Abstract

The success of the model-based infrared reflectrometry (MBIR) technique relies heavily on accurate modeling and fast calculation of the infrared metrology process, which continues to be a challenge, especially for three- dimensional (3D) trench structures. In this paper, we present a simplified formulation for effective medium approximation (EMA), determined by a fitting-based method for the modeling of 3D trench structures. Intensive investigations have been performed with an emphasis on the generality of the fitting-determined (FD)-EMA formulation in terms of trench depth, trench pitch, and incidence angle so that its application is not limited to a particular configuration. Simulations conducted on a taper trench structure have further verified the proposed FD-EMA and demonstrated that the MBIR metrology with the FD-EMA-based model achieves an accuracy one order higher than that of the conventional zeroth-order EMA-based model.

© 2011 Optical Society of America

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    [CrossRef]
  3. C. A. Durán, A. A. Maznev, G. T. Merklin, A. Mazurenko, and M. Gostein, “Infrared reflectometry for metrology of trenches in power devices,” in Proceedings of IEEE Conference on Advanced Semiconductor Manufacturing (IEEE, 2007), pp. 175–179.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. C. W. Zhang, S. Y. Liu, T. L. Shi, and Z. R. Tang, “Improved model-based infrared reflectrometry for measuring deep trench structures,” J. Opt. Soc. Am. A 26, 2327–2335 (2009).
    [CrossRef]
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    [CrossRef]
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  25. E. B. Grann, M. G. Moharam, and D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am. A 11, 2695–2703 (1994).
    [CrossRef]
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    [CrossRef]
  27. P. Y. Guittet, U. Mantz, and P. Weidner, “Infrared spectroscopic ellipsometry in semiconductor manufacturing,” Proc. SPIE 5375, 771–778 (2004).
    [CrossRef]
  28. S. Moon and D. Kim, “Fitting-based determination of an effective medium of a metallic periodic structure and application to photonic crystals,” J. Opt. Soc. Am. A 23, 199–207 (2006).
    [CrossRef]
  29. S. Moon and D. Kim, “Investigation of an effective medium theory for metallic periodic structure,” Proc. SPIE 6128, 61281M (2006).
    [CrossRef]
  30. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  31. P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

2010 (2)

2009 (1)

2007 (2)

I. Abdulhalim, “Simplified optical scatterometry for periodic nanoarrays in the near-quasi-static limit,” Appl. Opt. 46, 2219–2228(2007).
[CrossRef]

A. A. Maznev, A. Mazurenko, C. A. Durán, and M. Gostein, “Measuring trench structures for microelectronics with model-based infrared reflectometry,” AIP Conf. Proc. 931, 74–78 (2007).
[CrossRef]

2006 (2)

2005 (2)

P. A. Rosenthal, C. A. Durán, J. Tower, and A. Mazurenko, “Model-based infrared metrology for advanced technology nodes and 300 mm wafer processing,” AIP Conf. Proc. 788, 620–624 (2005).
[CrossRef]

P. Reinig, R. Dost, M. Mort, T. Hingst, U. Mantz, J. Moffitt, S. Shakya, C. J. Raymond, and M. Littau, “Metrology of deep trench etched memory structures using 3D scatterometry,” Proc. SPIE 5752, 559–569 (2005).
[CrossRef]

2004 (2)

A. Weidner, M. Slodowski, C. Halm, C. Schneider, and L. Pfitzner, “Effective-medium model for fast evaluation of scatterometric measurements on gratings,” Proc. SPIE 5375, 232–243 (2004).
[CrossRef]

P. Y. Guittet, U. Mantz, and P. Weidner, “Infrared spectroscopic ellipsometry in semiconductor manufacturing,” Proc. SPIE 5375, 771–778 (2004).
[CrossRef]

2001 (1)

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97–111 (2001).
[CrossRef]

2000 (1)

1998 (2)

1997 (4)

1996 (1)

P. Lalanne and D. L. Lalanne, “On the effective medium theory of subwavelength periodic structure,” J. Mod. Opt. 43, 2063–2085(1996).
[CrossRef]

1995 (1)

H. Kikuta, H. Yoshida, and K. Iwata, “Ability and limitation of effective medium theory for subwavelength gratings,” Opt. Rev. 2, 92–99 (1995).
[CrossRef]

1994 (1)

1993 (1)

1992 (1)

1968 (1)

J. L. Jackson and S. R. Coriell, “Transport coefficients of composite materials,” J. Appl. Phys. 39, 2349–2354 (1968).
[CrossRef]

1956 (1)

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

Abdulhalim, I.

Bao, J.

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97–111 (2001).
[CrossRef]

Bergner, B. C.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Choi, T. C.

T. C. Choi, Effective Medium Theory: Principles and Applications (Oxford University, 1999).

Coriell, S. R.

J. L. Jackson and S. R. Coriell, “Transport coefficients of composite materials,” J. Appl. Phys. 39, 2349–2354 (1968).
[CrossRef]

Dost, R.

P. Reinig, R. Dost, M. Mort, T. Hingst, U. Mantz, J. Moffitt, S. Shakya, C. J. Raymond, and M. Littau, “Metrology of deep trench etched memory structures using 3D scatterometry,” Proc. SPIE 5752, 559–569 (2005).
[CrossRef]

Durán, C. A.

A. A. Maznev, A. Mazurenko, C. A. Durán, and M. Gostein, “Measuring trench structures for microelectronics with model-based infrared reflectometry,” AIP Conf. Proc. 931, 74–78 (2007).
[CrossRef]

P. A. Rosenthal, C. A. Durán, J. Tower, and A. Mazurenko, “Model-based infrared metrology for advanced technology nodes and 300 mm wafer processing,” AIP Conf. Proc. 788, 620–624 (2005).
[CrossRef]

C. A. Durán, A. A. Maznev, G. T. Merklin, A. Mazurenko, and M. Gostein, “Infrared reflectometry for metrology of trenches in power devices,” in Proceedings of IEEE Conference on Advanced Semiconductor Manufacturing (IEEE, 2007), pp. 175–179.

Freese, W.

Germer, T. A.

Gostein, M.

A. A. Maznev, A. Mazurenko, C. A. Durán, and M. Gostein, “Measuring trench structures for microelectronics with model-based infrared reflectometry,” AIP Conf. Proc. 931, 74–78 (2007).
[CrossRef]

C. A. Durán, A. A. Maznev, G. T. Merklin, A. Mazurenko, and M. Gostein, “Infrared reflectometry for metrology of trenches in power devices,” in Proceedings of IEEE Conference on Advanced Semiconductor Manufacturing (IEEE, 2007), pp. 175–179.

Grann, E. B.

Guittet, P. Y.

P. Y. Guittet, U. Mantz, and P. Weidner, “Infrared spectroscopic ellipsometry in semiconductor manufacturing,” Proc. SPIE 5375, 771–778 (2004).
[CrossRef]

Gunning, W. J.

Halm, C.

A. Weidner, M. Slodowski, C. Halm, C. Schneider, and L. Pfitzner, “Effective-medium model for fast evaluation of scatterometric measurements on gratings,” Proc. SPIE 5375, 232–243 (2004).
[CrossRef]

Hingst, T.

P. Reinig, R. Dost, M. Mort, T. Hingst, U. Mantz, J. Moffitt, S. Shakya, C. J. Raymond, and M. Littau, “Metrology of deep trench etched memory structures using 3D scatterometry,” Proc. SPIE 5752, 559–569 (2005).
[CrossRef]

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. Vac. Sci. Technol. B 15, 361–368(1997).
[CrossRef]

Hugonin, J. P.

Ichioka, Y.

Iwata, K.

Jackson, J. L.

J. L. Jackson and S. R. Coriell, “Transport coefficients of composite materials,” J. Appl. Phys. 39, 2349–2354 (1968).
[CrossRef]

Jakatdar, N.

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97–111 (2001).
[CrossRef]

Kämpfe, T.

Kikuta, H.

Kim, D.

Kley, E. B.

Konishi, T.

Kubo, H.

Lalanne, D. L.

P. Lalanne and D. L. Lalanne, “On the effective medium theory of subwavelength periodic structure,” J. Mod. Opt. 43, 2063–2085(1996).
[CrossRef]

Lalanne, P.

Li, L.

Littau, M.

P. Reinig, R. Dost, M. Mort, T. Hingst, U. Mantz, J. Moffitt, S. Shakya, C. J. Raymond, and M. Littau, “Metrology of deep trench etched memory structures using 3D scatterometry,” Proc. SPIE 5752, 559–569 (2005).
[CrossRef]

Liu, S. Y.

Mantz, U.

P. Reinig, R. Dost, M. Mort, T. Hingst, U. Mantz, J. Moffitt, S. Shakya, C. J. Raymond, and M. Littau, “Metrology of deep trench etched memory structures using 3D scatterometry,” Proc. SPIE 5752, 559–569 (2005).
[CrossRef]

P. Y. Guittet, U. Mantz, and P. Weidner, “Infrared spectroscopic ellipsometry in semiconductor manufacturing,” Proc. SPIE 5375, 771–778 (2004).
[CrossRef]

Maznev, A. A.

A. A. Maznev, A. Mazurenko, C. A. Durán, and M. Gostein, “Measuring trench structures for microelectronics with model-based infrared reflectometry,” AIP Conf. Proc. 931, 74–78 (2007).
[CrossRef]

C. A. Durán, A. A. Maznev, G. T. Merklin, A. Mazurenko, and M. Gostein, “Infrared reflectometry for metrology of trenches in power devices,” in Proceedings of IEEE Conference on Advanced Semiconductor Manufacturing (IEEE, 2007), pp. 175–179.

Mazurenko, A.

A. A. Maznev, A. Mazurenko, C. A. Durán, and M. Gostein, “Measuring trench structures for microelectronics with model-based infrared reflectometry,” AIP Conf. Proc. 931, 74–78 (2007).
[CrossRef]

P. A. Rosenthal, C. A. Durán, J. Tower, and A. Mazurenko, “Model-based infrared metrology for advanced technology nodes and 300 mm wafer processing,” AIP Conf. Proc. 788, 620–624 (2005).
[CrossRef]

C. A. Durán, A. A. Maznev, G. T. Merklin, A. Mazurenko, and M. Gostein, “Infrared reflectometry for metrology of trenches in power devices,” in Proceedings of IEEE Conference on Advanced Semiconductor Manufacturing (IEEE, 2007), pp. 175–179.

McNeil, J. 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. Vac. Sci. Technol. B 15, 361–368(1997).
[CrossRef]

Merklin, G. T.

C. A. Durán, A. A. Maznev, G. T. Merklin, A. Mazurenko, and M. Gostein, “Infrared reflectometry for metrology of trenches in power devices,” in Proceedings of IEEE Conference on Advanced Semiconductor Manufacturing (IEEE, 2007), pp. 175–179.

Moffitt, J.

P. Reinig, R. Dost, M. Mort, T. Hingst, U. Mantz, J. Moffitt, S. Shakya, C. J. Raymond, and M. Littau, “Metrology of deep trench etched memory structures using 3D scatterometry,” Proc. SPIE 5752, 559–569 (2005).
[CrossRef]

Moharam, M. G.

Moon, S.

Morris, G. M.

Mort, M.

P. Reinig, R. Dost, M. Mort, T. Hingst, U. Mantz, J. Moffitt, S. Shakya, C. J. Raymond, and M. Littau, “Metrology of deep trench etched memory structures using 3D scatterometry,” Proc. SPIE 5752, 559–569 (2005).
[CrossRef]

Motamedi, M. E.

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. Vac. Sci. Technol. B 15, 361–368(1997).
[CrossRef]

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. Vac. Sci. Technol. B 15, 361–368(1997).
[CrossRef]

Niu, X.

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97–111 (2001).
[CrossRef]

Ohira, Y.

Petit, R.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).

Pfitzner, L.

A. Weidner, M. Slodowski, C. Halm, C. Schneider, and L. Pfitzner, “Effective-medium model for fast evaluation of scatterometric measurements on gratings,” Proc. SPIE 5375, 232–243 (2004).
[CrossRef]

Pommet, D. A.

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. Vac. Sci. Technol. B 15, 361–368(1997).
[CrossRef]

Raguin, D. H.

Raymond, C. J.

P. Reinig, R. Dost, M. Mort, T. Hingst, U. Mantz, J. Moffitt, S. Shakya, C. J. Raymond, and M. Littau, “Metrology of deep trench etched memory structures using 3D scatterometry,” Proc. SPIE 5752, 559–569 (2005).
[CrossRef]

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. Vac. Sci. Technol. B 15, 361–368(1997).
[CrossRef]

Reinig, P.

P. Reinig, R. Dost, M. Mort, T. Hingst, U. Mantz, J. Moffitt, S. Shakya, C. J. Raymond, and M. Littau, “Metrology of deep trench etched memory structures using 3D scatterometry,” Proc. SPIE 5752, 559–569 (2005).
[CrossRef]

Rosenthal, P. A.

P. A. Rosenthal, C. A. Durán, J. Tower, and A. Mazurenko, “Model-based infrared metrology for advanced technology nodes and 300 mm wafer processing,” AIP Conf. Proc. 788, 620–624 (2005).
[CrossRef]

Rytov, S. M.

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

Schneider, C.

A. Weidner, M. Slodowski, C. Halm, C. Schneider, and L. Pfitzner, “Effective-medium model for fast evaluation of scatterometric measurements on gratings,” Proc. SPIE 5375, 232–243 (2004).
[CrossRef]

Shakya, S.

P. Reinig, R. Dost, M. Mort, T. Hingst, U. Mantz, J. Moffitt, S. Shakya, C. J. Raymond, and M. Littau, “Metrology of deep trench etched memory structures using 3D scatterometry,” Proc. SPIE 5752, 559–569 (2005).
[CrossRef]

Shi, T. L.

Slodowski, M.

A. Weidner, M. Slodowski, C. Halm, C. Schneider, and L. Pfitzner, “Effective-medium model for fast evaluation of scatterometric measurements on gratings,” Proc. SPIE 5375, 232–243 (2004).
[CrossRef]

Southwell, W. H.

Spanos, C. J.

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97–111 (2001).
[CrossRef]

Suleski, T. J.

Takahara, K.

Tang, Z. R.

Tower, J.

P. A. Rosenthal, C. A. Durán, J. Tower, and A. Mazurenko, “Model-based infrared metrology for advanced technology nodes and 300 mm wafer processing,” AIP Conf. Proc. 788, 620–624 (2005).
[CrossRef]

Tünnermann, A.

Weidner, A.

A. Weidner, M. Slodowski, C. Halm, C. Schneider, and L. Pfitzner, “Effective-medium model for fast evaluation of scatterometric measurements on gratings,” Proc. SPIE 5375, 232–243 (2004).
[CrossRef]

Weidner, P.

P. Y. Guittet, U. Mantz, and P. Weidner, “Infrared spectroscopic ellipsometry in semiconductor manufacturing,” Proc. SPIE 5375, 771–778 (2004).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

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).
[CrossRef]

Yotsuya, T.

Yu, W.

Zhang, C. W.

AIP Conf. Proc. (2)

P. A. Rosenthal, C. A. Durán, J. Tower, and A. Mazurenko, “Model-based infrared metrology for advanced technology nodes and 300 mm wafer processing,” AIP Conf. Proc. 788, 620–624 (2005).
[CrossRef]

A. A. Maznev, A. Mazurenko, C. A. Durán, and M. Gostein, “Measuring trench structures for microelectronics with model-based infrared reflectometry,” AIP Conf. Proc. 931, 74–78 (2007).
[CrossRef]

Appl. Opt. (5)

IEEE Trans. Semicond. Manuf. (1)

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97–111 (2001).
[CrossRef]

J. Appl. Phys. (1)

J. L. Jackson and S. R. Coriell, “Transport coefficients of composite materials,” J. Appl. Phys. 39, 2349–2354 (1968).
[CrossRef]

J. Mod. Opt. (1)

P. Lalanne and D. L. Lalanne, “On the effective medium theory of subwavelength periodic structure,” J. Mod. Opt. 43, 2063–2085(1996).
[CrossRef]

J. Opt. Soc. Am. A (8)

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

H. Kikuta, Y. Ohira, H. Kubo, and K. Iwata, “Effective medium theory of two-dimensional subwavelength gratings in the non-quasi-static limit,” J. Opt. Soc. Am. A 15, 1577–1585 (1998).
[CrossRef]

P. Lalanne and J. P. Hugonin, “High-order effective-medium theory of subwavelength gratings in classical mounting: application to volume holograms,” J. Opt. Soc. Am. A 15, 1843–1851 (1998).
[CrossRef]

P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598(1997).
[CrossRef]

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
[CrossRef]

C. W. Zhang, S. Y. Liu, T. L. Shi, and Z. R. Tang, “Improved model-based infrared reflectrometry for measuring deep trench structures,” J. Opt. Soc. Am. A 26, 2327–2335 (2009).
[CrossRef]

S. Moon and D. Kim, “Fitting-based determination of an effective medium of a metallic periodic structure and application to photonic crystals,” J. Opt. Soc. Am. A 23, 199–207 (2006).
[CrossRef]

B. C. Bergner, T. A. Germer, and T. J. Suleski, “Effective medium approximations for modeling optical reflectance from gratings with rough edges,” J. Opt. Soc. Am. A 27, 1083–1090 (2010).
[CrossRef]

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

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. Vac. Sci. Technol. B 15, 361–368(1997).
[CrossRef]

Opt. Lett. (1)

Opt. Rev. (1)

H. Kikuta, H. Yoshida, and K. Iwata, “Ability and limitation of effective medium theory for subwavelength gratings,” Opt. Rev. 2, 92–99 (1995).
[CrossRef]

Proc. SPIE (4)

A. Weidner, M. Slodowski, C. Halm, C. Schneider, and L. Pfitzner, “Effective-medium model for fast evaluation of scatterometric measurements on gratings,” Proc. SPIE 5375, 232–243 (2004).
[CrossRef]

P. Reinig, R. Dost, M. Mort, T. Hingst, U. Mantz, J. Moffitt, S. Shakya, C. J. Raymond, and M. Littau, “Metrology of deep trench etched memory structures using 3D scatterometry,” Proc. SPIE 5752, 559–569 (2005).
[CrossRef]

P. Y. Guittet, U. Mantz, and P. Weidner, “Infrared spectroscopic ellipsometry in semiconductor manufacturing,” Proc. SPIE 5375, 771–778 (2004).
[CrossRef]

S. Moon and D. Kim, “Investigation of an effective medium theory for metallic periodic structure,” Proc. SPIE 6128, 61281M (2006).
[CrossRef]

Sov. Phys. JETP (1)

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

Other (5)

T. C. Choi, Effective Medium Theory: Principles and Applications (Oxford University, 1999).

C. A. Durán, A. A. Maznev, G. T. Merklin, A. Mazurenko, and M. Gostein, “Infrared reflectometry for metrology of trenches in power devices,” in Proceedings of IEEE Conference on Advanced Semiconductor Manufacturing (IEEE, 2007), pp. 175–179.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

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Figures (13)

Fig. 1
Fig. 1

(a) 3D trench structure and (b) its effective optical model. The incidence beam probes on the trench structure with the incidence angle θ and azimuthal angle φ.

Fig. 2
Fig. 2

Flow chart for determining the approximate expressions of n 0 and B.

Fig. 3
Fig. 3

FD discrete data (circles and squares) and the best-fitted polynomial curves (solid and dashed curves) of (a) zeroth-order effective refractive, (b) second-order term coefficient, and (c) second- order effective refractive in the FD-EMA formulation for TE and TM polarization.

Fig. 4
Fig. 4

Reflectance spectra calculated by RCWA, the FD-EMA-based model, the zeroth-order EMA-based model, and second-order EMA at the void fractions of (a) 20% in TE polarization, (b) 20% in TM polarization, (c) 40% in TE polarization, (d) 40% in TM polarization, (e) 60% in TE polarization, (f) 60% in TM polarization, (g) 80% in TE polarization, and (h) 80% in TM polarization. The trench parameters for the simulation: trench depth is 4 μm and trench pitch is 0.2 μm .

Fig. 5
Fig. 5

Zeroth-order refractive index n 0 and the second-order term coefficient B in different trench depths calculated by the fitting-based method. The void fraction f is set at 20%, 40%, 60%, and 80%, respectively.

Fig. 6
Fig. 6

Zeroth-order effective refractive index n 0 and the second-order term coefficient B in different trench pitches calculated by the fitting-based method. The void fraction f is set at 20%, 40%, 60%, and 80%, respectively.

Fig. 7
Fig. 7

Zeroth-order refractive index n 0 and the second-order term coefficient B with variable incidence angle calculated by the fitting-based method. The void fraction f is set at 20%, 40%, 60%, and 80%, respectively.

Fig. 8
Fig. 8

AARE of the FD-EMA-based model as a function of the trench depth at the void fractions of 20%, 40%, 60%, and 80%.

Fig. 9
Fig. 9

AARE of the FD-EMA-based model as a function of the trench pitch at the void fractions of 20%, 40%, 60%, and 80%.

Fig. 10
Fig. 10

AARE of the FD-EMA-based model as a function of the incidence angle at the void fractions of 20%, 40%, 60%, and 80%.

Fig. 11
Fig. 11

AARE of the FD-EMA-based model as a function of the void fraction for TE and TM polarization.

Fig. 12
Fig. 12

(a) Structure of the 3D taper trenches etched on the silicon substrate, (b) its effective medium model, and (c) its effective refractive index versus the depth.

Fig. 13
Fig. 13

Measured data of the taper trench simulated by RCWA and the best-fit spectra with zeroth-order EMA and the FD-EMA-based model. The trench parameters for the simulation: trench depth is 3 μm , top CD is 0.1 μm , bottom CD is 0.075 μm , and trench pitch is 0.18 μm .

Tables (3)

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Table 1 Polynomial Coefficients of the Best-Fitting Curve of the Zeroth-Order Effective Refractive Index in TE and TM Polarization

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Table 2 Polynomial Coefficients of the Best-Fitting Curve of the Second-Order Term Coefficient in TE and TM Polarization

Tables Icon

Table 3 Comparison of the Extraction Error of Zeroth-Order EMA and the FD-EMA-Based Model

Equations (12)

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ε 0 = ( 1 f ) ε g + f ε m ,
ε 0 = ε g ε m f ε g + ( 1 f ) ε m ,
ε 2 = ε 0 + π 2 3 f 2 ( 1 f ) 2 ( ε m ε g ) 2 ( Λ λ ) 2 ,
ε 2 = ε 0 + π 2 3 f 2 ( 1 f ) 2 ( 1 ε m 1 ε g ) 2 ( ε 0 ) 3 ε 0 ( Λ λ ) 2 ,
ε = ε 1 [ 1 f 2 ( ε g ε m ) ε g + ε m + f ( ε g ε m ) ] .
ε eff = ε 0 + ε 1 α 1 + ε 2 α 2 + ,
n eff = n 0 2 + B ( Λ λ ) 2 ,
R s = | r 1 e j δ + r 2 e j δ e j δ + r 1 r 2 e j δ | 2 ,
r 1 = n i cos θ 1 n eff cos θ 2 n i cos θ 1 + n eff cos θ 2 , r 2 = n eff cos θ 2 n s cos θ 3 n eff cos θ 2 + n s cos θ 3 ,
δ = 2 π H n eff cos θ 2 / λ .
n 0 = a 0 + a 1 f + a 2 f 2 + + a j f j B = b 0 + b 1 f + b 2 f 2 + + b k f k ,
AARE = i = 1 N | [ R r ( λ i ) R s ( λ i ) ] / R r ( λ i ) | N × 100 % ,

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