Abstract

The problem of detecting electric dipole emitters (such as a fluorescing molecule) through a microscope objective with a high numerical aperture is considered. Exact vector-wave-optics results of the field distribution of the electromagnetic field in image space are derived. On the basis of these results, the collection-efficiency function (CEF) is calculated. This calculation is compared with a semigeometric approximation of the CEF, for which a new analytical formula is presented. The derived results are important for applications such as fluorescence correlation spectroscopy and single-molecule fluorescence detection with a confocal microscope.

© 2000 Optical Society of America

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References

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  1. M. D. Barnes, W. B. Whitten, and J. M. Ramsey, Anal. Chem. 67, 418A (1995).
    [CrossRef]
  2. R. A. Keller, W. P. Ambrose, P. M. Goodwin, J. H. Jett, J. C. Martin, and M. Wu, Appl. Spectrosc. 50, 12A (1996).
    [CrossRef]
  3. S. M. Nie and R. N. Zare, Annu. Rev. Biophys. Biomol. Struct. 26, 567 (1997).
    [CrossRef]
  4. J. Enderlein, W. P. Ambrose, P. M. Goodwin, and R. A. Keller, in Microsystem Technology: A Powerful Tool for Biomolecular Studies, M. Köhler, T. Mejevaia, and H. P. Saluz, eds. (Birkhäuser, Basel, Switzerland, 1999), p. 311.
  5. S. Inoue, in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), p. 1.
    [CrossRef]
  6. R. H. Webb, Rep. Prog. Phys. 59, 427 (1996).
    [CrossRef]
  7. N. L. Thompson, in Topics in Fluorescence Spectroscopy, J. R. Lakowicz, ed. (Plenum, New York, 1991), Vol. 1, p. 337.
    [CrossRef]
  8. E. Wolf, Proc. R. Soc. London Ser. A 253, 349 (1959).
    [CrossRef]
  9. B. Richards and E. Wolf, Proc. R. Soc. London A 253, 358 (1959).
    [CrossRef]
  10. H. Qian and E. L. Elson, Appl. Opt. 30, 1185 (1991).
    [CrossRef] [PubMed]
  11. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Deutsch, Thun, Switzerland, 1984).

1997 (1)

S. M. Nie and R. N. Zare, Annu. Rev. Biophys. Biomol. Struct. 26, 567 (1997).
[CrossRef]

1996 (2)

1995 (1)

M. D. Barnes, W. B. Whitten, and J. M. Ramsey, Anal. Chem. 67, 418A (1995).
[CrossRef]

1991 (1)

1959 (2)

E. Wolf, Proc. R. Soc. London Ser. A 253, 349 (1959).
[CrossRef]

B. Richards and E. Wolf, Proc. R. Soc. London A 253, 358 (1959).
[CrossRef]

Ambrose, W. P.

R. A. Keller, W. P. Ambrose, P. M. Goodwin, J. H. Jett, J. C. Martin, and M. Wu, Appl. Spectrosc. 50, 12A (1996).
[CrossRef]

J. Enderlein, W. P. Ambrose, P. M. Goodwin, and R. A. Keller, in Microsystem Technology: A Powerful Tool for Biomolecular Studies, M. Köhler, T. Mejevaia, and H. P. Saluz, eds. (Birkhäuser, Basel, Switzerland, 1999), p. 311.

Barnes, M. D.

M. D. Barnes, W. B. Whitten, and J. M. Ramsey, Anal. Chem. 67, 418A (1995).
[CrossRef]

Elson, E. L.

Enderlein, J.

J. Enderlein, W. P. Ambrose, P. M. Goodwin, and R. A. Keller, in Microsystem Technology: A Powerful Tool for Biomolecular Studies, M. Köhler, T. Mejevaia, and H. P. Saluz, eds. (Birkhäuser, Basel, Switzerland, 1999), p. 311.

Goodwin, P. M.

R. A. Keller, W. P. Ambrose, P. M. Goodwin, J. H. Jett, J. C. Martin, and M. Wu, Appl. Spectrosc. 50, 12A (1996).
[CrossRef]

J. Enderlein, W. P. Ambrose, P. M. Goodwin, and R. A. Keller, in Microsystem Technology: A Powerful Tool for Biomolecular Studies, M. Köhler, T. Mejevaia, and H. P. Saluz, eds. (Birkhäuser, Basel, Switzerland, 1999), p. 311.

Inoue, S.

S. Inoue, in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), p. 1.
[CrossRef]

Jett, J. H.

Keller, R. A.

R. A. Keller, W. P. Ambrose, P. M. Goodwin, J. H. Jett, J. C. Martin, and M. Wu, Appl. Spectrosc. 50, 12A (1996).
[CrossRef]

J. Enderlein, W. P. Ambrose, P. M. Goodwin, and R. A. Keller, in Microsystem Technology: A Powerful Tool for Biomolecular Studies, M. Köhler, T. Mejevaia, and H. P. Saluz, eds. (Birkhäuser, Basel, Switzerland, 1999), p. 311.

Martin, J. C.

Nie, S. M.

S. M. Nie and R. N. Zare, Annu. Rev. Biophys. Biomol. Struct. 26, 567 (1997).
[CrossRef]

Qian, H.

Ramsey, J. M.

M. D. Barnes, W. B. Whitten, and J. M. Ramsey, Anal. Chem. 67, 418A (1995).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, Proc. R. Soc. London A 253, 358 (1959).
[CrossRef]

Thompson, N. L.

N. L. Thompson, in Topics in Fluorescence Spectroscopy, J. R. Lakowicz, ed. (Plenum, New York, 1991), Vol. 1, p. 337.
[CrossRef]

Webb, R. H.

R. H. Webb, Rep. Prog. Phys. 59, 427 (1996).
[CrossRef]

Whitten, W. B.

M. D. Barnes, W. B. Whitten, and J. M. Ramsey, Anal. Chem. 67, 418A (1995).
[CrossRef]

Wolf, E.

E. Wolf, Proc. R. Soc. London Ser. A 253, 349 (1959).
[CrossRef]

B. Richards and E. Wolf, Proc. R. Soc. London A 253, 358 (1959).
[CrossRef]

Wu, M.

Zare, R. N.

S. M. Nie and R. N. Zare, Annu. Rev. Biophys. Biomol. Struct. 26, 567 (1997).
[CrossRef]

Anal. Chem. (1)

M. D. Barnes, W. B. Whitten, and J. M. Ramsey, Anal. Chem. 67, 418A (1995).
[CrossRef]

Annu. Rev. Biophys. Biomol. Struct. (1)

S. M. Nie and R. N. Zare, Annu. Rev. Biophys. Biomol. Struct. 26, 567 (1997).
[CrossRef]

Appl. Opt. (1)

Appl. Spectrosc. (1)

Proc. R. Soc. London A (1)

B. Richards and E. Wolf, Proc. R. Soc. London A 253, 358 (1959).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

E. Wolf, Proc. R. Soc. London Ser. A 253, 349 (1959).
[CrossRef]

Rep. Prog. Phys. (1)

R. H. Webb, Rep. Prog. Phys. 59, 427 (1996).
[CrossRef]

Other (4)

N. L. Thompson, in Topics in Fluorescence Spectroscopy, J. R. Lakowicz, ed. (Plenum, New York, 1991), Vol. 1, p. 337.
[CrossRef]

J. Enderlein, W. P. Ambrose, P. M. Goodwin, and R. A. Keller, in Microsystem Technology: A Powerful Tool for Biomolecular Studies, M. Köhler, T. Mejevaia, and H. P. Saluz, eds. (Birkhäuser, Basel, Switzerland, 1999), p. 311.

S. Inoue, in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), p. 1.
[CrossRef]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Deutsch, Thun, Switzerland, 1984).

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Figures (2)

Fig. 1
Fig. 1

Schematic of the imaging geometry of the dipole.

Fig. 2
Fig. 2

CEF as computed by (a) the full wave-vector approach and (b) the semigeometric approximation of Qian and Elson.

Equations (13)

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E=0ηmaxdη sin η02πdψcos ηcos ηeˆpeˆ+eˆ|peˆ|expikR/Rexpiksˆr,
eˆ=cos ψ cos η,sin ψ cos η,-sin η,  eˆ|=-sin ψ,cos ψ,0,  eˆ=cos ψ cosη,sin ψ cosη,sin η,  sˆ=-sin η cos ψ,-sin η sin ψ,cos η,
ηmax=arctanNAMn2-NA2.
B=n0ηmaxdη sin η02πdψcos ηcos ηeˆ|peˆ-eˆpeˆ|expikR/Rexpiksˆr,
sˆr=sˆR+ρ+z=R-ρ sin η cosψ-ϕ+z cos η,
Ea=2πexpikR+ikRR0ηmaxdη sin ηcos ηcos η×GabEη,ϕ,ρpb expikz cos η,
GxxE=121+cos η cos ηJ0+121-cos η cos ηJ2 cos 2ϕ,  GxyE=GyxE=121-cos η cos ηJ2 sin 2ϕ,  GxzE=-i cos η sin ηJ1 cos ϕ,  GyyE=121+cos η cos ηJ0-121-cos η cos ηJ2 cos2 ϕ,  GyzE=-i cos η sin ηJ1 sin ϕ,  GzxE=i sin η cos ηJ1 cos ϕ,  GzyE=i sin η cos ηJ1 sin ϕ,  GzzE=-sin η sin ηJ0,
GxxB=-GyyB=12cos η-cos ηJ2 sin 2ϕ,  GxyB=-12cos η+cos ηJ0-12cos η-cos ηJ2 cos 2ϕ,  GxzB=iJ1 sin η sin ϕ,  GyxB=12cos η+cos ηJ0-12cos η-cos ηJ2 cos 2ϕ,  GyzB=-iJ1 sin η cos ϕ,  GzxB=-iJ1 sin η sin ϕ,  GzyB=iJ1 sin η cos ϕ,  GzzB=0.
CEFρ,z=circρa/acircρa-ρ/uzπu2zdρa,
CEFρ, z=0,ρu+a  CEFρ,z=maxa,w02θ1a2+θ2u2-Δ/πa2u2,u-a<ρ<u+a  CEFρ,z=maxa,w02/maxa,u2, ρu-a,
θ1=arccosa2+ρ2-u2/2aρ,  θ2=arccosu2+ρ2-a2/2uρ,
Δ=a+ρ+u-a+ρ+ua-ρ+ua+ρ-u1/22
V=0dρρ-dzCEFρ,z,

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