Abstract

The resolution of coherent and incoherent imaging systems is usually evaluated in terms of classical resolution criteria, such as Rayleigh’s. Based on these criteria, incoherent imaging is generally concluded to be ‘better’ than coherent imaging. However, this paper reveals some misconceptions in the application of the classical criteria, which may lead to wrong conclusions. Furthermore, it is shown that classical resolution criteria are no longer appropriate if images are interpreted quantitatively instead of qualitatively. Then one needs an alternative criterion to compare coherent and incoherent imaging systems objectively. Such a criterion, which relates resolution to statistical measurement precision, is proposed in this paper. It is applied in the field of electron microscopy, where the question whether coherent high resolution transmission electron microscopy (HRTEM) or incoherent annular dark field scanning transmission electron microscopy (ADF STEM) is preferable has been an issue of considerable debate.

© 2006 Optical Society of America

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References

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  1. L. Rayleigh, "Wave theory of light," in Scientific papers by Lord Rayleigh, John William Strutt, (Cambridge University Press, Cambridge, 1902), Vol. 3, pp. 47-189.
  2. J. W. Goodman, Introduction to fourier optics (McGraw-Hill, San Francisco, 1968).
  3. A. J. den Dekker and A. van den Bos, "Resolution: A survey," J. Opt. Soc. Am. A 14,547-557 (1997).
    [CrossRef]
  4. V. Ronchi, "Resolving power of calculated and detected images," J. Opt. Soc. Am. 51,458-460 (1961).
    [CrossRef]
  5. L. Rayleigh, "On the theory of optical images, with special reference to the microscope," in Scientific papers by Lord Rayleigh, John William Strutt, (Cambridge University Press, Cambridge, 1903), Vol. 4, pp. 235-260.
  6. J. C. H. Spence, High-resolution electron microscopy, 3rd edition (Oxford University Press, New York, 2003).
  7. S. J. Pennycook and Y. Yan, "Z-contrast imaging in the scanning transmission electron microscope," in Progress in transmission electron microscopy 1 - Concepts and techniques, X.-F. Zhang and Z. Zhang, eds. (Springer-Verlag, Berlin, 2001), pp. 81-111.
  8. P. D. Nellist and S. J. Pennycook, "Accurate structure determination from image reconstruction in ADF STEM," J. Microsc. 190,159-170 (1998).
    [CrossRef]
  9. D. Van Dyck and M. Op de Beeck, "A simple intuitive theory for electron diffraction," Ultramicroscopy 64,99-107 (1996).
    [CrossRef]
  10. S. Van Aert, A. J. den Dekker, A. van den Bos, and D. Van Dyck, "Statistical experimental design for quantitative atomic resolution transmission electron microscopy," in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic Press, San Diego, 2004), Vol. 130, pp. 1-164.
    [CrossRef]
  11. A. J. den Dekker, S. Van Aert, D. Van Dyck, A. van den Bos, and P. Geuens, "Does a monochromator improve the precision in quantitative HRTEM?," Ultramicroscopy 89,275-290 (2001).
    [CrossRef]
  12. S. Van Aert, A. J. den Dekker, D. Van Dyck, and A. van den Bos, "Optimal experimental design of STEM measurement of atom column positions," Ultramicroscopy 90,273-289 (2002).
    [CrossRef] [PubMed]
  13. S. J. Pennycook, B. Rafferty, and P. D. Nellist, "Z-contrast imaging in an aberration-corrected scanning transmission electron microscope," Microsc. Microanal. 6,343-352 (2000).
    [PubMed]
  14. O. Scherzer, "The theoretical resolution limit of the electron microscope," J. Appl. Phys. 20,20-28 (1949).
    [CrossRef]
  15. L. J. van Vliet, F. R. Boddeke, D. Sudar, and I. T. Young, "Image detectors for digital image microscopy," in Digital image analysis of microbes; Imaging, morphometry, fluorometry and motility techniques and applications, modern microbiological methods, M. H. F.Wilkinson and F. Schut, eds. (JohnWiley and Sons, Chichester (UK), 1998), pp. 37-64.
  16. A. van den Bos and A. J. den Dekker, "Resolution reconsidered - Conventional approaches and an alternative," in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic Press, San Diego, 2001), Vol. 117, pp. 241-360.
    [CrossRef]
  17. A. J. den Dekker, S. Van Aert, A. van den Bos, and D. Van Dyck, "Maximum likelihood estimation of structure parameters from high resolution electron microscopy images. Part I: A theoretical framework," Ultramicroscopy 104,83-106 (2005).
    [CrossRef]

2005

A. J. den Dekker, S. Van Aert, A. van den Bos, and D. Van Dyck, "Maximum likelihood estimation of structure parameters from high resolution electron microscopy images. Part I: A theoretical framework," Ultramicroscopy 104,83-106 (2005).
[CrossRef]

2002

S. Van Aert, A. J. den Dekker, D. Van Dyck, and A. van den Bos, "Optimal experimental design of STEM measurement of atom column positions," Ultramicroscopy 90,273-289 (2002).
[CrossRef] [PubMed]

2001

A. J. den Dekker, S. Van Aert, D. Van Dyck, A. van den Bos, and P. Geuens, "Does a monochromator improve the precision in quantitative HRTEM?," Ultramicroscopy 89,275-290 (2001).
[CrossRef]

2000

S. J. Pennycook, B. Rafferty, and P. D. Nellist, "Z-contrast imaging in an aberration-corrected scanning transmission electron microscope," Microsc. Microanal. 6,343-352 (2000).
[PubMed]

1998

P. D. Nellist and S. J. Pennycook, "Accurate structure determination from image reconstruction in ADF STEM," J. Microsc. 190,159-170 (1998).
[CrossRef]

1997

1996

D. Van Dyck and M. Op de Beeck, "A simple intuitive theory for electron diffraction," Ultramicroscopy 64,99-107 (1996).
[CrossRef]

1961

1949

O. Scherzer, "The theoretical resolution limit of the electron microscope," J. Appl. Phys. 20,20-28 (1949).
[CrossRef]

den Dekker, A. J.

A. J. den Dekker, S. Van Aert, A. van den Bos, and D. Van Dyck, "Maximum likelihood estimation of structure parameters from high resolution electron microscopy images. Part I: A theoretical framework," Ultramicroscopy 104,83-106 (2005).
[CrossRef]

S. Van Aert, A. J. den Dekker, D. Van Dyck, and A. van den Bos, "Optimal experimental design of STEM measurement of atom column positions," Ultramicroscopy 90,273-289 (2002).
[CrossRef] [PubMed]

A. J. den Dekker, S. Van Aert, D. Van Dyck, A. van den Bos, and P. Geuens, "Does a monochromator improve the precision in quantitative HRTEM?," Ultramicroscopy 89,275-290 (2001).
[CrossRef]

A. J. den Dekker and A. van den Bos, "Resolution: A survey," J. Opt. Soc. Am. A 14,547-557 (1997).
[CrossRef]

Geuens, P.

A. J. den Dekker, S. Van Aert, D. Van Dyck, A. van den Bos, and P. Geuens, "Does a monochromator improve the precision in quantitative HRTEM?," Ultramicroscopy 89,275-290 (2001).
[CrossRef]

Nellist, P. D.

S. J. Pennycook, B. Rafferty, and P. D. Nellist, "Z-contrast imaging in an aberration-corrected scanning transmission electron microscope," Microsc. Microanal. 6,343-352 (2000).
[PubMed]

P. D. Nellist and S. J. Pennycook, "Accurate structure determination from image reconstruction in ADF STEM," J. Microsc. 190,159-170 (1998).
[CrossRef]

Op de Beeck, M.

D. Van Dyck and M. Op de Beeck, "A simple intuitive theory for electron diffraction," Ultramicroscopy 64,99-107 (1996).
[CrossRef]

Pennycook, S. J.

S. J. Pennycook, B. Rafferty, and P. D. Nellist, "Z-contrast imaging in an aberration-corrected scanning transmission electron microscope," Microsc. Microanal. 6,343-352 (2000).
[PubMed]

P. D. Nellist and S. J. Pennycook, "Accurate structure determination from image reconstruction in ADF STEM," J. Microsc. 190,159-170 (1998).
[CrossRef]

Rafferty, B.

S. J. Pennycook, B. Rafferty, and P. D. Nellist, "Z-contrast imaging in an aberration-corrected scanning transmission electron microscope," Microsc. Microanal. 6,343-352 (2000).
[PubMed]

Ronchi, V.

Scherzer, O.

O. Scherzer, "The theoretical resolution limit of the electron microscope," J. Appl. Phys. 20,20-28 (1949).
[CrossRef]

Van Aert, S.

A. J. den Dekker, S. Van Aert, A. van den Bos, and D. Van Dyck, "Maximum likelihood estimation of structure parameters from high resolution electron microscopy images. Part I: A theoretical framework," Ultramicroscopy 104,83-106 (2005).
[CrossRef]

S. Van Aert, A. J. den Dekker, D. Van Dyck, and A. van den Bos, "Optimal experimental design of STEM measurement of atom column positions," Ultramicroscopy 90,273-289 (2002).
[CrossRef] [PubMed]

A. J. den Dekker, S. Van Aert, D. Van Dyck, A. van den Bos, and P. Geuens, "Does a monochromator improve the precision in quantitative HRTEM?," Ultramicroscopy 89,275-290 (2001).
[CrossRef]

van den Bos, A.

A. J. den Dekker, S. Van Aert, A. van den Bos, and D. Van Dyck, "Maximum likelihood estimation of structure parameters from high resolution electron microscopy images. Part I: A theoretical framework," Ultramicroscopy 104,83-106 (2005).
[CrossRef]

S. Van Aert, A. J. den Dekker, D. Van Dyck, and A. van den Bos, "Optimal experimental design of STEM measurement of atom column positions," Ultramicroscopy 90,273-289 (2002).
[CrossRef] [PubMed]

A. J. den Dekker, S. Van Aert, D. Van Dyck, A. van den Bos, and P. Geuens, "Does a monochromator improve the precision in quantitative HRTEM?," Ultramicroscopy 89,275-290 (2001).
[CrossRef]

A. J. den Dekker and A. van den Bos, "Resolution: A survey," J. Opt. Soc. Am. A 14,547-557 (1997).
[CrossRef]

Van Dyck, D.

A. J. den Dekker, S. Van Aert, A. van den Bos, and D. Van Dyck, "Maximum likelihood estimation of structure parameters from high resolution electron microscopy images. Part I: A theoretical framework," Ultramicroscopy 104,83-106 (2005).
[CrossRef]

S. Van Aert, A. J. den Dekker, D. Van Dyck, and A. van den Bos, "Optimal experimental design of STEM measurement of atom column positions," Ultramicroscopy 90,273-289 (2002).
[CrossRef] [PubMed]

A. J. den Dekker, S. Van Aert, D. Van Dyck, A. van den Bos, and P. Geuens, "Does a monochromator improve the precision in quantitative HRTEM?," Ultramicroscopy 89,275-290 (2001).
[CrossRef]

D. Van Dyck and M. Op de Beeck, "A simple intuitive theory for electron diffraction," Ultramicroscopy 64,99-107 (1996).
[CrossRef]

J. Appl. Phys.

O. Scherzer, "The theoretical resolution limit of the electron microscope," J. Appl. Phys. 20,20-28 (1949).
[CrossRef]

J. Microsc.

P. D. Nellist and S. J. Pennycook, "Accurate structure determination from image reconstruction in ADF STEM," J. Microsc. 190,159-170 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Microsc. Microanal.

S. J. Pennycook, B. Rafferty, and P. D. Nellist, "Z-contrast imaging in an aberration-corrected scanning transmission electron microscope," Microsc. Microanal. 6,343-352 (2000).
[PubMed]

Ultramicroscopy

A. J. den Dekker, S. Van Aert, D. Van Dyck, A. van den Bos, and P. Geuens, "Does a monochromator improve the precision in quantitative HRTEM?," Ultramicroscopy 89,275-290 (2001).
[CrossRef]

S. Van Aert, A. J. den Dekker, D. Van Dyck, and A. van den Bos, "Optimal experimental design of STEM measurement of atom column positions," Ultramicroscopy 90,273-289 (2002).
[CrossRef] [PubMed]

A. J. den Dekker, S. Van Aert, A. van den Bos, and D. Van Dyck, "Maximum likelihood estimation of structure parameters from high resolution electron microscopy images. Part I: A theoretical framework," Ultramicroscopy 104,83-106 (2005).
[CrossRef]

D. Van Dyck and M. Op de Beeck, "A simple intuitive theory for electron diffraction," Ultramicroscopy 64,99-107 (1996).
[CrossRef]

Other

S. Van Aert, A. J. den Dekker, A. van den Bos, and D. Van Dyck, "Statistical experimental design for quantitative atomic resolution transmission electron microscopy," in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic Press, San Diego, 2004), Vol. 130, pp. 1-164.
[CrossRef]

L. Rayleigh, "Wave theory of light," in Scientific papers by Lord Rayleigh, John William Strutt, (Cambridge University Press, Cambridge, 1902), Vol. 3, pp. 47-189.

J. W. Goodman, Introduction to fourier optics (McGraw-Hill, San Francisco, 1968).

L. Rayleigh, "On the theory of optical images, with special reference to the microscope," in Scientific papers by Lord Rayleigh, John William Strutt, (Cambridge University Press, Cambridge, 1903), Vol. 4, pp. 235-260.

J. C. H. Spence, High-resolution electron microscopy, 3rd edition (Oxford University Press, New York, 2003).

S. J. Pennycook and Y. Yan, "Z-contrast imaging in the scanning transmission electron microscope," in Progress in transmission electron microscopy 1 - Concepts and techniques, X.-F. Zhang and Z. Zhang, eds. (Springer-Verlag, Berlin, 2001), pp. 81-111.

L. J. van Vliet, F. R. Boddeke, D. Sudar, and I. T. Young, "Image detectors for digital image microscopy," in Digital image analysis of microbes; Imaging, morphometry, fluorometry and motility techniques and applications, modern microbiological methods, M. H. F.Wilkinson and F. Schut, eds. (JohnWiley and Sons, Chichester (UK), 1998), pp. 37-64.

A. van den Bos and A. J. den Dekker, "Resolution reconsidered - Conventional approaches and an alternative," in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic Press, San Diego, 2001), Vol. 117, pp. 241-360.
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

Cross sections of HRTEM and ADF STEM intensity distributions for different thicknesses. The height of the central dip relative to the maximum or minimum intensity above or below the background is indicated.

Fig. 2.
Fig. 2.

Lower bound on the standard deviation of the distance for two point sources separated by the Rayleigh distance as a function of the relative phase ϕ.

Fig. 3.
Fig. 3.

Lower bound on the standard deviation of the distance between two Si[110] columns for HRTEM and ADF STEM as a function of the field of view. The recording time and pixel size are kept constant in this evaluation.

Equations (16)

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

I ( r ) = δ ( r β 1 ) * t ( r β 1 ) + exp ( ) δ ( r β 2 ) * t ( r β 2 ) 2
= t ( r β 1 ) + exp ( ) t ( r β 2 ) 2
I ( r ) = δ ( r β 1 ) * t ( r β 1 ) | 2 + | exp ( ) δ ( r β 2 ) * t ( r β 2 ) 2
= t ( r β 1 ) 2 + ∣t ( r β 2 ) 2 .
I ( r ) = 1 + n = 1 n c a n φ 1 s , n ( r β n ) * t ( r β n ) 2 ,
I ( r ) = 1 + n = 1 n c A n φ 1 s , n ( r β n ) * t ( r β n ) 2 ,
P ( ω ; θ ) = m = 1 M λ m ω m ω m ! exp ( λ m )
λ m = N C S m I ( r ) d r N C I ( r m ) S m
F = E [ 2 In P ( W ; θ ) θ θ T ] ,
F rs = m = 1 M 1 λ m λ m θ r λ m θ s .
F rs = N S m C m = 1 M 1 I ( r m ) I ( r m ) θ r I ( r m ) θ s
coν ( θ ̂ ) F 1
νar ( θ ̂ r ) [ F 1 ] rr ,
cov ( γ ̂ ) γ θ T F 1 γ T θ
var ( δ ̂ ) δ θ T F 1 δ T θ
δ θ T = 1 δ ( β x 1 β x 2 β x 2 β x 1 β y 1 β y 2 β y 2 β y 1 ) .

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