Abstract

A new class of superresolution pairs of pupil filters for three-dimensional, two-pupil confocal imaging is proposed. A distinctive feature of these filters is the asymmetry of their impulse response. For synthesizing the amplitude transmittance of such filters the Fourier transform properties of Hermitian functions are employed. It is shown that, with simple phase-only filters that belong to the class in question, either axial or unidirectional lateral superresolution is achieved.

© 1998 Optical Society of America

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  1. L. Cheng, G. G. Siu, “Asymmetric apodization,” Maes. Sci. Technol. 2, 198–202 (1991).
    [CrossRef]
  2. G. G. Siu, L. Cheng, D. S. Chiu, K. S. Chang, “Improved side-lobe suppression in asymmetric apodization,” J. Phys. D: Appl. Phys. 27, 459–463 (1994).
  3. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  4. T. Wilson, “Confocal microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990), pp. 1–64.
    [CrossRef]
  5. T. Wilson, C. Sheppard, Theory and Practice of Confocal Scanning Microscopy (Academic, London, 1984).
  6. Z. Hegedus, “Pupil filters in confocal imaging,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990), pp. 171–183.
  7. T. Wilson, “Optical aspects of confocal microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990), pp. 93–141.
  8. A. Barna, “Fraunhofer diffraction by semicircular apertures,” J. Opt. Soc. Am. 67, 122–123 (1977).
    [CrossRef]
  9. M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, Singapore, 1996).
  10. M. Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, G. Saavedra, “Tunable axial resolution by annular binary filters. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
    [CrossRef]
  11. C. J. R. Sheppard, “Leaky annular pupils for improved axial imaging,” Optik (Stuttgart) 99, 32–34 (1995).
  12. A. Magiera, “Transversal and axial gains in the confocal scanning microscope of leaky annular pupils,” Opt. Appl. 26, 57–60 (1996).
  13. M. Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, “On-axis diffractional behavior of two-dimensional pupils,” Appl. Opt. 33, 2223–2229 (1994).
    [CrossRef] [PubMed]
  14. C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional imaging in confocal microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990), pp. 143–169.
  15. T. Wilson, “Optical sectioning in confocal fluorescent microscopes,” J. Microsc. 154, 143–156 (1988).
    [CrossRef]
  16. J. P. Mills, B. J. Thompson, “Effect of aberrations and apodization on the performance of coherent optical systems I. The amplitude impulse response,” J. Opt. Soc. Am. A 3, 694–703 (1986).
    [CrossRef]
  17. H. Osterberg, L. W. Smith, “Defocusing images to increase resolution,” Science 134, 1193–1196 (1961).
    [CrossRef] [PubMed]
  18. T. R. M. Sales, G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14, 1637–1646(1997).
    [CrossRef]

1997 (1)

1996 (1)

A. Magiera, “Transversal and axial gains in the confocal scanning microscope of leaky annular pupils,” Opt. Appl. 26, 57–60 (1996).

1995 (2)

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

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

1994 (2)

G. G. Siu, L. Cheng, D. S. Chiu, K. S. Chang, “Improved side-lobe suppression in asymmetric apodization,” J. Phys. D: Appl. Phys. 27, 459–463 (1994).

M. Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, “On-axis diffractional behavior of two-dimensional pupils,” Appl. Opt. 33, 2223–2229 (1994).
[CrossRef] [PubMed]

1991 (1)

L. Cheng, G. G. Siu, “Asymmetric apodization,” Maes. Sci. Technol. 2, 198–202 (1991).
[CrossRef]

1988 (1)

T. Wilson, “Optical sectioning in confocal fluorescent microscopes,” J. Microsc. 154, 143–156 (1988).
[CrossRef]

1986 (1)

1977 (1)

1961 (1)

H. Osterberg, L. W. Smith, “Defocusing images to increase resolution,” Science 134, 1193–1196 (1961).
[CrossRef] [PubMed]

Andrés, P.

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

M. Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, “On-axis diffractional behavior of two-dimensional pupils,” Appl. Opt. 33, 2223–2229 (1994).
[CrossRef] [PubMed]

Barna, A.

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

Chang, K. S.

G. G. Siu, L. Cheng, D. S. Chiu, K. S. Chang, “Improved side-lobe suppression in asymmetric apodization,” J. Phys. D: Appl. Phys. 27, 459–463 (1994).

Cheng, L.

G. G. Siu, L. Cheng, D. S. Chiu, K. S. Chang, “Improved side-lobe suppression in asymmetric apodization,” J. Phys. D: Appl. Phys. 27, 459–463 (1994).

L. Cheng, G. G. Siu, “Asymmetric apodization,” Maes. Sci. Technol. 2, 198–202 (1991).
[CrossRef]

Chiu, D. S.

G. G. Siu, L. Cheng, D. S. Chiu, K. S. Chang, “Improved side-lobe suppression in asymmetric apodization,” J. Phys. D: Appl. Phys. 27, 459–463 (1994).

Cogswell, C. J.

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional imaging in confocal microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990), pp. 143–169.

Gu, M.

M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, Singapore, 1996).

Hegedus, Z.

Z. Hegedus, “Pupil filters in confocal imaging,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990), pp. 171–183.

Magiera, A.

A. Magiera, “Transversal and axial gains in the confocal scanning microscope of leaky annular pupils,” Opt. Appl. 26, 57–60 (1996).

Martínez-Corral, M.

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

M. Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, “On-axis diffractional behavior of two-dimensional pupils,” Appl. Opt. 33, 2223–2229 (1994).
[CrossRef] [PubMed]

Mills, J. P.

Morris, G. M.

Ojeda-Castañeda, J.

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

M. Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, “On-axis diffractional behavior of two-dimensional pupils,” Appl. Opt. 33, 2223–2229 (1994).
[CrossRef] [PubMed]

Osterberg, H.

H. Osterberg, L. W. Smith, “Defocusing images to increase resolution,” Science 134, 1193–1196 (1961).
[CrossRef] [PubMed]

Saavedra, G.

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

Sales, T. R. M.

Sheppard, C.

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

Sheppard, C. J. R.

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

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional imaging in confocal microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990), pp. 143–169.

Siu, G. G.

G. G. Siu, L. Cheng, D. S. Chiu, K. S. Chang, “Improved side-lobe suppression in asymmetric apodization,” J. Phys. D: Appl. Phys. 27, 459–463 (1994).

L. Cheng, G. G. Siu, “Asymmetric apodization,” Maes. Sci. Technol. 2, 198–202 (1991).
[CrossRef]

Smith, L. W.

H. Osterberg, L. W. Smith, “Defocusing images to increase resolution,” Science 134, 1193–1196 (1961).
[CrossRef] [PubMed]

Thompson, B. J.

Wilson, T.

T. Wilson, “Optical sectioning in confocal fluorescent microscopes,” J. Microsc. 154, 143–156 (1988).
[CrossRef]

T. Wilson, “Confocal microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990), pp. 1–64.
[CrossRef]

T. Wilson, “Optical aspects of confocal microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990), pp. 93–141.

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

Appl. Opt. (1)

J. Microsc. (1)

T. Wilson, “Optical sectioning in confocal fluorescent microscopes,” J. Microsc. 154, 143–156 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Phys. D: Appl. Phys. (1)

G. G. Siu, L. Cheng, D. S. Chiu, K. S. Chang, “Improved side-lobe suppression in asymmetric apodization,” J. Phys. D: Appl. Phys. 27, 459–463 (1994).

Maes. Sci. Technol. (1)

L. Cheng, G. G. Siu, “Asymmetric apodization,” Maes. Sci. Technol. 2, 198–202 (1991).
[CrossRef]

Opt. Appl. (1)

A. Magiera, “Transversal and axial gains in the confocal scanning microscope of leaky annular pupils,” Opt. Appl. 26, 57–60 (1996).

Opt. Commun. (1)

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

Optik (Stuttgart) (1)

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

Science (1)

H. Osterberg, L. W. Smith, “Defocusing images to increase resolution,” Science 134, 1193–1196 (1961).
[CrossRef] [PubMed]

Other (7)

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional imaging in confocal microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990), pp. 143–169.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

T. Wilson, “Confocal microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990), pp. 1–64.
[CrossRef]

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

Z. Hegedus, “Pupil filters in confocal imaging,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990), pp. 171–183.

T. Wilson, “Optical aspects of confocal microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990), pp. 93–141.

M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, Singapore, 1996).

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Figures (7)

Fig. 1
Fig. 1

Optical scheme of a transmission-mode confocal scanning microscope: L i1, L i2, L c1, L c2: lenses of focal length f; P 1, P 2: pupil filters; O: object, D: point detector; S: point source.

Fig. 2
Fig. 2

General scheme of the lateral-resolution pupil filters.

Fig. 3
Fig. 3

Nonlinear gray-scale representations of (a) N 1(∊)h 1 2(x s , 0; π/3, ∊) and (b) N 2(∊)h 1 2(x s , 0; π/3, ∊) h 2 2(x s , 0; π/3, ∊). N 1, N 2: normalization factors. A one-sided suppression of the sidelobes, as described in Ref. 1, is observed in (a) for ∊ ≈ 0.9.

Fig. 4
Fig. 4

Main cross sections of normalized h 1 2 and (h 1 h 2)2 along (a) and (b) the good and (c) and (d) the bad directions for ϕ = π/3 and ∊ = 0.7. The dashed curves correspond to a nonapodized system.

Fig. 5
Fig. 5

Images of two fluorescent point sources separated by a normalized distance d for points placed along (a), (b), and (c) the good and (d) the bad directions. Main cross sections along the good, i.e., x s , direction for (a) d = 0.610 and (b) d = 0.375. Pseudo 3-D plots for (c) and (d) d = 0.375. The dashed curves correspond to a nonapodized system.

Fig. 6
Fig. 6

Superresolution pair of complex-conjugate three-zone pupils.

Fig. 7
Fig. 7

Normalized axial distributions in the image of a point source: (a) the light amplitude in a conventional imaging system apodized with the pupil function P 1′ and (b) the light intensity in a two-pupil confocal scanning system apodized with P 1′ and (P 1′)* (reflection mode) or with either two pupil functions P 1′ or two pupil functions (P 1′)* (transmission mode); (c) integrated intensity for an axially apodized system. In (a)–(c), the dashed curves indicate the corresponding distributions for a nonapodized system.

Equations (29)

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h ˜ 1 x 0 ,   y 0 = - -   P 1 ξ 1 ,   η 1 × exp i   k f ξ 1 x 0 + η 1 y 0 d ξ 1 d η 1 ,
h 2 x 2 ,   y 2 = - -   P 2 ξ 2 ,   η 2 × exp - i   k f ξ 2 x 2 + η 2 y 2 d ξ 2 d η 2 ,
U x 2 ,   y 2 ;   x s ,   y s = - -   h ˜ 1 x 0 ,   y 0 t x 0 + x s ,   y 0 + y s h 2 x 2 - x 0 ,   y 2 - y 0 d x 0 d y 0 ,
I x s ,   y s = - - h ˜ 1 - x 0 ,   - y 0 t - x 0 + x s ,   - y 0 + y s ×   h 2 x 0 ,   y 0 d x 0 d y 0 2 .
h 1 x 0 ,   y 0 h ˜ 1 - x 0 ,   - y 0 = - -   P 1 ξ 1 ,   η 1 exp - i   k f ξ 1 x 0 + η 1 y 0 × d ξ 1 d η 1 ,
I x s ,   y s = - -   h 1 x 0 ,   y 0 t x s - x 0 ,   y s - y 0 × h 2 x 0 ,   y 0 d x 0 d y 0 2 = | h 1 x s ,   y s h 2 x s ,   y s t x s ,   y s | 2 ,
PSF x s ,   y s = | h 1 x s ,   y s h 2 x s ,   y s | 2 .
I x s ,   y s = | h 1 x s ,   y s h 2 x s ,   y s | 2 F x s ,   y s ,
h 1 x s ,   y s h 2 x s ,   y s = h 1 x s ,   y s h 1 - x s ,   - y s = h 2 - x s ,   - y s h 2 x s ,   y s ,
P q ξ q ,   η q = H ξ q exp - 1 q i ϕ + H - ξ q × exp - 1 q + 1 i ϕ circ r q r 0 - circ r q r 0 + circ r q r 0 ,
ρ q r q / r 0 ,
v k x s 2 + y s 2 1 / 2   sin a ,
u 4 kz s sin 2 a / 2 ,
h 1 v ,   u = 2   0 1   P 1 ρ 1 exp 1 2   iu ρ 1 2 J 0 v ρ 1 ρ 1 d ρ 1 .
h 2 v ,   u = 2   0 1   P 2 ρ 2 exp + 1 2   iu ρ 2 2 J 0 v ρ 2 ρ 2 d ρ 2
h 2 v ,   u = 2   0 1   P 2 ρ 2 exp - 1 2   iu ρ 2 2 J 0 v ρ 2 ρ 2 d ρ 2
PSF v ,   u = | h 1 v ,   u h 2 v ,   u | 2 .
PSF r v ,   u = | h 1 v ,   u h 1 * v ,   - u | 2 = | h 1 v ,   u h 1 v ,   - u | 2 .
PSF t v ,   u = | h 1 v ,   u h 1 v ,   - u | 2 ,
PSF r u = PSF t u = | h 1 u h 1 - u | 2 ,
ρ q 2 - 1 2 = s q .
h 1 u = - 0.5 0.5   g 1 s 1 exp ius 1 / 2 d s 1 ,
PSF r u = PSF t u = - 0.5 0.5   g 1 s 1 exp + ius 1 / 2 d s 1 × - 0.5 0.5   g 1 s 2 exp - ius 2 / 2 d s 2 2 .
h 0 = - g e s d s ,
h u u u = 0 = - 1 2 -   g o s s d s ,
g e 2 s + g o 2 s 1 .
g 1 s 1 = i - 0.5 s 1 < - 0.25 1 - 0.25 s 1 < 0.25 - i 0.25 s 1 0.5 0 otherwise .
I int u = 2   0   | h 1 v ,   u h 2 v ,   u | 2 v d v .
I u = - 0.5 0.5   g 1 s g 2 s exp ius d s 2 .

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