Abstract

Image modeling establishes the relation between an object and its image when an optical microscope is used to measure the dimensions of an object of size comparable to the illumination wavelength. It accounts for the influence of all of the parameters that can affect the image and relates the apparent feature width (FW) in the image to the true FW of the object. The values of these parameters, however, have uncertainties, and these uncertainties propagate through the model and lead to parametric uncertainty in the FW measurement, a key component of the combined measurement uncertainty. The combined uncertainty is required in order to decide if the result is adequate for its intended purpose and to ascertain if it is consistent with other results. The parametric uncertainty for optical photomask measurements derived using an edge intensity threshold approach has been described previously; this paper describes an image library approach to this issue and shows results for optical photomask metrology over a FW range of 10 nm to 8 µm using light of wavelength 365 nm. The principles will be described; a one-dimensional image library will be used; the method of comparing images, along with a simple interpolation method, will be explained; and results will be presented. This method is easily extended to any kind of imaging microscope and to more dimensions in parameter space. It is more general than the edge threshold method and leads to markedly different uncertainties for features smaller than the wavelength.

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References

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  1. NIST SRM 2059 Photomask Linewidth Standard is available from the Office of Standard Reference Materials, NIST, EM 205, Gaithersburg, MD 20899. See sample certificate at https://www-s.nist.gov/srmors/certificates/view_certGIF.cfm?certificate=2059 .
  2. J. Potzick, “Metrology and process control: dealing with measurement uncertainty,” Proc. SPIE 7638, 76381U (2010).
    [CrossRef]
  3. D. Nyyssonen, “Linewidth measurement with an optical microscope: the effect of operating conditions on the image profile,” Appl. Opt. 16, 2223–2230 (1977).
    [CrossRef]
  4. J. Villarrubia, A. Vladar, and M. Postek, “Scanning electron microscope dimensional metrology using a model-based library,” Surf. Interface Anal. 37, 951–958, doi:10.1002/sia.2087 (2005).
    [CrossRef]
  5. J. Villarrubia, “Morphological estimation of tip geometry for scanned probe microscopy,” Surf. Sci. 321, 287–300 (1994).
    [CrossRef]
  6. NIST SRM 2800 Microscope Pitch Standard is available from the Office of Standard Reference Materials, NIST, EM 205, Gaithersburg, MD 20899. For details see sample certificate at https://www-s.nist.gov/srmors/certificates/view_certGIF.cfm?certificate=2800 .
  7. M. Davidson, “A modal model for diffraction gratings,” J. Mod. Opt. 50, 1817–1834 (2003).
    [CrossRef]
  8. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).
  9. R. F. Harrington, Field Computations by Moment Methods (Krieger, 1982).
  10. A. Taflove and S. C. Hagness, Computational Electrodynamics, the Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  11. E. Marx and J. Potzick, “Simulation of optical microscope images for photomask feature size measurements,” in Proceedings of 2005 IEEE Antennas and Propagation Society International Symposium (IEEE, 2005), pp. 2116–2119.
  12. E. Marx, “Images of strips on and trenches in substrates,” Appl. Opt. 46, 5571–5587 (2007).
    [CrossRef]
  13. SEMI Standard P35, “Terminology for microlithography metrology,” SEMI International Standards, San Jose, California.
  14. R. Attota, R. M. Silver, and J. Potzick, “Optical illumination and critical dimension analysis using the through-focus focus metric,” Proc. SPIE 6289, 62890Q (2006).
    [CrossRef]
  15. Joint Committee for Guides in Metrology, JCGM 100:2008, Evaluation of Measurement Data—Guide to the Expression of Uncertainty in Measurement (JCGM, 2008), http://www.bipm.org/en/publications/guides/gum.html .
  16. B. Taylor and C. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Technical Note 1297 (National Institute of Standards and Technology, 1994).
  17. R. Silver, T. Germer, R. Attota, B. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
    [CrossRef]
  18. R. M. Silver, N. F. Zhang, B. Barnes, H. Zhou, A. Heckert, R. Dixson, T. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
    [CrossRef]
  19. R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, J. Qin, and R. Dixson, “Nested uncertainties and hybrid metrology to improve measurement accuracy,” Proc. SPIE 7971, 797116 (2011).
    [CrossRef]
  20. J. Potzick, E. Marx, and M. Davidson, “Parametric uncertainty in optical image modeling,” Proc. SPIE 6349, 63494U (2006).
    [CrossRef]
  21. International Organization for Standardization, ISO/IEC Guide 99:2007, International Vocabulary of Metrology—Basic and General Concepts and Associated Terms (VIM), 3rd ed. (ISO, 2007), http://www.bipm.org/en/publications/guides/vim.html .
  22. C. Frase, D. Gnieser, and H. Bosse, “Model-based SEM for dimensional metrology tasks in semiconductor and mask industry,” J. Phys. D 42, 183001 (2009).
    [CrossRef]
  23. J. Potzick, “Accuracy and traceability in dimensional measurements,” Proc. SPIE 3332, 471–479 (1998).
    [CrossRef]
  24. E. Marx and J. Potzick, “Computational parameters in simulation of microscope images,” PIERS Online 5, 11–15 (2009).
    [CrossRef]

2011 (1)

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, J. Qin, and R. Dixson, “Nested uncertainties and hybrid metrology to improve measurement accuracy,” Proc. SPIE 7971, 797116 (2011).
[CrossRef]

2010 (1)

J. Potzick, “Metrology and process control: dealing with measurement uncertainty,” Proc. SPIE 7638, 76381U (2010).
[CrossRef]

2009 (3)

C. Frase, D. Gnieser, and H. Bosse, “Model-based SEM for dimensional metrology tasks in semiconductor and mask industry,” J. Phys. D 42, 183001 (2009).
[CrossRef]

R. M. Silver, N. F. Zhang, B. Barnes, H. Zhou, A. Heckert, R. Dixson, T. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

E. Marx and J. Potzick, “Computational parameters in simulation of microscope images,” PIERS Online 5, 11–15 (2009).
[CrossRef]

2007 (2)

E. Marx, “Images of strips on and trenches in substrates,” Appl. Opt. 46, 5571–5587 (2007).
[CrossRef]

R. Silver, T. Germer, R. Attota, B. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

2006 (2)

R. Attota, R. M. Silver, and J. Potzick, “Optical illumination and critical dimension analysis using the through-focus focus metric,” Proc. SPIE 6289, 62890Q (2006).
[CrossRef]

J. Potzick, E. Marx, and M. Davidson, “Parametric uncertainty in optical image modeling,” Proc. SPIE 6349, 63494U (2006).
[CrossRef]

2005 (1)

J. Villarrubia, A. Vladar, and M. Postek, “Scanning electron microscope dimensional metrology using a model-based library,” Surf. Interface Anal. 37, 951–958, doi:10.1002/sia.2087 (2005).
[CrossRef]

2003 (1)

M. Davidson, “A modal model for diffraction gratings,” J. Mod. Opt. 50, 1817–1834 (2003).
[CrossRef]

1998 (1)

J. Potzick, “Accuracy and traceability in dimensional measurements,” Proc. SPIE 3332, 471–479 (1998).
[CrossRef]

1994 (1)

J. Villarrubia, “Morphological estimation of tip geometry for scanned probe microscopy,” Surf. Sci. 321, 287–300 (1994).
[CrossRef]

1977 (1)

Allgair, J.

R. Silver, T. Germer, R. Attota, B. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

Attota, R.

R. Silver, T. Germer, R. Attota, B. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

R. Attota, R. M. Silver, and J. Potzick, “Optical illumination and critical dimension analysis using the through-focus focus metric,” Proc. SPIE 6289, 62890Q (2006).
[CrossRef]

Barnes, B.

R. M. Silver, N. F. Zhang, B. Barnes, H. Zhou, A. Heckert, R. Dixson, T. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

R. Silver, T. Germer, R. Attota, B. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

Barnes, B. M.

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, J. Qin, and R. Dixson, “Nested uncertainties and hybrid metrology to improve measurement accuracy,” Proc. SPIE 7971, 797116 (2011).
[CrossRef]

Bosse, H.

C. Frase, D. Gnieser, and H. Bosse, “Model-based SEM for dimensional metrology tasks in semiconductor and mask industry,” J. Phys. D 42, 183001 (2009).
[CrossRef]

Bunday, B.

R. M. Silver, N. F. Zhang, B. Barnes, H. Zhou, A. Heckert, R. Dixson, T. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

R. Silver, T. Germer, R. Attota, B. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

Davidson, M.

J. Potzick, E. Marx, and M. Davidson, “Parametric uncertainty in optical image modeling,” Proc. SPIE 6349, 63494U (2006).
[CrossRef]

M. Davidson, “A modal model for diffraction gratings,” J. Mod. Opt. 50, 1817–1834 (2003).
[CrossRef]

Dixson, R.

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, J. Qin, and R. Dixson, “Nested uncertainties and hybrid metrology to improve measurement accuracy,” Proc. SPIE 7971, 797116 (2011).
[CrossRef]

R. M. Silver, N. F. Zhang, B. Barnes, H. Zhou, A. Heckert, R. Dixson, T. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

Frase, C.

C. Frase, D. Gnieser, and H. Bosse, “Model-based SEM for dimensional metrology tasks in semiconductor and mask industry,” J. Phys. D 42, 183001 (2009).
[CrossRef]

Germer, T.

R. M. Silver, N. F. Zhang, B. Barnes, H. Zhou, A. Heckert, R. Dixson, T. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

R. Silver, T. Germer, R. Attota, B. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

Gnieser, D.

C. Frase, D. Gnieser, and H. Bosse, “Model-based SEM for dimensional metrology tasks in semiconductor and mask industry,” J. Phys. D 42, 183001 (2009).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics, the Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

R. F. Harrington, Field Computations by Moment Methods (Krieger, 1982).

Heckert, A.

R. M. Silver, N. F. Zhang, B. Barnes, H. Zhou, A. Heckert, R. Dixson, T. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

Jun, J.

R. Silver, T. Germer, R. Attota, B. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

Kuyatt, C.

B. Taylor and C. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Technical Note 1297 (National Institute of Standards and Technology, 1994).

Marx, E.

E. Marx and J. Potzick, “Computational parameters in simulation of microscope images,” PIERS Online 5, 11–15 (2009).
[CrossRef]

R. Silver, T. Germer, R. Attota, B. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

E. Marx, “Images of strips on and trenches in substrates,” Appl. Opt. 46, 5571–5587 (2007).
[CrossRef]

J. Potzick, E. Marx, and M. Davidson, “Parametric uncertainty in optical image modeling,” Proc. SPIE 6349, 63494U (2006).
[CrossRef]

E. Marx and J. Potzick, “Simulation of optical microscope images for photomask feature size measurements,” in Proceedings of 2005 IEEE Antennas and Propagation Society International Symposium (IEEE, 2005), pp. 2116–2119.

Nyyssonen, D.

Postek, M.

J. Villarrubia, A. Vladar, and M. Postek, “Scanning electron microscope dimensional metrology using a model-based library,” Surf. Interface Anal. 37, 951–958, doi:10.1002/sia.2087 (2005).
[CrossRef]

Potzick, J.

J. Potzick, “Metrology and process control: dealing with measurement uncertainty,” Proc. SPIE 7638, 76381U (2010).
[CrossRef]

E. Marx and J. Potzick, “Computational parameters in simulation of microscope images,” PIERS Online 5, 11–15 (2009).
[CrossRef]

J. Potzick, E. Marx, and M. Davidson, “Parametric uncertainty in optical image modeling,” Proc. SPIE 6349, 63494U (2006).
[CrossRef]

R. Attota, R. M. Silver, and J. Potzick, “Optical illumination and critical dimension analysis using the through-focus focus metric,” Proc. SPIE 6289, 62890Q (2006).
[CrossRef]

J. Potzick, “Accuracy and traceability in dimensional measurements,” Proc. SPIE 3332, 471–479 (1998).
[CrossRef]

E. Marx and J. Potzick, “Simulation of optical microscope images for photomask feature size measurements,” in Proceedings of 2005 IEEE Antennas and Propagation Society International Symposium (IEEE, 2005), pp. 2116–2119.

Qin, J.

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, J. Qin, and R. Dixson, “Nested uncertainties and hybrid metrology to improve measurement accuracy,” Proc. SPIE 7971, 797116 (2011).
[CrossRef]

Silver, R.

R. Silver, T. Germer, R. Attota, B. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

Silver, R. M.

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, J. Qin, and R. Dixson, “Nested uncertainties and hybrid metrology to improve measurement accuracy,” Proc. SPIE 7971, 797116 (2011).
[CrossRef]

R. M. Silver, N. F. Zhang, B. Barnes, H. Zhou, A. Heckert, R. Dixson, T. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

R. Attota, R. M. Silver, and J. Potzick, “Optical illumination and critical dimension analysis using the through-focus focus metric,” Proc. SPIE 6289, 62890Q (2006).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics, the Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Taylor, B.

B. Taylor and C. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Technical Note 1297 (National Institute of Standards and Technology, 1994).

Villarrubia, J.

J. Villarrubia, A. Vladar, and M. Postek, “Scanning electron microscope dimensional metrology using a model-based library,” Surf. Interface Anal. 37, 951–958, doi:10.1002/sia.2087 (2005).
[CrossRef]

J. Villarrubia, “Morphological estimation of tip geometry for scanned probe microscopy,” Surf. Sci. 321, 287–300 (1994).
[CrossRef]

Vladar, A.

J. Villarrubia, A. Vladar, and M. Postek, “Scanning electron microscope dimensional metrology using a model-based library,” Surf. Interface Anal. 37, 951–958, doi:10.1002/sia.2087 (2005).
[CrossRef]

Zhang, N. F.

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, J. Qin, and R. Dixson, “Nested uncertainties and hybrid metrology to improve measurement accuracy,” Proc. SPIE 7971, 797116 (2011).
[CrossRef]

R. M. Silver, N. F. Zhang, B. Barnes, H. Zhou, A. Heckert, R. Dixson, T. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

Zhou, H.

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, J. Qin, and R. Dixson, “Nested uncertainties and hybrid metrology to improve measurement accuracy,” Proc. SPIE 7971, 797116 (2011).
[CrossRef]

R. M. Silver, N. F. Zhang, B. Barnes, H. Zhou, A. Heckert, R. Dixson, T. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

Appl. Opt. (2)

J. Mod. Opt. (1)

M. Davidson, “A modal model for diffraction gratings,” J. Mod. Opt. 50, 1817–1834 (2003).
[CrossRef]

J. Phys. D (1)

C. Frase, D. Gnieser, and H. Bosse, “Model-based SEM for dimensional metrology tasks in semiconductor and mask industry,” J. Phys. D 42, 183001 (2009).
[CrossRef]

Proc. SPIE (7)

J. Potzick, “Accuracy and traceability in dimensional measurements,” Proc. SPIE 3332, 471–479 (1998).
[CrossRef]

R. Attota, R. M. Silver, and J. Potzick, “Optical illumination and critical dimension analysis using the through-focus focus metric,” Proc. SPIE 6289, 62890Q (2006).
[CrossRef]

J. Potzick, “Metrology and process control: dealing with measurement uncertainty,” Proc. SPIE 7638, 76381U (2010).
[CrossRef]

R. Silver, T. Germer, R. Attota, B. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

R. M. Silver, N. F. Zhang, B. Barnes, H. Zhou, A. Heckert, R. Dixson, T. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, J. Qin, and R. Dixson, “Nested uncertainties and hybrid metrology to improve measurement accuracy,” Proc. SPIE 7971, 797116 (2011).
[CrossRef]

J. Potzick, E. Marx, and M. Davidson, “Parametric uncertainty in optical image modeling,” Proc. SPIE 6349, 63494U (2006).
[CrossRef]

Surf. Interface Anal. (1)

J. Villarrubia, A. Vladar, and M. Postek, “Scanning electron microscope dimensional metrology using a model-based library,” Surf. Interface Anal. 37, 951–958, doi:10.1002/sia.2087 (2005).
[CrossRef]

Surf. Sci. (1)

J. Villarrubia, “Morphological estimation of tip geometry for scanned probe microscopy,” Surf. Sci. 321, 287–300 (1994).
[CrossRef]

Other (11)

NIST SRM 2800 Microscope Pitch Standard is available from the Office of Standard Reference Materials, NIST, EM 205, Gaithersburg, MD 20899. For details see sample certificate at https://www-s.nist.gov/srmors/certificates/view_certGIF.cfm?certificate=2800 .

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

R. F. Harrington, Field Computations by Moment Methods (Krieger, 1982).

A. Taflove and S. C. Hagness, Computational Electrodynamics, the Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

E. Marx and J. Potzick, “Simulation of optical microscope images for photomask feature size measurements,” in Proceedings of 2005 IEEE Antennas and Propagation Society International Symposium (IEEE, 2005), pp. 2116–2119.

International Organization for Standardization, ISO/IEC Guide 99:2007, International Vocabulary of Metrology—Basic and General Concepts and Associated Terms (VIM), 3rd ed. (ISO, 2007), http://www.bipm.org/en/publications/guides/vim.html .

NIST SRM 2059 Photomask Linewidth Standard is available from the Office of Standard Reference Materials, NIST, EM 205, Gaithersburg, MD 20899. See sample certificate at https://www-s.nist.gov/srmors/certificates/view_certGIF.cfm?certificate=2059 .

SEMI Standard P35, “Terminology for microlithography metrology,” SEMI International Standards, San Jose, California.

Joint Committee for Guides in Metrology, JCGM 100:2008, Evaluation of Measurement Data—Guide to the Expression of Uncertainty in Measurement (JCGM, 2008), http://www.bipm.org/en/publications/guides/gum.html .

B. Taylor and C. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Technical Note 1297 (National Institute of Standards and Technology, 1994).

E. Marx and J. Potzick, “Computational parameters in simulation of microscope images,” PIERS Online 5, 11–15 (2009).
[CrossRef]

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Figures (14)

Fig. 1.
Fig. 1.

Simulated microscope grayscale images of isolated 0.125 µm and 1.000 µm lines and 0.125 µm and 1.000 µm spaces viewed in transmission. Image intensity is overlaid as a graph.

Fig. 2.
Fig. 2.

FW error as function of FW. Traditional two-point (linear) calibration works well for measuring isolated features larger than the wavelength, but generates large errors for smaller objects. Note the logarithmic FW scale used throughout this paper.

Fig. 3.
Fig. 3.

Cross section of a typical chrome-on-quartz photomask line feature.

Fig. 4.
Fig. 4.

Image best focus as function of FW for the photomask features in Fig. 5. Best focus is defined here as maximum acutance.

Fig. 5.
Fig. 5.

Image intensity at the edge of an isolated line or space as function of FW. Lines and spaces behave differently for FWs smaller than the wavelength.

Fig. 6.
Fig. 6.

Calibration FW errors (a) replotted from Fig. 2 for comparison with Figs. 6(b) and 6(c), (b) using the variable threshold method of image edge identification (errors are largely due to interpolation of sampled data), and (c) using the image library method of extracting data from a microscope image. All plotted to the same scale.

Fig. 7.
Fig. 7.

Image acutance at feature edge of isolated lines and spaces.

Fig. 8.
Fig. 8.

Combined parametric standard uncertainty using the variable edge threshold method and the parameter values and their uncertainties listed in Table 1.

Fig. 9.
Fig. 9.

Shape of image-matching metric Q surface for identifying the FW in an image; minimum Q points to the best library image match. (a) Isolated lines, (b) isolated spaces. Q is small at small spacewidths because the image intensity is small; see Fig. 11(b).

Fig. 10.
Fig. 10.

Shape of image-matching metric Q surface for finding the apparent FW in an image when the illumination NA is in error by δ NA ; minimum Q points to the best library image match. (a) 125 nm line and (b) 125 nm space.

Fig. 11.
Fig. 11.

Combined parametric standard uncertainty for linewidths and spacewidths using the parameter values and uncertainties in Table 1 and (a) the library-matching method of finding the best match feature width; (b) the variable edge threshold method of extracting feature width data (the same data as Fig. 8, rescaled for comparison).

Fig. 12.
Fig. 12.

Image contrast for photomask lines and spaces. The contrast is nearly independent of feature width for features larger than the wavelength but varies monotonically for smaller features.

Fig. 13.
Fig. 13.

Separate parametric uncertainty components contributing to the combined parametric feature width uncertainty in Fig. 11(a). (a) Isolated lines; the major components appear to be microscope illumination and objective NA. (b) Isolated spaces; the major component appears to be microscope illumination NA.

Fig. 14.
Fig. 14.

Scattering by a Cr strip on a quartz layer on a substrate, incidence from below the layer. Unknown boundary functions η i j and η i j are jumps in the normal derivatives of the fields in V i across the boundary C j .

Tables (1)

Tables Icon

Table 1. Imaging Parameters Used Here and Their Uncertainties

Equations (18)

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

object size = scale factor × image size + offset .
( 2 + k 2 ) E z ( x , y ) = 0 , ( 2 + k 2 ) H z ( x , y ) = 0 ,
= e ^ x / x + e ^ y / y , k⃗ = e ^ x k x + e ^ y k y ,
δ FW i = FW / P i δ P i
u p i ( FW ) = u ( P i ) | FW / P i | ,
u param ( FW ) = i [ u ( P i ) FW / P i ] 2 ,
u ( FW ) = [ σ 2 + u param 2 + u other 2 ] ,
u param ( P k ) = i ( P k / P i ) 2 u 2 ( P i ) , i k .
Q j k = 1 n i = 1 n [ I j ( x i ) I k ( x i ) ] 2 ,
U 1 ( ξ⃗ ) = G 11 { η 11 } ( ξ⃗ ) + G 12 { η 12 } ( ξ⃗ ) ,
U 2 ( ξ⃗ ) = G 21 { η 21 } ( ξ⃗ ) + G 23 { η 23 } ( ξ⃗ ) ,
U 3 ( ξ⃗ ) = G 32 { Δ 2 ( U 3 / n ) } ( ξ⃗ ) + N 32 { Δ 2 U 3 } ( ξ⃗ ) + G 33 { Δ 3 ( U 3 / n ) } ( ξ⃗ ) + N 33 { Δ 3 U 3 } ( ξ⃗ ) ,
U 7 ( ξ⃗ ) = G 71 { η 71 } ( ξ⃗ ) + G 73 { η 73 } ( ξ⃗ ) + G 70 { η 70 } ( ξ⃗ ) ,
G { η } ( ξ⃗ ) = i 4 C d s η ( s ) H 0 ( 1 ) ( k R ) ,
N { ϕ } ( ξ⃗ ) = i 4 C d s ϕ ( s ) H 1 ( 1 ) ( k R ) k n ^ · R ^ ,
G 11 1 { η 11 } + G 12 1 { η 12 } G 71 1 { η 71 } G 73 1 { η 73 } G 70 1 { η 70 } = 0 ,
1 2 η 11 + N 12 1 { η 12 } + 1 2 b ¯ 1 η 71 b ¯ 1 N 70 1 { η 70 } β ¯ 1 N 11 1 { η 11 } β ¯ 1 N 12 1 { η 12 } = 0 ,
[ ( 1 / 2 + N 32 2 ) G 11 2 + b 2 G 32 2 N 11 2 ] { η 11 } + [ ( 1 / 2 + N 32 2 ) G 12 2 + b 2 G 32 2 ( 1 / 2 + N 12 2 ) ] { η 12 } N 33 2 G 71 3 { η 71 } + ( 1 / 2 b ¯ 3 G 33 2 N 33 2 G 73 3 ) { η 73 } ( b ¯ 3 G 33 2 N 70 3 + N 33 2 G 70 3 ) { η 70 } + β 2 G 32 2 N 11 2 { η 11 } + β 2 G 32 2 N 12 2 { η 12 } β ¯ 3 G 33 2 N 71 3 { η 71 } β ¯ 3 G 33 2 N 73 3 { η 73 } β ¯ 3 G 33 2 N 70 3 { η 70 } = ( 1 / 2 + N 32 2 ) { E z h 1 } b 2 G 32 2 { E z h 1 / n } + N 33 2 { E z h 7 } + b ¯ 3 G 33 2 { E z h 7 / n } β 2 G 32 2 { H z h 1 / s } + β ¯ 3 G 33 2 { H z h 7 / s } ,

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