Abstract

Eigenvalue computation is an important part of many modal diffraction methods, including the rigorous coupled wave approach (RCWA) and the Chandezon method. This procedure is known to be computationally intensive, accounting for a large proportion of the overall run time. However, in many cases, eigenvalue information is already available from previous calculations. Some of the examples include adjacent slices in the RCWA, spectral- or angle-resolved scans in optical scatterometry and parameter derivatives in optimization. In this paper, we present a new technique that provides accurate and highly reliable solutions with significant improvements in computational time. The proposed method takes advantage of known eigensolution information and is based on perturbation method.

© 2011 Optical Society of America

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  1. M. G. Moharam, D. A. Pommet, and E. B. Grann, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  2. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767(1997).
    [CrossRef]
  3. J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235–241 (1980).
    [CrossRef]
  4. G. Granet, “Analysis of diffraction by surface-relief crossed gratings with the use of the Chandezon method: application to multilayer crossed gratings,” J. Opt. Soc. Am. A 15, 1121–1130(1998).
    [CrossRef]
  5. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  6. C. Zhou and L. Li, “Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings,” J. Opt. A 6, 43–50 (2004).
    [CrossRef]
  7. B. Bai and L. Li, “Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with square symmetry,” J. Opt. Soc. Am. A 23, 572–580 (2006).
    [CrossRef]
  8. E. Popov and M. Nevière, “Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894(2001).
    [CrossRef]
  9. T. Schuster, J. Ruoff, N. Kerwien, S. Raffler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A 24, 2880–2890 (2007).
    [CrossRef]
  10. J. Bischoff, “Formulation of the normal vector RCWA for symmetric crossed gratings in symmetric mountings,” J. Opt. Soc. Am. A 27, pp. 1024–1031 (2010).
    [CrossRef]
  11. C. J. Raymond, S. S. H. Naqvi, and J. R. McNeil, “Resist and etched line profile characterization using scatterometry,” Proc. SPIE 3050, 476–486 (1997).
    [CrossRef]
  12. X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE 3677, 159–168 (1999).
    [CrossRef]
  13. C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, and J. R. McNeil, “Multiparameter grating metrology using optical scatterometry,” J. Vac. Sci. Technol. B 15, 361–368 (1997).
    [CrossRef]
  14. N. P. van der Aa and R. M. M. Mattheij, “Computing shape parameter sensitivity of the field of one-dimensional surface-relief gratings by using an analytical approach based on RCWA,” J. Opt. Soc. Am. A 24, 2692–2700 (2007).
    [CrossRef]
  15. J. Bischoff, K. Hehl, X. Niu, and W. Jin, “Approximating eigensolutions for use in determining the profile of a structure formed on a semiconductor wafer,” U.S. Patent 7,630,873 (8 December 2009).
  16. E. J. Hinch, Perturbation Methods, Cambridge Texts in Applied Mathematics (Cambridge University, 1992).
  17. K. Edee, J. P. Plumey, G. Granet, and J. Hazart, “Perturbation method for the rigorous coupled wave analysis of grating diffraction,” Opt. Express 18, 26274–26284 (2010).
    [CrossRef] [PubMed]

2010

2007

2006

2004

C. Zhou and L. Li, “Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings,” J. Opt. A 6, 43–50 (2004).
[CrossRef]

2001

1999

X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE 3677, 159–168 (1999).
[CrossRef]

1998

1997

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

C. J. Raymond, S. S. H. Naqvi, and J. R. McNeil, “Resist and etched line profile characterization using scatterometry,” Proc. SPIE 3050, 476–486 (1997).
[CrossRef]

1996

1995

1992

E. J. Hinch, Perturbation Methods, Cambridge Texts in Applied Mathematics (Cambridge University, 1992).

1980

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

Bai, B.

Bao, J.

X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE 3677, 159–168 (1999).
[CrossRef]

Bischoff, J.

J. Bischoff, K. Hehl, X. Niu, and W. Jin, “Approximating eigensolutions for use in determining the profile of a structure formed on a semiconductor wafer,” U.S. Patent 7,630,873 (8 December 2009).

J. Bischoff, “Formulation of the normal vector RCWA for symmetric crossed gratings in symmetric mountings,” J. Opt. Soc. Am. A 27, pp. 1024–1031 (2010).
[CrossRef]

Chandezon, J.

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

Edee, K.

Granet, G.

Grann, E. B.

Hazart, J.

Hehl, K.

J. Bischoff, K. Hehl, X. Niu, and W. Jin, “Approximating eigensolutions for use in determining the profile of a structure formed on a semiconductor wafer,” U.S. Patent 7,630,873 (8 December 2009).

Hinch, E. J.

E. J. Hinch, Perturbation Methods, Cambridge Texts in Applied Mathematics (Cambridge University, 1992).

Jakatdar, N.

X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE 3677, 159–168 (1999).
[CrossRef]

Jin, W.

J. Bischoff, K. Hehl, X. Niu, and W. Jin, “Approximating eigensolutions for use in determining the profile of a structure formed on a semiconductor wafer,” U.S. Patent 7,630,873 (8 December 2009).

Kerwien, N.

Li, L.

Mattheij, R. M. M.

Maystre, D.

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

McNeil, J. R.

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

C. J. Raymond, S. S. H. Naqvi, and J. R. McNeil, “Resist and etched line profile characterization using scatterometry,” Proc. SPIE 3050, 476–486 (1997).
[CrossRef]

Moharam, M. G.

Murnane, M. R.

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

C. J. Raymond, S. S. H. Naqvi, and J. R. McNeil, “Resist and etched line profile characterization using scatterometry,” Proc. SPIE 3050, 476–486 (1997).
[CrossRef]

Nevière, M.

Niu, X.

J. Bischoff, K. Hehl, X. Niu, and W. Jin, “Approximating eigensolutions for use in determining the profile of a structure formed on a semiconductor wafer,” U.S. Patent 7,630,873 (8 December 2009).

X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE 3677, 159–168 (1999).
[CrossRef]

Osten, W.

Plumey, J. P.

Pommet, D. A.

Popov, E.

Prins, S. L.

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

Raffler, S.

Raoult, G.

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

Raymond, C. J.

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

C. J. Raymond, S. S. H. Naqvi, and J. R. McNeil, “Resist and etched line profile characterization using scatterometry,” Proc. SPIE 3050, 476–486 (1997).
[CrossRef]

Ruoff, J.

Schuster, T.

Spanos, C.

X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE 3677, 159–168 (1999).
[CrossRef]

van der Aa, N. P.

Yedur, S.

X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE 3677, 159–168 (1999).
[CrossRef]

Zhou, C.

C. Zhou and L. Li, “Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings,” J. Opt. A 6, 43–50 (2004).
[CrossRef]

J. Opt.

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

J. Opt. A

C. Zhou and L. Li, “Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings,” J. Opt. A 6, 43–50 (2004).
[CrossRef]

J. Opt. Soc. Am. A

G. Granet, “Analysis of diffraction by surface-relief crossed gratings with the use of the Chandezon method: application to multilayer crossed gratings,” J. Opt. Soc. Am. A 15, 1121–1130(1998).
[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]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

M. G. Moharam, D. A. Pommet, and E. B. Grann, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
[CrossRef]

E. Popov and M. Nevière, “Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894(2001).
[CrossRef]

B. Bai and L. Li, “Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with square symmetry,” J. Opt. Soc. Am. A 23, 572–580 (2006).
[CrossRef]

N. P. van der Aa and R. M. M. Mattheij, “Computing shape parameter sensitivity of the field of one-dimensional surface-relief gratings by using an analytical approach based on RCWA,” J. Opt. Soc. Am. A 24, 2692–2700 (2007).
[CrossRef]

T. Schuster, J. Ruoff, N. Kerwien, S. Raffler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A 24, 2880–2890 (2007).
[CrossRef]

J. Bischoff, “Formulation of the normal vector RCWA for symmetric crossed gratings in symmetric mountings,” J. Opt. Soc. Am. A 27, pp. 1024–1031 (2010).
[CrossRef]

J. Vac. Sci. Technol. B

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

Opt. Express

Proc. SPIE

C. J. Raymond, S. S. H. Naqvi, and J. R. McNeil, “Resist and etched line profile characterization using scatterometry,” Proc. SPIE 3050, 476–486 (1997).
[CrossRef]

X. Niu, N. Jakatdar, J. Bao, C. Spanos, and S. Yedur, “Specular spectroscopic scatterometry in DUV lithography,” Proc. SPIE 3677, 159–168 (1999).
[CrossRef]

Other

J. Bischoff, K. Hehl, X. Niu, and W. Jin, “Approximating eigensolutions for use in determining the profile of a structure formed on a semiconductor wafer,” U.S. Patent 7,630,873 (8 December 2009).

E. J. Hinch, Perturbation Methods, Cambridge Texts in Applied Mathematics (Cambridge University, 1992).

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Figures (14)

Fig. 1
Fig. 1

Crossed grating setup for the numerical examples.

Fig. 2
Fig. 2

Zero-order reflection for a crossed grating versus the AOI. The graph shows the exact RCWA results in comparison with perturbation computations with and without degeneration healing for TE and TM polarization.

Fig. 3
Fig. 3

Zero-order reflection for a crossed grating with square holes versus hole width. The graph shows the exact RCWA result in comparison with perturbation computations with and without degeneration healing. TE- and TM-polarization results coincide due to the symmetry.

Fig. 4
Fig. 4

Zero-order reflection for a dielectric crossed grating with rectangular holes versus wavelength at normal incidence. The diagram shows the exact RCWA result in comparison with perturbation computations with and without degeneration healing.

Fig. 5
Fig. 5

Zero-order reflection (TE polarization) for a dielectric ( n = 1.5 ) and an absorbing crossed grating ( n = 3.72 j 0.0061 ) with rectangular holes versus grating period perturbation for oblique incidence ( angle = 30 ° , λ = 400 nm ). The diagram shows the exact RCWA result in comparison with perturbation computations with and without degeneration healing.

Fig. 6
Fig. 6

Zero-order reflection (TE polarization) for a crossed grating with rectangular holes versus perturbation of the refractive index n (k fixed at 0.0061 ) for oblique incidence ( angle = 30 ° , λ = 750 nm ). The diagram shows the exact RCWA result in comparison with perturbation computations with and without degeneration healing.

Fig. 7
Fig. 7

Zero-order reflection (TM polarization) for a metallic crossed grating with rectangular holes versus perturbation of the complex refractive index (n, k ranging from 0.5 j 5 ± 2 % ) for oblique incidence ( angle = 30 ° , λ = 750 nm ). The diagram shows the exact RCWA result in comparison with perturbation computations with and without degeneration healing.

Fig. 8
Fig. 8

Simulation of a spectral scan (step width = 5 nm ). The incremental perturbation approach (second order with healing) is compared with a regular RCWA result. The light is incident normal on a slightly absorbing grating made of square holes etched 50 nm into a material with n = 3.721087 0.006151 .

Fig. 9
Fig. 9

Relative error of the perturbation solution related to the regular solution for the example in Fig. 8. Parameter is the wavelength step width for spectrum calculation. The visible deviations in Fig. 8 are related to significant relative errors at about 320 and 380 nm . A decreased step size can help to reduce these errors drastically.

Fig. 10
Fig. 10

Relative error of the perturbation solution related to the regular solution for the example in Fig. 8. Parameter is the threshold value ε for the cluster healing. The wavelength step size was 5 nm .

Fig. 11
Fig. 11

Relative error of the perturbation solution related to the regular solution for the simulation of an angular scan. Starting reference was normal incidence. Parameter is the angular step width ( 1 ° , 2 ° , and 4 ° ).

Fig. 12
Fig. 12

Relative cost (time ratio) R CE [see Eq. (22)] for cluster diagonalization versus full diagonalization for the incidence angle perturbation (example of Subsection 3A) and for the hole width perturbation (example of Subsection 3B).

Fig. 13
Fig. 13

Relative error and modified relative cost R CE for the perturbation of the grating period (top) (cf. Subsection 3D) and the incidence angle (cf. Subsection 3A).

Fig. 14
Fig. 14

Relative cost (time ratio) R CE (22) for the example of Subsection 3G. Parameter is the angular increment.

Equations (22)

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

D = W 1 · A · W .
D 0 = W 0 1 · A 0 · W 0 .
A = A 0 + Δ A .
W = W 0 · Δ W .
D = Δ W 1 · D 0 · Δ W + Δ W 1 · W 0 1 · Δ A · W 0 · Δ W .
Δ W = exp ( X ) .
Δ W 1 + X ,
D D 1 = N [ D 0 , X ] = N X · D 0 + D 0 · X ,
N = D 0 + W 0 1 · Δ A · W 0 .
D 1 , i i = N i i = D 0 , i i + Δ D i i = D 0 , i i + ( W 0 1 · Δ A · W 0 ) i i ,
N i j ( [ D 0 , X ] ) i j = 0 ,
X i j = N i j / ( D 0 , i i D 0 , j j ) .
Δ W 1 + X + X 2 / 2 ,
| N i j | < ε · | D 0 , i i D 0 , j j |
Δ W · D = N · Δ W .
N = ( N i i N i j N j i N j j ) , Δ W = ( Δ W i i Δ W i j Δ W j i Δ W j j ) .
D 11 , 22 = N 11 + N 22 2 ± ( N 11 N 22 ) 2 4 + N 12 N 21 N ¯ ± Δ .
Δ = δ 2 + N 12 · N 21 , δ = N 11 N 22 2 .
( Δ W 11 Δ W 12 Δ W 21 Δ W 22 ) = ( 1 ( δ Δ ) / N 21 N 21 / ( δ + Δ ) 1 ) .
( Δ W 11 Δ W 12 Δ W 21 Δ W 22 ) 1 = 1 1 + ( Δ δ ) / ( Δ + δ ) ( 1 ( Δ δ ) / N 21 N 21 / ( Δ + δ ) 1 ) .
gain = t t R / t t P = 1 + t O t E t P t E + t O t E .
R CE = t CE / t E = i = 1 N M ci 3 M 3 ,

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