Abstract

An operator algebra description of Fourier optics is used to examine the imaging properties of transmission electron microscopy when applied to the study of weak specimens. Effects due to the curvature of the incident beam, the finite extent of the source, beam tilt, and objective aperture shift are examined. An expression for the contrast transfer function is derived that can account for either beam tilt in conjunction with a centered aperture or a shifted aperture in conjunction with an aligned beam. It shows that high phase contrast over a broad spatial-frequency range can be achieved by laterally shifting the objective aperture rather than defocusing the specimen, as is normally done.

© 2003 Optical Society of America

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  1. J. Frank, P. Penczek, R. K. Agrawal, R. A. Grassucci, A. B. Heagle, “Three-dimensional cryoelectron microscopy of ribosomes.,” Methods Enzymol. 317, 276–291 (2000).
    [CrossRef]
  2. M. van Heel, B. Gowen, R. Matadeen, E. Orlova, R. Finn, T. Pape, D. Cohen, H. Stark, R. Schmidt, M. Schatz, A. Patwardhan, “Single-particle electron cryomicroscopy: towards atomic resolution,” Quart. Rev. Biophys. 33, 307–369 (2000).
    [CrossRef]
  3. P. W. Hawkes, E. Kasper, Principles of Electron Optics: Wave Optics (Academic, London, 1994), Vol. 3.
  4. L. Reimer, Transmission Electron Microscopy: Physics of Image Formation and Microanalysis (Springer-Verlag, New York, 1997).
  5. K.-J. Hanzen, “The optical transfer theory of the electron microscope: fundamental principles and applications,” Adv. Opt. Electron Microsc. 4, 1–84 (1971).
  6. O. Scherzer, “The theoretical resolution limit of the electron microscope,” J. Appl. Phys. 20, 20–29 (1949).
    [CrossRef]
  7. W. Coene, D. Van Dyck, J. Van Landuyt, “An extension of the standard theory of partial coherence for the effect of beam convergence in high resolution electron microscopy,” Optik 73, 13–18 (1986).
  8. J. Frank, “The envelope of electron microscopic transfer functions for partially coherent illumination,” Optik 38, 519–536 (1973).
  9. C. J. Humphreys, J. C. H. Spence, “Resolution and illumination coherence in electron microscopy,” Optik 58, 125–144 (1981).
  10. R. H. Wade, J. Frank, “Electron microscope transfer functions for partially coherent axial illumination and chromatic defocus spread,” Optik 49, 81–92 (1977).
  11. P. W. Hawkes, “Electron microscope transfer functions in closed form with tilted illumination,” Optik 55, 207–212 (1980).
  12. S. C. McFarlane, “The imaging of amorphous specimens in a tilted-beam electron microscope,” J. Phys. C 8, 2819–2836 (1975).
    [CrossRef]
  13. R. H. Wade, “Concerning tilted beam electron microscope transfer functions,” Optik 45, 87–91 (1976).
  14. R. H. Wade, W. K. Jenkins, “Tilted beam electron microscopy: the effective coherent aperture,” Optik 50, 1–17 (1978).
  15. F. Zemlin, K. Weiss, P. Schiske, W. Kunath, K.-H. Herrmann, “Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms,” Ultramicroscopy 3, 49–60 (1978).
    [CrossRef]
  16. A. Patwardhan, “Coherent non-planar illumination of a defocused specimen: consequences for transmission electron microscopy,” Optik 113, 4–12 (2002).
    [CrossRef]
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, Singapore, 1996).
  18. H. J. Butterweck, “General theory of linear, coherent optical data-processing systems,” J. Opt. Soc. Am. 67, 60–70 (1977).
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  19. H. J. Butterweck, “Principles of optical data processing,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1981), pp. 213–280.
  20. V. P. Ivanchenkov, P. V. Orlov, “Operator description of optical computation systems with a partially coherent light source,” Avtometriya 5, 63–70 (1985).
  21. M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
    [CrossRef]
  22. M. Nazarathy, J. Shamir, “Holography described by operator algebra,” J. Opt. Soc. Am. 71, 529–541 (1981).
    [CrossRef]
  23. J. Shamir, Optical Systems and Processes (SPIE Press, Bellingham, Wash., 1999).
  24. D. A. Tichenor, J. W. Goodman, “Coherent transfer function,” J. Opt. Soc. Am. 62, 293–295 (1972).
    [CrossRef]
  25. D. L. Misell, “On the validity of the weak-phase and other approximations in the analysis of electron microscope images,” J. Phys. D 9(13), 1849–1866 (1976).
    [CrossRef]
  26. A. Patwardhan, “Verification of intensity expressions using Maple” (2002), retrieved http://www.cbem.ic.ac.uk/ardan/opalg/opalg1.html .
  27. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  28. O. Bryngdahl, A. Lohmann, “Single-sideband holography,” J. Opt. Soc. Am. 58, 620–624 (1968).
    [CrossRef]
  29. K. H. Downing, B. M. Siegel, “Phase shift determination in single-sideband holography,” Optik 38, 21–28 (1973).
  30. J. W. Goodman, Statistical Optics (Wiley, New York, 2000), Sec. 7.2.1, pp. 303–304.
  31. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).
  32. S. Uhlemann, M. Haider, “Residual wave aberrations in the first spherical aberration corrected transmission electron microscope,” Ultramicroscopy 72, 109–119 (1998).
    [CrossRef]
  33. P. W. Hawkes, “The dependence of the spherical aberration coefficient of an electron-optical objective lens on object position and magnification,” Br. J. Appl. Phys. 1, 131–133 (1968).
  34. A. Patwardhan, “Verification of aberration terms using Maple” (2002), retrieved http://www.cbem.ic.ac.uk/ardan/opalg/opalg3.html .
  35. A. Patwardhan, “Maple worksheet for the numerical integration of contrast transfer functions” (2002), retrieved http://www.cbem.ic.ac.uk/ardan/opalg/opalg2.mws .
  36. M. Haider, H. Rose, S. Uhlemann, E. Schwan, B. Kabius, K. Urban, “A spherical-aberration-corrected 200 kV transmission electron microscope,” Ultramicroscopy 75, 53–60 (1998).
    [CrossRef]
  37. D. B. Williams, C. B. Carter, Transmission Electron Microscopy: a Textbook for Materials Science (Plenum, New York, 1996), Sec. 9, p. 133.

2002

A. Patwardhan, “Coherent non-planar illumination of a defocused specimen: consequences for transmission electron microscopy,” Optik 113, 4–12 (2002).
[CrossRef]

2000

J. Frank, P. Penczek, R. K. Agrawal, R. A. Grassucci, A. B. Heagle, “Three-dimensional cryoelectron microscopy of ribosomes.,” Methods Enzymol. 317, 276–291 (2000).
[CrossRef]

M. van Heel, B. Gowen, R. Matadeen, E. Orlova, R. Finn, T. Pape, D. Cohen, H. Stark, R. Schmidt, M. Schatz, A. Patwardhan, “Single-particle electron cryomicroscopy: towards atomic resolution,” Quart. Rev. Biophys. 33, 307–369 (2000).
[CrossRef]

1998

S. Uhlemann, M. Haider, “Residual wave aberrations in the first spherical aberration corrected transmission electron microscope,” Ultramicroscopy 72, 109–119 (1998).
[CrossRef]

M. Haider, H. Rose, S. Uhlemann, E. Schwan, B. Kabius, K. Urban, “A spherical-aberration-corrected 200 kV transmission electron microscope,” Ultramicroscopy 75, 53–60 (1998).
[CrossRef]

1986

W. Coene, D. Van Dyck, J. Van Landuyt, “An extension of the standard theory of partial coherence for the effect of beam convergence in high resolution electron microscopy,” Optik 73, 13–18 (1986).

1985

V. P. Ivanchenkov, P. V. Orlov, “Operator description of optical computation systems with a partially coherent light source,” Avtometriya 5, 63–70 (1985).

1981

C. J. Humphreys, J. C. H. Spence, “Resolution and illumination coherence in electron microscopy,” Optik 58, 125–144 (1981).

M. Nazarathy, J. Shamir, “Holography described by operator algebra,” J. Opt. Soc. Am. 71, 529–541 (1981).
[CrossRef]

1980

M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
[CrossRef]

P. W. Hawkes, “Electron microscope transfer functions in closed form with tilted illumination,” Optik 55, 207–212 (1980).

1978

R. H. Wade, W. K. Jenkins, “Tilted beam electron microscopy: the effective coherent aperture,” Optik 50, 1–17 (1978).

F. Zemlin, K. Weiss, P. Schiske, W. Kunath, K.-H. Herrmann, “Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms,” Ultramicroscopy 3, 49–60 (1978).
[CrossRef]

1977

R. H. Wade, J. Frank, “Electron microscope transfer functions for partially coherent axial illumination and chromatic defocus spread,” Optik 49, 81–92 (1977).

H. J. Butterweck, “General theory of linear, coherent optical data-processing systems,” J. Opt. Soc. Am. 67, 60–70 (1977).
[CrossRef]

1976

R. H. Wade, “Concerning tilted beam electron microscope transfer functions,” Optik 45, 87–91 (1976).

D. L. Misell, “On the validity of the weak-phase and other approximations in the analysis of electron microscope images,” J. Phys. D 9(13), 1849–1866 (1976).
[CrossRef]

1975

S. C. McFarlane, “The imaging of amorphous specimens in a tilted-beam electron microscope,” J. Phys. C 8, 2819–2836 (1975).
[CrossRef]

1973

K. H. Downing, B. M. Siegel, “Phase shift determination in single-sideband holography,” Optik 38, 21–28 (1973).

J. Frank, “The envelope of electron microscopic transfer functions for partially coherent illumination,” Optik 38, 519–536 (1973).

1972

1971

K.-J. Hanzen, “The optical transfer theory of the electron microscope: fundamental principles and applications,” Adv. Opt. Electron Microsc. 4, 1–84 (1971).

1968

P. W. Hawkes, “The dependence of the spherical aberration coefficient of an electron-optical objective lens on object position and magnification,” Br. J. Appl. Phys. 1, 131–133 (1968).

O. Bryngdahl, A. Lohmann, “Single-sideband holography,” J. Opt. Soc. Am. 58, 620–624 (1968).
[CrossRef]

1949

O. Scherzer, “The theoretical resolution limit of the electron microscope,” J. Appl. Phys. 20, 20–29 (1949).
[CrossRef]

Agrawal, R. K.

J. Frank, P. Penczek, R. K. Agrawal, R. A. Grassucci, A. B. Heagle, “Three-dimensional cryoelectron microscopy of ribosomes.,” Methods Enzymol. 317, 276–291 (2000).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).

Bryngdahl, O.

Butterweck, H. J.

H. J. Butterweck, “General theory of linear, coherent optical data-processing systems,” J. Opt. Soc. Am. 67, 60–70 (1977).
[CrossRef]

H. J. Butterweck, “Principles of optical data processing,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1981), pp. 213–280.

Carter, C. B.

D. B. Williams, C. B. Carter, Transmission Electron Microscopy: a Textbook for Materials Science (Plenum, New York, 1996), Sec. 9, p. 133.

Coene, W.

W. Coene, D. Van Dyck, J. Van Landuyt, “An extension of the standard theory of partial coherence for the effect of beam convergence in high resolution electron microscopy,” Optik 73, 13–18 (1986).

Cohen, D.

M. van Heel, B. Gowen, R. Matadeen, E. Orlova, R. Finn, T. Pape, D. Cohen, H. Stark, R. Schmidt, M. Schatz, A. Patwardhan, “Single-particle electron cryomicroscopy: towards atomic resolution,” Quart. Rev. Biophys. 33, 307–369 (2000).
[CrossRef]

Downing, K. H.

K. H. Downing, B. M. Siegel, “Phase shift determination in single-sideband holography,” Optik 38, 21–28 (1973).

Finn, R.

M. van Heel, B. Gowen, R. Matadeen, E. Orlova, R. Finn, T. Pape, D. Cohen, H. Stark, R. Schmidt, M. Schatz, A. Patwardhan, “Single-particle electron cryomicroscopy: towards atomic resolution,” Quart. Rev. Biophys. 33, 307–369 (2000).
[CrossRef]

Frank, J.

J. Frank, P. Penczek, R. K. Agrawal, R. A. Grassucci, A. B. Heagle, “Three-dimensional cryoelectron microscopy of ribosomes.,” Methods Enzymol. 317, 276–291 (2000).
[CrossRef]

R. H. Wade, J. Frank, “Electron microscope transfer functions for partially coherent axial illumination and chromatic defocus spread,” Optik 49, 81–92 (1977).

J. Frank, “The envelope of electron microscopic transfer functions for partially coherent illumination,” Optik 38, 519–536 (1973).

Goodman, J. W.

D. A. Tichenor, J. W. Goodman, “Coherent transfer function,” J. Opt. Soc. Am. 62, 293–295 (1972).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, Singapore, 1996).

J. W. Goodman, Statistical Optics (Wiley, New York, 2000), Sec. 7.2.1, pp. 303–304.

Gowen, B.

M. van Heel, B. Gowen, R. Matadeen, E. Orlova, R. Finn, T. Pape, D. Cohen, H. Stark, R. Schmidt, M. Schatz, A. Patwardhan, “Single-particle electron cryomicroscopy: towards atomic resolution,” Quart. Rev. Biophys. 33, 307–369 (2000).
[CrossRef]

Grassucci, R. A.

J. Frank, P. Penczek, R. K. Agrawal, R. A. Grassucci, A. B. Heagle, “Three-dimensional cryoelectron microscopy of ribosomes.,” Methods Enzymol. 317, 276–291 (2000).
[CrossRef]

Haider, M.

S. Uhlemann, M. Haider, “Residual wave aberrations in the first spherical aberration corrected transmission electron microscope,” Ultramicroscopy 72, 109–119 (1998).
[CrossRef]

M. Haider, H. Rose, S. Uhlemann, E. Schwan, B. Kabius, K. Urban, “A spherical-aberration-corrected 200 kV transmission electron microscope,” Ultramicroscopy 75, 53–60 (1998).
[CrossRef]

Hanzen, K.-J.

K.-J. Hanzen, “The optical transfer theory of the electron microscope: fundamental principles and applications,” Adv. Opt. Electron Microsc. 4, 1–84 (1971).

Hawkes, P. W.

P. W. Hawkes, “Electron microscope transfer functions in closed form with tilted illumination,” Optik 55, 207–212 (1980).

P. W. Hawkes, “The dependence of the spherical aberration coefficient of an electron-optical objective lens on object position and magnification,” Br. J. Appl. Phys. 1, 131–133 (1968).

P. W. Hawkes, E. Kasper, Principles of Electron Optics: Wave Optics (Academic, London, 1994), Vol. 3.

Heagle, A. B.

J. Frank, P. Penczek, R. K. Agrawal, R. A. Grassucci, A. B. Heagle, “Three-dimensional cryoelectron microscopy of ribosomes.,” Methods Enzymol. 317, 276–291 (2000).
[CrossRef]

Herrmann, K.-H.

F. Zemlin, K. Weiss, P. Schiske, W. Kunath, K.-H. Herrmann, “Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms,” Ultramicroscopy 3, 49–60 (1978).
[CrossRef]

Humphreys, C. J.

C. J. Humphreys, J. C. H. Spence, “Resolution and illumination coherence in electron microscopy,” Optik 58, 125–144 (1981).

Ivanchenkov, V. P.

V. P. Ivanchenkov, P. V. Orlov, “Operator description of optical computation systems with a partially coherent light source,” Avtometriya 5, 63–70 (1985).

Jenkins, W. K.

R. H. Wade, W. K. Jenkins, “Tilted beam electron microscopy: the effective coherent aperture,” Optik 50, 1–17 (1978).

Kabius, B.

M. Haider, H. Rose, S. Uhlemann, E. Schwan, B. Kabius, K. Urban, “A spherical-aberration-corrected 200 kV transmission electron microscope,” Ultramicroscopy 75, 53–60 (1998).
[CrossRef]

Kasper, E.

P. W. Hawkes, E. Kasper, Principles of Electron Optics: Wave Optics (Academic, London, 1994), Vol. 3.

Kunath, W.

F. Zemlin, K. Weiss, P. Schiske, W. Kunath, K.-H. Herrmann, “Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms,” Ultramicroscopy 3, 49–60 (1978).
[CrossRef]

Lohmann, A.

Matadeen, R.

M. van Heel, B. Gowen, R. Matadeen, E. Orlova, R. Finn, T. Pape, D. Cohen, H. Stark, R. Schmidt, M. Schatz, A. Patwardhan, “Single-particle electron cryomicroscopy: towards atomic resolution,” Quart. Rev. Biophys. 33, 307–369 (2000).
[CrossRef]

McFarlane, S. C.

S. C. McFarlane, “The imaging of amorphous specimens in a tilted-beam electron microscope,” J. Phys. C 8, 2819–2836 (1975).
[CrossRef]

Misell, D. L.

D. L. Misell, “On the validity of the weak-phase and other approximations in the analysis of electron microscope images,” J. Phys. D 9(13), 1849–1866 (1976).
[CrossRef]

Nazarathy, M.

Orlov, P. V.

V. P. Ivanchenkov, P. V. Orlov, “Operator description of optical computation systems with a partially coherent light source,” Avtometriya 5, 63–70 (1985).

Orlova, E.

M. van Heel, B. Gowen, R. Matadeen, E. Orlova, R. Finn, T. Pape, D. Cohen, H. Stark, R. Schmidt, M. Schatz, A. Patwardhan, “Single-particle electron cryomicroscopy: towards atomic resolution,” Quart. Rev. Biophys. 33, 307–369 (2000).
[CrossRef]

Pape, T.

M. van Heel, B. Gowen, R. Matadeen, E. Orlova, R. Finn, T. Pape, D. Cohen, H. Stark, R. Schmidt, M. Schatz, A. Patwardhan, “Single-particle electron cryomicroscopy: towards atomic resolution,” Quart. Rev. Biophys. 33, 307–369 (2000).
[CrossRef]

Patwardhan, A.

A. Patwardhan, “Coherent non-planar illumination of a defocused specimen: consequences for transmission electron microscopy,” Optik 113, 4–12 (2002).
[CrossRef]

M. van Heel, B. Gowen, R. Matadeen, E. Orlova, R. Finn, T. Pape, D. Cohen, H. Stark, R. Schmidt, M. Schatz, A. Patwardhan, “Single-particle electron cryomicroscopy: towards atomic resolution,” Quart. Rev. Biophys. 33, 307–369 (2000).
[CrossRef]

Penczek, P.

J. Frank, P. Penczek, R. K. Agrawal, R. A. Grassucci, A. B. Heagle, “Three-dimensional cryoelectron microscopy of ribosomes.,” Methods Enzymol. 317, 276–291 (2000).
[CrossRef]

Reimer, L.

L. Reimer, Transmission Electron Microscopy: Physics of Image Formation and Microanalysis (Springer-Verlag, New York, 1997).

Rose, H.

M. Haider, H. Rose, S. Uhlemann, E. Schwan, B. Kabius, K. Urban, “A spherical-aberration-corrected 200 kV transmission electron microscope,” Ultramicroscopy 75, 53–60 (1998).
[CrossRef]

Saxton, W. O.

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

Schatz, M.

M. van Heel, B. Gowen, R. Matadeen, E. Orlova, R. Finn, T. Pape, D. Cohen, H. Stark, R. Schmidt, M. Schatz, A. Patwardhan, “Single-particle electron cryomicroscopy: towards atomic resolution,” Quart. Rev. Biophys. 33, 307–369 (2000).
[CrossRef]

Scherzer, O.

O. Scherzer, “The theoretical resolution limit of the electron microscope,” J. Appl. Phys. 20, 20–29 (1949).
[CrossRef]

Schiske, P.

F. Zemlin, K. Weiss, P. Schiske, W. Kunath, K.-H. Herrmann, “Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms,” Ultramicroscopy 3, 49–60 (1978).
[CrossRef]

Schmidt, R.

M. van Heel, B. Gowen, R. Matadeen, E. Orlova, R. Finn, T. Pape, D. Cohen, H. Stark, R. Schmidt, M. Schatz, A. Patwardhan, “Single-particle electron cryomicroscopy: towards atomic resolution,” Quart. Rev. Biophys. 33, 307–369 (2000).
[CrossRef]

Schwan, E.

M. Haider, H. Rose, S. Uhlemann, E. Schwan, B. Kabius, K. Urban, “A spherical-aberration-corrected 200 kV transmission electron microscope,” Ultramicroscopy 75, 53–60 (1998).
[CrossRef]

Shamir, J.

Siegel, B. M.

K. H. Downing, B. M. Siegel, “Phase shift determination in single-sideband holography,” Optik 38, 21–28 (1973).

Spence, J. C. H.

C. J. Humphreys, J. C. H. Spence, “Resolution and illumination coherence in electron microscopy,” Optik 58, 125–144 (1981).

Stark, H.

M. van Heel, B. Gowen, R. Matadeen, E. Orlova, R. Finn, T. Pape, D. Cohen, H. Stark, R. Schmidt, M. Schatz, A. Patwardhan, “Single-particle electron cryomicroscopy: towards atomic resolution,” Quart. Rev. Biophys. 33, 307–369 (2000).
[CrossRef]

Tichenor, D. A.

Uhlemann, S.

S. Uhlemann, M. Haider, “Residual wave aberrations in the first spherical aberration corrected transmission electron microscope,” Ultramicroscopy 72, 109–119 (1998).
[CrossRef]

M. Haider, H. Rose, S. Uhlemann, E. Schwan, B. Kabius, K. Urban, “A spherical-aberration-corrected 200 kV transmission electron microscope,” Ultramicroscopy 75, 53–60 (1998).
[CrossRef]

Urban, K.

M. Haider, H. Rose, S. Uhlemann, E. Schwan, B. Kabius, K. Urban, “A spherical-aberration-corrected 200 kV transmission electron microscope,” Ultramicroscopy 75, 53–60 (1998).
[CrossRef]

Van Dyck, D.

W. Coene, D. Van Dyck, J. Van Landuyt, “An extension of the standard theory of partial coherence for the effect of beam convergence in high resolution electron microscopy,” Optik 73, 13–18 (1986).

van Heel, M.

M. van Heel, B. Gowen, R. Matadeen, E. Orlova, R. Finn, T. Pape, D. Cohen, H. Stark, R. Schmidt, M. Schatz, A. Patwardhan, “Single-particle electron cryomicroscopy: towards atomic resolution,” Quart. Rev. Biophys. 33, 307–369 (2000).
[CrossRef]

Van Landuyt, J.

W. Coene, D. Van Dyck, J. Van Landuyt, “An extension of the standard theory of partial coherence for the effect of beam convergence in high resolution electron microscopy,” Optik 73, 13–18 (1986).

Wade, R. H.

R. H. Wade, W. K. Jenkins, “Tilted beam electron microscopy: the effective coherent aperture,” Optik 50, 1–17 (1978).

R. H. Wade, J. Frank, “Electron microscope transfer functions for partially coherent axial illumination and chromatic defocus spread,” Optik 49, 81–92 (1977).

R. H. Wade, “Concerning tilted beam electron microscope transfer functions,” Optik 45, 87–91 (1976).

Weiss, K.

F. Zemlin, K. Weiss, P. Schiske, W. Kunath, K.-H. Herrmann, “Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms,” Ultramicroscopy 3, 49–60 (1978).
[CrossRef]

Williams, D. B.

D. B. Williams, C. B. Carter, Transmission Electron Microscopy: a Textbook for Materials Science (Plenum, New York, 1996), Sec. 9, p. 133.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).

Zemlin, F.

F. Zemlin, K. Weiss, P. Schiske, W. Kunath, K.-H. Herrmann, “Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms,” Ultramicroscopy 3, 49–60 (1978).
[CrossRef]

Adv. Opt. Electron Microsc.

K.-J. Hanzen, “The optical transfer theory of the electron microscope: fundamental principles and applications,” Adv. Opt. Electron Microsc. 4, 1–84 (1971).

Avtometriya

V. P. Ivanchenkov, P. V. Orlov, “Operator description of optical computation systems with a partially coherent light source,” Avtometriya 5, 63–70 (1985).

Br. J. Appl. Phys.

P. W. Hawkes, “The dependence of the spherical aberration coefficient of an electron-optical objective lens on object position and magnification,” Br. J. Appl. Phys. 1, 131–133 (1968).

J. Appl. Phys.

O. Scherzer, “The theoretical resolution limit of the electron microscope,” J. Appl. Phys. 20, 20–29 (1949).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. C

S. C. McFarlane, “The imaging of amorphous specimens in a tilted-beam electron microscope,” J. Phys. C 8, 2819–2836 (1975).
[CrossRef]

J. Phys. D

D. L. Misell, “On the validity of the weak-phase and other approximations in the analysis of electron microscope images,” J. Phys. D 9(13), 1849–1866 (1976).
[CrossRef]

Methods Enzymol.

J. Frank, P. Penczek, R. K. Agrawal, R. A. Grassucci, A. B. Heagle, “Three-dimensional cryoelectron microscopy of ribosomes.,” Methods Enzymol. 317, 276–291 (2000).
[CrossRef]

Optik

W. Coene, D. Van Dyck, J. Van Landuyt, “An extension of the standard theory of partial coherence for the effect of beam convergence in high resolution electron microscopy,” Optik 73, 13–18 (1986).

J. Frank, “The envelope of electron microscopic transfer functions for partially coherent illumination,” Optik 38, 519–536 (1973).

C. J. Humphreys, J. C. H. Spence, “Resolution and illumination coherence in electron microscopy,” Optik 58, 125–144 (1981).

R. H. Wade, J. Frank, “Electron microscope transfer functions for partially coherent axial illumination and chromatic defocus spread,” Optik 49, 81–92 (1977).

P. W. Hawkes, “Electron microscope transfer functions in closed form with tilted illumination,” Optik 55, 207–212 (1980).

R. H. Wade, “Concerning tilted beam electron microscope transfer functions,” Optik 45, 87–91 (1976).

R. H. Wade, W. K. Jenkins, “Tilted beam electron microscopy: the effective coherent aperture,” Optik 50, 1–17 (1978).

A. Patwardhan, “Coherent non-planar illumination of a defocused specimen: consequences for transmission electron microscopy,” Optik 113, 4–12 (2002).
[CrossRef]

K. H. Downing, B. M. Siegel, “Phase shift determination in single-sideband holography,” Optik 38, 21–28 (1973).

Quart. Rev. Biophys.

M. van Heel, B. Gowen, R. Matadeen, E. Orlova, R. Finn, T. Pape, D. Cohen, H. Stark, R. Schmidt, M. Schatz, A. Patwardhan, “Single-particle electron cryomicroscopy: towards atomic resolution,” Quart. Rev. Biophys. 33, 307–369 (2000).
[CrossRef]

Ultramicroscopy

F. Zemlin, K. Weiss, P. Schiske, W. Kunath, K.-H. Herrmann, “Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms,” Ultramicroscopy 3, 49–60 (1978).
[CrossRef]

M. Haider, H. Rose, S. Uhlemann, E. Schwan, B. Kabius, K. Urban, “A spherical-aberration-corrected 200 kV transmission electron microscope,” Ultramicroscopy 75, 53–60 (1998).
[CrossRef]

S. Uhlemann, M. Haider, “Residual wave aberrations in the first spherical aberration corrected transmission electron microscope,” Ultramicroscopy 72, 109–119 (1998).
[CrossRef]

Other

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A. Patwardhan, “Maple worksheet for the numerical integration of contrast transfer functions” (2002), retrieved http://www.cbem.ic.ac.uk/ardan/opalg/opalg2.mws .

J. W. Goodman, Statistical Optics (Wiley, New York, 2000), Sec. 7.2.1, pp. 303–304.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999).

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H. J. Butterweck, “Principles of optical data processing,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1981), pp. 213–280.

A. Patwardhan, “Verification of intensity expressions using Maple” (2002), retrieved http://www.cbem.ic.ac.uk/ardan/opalg/opalg1.html .

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

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Figures (4)

Fig. 1
Fig. 1

Depiction of the two lens model examined in this study. The distances between the source and the condenser, d12, between the condenser and objective lenses, d23+d34, and between the objective lens and the image plane, d45 are assumed to be constant. The specimen is assumed to be neither in condenser focus nor in objective focus.

Fig. 2
Fig. 2

Example of a rectangular aperture and various corresponding aperture functions when the beam is tilted by ν=(νmax/2, 0): (a) Ps(ν), (b) Ps(νx-νmax/2, νy), (c) Pe, and (d) Po.

Fig. 3
Fig. 3

Comparison of the numerical integration of Eq. (51) (solid curves) and Eq. (55) (dashed curves) under the assumptions that a phase specimen is imaged at Δd=0, that the objective aperture is infinitely large, and that the source is described by Eqs. (56)–(58) with νs=0.01 Å-1. The following parameters are also assumed constant: Csph=2 mm, λ=2 pm, χ=1, and θ=0°. The integration was performed with Maple 7.0 (Ref. 35) for a spatial-frequency range of 0.01 to 0.5 Å-1 with an increment of 0.00245 Å-1. The pairs of curves (a), (b); (c), (d); (e), (f); (g), (h); and (i), (j) represent the (amplitude, phase) of the integrals for (ν, ω) values of (0, 0°), (0.03, 0°), (0.2, 0°), (0.2, 90°), and (0.5, 0°), respectively, where ν is given in units of inverse angstroms. The x axis represents ν in terms of inverse angstroms.

Fig. 4
Fig. 4

Conditions and Parameters are identical to those used in Fig. 3 except that the specimen defocus is Δd=-1 μm.

Equations (98)

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u2(ρ2)=A(ρ1)l12exp(jkl12),
u2(ρ2)=A(ρ1)d12expjkρ122d12expjkρ222d12exp-jk(ρ1  ρ2)d12,
u2(ρ2)=A(ρ1)d12expjkρ122d12Qρ21d12Gρ2-ρ1d12.
u3(ρ3)=Rρ3[d23]Pρ2cQρ2[-1/fc]u2.
u5(ρ5)=Rρ5[d45]Pρ4oWρ4Qρ4[-1/fo]Rρ4[d34]Tρ3u3.
u5(ρ5)=C1(ρ1)Q1d45V1λd45FP oWQ1dαV1λd34×FTQ1dβV1λd23FQ1dγPcG-ρ1d12,
C1(ρ1)=1λ3d12d23d34d45expjkρ122d12A(ρ1),
1dα=1d34+1d45-1fo,
1dβ=1d23+1d34,
1dγ=1d12+1d23-1fc.
λdγQρ31dβVρ31λd23Qρ3[-λ2dγ]Vρ3[-λdγ]
×Fρ3Qνc[-λ2dγ]FνcPρ2cGρ2-ρ1d12,
Qρ1-dγd122Sνc-ρ1λd12Gνcλdγρ1d12Qνc[-λ2dγ]FνcPρ2c.
πλ λ2dγ 1.4wc21|Δc|1.4πλ dicd23wc2.
2πλ  λdγρ1d12  1.2wc1|Δc|2.4πρ1Mcd23wc,
λdγQρ1-dγd122Q1dVdγd23PcG-ρ1d12,
1d=1dβ-dγd232.
u5(ρ5)=λdγQρ1-dγd122C1(ρ1)Q1d45V1λd45×FP oWQ1dα×V1λd34F TQ1dVdγd23P cG-ρ1d12.
u5=λ2dγd342dQρ1-dγd122C1(ρ1)Qρ51d45Vρ5-d34λdd45×Fρ5Vρ4-d34dPρ4oWρ4Qρ41dϕ×Fρ4Qν[-λ2d]FνTρ3Vρ3dγd23Pρ3cGρ3-ρ1d12,
1dϕ=d342d2dα-1d.
Fρ5gρ4Fρ4hν=Iν{(hν)(Fρ5gρ4Gρ4[-λν])}.
Fρ5gρ4Gρ4[-λν]
=Fρ5Vρ4-d34dPρ4oWρ4Qρ41dϕGρ4[-λν],
Fρ5Qρ41dϕGρ4[-λν]  Fρ5Vρ4-d34dPρ4oWρ4.
(λdϕQν[-λ2dϕ])Gρ5[-λ2dϕν]Qρ5[-λ2dϕ]
λdϕ(Gρ5[-λ2dϕν]Qρ5[-λ2dϕ])Qν[-λ2dϕ]
×Vν-λdϕd34dPνoWν,
u5=λ3dγdϕd342dQρ1-dγd122C1(ρ1)Qρ51d45-d342dϕd2d452×Vρ5-d34λdd45×IνQν[-λ2d]FνTρ3Vρ3dγd23Pρ3cGρ3-ρ1d12×Qν[-λ2dϕ]Vν-λdϕd34dPνoWνGρ5[-λ2dϕν].
Iν(gνGρ5[-λ2dϕν])=Vρ5[λdϕ]Fρ5gν,
u5=λ3dϕdγd342dQρ1-dγd122C1(ρ1)Qρ51d45-do2dϕχ2d452×Vρ5doχd45Fρ5VνλdoχQνχd34dodαPνoWν×FνTρ3Vρ3dγd23Pρ3cGρ3-ρ1d12,
χ=-dod34ddϕ
u5=λ3dϕdγd342dQρ1-dγd122C1(ρ1)×Qρ51d45-do2dϕχ2d452Gρ5dγdoρ1χd12d23d45×Vρ5doχd45Fρ5Sνdγρ1λd12d23×VνλdoχQνχd34dodαPνoWν×FνTρ3Vρ3dγd23Pρ3c.
Tρ3Vρ3dγd23Pρ3c=exp[-E(ρ3)+jH(ρ3)]1-E(ρ3)+jH(ρ3),
δ(ν)-(ν)+jη(ν),
I=C2*C2|Fdoρ5/d45χ{Ps(ν-ν)exp[-jkW(ν-ν)]×[δ(ν)-(ν)+jη(ν)]}|2,
C2=dϕdγd34d12d23d45d A(ρ1),
ν=dγρ1λd12d23,
Ps(ν)=VνλdoχPνo,
exp[-jkW(ν)]=VνλdoχQνχd34dodαWν.
Pe=12[Ps(ν-ν)+Ps(ν+ν)],
Po=12[Ps(ν-ν)-Ps(ν+ν)].
Z=(Pe+Po)[δ-(r+ji)+j(ηr+jηi)]×(Ce-jSe)(Co-jSo)(Cf+jSf).
X=Pe{[(δ-r)Ce+ηrSe]Cf-o+(iCe-ηiSe)Sf-o}+Po{(-iSe-ηiCe)Cf-o+[(δ-r)Se-ηrCe]Sf-o},
Y=Pe{[(-δ+r)Se+ηrCe]Cf-o+(-iSe-ηiCe)Sf-o}+Po{(-iCe+ηiSe)Cf-o+[(δ-r)Ce+ηrSe]Sf-o},
IνX=Ps(ν)Ce0+IνPe[(-rCe+ηrSe)Cf-o+(iCe-ηiSe)Sf-o]+IνPo[(-iSe-ηiCe)Cf-o+(-rSe-ηrCe)Sf-o],
IνY=-Ps(ν)Se0+IνPe[(rSe+ηrCe)Cf-o+(-iSe-ηiCe)Sf-o]+IνPo[(-iCe+ηiSe)×Cf-o+(-rCe+ηrSe)Sf-o].
(IνX)2Ps(ν)2Ce02+2Ps(ν)Ce0IνPe[(-rCe+ηrSe)Cf-o+(iCe-ηiSe)Sf-o]+2Ps(ν)CeoIνPo[(-iSe-ηiCe)Cf-o+(-rSe-ηrCe)Sf-o],
(IνY)2Ps(ν)2Seo2-2Ps(ν)Se0IνPe[(rSe+ηrCe)Cf-o+(-iSe-ηiCe)Sf-o]-2Ps(ν)Se0IνPo[(-iCe+ηiSe)Cf-o+(-rCe+ηrSe)Sf-o].
I=|C2|2Ps(ν)2+2Ps(ν)Vρ5doχd45Fρ5(Pe{- cos(kWe)+η sin(kWe)}exp(-jkWo))+2jPs(ν)Vρ5doχd45Fρ5(Po{ sin(kWe)+η cos(kWe)}exp(-jkWo)),
FνI-I0I0=2χd45do2Vν-d45χdo(Peexp(-jkWo)×{- cos(kWe)+η sin(kWe)}+jPoexp(-jkWo){+ sin(kWe)+η cos(kWe)}).
FνI-I0I0=2χd45do2Vν-d45χdo(-Pe+jPoη).
η(ν)=12doχd452exp(jkWo)Vν-dod45χFνI-I0I0{Pesin(kWe)+jPocos(kWe)},
dI=|C2|2g(ρ1-ρ10)Ps(ν)2+2Ps(ν)Vρ5doχd45Fρ5{Pe(-Ce+ηSe)exp(-jkWo)}+2jPs(ν)Vρ5doχd45Fρ5{Po(Se+ηCe)exp(-jkWo)}dρ1,
FνI-I0I0=2 |C2|2I0λd12d23dγχd45do2Vν-χd45do×{Pe(-Ce+ηSe)+jPo(Se+ηCe)}Ps(ν10+ν)×exp(-jkWo)gλd12d23νdγdν,
I0=|C2|2Ps(ν)2g(ρ1-ρ10)dρ1,
ν10=dγρ10λd12d23,
Wo=W(ν-ν)o=W(ν-ν10-ν)oW(ν-ν10)o-ννW(ν-ν10)e=Wo-ννWe.
Fν×I-I0I0=2 |C2|2I0χd45do2Vν-χd45do×Ps(ν10)exp(-jkWo)×{Pe(-Ce+ηSe)+jPo(Se+ηCe)}×G˜-dγλ2d12d23 νWe,
g(ρ)=Bπρs2exp-ρ2ρs2,
G˜-dγλ2d12d23 νWe=B exp-π2νWeλ2νs2,
νs=ρsdγλd12d23=ρsλdicd121dic-d23.
Wρ4=14 Csphρ4+Castκ4+12 Ccofρ302-χdod342dαρ2-Cdisρ302κ2-Ccomρ2κ2,
ρ2=|ρ4|2do2,κ2=ρ30ρ4do.
ρ2=λ2χ2 |ν-ν|2,κ2=λχρ30  (ν-ν).
We=14 Csphλ4χ4 [ν4+4ν2ν2cos2(θ-ω)+2ν2ν2+ν4]+Castλ2χ2 ρ302[ν2cos2(θ-φ)+ν2cos2(ω-φ)]+12 Ccofρ302-χdod342dαλ2χ2 (ν2+ν2)+Cdisλχ ρ303νcos(ω-φ)+Ccomλ3χ3 ρ30νν22 cos(ω-φ)-1+2 cos2θ-(ω+φ)2+ν3cos(ω-φ),
Wo=-Csphλ4χ4 [(νν3+ν3ν)cos(θ-ω)]-2 Castλ2χ2 ρ302νν cos(ω-φ)cos(θ-φ)-212 Ccofρ302-χdod342dαλ2χ2νν cos(θ-ω)-Cdisλχ ρ303ν cos(θ-φ)-Ccomλ3χ3 ρ30{ν3cos(θ-φ)+ν2ν[2 cos(θ-φ)+cos(θ-2ω+φ)]}.
We=14 Csphλ4χ4 [ν4+4ν2ν2cos2(θ-ω)+2ν2ν2]+Castλ2χ2 ρ302ν2cos2(θ-φ)+12 Ccofρ302-χdod342dαλ2χ2ν2+Ccomλ3χ3 ρ30νν22 cos(ω-φ)-1+2 cos2θ-(ω+φ)2.
We=Csphλ4χ4ν3+2Csphλ4χ4ν102-λ2dod34χdαν×cos(θ), sin(θ)+Csphλ4χ4ν102ν×cos(2ω-θ), sin(2ω-θ).
1dα=-Δddod34,
χd=1/1-Δddc-do,
V[s]{u(ρ)}=u(sρ).
S [m]{u(ρ)}=u(ρ-m).
G [s]{u(ρ)}=exp(jks  ρ)u(ρ).
Q[a]{u(ρ)}=expjk2aρ2u(ρ).
δ[m]{u(ρ)}=δ(ρ-m)u(ρ).
F{u(ρ)}=u(ρ)exp(-j2πν  ρ)dρ.
R[d]{u(ρ)}=exp(jkd)jλdQ1dV1λdFQ1du(ρ).
Q[a1]Q[a2]=Q[a1+a2],
FQ[c]=expjkcλ2R-cλ2F,
FG [s]=SsλF,
gS [m]=S[m](S[-m]g),
(S [m]Qρ[a])=(Qm[a])Gρ[-am]Qρ[a],
FF=V [-1],
V [s1]V [s2]=V [s1s2],
V [s]Q[a]=Q[s2a]V [s],
gV[b]=V [b]V1bg,
FV [b]=1b2V1bF,
Fg=(Fg)  F,
(FQ[a])=jλaQ-λ2a,
(Fδ[m])=G [-λm],
δ[m]g(ρ)=δ [m]g(m),
V [b]G[s]=G [bs]V [b],
FS [m]=G [-λm]F.
Fρ3gρ2Fρ2hρ1=g(ρ2)exp(-j2πρ3ρ2)×h(ρ1)exp(-j2πρ2ρ1)dρ1dρ2,
h(ρ1)g(ρ2)exp(-j2πρ1ρ2)
×exp(-j2πρ3ρ2)dρ2dρ1.
h(ρ1)(Fρ3gρ2Gρ2[-λρ1])dρ1
=Iρ1{(hρ1)(Fρ3gρ2Gρ2[-λρ1])},

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