Abstract

With the use of an approach that is equivalent to Heisenberg’s uncertainty principle, it is possible to derive the axial and lateral gain factors of optical systems consisting of rotationally symmetric pupil plane filters with a large solid angle. We discuss the two- and three-dimensional cases. In contrast to previously published approaches, our results are valid for solid angles as large as 4π. The annular circular aperture is discussed as an example. Resolution gains for annular binary filters consisting of s transmitting rings are calculated, allowing the discussion of the optical properties of all rotationally symmetric binary filters in terms of gain factors. Optical systems with binary two-ring apertures are discussed; a special case is the 4Pi setup. Our novel method yields simple analytical formulas that are functions of the various aperture parameters.

© 1999 Optical Society of America

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References

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  1. E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. Mikrosc. Anat. Entwicklungsmech. 9, 413–468 (1873).
    [CrossRef]
  2. M. Bern, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1997), Chap. VIII.
  3. T. R. M. Sales, “Smallest focal spot,” Phys. Rev. Lett. 81, 3844–3847 (1998).
    [CrossRef]
  4. C. J. R. Sheppard, Z. S. Hegedus, “Axial behavior of pupil-plane filters,” J. Opt. Soc. Am. A 5, 643–647 (1988).
    [CrossRef]
  5. M. Martı́nez-Corral, P. Andrés, J. Ojeda-Castañeda, G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to Confocal Microscopy,” Opt. Commun. 119, 491–498 (1995).
    [CrossRef]
  6. S. W. Hell, E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A 9, 2159–2166 (1992).
    [CrossRef]
  7. Min Gu, C. J. R. Sheppard, “Three dimensional transfer functions in 4Pi confocal microscopes,” J. Opt. Soc. Am. A 11, 1619–1627 (1994).
    [CrossRef]
  8. S. W. Hell, S. Lindek, C. Cremer, E. H. K. Stelzer, “Measurement of the 4Pi-confocal point spread function proves 75 nm axial resolution,” Appl. Phys. Lett. 64, 1335–1337 (1994).
    [CrossRef]
  9. J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1995), pp. 13ff.
  10. W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Phys. 45, 172–198 (1927).
    [CrossRef]
  11. Z. S. Hegedus, “Annular pupil arrays, application to confocal scanning,” Opt. Acta 32, 815–826 (1985).
    [CrossRef]
  12. Z. S. Hegedus, V. Sarafis, “Superresolving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A 3, 1892–1896 (1986).
    [CrossRef]
  13. M. Kowalczyk, M. Martı́nez-Corral, T. Cichocki, P. Andrés, “One-dimensional error-diffusion technique adapted for binarization of rotationally symmetric pupil filters,” Opt. Commun. 114, 211–218 (1995).
    [CrossRef]
  14. S. Grill, “Klassische und quantenmechanische Beschreibung des Auflösungsvermögens in der Mikroskopie,” Diploma thesis (Fakultät für Physik und Astronomie, Ruprecht-Karls-Universität, Heidelberg, Germany, 1998).
  15. E. H. K. Stelzer, S. Lindek, “Fundamental reduction of the observation volume in far-field light microscopy by detection orthogonal to the illumination axis: confocal theta microscopy,” Opt. Commun. 111, 536–547 (1994).
    [CrossRef]
  16. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]

1998 (1)

T. R. M. Sales, “Smallest focal spot,” Phys. Rev. Lett. 81, 3844–3847 (1998).
[CrossRef]

1995 (2)

M. Martı́nez-Corral, P. Andrés, J. Ojeda-Castañeda, G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to Confocal Microscopy,” Opt. Commun. 119, 491–498 (1995).
[CrossRef]

M. Kowalczyk, M. Martı́nez-Corral, T. Cichocki, P. Andrés, “One-dimensional error-diffusion technique adapted for binarization of rotationally symmetric pupil filters,” Opt. Commun. 114, 211–218 (1995).
[CrossRef]

1994 (3)

E. H. K. Stelzer, S. Lindek, “Fundamental reduction of the observation volume in far-field light microscopy by detection orthogonal to the illumination axis: confocal theta microscopy,” Opt. Commun. 111, 536–547 (1994).
[CrossRef]

Min Gu, C. J. R. Sheppard, “Three dimensional transfer functions in 4Pi confocal microscopes,” J. Opt. Soc. Am. A 11, 1619–1627 (1994).
[CrossRef]

S. W. Hell, S. Lindek, C. Cremer, E. H. K. Stelzer, “Measurement of the 4Pi-confocal point spread function proves 75 nm axial resolution,” Appl. Phys. Lett. 64, 1335–1337 (1994).
[CrossRef]

1992 (1)

1988 (1)

1986 (1)

1985 (1)

Z. S. Hegedus, “Annular pupil arrays, application to confocal scanning,” Opt. Acta 32, 815–826 (1985).
[CrossRef]

1964 (1)

1927 (1)

W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Phys. 45, 172–198 (1927).
[CrossRef]

1873 (1)

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. Mikrosc. Anat. Entwicklungsmech. 9, 413–468 (1873).
[CrossRef]

Abbe, E.

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. Mikrosc. Anat. Entwicklungsmech. 9, 413–468 (1873).
[CrossRef]

Andrés, P.

M. Kowalczyk, M. Martı́nez-Corral, T. Cichocki, P. Andrés, “One-dimensional error-diffusion technique adapted for binarization of rotationally symmetric pupil filters,” Opt. Commun. 114, 211–218 (1995).
[CrossRef]

M. Martı́nez-Corral, P. Andrés, J. Ojeda-Castañeda, G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to Confocal Microscopy,” Opt. Commun. 119, 491–498 (1995).
[CrossRef]

Bern, M.

M. Bern, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1997), Chap. VIII.

Cichocki, T.

M. Kowalczyk, M. Martı́nez-Corral, T. Cichocki, P. Andrés, “One-dimensional error-diffusion technique adapted for binarization of rotationally symmetric pupil filters,” Opt. Commun. 114, 211–218 (1995).
[CrossRef]

Cremer, C.

S. W. Hell, S. Lindek, C. Cremer, E. H. K. Stelzer, “Measurement of the 4Pi-confocal point spread function proves 75 nm axial resolution,” Appl. Phys. Lett. 64, 1335–1337 (1994).
[CrossRef]

Grill, S.

S. Grill, “Klassische und quantenmechanische Beschreibung des Auflösungsvermögens in der Mikroskopie,” Diploma thesis (Fakultät für Physik und Astronomie, Ruprecht-Karls-Universität, Heidelberg, Germany, 1998).

Gu, Min

Hegedus, Z. S.

Heisenberg, W.

W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Phys. 45, 172–198 (1927).
[CrossRef]

Hell, S. W.

S. W. Hell, S. Lindek, C. Cremer, E. H. K. Stelzer, “Measurement of the 4Pi-confocal point spread function proves 75 nm axial resolution,” Appl. Phys. Lett. 64, 1335–1337 (1994).
[CrossRef]

S. W. Hell, E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A 9, 2159–2166 (1992).
[CrossRef]

Kowalczyk, M.

M. Kowalczyk, M. Martı́nez-Corral, T. Cichocki, P. Andrés, “One-dimensional error-diffusion technique adapted for binarization of rotationally symmetric pupil filters,” Opt. Commun. 114, 211–218 (1995).
[CrossRef]

Lindek, S.

E. H. K. Stelzer, S. Lindek, “Fundamental reduction of the observation volume in far-field light microscopy by detection orthogonal to the illumination axis: confocal theta microscopy,” Opt. Commun. 111, 536–547 (1994).
[CrossRef]

S. W. Hell, S. Lindek, C. Cremer, E. H. K. Stelzer, “Measurement of the 4Pi-confocal point spread function proves 75 nm axial resolution,” Appl. Phys. Lett. 64, 1335–1337 (1994).
[CrossRef]

Marti´nez-Corral, M.

M. Kowalczyk, M. Martı́nez-Corral, T. Cichocki, P. Andrés, “One-dimensional error-diffusion technique adapted for binarization of rotationally symmetric pupil filters,” Opt. Commun. 114, 211–218 (1995).
[CrossRef]

M. Martı́nez-Corral, P. Andrés, J. Ojeda-Castañeda, G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to Confocal Microscopy,” Opt. Commun. 119, 491–498 (1995).
[CrossRef]

McCutchen, C. W.

Ojeda-Castañeda, J.

M. Martı́nez-Corral, P. Andrés, J. Ojeda-Castañeda, G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to Confocal Microscopy,” Opt. Commun. 119, 491–498 (1995).
[CrossRef]

Saavedra, G.

M. Martı́nez-Corral, P. Andrés, J. Ojeda-Castañeda, G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to Confocal Microscopy,” Opt. Commun. 119, 491–498 (1995).
[CrossRef]

Sales, T. R. M.

T. R. M. Sales, “Smallest focal spot,” Phys. Rev. Lett. 81, 3844–3847 (1998).
[CrossRef]

Sarafis, V.

Sheppard, C. J. R.

Stelzer, E. H. K.

S. W. Hell, S. Lindek, C. Cremer, E. H. K. Stelzer, “Measurement of the 4Pi-confocal point spread function proves 75 nm axial resolution,” Appl. Phys. Lett. 64, 1335–1337 (1994).
[CrossRef]

E. H. K. Stelzer, S. Lindek, “Fundamental reduction of the observation volume in far-field light microscopy by detection orthogonal to the illumination axis: confocal theta microscopy,” Opt. Commun. 111, 536–547 (1994).
[CrossRef]

S. W. Hell, E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A 9, 2159–2166 (1992).
[CrossRef]

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1995), pp. 13ff.

Wolf, E.

M. Bern, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1997), Chap. VIII.

Appl. Phys. Lett. (1)

S. W. Hell, S. Lindek, C. Cremer, E. H. K. Stelzer, “Measurement of the 4Pi-confocal point spread function proves 75 nm axial resolution,” Appl. Phys. Lett. 64, 1335–1337 (1994).
[CrossRef]

Arch. Mikrosc. Anat. Entwicklungsmech. (1)

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. Mikrosc. Anat. Entwicklungsmech. 9, 413–468 (1873).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Acta (1)

Z. S. Hegedus, “Annular pupil arrays, application to confocal scanning,” Opt. Acta 32, 815–826 (1985).
[CrossRef]

Opt. Commun. (3)

M. Kowalczyk, M. Martı́nez-Corral, T. Cichocki, P. Andrés, “One-dimensional error-diffusion technique adapted for binarization of rotationally symmetric pupil filters,” Opt. Commun. 114, 211–218 (1995).
[CrossRef]

E. H. K. Stelzer, S. Lindek, “Fundamental reduction of the observation volume in far-field light microscopy by detection orthogonal to the illumination axis: confocal theta microscopy,” Opt. Commun. 111, 536–547 (1994).
[CrossRef]

M. Martı́nez-Corral, P. Andrés, J. Ojeda-Castañeda, G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to Confocal Microscopy,” Opt. Commun. 119, 491–498 (1995).
[CrossRef]

Phys. Rev. Lett. (1)

T. R. M. Sales, “Smallest focal spot,” Phys. Rev. Lett. 81, 3844–3847 (1998).
[CrossRef]

Z. Phys. (1)

W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Phys. 45, 172–198 (1927).
[CrossRef]

Other (3)

S. Grill, “Klassische und quantenmechanische Beschreibung des Auflösungsvermögens in der Mikroskopie,” Diploma thesis (Fakultät für Physik und Astronomie, Ruprecht-Karls-Universität, Heidelberg, Germany, 1998).

M. Bern, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1997), Chap. VIII.

J. T. Verdeyen, Laser Electronics (Prentice-Hall, Englewood Cliffs, N.J., 1995), pp. 13ff.

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Figures (9)

Fig. 1
Fig. 1

(a) Double-slit aperture with an opening angle of αmax=π/3 and an obstruction factor of =0.6. (b) Corresponding annular aperture for the three-dimensional case. The shaded area represents the region through which light can reach the focus. If is set to zero, this aperture corresponds to a circular aperture or, in the two-dimensional case in (a), to a slit aperture.

Fig. 2
Fig. 2

Two-dimensional approach. Shown are the minimal standard deviations of the intensity distribution in the focal region in the lateral (solid curves) and axial (dashed curves) directions as a function of αmax with the use of a slit and a double-slit aperture. The slit aperture is equivalent to =0. The double-slit aperture has a value of =0.8. The wavelength is λ=633 nm.

Fig. 3
Fig. 3

(a) Minimal standard deviations of the intensity distribution in the focal region in the lateral (ΔxCC,3D, solid curve) and axial (ΔzCC,3D, dashed curve) directions as a function of αmax with the use of a clear aperture. (b) Minimal focal volume FV=(ΔxCC,3D)2(ΔzCC,3D) as a function of αmax with the use of a clear aperture. The focal volume reaches a minimum at αmax=π, representing a spherical wave. The wavelength is λ=633 nm.

Fig. 4
Fig. 4

Lateral gain factors of an annular aperture as a function of the obstruction factor for different αmax.

Fig. 5
Fig. 5

Axial gain factors of an annular aperture as a function of the obstruction factor for different αmax.

Fig. 6
Fig. 6

(a) Binary annular aperture consisting of two rings described by γ=0.4, =0.8, and αmax=π/3. The shaded area represents the regions through which light can reach the focus. The projections onto the focal plane and onto a plane perpendicular to the focal plane are shown. (b) Corresponding lateral and axial gain factors as a function of the opening angle αmax.

Fig. 7
Fig. 7

Lateral gain factors of a two-ring aperture with identical inner and outer transmission areas as a function of the obscuration parameter μ for different opening angles αmax. The lateral gain decreases for setups with a higher αmax.

Fig. 8
Fig. 8

(a) 4Pi setup where each of the objective lenses has an opening angle of αmaxOL=π/4. The shaded area represents the regions through which light can reach the focus. The projections onto the focal plane and onto a plane perpendicular to the focal plane are shown. (b) Corresponding lateral and axial gain factors as a function of the opening angle αmaxOL.

Fig. 9
Fig. 9

Minimal standard deviations of the focal intensity distribution in the lateral and axial directions of a 4Pi setup as a function of the opening angle αmaxOL of each of the objective lenses. The wavelength is λ=633 nm.

Equations (47)

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ΔpjΔ j2  j{x, y, z},
px=|p|sin α,pz=|p|cos α,
mn, j=-ππ[pj(α)]nDp(α)dα  j{x, y, z}.
Δpj=m2,jm0,j-m1, jm0, j21/2  j{x, y, z}.
Δj2m2,jm0,j-m1,jm0,j21/2  j{x, y, z}.
DpS(α)=Θ(α+αmax)-Θ(α-αmax),
 ΔxS,2D=αmax λ2 π(2 αmax-sin 2 αmax)1/2,
ΔzS,2D=αmax λ2 π[2 αmax2-4(sin αmax)2+αmax sin 2 αmax]1/2,
DpDS(α)=Θ(α+αmax)-Θ(α-αmax)-Θ(α+αmax)+Θ(α-αmax).
ΔxDS,2D=[(1-)αmax]1/2λ2 π[2 αmax(1-)-sin 2 αmax+sin 2αmax]1/2,
ΔzDS,2D=(1-)αmaxλ2 π{αmax(1-)[2 αmax(1-)+sin 2 αmax-sin 2αmax]-4(sin αmax-sin αmax)2}1/2.
Glat,2DDS=2 αmax(1-)-sin 2 αmax+sin 2αmax(1-)(2 αmax-sin 2 αmax)1/2,
Gax,2DDS=11- αmax(1-)[2 αmax(1-)+sin 2 αmax-sin 2αmax]-4(sin αmax-sin αmax)22 αmax2-4 sin αmax2+αmax sin 2 αmax1/2.
mn, j=02 π0π[pj(α, φ)]nDp(α, φ)×sin α dαdφ  j{x, y, z}.
px(α, φ)=|p|sin α cos φ,pz(α, φ)=|p|cos α.
DpCC(α, φ)=Θ(α)-Θ(α-αmax),
ΔxCC=3λ2 π(3-2 cos αmax-cos 2 αmax)1/2,
ΔzCC=3λ2 π(1-cos αmax).
DpAA(α, φ)=Θ(α-αmax)-Θ(α-αmax).
 ΔxAA=3λ2 π[4-cos 2 αmax-cos 2αmax-cos(αmax-amax)-cos(αmax+amax)]1/2,
ΔzAA=3λ2 π(cos amax-cos amax).
GlatAA=4-cos 2 αmax-cos 2αmax-cos(αmax-amax)-cos(αmax+amax)3-2 cos αmax-cos 2 αmax1/2,
GaxAA=cos amax-cos amax1-cos αmax.
limαmax0 GlatAA=1+2,limαmax0 GaxAA=1-2.
Ti(α)=1,2iαmaxα2i+1αmax0,otherwise.
0k1  k{0, 1,, 2s-1}  kl  k<l.
DpSRA(α, φ)=i=0s-1[Θ(α-2iαmax)-Θ(α-2i+1αmax)].
ΔxSRA=6λ2 π i=0s-1(cos 2iαmax-cos 2i+1αmax)i=0s-1(9 cos 2iαmax-cos 32iαmax-9 cos 2i+1αmax+cos 32i+1αmax)1/2,
ΔzSRA
=3λi=0s-1(cos 2iαmax-cos 2i+1αmax)2 π-3i=0s-1[(cos 2iαmax)2-(cos 2i+1αmax)2]2+4i=0s-1(cos 2iαmax-cos 2i+1αmax)i=0s-1[(cos 2iαmax)3-(cos 2i+1αmax)3]1/2.
GlatSRA=ΔxCCΔxSRA,GaxSRA=ΔzCCΔzSRA.
Glat,app.SRA=k=02s-1(-1)k+1k4k=02s-1(-1)k+1k21/2,
Gax,app.SRA=-3k=02s-1(-1)k+1k42+4k=02s-1(-1)k+1k2k=02s-1(-1)k+1k61/2k=02s-1(-1)k+1k2.
s=2,0=0,1=γ,2=,3=1.
GlatB2R=12 8-9 cos αmax+cos 3αmax-9 cos γαmax+cos 3γαmax+9 cos αmax-cos 3αmax(1-cos αmax-cos γαmax+cos αmax)(3-2 cos αmax-cos 2 αmax)1/2,
GaxB2R
={-3[1-(cos αmax)2-(cos γαmax)2+(cos αmax)2]2+4(1-cos αmax-cos γαmax+cos αmax)[1-(cos αmax)3-(cos γαmax)3+(cos αmax)3]}1/2(1-cos αmax-cos γαmax+cos αmax)(1-cos αmax).
=1αmax arccos12 (1-μ+cos αmax+μ cos αmax),
γ=1αmax arccos12 (1+μ+cos αmax-μ cos αmax)
GaxB2R=(1+μ+μ2).
limαmax0 GlatB2R=1.
s=2,αmax=π,0=0,
1=αmaxOLπ,2=1-αmaxOLπ,3=1
Δx4Pi=3λ2 π(3-2 cos αmaxOL-cos 2 αmaxOL)1/2,
Δz4Pi=3λ2 π2(3+2 cos αmaxOL+cos 2 αmaxOL)1/2.
Glat4Pi=1,
Gax4Pi=2(3+2 cos αmaxOL+cos 2 αmaxOL)1/21-cos αmaxOL.

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