Abstract

An investigation of the imaging of spheres can aid in understanding details of the surface profiling method of confocal microscopy. The use of semicircular masks to eliminate artifacts in confocal profiling is investigated experimentally. A theoretical treatment of image formation for a spherical object in reflection confocal microscopy is presented. For large spheres a simple approximate theory is described and is shown to be equivalent to application of the Kirchhoff theory for surface scattering.

© 2000 Optical Society of America

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References

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  1. W. Weise, P. Zinin, T. Wilson, A. Briggs, S. Boseck, “Imaging of spheres with the confocal scanning optical microscope,” Opt. Lett. 21, 1800–1802 (1996).
    [CrossRef] [PubMed]
  2. D. M. Maurice, “Cellular membrane activity in the corneal endothelium of the intact eye,” Experientia 15, 1094–1095 (1968).
    [CrossRef]
  3. C. J. Koester, “A scanning mirror microscope with optical sectioning characteristics: applications in ophthalmology,” Appl. Opt. 19, 1749–1757 (1980).
    [CrossRef] [PubMed]
  4. C. J. R. Sheppard, D. K. Hamilton, “High resolution stereoscopic imaging,” Appl. Opt. 22, 886–887 (1983).
    [CrossRef] [PubMed]
  5. P. Török, C. J. R. Sheppard, Z. Laczik, “Dark-field and differential phase contrast imaging modes in confocal microscopy using a half-aperture stop,” Optik 103, 101–106 (1996).
  6. P. Török, C. J. R. Sheppard, Z. Laczik, “The effect of half-stop lateral misalignment on imaging of dark-field and stereoscopic confocal microscopes,” Appl. Opt. 35, 6732–6739 (1996).
    [CrossRef] [PubMed]
  7. C. J. R. Sheppard, J. T. Sheridan, “Micrometrology of thick structures,” in Optical Storage and Scanning Technology, T. Wilson, ed., Proc. SPIE1139, 32–39 (1989).
    [CrossRef]
  8. C. J. R. Sheppard, “General considerations of diffraction theory of 3-D imaging,” Eur. J. Cell Biol. Suppl. 25, 29–32 (1989).
  9. J. T. Sheridan, C. J. R. Sheppard, “Diffraction by striated muscle fibres: application to image modelling,” Bioimaging 1, 214–227 (1993).
    [CrossRef]
  10. J. T. Sheridan, C. J. R. Sheppard, “The coherent imaging of periodic thick fine isolated structures,” J. Opt. Soc. Am. A 10, 614–632 (1993).
    [CrossRef]
  11. J. T. Sheridan, C. J. R. Sheppard, “Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory,” Opt. Commun. 105, 367–378 (1994).
    [CrossRef]
  12. C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
    [CrossRef] [PubMed]
  13. N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
    [CrossRef]
  14. C. J. R. Sheppard, M. Gu, “Modeling of 3-D brightfield microscope systems,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 352–358 (1994).
    [CrossRef]
  15. C. J. R. Sheppard, T. J. Connolly, M. Gu, “Imaging and reconstruction for rough surface scattering in the Kirchhoff approximation by confocal microscopy,” J. Mod. Opt. 40, 2407–2421 (1993).
    [CrossRef]
  16. C. J. R. Sheppard, T. J. Connolly, M. Gu, “The scattering potential for imaging in the reflection geometry,” Opt. Commun. 117, 16–19 (1995).
    [CrossRef]
  17. C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
    [CrossRef]
  18. C. J. R. Sheppard, “The spatial frequency cut-off in three dimensional imaging. II,” Optik 74, 128–129 (1986).
  19. C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
    [CrossRef]
  20. C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 11, 593–598 (1994).
    [CrossRef]
  21. L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Nonrelativistic Theory) (Pergamon, Oxford, 1959).
  22. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1978).
  23. J. W. Nicholson, “Diffraction of short waves by a rigid sphere,” Philos. Mag. 11, 193–205 (1906).
    [CrossRef]
  24. J. W. Nicholson, “The scattering of light by a large conducting sphere,” Proc. Math. Soc. London 9, 67–80 (1910).
  25. J. W. Nicholson, “The scattering of light by a large conducting sphere,” Proc. Math. Soc. London 11, 277–284 (1912).
  26. T. J. l’A. Bromwich, “The scattering of plane electric waves by a sphere,” Phil. Trans. R. Soc. London 220, 189–206 (1920).
  27. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).
  28. C. J. R. Sheppard, K. G. Larkin, “Effect of numerical aperture on interference fringe spacing,” Appl. Opt. 34, 4731–4734 (1995).
    [CrossRef] [PubMed]
  29. C. J. R. Sheppard, C. J. Cogswell, “Reflection and transmission confocal microscopy,” in Optics in Medicine, Biology and Environmental Research, G. von Bally, S. Khanna, eds., Proceedings of the First International Conference on Optics within Life Sciences, Garmisch-Partenkirchen, Germany, 1990 (Elsevier, Amsterdam, 1993), pp. 310–315.

1996 (3)

1995 (2)

C. J. R. Sheppard, T. J. Connolly, M. Gu, “The scattering potential for imaging in the reflection geometry,” Opt. Commun. 117, 16–19 (1995).
[CrossRef]

C. J. R. Sheppard, K. G. Larkin, “Effect of numerical aperture on interference fringe spacing,” Appl. Opt. 34, 4731–4734 (1995).
[CrossRef] [PubMed]

1994 (2)

J. T. Sheridan, C. J. R. Sheppard, “Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory,” Opt. Commun. 105, 367–378 (1994).
[CrossRef]

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 11, 593–598 (1994).
[CrossRef]

1993 (4)

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Imaging and reconstruction for rough surface scattering in the Kirchhoff approximation by confocal microscopy,” J. Mod. Opt. 40, 2407–2421 (1993).
[CrossRef]

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

J. T. Sheridan, C. J. R. Sheppard, “Diffraction by striated muscle fibres: application to image modelling,” Bioimaging 1, 214–227 (1993).
[CrossRef]

J. T. Sheridan, C. J. R. Sheppard, “The coherent imaging of periodic thick fine isolated structures,” J. Opt. Soc. Am. A 10, 614–632 (1993).
[CrossRef]

1991 (1)

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

1990 (1)

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
[CrossRef]

1989 (1)

C. J. R. Sheppard, “General considerations of diffraction theory of 3-D imaging,” Eur. J. Cell Biol. Suppl. 25, 29–32 (1989).

1986 (1)

C. J. R. Sheppard, “The spatial frequency cut-off in three dimensional imaging. II,” Optik 74, 128–129 (1986).

1985 (1)

1983 (1)

1980 (1)

1968 (1)

D. M. Maurice, “Cellular membrane activity in the corneal endothelium of the intact eye,” Experientia 15, 1094–1095 (1968).
[CrossRef]

1920 (1)

T. J. l’A. Bromwich, “The scattering of plane electric waves by a sphere,” Phil. Trans. R. Soc. London 220, 189–206 (1920).

1912 (1)

J. W. Nicholson, “The scattering of light by a large conducting sphere,” Proc. Math. Soc. London 11, 277–284 (1912).

1910 (1)

J. W. Nicholson, “The scattering of light by a large conducting sphere,” Proc. Math. Soc. London 9, 67–80 (1910).

1906 (1)

J. W. Nicholson, “Diffraction of short waves by a rigid sphere,” Philos. Mag. 11, 193–205 (1906).
[CrossRef]

Boseck, S.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).

Briggs, A.

Bromwich, T. J. l’A.

T. J. l’A. Bromwich, “The scattering of plane electric waves by a sphere,” Phil. Trans. R. Soc. London 220, 189–206 (1920).

Cogswell, C. J.

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
[CrossRef]

C. J. R. Sheppard, C. J. Cogswell, “Reflection and transmission confocal microscopy,” in Optics in Medicine, Biology and Environmental Research, G. von Bally, S. Khanna, eds., Proceedings of the First International Conference on Optics within Life Sciences, Garmisch-Partenkirchen, Germany, 1990 (Elsevier, Amsterdam, 1993), pp. 310–315.

Connolly, T. J.

C. J. R. Sheppard, T. J. Connolly, M. Gu, “The scattering potential for imaging in the reflection geometry,” Opt. Commun. 117, 16–19 (1995).
[CrossRef]

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Imaging and reconstruction for rough surface scattering in the Kirchhoff approximation by confocal microscopy,” J. Mod. Opt. 40, 2407–2421 (1993).
[CrossRef]

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1978).

Gu, M.

C. J. R. Sheppard, T. J. Connolly, M. Gu, “The scattering potential for imaging in the reflection geometry,” Opt. Commun. 117, 16–19 (1995).
[CrossRef]

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 11, 593–598 (1994).
[CrossRef]

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Imaging and reconstruction for rough surface scattering in the Kirchhoff approximation by confocal microscopy,” J. Mod. Opt. 40, 2407–2421 (1993).
[CrossRef]

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

C. J. R. Sheppard, M. Gu, “Modeling of 3-D brightfield microscope systems,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 352–358 (1994).
[CrossRef]

Hamilton, D. K.

Kawata, S.

Kawata, Y.

Koester, C. J.

Laczik, Z.

P. Török, C. J. R. Sheppard, Z. Laczik, “Dark-field and differential phase contrast imaging modes in confocal microscopy using a half-aperture stop,” Optik 103, 101–106 (1996).

P. Török, C. J. R. Sheppard, Z. Laczik, “The effect of half-stop lateral misalignment on imaging of dark-field and stereoscopic confocal microscopes,” Appl. Opt. 35, 6732–6739 (1996).
[CrossRef] [PubMed]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Nonrelativistic Theory) (Pergamon, Oxford, 1959).

Larkin, K. G.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Nonrelativistic Theory) (Pergamon, Oxford, 1959).

Mao, X. Q.

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

Maurice, D. M.

D. M. Maurice, “Cellular membrane activity in the corneal endothelium of the intact eye,” Experientia 15, 1094–1095 (1968).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1978).

Nicholson, J. W.

J. W. Nicholson, “The scattering of light by a large conducting sphere,” Proc. Math. Soc. London 11, 277–284 (1912).

J. W. Nicholson, “The scattering of light by a large conducting sphere,” Proc. Math. Soc. London 9, 67–80 (1910).

J. W. Nicholson, “Diffraction of short waves by a rigid sphere,” Philos. Mag. 11, 193–205 (1906).
[CrossRef]

Sheppard, C. J. R.

P. Török, C. J. R. Sheppard, Z. Laczik, “Dark-field and differential phase contrast imaging modes in confocal microscopy using a half-aperture stop,” Optik 103, 101–106 (1996).

P. Török, C. J. R. Sheppard, Z. Laczik, “The effect of half-stop lateral misalignment on imaging of dark-field and stereoscopic confocal microscopes,” Appl. Opt. 35, 6732–6739 (1996).
[CrossRef] [PubMed]

C. J. R. Sheppard, T. J. Connolly, M. Gu, “The scattering potential for imaging in the reflection geometry,” Opt. Commun. 117, 16–19 (1995).
[CrossRef]

C. J. R. Sheppard, K. G. Larkin, “Effect of numerical aperture on interference fringe spacing,” Appl. Opt. 34, 4731–4734 (1995).
[CrossRef] [PubMed]

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 11, 593–598 (1994).
[CrossRef]

J. T. Sheridan, C. J. R. Sheppard, “Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory,” Opt. Commun. 105, 367–378 (1994).
[CrossRef]

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

J. T. Sheridan, C. J. R. Sheppard, “The coherent imaging of periodic thick fine isolated structures,” J. Opt. Soc. Am. A 10, 614–632 (1993).
[CrossRef]

J. T. Sheridan, C. J. R. Sheppard, “Diffraction by striated muscle fibres: application to image modelling,” Bioimaging 1, 214–227 (1993).
[CrossRef]

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Imaging and reconstruction for rough surface scattering in the Kirchhoff approximation by confocal microscopy,” J. Mod. Opt. 40, 2407–2421 (1993).
[CrossRef]

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
[CrossRef]

C. J. R. Sheppard, “General considerations of diffraction theory of 3-D imaging,” Eur. J. Cell Biol. Suppl. 25, 29–32 (1989).

C. J. R. Sheppard, “The spatial frequency cut-off in three dimensional imaging. II,” Optik 74, 128–129 (1986).

C. J. R. Sheppard, D. K. Hamilton, “High resolution stereoscopic imaging,” Appl. Opt. 22, 886–887 (1983).
[CrossRef] [PubMed]

C. J. R. Sheppard, J. T. Sheridan, “Micrometrology of thick structures,” in Optical Storage and Scanning Technology, T. Wilson, ed., Proc. SPIE1139, 32–39 (1989).
[CrossRef]

C. J. R. Sheppard, M. Gu, “Modeling of 3-D brightfield microscope systems,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 352–358 (1994).
[CrossRef]

C. J. R. Sheppard, C. J. Cogswell, “Reflection and transmission confocal microscopy,” in Optics in Medicine, Biology and Environmental Research, G. von Bally, S. Khanna, eds., Proceedings of the First International Conference on Optics within Life Sciences, Garmisch-Partenkirchen, Germany, 1990 (Elsevier, Amsterdam, 1993), pp. 310–315.

Sheridan, J. T.

J. T. Sheridan, C. J. R. Sheppard, “Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory,” Opt. Commun. 105, 367–378 (1994).
[CrossRef]

J. T. Sheridan, C. J. R. Sheppard, “Diffraction by striated muscle fibres: application to image modelling,” Bioimaging 1, 214–227 (1993).
[CrossRef]

J. T. Sheridan, C. J. R. Sheppard, “The coherent imaging of periodic thick fine isolated structures,” J. Opt. Soc. Am. A 10, 614–632 (1993).
[CrossRef]

C. J. R. Sheppard, J. T. Sheridan, “Micrometrology of thick structures,” in Optical Storage and Scanning Technology, T. Wilson, ed., Proc. SPIE1139, 32–39 (1989).
[CrossRef]

Streibl, N.

Török, P.

P. Török, C. J. R. Sheppard, Z. Laczik, “The effect of half-stop lateral misalignment on imaging of dark-field and stereoscopic confocal microscopes,” Appl. Opt. 35, 6732–6739 (1996).
[CrossRef] [PubMed]

P. Török, C. J. R. Sheppard, Z. Laczik, “Dark-field and differential phase contrast imaging modes in confocal microscopy using a half-aperture stop,” Optik 103, 101–106 (1996).

Weise, W.

Wilson, T.

Zinin, P.

Appl. Opt. (4)

Bioimaging (1)

J. T. Sheridan, C. J. R. Sheppard, “Diffraction by striated muscle fibres: application to image modelling,” Bioimaging 1, 214–227 (1993).
[CrossRef]

Eur. J. Cell Biol. Suppl. (1)

C. J. R. Sheppard, “General considerations of diffraction theory of 3-D imaging,” Eur. J. Cell Biol. Suppl. 25, 29–32 (1989).

Experientia (1)

D. M. Maurice, “Cellular membrane activity in the corneal endothelium of the intact eye,” Experientia 15, 1094–1095 (1968).
[CrossRef]

J. Microsc. (1)

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
[CrossRef]

J. Mod. Opt. (1)

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Imaging and reconstruction for rough surface scattering in the Kirchhoff approximation by confocal microscopy,” J. Mod. Opt. 40, 2407–2421 (1993).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (3)

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

J. T. Sheridan, C. J. R. Sheppard, “Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory,” Opt. Commun. 105, 367–378 (1994).
[CrossRef]

C. J. R. Sheppard, T. J. Connolly, M. Gu, “The scattering potential for imaging in the reflection geometry,” Opt. Commun. 117, 16–19 (1995).
[CrossRef]

Opt. Lett. (1)

Optik (2)

P. Török, C. J. R. Sheppard, Z. Laczik, “Dark-field and differential phase contrast imaging modes in confocal microscopy using a half-aperture stop,” Optik 103, 101–106 (1996).

C. J. R. Sheppard, “The spatial frequency cut-off in three dimensional imaging. II,” Optik 74, 128–129 (1986).

Phil. Trans. R. Soc. London (1)

T. J. l’A. Bromwich, “The scattering of plane electric waves by a sphere,” Phil. Trans. R. Soc. London 220, 189–206 (1920).

Philos. Mag. (1)

J. W. Nicholson, “Diffraction of short waves by a rigid sphere,” Philos. Mag. 11, 193–205 (1906).
[CrossRef]

Phys. Rev. Lett. (1)

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

Proc. Math. Soc. London (2)

J. W. Nicholson, “The scattering of light by a large conducting sphere,” Proc. Math. Soc. London 9, 67–80 (1910).

J. W. Nicholson, “The scattering of light by a large conducting sphere,” Proc. Math. Soc. London 11, 277–284 (1912).

Other (6)

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Nonrelativistic Theory) (Pergamon, Oxford, 1959).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1978).

C. J. R. Sheppard, C. J. Cogswell, “Reflection and transmission confocal microscopy,” in Optics in Medicine, Biology and Environmental Research, G. von Bally, S. Khanna, eds., Proceedings of the First International Conference on Optics within Life Sciences, Garmisch-Partenkirchen, Germany, 1990 (Elsevier, Amsterdam, 1993), pp. 310–315.

C. J. R. Sheppard, M. Gu, “Modeling of 3-D brightfield microscope systems,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 352–358 (1994).
[CrossRef]

C. J. R. Sheppard, J. T. Sheridan, “Micrometrology of thick structures,” in Optical Storage and Scanning Technology, T. Wilson, ed., Proc. SPIE1139, 32–39 (1989).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Confocal optical microscope used to investigate imaging of spheres. Light from a laser is beam expanded and focused onto the object, in this case a ball bearing, and the reflected light is collected, collimated, and focused onto a pinhole in front of a photomultiplier tube (PMT). In practice a pinhole spatial filter was used in the illumination path, but this had little effect on image quality.

Fig. 2
Fig. 2

xz image of a ball bearing of 2-mm diameter. The image consists of a strong reflection from the surface of the sphere, with fringes, and a bright peak from the center of the sphere. The z scan could not permit imaging of both the surface and the center of the sphere in one scan, but the images are arranged to scale.

Fig. 3
Fig. 3

Optical configurations that use two half-stops: A (corresponding to dark-field or stereoscopic imaging) and B (divided aperture).

Fig. 4
Fig. 4

xz image acquired with configuration A of Fig. 3. There was no reflection from the surface of the sphere. The bright peak at the center remained as illustrated but was weaker, stretched axially because of the reduced aperture, and tilted.

Fig. 5
Fig. 5

xz image acquired with configuration B of Fig. 3. The image of the surface is present. The intensity at the center is dark but with weak components above and below focus. Again the z scan could not permit imaging of both the surface and the center of the sphere in one scan, but the images are arranged to scale.

Fig. 6
Fig. 6

(a) Confocal reflection image calculated with our approach for a sphere of 20-µm radius. The numerical aperture is 0.5, and the wavelength is 0.488 µm. The coordinates v = kR and u = kZ are dimensionless. The 3-D CTF is considered constant over its passband. (b) Lateral view of the image. The presence of two maxima, one that results when the system is focused on the surface and the other when the focus is on the center of curvature, is emphasized. It is clear that central peak is bigger than the one from the surface. (c) Isophotes of the image. The central distribution of intensity is similar to the point-spread function.

Fig. 7
Fig. 7

(a) Calculated image of the same object as that in Fig. 1. The CTF assumed is that for uniform angular illumination. (b) In this lateral view of the image the small difference between the two peaks is shown, but the central peak is still the larger. (c) Contour curves of the image. It is possible to observe that the maximum on the surface is narrower compared with the corresponding maximum obtained with a constant CTF.

Fig. 8
Fig. 8

(a) Image calculated with a 3-D CTF that includes the aplanatic factor of a system that satisfies the sign condition. The differences from the image shown in Fig. 2 are minor. (b) Lateral view of the image. The sidelobes of both maxima become larger when this CTF is assumed. (c) Isophotes of the image, showing the structure of the secondary maxima.

Equations (21)

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

UX, Y, Z=-+ P2kx, kyP1kx, ky×Skx, ky, kx, ky×exp-ikm·Rdkxdkydkxdky,
k=kxi+kyj+kzk,  k=kxi+kyj+kzk,
R=Xi+Yj+Zk
km=kmi+nj+sk=k-k.
Q1kx, ky, kz=P1kx, ky×δkz-k2-kx2-ky21/2,  Q2kx, ky, kz=P2kx, kyδkz+k2+kx2-ky21/2,
P1kx, ky=-+ Q1kx, ky, kzdkz,  P2kx, ky=-+ Q2kx, ky, kzdkz.
Tm, n, s=Skx, ky, kx, ky,
UR=k6-+ cmTmexp-ikm·Rd3m,
cm=1k3-+ Q2kQ1kd3k.
Tm=12πk2n=02n+1AnPncos γ,
sinγ2=|k-k|2k=|m|2,
cos γ=1-m2/2.
l=m2+n21/2.
UR= cmTmJ0klRexp-iksZldlds= cmTmJ0k|m|R sin θ×exp-ik|m|Z cos θm2d|m|sin θdθ,
UR=n=0-1n2n+1An× Pnm22-1cmJ0k|m|R sin θ×exp-ik|m|Z cos θm2d|m|sin θdθ.
Tm=-ka2exp-ika|m|+2i 1-m2/41/2|m|×J1ka|m|1-m2/41/2.
Tm=-ka2 exp-ika|m|.
Tm=k4im2F3ln nr,
Tm=k4im2F3br,
Tm=sinka|m|ka|m|-coska|m|.
s=2-σ,

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