Abstract

Using an electromagnetic approach, we calculate the properties of a confocal fluorescence microscope. It is expected that the results will be more reliable than those obtained by conventional scalar theory, the results of which differ significantly from ours. We calculate the point-spread function and the optical transfer function and study the influence of detector size and fluorescence wavelength on the optical sectioning capability. Our calculations are based on electromagnetic diffraction theory in the Debye approximation. The recently noted asymmetry between the illumination and the detection sensitivity distribution is also taken into account.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. J. R. Sheppard, T. Wilson, “Image formation in scanning microscopes with partially coherent source and detector,” Opt. Acta 25, 315–325 (1978).
    [CrossRef]
  2. T. Wilson, “Optical sectioning in confocal fluorescent microscopes,”J. Micros. 154, 143–156 (1989).
    [CrossRef]
  3. T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).
  4. M. Gu, C. J. R. Sheppard, “Confocal fluorescent microscopy with a finite-sized circular detector,” J. Opt. Soc. Am. A 9, 151–153 (1992).
    [CrossRef]
  5. T. Wilson, “The role of the pinhole in confocal imaging systems,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1990).
    [CrossRef]
  6. C. J. R. Sheppard, “Axial resolution of confocal fluorescence microscopy,” J. Microsc. 154, 237–241 (1989).
    [CrossRef]
  7. S. Kimura, C. Munakata, “Calculation of three-dimensional optical transfer function for a confocal scanning fluorescent microscope,” J. Opt. Soc. Am. A 6, 1015–1019 (1989).
    [CrossRef]
  8. S. Kawata, R. Arimoto, O. Nakamura, “Three-dimensional optical-transfer function analysis for a laser-scan fluorescence microscope with an extended detector,”J. Opt. Soc. Am A 8, 171–175 (1991).
    [CrossRef]
  9. E. Wolf, “Electromagnetic diffraction in optical systems I,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
    [CrossRef]
  10. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  11. T. D. Visser, G. J. Brakenhoff, F. C. A. Groen, “The fluorescence point response in confocal microscopy,” Optik 87, 39–40 (1991).
  12. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
  13. P. J. W. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
    [CrossRef]
  14. T. D. Visser, S. H. Wiersma, “Diffraction of converging electromagnetic beams,” J. Opt. Soc. Am. A 9, 2034–2047 (1992).
    [CrossRef]
  15. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).
  16. H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. 55, 116–128 (1943).
    [CrossRef]
  17. R. Barakat, D. Lev, “Transfer functions and total illuminance of high numerical aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 53, 324–332 (1963).
    [CrossRef]
  18. T. D. Visser, S. H. Wiersma, “Spherical aberration and the electromagnetic field in high-aperture systems,” J. Opt. Soc. Am. A 8, 1404–1410 (1991).
    [CrossRef]
  19. An alternative definition is given by Sheppard and Matthews [J. Opt. Soc. Am. A 4, 1354–1360 (1987)] that has the advantage that the axial intensity distribution is less sensitive to the aperture angle. Here, however, we follow the definition given by Richards and Wolf.
  20. C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London Ser. A 379, 145–158 (1982).
    [CrossRef]
  21. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1986).
  22. C. J. R. Sheppard, M. Gu, “The significance of 3D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
    [CrossRef]
  23. A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
    [CrossRef]
  24. T. D. Visser, F. C. A. Groen, G. J. Brakenhoff, “Absorption and scattering correction in fluorescence confocal microscopy,” J. Microsc. 163, 189–200 (1991).
    [CrossRef]
  25. C. J. R. Sheppard, C. Cogswell, “Effect of aberrating layers and tube length on confocal imaging properties,” Optik 87, 34–38 (1991).
  26. T. D. Visser, J. L. Oud, G. J. Brakenhoff, “Refractive index and distance measurements in 3-D microscopy,” Optik 90, 17–19 (1992).
  27. Numerical Algorithm Group fortran Library Manual, Mark 14 (NAG Ltd., Oxford, UK, 1991).
  28. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [CrossRef]
  29. Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
    [CrossRef]

1992 (4)

C. J. R. Sheppard, M. Gu, “The significance of 3D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

T. D. Visser, J. L. Oud, G. J. Brakenhoff, “Refractive index and distance measurements in 3-D microscopy,” Optik 90, 17–19 (1992).

M. Gu, C. J. R. Sheppard, “Confocal fluorescent microscopy with a finite-sized circular detector,” J. Opt. Soc. Am. A 9, 151–153 (1992).
[CrossRef]

T. D. Visser, S. H. Wiersma, “Diffraction of converging electromagnetic beams,” J. Opt. Soc. Am. A 9, 2034–2047 (1992).
[CrossRef]

1991 (5)

T. D. Visser, S. H. Wiersma, “Spherical aberration and the electromagnetic field in high-aperture systems,” J. Opt. Soc. Am. A 8, 1404–1410 (1991).
[CrossRef]

T. D. Visser, F. C. A. Groen, G. J. Brakenhoff, “Absorption and scattering correction in fluorescence confocal microscopy,” J. Microsc. 163, 189–200 (1991).
[CrossRef]

C. J. R. Sheppard, C. Cogswell, “Effect of aberrating layers and tube length on confocal imaging properties,” Optik 87, 34–38 (1991).

S. Kawata, R. Arimoto, O. Nakamura, “Three-dimensional optical-transfer function analysis for a laser-scan fluorescence microscope with an extended detector,”J. Opt. Soc. Am A 8, 171–175 (1991).
[CrossRef]

T. D. Visser, G. J. Brakenhoff, F. C. A. Groen, “The fluorescence point response in confocal microscopy,” Optik 87, 39–40 (1991).

1989 (3)

T. Wilson, “Optical sectioning in confocal fluorescent microscopes,”J. Micros. 154, 143–156 (1989).
[CrossRef]

C. J. R. Sheppard, “Axial resolution of confocal fluorescence microscopy,” J. Microsc. 154, 237–241 (1989).
[CrossRef]

S. Kimura, C. Munakata, “Calculation of three-dimensional optical transfer function for a confocal scanning fluorescent microscope,” J. Opt. Soc. Am. A 6, 1015–1019 (1989).
[CrossRef]

1987 (1)

1982 (2)

C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London Ser. A 379, 145–158 (1982).
[CrossRef]

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

1981 (1)

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

1978 (1)

C. J. R. Sheppard, T. Wilson, “Image formation in scanning microscopes with partially coherent source and detector,” Opt. Acta 25, 315–325 (1978).
[CrossRef]

1965 (1)

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
[CrossRef]

1963 (1)

R. Barakat, D. Lev, “Transfer functions and total illuminance of high numerical aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 53, 324–332 (1963).
[CrossRef]

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems I,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

1943 (1)

H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. 55, 116–128 (1943).
[CrossRef]

1909 (1)

P. J. W. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
[CrossRef]

Arimoto, R.

S. Kawata, R. Arimoto, O. Nakamura, “Three-dimensional optical-transfer function analysis for a laser-scan fluorescence microscope with an extended detector,”J. Opt. Soc. Am A 8, 171–175 (1991).
[CrossRef]

Barakat, R.

R. Barakat, D. Lev, “Transfer functions and total illuminance of high numerical aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 53, 324–332 (1963).
[CrossRef]

Boivin, A.

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1986).

Brakenhoff, G. J.

T. D. Visser, J. L. Oud, G. J. Brakenhoff, “Refractive index and distance measurements in 3-D microscopy,” Optik 90, 17–19 (1992).

T. D. Visser, F. C. A. Groen, G. J. Brakenhoff, “Absorption and scattering correction in fluorescence confocal microscopy,” J. Microsc. 163, 189–200 (1991).
[CrossRef]

T. D. Visser, G. J. Brakenhoff, F. C. A. Groen, “The fluorescence point response in confocal microscopy,” Optik 87, 39–40 (1991).

Cogswell, C.

C. J. R. Sheppard, C. Cogswell, “Effect of aberrating layers and tube length on confocal imaging properties,” Optik 87, 34–38 (1991).

Debye, P. J. W.

P. J. W. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
[CrossRef]

Groen, F. C. A.

T. D. Visser, G. J. Brakenhoff, F. C. A. Groen, “The fluorescence point response in confocal microscopy,” Optik 87, 39–40 (1991).

T. D. Visser, F. C. A. Groen, G. J. Brakenhoff, “Absorption and scattering correction in fluorescence confocal microscopy,” J. Microsc. 163, 189–200 (1991).
[CrossRef]

Gu, M.

C. J. R. Sheppard, M. Gu, “The significance of 3D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Confocal fluorescent microscopy with a finite-sized circular detector,” J. Opt. Soc. Am. A 9, 151–153 (1992).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. 55, 116–128 (1943).
[CrossRef]

Kawata, S.

S. Kawata, R. Arimoto, O. Nakamura, “Three-dimensional optical-transfer function analysis for a laser-scan fluorescence microscope with an extended detector,”J. Opt. Soc. Am A 8, 171–175 (1991).
[CrossRef]

Kimura, S.

Lev, D.

R. Barakat, D. Lev, “Transfer functions and total illuminance of high numerical aperture systems obeying the sine condition,” J. Opt. Soc. Am. A 53, 324–332 (1963).
[CrossRef]

Li, Y.

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

Munakata, C.

Nakamura, O.

S. Kawata, R. Arimoto, O. Nakamura, “Three-dimensional optical-transfer function analysis for a laser-scan fluorescence microscope with an extended detector,”J. Opt. Soc. Am A 8, 171–175 (1991).
[CrossRef]

Oud, J. L.

T. D. Visser, J. L. Oud, G. J. Brakenhoff, “Refractive index and distance measurements in 3-D microscopy,” Optik 90, 17–19 (1992).

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, M. Gu, “The significance of 3D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Confocal fluorescent microscopy with a finite-sized circular detector,” J. Opt. Soc. Am. A 9, 151–153 (1992).
[CrossRef]

C. J. R. Sheppard, C. Cogswell, “Effect of aberrating layers and tube length on confocal imaging properties,” Optik 87, 34–38 (1991).

C. J. R. Sheppard, “Axial resolution of confocal fluorescence microscopy,” J. Microsc. 154, 237–241 (1989).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London Ser. A 379, 145–158 (1982).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Image formation in scanning microscopes with partially coherent source and detector,” Opt. Acta 25, 315–325 (1978).
[CrossRef]

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

Visser, T. D.

T. D. Visser, J. L. Oud, G. J. Brakenhoff, “Refractive index and distance measurements in 3-D microscopy,” Optik 90, 17–19 (1992).

T. D. Visser, S. H. Wiersma, “Diffraction of converging electromagnetic beams,” J. Opt. Soc. Am. A 9, 2034–2047 (1992).
[CrossRef]

T. D. Visser, S. H. Wiersma, “Spherical aberration and the electromagnetic field in high-aperture systems,” J. Opt. Soc. Am. A 8, 1404–1410 (1991).
[CrossRef]

T. D. Visser, G. J. Brakenhoff, F. C. A. Groen, “The fluorescence point response in confocal microscopy,” Optik 87, 39–40 (1991).

T. D. Visser, F. C. A. Groen, G. J. Brakenhoff, “Absorption and scattering correction in fluorescence confocal microscopy,” J. Microsc. 163, 189–200 (1991).
[CrossRef]

Wiersma, S. H.

Wilson, T.

T. Wilson, “Optical sectioning in confocal fluorescent microscopes,”J. Micros. 154, 143–156 (1989).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London Ser. A 379, 145–158 (1982).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Image formation in scanning microscopes with partially coherent source and detector,” Opt. Acta 25, 315–325 (1978).
[CrossRef]

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

T. Wilson, “The role of the pinhole in confocal imaging systems,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1990).
[CrossRef]

Wolf, E.

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems I,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Ann. Phys. (1)

P. J. W. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
[CrossRef]

J. Micros. (1)

T. Wilson, “Optical sectioning in confocal fluorescent microscopes,”J. Micros. 154, 143–156 (1989).
[CrossRef]

J. Microsc. (3)

C. J. R. Sheppard, “Axial resolution of confocal fluorescence microscopy,” J. Microsc. 154, 237–241 (1989).
[CrossRef]

C. J. R. Sheppard, M. Gu, “The significance of 3D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

T. D. Visser, F. C. A. Groen, G. J. Brakenhoff, “Absorption and scattering correction in fluorescence confocal microscopy,” J. Microsc. 163, 189–200 (1991).
[CrossRef]

J. Opt. Soc. Am A (1)

S. Kawata, R. Arimoto, O. Nakamura, “Three-dimensional optical-transfer function analysis for a laser-scan fluorescence microscope with an extended detector,”J. Opt. Soc. Am A 8, 171–175 (1991).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Acta (1)

C. J. R. Sheppard, T. Wilson, “Image formation in scanning microscopes with partially coherent source and detector,” Opt. Acta 25, 315–325 (1978).
[CrossRef]

Opt. Commun. (2)

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

Optik (3)

C. J. R. Sheppard, C. Cogswell, “Effect of aberrating layers and tube length on confocal imaging properties,” Optik 87, 34–38 (1991).

T. D. Visser, J. L. Oud, G. J. Brakenhoff, “Refractive index and distance measurements in 3-D microscopy,” Optik 90, 17–19 (1992).

T. D. Visser, G. J. Brakenhoff, F. C. A. Groen, “The fluorescence point response in confocal microscopy,” Optik 87, 39–40 (1991).

Phys. Rev. (1)

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. 138, B1561–B1565 (1965).
[CrossRef]

Proc. Phys. Soc. (1)

H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. 55, 116–128 (1943).
[CrossRef]

Proc. R. Soc. London Ser. A (3)

C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London Ser. A 379, 145–158 (1982).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems I,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Other (6)

T. Wilson, “The role of the pinhole in confocal imaging systems,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1990).
[CrossRef]

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1986).

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Numerical Algorithm Group fortran Library Manual, Mark 14 (NAG Ltd., Oxford, UK, 1991).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Model of the confocal fluorescence microscope. An expanded plane laser beam, which is approximately homogeneous, is focused by L1 onto a fluorescent object that can be scanned mechanically. The focus of L1 is the origin of the Cartesian coordinates (x, y, z) and the polar coordinates θ and ϕ. L1 has focal length f1 and semiaperture angle Ω1. The fluorescent light originating from the object is collimated by L1 and deflected by a dichroic mirror onto L2, which in turn focuses the light onto the pinhole detector. L2 has focal length f2 and semiaperture angle Ω2. The detector, which is placed at the focus of L2, has a radius of vd optical coordinates. The center of the detector coincides with the center of a second set of coordinates ( x ˜ , y ˜ , z ˜) and θ ˜ and ϕ ˜. The wave fronts S0, S1, and S2 are discussed in the text.

Fig. 2
Fig. 2

(Normalized) lateral variation C(q) of the three-dimensional OTF as p = 0 for different radii vd of the detector. Ω1 = 60°, M = 10×, β = 1.2. In this and subsequent figures the radii of the pinhole are taken as the size that they have in the confocal region (i.e., true radius divided by M).

Fig. 3
Fig. 3

(Normalized) axial variation C(p) of the three-dimensional OTF as q = 0 for different radii vd of the detector. Ω1 = 60°, M = 10×, β = 1.2. The curves for a point detector and vd = 2.5 are very close together on the scale of this figure.

Fig. 4
Fig. 4

Detected signal for a point object, axially scanned through focus, for different pinhole radii vd. The semiaperture angle of the illumination lens L1 is 60°, the magnification M is 10×, and β = λfl/λex = 1.2. The response is normalized to F(0).

Fig. 5
Fig. 5

Response of a confocal fluorescence microscope with a point detector for two point objects on the axis that are a distance δu apart. All curves are normalized to unity. Ω1 = 60°, M = 10×, β = 1.2. For clarity the curves have been displaced with respect to one another.

Fig. 6
Fig. 6

Detected signal for a perfect planar fluorescent object, axially scanned through the focus of a confocal fluorescence microscope for different values of the detection pinhole radius vd. Ω1 = 600, M = 10×, β = 1.2.

Fig. 7
Fig. 7

Response of a confocal fluorescence microscope with a point detector imaging a point object that is axially scanned through focus for different values of β = λfl/λex. Notice the difference between the predictions of electromagnetic theory (solid curves) and of scalar theory (dashed curve). Ω1 = 60° and M = 10×.

Fig. 8
Fig. 8

Detected signal for a perfect planar fluorescent object, axially scanned through the focus of a confocal fluorescence microscope with a point detector for different values of β = λfl/λex. Ω1 = 60° and M = 10×. Dashed curve, the prediction of scalar theory; solid curves, the results according to electromagnetic diffraction theory. Note the large difference between them.

Fig. 9
Fig. 9

Response of the confocal fluorescence microscope’s imaging of two planes, both of which are perpendicular to the central axis. The distance between the planes is δu optical coordinates. For clarity the curves have been displaced with respect to each other.

Fig. 10
Fig. 10

Region of integration for the function H(v, u; M) for the case in which (a) vdMv and (b) vd < Mv.

Equations (63)

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

E ( x ) = - i k f 2 π exp ( i k f ) Γ E S exp [ - i k ( q ^ · x ) ] d Γ ,
r = f 1 sin θ .
δ S 0 = δ S 1 cos θ ,             0 θ Ω .
A S 1 ( θ ) 2 δ S 1 cos θ ,             0 θ Ω .
A S 1 ( θ ) = cos 1 / 2 θ .
A S 0 ( θ ) = cos - 1 / 2 θ .
A S 0 ( r ) = ( 1 - r 2 f 1 2 ) - 1 / 4 ,             0 r f 1 sin Ω 1 .
A S 2 ( θ ˜ ) = [ 1 - ( f 2 f 1 ) 2 sin 2 θ ˜ ] - 1 / 4 cos 1 / 2 θ ˜ ,             0 θ ˜ Ω 2 ,
f 2 sin θ ˜ = f 1 sin θ ,
( f 2 f 1 ) 2 sin 2 θ ˜ = sin 2 θ 1 ,
A S 2 ; M ( θ ˜ ) = ( 1 - M 2 sin 2 θ ˜ ) - 1 / 4 cos 1 / 2 θ ˜ ,             0 θ ˜ Ω 2 ,
M = f 2 f 1 = sin Ω 1 sin Ω 2 .
E S ; α ( θ , ϕ ) = A S ( θ ) ( E ^ 1 cos α + E ^ 2 sin α ) ,
E ^ 1 = [ sin 2 ϕ + cos θ cos 2 ϕ cos ϕ sin ϕ ( cos θ - 1 ) - sin θ cos ϕ ] ,
E ^ 2 = [ cos ϕ sin ϕ ( cos θ - 1 ) cos 2 ϕ + cos θ sin 2 ϕ - sin θ sin ϕ ] .
I e x ( v , u ) = I 0 2 + 2 I 1 2 + I 2 2 ,
I 0 ( v , u ) = 0 Ω 1 cos 1 / 2 θ sin θ ( 1 + cos θ ) J 0 ( v sin θ sin Ω 1 ) × exp ( i u cos θ sin 2 Ω 1 ) d θ ,
I 1 ( v , u ) = 0 Ω 1 cos 1 / 2 θ sin 2 θ J 1 ( v sin θ sin Ω 1 ) exp ( i u cos θ sin 2 Ω 1 ) d θ ,
I 2 ( v , u ) = 0 Ω 1 cos 1 / 2 θ sin θ ( 1 - cos θ ) J 2 ( v sin θ sin Ω 1 ) × exp ( i u cos θ sin 2 Ω 1 ) d θ ;
u = 2 π λ ex z sin 2 Ω 1 ,             v = 2 π λ ex ( x 2 + y 2 ) 1 / 2 sin Ω 1 ,
I ˜ fl ( v ˜ , u ˜ ) = I ˜ 0 2 + 2 I ˜ 1 2 + I ˜ 2 2 ,
I ˜ 0 ( v ˜ , u ˜ ) = 0 Ω 2 A S ; M ( θ ˜ ) sin θ ˜ ( 1 + cos θ ˜ ) J 0 ( v ˜ sin θ ˜ β sin Ω 1 ) × exp ( i u ˜ cos θ ˜ β sin 2 Ω 1 ) d θ ˜ ,
I ˜ 1 ( v ˜ , u ˜ ) = 0 Ω 2 A S ; M ( θ ˜ ) sin 2 θ ˜ J 1 ( v ˜ sin θ ˜ β sin Ω 1 ) × exp ( i u ˜ cos θ ˜ β sin 2 Ω 1 ) d θ ˜ ,
I ˜ 2 ( v ˜ , u ˜ ) = 0 Ω 2 A S ; M ( θ ˜ ) sin θ ˜ ( 1 - cos θ ˜ ) J 2 ( v ˜ sin θ ˜ β sin Ω 1 ) × exp ( i u ˜ cos θ ˜ β sin 2 Ω 1 ) d θ ˜ .
β = λ fl λ ex
u ˜ = 2 π λ ex z ˜ sin 2 Ω 1 ,             v ˜ = 2 π λ ex ( x ˜ 2 + y ˜ 2 ) 1 / 2 sin Ω 1 .
Ω 2 = sin - 1 ( sin Ω 1 M ) .
I d ( v ˜ x , v ˜ y ; x ) = I ˜ fl ( v ˜ x , v ˜ y , 0 ) ,             x = ( 0 , 0 , 0 ) .
I d ( v ˜ x , v ˜ y , x ) = I ˜ fl ( v ˜ x - M v x , v ˜ y - M v y , 0 ) ,             x = ( v x , v y , 0 ) .
b = ( 1 f - 1 d ) - 1 .
d b d d = - ( 1 f - 1 d ) - 2 1 d 2 = - b 2 d 2 = - M 2 .
I d ( v ˜ x , v ˜ y ; v x , v y , u ) = I ˜ fl ( v ˜ x - M v x , v ˜ y - M v y , M 2 u ) .
I d ( v ˜ x , v ˜ y ; x s ) = - - - I ex ( x ) o ( x - x s ) × I ˜ fl ( v ˜ x - M v x , v ˜ y - M v y , M 2 u ) d v x d v y d u ,
F ( x s ) = - + - + - + - + - + I ex ( x ) o ( x - x s ) × I ˜ fl ( v ˜ x - M v x , v ˜ y - M v y , M 2 u ) × D ( v ˜ x , v ˜ y ) d v x d v y d u d v ˜ x d v ˜ y .
D ( v ˜ x , v ˜ y ) = { 1 if v ˜ x 2 + v ˜ y 2 v d 2 0 otherwise ,
o ( x ) = 1 ( 2 π ) 3 / 2 O ( m , n , p ) × exp [ 2 π i ( m v x + n v y + p u ) ] d m d n d p ,
F ( x s ) = O ( m , n , p ) C ( m , n , p ) × exp [ - 2 π i ( m v x , s + n v y , s + p u s ) ] d m d n d p ,
C ( m , n , p ) = - - - - - I ex ( x ) I ˜ fl ( v ˜ x - M v x , v ˜ y - M v x , M 2 u ) × D ( v ˜ x , v ˜ y ) exp [ 2 π i ( m v x + n v y + p u ) ] d v x d v y d u d v ˜ x d v ˜ y .
H ( v x , v y , u ; M ) = - v d - M v y v d - M v y - ( v d 2 - v ˜ y 2 ) 1 / 2 - M v x ( v d 2 - v ˜ y 2 ) 1 / 2 - M v x I ˜ fl ( w ˜ x , w ˜ y , M 2 u ) × d w ˜ x d w ˜ y ,
C ( m , n , p ) = - - - I ex ( v x , v y , u ) H ( v x , v y , u ; M ) × exp [ 2 π i ( m v x + n v y + p u ) ] d v x d v y d u .
C ( q , p ) = - 0 I ex ( v , u ) H ( v , u ; M ) × exp ( 2 π i p u ) v J 0 ( 2 π v q ) d v d u ,
C ( q , p ) q | q = 0 = lim [ - 0 I ex ( v , u ) H ( v , u ; M ) × exp ( 2 π i p u ) v J 0 ( 2 π v q ) q d v d u ] .
C ( q = 0 , p ) q = 0.
C ( m , n , p ) = - - - I ex ( x ) I ˜ fl ( - M v x - M v y , M 2 u ) × exp [ 2 π i ( m v x + m v y + p u ) ] d v x d v y d u .
C ( q , p ) = - 0 I ex ( v , u ) I ˜ fl ( M v , M 2 u ) × exp ( 2 π i p u ) v J 0 ( 2 π v q ) d v d u .
C ( q ) = 0 v J 0 ( 2 π v q ) [ 0 I ex ( v , u ) H ( v , u ; M ) d u ] d v .
C ( p ) = 0 0 I ex ( v , u ) H ( v , u ; M ) v cos [ 2 π p u ] d v d u .
o ( x ) = δ ( v x ) δ ( v y ) δ ( u - u ) .
F ( v ˜ ; 0 , 0 , u ) = I 0 ( 0 , 0 , u ) 2 0 v d I ˜ fl ( v ˜ , M 1 u ) v ˜ d v ˜ .
F ( 0 , 0 , u ) = I 0 ( 0 , 0 , u ) I ˜ 0 ( 0 , M 2 u ) 2 .
o ( x ) = δ ( u - u ) .
F ( u ) = - - - v d - M v y v d - M v y - ( v d 2 - v ˜ y 2 ) 1 / 2 - M v x ( v d 2 - v ˜ y 2 ) 1 / 2 - M v x I ex ( v x , v y , u ) × I ˜ fl ( w ˜ x , w ˜ y , M 2 u ) d w ˜ x d w ˜ y d v x d v y ,
F ( u ) = 0 I ex ( v , u ) H ( v , u ) v d u .
F ( u ) = 0 I ex ( v , u ) I ˜ fl ( - M v , M 2 u ) v d u .
H ( v x , v y , u ; M ) = - v d - M v y v d - M v y - ( v d 2 - v ˜ y 2 ) 1 / 2 - M v x ( v d 2 - v ˜ y 2 ) 1 / 2 - M v x I ˜ fl ( w ˜ x , w ˜ y , M 2 u ) d w ˜ x d w ˜ y .
H ( v , 0 , u ; M ) v d M v = 0 M v + v d - Φ ( r ) Φ ( r ) I ˜ f l ( r ) r d ϕ d r
= 0 M v + v d 2 I ˜ fl ( r ) r Φ ( r ) d r ,
w ˜ x v d 2 - r 2 - M 2 v 2 - 2 M v .
ϕ = cos - 1 ( v d 2 - r 2 - M 2 v 2 - 2 M v r ) .
Φ ( r ) v d M v = { π if 0 r v d - M v cos - 1 [ ( v d 2 - r 2 - M 2 v 2 ) / - 2 M v r ] otherwise .
H ( v , 0 , u ; M ) v d M v = m v - v d M v + v d - Φ ( r ) Φ ( r ) I ˜ fl ( r ) r d ϕ d r
= M v - v d M v + v d 2 I ˜ fl ( r ) r Φ ( r ) d r .
Φ ( r ) v d < M v = cos - 1 ( v d 2 - r 2 - M 2 v 2 - 2 M v r ) .

Metrics