Abstract

A polarization interferometric method is presented for the quantitative microscopy of topographical structures with subwavelength linewidths. A liquid-crystal phase shifter is inserted into the imaging optics of a reflected-light microscope, and the principles of phase-shifting interferometry are applied to measuring the phase and the contrast of the TE-polarized image (E parallel edge) with the TM-polarized image (E perpendicular edge) as the reference. This common-path interferometric method provides selective edge detection for line structures because the polarization difference is localized at the structure edges. Two different threshold criteria for linewidth determination are discussed: distance of the contrast minima and distance of the points of the steepest phase change. Linewidths as small as 300 nm were measured at a 635-nm wavelength. The dependence on the illumination numerical aperture, as well as on the material, the width, and the depth of the structure, is investigated both experimentally and by rigorous numerical simulations.

© 2000 Optical Society of America

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References

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  1. D. Nyyssonen, “Linewidth measurement with an optical microscope: the effect of operating conditions on the image profile,” Appl. Opt. 16, 2223–2230 (1977).
    [CrossRef] [PubMed]
  2. D. Nyyssonen, “Practical method for edge detection and focusing for linewidth measurements on wafers,” Opt. Eng. 26, 81–85 (1987).
    [CrossRef]
  3. D. Nyyssonen, R. D. Larrabee, “Submicrometer linewidth metrology in the optical microscope,” J. Res. Nat. Bur. Stand. 92, 187–204 (1987).
    [CrossRef]
  4. K.-P. Schröder, W. Mirandé, H. Geuther, C. Herrmann, “In quest of nanometer accuracy: supporting optical metrology by rigorous diffraction theory and AFM topography,” Opt. Commun. 115, 568–575 (1995).
    [CrossRef]
  5. For instance, the Leica Model LMS IPRO.
  6. W. Mirandé, C. G. Fraase, “Comparison of linewidth measurements on Si structures performed by atomic force microscopy (AFM) and low voltage scanning electron microscopy (SEM),” in Proceedings of the Third Seminar on Quantitative Microscopy, K. Hasche, W. Mirandé, G. Wilkening, eds. (Physikalisch-Technischen-Bundesanstalt, Braunschweig, Germany, 1998), PTB-Bericht F-34, pp. 89–96.
  7. D. W. Pohl, “Nano-optics and scanning near-field optical microscopy,” in Scanning Tunneling Microscopy II, 2nd. ed., R. Wiesendanger, H.-J. Güntherodt, eds., Vol. 28 of Springer Series in Surface Sciences (Springer-Verlag, Berlin, 1995), pp. 235–271.
  8. D. Sarid, Scanning Force Microscopy, Oxford Series in Optical and Imaging Sciences (Oxford U. Press, Oxford, 1994).
  9. M. Totzeck, H. J. Tiziani, “Interference microscopy of sub-λ structures: a rigorous computation method and measurements,” Opt. Commun. 136, 61–74 (1997).
    [CrossRef]
  10. K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
    [CrossRef]
  11. R. Barakat, “Optical linewidth measurements using a polarized microscope with crossed polarizers,” Appl. Opt. 29, 5038–5039 (1990).
    [CrossRef] [PubMed]
  12. T. R. Corle, G. S. Kino, Confocal Scanning Optical Microscopy (Academic, San Diego, 1996), pp. 286–294.
  13. S. Kimura, T. Wilson, “Confocal scanning dark-field polarization microscopy,” Appl. Opt. 33, 1274–1278 (1994).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  17. K. Leonhardt, H. J. Jordan, H. J. Tiziani, “Micro-ellipso-height-profilometry,” Opt. Commun. 80, 205–209 (1991).
    [CrossRef]
  18. K. Ramesh, V. Ganapathy, “Phase-shifting methodologies in photoelastic analysis—the application of the Jones calculus,” J. Strain Anal. 31, 423–432 (1996).
    [CrossRef]
  19. A. Asundi, L. Tong, C. G. Boay, “Phase-shifting method with a normal polariscope,” Appl. Opt. 38, 5931–5935 (1999).
    [CrossRef]
  20. J. W. Jaronski, H. T. Kasprzak, “Generalized algorithm for photoelastic measurements based on phase-stepping imaging polarimetry,” Appl. Opt. 38, 7018–7025 (1999).
    [CrossRef]
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    [CrossRef]
  22. D. S. Marx, D. Psaltis, “Polarization quadrature measurement of subwavelength diffracting structures,” Appl. Opt. 36, 6434–6440 (1997).
    [CrossRef]
  23. R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
    [CrossRef] [PubMed]
  24. R. Oldenbourg, “Analysis of edge birefringence,” Biophys. J. 60, 629–641 (1991).
    [CrossRef] [PubMed]
  25. M. Totzeck, H. J. Tiziani, “Phase-shifting polarization interferometry for microstructure linewidth measurements,” Opt. Lett. 24, 294–296 (1999).
    [CrossRef]
  26. M. Totzeck, H. Jacobsen, H. J. Tiziani, “Phase-shifting polarization interferometry for microstructure inspection,” in Interferometry ’99: Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 75–85 (1999).
  27. J. Schmit, K. Creath, “Window function influence on phase error in phase-shifting algorithms,” Appl. Opt. 35, 5642–5649 (1996).
    [CrossRef] [PubMed]
  28. M. Totzeck, M. A. Krumbügel, “Lateral resolution in the near-field and far-field phase images of π-phase-shifting structures,” Opt. Commun. 112, 189–200 (1994).
    [CrossRef]
  29. 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]
  30. M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “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]
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    [CrossRef]
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    [CrossRef]
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1999

1997

1996

1995

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]

M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “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]

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

K.-P. Schröder, W. Mirandé, H. Geuther, C. Herrmann, “In quest of nanometer accuracy: supporting optical metrology by rigorous diffraction theory and AFM topography,” Opt. Commun. 115, 568–575 (1995).
[CrossRef]

1994

S. Kimura, T. Wilson, “Confocal scanning dark-field polarization microscopy,” Appl. Opt. 33, 1274–1278 (1994).
[CrossRef] [PubMed]

M. Totzeck, M. A. Krumbügel, “Lateral resolution in the near-field and far-field phase images of π-phase-shifting structures,” Opt. Commun. 112, 189–200 (1994).
[CrossRef]

1992

E. R. Cochran, C. Ai, “Interferometric stress birefringence measurement,” Appl. Opt. 31, 6072–6076 (1992).
[CrossRef]

1991

R. Oldenbourg, “Analysis of edge birefringence,” Biophys. J. 60, 629–641 (1991).
[CrossRef] [PubMed]

K. Leonhardt, H. J. Jordan, H. J. Tiziani, “Micro-ellipso-height-profilometry,” Opt. Commun. 80, 205–209 (1991).
[CrossRef]

1990

1988

1987

D. Nyyssonen, “Practical method for edge detection and focusing for linewidth measurements on wafers,” Opt. Eng. 26, 81–85 (1987).
[CrossRef]

D. Nyyssonen, R. D. Larrabee, “Submicrometer linewidth metrology in the optical microscope,” J. Res. Nat. Bur. Stand. 92, 187–204 (1987).
[CrossRef]

1977

Ai, C.

E. R. Cochran, C. Ai, “Interferometric stress birefringence measurement,” Appl. Opt. 31, 6072–6076 (1992).
[CrossRef]

Asundi, A.

Barakat, R.

Boay, C. G.

Cochran, E. R.

E. R. Cochran, C. Ai, “Interferometric stress birefringence measurement,” Appl. Opt. 31, 6072–6076 (1992).
[CrossRef]

Cohn, R. F.

Corle, T. R.

T. R. Corle, G. S. Kino, Confocal Scanning Optical Microscopy (Academic, San Diego, 1996), pp. 286–294.

Creath, K.

Fraase, C. G.

W. Mirandé, C. G. Fraase, “Comparison of linewidth measurements on Si structures performed by atomic force microscopy (AFM) and low voltage scanning electron microscopy (SEM),” in Proceedings of the Third Seminar on Quantitative Microscopy, K. Hasche, W. Mirandé, G. Wilkening, eds. (Physikalisch-Technischen-Bundesanstalt, Braunschweig, Germany, 1998), PTB-Bericht F-34, pp. 89–96.

Ganapathy, V.

K. Ramesh, V. Ganapathy, “Phase-shifting methodologies in photoelastic analysis—the application of the Jones calculus,” J. Strain Anal. 31, 423–432 (1996).
[CrossRef]

Gaylord, T. K.

Geuther, H.

K.-P. Schröder, W. Mirandé, H. Geuther, C. Herrmann, “In quest of nanometer accuracy: supporting optical metrology by rigorous diffraction theory and AFM topography,” Opt. Commun. 115, 568–575 (1995).
[CrossRef]

Grann, E. B.

Herrmann, C.

K.-P. Schröder, W. Mirandé, H. Geuther, C. Herrmann, “In quest of nanometer accuracy: supporting optical metrology by rigorous diffraction theory and AFM topography,” Opt. Commun. 115, 568–575 (1995).
[CrossRef]

Holmes, R. D.

Hopkins, H. H.

H. H. Hopkins, “Image formation with partially coherent light,” Photogr. Sci. Eng. 21, 114–123 (1977).

Jacobsen, H.

M. Totzeck, H. Jacobsen, H. J. Tiziani, “Phase-shifting polarization interferometry for microstructure inspection,” in Interferometry ’99: Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 75–85 (1999).

Jaronski, J. W.

Jordan, H. J.

K. Leonhardt, H. J. Jordan, H. J. Tiziani, “Micro-ellipso-height-profilometry,” Opt. Commun. 80, 205–209 (1991).
[CrossRef]

Kasprzak, H. T.

Kimura, S.

Kino, G. S.

T. R. Corle, G. S. Kino, Confocal Scanning Optical Microscopy (Academic, San Diego, 1996), pp. 286–294.

Kruger, J.

Krumbügel, M. A.

M. Totzeck, M. A. Krumbügel, “Lateral resolution in the near-field and far-field phase images of π-phase-shifting structures,” Opt. Commun. 112, 189–200 (1994).
[CrossRef]

Lalanne, P.

Larrabee, R. D.

D. Nyyssonen, R. D. Larrabee, “Submicrometer linewidth metrology in the optical microscope,” J. Res. Nat. Bur. Stand. 92, 187–204 (1987).
[CrossRef]

Law, B. M.

Leonhardt, K.

K. Leonhardt, H. J. Jordan, H. J. Tiziani, “Micro-ellipso-height-profilometry,” Opt. Commun. 80, 205–209 (1991).
[CrossRef]

Li, L.

Marx, D. S.

Mei, G.

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

Mirandé, W.

K.-P. Schröder, W. Mirandé, H. Geuther, C. Herrmann, “In quest of nanometer accuracy: supporting optical metrology by rigorous diffraction theory and AFM topography,” Opt. Commun. 115, 568–575 (1995).
[CrossRef]

W. Mirandé, C. G. Fraase, “Comparison of linewidth measurements on Si structures performed by atomic force microscopy (AFM) and low voltage scanning electron microscopy (SEM),” in Proceedings of the Third Seminar on Quantitative Microscopy, K. Hasche, W. Mirandé, G. Wilkening, eds. (Physikalisch-Technischen-Bundesanstalt, Braunschweig, Germany, 1998), PTB-Bericht F-34, pp. 89–96.

Moharam, M. G.

Morris, G. M.

Nyyssonen, D.

D. Nyyssonen, “Practical method for edge detection and focusing for linewidth measurements on wafers,” Opt. Eng. 26, 81–85 (1987).
[CrossRef]

D. Nyyssonen, R. D. Larrabee, “Submicrometer linewidth metrology in the optical microscope,” J. Res. Nat. Bur. Stand. 92, 187–204 (1987).
[CrossRef]

D. Nyyssonen, “Linewidth measurement with an optical microscope: the effect of operating conditions on the image profile,” Appl. Opt. 16, 2223–2230 (1977).
[CrossRef] [PubMed]

Oldenbourg, R.

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

R. Oldenbourg, “Analysis of edge birefringence,” Biophys. J. 60, 629–641 (1991).
[CrossRef] [PubMed]

Pak, H. K.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985), p. 565.

Pohl, D. W.

D. W. Pohl, “Nano-optics and scanning near-field optical microscopy,” in Scanning Tunneling Microscopy II, 2nd. ed., R. Wiesendanger, H.-J. Güntherodt, eds., Vol. 28 of Springer Series in Surface Sciences (Springer-Verlag, Berlin, 1995), pp. 235–271.

Pommet, D. A.

Psaltis, D.

Ramesh, K.

K. Ramesh, V. Ganapathy, “Phase-shifting methodologies in photoelastic analysis—the application of the Jones calculus,” J. Strain Anal. 31, 423–432 (1996).
[CrossRef]

Sarid, D.

D. Sarid, Scanning Force Microscopy, Oxford Series in Optical and Imaging Sciences (Oxford U. Press, Oxford, 1994).

Schmit, J.

Schröder, K.-P.

K.-P. Schröder, W. Mirandé, H. Geuther, C. Herrmann, “In quest of nanometer accuracy: supporting optical metrology by rigorous diffraction theory and AFM topography,” Opt. Commun. 115, 568–575 (1995).
[CrossRef]

See, C. W.

Somekh, M. G.

Tiziani, H. J.

M. Totzeck, H. J. Tiziani, “Phase-shifting polarization interferometry for microstructure linewidth measurements,” Opt. Lett. 24, 294–296 (1999).
[CrossRef]

M. Totzeck, H. J. Tiziani, “Interference microscopy of sub-λ structures: a rigorous computation method and measurements,” Opt. Commun. 136, 61–74 (1997).
[CrossRef]

K. Leonhardt, H. J. Jordan, H. J. Tiziani, “Micro-ellipso-height-profilometry,” Opt. Commun. 80, 205–209 (1991).
[CrossRef]

M. Totzeck, H. Jacobsen, H. J. Tiziani, “Phase-shifting polarization interferometry for microstructure inspection,” in Interferometry ’99: Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 75–85 (1999).

Tong, L.

Totzeck, M.

M. Totzeck, H. J. Tiziani, “Phase-shifting polarization interferometry for microstructure linewidth measurements,” Opt. Lett. 24, 294–296 (1999).
[CrossRef]

M. Totzeck, H. J. Tiziani, “Interference microscopy of sub-λ structures: a rigorous computation method and measurements,” Opt. Commun. 136, 61–74 (1997).
[CrossRef]

M. Totzeck, M. A. Krumbügel, “Lateral resolution in the near-field and far-field phase images of π-phase-shifting structures,” Opt. Commun. 112, 189–200 (1994).
[CrossRef]

M. Totzeck, H. Jacobsen, H. J. Tiziani, “Phase-shifting polarization interferometry for microstructure inspection,” in Interferometry ’99: Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 75–85 (1999).

Wagner, J. W.

Wilson, T.

Appl. Opt.

D. Nyyssonen, “Linewidth measurement with an optical microscope: the effect of operating conditions on the image profile,” Appl. Opt. 16, 2223–2230 (1977).
[CrossRef] [PubMed]

R. Barakat, “Optical linewidth measurements using a polarized microscope with crossed polarizers,” Appl. Opt. 29, 5038–5039 (1990).
[CrossRef] [PubMed]

S. Kimura, T. Wilson, “Confocal scanning dark-field polarization microscopy,” Appl. Opt. 33, 1274–1278 (1994).
[CrossRef] [PubMed]

R. F. Cohn, J. W. Wagner, J. Kruger, “Dynamic imaging microellipsometry: theory, system design, and feasibility demonstration,” Appl. Opt. 27, 4664–4671 (1988).
[CrossRef] [PubMed]

A. Asundi, L. Tong, C. G. Boay, “Phase-shifting method with a normal polariscope,” Appl. Opt. 38, 5931–5935 (1999).
[CrossRef]

J. W. Jaronski, H. T. Kasprzak, “Generalized algorithm for photoelastic measurements based on phase-stepping imaging polarimetry,” Appl. Opt. 38, 7018–7025 (1999).
[CrossRef]

E. R. Cochran, C. Ai, “Interferometric stress birefringence measurement,” Appl. Opt. 31, 6072–6076 (1992).
[CrossRef]

D. S. Marx, D. Psaltis, “Polarization quadrature measurement of subwavelength diffracting structures,” Appl. Opt. 36, 6434–6440 (1997).
[CrossRef]

C. W. See, M. G. Somekh, R. D. Holmes, “Scanning optical microellipsometer for pure surface profiling,” Appl. Opt. 35, 6663–6668 (1996).
[CrossRef] [PubMed]

J. Schmit, K. Creath, “Window function influence on phase error in phase-shifting algorithms,” Appl. Opt. 35, 5642–5649 (1996).
[CrossRef] [PubMed]

Biophys. J.

R. Oldenbourg, “Analysis of edge birefringence,” Biophys. J. 60, 629–641 (1991).
[CrossRef] [PubMed]

J. Microsc.

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

J. Res. Nat. Bur. Stand.

D. Nyyssonen, R. D. Larrabee, “Submicrometer linewidth metrology in the optical microscope,” J. Res. Nat. Bur. Stand. 92, 187–204 (1987).
[CrossRef]

J. Strain Anal.

K. Ramesh, V. Ganapathy, “Phase-shifting methodologies in photoelastic analysis—the application of the Jones calculus,” J. Strain Anal. 31, 423–432 (1996).
[CrossRef]

Opt. Commun.

M. Totzeck, M. A. Krumbügel, “Lateral resolution in the near-field and far-field phase images of π-phase-shifting structures,” Opt. Commun. 112, 189–200 (1994).
[CrossRef]

K. Leonhardt, H. J. Jordan, H. J. Tiziani, “Micro-ellipso-height-profilometry,” Opt. Commun. 80, 205–209 (1991).
[CrossRef]

K.-P. Schröder, W. Mirandé, H. Geuther, C. Herrmann, “In quest of nanometer accuracy: supporting optical metrology by rigorous diffraction theory and AFM topography,” Opt. Commun. 115, 568–575 (1995).
[CrossRef]

M. Totzeck, H. J. Tiziani, “Interference microscopy of sub-λ structures: a rigorous computation method and measurements,” Opt. Commun. 136, 61–74 (1997).
[CrossRef]

Opt. Eng.

D. Nyyssonen, “Practical method for edge detection and focusing for linewidth measurements on wafers,” Opt. Eng. 26, 81–85 (1987).
[CrossRef]

Opt. Lett.

Photogr. Sci. Eng.

H. H. Hopkins, “Image formation with partially coherent light,” Photogr. Sci. Eng. 21, 114–123 (1977).

Prog. Opt.

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[CrossRef]

Other

For instance, the Leica Model LMS IPRO.

W. Mirandé, C. G. Fraase, “Comparison of linewidth measurements on Si structures performed by atomic force microscopy (AFM) and low voltage scanning electron microscopy (SEM),” in Proceedings of the Third Seminar on Quantitative Microscopy, K. Hasche, W. Mirandé, G. Wilkening, eds. (Physikalisch-Technischen-Bundesanstalt, Braunschweig, Germany, 1998), PTB-Bericht F-34, pp. 89–96.

D. W. Pohl, “Nano-optics and scanning near-field optical microscopy,” in Scanning Tunneling Microscopy II, 2nd. ed., R. Wiesendanger, H.-J. Güntherodt, eds., Vol. 28 of Springer Series in Surface Sciences (Springer-Verlag, Berlin, 1995), pp. 235–271.

D. Sarid, Scanning Force Microscopy, Oxford Series in Optical and Imaging Sciences (Oxford U. Press, Oxford, 1994).

T. R. Corle, G. S. Kino, Confocal Scanning Optical Microscopy (Academic, San Diego, 1996), pp. 286–294.

M. Totzeck, H. Jacobsen, H. J. Tiziani, “Phase-shifting polarization interferometry for microstructure inspection,” in Interferometry ’99: Techniques and Technologies, M. Kujawinska, M. Takeda, eds., Proc. SPIE3744, 75–85 (1999).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985), p. 565.

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Figures (11)

Fig. 1
Fig. 1

Basic setup for phase-shifting polarization interferometry: BS, beam splitter; T, tube lens; C, adjustable compensator. The inset on the left-hand side shows the orientations of the polarizing components.

Fig. 2
Fig. 2

Setup for microscopical polarization interferometry: G, rotating ground glass; L, microscope lens; T, tube lens; M, additional magnification; BS, nonpolarizing beam splitter; LCPS, liquid-crystal phase shifter.

Fig. 3
Fig. 3

Two measurement examples: (a) intensity, (b) contrast, and (c) phase images of a small trench with a width of 330 nm and a depth of 160 nm in silicon between two bars. (d) Intensity, (e) contrast, and (f) phase images of five trenches with a depth of 120 nm and a pitch of 800 nm in silicon.

Fig. 4
Fig. 4

Line plots of the intensity, the contrast, and the phase of the narrow trench (straight, solid, vertical line) shown in Fig. 3(c). An offset of 3 rad was added to the phase curve.

Fig. 5
Fig. 5

Positions of the minima of the contrast images shown in (a) Fig. 3(b) and (b) Fig. 3(e) as obtained by use of the extreme-value algorithm. Positions of the steepest phase in the phase images shown in (c) Fig. 3(c) and (d) Fig. 3(f) as obtained by use of the extreme-value algorithm. White lines represent negative gradients, and black lines positive gradients.

Fig. 6
Fig. 6

Phase distribution and the corresponding gradient for the 160-nm-deep trench in silicon shown in Fig. 3(c). The dashed vertical line on the left-hand side of Fig. 3(c) represents the cross section.

Fig. 7
Fig. 7

Computed (a) contrast and (b) phase images and measured (c) contrast and (d) phase images at different focus heights for a silicon bar with a width of 270 nm and a height of 160 nm on silicon with a refractive index of n = 3.8 + i0.3.

Fig. 8
Fig. 8

Computed contrast- and phase-image plots of a silicon bar with a width of 500 nm and a depth of 120 nm at λ = 635 nm for an increasing illumination NA. The NA goes from zero (bottom) to 0.9 (top) in 0.1-increment steps. The plots are displaced in equal segments along the ordinate to facilitate comparison.

Fig. 9
Fig. 9

(a) Depth of the contrast minimum plotted as a function of structure height (the x axis) for widths of 0.2λ (×), 0.4λ (○), 0.6λ (+), 0.8λ (□), λ (⋄), and 1.8λ (▽) in silicon at a wavelength of λ = 500 nm and a refractive index34 of n = 4.298 + i0.073. (b) Linewidth computed from the contrast image. (c) Linewidth computed from the phase image. For comparison, the actual linewidth is shown in (b) and (c) at a height of 0.

Fig. 10
Fig. 10

(a), (b), (c) Same as for Fig. 9 but with glass (n = 1.5).

Fig. 11
Fig. 11

(a) Computed linewidths from the contrast image (○) and the phase image (×) for a trench with a depth of 0.2λ in silicon plotted as functions of the linewidth at λ = 500 nm. (b) The corresponding depths of the contrast minima.

Equations (18)

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

Ei=ExiEyi=|Exi||Eyi|expiΔψi,
E=MtRMrEi.
I=|PCE|2.
Rx, y=Rxx, yexpiφxx, y×100R¯x, yexpiΔφx, y,
M=MtMr=Mx100M¯ expiψm,
ExEy=MxRxx, yexpiφxx, y×|Exi||Eyi|M¯R¯x, yexpiψm+Δψi+Δφx, y.
C=expiβ001,  P=1100,
tan ϕx, y=ZN=argEy-argEx,
I0x, y=18j=18 Ij=|Ex|2+|Ey|2,
γx, y=Z2+N21/232I0=2|Ex||Ey||Ex|2+|Ey|2,
Z=5I2-15I4+11I6-I8,  N=I1-11I3+15I5-5I7.
ϕx, y=Δφx, y+ψm+Δψi
I0x, y=|MxExiRxx, y|2+|MyEyiRyx, y|2
γx, y=|MxMyExiEyiRxix, yRyix, y|I0x, y
ϕcx, y=ψmx, y+Δψi,
Δφx, y=ϕx, y-ϕcx, y.
Ix, y=I0x, y/Icx, y.
Ix, y, z=μ Lkμtν HkνtE˜xkνtcos α+E˜ykνtsin α expiΘexpikν·rcos1/2ϑ2.

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