Abstract

A novel procedure to design axial and transverse superresolving pupil filters for the 4Pi-confocal microscope is presented. The method is based on the use of a series of figures of merit developed to describe the effect of inserting two identical filters in the two arms of the illumination path of the microscope. As a practical implementation, we have applied our method to obtain superresolving continuous phase-only filters. Different resolution-improving phase functions are shown for the transverse and the axial direction. These filters provided axial gain up to 1.3 and transverse gain up to 1.4 without an increase in sidelobes.

© 2003 Optical Society of America

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  1. S. Hell, E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A 9, 2159–2166 (1992).
    [CrossRef]
  2. S. Hell, E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. 93, 277–282 (1992).
    [CrossRef]
  3. M. Schrader, S. W. Hell, H. T. M. van der Voort, “Three-dimensional super-resolution with a 4Pi-confocal microscope using image restoration,” J. Appl. Phys. 84, 4033–4042 (1998).
    [CrossRef]
  4. Z. S. Hegedus, V. Sarafis, “Superresolving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A 3, 1892–1896 (1986).
    [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. M. Martı́nez-Corral, P. Andrés, C. J. Zapata-Rodrı́guez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).
  7. C. J. R. Sheppard, “Leaky annular pupils for improved axial imaging,” Optik (Stuttgart) 99, 32–34 (1995).
  8. M. Gu, T. Tannous, J. R. Sheppard, “Effect of an annular pupil on confocal imaging through highly scattering media,” Opt. Lett. 21, 312–314 (1995).
    [CrossRef]
  9. G. Boyer, “New class of axially apodizing filters for confocal scanning microscopy,” J. Opt. Soc. Am. A 19, 584–589 (2002).
    [CrossRef]
  10. G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–435 (1952).
    [CrossRef]
  11. M. Martı́nez-Corral, A. Pons, M. T. Caballero, “Axial apodization in 4Pi-confocal microscopy by annular binary filters,” J. Opt. Soc. Am. A 19, 1532–1536 (2002).
    [CrossRef]
  12. C. M. Blanca, J. Bewersdorf, S. W. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79, 2321–2323 (2001).
    [CrossRef]
  13. D. M. de Juana, J. E. Oti, V. F. Canales, M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28, 607–609 (2003).
    [CrossRef] [PubMed]
  14. C. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
    [CrossRef]
  15. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  16. M. Gu, C. J. R. Sheppard, “Three-dimensional transfer functions in 4Pi confocal microscopes,” J. Opt. Soc. Am. A 11, 1619–1627 (1994).
    [CrossRef]
  17. S. Hell, P. Hänninen, A. Kuusisto, M. Shrader, E. Soini, “Annular aperture two-photon excitation microscopy,” Opt. Commun. 117, 20–24 (1995).
    [CrossRef]

2003

2002

2001

C. M. Blanca, J. Bewersdorf, S. W. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79, 2321–2323 (2001).
[CrossRef]

1998

M. Martı́nez-Corral, P. Andrés, C. J. Zapata-Rodrı́guez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

M. Schrader, S. W. Hell, H. T. M. van der Voort, “Three-dimensional super-resolution with a 4Pi-confocal microscope using image restoration,” J. Appl. Phys. 84, 4033–4042 (1998).
[CrossRef]

1995

C. J. R. Sheppard, “Leaky annular pupils for improved axial imaging,” Optik (Stuttgart) 99, 32–34 (1995).

S. Hell, P. Hänninen, A. Kuusisto, M. Shrader, E. Soini, “Annular aperture two-photon excitation microscopy,” Opt. Commun. 117, 20–24 (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]

M. Gu, T. Tannous, J. R. Sheppard, “Effect of an annular pupil on confocal imaging through highly scattering media,” Opt. Lett. 21, 312–314 (1995).
[CrossRef]

1994

1992

S. Hell, E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. 93, 277–282 (1992).
[CrossRef]

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

1987

1986

1959

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

1952

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–435 (1952).
[CrossRef]

Andrés, P.

M. Martı́nez-Corral, P. Andrés, C. J. Zapata-Rodrı́guez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

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]

Bewersdorf, J.

C. M. Blanca, J. Bewersdorf, S. W. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79, 2321–2323 (2001).
[CrossRef]

Blanca, C. M.

C. M. Blanca, J. Bewersdorf, S. W. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79, 2321–2323 (2001).
[CrossRef]

Boyer, G.

Caballero, M. T.

Cagigal, M. P.

Canales, V. F.

de Juana, D. M.

Gu, M.

Hänninen, P.

S. Hell, P. Hänninen, A. Kuusisto, M. Shrader, E. Soini, “Annular aperture two-photon excitation microscopy,” Opt. Commun. 117, 20–24 (1995).
[CrossRef]

Hegedus, Z. S.

Hell, S.

S. Hell, P. Hänninen, A. Kuusisto, M. Shrader, E. Soini, “Annular aperture two-photon excitation microscopy,” Opt. Commun. 117, 20–24 (1995).
[CrossRef]

S. Hell, E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. 93, 277–282 (1992).
[CrossRef]

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

Hell, S. W.

C. M. Blanca, J. Bewersdorf, S. W. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79, 2321–2323 (2001).
[CrossRef]

M. Schrader, S. W. Hell, H. T. M. van der Voort, “Three-dimensional super-resolution with a 4Pi-confocal microscope using image restoration,” J. Appl. Phys. 84, 4033–4042 (1998).
[CrossRef]

Kuusisto, A.

S. Hell, P. Hänninen, A. Kuusisto, M. Shrader, E. Soini, “Annular aperture two-photon excitation microscopy,” Opt. Commun. 117, 20–24 (1995).
[CrossRef]

Marti´nez-Corral, M.

M. Martı́nez-Corral, A. Pons, M. T. Caballero, “Axial apodization in 4Pi-confocal microscopy by annular binary filters,” J. Opt. Soc. Am. A 19, 1532–1536 (2002).
[CrossRef]

M. Martı́nez-Corral, P. Andrés, C. J. Zapata-Rodrı́guez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

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]

Matthews, H. J.

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]

Oti, J. E.

Pons, A.

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[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]

Sarafis, V.

Schrader, M.

M. Schrader, S. W. Hell, H. T. M. van der Voort, “Three-dimensional super-resolution with a 4Pi-confocal microscope using image restoration,” J. Appl. Phys. 84, 4033–4042 (1998).
[CrossRef]

Sheppard, C. J. R.

M. Martı́nez-Corral, P. Andrés, C. J. Zapata-Rodrı́guez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

C. J. R. Sheppard, “Leaky annular pupils for improved axial imaging,” Optik (Stuttgart) 99, 32–34 (1995).

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

C. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[CrossRef]

Sheppard, J. R.

Shrader, M.

S. Hell, P. Hänninen, A. Kuusisto, M. Shrader, E. Soini, “Annular aperture two-photon excitation microscopy,” Opt. Commun. 117, 20–24 (1995).
[CrossRef]

Soini, E.

S. Hell, P. Hänninen, A. Kuusisto, M. Shrader, E. Soini, “Annular aperture two-photon excitation microscopy,” Opt. Commun. 117, 20–24 (1995).
[CrossRef]

Stelzer, E. H. K.

S. Hell, E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. 93, 277–282 (1992).
[CrossRef]

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

Tannous, T.

Toraldo di Francia, G.

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–435 (1952).
[CrossRef]

van der Voort, H. T. M.

M. Schrader, S. W. Hell, H. T. M. van der Voort, “Three-dimensional super-resolution with a 4Pi-confocal microscope using image restoration,” J. Appl. Phys. 84, 4033–4042 (1998).
[CrossRef]

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Zapata-Rodri´guez, C. J.

M. Martı́nez-Corral, P. Andrés, C. J. Zapata-Rodrı́guez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

Appl. Phys. Lett.

C. M. Blanca, J. Bewersdorf, S. W. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79, 2321–2323 (2001).
[CrossRef]

J. Appl. Phys.

M. Schrader, S. W. Hell, H. T. M. van der Voort, “Three-dimensional super-resolution with a 4Pi-confocal microscope using image restoration,” J. Appl. Phys. 84, 4033–4042 (1998).
[CrossRef]

J. Opt. Soc. Am. A

Nuovo Cimento Suppl.

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–435 (1952).
[CrossRef]

Opt. Commun.

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]

S. Hell, E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. 93, 277–282 (1992).
[CrossRef]

S. Hell, P. Hänninen, A. Kuusisto, M. Shrader, E. Soini, “Annular aperture two-photon excitation microscopy,” Opt. Commun. 117, 20–24 (1995).
[CrossRef]

Opt. Lett.

Optik (Stuttgart)

M. Martı́nez-Corral, P. Andrés, C. J. Zapata-Rodrı́guez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

C. J. R. Sheppard, “Leaky annular pupils for improved axial imaging,” Optik (Stuttgart) 99, 32–34 (1995).

Proc. R. Soc. London Ser. A

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Normalized axial intensity PSF for the phase filter ϕ(ρ, a, b)=a sin(2πbρ2) with parameters a=1.997 and b=1.318 (solid curve) compared with the unapodized case (dashed curve).

Fig. 2
Fig. 2

Normalized axial intensity PSF for the phase filter ϕ(ρ, a, b)=a sin(2πbρ3) with parameters a=2.004 and b=1.310 (solid curve) compared with the unapodized case (dashed curve).

Fig. 3
Fig. 3

Normalized axial intensity PSF for the phase filter ϕ(ρ, a, b)=aρ3+bρ5 with parameters a=4.759 and b=3.399 (solid curve) compared with the unapodized case (dashed curve).

Fig. 4
Fig. 4

Normalized z response to an infinitely thin fluorescent layer for the phase filter ϕ(ρ, a, b)=a sin(2πbρ3) with parameters a=2.004 and b=1.310 (solid curve) compared with the unapodized case (dashed curve).

Fig. 5
Fig. 5

Transverse intensity PSF for three phase filters: ϕ(ρ, a, b)=a sin(2πbρ) with a=22.790 and b=0.310 (dotted curve), ϕ(ρ, a, b)=a sin(2πρ)+b sin(4πρ) with a=-0.993 and b=1.400 (dashed curve), and ϕ(ρ, a, b)=aρ3+bρ5 with a=61.453 and b=-42.322 (dashed–dotted curve) compared with the unapodized case (solid curve).

Tables (2)

Tables Icon

Table 1 Axial Superresolution Performance for Three Phase Filtersa

Tables Icon

Table 2 Transverse Superresolution Performance for Six Phase Filtersa

Equations (28)

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

Ex=I0(u, v)+I2(u, v)cos 2φ,
Ey=I2(u, v)sin 2φ,
Ez=-2iI1(u, v)cos φ,
u=4nkz sin2 α/2,
v=nkr sin α,
I0(u, v)=0αA(θ)sin θ(1+cos θ)J0v sin θsin α×expiu cos θ4 sin2(α/2)dθ,
I1(u, v)=0αA(θ)sin2 θJ1v sin θsin αexpiu cos θ4 sin2(α/2)dθ,
I2(u, v)=0αA(θ)sin θ(1-cos θ)J2v sin θsin α×expiu cos θ4 sin2(α/2)dθ.
H(u, v, φ)=|Eill(u, v, φ)+Eill(-u, v, φ)|2×|Edet(u, v)+Edet(-u, v)|2,
H(u, v=0)=|I0ill(u)+I0ill(-u)|2×|I0det(u)+I0det(-u)|2,
I0ill(u)=0αA(θ)(1+cos θ)expiu cos θ4 sin2(α/2)sin θdθ.
E(u, v)=20αA(θ)J0v sin θsin α×exp-iu sin2(θ/2)2 sin2(α/2)sin θdθ.
H(u=0, v)=|E(u=0, v)|2|E(u=0, v, A(θ)1)|2.
Hill(u, v=0)=|C0|2-Re(C0*C2)16 sin4(α/2)u2+O(u4)(32-cos α-12 cos2 α)2,
Cn=cos α1A(x)(1+x)xndx.
S4Pi=|C0|2(32-cos α-12 cos2 α)2,
GA4Pi=6(cos α+3)3 cos3 α+7 cos2 α+7 cos α+7Re(C0*C2)|C0|2.
Hill(u=0, v)=|Q0|2(1-cos α)2-Re(Q0*Q2)2 sin2 α(1-cos α)2v2+O(v4),
Qn=cos α1A(x)(1-x2)n/2dx.
GT4Pi=1-cos α23-cos α+13 cos3 αRe(Q0*Q2)|Q0|2.
GA4Pi(a, b)-GA0=0,
GT4Pi(a, b)-GT0=0,
S4Pi(a, b)-S0=0,
Hill(u, v=0)=|I0ill(u)+I0ill(-u)|2,
Hill(u, 0)=2cos α1A(x)(1+x)×1-x22[4 sin2(α/2)]2u2+O(u4)dx2,
Hill(u, 0)2C0-C2[4 sin2(α/2)]2u224|C0|2-4 Re(C0*C2)[4 sin2(α/2)]2u2,
0=4|C0|2-4 Re(C0*C2)[4 sin2(α/2)]21GA2GARe(C0*C2)|C0|2.
GA=32-cos α-cos2 α/2712-cos3 α/3-cos4 α/4Re(C0*C2)|C0|2.

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