Abstract

We propose a new, to our knowledge, method for determining the two main critical parameters of periodic one-dimensional lamellar structures, namely, linewidths and etched depths. The method is simple and requires only two measurements for the phase of the zero-transmitted order under two orthogonal polarizations. It is inspired by the analogy between subwavelength gratings and anisotropic homogeneous thin films. The method is tested with experimental data obtained with a Mach–Zehnder interferometer. Etched depths and linewidths derived from the interferograms and electromagnetic theory are compared with scanning-electron-microscope observations.

© 1999 Optical Society of America

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  1. A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).
  2. A. Roger, M. Breidne, “Grating profile reconstruction by an inverse scattering method,” Opt. Commun. 35, 299–302 (1980).
    [CrossRef]
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    [CrossRef]
  4. N. Chateau, J. C. Saget, P. Chavel, “Diffraction analysis and experimental investigation of reflection-free holographic phase gratings,” Pure Appl. Opt. 2, 299–314 (1993).
    [CrossRef]
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  6. H. Giovannini, C. Deumié, H. Akhouayri, C. Amra, “Angle-resolved polarimetric phase measurement for the characterization of gratings,” Opt. Lett. 21, 1619–1621 (1996).
    [CrossRef] [PubMed]
  7. S. S. H. Naqvi, J. R. McNeil, R. H. Krukar, K. P. Bishop, “Scatterometry and the simulation of diffraction based metrology,” Microlith. World 2, 5–16 (1993).
  8. K. P. Giapis, R. A. Gottscho, L. A. Clark, J. B. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
    [CrossRef]
  9. N. Blayo, R. A. Cirelli, F. P. Klemens, J. T. C. Lee, “Ultraviolet-visible ellipsometry for process control during the etching of submicrometer features,” J. Opt. Soc. Am. A 12, 591–599 (1995).
    [CrossRef]
  10. M. K. Minhas, S. A. Coulombe, S. S. H. Naqvi, J. R. McNeil, “Ellipsometric scatterometry for the metrology of sub-0.1-µm-linewidth structures,” Appl. Opt. 37, 5112–5115 (1998).
    [CrossRef]
  11. G. Bouchitté, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
    [CrossRef]
  12. Strictly speaking, the Airy formula for thin films has to be used.
  13. Ph. Lalanne, S. Astilean, P. Chavel, E. Cambril, H. Launois, “Blazed-binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings,” Opt. Lett. 23, 1081–1083 (1998).
    [CrossRef]
  14. For the sake of illustration, see Fig. 1 in Ref. 13, where it is shown that, for periods larger than the structural cutoff, the phase of the zero order is not a monotonous function of the fill factor and exhibits a rather chaotic behavior.
  15. Ph. Lalanne, D. Lemercier-Lalanne, “Depth dependence of the effective properties of subwavelength gratings,” J. Opt. Soc. Am. A 14, 450–458 (1997).
    [CrossRef]
  16. Ph. Lalanne, J. Hazart, P. Chavel, E. Cambril, H. Launois, “A transmission polarizing beam splitter grating,” J. Opt. A: Pure Appl. Opt. 1, 215–219 (1999).
    [CrossRef]
  17. Strictly speaking, the phase shifts obtained from the interferograms are known modulo 2π. The indetermination is removed in practice by consideration of the small finite set of possible 2π phase jumps and retention of only physical solutions. If one knows approximately the etched depth (as is the case for most characterization problems) or if in situ monitoring is considered, there is no ambiguity. In the latter case the absolute phase shift is observed in real time.
  18. M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  19. Ph. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  20. For the SEM linewidth measurements the Au film deposited on the sample is a source of error. The additional thickness is difficult to estimate. However, because the gold coating of the vertical walls appears brigther and scummy in the SEM photographs, the additional thickness can be partially removed during estimation of the linewidth.
  21. F. C. Chen, W. C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett. 72, 3080–3082 (1998).
    [CrossRef]

1999

Ph. Lalanne, J. Hazart, P. Chavel, E. Cambril, H. Launois, “A transmission polarizing beam splitter grating,” J. Opt. A: Pure Appl. Opt. 1, 215–219 (1999).
[CrossRef]

1998

1997

1996

1995

1994

1993

N. Chateau, J. C. Saget, P. Chavel, “Diffraction analysis and experimental investigation of reflection-free holographic phase gratings,” Pure Appl. Opt. 2, 299–314 (1993).
[CrossRef]

S. S. H. Naqvi, J. R. McNeil, R. H. Krukar, K. P. Bishop, “Scatterometry and the simulation of diffraction based metrology,” Microlith. World 2, 5–16 (1993).

1991

K. P. Giapis, R. A. Gottscho, L. A. Clark, J. B. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

1985

G. Bouchitté, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
[CrossRef]

1980

A. Roger, M. Breidne, “Grating profile reconstruction by an inverse scattering method,” Opt. Commun. 35, 299–302 (1980).
[CrossRef]

Akhouayri, H.

Amra, C.

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Astilean, S.

Azzam, R. M. A.

Bishop, K. P.

S. S. H. Naqvi, J. R. McNeil, R. H. Krukar, K. P. Bishop, “Scatterometry and the simulation of diffraction based metrology,” Microlith. World 2, 5–16 (1993).

Blayo, N.

Bouchitté, G.

G. Bouchitté, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
[CrossRef]

Breidne, M.

A. Roger, M. Breidne, “Grating profile reconstruction by an inverse scattering method,” Opt. Commun. 35, 299–302 (1980).
[CrossRef]

Cambril, E.

Ph. Lalanne, J. Hazart, P. Chavel, E. Cambril, H. Launois, “A transmission polarizing beam splitter grating,” J. Opt. A: Pure Appl. Opt. 1, 215–219 (1999).
[CrossRef]

Ph. Lalanne, S. Astilean, P. Chavel, E. Cambril, H. Launois, “Blazed-binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings,” Opt. Lett. 23, 1081–1083 (1998).
[CrossRef]

Chateau, N.

N. Chateau, J. C. Saget, P. Chavel, “Diffraction analysis and experimental investigation of reflection-free holographic phase gratings,” Pure Appl. Opt. 2, 299–314 (1993).
[CrossRef]

Chavel, P.

Ph. Lalanne, J. Hazart, P. Chavel, E. Cambril, H. Launois, “A transmission polarizing beam splitter grating,” J. Opt. A: Pure Appl. Opt. 1, 215–219 (1999).
[CrossRef]

Ph. Lalanne, S. Astilean, P. Chavel, E. Cambril, H. Launois, “Blazed-binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings,” Opt. Lett. 23, 1081–1083 (1998).
[CrossRef]

N. Chateau, J. C. Saget, P. Chavel, “Diffraction analysis and experimental investigation of reflection-free holographic phase gratings,” Pure Appl. Opt. 2, 299–314 (1993).
[CrossRef]

Chen, F. C.

F. C. Chen, W. C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett. 72, 3080–3082 (1998).
[CrossRef]

Chew, W. C.

F. C. Chen, W. C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett. 72, 3080–3082 (1998).
[CrossRef]

Cirelli, R. A.

Clark, L. A.

K. P. Giapis, R. A. Gottscho, L. A. Clark, J. B. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

Coulombe, S. A.

Cui, Y.

Deumié, C.

Franke, J. E.

Gaylord, T. K.

Giapis, K. P.

K. P. Giapis, R. A. Gottscho, L. A. Clark, J. B. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

Giovannini, H.

Gottscho, R. A.

S. S. H. Naqvi, R. H. Krukar, J. R. McNeil, J. E. Franke, T. M. Niemczyk, D. M. Haaland, R. A. Gottscho, A. Kornblit, “Etch depth estimation of large-period silicon gratings with multivariate calibration of rigorously simulated diffraction gratings,” J. Opt. Soc. Am. A 11, 2485–2493 (1994).
[CrossRef]

K. P. Giapis, R. A. Gottscho, L. A. Clark, J. B. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

Grann, E. B.

Haaland, D. M.

Hazart, J.

Ph. Lalanne, J. Hazart, P. Chavel, E. Cambril, H. Launois, “A transmission polarizing beam splitter grating,” J. Opt. A: Pure Appl. Opt. 1, 215–219 (1999).
[CrossRef]

Klemens, F. P.

Kornblit, A.

S. S. H. Naqvi, R. H. Krukar, J. R. McNeil, J. E. Franke, T. M. Niemczyk, D. M. Haaland, R. A. Gottscho, A. Kornblit, “Etch depth estimation of large-period silicon gratings with multivariate calibration of rigorously simulated diffraction gratings,” J. Opt. Soc. Am. A 11, 2485–2493 (1994).
[CrossRef]

K. P. Giapis, R. A. Gottscho, L. A. Clark, J. B. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

Krukar, R. H.

Kruskal, J. B.

K. P. Giapis, R. A. Gottscho, L. A. Clark, J. B. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

Lalanne, Ph.

Lambert, D.

K. P. Giapis, R. A. Gottscho, L. A. Clark, J. B. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

Launois, H.

Ph. Lalanne, J. Hazart, P. Chavel, E. Cambril, H. Launois, “A transmission polarizing beam splitter grating,” J. Opt. A: Pure Appl. Opt. 1, 215–219 (1999).
[CrossRef]

Ph. Lalanne, S. Astilean, P. Chavel, E. Cambril, H. Launois, “Blazed-binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings,” Opt. Lett. 23, 1081–1083 (1998).
[CrossRef]

Lee, J. T. C.

Lemercier-Lalanne, D.

McNeil, J. R.

Minhas, M. K.

Moharam, M. G.

Morris, G. M.

Naqvi, S. S. H.

Niemczyk, T. M.

Petit, R.

G. Bouchitté, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
[CrossRef]

Pommet, D. A.

Roger, A.

A. Roger, M. Breidne, “Grating profile reconstruction by an inverse scattering method,” Opt. Commun. 35, 299–302 (1980).
[CrossRef]

Saget, J. C.

N. Chateau, J. C. Saget, P. Chavel, “Diffraction analysis and experimental investigation of reflection-free holographic phase gratings,” Pure Appl. Opt. 2, 299–314 (1993).
[CrossRef]

Sinatore, D.

K. P. Giapis, R. A. Gottscho, L. A. Clark, J. B. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Appl. Opt.

Appl. Phys. Lett.

F. C. Chen, W. C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett. 72, 3080–3082 (1998).
[CrossRef]

Electromagnetics

G. Bouchitté, R. Petit, “Homogenization techniques as applied in the electromagnetic theory of gratings,” Electromagnetics 5, 17–36 (1985).
[CrossRef]

J. Opt. A: Pure Appl. Opt.

Ph. Lalanne, J. Hazart, P. Chavel, E. Cambril, H. Launois, “A transmission polarizing beam splitter grating,” J. Opt. A: Pure Appl. Opt. 1, 215–219 (1999).
[CrossRef]

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. A

K. P. Giapis, R. A. Gottscho, L. A. Clark, J. B. Kruskal, D. Lambert, A. Kornblit, D. Sinatore, “Use of light scattering in characterizing reactively ion etched profiles,” J. Vac. Sci. Technol. A 9, 664–668 (1991).
[CrossRef]

Microlith. World

S. S. H. Naqvi, J. R. McNeil, R. H. Krukar, K. P. Bishop, “Scatterometry and the simulation of diffraction based metrology,” Microlith. World 2, 5–16 (1993).

Opt. Commun.

A. Roger, M. Breidne, “Grating profile reconstruction by an inverse scattering method,” Opt. Commun. 35, 299–302 (1980).
[CrossRef]

Opt. Lett.

Pure Appl. Opt.

N. Chateau, J. C. Saget, P. Chavel, “Diffraction analysis and experimental investigation of reflection-free holographic phase gratings,” Pure Appl. Opt. 2, 299–314 (1993).
[CrossRef]

Other

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).

For the sake of illustration, see Fig. 1 in Ref. 13, where it is shown that, for periods larger than the structural cutoff, the phase of the zero order is not a monotonous function of the fill factor and exhibits a rather chaotic behavior.

Strictly speaking, the Airy formula for thin films has to be used.

Strictly speaking, the phase shifts obtained from the interferograms are known modulo 2π. The indetermination is removed in practice by consideration of the small finite set of possible 2π phase jumps and retention of only physical solutions. If one knows approximately the etched depth (as is the case for most characterization problems) or if in situ monitoring is considered, there is no ambiguity. In the latter case the absolute phase shift is observed in real time.

For the SEM linewidth measurements the Au film deposited on the sample is a source of error. The additional thickness is difficult to estimate. However, because the gold coating of the vertical walls appears brigther and scummy in the SEM photographs, the additional thickness can be partially removed during estimation of the linewidth.

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Figures (4)

Fig. 1
Fig. 1

TE–TM phase-shift ratio in the static limit as a function of the fill factor for a grating etched in a material with a relative permittivity ∊ = 5.29.

Fig. 2
Fig. 2

Interferograms obtained for one of the gratings (a) TE polarization (b) TM polarization. Interferograms are approximately 500 µm × 500 µm square. The phase shift observed between the two interferograms is a direct observation of the form birefringence of the subwavelength grating. For ease of interferogram exploitation note that nonlinear postprocessing was applied to the CCD pictures, which otherwise show regular sinusoidal patterns.

Fig. 3
Fig. 3

Graphical resolution for the inverse problem and for the grating fabricated with a dose deviation of 60%. Note the dashed-curve bend near f = 0.55. This nonmonotonous behavior may be responsible for nonunique solutions. We believe that this is because the grating period considered here is larger than the cutoff and the structural cutoff values. Dashed curve, ΦTE(f, h) - ΦTEmeas = 0. Solid curve, ΦTM(f, h) - ΦTMmeas = 0.

Fig. 4
Fig. 4

SEM picture for estimating the grating depth and for validating the lamellar grating assumption.

Tables (1)

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Table 1 Experimental Dataa

Equations (4)

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ϕTE=2π/λnTE-1h
ϕTM=2π/λnTM-1h.
ΦTEf, h-ΦTEmeas=0,
ΦTMf, h-ΦTMmeas=0,

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