

Lu, T.

C. Zhang, B. Yang, X. Wu, T. Lu, Y. Zheng, and W. Su, "Calculation of the effective dielectric function of composites with periodic geometry," Physica B 293, 16-32 (2000).

Mammana, A. P.

G. F. Mendes, L. Cescato, J. Frejlich, E. S. Braga, and A. P. Mammana, "Continuous optical measurement of the dry etching of silicon using the diffraction of a lamellar grating," J. Electrochem. Soc. 132, 190-193 (1985).

McNeil, J. R.

B. K. Minhas, S. A. Coulombe, S. SohailH. Naqvi, and J. R. McNeil, "Ellipsometric scatterometry for the metrology of sub-0.10 μm linewidth structure," Appl. Opt. 37, 5112-5115 (1998).

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. W. Hosch, and J. R. McNeil, "Multiparameter grating metrology using optical scatterometry," J. Vacuum Sci. Technol. B 15, 361-368 (1997).

B. K. Minhas, S. L. Prins, S. S. H. Naqvi, and J. R. McNeil, "Toward sub-0.1-mm CD measurements using scatterometry," in Integrated Circuit Metrology, Inspection, and Process Control X, S. K. Jones, ed., Proc. SPIE 2725, 729-739 (1996).

C. J. Raymond, M. R. Murnane, S. S. H. Naqvi, and J. R. McNeil, "Metrology of subwavelength photoresist gratings using optical scatterometry," J. Vacuum Sci. Technol. B 13, 1484-1495 (1995).

S. S. H. Naqvi, R. H. Krukar, J. R. McNeil, J. E. Franke, T. M. Niemszyk, D. M. Haaland, R. A. Gottscho, and A. Kornblit, "Etch depth estimation of large-period silicon gratings with mulivariate calibration of rigorously simulated diffraction profiles," J. Opt. Soc. Am. A 11, 2485-2492 (1994).

S. S. H. Naqvi, S. H. Zaidi, S. R. Brueck, and J. R. McNeil, "Diffractive techniques for lithographic process monitoring and control," J. Vac. Sci. Technol. B 12, 3600-3606 (1994).

S. S. H. Naqvi, J. R. McNeil, R. H. Krukar, and K. P. Bishop, "Scatterometry and the simulation of diffraction-based metrology," Microlithogr. World 2, 5-16 (1993).

Meier, H.

H. P. Kleinknecht and H. Meier, "Optical monitoring of the etching of SiO2 and Si3N4 by the use of grating test pattern," J. Electrochem. Soc. 125, 798-803 (1978).

Mendes, G. F.

G. F. Mendes, L. Cescato, J. Frejlich, E. S. Braga, and A. P. Mammana, "Continuous optical measurement of the dry etching of silicon using the diffraction of a lamellar grating," J. Electrochem. Soc. 132, 190-193 (1985).

G. F. Mendes, L. Cescato, and J. Frejlich, "Gratings for metrology and process control. 2: Thin film thickness measurement," Appl. Opt. 23, 576-583 (1984).
[PubMed]

Minhas, B. K.

B. K. Minhas, S. A. Coulombe, S. SohailH. Naqvi, and J. R. McNeil, "Ellipsometric scatterometry for the metrology of sub-0.10 μm linewidth structure," Appl. Opt. 37, 5112-5115 (1998).

B. K. Minhas, S. L. Prins, S. S. H. Naqvi, and J. R. McNeil, "Toward sub-0.1-mm CD measurements using scatterometry," in Integrated Circuit Metrology, Inspection, and Process Control X, S. K. Jones, ed., Proc. SPIE 2725, 729-739 (1996).

Moharam, M. G.

Morris, G. M.

P. Lalanne and G. M. Morris, "Highly improved convergence of the coupled wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).

D. H. Raguin and G. M. Morris, "Antireflection structured surfaces for the infrared spectral region," J. Opt. Soc. Am. A 32, 1154-1167 (1993).

Murnane, M. R.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. W. Hosch, and J. R. McNeil, "Multiparameter grating metrology using optical scatterometry," J. Vacuum Sci. Technol. B 15, 361-368 (1997).

C. J. Raymond, M. R. Murnane, S. S. H. Naqvi, and J. R. McNeil, "Metrology of subwavelength photoresist gratings using optical scatterometry," J. Vacuum Sci. Technol. B 13, 1484-1495 (1995).

Naqvi, H.

Naqvi, S. S. H.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. W. Hosch, and J. R. McNeil, "Multiparameter grating metrology using optical scatterometry," J. Vacuum Sci. Technol. B 15, 361-368 (1997).

B. K. Minhas, S. L. Prins, S. S. H. Naqvi, and J. R. McNeil, "Toward sub-0.1-mm CD measurements using scatterometry," in Integrated Circuit Metrology, Inspection, and Process Control X, S. K. Jones, ed., Proc. SPIE 2725, 729-739 (1996).

C. J. Raymond, M. R. Murnane, S. S. H. Naqvi, and J. R. McNeil, "Metrology of subwavelength photoresist gratings using optical scatterometry," J. Vacuum Sci. Technol. B 13, 1484-1495 (1995).

S. S. H. Naqvi, R. H. Krukar, J. R. McNeil, J. E. Franke, T. M. Niemszyk, D. M. Haaland, R. A. Gottscho, and A. Kornblit, "Etch depth estimation of large-period silicon gratings with mulivariate calibration of rigorously simulated diffraction profiles," J. Opt. Soc. Am. A 11, 2485-2492 (1994).

S. S. H. Naqvi, S. H. Zaidi, S. R. Brueck, and J. R. McNeil, "Diffractive techniques for lithographic process monitoring and control," J. Vac. Sci. Technol. B 12, 3600-3606 (1994).

S. S. H. Naqvi, J. R. McNeil, R. H. Krukar, and K. P. Bishop, "Scatterometry and the simulation of diffraction-based metrology," Microlithogr. World 2, 5-16 (1993).

Neviere, M.

Niemszyk, T. M.

Nikoonahad, M.

A. Levy, K. A. Brown, R. Smedt, G. Bultman, M. Nikoonahad, D. Wack, J. Fielden, and I. Abdulhalim, "Methods and systems for determining a critical dimension and overlay of a specimen," U.S. patent application no. 2004/0235205 (25 November 2004).

Oja, E.

Opsal, J.

J. Opsal, H. Chu, Y. Wen, G. Li, and Y. C. Chang, "Contact hole inspection by real-time optical CD metrology," in Metrology, Inspection, and Process Control for Lithography XVII,, D. J. Herr, ed., Proc. SPIE 5038, 597-607 (2003).

J. Opsal, H. Chu, Y. Wen, Y. C. Chang, and G. Li, "Fundamental solutions for real-time optical CD metrology," in Metrology, Inspection, and Process Control for Lithography XVI, D. J. Herr, ed., Proc. SPIE 4689, 163-176 (2002).

Pendry, J. B.

F. García-Vidal, J. M. Pitarke, and J. B. Pendry, "Effective medium theory of the optical properties of aligned carbon nanotubes," Phys. Rev. B 78, 4289-4292 (1997).

Pitarke, J. M.

F. García-Vidal, J. M. Pitarke, and J. B. Pendry, "Effective medium theory of the optical properties of aligned carbon nanotubes," Phys. Rev. B 78, 4289-4292 (1997).

Plumey, J. P.

Pommet, D. A.

Popov, E.

Prins, S. L.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. W. Hosch, and J. R. McNeil, "Multiparameter grating metrology using optical scatterometry," J. Vacuum Sci. Technol. B 15, 361-368 (1997).

B. K. Minhas, S. L. Prins, S. S. H. Naqvi, and J. R. McNeil, "Toward sub-0.1-mm CD measurements using scatterometry," in Integrated Circuit Metrology, Inspection, and Process Control X, S. K. Jones, ed., Proc. SPIE 2725, 729-739 (1996).

Raguin, D. H.

D. H. Raguin and G. M. Morris, "Antireflection structured surfaces for the infrared spectral region," J. Opt. Soc. Am. A 32, 1154-1167 (1993).

Raymond, C. J.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. W. Hosch, and J. R. McNeil, "Multiparameter grating metrology using optical scatterometry," J. Vacuum Sci. Technol. B 15, 361-368 (1997).

C. J. Raymond, M. R. Murnane, S. S. H. Naqvi, and J. R. McNeil, "Metrology of subwavelength photoresist gratings using optical scatterometry," J. Vacuum Sci. Technol. B 13, 1484-1495 (1995).

Robinson, J. C.

J. Allgair, R. R. Hershey, L. C. Litt, D. C. Benoit, P. Herrera, A. Levy, Y. Xu, U. K. Whitney, J. C. Robinson, B. Braymer, I. Abdulhalim, and M. Faeyrman, "Spectroscopic CD offers higher precision metrology for sub-0.18 μm linewidth control," KLA-Tencor Magazine on Yield Management Solutions, 8-13 (2002), www.kla-tencor.com/company/magazine/fall01.

I. J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Seligson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. SPIE 3998, 125-134 (2000).

Rytov, S. M.

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Saarinen, J.

Schubert, M.

M. Schubert, "Generalized ellipsometry and complex optical systems," Thin Solid Films 313-314, 323-332 (1998).

Seligson, J.

I. J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Seligson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. SPIE 3998, 125-134 (2000).

Shin, S.-C.

C.-Y. You, S.-C. Shin, and S.-Y. Kim, "Modified effective-medium theory for magneto-optical spectra of magnetic materials," Phys. Rev. B 55, 5953-5958 (1997).

Smedt, R.

A. Levy, K. A. Brown, R. Smedt, G. Bultman, M. Nikoonahad, D. Wack, J. Fielden, and I. Abdulhalim, "Methods and systems for determining a critical dimension and overlay of a specimen," U.S. patent application no. 2004/0235205 (25 November 2004).

Sohail, S.

Stutzman, B. S.

B. S. Stutzman, H.-T. Huang, and F. L. Terry, Jr., "Two-channel spectroscopic reflectometry for in situ monitoring of blanket and patterned structures during reactive ion etching," J. Vac. Sci. Technol. B 18, 2785-2793 (2000).

Su, W.

C. Zhang, B. Yang, X. Wu, T. Lu, Y. Zheng, and W. Su, "Calculation of the effective dielectric function of composites with periodic geometry," Physica B 293, 16-32 (2000).

Terry, F. L.

H.-T. Huang and F. L. Terry, Jr., "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).

H.-T. Huang, W. Kong, and F. L. Terry, Jr., "Normal incidence spectroscopic ellipsometry for critical dimension monitoring," Appl. Phys. Lett. 78, 3983-3985 (2001).

B. S. Stutzman, H.-T. Huang, and F. L. Terry, Jr., "Two-channel spectroscopic reflectometry for in situ monitoring of blanket and patterned structures during reactive ion etching," J. Vac. Sci. Technol. B 18, 2785-2793 (2000).

Wack, D.

A. Levy, K. A. Brown, R. Smedt, G. Bultman, M. Nikoonahad, D. Wack, J. Fielden, and I. Abdulhalim, "Methods and systems for determining a critical dimension and overlay of a specimen," U.S. patent application no. 2004/0235205 (25 November 2004).

Wen, Y.

J. Opsal, H. Chu, Y. Wen, G. Li, and Y. C. Chang, "Contact hole inspection by real-time optical CD metrology," in Metrology, Inspection, and Process Control for Lithography XVII,, D. J. Herr, ed., Proc. SPIE 5038, 597-607 (2003).

J. Opsal, H. Chu, Y. Wen, Y. C. Chang, and G. Li, "Fundamental solutions for real-time optical CD metrology," in Metrology, Inspection, and Process Control for Lithography XVI, D. J. Herr, ed., Proc. SPIE 4689, 163-176 (2002).

Whitney, U.

I. J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Seligson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. SPIE 3998, 125-134 (2000).

Whitney, U. K.

J. Allgair, R. R. Hershey, L. C. Litt, D. C. Benoit, P. Herrera, A. Levy, Y. Xu, U. K. Whitney, J. C. Robinson, B. Braymer, I. Abdulhalim, and M. Faeyrman, "Spectroscopic CD offers higher precision metrology for sub-0.18 μm linewidth control," KLA-Tencor Magazine on Yield Management Solutions, 8-13 (2002), www.kla-tencor.com/company/magazine/fall01.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Wu, X.

C. Zhang, B. Yang, X. Wu, T. Lu, Y. Zheng, and W. Su, "Calculation of the effective dielectric function of composites with periodic geometry," Physica B 293, 16-32 (2000).

Xu, Y.

J. Allgair, R. R. Hershey, L. C. Litt, D. C. Benoit, P. Herrera, A. Levy, Y. Xu, U. K. Whitney, J. C. Robinson, B. Braymer, I. Abdulhalim, and M. Faeyrman, "Spectroscopic CD offers higher precision metrology for sub-0.18 μm linewidth control," KLA-Tencor Magazine on Yield Management Solutions, 8-13 (2002), www.kla-tencor.com/company/magazine/fall01.

I. J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Seligson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. SPIE 3998, 125-134 (2000).

Y. Xu and I. Abdulhalim, "Spectroscopic scatterometer system," U.S. patent 6,483,080, 19 November 2002.

Yang, B.

C. Zhang, B. Yang, X. Wu, T. Lu, Y. Zheng, and W. Su, "Calculation of the effective dielectric function of composites with periodic geometry," Physica B 293, 16-32 (2000).

Yoshida, H.

H. Kikuta, H. Yoshida, and K. Iwata, "Ability and limitation of effective medium theory for subwavelength gratings," Opt. Rev. 2, 92-99 (1995).

You, C.-Y.

C.-Y. You, S.-C. Shin, and S.-Y. Kim, "Modified effective-medium theory for magneto-optical spectra of magnetic materials," Phys. Rev. B 55, 5953-5958 (1997).

Zaidi, S. H.

S. S. H. Naqvi, S. H. Zaidi, S. R. Brueck, and J. R. McNeil, "Diffractive techniques for lithographic process monitoring and control," J. Vac. Sci. Technol. B 12, 3600-3606 (1994).

Zalicki, P.

I. J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Seligson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. SPIE 3998, 125-134 (2000).

Zhang, C.

C. Zhang, B. Yang, X. Wu, T. Lu, Y. Zheng, and W. Su, "Calculation of the effective dielectric function of composites with periodic geometry," Physica B 293, 16-32 (2000).

Zheng, Y.

C. Zhang, B. Yang, X. Wu, T. Lu, Y. Zheng, and W. Su, "Calculation of the effective dielectric function of composites with periodic geometry," Physica B 293, 16-32 (2000).

Appl. Opt. (4)

Appl. Phys. Lett. (1)

H.-T. Huang, W. Kong, and F. L. Terry, Jr., "Normal incidence spectroscopic ellipsometry for critical dimension monitoring," Appl. Phys. Lett. 78, 3983-3985 (2001).

Infrared Phys. Technol. (1)

S. Hava and M. Auslender, "Groove depth dependence of IR transmission spectra through silicon gratings: experiment versus theory," Infrared Phys. Technol. 41, 149-154 (2000).

J. Electrochem. Soc. (2)

G. F. Mendes, L. Cescato, J. Frejlich, E. S. Braga, and A. P. Mammana, "Continuous optical measurement of the dry etching of silicon using the diffraction of a lamellar grating," J. Electrochem. Soc. 132, 190-193 (1985).

H. P. Kleinknecht and H. Meier, "Optical monitoring of the etching of SiO2 and Si3N4 by the use of grating test pattern," J. Electrochem. Soc. 125, 798-803 (1978).

J. Mod. Opt. (1)

L. Li, "A modal analysis of lamellar diffraction gratings in conical mountings," J. Mod. Opt. 40, 553-573 (1993).

J. Opt. A (1)

I. Abdulhalim, "Analytic propagation matrix method for linear optics of arbitrary biaxial layered media," J. Opt. A 1, 646-653 (1999).

J. Opt. Soc. Am. A (14)

D. H. Raguin and G. M. Morris, "Antireflection structured surfaces for the infrared spectral region," J. Opt. Soc. Am. A 32, 1154-1167 (1993).

E. B. Grann, M. G. Moharam, and D. A. Pommet, "Artificial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings," J. Opt. Soc. Am. A 11, 2695-2703 (1994).

L. Li, "Multilayer modal method for diffraction gratings of arbitary profile, depth, and permittivity," J. Opt. Soc. Am. A 10, 2583-2591 (1993).

L. Li, "Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity:addendum," J. Opt. Soc. Am. A 11, 1685 (1994).

L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996).

L. Li, "Use of Fourier series in the analysis of the discontinuous structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).

E. Popov and M. Neviere, "Grating theory: new equations in Fourier space leading to fast converging results for TM polarization," J. Opt. Soc. Am. A 17, 1773-1784 (2000).

M. G. Moharam, E. B. Grann, and D. A. Pommet, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995).

P. Lalanne and G. M. Morris, "Highly improved convergence of the coupled wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).

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Figures (16)

Fig. 1
Fig. 1

Schematic showing the geometry of the problem.

Fig. 2
Fig. 2

Uniaxial film on substrate as the equivalent of the grating on substrate in the NQS limit.

Fig. 3
Fig. 3

(a) Comparison of the ordinary and extraordinary refractive indices versus the wavelength in the NQSA using exact solution of the dispersion equation and using the second-order approximation, calculated for photoresist with p = 360   nm and f = 0.5 . (b) Variation of the ordinary and extraordinary indices with the fill factor for λ = 7 p ; p = 360   nm calculated with the second-order approximation.

Fig. 4
Fig. 4

Reflectivity versus the incidence angle calculated using exact numerical calculation with the eigenfunctions approach and using the analytic NQSA for λ = 1.25   μm , p = 180   nm , f = 0.475 , d = 1   μm for the same photoresist on Si.

Fig. 5
Fig. 5

Same as Fig. 4 for the ellipsometric parameters.

Fig. 6
Fig. 6

Reflectivity spectra for the same structure of Fig. 4 but θ i = 38 ° , f = 0.5 showing the excellent agreement between the exact calculation and the analytic NQSA.

Fig. 7
Fig. 7

Same as Fig. 6 for the ellipsometric parameters.

Fig. 8
Fig. 8

Same as in Fig. 6 except that p = 360   nm showing the good agreement between the exact and NQSA calculations at least for the TE reflectivity.

Fig. 9
Fig. 9

Same as in Fig. 8 for the ellipsometric parameters.

Fig. 10
Fig. 10

Reflectivity spectra at different CD values calculated using θ i = 76 ° in 0.9 nm steps demonstrating the high sensitivity to the CD within the NQSA (a) TM, (b) TE. Other parameters are indicated in the figure.

Fig. 11
Fig. 11

Same as Fig. 10 for the ellipsometric parameters (a) tan ( ψ ) , (b) cos ( Δ ) .

Fig. 12
Fig. 12

Same as Fig. 10 for smaller pitch: p = 50   nm and lower height d = 100   nm in 1 nm steps of the CD corresponding to the present design rule of the nanoelectronics industry.

Fig. 13
Fig. 13

Same as Fig. 12 for the ellipsometric parameters.

Fig. 14
Fig. 14

Variation of the phase spectrum with the CD for the same structure of Figs. 12 and 13 showing that the TM phase (a) is much more sensitive to the CD than the TE phase.

Fig. 15
Fig. 15

Schematic of possible nanoarrayed structure used in the nanoelectronics industry consisting of dielectric multilayers and a grating with trapezoidal profile. This structure can be simulated with the 4 × 4 matrix approach by dividing the gratings trapezoidal shape into thin rectangular slices.

Fig. 16
Fig. 16

Variation of the reflected spectra with the wall angle for trapezoidal resist lines on Si.

Equations (16)

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m λ / p = n d sin θ m n i sin θ i .
p < λ max ( n s , n i ) + n i sin θ i .
n m 2 n T E 2 tan ( π p ( 1 f ) n m 2 n T E 2 λ ) = n g 2 n T E 2 tan ( π p f n g 2 n T E 2 λ ) ,
n m 2 n T M 2 tan ( π p ( 1 f ) n m 2 n T M 2 λ ) = n m 2 n g 2 n g 2 n T M 2 tan ( π p f n g 2 n T M 2 λ ) ,
n T E 0 = n m 2 ( 1 f ) + f n g 2 ,     n T M 0 = n m n g n g 2 ( 1 f ) + f n m 2 ,
n T E 2 = { n T E 0 2 + 1 3 [ π f ( 1 f ) p λ ] 2 ( n g 2 n m 2 ) 2 } 1 / 2 ,
n T M 2 = { n T M 0 2 + 1 3 [ π f ( 1 f ) p λ ] 2 ( 1 n g 2 1 n m 2 ) 2 n T M 0 6 n T E 0 2 } 1 / 2 .
r p p = r i g p p + r g s p p exp ( i 2 β T M ) 1 + r i g p p r g s p p exp ( i 2 β T M ) ,
r s s = r i g s s + r g s s s exp ( i 2 β T E ) 1 + r i g s s r g s s s exp ( i 2 β T E ) ,
r i g p p = n T M n T E cos θ i n i ν T E n T M n T E cos θ i + n i ν T E ,
r g s p p = n T M n T E cos θ s + n s ν T E n T M n T E cos θ s + n s ν T E ,
r i g s s = n i cos θ i ν T E n i cos θ i + ν T E ,
r g s s s = n s cos θ s T E n s cos θ s + ν T E ,
P g = [ cos ( k 0 h ν T M ) i ν T M sin ( k 0 h ν T M ) n T M 2 0 0 i n T M 2 sin ( k 0 h ν T M ) ν T M cos ( k 0 h ν T M ) 0 0 0 0 cos ( k 0 h ν T E ) i sin ( k 0 h ν T E ) ν T E 0 0 i ν T E sin ( k 0 h ν T E ) cos ( k 0 h ν T E ) ] ,
P i s o = [ cos ( k 0 h ν z ) i ν z sin ( k 0 h ν z ) ε 0 0 i ε   sin ( k 0 h ν z ) ν z cos ( k 0 h ν z ) 0 0 0 0 cos ( k 0 h ν z ) i sin ( k 0 h ν z ) ν z 0 0 i ν z sin ( k 0 h ν z ) cos ( k 0 h ν z ) ] ,
r = 1 a 2 a 8 [ a 1 a 8 0 0 a 2 a 7 ] , t = [ b 1 + b 2 r p p 0 0 b 8 + b 7 r s s ] ,

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