Abstract

The three-dimensional coherent transfer function for D-shaped pupils in reflection-mode confocal scanning microscopy is analytically derived under the paraxial approximation. Three-dimensional numerical plots are presented, showing the dependence of the transfer functions on the width of a central divider. The applications in fiber-optical confocal scanning microscopy are also discussed.

© 2009 Optical Society of America

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References

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  1. P. J. Dwyer and C. A. DiMarzio, “Confocal reflectance theta line scanning microscope for imaging human skin in vivo,” Opt. Lett. 31, 942-944 (2006).
    [CrossRef] [PubMed]
  2. P. J. Dwyer, C. A. DiMarzio, and M. Rajadhyaksha, “Confocal theta line-scanning microscope for imaging human tissues,” Appl. Opt. 46, 1843-1851 (2007).
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  3. C. J. R. Sheppard, W. Gong, and K. Si, “The divided aperture technique for microscopy through scattering media,” Opt. Express 16, 17031-17038 (2008).
    [CrossRef] [PubMed]
  4. C. J. Koester, “A scanning mirror microscope with optical sectioning characteristics: applications in ophthalmology,” Appl. Opt. 19, 1749-1757 (1980).
    [CrossRef] [PubMed]
  5. H. Siedentopf and R. Zsigmondy, “Uber Sichtbarmachung und Grössenbestimmung ultramikroskopischer Teilchen, mit besonderer Anwendung auf Goldrubingläser,” Ann. Phys. (Leipzig) 10, 1-39 (1903).
  6. H. Goldman, “Spaltlampenphotographie und -photometrie,” Ophthalmologica 98, 257-270 (1940).
    [CrossRef]
  7. D. M. Maurice, “Cellular membrane activity in the corneal endothelium of the intact eye,” Experientia 24, 1094-1095(1968).
    [CrossRef] [PubMed]
  8. C. J. Koester, “Comparison of optical sectioning methods: the scanning slit confocal microscope,” in Handbook of Confocal Microscopy, J. Pawley, ed. (Plenum, 1990).
  9. E. H. K. Stelzer, S. Lindek, S. Albrecht, R. Pick, G. Ritter, N. J. Salmon, and R. Stricker, “A new tool for the observation of embryos and other large specimens--confocal theta-fluorescence microscopy,” J. Microsc. (Oxford) 179, 1-10 (1995).
    [CrossRef]
  10. T. D. Wang, M. J. Mandella, C. H. Contag, and G. S. Kino, “Dual-axis confocal microscope for high-resolution in vivo imaging,” Opt. Lett. 28, 414-416 (2003).
    [CrossRef] [PubMed]
  11. A. H. Voie, D. H. Burns, and F. A. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. (Oxford) 170, 229-236 (1993).
    [CrossRef]
  12. J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007-1009(2004).
    [CrossRef] [PubMed]
  13. P. Török, C. J. R. Sheppard, and Z. Laczik, “Dark-field and differential phase contrast imaging modes in confocal microscopy using a half-aperture stop,” Optik (Jena) 103, 101-106 (1996).
  14. P. Török, C. J. R. Sheppard, and 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]
  15. C. J. R. Sheppard, M. Gu, and X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281-284 (1991).
    [CrossRef]
  16. M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, 1996).
    [CrossRef]
  17. C. J. R. Sheppard and K. G. Larkin, “Effect of numerical aperture on interference fringe spacing,” Appl. Opt. 34, 4731-4734 (1995).
    [CrossRef] [PubMed]
  18. C. J. R. Sheppard and X. Q. Mao, “Three dimensional imaging in a microscope,” J. Opt. Soc. Am. A 6, 1260-1269 (1989).
    [CrossRef]
  19. L. Giniunas, R. Juskaitis, and S. V. Shatalin, “Scanning fiber-optic microscope,” Electron. Lett. 27, 724-726 (1991).
    [CrossRef]
  20. M. Gu, C. J. R. Sheppard, and X. Gan, “Image formation in a fiber-optical confocal scanning microscope,” J. Opt. Soc. Am. A 8, 1755-1761 (1991).
    [CrossRef]
  21. M. Gu and C. J. R. Sheppard, “Signal level of the fibre optical confocal scanning microscope,” J. Mod. Opt. 38, 1621-1630(1991).
    [CrossRef]
  22. S. Kimura and T. Wilson, “Confocal scanning optical microscope using single-mode fiber for signal detection,” Appl. Opt. 30, 2143-2150 (1991).
    [CrossRef] [PubMed]
  23. T. Dabbs and M. Glass, “Single-mode fibers used as confocal microscope pinholes,” Appl. Opt. 31, 705-706 (1992).
    [CrossRef] [PubMed]
  24. P. Delaney and M. Harris, “Fiber optics in confocal microscopy,” in Handbook of Biological Confocal Microscopy, 2nd ed, J.Pawley, ed. (Plenum, 1995), pp. 515-523.
  25. M. Gu and C. J. R. Sheppard, “Three-dimensional coherent transfer function in reflection-mode confocal microscopy using annular lenses,” J. Mod. Opt. 39, 783-793 (1992).
    [CrossRef]
  26. M. Gu and C. J. R. Sheppard, “Three-dimensional optical transfer function in the fiber-optical confocal fluorescent microscope using annular lenses,” J. Opt. Soc. Am. A 9, 1991-1999 (1992).
    [CrossRef]

2008 (1)

2007 (1)

2006 (1)

2004 (1)

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007-1009(2004).
[CrossRef] [PubMed]

2003 (1)

1996 (3)

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

M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, 1996).
[CrossRef]

1995 (3)

P. Delaney and M. Harris, “Fiber optics in confocal microscopy,” in Handbook of Biological Confocal Microscopy, 2nd ed, J.Pawley, ed. (Plenum, 1995), pp. 515-523.

E. H. K. Stelzer, S. Lindek, S. Albrecht, R. Pick, G. Ritter, N. J. Salmon, and R. Stricker, “A new tool for the observation of embryos and other large specimens--confocal theta-fluorescence microscopy,” J. Microsc. (Oxford) 179, 1-10 (1995).
[CrossRef]

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

1993 (1)

A. H. Voie, D. H. Burns, and F. A. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. (Oxford) 170, 229-236 (1993).
[CrossRef]

1992 (3)

1991 (5)

M. Gu, C. J. R. Sheppard, and X. Gan, “Image formation in a fiber-optical confocal scanning microscope,” J. Opt. Soc. Am. A 8, 1755-1761 (1991).
[CrossRef]

S. Kimura and T. Wilson, “Confocal scanning optical microscope using single-mode fiber for signal detection,” Appl. Opt. 30, 2143-2150 (1991).
[CrossRef] [PubMed]

L. Giniunas, R. Juskaitis, and S. V. Shatalin, “Scanning fiber-optic microscope,” Electron. Lett. 27, 724-726 (1991).
[CrossRef]

M. Gu and C. J. R. Sheppard, “Signal level of the fibre optical confocal scanning microscope,” J. Mod. Opt. 38, 1621-1630(1991).
[CrossRef]

C. J. R. Sheppard, M. Gu, and 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. Koester, “Comparison of optical sectioning methods: the scanning slit confocal microscope,” in Handbook of Confocal Microscopy, J. Pawley, ed. (Plenum, 1990).

1989 (1)

1980 (1)

1968 (1)

D. M. Maurice, “Cellular membrane activity in the corneal endothelium of the intact eye,” Experientia 24, 1094-1095(1968).
[CrossRef] [PubMed]

1940 (1)

H. Goldman, “Spaltlampenphotographie und -photometrie,” Ophthalmologica 98, 257-270 (1940).
[CrossRef]

1903 (1)

H. Siedentopf and R. Zsigmondy, “Uber Sichtbarmachung und Grössenbestimmung ultramikroskopischer Teilchen, mit besonderer Anwendung auf Goldrubingläser,” Ann. Phys. (Leipzig) 10, 1-39 (1903).

Gong, W.

Sheppard, C. J. R.

Albrecht, S.

E. H. K. Stelzer, S. Lindek, S. Albrecht, R. Pick, G. Ritter, N. J. Salmon, and R. Stricker, “A new tool for the observation of embryos and other large specimens--confocal theta-fluorescence microscopy,” J. Microsc. (Oxford) 179, 1-10 (1995).
[CrossRef]

Burns, D. H.

A. H. Voie, D. H. Burns, and F. A. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. (Oxford) 170, 229-236 (1993).
[CrossRef]

Contag, C. H.

Dabbs, T.

Del Bene, F.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007-1009(2004).
[CrossRef] [PubMed]

Delaney, P.

P. Delaney and M. Harris, “Fiber optics in confocal microscopy,” in Handbook of Biological Confocal Microscopy, 2nd ed, J.Pawley, ed. (Plenum, 1995), pp. 515-523.

DiMarzio, C. A.

Dwyer, P. J.

Gan, X.

Giniunas, L.

L. Giniunas, R. Juskaitis, and S. V. Shatalin, “Scanning fiber-optic microscope,” Electron. Lett. 27, 724-726 (1991).
[CrossRef]

Glass, M.

Goldman, H.

H. Goldman, “Spaltlampenphotographie und -photometrie,” Ophthalmologica 98, 257-270 (1940).
[CrossRef]

Gu, M.

M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, 1996).
[CrossRef]

M. Gu and C. J. R. Sheppard, “Three-dimensional coherent transfer function in reflection-mode confocal microscopy using annular lenses,” J. Mod. Opt. 39, 783-793 (1992).
[CrossRef]

M. Gu and C. J. R. Sheppard, “Three-dimensional optical transfer function in the fiber-optical confocal fluorescent microscope using annular lenses,” J. Opt. Soc. Am. A 9, 1991-1999 (1992).
[CrossRef]

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

M. Gu and C. J. R. Sheppard, “Signal level of the fibre optical confocal scanning microscope,” J. Mod. Opt. 38, 1621-1630(1991).
[CrossRef]

M. Gu, C. J. R. Sheppard, and X. Gan, “Image formation in a fiber-optical confocal scanning microscope,” J. Opt. Soc. Am. A 8, 1755-1761 (1991).
[CrossRef]

Harris, M.

P. Delaney and M. Harris, “Fiber optics in confocal microscopy,” in Handbook of Biological Confocal Microscopy, 2nd ed, J.Pawley, ed. (Plenum, 1995), pp. 515-523.

Huisken, J.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007-1009(2004).
[CrossRef] [PubMed]

Juskaitis, R.

L. Giniunas, R. Juskaitis, and S. V. Shatalin, “Scanning fiber-optic microscope,” Electron. Lett. 27, 724-726 (1991).
[CrossRef]

Kimura, S.

Kino, G. S.

Koester, C. J.

C. J. Koester, “Comparison of optical sectioning methods: the scanning slit confocal microscope,” in Handbook of Confocal Microscopy, J. Pawley, ed. (Plenum, 1990).

C. J. Koester, “A scanning mirror microscope with optical sectioning characteristics: applications in ophthalmology,” Appl. Opt. 19, 1749-1757 (1980).
[CrossRef] [PubMed]

Laczik, Z.

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

Larkin, K. G.

Lindek, S.

E. H. K. Stelzer, S. Lindek, S. Albrecht, R. Pick, G. Ritter, N. J. Salmon, and R. Stricker, “A new tool for the observation of embryos and other large specimens--confocal theta-fluorescence microscopy,” J. Microsc. (Oxford) 179, 1-10 (1995).
[CrossRef]

Mandella, M. J.

Mao, X. Q.

C. J. R. Sheppard, M. Gu, and 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 and X. Q. Mao, “Three dimensional imaging in a microscope,” J. Opt. Soc. Am. A 6, 1260-1269 (1989).
[CrossRef]

Maurice, D. M.

D. M. Maurice, “Cellular membrane activity in the corneal endothelium of the intact eye,” Experientia 24, 1094-1095(1968).
[CrossRef] [PubMed]

Pick, R.

E. H. K. Stelzer, S. Lindek, S. Albrecht, R. Pick, G. Ritter, N. J. Salmon, and R. Stricker, “A new tool for the observation of embryos and other large specimens--confocal theta-fluorescence microscopy,” J. Microsc. (Oxford) 179, 1-10 (1995).
[CrossRef]

Rajadhyaksha, M.

Ritter, G.

E. H. K. Stelzer, S. Lindek, S. Albrecht, R. Pick, G. Ritter, N. J. Salmon, and R. Stricker, “A new tool for the observation of embryos and other large specimens--confocal theta-fluorescence microscopy,” J. Microsc. (Oxford) 179, 1-10 (1995).
[CrossRef]

Salmon, N. J.

E. H. K. Stelzer, S. Lindek, S. Albrecht, R. Pick, G. Ritter, N. J. Salmon, and R. Stricker, “A new tool for the observation of embryos and other large specimens--confocal theta-fluorescence microscopy,” J. Microsc. (Oxford) 179, 1-10 (1995).
[CrossRef]

Shatalin, S. V.

L. Giniunas, R. Juskaitis, and S. V. Shatalin, “Scanning fiber-optic microscope,” Electron. Lett. 27, 724-726 (1991).
[CrossRef]

Sheppard, C. J. R.

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

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

M. Gu and C. J. R. Sheppard, “Three-dimensional coherent transfer function in reflection-mode confocal microscopy using annular lenses,” J. Mod. Opt. 39, 783-793 (1992).
[CrossRef]

M. Gu and C. J. R. Sheppard, “Three-dimensional optical transfer function in the fiber-optical confocal fluorescent microscope using annular lenses,” J. Opt. Soc. Am. A 9, 1991-1999 (1992).
[CrossRef]

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

M. Gu, C. J. R. Sheppard, and X. Gan, “Image formation in a fiber-optical confocal scanning microscope,” J. Opt. Soc. Am. A 8, 1755-1761 (1991).
[CrossRef]

M. Gu and C. J. R. Sheppard, “Signal level of the fibre optical confocal scanning microscope,” J. Mod. Opt. 38, 1621-1630(1991).
[CrossRef]

C. J. R. Sheppard and X. Q. Mao, “Three dimensional imaging in a microscope,” J. Opt. Soc. Am. A 6, 1260-1269 (1989).
[CrossRef]

Siedentopf, H.

H. Siedentopf and R. Zsigmondy, “Uber Sichtbarmachung und Grössenbestimmung ultramikroskopischer Teilchen, mit besonderer Anwendung auf Goldrubingläser,” Ann. Phys. (Leipzig) 10, 1-39 (1903).

Spelman, F. A.

A. H. Voie, D. H. Burns, and F. A. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. (Oxford) 170, 229-236 (1993).
[CrossRef]

Stelzer, E. H. K.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007-1009(2004).
[CrossRef] [PubMed]

E. H. K. Stelzer, S. Lindek, S. Albrecht, R. Pick, G. Ritter, N. J. Salmon, and R. Stricker, “A new tool for the observation of embryos and other large specimens--confocal theta-fluorescence microscopy,” J. Microsc. (Oxford) 179, 1-10 (1995).
[CrossRef]

Stricker, R.

E. H. K. Stelzer, S. Lindek, S. Albrecht, R. Pick, G. Ritter, N. J. Salmon, and R. Stricker, “A new tool for the observation of embryos and other large specimens--confocal theta-fluorescence microscopy,” J. Microsc. (Oxford) 179, 1-10 (1995).
[CrossRef]

Swoger, J.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007-1009(2004).
[CrossRef] [PubMed]

Török, P.

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

Voie, A. H.

A. H. Voie, D. H. Burns, and F. A. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. (Oxford) 170, 229-236 (1993).
[CrossRef]

Wang, T. D.

Wilson, T.

Wittbrodt, J.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007-1009(2004).
[CrossRef] [PubMed]

Zsigmondy, R.

H. Siedentopf and R. Zsigmondy, “Uber Sichtbarmachung und Grössenbestimmung ultramikroskopischer Teilchen, mit besonderer Anwendung auf Goldrubingläser,” Ann. Phys. (Leipzig) 10, 1-39 (1903).

Ann. Phys. (Leipzig) (1)

H. Siedentopf and R. Zsigmondy, “Uber Sichtbarmachung und Grössenbestimmung ultramikroskopischer Teilchen, mit besonderer Anwendung auf Goldrubingläser,” Ann. Phys. (Leipzig) 10, 1-39 (1903).

Appl. Opt. (6)

Electron. Lett. (1)

L. Giniunas, R. Juskaitis, and S. V. Shatalin, “Scanning fiber-optic microscope,” Electron. Lett. 27, 724-726 (1991).
[CrossRef]

Experientia (1)

D. M. Maurice, “Cellular membrane activity in the corneal endothelium of the intact eye,” Experientia 24, 1094-1095(1968).
[CrossRef] [PubMed]

J. Microsc. (Oxford) (2)

E. H. K. Stelzer, S. Lindek, S. Albrecht, R. Pick, G. Ritter, N. J. Salmon, and R. Stricker, “A new tool for the observation of embryos and other large specimens--confocal theta-fluorescence microscopy,” J. Microsc. (Oxford) 179, 1-10 (1995).
[CrossRef]

A. H. Voie, D. H. Burns, and F. A. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. (Oxford) 170, 229-236 (1993).
[CrossRef]

J. Mod. Opt. (2)

M. Gu and C. J. R. Sheppard, “Three-dimensional coherent transfer function in reflection-mode confocal microscopy using annular lenses,” J. Mod. Opt. 39, 783-793 (1992).
[CrossRef]

M. Gu and C. J. R. Sheppard, “Signal level of the fibre optical confocal scanning microscope,” J. Mod. Opt. 38, 1621-1630(1991).
[CrossRef]

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

Ophthalmologica (1)

H. Goldman, “Spaltlampenphotographie und -photometrie,” Ophthalmologica 98, 257-270 (1940).
[CrossRef]

Opt. Commun. (1)

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

Opt. Express (1)

Opt. Lett. (2)

Optik (Jena) (1)

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

Science (1)

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007-1009(2004).
[CrossRef] [PubMed]

Other (3)

M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, 1996).
[CrossRef]

C. J. Koester, “Comparison of optical sectioning methods: the scanning slit confocal microscope,” in Handbook of Confocal Microscopy, J. Pawley, ed. (Plenum, 1990).

P. Delaney and M. Harris, “Fiber optics in confocal microscopy,” in Handbook of Biological Confocal Microscopy, 2nd ed, J.Pawley, ed. (Plenum, 1995), pp. 515-523.

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Figures (8)

Fig. 1
Fig. 1

Geometry of the confocal microscope with two centro symmetric D-shaped pupils.

Fig. 2
Fig. 2

2D convolution of two D-shaped pupils P ( ρ 1 ) and P ( ρ 2 ) . Q is an arbitrary point on the boundary of the overlapping region. The lengths of O 1 Q and O 2 Q are ρ 1 and ρ 2 , respectively. The distance between O 1 and O 2 is l.

Fig. 3
Fig. 3

Effective region of l in a confocal microscope with two D-shaped pupils compared with the conventional confocal microscope with two circular pupils.

Fig. 4
Fig. 4

3D CTFs with different distance parameter d and different angle ψ. For d = 0 and ψ = π / 2 , the 3D CTF is the same as for the conventional confocal microscope with two circular pupils.

Fig. 5
Fig. 5

3D CTF c ( l = 0 , s ) as a function of the distance parameter d.

Fig. 6
Fig. 6

Transverse cross sections (a) for different angular parameter ψ with d = 0.1 and (b) for different distance parameter d with ψ = π / 4 .

Fig. 7
Fig. 7

2D in-focus CTF in a confocal microscope with two centrosymmetric D-shaped pupils.

Fig. 8
Fig. 8

3D CTFs for FOCSM at d = 0.1 , with different parameter A and different angle ψ.

Equations (17)

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I ( r s ) | c ( m ) T ( m ) exp ( 2 π i m · r s ) d m | 2 ,
P ( ρ , θ , u ) = { exp [ i ( u / 2 ) ρ 2 ] , d ρ 1 ,       cos 1 ( ρ / d ) θ cos 1 ( ρ / d ) , 0 , otherwise
u = ( 8 π n / λ ) z sin 2 ( α / 2 ) .
h ( v x , v y , u ) = h 1 ( v x , v y , u ) h 2 ( v x , v y , u ) ,
h 1 ( v x , v y , u ) = h 2 ( v x , v y , u ) = P ( ρ , θ , u ) exp [ i ( v x ρ cos θ + v y ρ sin θ ) ] ρ d ρ d θ ,
v x 2 + v y 2 = ( ( 2 π n / λ ) r sin α ) 2 .
c ( l , ψ , s ) = F 3 [ h 1 ( v x , v y , u ) h 2 ( v x , v y , u ) ] = F 3 [ h 1 ( v x , v y , u ) h 1 ( v x , v y , u ) ] ,
c ( l , ψ , s ) = [ P ( ρ , θ , u ) 2 P ( ρ , θ + π , u ) ] exp ( i u s ) d u ,
c ( l , ψ , s ) = σ δ ( s ρ 2 l 2 / 4 ) ρ d ρ d θ ,
c A ( l , ψ , s ) = 1 2 π ϑ 0 ϑ 1 ( d + m / 2 ) / cos θ ρ 0 δ ( s ρ 2 l 2 / 4 ) ρ d ρ d θ ;
c B ( l , ψ , s ) = 1 2 π ϑ 0 ϑ 2 ( d + m / 2 ) / cos θ ρ 0 δ ( s ρ 2 l 2 / 4 ) ρ d ρ d θ ,
{ ϑ 0 = tan 1 { [ 1 ( d + m ) 2 n / 2 ] / ( d + m / 2 ) } ϑ 1 = tan 1 { [ 1 d 2 n / 2 ] / ( d + m / 2 ) } ϑ 2 = tan 1 { [ 1 ( d + m ) 2 + n / 2 ] / ( d + m / 2 ) } ρ 0 = l | cos ( ψ + θ ) | / 2 + 1 l 2 sin 2 ( ψ + θ ) / 4 .
c ( l , ψ , s ) = { c A ( l , ψ , s ) , when     tan 1 [ 1 d 2 / ( 1 d ) ] ψ π / 2 c A ( l , ψ , s ) + c B ( l , ψ , s ) , when     0 < ψ < tan 1 [ 1 d 2 / ( 1 d ) ] c B ( l , ψ , s ) , when     ψ = 0 .
s 0 = 1 2 sin 2 ( α / 2 ) .
c ( l , 0 , s ) = { 1 / π cos 1 ( d + l / 2 s l 2 / 4 ) , d 2 + l 2 / 2 + d l s 1 + l 2 / 2 l 1 / π [ cos 1 ( d + l / 2 s l 2 / 4 ) cos 1 ( 1 s l s l 2 / 4 ) ] , 1 + l 2 / 2 l s 1 l 2 / 2 d l .
c f ( l , ψ , s ) = K exp ( A l 2 / 4 ) σ exp ( A ρ 2 ) δ ( s ρ 2 l 2 / 4 ) ρ d ρ d θ ,
c f ( l , ψ , s ) = exp ( A s ) c ( l , ψ , s ) ,

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