Abstract

A novel function, the point-spread autocorrelation function (PSAF), which is closely related to the point-spread function, of a high numerical aperture microscope objective is introduced. The function is both experimentally measured and theoretically modeled for various apodization conditions. These include varying the effective numerical aperture of the objective, applying annuli of different size, and illuminating the objective with a spatially nonuniform intensity distribution. An excellent agreement between experimental data and theoretical modeling is obtained without the use of any fitting parameters. The PSAF technique is sensitive to the various apodization conditions, affecting both the width of the PSAF signal and the amplitude of the sidelobes. A potential use of the technique is the measurement of the effective numerical aperture and the optimization of the illumination conditions in complex microscopical systems.

© 1997 Optical Society of America

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References

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  1. G. J. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
    [CrossRef]
  2. C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
    [CrossRef]
  3. H. Jacobsen, S. W. Hell, “Effect of the specimen refractive index on the imaging of a confocal fluorescence microscope employing high aperture oil immersion lenses,” Bioimaging 3, 39–47 (1995).
    [CrossRef]
  4. G. J. Brakenhoff, H. T. M. v. d. Voort, E. A. v. Spronsen, N. Nanninga, “Three-dimensional imaging by confocal scanning fluorescence microscopy,” Ann. N. Y. Acad. Sci. 483, 405–415 (1986).
    [CrossRef] [PubMed]
  5. S. F. Gibson, F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A 8, 1601–1613 (1991).
    [CrossRef]
  6. H. T. M. v. d. Voort, G. J. Brakenhoff, G. C. A. M. Janssen, “Determination of the 3-dimensional optical properties of a confocal scanning laser microscope,” Optik 78, 48–53 (1988).
  7. M. Müller, G. J. Brakenhoff, “Characterization of high-numerical-aperture lenses by spatial autocorrelation of the focal field,” Opt. Lett. 20, 2159–2161 (1995).
    [CrossRef] [PubMed]
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).
  9. J. J. Stamnes, Waves in Focal Regions (Institute of Physics, Bristol, 1986).
  10. J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
    [CrossRef]
  11. M. Gu, C. J. R. Sheppard, “Three-dimensional imaging in confocal fluorescent microscopy with annular lenses,” J. Modern Opt. 38, 2247–2263 (1991).
    [CrossRef]
  12. M. Kempe, U. Stamm, B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89, 119–125 (1992).
    [CrossRef]

1995 (2)

H. Jacobsen, S. W. Hell, “Effect of the specimen refractive index on the imaging of a confocal fluorescence microscope employing high aperture oil immersion lenses,” Bioimaging 3, 39–47 (1995).
[CrossRef]

M. Müller, G. J. Brakenhoff, “Characterization of high-numerical-aperture lenses by spatial autocorrelation of the focal field,” Opt. Lett. 20, 2159–2161 (1995).
[CrossRef] [PubMed]

1992 (1)

M. Kempe, U. Stamm, B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89, 119–125 (1992).
[CrossRef]

1991 (2)

S. F. Gibson, F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A 8, 1601–1613 (1991).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Three-dimensional imaging in confocal fluorescent microscopy with annular lenses,” J. Modern Opt. 38, 2247–2263 (1991).
[CrossRef]

1988 (1)

H. T. M. v. d. Voort, G. J. Brakenhoff, G. C. A. M. Janssen, “Determination of the 3-dimensional optical properties of a confocal scanning laser microscope,” Optik 78, 48–53 (1988).

1986 (1)

G. J. Brakenhoff, H. T. M. v. d. Voort, E. A. v. Spronsen, N. Nanninga, “Three-dimensional imaging by confocal scanning fluorescence microscopy,” Ann. N. Y. Acad. Sci. 483, 405–415 (1986).
[CrossRef] [PubMed]

1983 (1)

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

1979 (1)

G. J. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

1977 (1)

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

Barends, P.

G. J. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

Blom, P.

G. J. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Brakenhoff, G. J.

M. Müller, G. J. Brakenhoff, “Characterization of high-numerical-aperture lenses by spatial autocorrelation of the focal field,” Opt. Lett. 20, 2159–2161 (1995).
[CrossRef] [PubMed]

H. T. M. v. d. Voort, G. J. Brakenhoff, G. C. A. M. Janssen, “Determination of the 3-dimensional optical properties of a confocal scanning laser microscope,” Optik 78, 48–53 (1988).

G. J. Brakenhoff, H. T. M. v. d. Voort, E. A. v. Spronsen, N. Nanninga, “Three-dimensional imaging by confocal scanning fluorescence microscopy,” Ann. N. Y. Acad. Sci. 483, 405–415 (1986).
[CrossRef] [PubMed]

G. J. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

Gibson, S. F.

Gu, M.

M. Gu, C. J. R. Sheppard, “Three-dimensional imaging in confocal fluorescent microscopy with annular lenses,” J. Modern Opt. 38, 2247–2263 (1991).
[CrossRef]

Hell, S. W.

H. Jacobsen, S. W. Hell, “Effect of the specimen refractive index on the imaging of a confocal fluorescence microscope employing high aperture oil immersion lenses,” Bioimaging 3, 39–47 (1995).
[CrossRef]

Jacobsen, H.

H. Jacobsen, S. W. Hell, “Effect of the specimen refractive index on the imaging of a confocal fluorescence microscope employing high aperture oil immersion lenses,” Bioimaging 3, 39–47 (1995).
[CrossRef]

Janssen, G. C. A. M.

H. T. M. v. d. Voort, G. J. Brakenhoff, G. C. A. M. Janssen, “Determination of the 3-dimensional optical properties of a confocal scanning laser microscope,” Optik 78, 48–53 (1988).

Kempe, M.

M. Kempe, U. Stamm, B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89, 119–125 (1992).
[CrossRef]

Lanni, F.

Müller, M.

Nanninga, N.

G. J. Brakenhoff, H. T. M. v. d. Voort, E. A. v. Spronsen, N. Nanninga, “Three-dimensional imaging by confocal scanning fluorescence microscopy,” Ann. N. Y. Acad. Sci. 483, 405–415 (1986).
[CrossRef] [PubMed]

Pedersen, H. M.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Sheppard, C. J. R.

M. Gu, C. J. R. Sheppard, “Three-dimensional imaging in confocal fluorescent microscopy with annular lenses,” J. Modern Opt. 38, 2247–2263 (1991).
[CrossRef]

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Spronsen, E. A. v.

G. J. Brakenhoff, H. T. M. v. d. Voort, E. A. v. Spronsen, N. Nanninga, “Three-dimensional imaging by confocal scanning fluorescence microscopy,” Ann. N. Y. Acad. Sci. 483, 405–415 (1986).
[CrossRef] [PubMed]

Stamm, U.

M. Kempe, U. Stamm, B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89, 119–125 (1992).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Institute of Physics, Bristol, 1986).

Voort, H. T. M. v. d.

H. T. M. v. d. Voort, G. J. Brakenhoff, G. C. A. M. Janssen, “Determination of the 3-dimensional optical properties of a confocal scanning laser microscope,” Optik 78, 48–53 (1988).

G. J. Brakenhoff, H. T. M. v. d. Voort, E. A. v. Spronsen, N. Nanninga, “Three-dimensional imaging by confocal scanning fluorescence microscopy,” Ann. N. Y. Acad. Sci. 483, 405–415 (1986).
[CrossRef] [PubMed]

Wilhelmi, B.

M. Kempe, U. Stamm, B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89, 119–125 (1992).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Ann. N. Y. Acad. Sci. (1)

G. J. Brakenhoff, H. T. M. v. d. Voort, E. A. v. Spronsen, N. Nanninga, “Three-dimensional imaging by confocal scanning fluorescence microscopy,” Ann. N. Y. Acad. Sci. 483, 405–415 (1986).
[CrossRef] [PubMed]

Bioimaging (1)

H. Jacobsen, S. W. Hell, “Effect of the specimen refractive index on the imaging of a confocal fluorescence microscope employing high aperture oil immersion lenses,” Bioimaging 3, 39–47 (1995).
[CrossRef]

J. Microsc. (1)

G. J. Brakenhoff, P. Blom, P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

J. Modern Opt. (1)

M. Gu, C. J. R. Sheppard, “Three-dimensional imaging in confocal fluorescent microscopy with annular lenses,” J. Modern Opt. 38, 2247–2263 (1991).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Acta (2)

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

Opt. Commun. (1)

M. Kempe, U. Stamm, B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89, 119–125 (1992).
[CrossRef]

Opt. Lett. (1)

Optik (1)

H. T. M. v. d. Voort, G. J. Brakenhoff, G. C. A. M. Janssen, “Determination of the 3-dimensional optical properties of a confocal scanning laser microscope,” Optik 78, 48–53 (1988).

Other (2)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

J. J. Stamnes, Waves in Focal Regions (Institute of Physics, Bristol, 1986).

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Figures (7)

Fig. 1
Fig. 1

Schematic of the experimental setup for PSAF measurements: BS’s, 50% beamsplitters; L1, 200-mm lens; L2, 80-mm lens; PAR, lock-in amplifier to rectify the PMT output.

Fig. 2
Fig. 2

Amplitude and phase of the (complex) focal field distribution, as a function of the lateral coordinate in the geometrical focal plane, of a high-NA lens. The focal field distribution is calculated for a circular aperture and NA = 1.3.

Fig. 3
Fig. 3

(a) Typical functional dependence of the intensity PSF and PSAF for equal focusing conditions: NA = 1.3 and a circular aperture. The inset is an enlargement showing that the nodes of the PSAF are at lateral focal shifts close but not equal to the lateral positions of the nodes of the intensity PSF. (b) The (space-integrated) value of the fluorescence as a function of the lateral focal shift for various values of the induced temporal delay. At a certain lateral focal shift all fluorescence values for the different temporal delays coincide, implying a zero magnitude in the difference between the maximum and minimum values of the total fluorescence. The focusing conditions are the same as in (a).

Fig. 4
Fig. 4

Comparison of the experimental PSAF signal with the theoretically expected PSAF response for three different values of the NA of the objective: (a) NA = 1.3, (b) NA = 0.9, (c) NA = 0.6. The experimental data represent an average over five subsequent scans of the induced lateral focal shift. The theoretical curves are calculated for the given NA and excitation wavelength without any fitting parameters.

Fig. 5
Fig. 5

PSAF signal for different sizes of annuli, with (a) 0%, (b) 16.7%, (c) 50%, (d) 66.7% of the central part of the aperture (NA = 1.3) blocked. The experimental data represent an average over five subsequent scans of the induced lateral focal shift. The theoretical curves are calculated for the given NA and excitation wavelength without any fitting parameters.

Fig. 6
Fig. 6

(a) Geometry of the displacement of the illumination profile across the aperture of the objective with respect to the optical axis. The displacement is in the same direction as the object beam is shifted with respect to the reference and is expressed as the ratio of the shift of the center of the illumination distribution with respect to the full aperture of the objective. (b) Experimental PSAF signal for several displacements varying from zero to 50%. The experimental data represent an average over five subsequent scans of the induced lateral focal shift. (c) Drop in intensity of the PSAF signal at a zero lateral focal shift as a function of the displacement.

Fig. 7
Fig. 7

(a) Geometry of the displacement of an annulus within the (stationary) illumination profile across the aperture of the objective with respect to the optical axis. The displacement is in the same direction as the object beam is shifted with respect to the reference and is expressed as the ratio of the shift of the center of the annulus with respect to the full aperture of the objective. The experimental PSAF signal is shown for several displacements of the 50% annulus, at (b) 0%, (c) 3.5%, (d) 6.5%. The experimental data represent an average over five subsequent scans of the induced lateral focal shift.

Equations (3)

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GΔr, τ =-dt-drOrur, t+ur+Δr, t+τ2.
IPSAFΔr=maxGΔr, τ; τ0, λ/c-minGΔr, τ; τ0, λ/c,
rlat=0.61λ/NA.

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