Abstract

Three-dimensional fluorescent image formation is analyzed for a fiber-optical confocal scanning microscope that uses annular lenses. The three-dimensional optical transfer function is derived, including the effects of the central obstruction radius and of the fiber spot size. Numerical calculations of the three-dimensional optical transfer functions have been performed for various cases. No negative values exist in the transfer functions, and no missing cone of spatial frequencies appears if the illumination fiber is identical to the collection fiber. It is found that using annular lenses in a system with optical fibers can result in improved resolution in both transverse and axial directions.

© 1992 Optical Society of America

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References

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  1. C. J. R. Sheppard, “Scanning optical microscopy,” in Advances in Optical and Electron Microscopy, R. Barer, V. E. Cosslett, eds. (Academic, London, 1987), Vol. 10, pp. 1–98.
  2. C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik 72, 131–133 (1986).
  3. C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging II,” Optik 74, 128–129 (1986).
  4. S. Kimura, C. Munakata, “Calculation of three-dimensional optical transfer function for a confocal scanning fluorescent microscope,” J. Opt. Soc. Am. A 6, 1015–1019 (1989).
    [CrossRef]
  5. S. Kimura, C. Munakata, “Three-dimensional optical transfer function for the fluorescent scanning optical microscope with a slit,” Appl. Opt. 29, 1000–1007 (1990).
    [CrossRef]
  6. S. Kimura, C. Munakata, “Dependence of 3-D optical transfer function functions on the pinhole radius in a fluorescent confocal optical microscope,” Appl. Opt. 29, 3007–3011 (1990).
    [CrossRef] [PubMed]
  7. O. Nakamura, S. Kawata, “Three-dimensional transfer function analysis of the tomographic capability of a confocal fluorescent microscope,” J. Opt. Soc. Am. A 7, 522–526 (1990).
    [CrossRef] [PubMed]
  8. S. Kawata, R. Arimoto, O. Nakamura, “Three-dimensional optical-transfer function analysis for laser-scan fluorescence microscope with an extended detector,” J. Opt. Soc. Am. A 8, 171–175 (1991).
    [CrossRef]
  9. R. Arimoto, S. Kawata, “Laser-scan fluorescence microscope with annular excitation,” Optik 86, 7–10 (1990).
  10. M. Gu, C. J. R. Sheppard, “Confocal fluorescent microscopy with a finite-sized circular detector,” J. Opt. Soc. Am. A 9, 151–153 (1992).
    [CrossRef]
  11. M. Gu, C. J. R. Sheppard, “Three-dimensional imaging in confocal fluorescent microscopy with annular lenses,” J. Mod. Opt. 38, 2247–2763 (1991).
    [CrossRef]
  12. M. Gu, C. J. R. Sheppard, X. Gan, “Image formation in a fibre-optical confocal scanning microscope,” J. Opt. Soc. Am. A 8, 1755–1761 (1991).
    [CrossRef]
  13. M. Gu, X. Gan, C. J. R. Sheppard, “Three-dimensional coherent transfer functions in fibre-optical confocal scanning microscopes,” J. Opt. Soc. Am. A 8, 1019–1025 (1991).
    [CrossRef]
  14. M. Gu, C. J. R. Sheppard, “Signal level of the fibre-optical confocal scanning microscope,” J. Mod. Opt. 38, 1621–1630 (1991).
    [CrossRef]
  15. S. Kimura, T. Wilson, “Confocal scanning optical microscope using single-mode fiber for signal detection,” Appl. Opt. 30, 2143–2150 (1991).
    [CrossRef] [PubMed]
  16. J. Benshchop, G. van Rosmalen, “Confocal compact scanning optical microscope based on compact disk technology,” Appl. Opt. 30, 1179–1184 (1991).
    [CrossRef]
  17. M. Gu, C. J. R. Sheppard, “Axial resolution in the fibre-optical confocal scanning microscope using annular lenses,” Opt. Commun. 88, 27–32 (1992).
    [CrossRef]
  18. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  19. C. J. R. Sheppard, X. Q. Mao, “Three-dimensional imaging in a microscope,” J. Opt. Soc. Am. A 6, 1260–1269 (1989).
    [CrossRef]
  20. B. R. Frieden, “Optical transfer of a three-dimensional object,”J. Opt. Soc. Am. 57, 56–66 (1967).
    [CrossRef]
  21. C. J. R. Sheppard, M. Gu, “Three-dimensional optical transfer function for an annular lens,” Opt. Commun. 81, 276–280 (1991).
    [CrossRef]
  22. I. S. Gradstein, I. M. Ryshik, Tables of Series, Products, and Integrals (Verlag Harri Deutsch, Frankfurt, 1981).
  23. C. J. R. Sheppard, M. Gu, “The significance of the three-dimensional transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
    [CrossRef]
  24. C. J. R. Sheppard, M. Gu, “Approximation to the three-dimensional optical transfer function,” J. Opt. Soc. Am. A 8, 692–694 (1991).
    [CrossRef]
  25. M. Gu, C. J. R. Sheppard, “Effects of the finite-sized detector on the OTF in confocal fluorescent microscopy,” Optik 89, 65–69 (1991).

1992

M. Gu, C. J. R. Sheppard, “Confocal fluorescent microscopy with a finite-sized circular detector,” J. Opt. Soc. Am. A 9, 151–153 (1992).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Axial resolution in the fibre-optical confocal scanning microscope using annular lenses,” Opt. Commun. 88, 27–32 (1992).
[CrossRef]

C. J. R. Sheppard, M. Gu, “The significance of the three-dimensional transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

1991

C. J. R. Sheppard, M. Gu, “Approximation to the three-dimensional optical transfer function,” J. Opt. Soc. Am. A 8, 692–694 (1991).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Effects of the finite-sized detector on the OTF in confocal fluorescent microscopy,” Optik 89, 65–69 (1991).

C. J. R. Sheppard, M. Gu, “Three-dimensional optical transfer function for an annular lens,” Opt. Commun. 81, 276–280 (1991).
[CrossRef]

S. Kawata, R. Arimoto, O. Nakamura, “Three-dimensional optical-transfer function analysis for laser-scan fluorescence microscope with an extended detector,” J. Opt. Soc. Am. A 8, 171–175 (1991).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Three-dimensional imaging in confocal fluorescent microscopy with annular lenses,” J. Mod. Opt. 38, 2247–2763 (1991).
[CrossRef]

M. Gu, C. J. R. Sheppard, X. Gan, “Image formation in a fibre-optical confocal scanning microscope,” J. Opt. Soc. Am. A 8, 1755–1761 (1991).
[CrossRef]

M. Gu, X. Gan, C. J. R. Sheppard, “Three-dimensional coherent transfer functions in fibre-optical confocal scanning microscopes,” J. Opt. Soc. Am. A 8, 1019–1025 (1991).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Signal level of the fibre-optical confocal scanning microscope,” J. Mod. Opt. 38, 1621–1630 (1991).
[CrossRef]

S. Kimura, T. Wilson, “Confocal scanning optical microscope using single-mode fiber for signal detection,” Appl. Opt. 30, 2143–2150 (1991).
[CrossRef] [PubMed]

J. Benshchop, G. van Rosmalen, “Confocal compact scanning optical microscope based on compact disk technology,” Appl. Opt. 30, 1179–1184 (1991).
[CrossRef]

1990

S. Kimura, C. Munakata, “Three-dimensional optical transfer function for the fluorescent scanning optical microscope with a slit,” Appl. Opt. 29, 1000–1007 (1990).
[CrossRef]

S. Kimura, C. Munakata, “Dependence of 3-D optical transfer function functions on the pinhole radius in a fluorescent confocal optical microscope,” Appl. Opt. 29, 3007–3011 (1990).
[CrossRef] [PubMed]

O. Nakamura, S. Kawata, “Three-dimensional transfer function analysis of the tomographic capability of a confocal fluorescent microscope,” J. Opt. Soc. Am. A 7, 522–526 (1990).
[CrossRef] [PubMed]

R. Arimoto, S. Kawata, “Laser-scan fluorescence microscope with annular excitation,” Optik 86, 7–10 (1990).

1989

1986

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik 72, 131–133 (1986).

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging II,” Optik 74, 128–129 (1986).

1967

Arimoto, R.

Benshchop, J.

Frieden, B. R.

Gan, X.

Gradstein, I. S.

I. S. Gradstein, I. M. Ryshik, Tables of Series, Products, and Integrals (Verlag Harri Deutsch, Frankfurt, 1981).

Gu, M.

C. J. R. Sheppard, M. Gu, “The significance of the three-dimensional transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Axial resolution in the fibre-optical confocal scanning microscope using annular lenses,” Opt. Commun. 88, 27–32 (1992).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Confocal fluorescent microscopy with a finite-sized circular detector,” J. Opt. Soc. Am. A 9, 151–153 (1992).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Signal level of the fibre-optical confocal scanning microscope,” J. Mod. Opt. 38, 1621–1630 (1991).
[CrossRef]

M. Gu, X. Gan, C. J. R. Sheppard, “Three-dimensional coherent transfer functions in fibre-optical confocal scanning microscopes,” J. Opt. Soc. Am. A 8, 1019–1025 (1991).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Three-dimensional imaging in confocal fluorescent microscopy with annular lenses,” J. Mod. Opt. 38, 2247–2763 (1991).
[CrossRef]

M. Gu, C. J. R. Sheppard, X. Gan, “Image formation in a fibre-optical confocal scanning microscope,” J. Opt. Soc. Am. A 8, 1755–1761 (1991).
[CrossRef]

C. J. R. Sheppard, M. Gu, “Approximation to the three-dimensional optical transfer function,” J. Opt. Soc. Am. A 8, 692–694 (1991).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Effects of the finite-sized detector on the OTF in confocal fluorescent microscopy,” Optik 89, 65–69 (1991).

C. J. R. Sheppard, M. Gu, “Three-dimensional optical transfer function for an annular lens,” Opt. Commun. 81, 276–280 (1991).
[CrossRef]

Kawata, S.

Kimura, S.

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Mao, X. Q.

Munakata, C.

Nakamura, O.

Ryshik, I. M.

I. S. Gradstein, I. M. Ryshik, Tables of Series, Products, and Integrals (Verlag Harri Deutsch, Frankfurt, 1981).

Sheppard, C. J. R.

C. J. R. Sheppard, M. Gu, “The significance of the three-dimensional transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Confocal fluorescent microscopy with a finite-sized circular detector,” J. Opt. Soc. Am. A 9, 151–153 (1992).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Axial resolution in the fibre-optical confocal scanning microscope using annular lenses,” Opt. Commun. 88, 27–32 (1992).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Signal level of the fibre-optical confocal scanning microscope,” J. Mod. Opt. 38, 1621–1630 (1991).
[CrossRef]

M. Gu, C. J. R. Sheppard, X. Gan, “Image formation in a fibre-optical confocal scanning microscope,” J. Opt. Soc. Am. A 8, 1755–1761 (1991).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Three-dimensional imaging in confocal fluorescent microscopy with annular lenses,” J. Mod. Opt. 38, 2247–2763 (1991).
[CrossRef]

M. Gu, X. Gan, C. J. R. Sheppard, “Three-dimensional coherent transfer functions in fibre-optical confocal scanning microscopes,” J. Opt. Soc. Am. A 8, 1019–1025 (1991).
[CrossRef]

C. J. R. Sheppard, M. Gu, “Three-dimensional optical transfer function for an annular lens,” Opt. Commun. 81, 276–280 (1991).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Effects of the finite-sized detector on the OTF in confocal fluorescent microscopy,” Optik 89, 65–69 (1991).

C. J. R. Sheppard, M. Gu, “Approximation to the three-dimensional optical transfer function,” J. Opt. Soc. Am. A 8, 692–694 (1991).
[CrossRef]

C. J. R. Sheppard, X. Q. Mao, “Three-dimensional imaging in a microscope,” J. Opt. Soc. Am. A 6, 1260–1269 (1989).
[CrossRef]

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik 72, 131–133 (1986).

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging II,” Optik 74, 128–129 (1986).

C. J. R. Sheppard, “Scanning optical microscopy,” in Advances in Optical and Electron Microscopy, R. Barer, V. E. Cosslett, eds. (Academic, London, 1987), Vol. 10, pp. 1–98.

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

van Rosmalen, G.

Wilson, T.

Appl. Opt.

J. Microsc.

C. J. R. Sheppard, M. Gu, “The significance of the three-dimensional transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

J. Mod. Opt.

M. Gu, C. J. R. Sheppard, “Three-dimensional imaging in confocal fluorescent microscopy with annular lenses,” J. Mod. Opt. 38, 2247–2763 (1991).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Signal level of the fibre-optical confocal scanning microscope,” J. Mod. Opt. 38, 1621–1630 (1991).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

M. Gu, C. J. R. Sheppard, “Axial resolution in the fibre-optical confocal scanning microscope using annular lenses,” Opt. Commun. 88, 27–32 (1992).
[CrossRef]

C. J. R. Sheppard, M. Gu, “Three-dimensional optical transfer function for an annular lens,” Opt. Commun. 81, 276–280 (1991).
[CrossRef]

Optik

M. Gu, C. J. R. Sheppard, “Effects of the finite-sized detector on the OTF in confocal fluorescent microscopy,” Optik 89, 65–69 (1991).

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik 72, 131–133 (1986).

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging II,” Optik 74, 128–129 (1986).

R. Arimoto, S. Kawata, “Laser-scan fluorescence microscope with annular excitation,” Optik 86, 7–10 (1990).

Other

C. J. R. Sheppard, “Scanning optical microscopy,” in Advances in Optical and Electron Microscopy, R. Barer, V. E. Cosslett, eds. (Academic, London, 1987), Vol. 10, pp. 1–98.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

I. S. Gradstein, I. M. Ryshik, Tables of Series, Products, and Integrals (Verlag Harri Deutsch, Frankfurt, 1981).

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of the fiber-optical confocal scanning microscope in fluorescence.

Fig. 2
Fig. 2

3-D OTF’s for the FOCSM for A = 1 when both lenses have the same radii of the central obstruction: (a) 1 = 2 = 0, (b) 1 = 2 = 0.5, (c) 1 = 2 = 0.75.

Fig. 3
Fig. 3

3-D OTF’s for the FOCSM for A = 10 when both lenses have the same radii of the central obstruction: (a) 1 = 2 = 0, (b) 1 = 2 = 0.5, (c) 1 = 2 = 0.75.

Fig. 4
Fig. 4

Variations of (a) transverse and (b) axial sections of the 3-D OTF’s for the FOCSM with different values of A when 1 = 2 = 0.5.

Fig. 5
Fig. 5

Variations of (a) transverse and (b) axial sections of the 3-D OTF’s for the FOCSM when A = 10: 1, 1 = 2 = 0; 2, 1 = 2 = 0.5; 3, 1 = 2 = 0.75; 4, 1 = 0.5 and 2 = 0 (or 1 = 0 and 2 = 0.5).

Fig. 6
Fig. 6

Half-width Δs of the axial section of the 3-D OTF as a function of A: 1, 1 = 2 = 0; 2, 1 = 2 = 0.5.

Fig. 7
Fig. 7

3-D OTF’s for the FOCSM for A = 10 when one of the lenses is an annular pupil: (a) 1 = 0.5 and 2 = 0 (or 1 = 0 and 2 = 0.5), (b) 1 = 0.75 and 2 = 0 (or 1 = 0 and 2 = 0.75).

Fig. 8
Fig. 8

3-D CTF for the FOCSM in transmission when both lenses have the same radii of the central obstruction (1 = 2 = 0.5) for (a) A = 2 and (b) A = 5.

Fig. 9
Fig. 9

3-D OTF’s for the FOCSM for A1 = 1 and A2 = 20 when both lenses are circular pupils (i.e., 1 = 2 = 0).

Equations (32)

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h k ( x , y , z ) = - P k ( ξ , η , z ) exp [ - i 2 π d k λ ( ξ x + η y ) ] d ξ d η             ( k = 1 , 2 ) ,
I o ( r 1 ) = | - U 2 ( x 0 , y 0 ) δ ( z 0 ) h 1 ( r 0 + r 1 M 1 ) d r 0 | 2 .
I 1 ( r 1 ) = | - U 1 ( x 0 , y 0 ) δ ( z 0 ) h 1 ( r 0 + r 1 M 1 ) d r 0 | 2 o ( r s - r 1 ) ,
I 2 ( r 1 ) = | - U 2 * ( x 2 , y 2 ) δ ( z 2 ) h 2 ( r 1 + r 2 M 2 ) d r 2 | 2 ,
I ( r s ) = - [ | - U 1 ( x 0 , y 0 ) δ ( z 0 ) h 1 ( r 0 + r 1 M 1 ) d r 0 | 2 × o ( r s - r 1 ) × | - U 2 * ( x 2 , y 2 ) δ ( z 2 ) h 2 ( r 1 + r 2 M 2 ) d r 2 | 2 ] d r 1 ,
I ( r s ) = h eff ( r s ) 3 o ( r s ) ,
h eff ( r ) = | U 1 ( x s M 1 , y s M 1 ) 2 h 1 ( r s M 1 ) | 2 × | U 2 * ( x s M 1 , y s M 1 ) 2 h 2 ( r s ) | 2 .
C ( m ) = F 3 { | U 1 ( x 2 M 1 , y s M 1 ) 2 h 1 ( r s M 1 ) | 2 × | U 2 * ( x s M 1 , y s M 1 ) 2 h 2 ( r s ) | 2 } ,
C ( m ) = C 1 ( m ) 3 C 2 ( m ) ,
C 1 ( m ) = F 3 { | U 1 ( x s M 1 , y s M 1 ) 2 h 1 ( r s M 1 ) | 2 } ,
C 2 ( m ) = F 3 { | U 2 * ( x s M 1 , y s M 1 ) 2 h 2 ( r s ) | 2 } ,
C 1 ( l , s ) = - C 1 ( l , z ) exp ( - i 2 π s z ) d z ,
C 1 ( l , z ) = [ U ˜ 1 ( M 1 l ) P 1 ( λ d 2 l , z ) ] 2 [ U ˜ 1 * ( M 1 l ) P 1 * ( λ d 2 l , z ) ] ,
P 1 ( r , u ) = exp [ i u 2 ( r a 0 ) 2 ] 1 a 0 < r < a 0 = 0 otherwise ,
u = ( 8 π / λ ) z sin 2 ( α / 2 ) .
U 1 ( r ) = exp [ - 1 2 ( r r 01 ) 2 ] ,
U ˜ 1 ( l ) = 2 π r 01 2 exp [ - 1 2 ( 2 π l r 01 ) 2 ] .
C 1 ( l , s ) = exp { - A 1 [ l 2 4 + ( s l ) 2 ] } l A 1 × [ Erf ( y 1 A 1 ) - Erf ( y 2 A 1 ) ] ( s ( 1 - 1 2 ) / 2 ) ,
A 1 = ( 2 π a 0 r 01 λ d 1 ) 2 ,
y 1 = Re { [ 1 - ( l 2 - s l ) 2 ] 1 / 2 } ,
y 2 = Re { [ 1 2 - ( l 2 - s l ) 2 ] 1 / 2 } ,
Erf ( x ) = 2 π 0 x exp ( - t 2 ) d t .
C 2 ( l , s ) = exp { - A 2 [ l 2 4 + ( s l ) 2 ] } l A 2 × [ Erf ( y 1 A 2 ) - Erf ( y 3 A 2 ) ] ( s ( 1 - 2 2 ) / 2 ) ,
A 2 = ( 2 π a 0 r 02 λ d 1 ) 2 ,
y 3 = Re { [ 2 2 - ( l 2 - s l ) 2 ] 1 / 2 } .
C ( l , s ) = ν 1 A 1 A 2 1 l 1 l 2 × { exp [ - A 1 ( l 1 2 4 + ( s - s / 2 ) 2 l 1 2 ) ] × [ Erf ( A 1 y 1 ) - Erf ( A 1 y 2 ) ] } × { exp [ - A 2 ( l 2 2 4 + ( s + s / 2 ) 2 l 2 2 ) ] × [ Erf ( A 2 y 3 ) - Erf ( A 2 y 4 ) ] } d m d n d s [ s 1 - ( 1 2 + 2 2 ) / 2 ] ,
y 1 = Re { [ 1 - ( s - s / 2 l 1 + l 1 2 ) 2 ] 1 / 2 } ,
y 2 = Re { [ 1 2 - ( s - s / 2 l 1 - l 1 2 ) 2 ] 1 / 2 } ,
y 3 = Re { [ 1 - ( s + s / 2 l 2 + l 2 2 ) 2 ] 1 / 2 } ,
y 4 = Re { [ 2 2 - ( s + s / 2 l 2 - l 2 2 ) 2 ] 1 / 2 } ,
l 1 = [ ( m - l / 2 ) 2 + n 2 ] 1 / 2 ,
l 2 = [ ( m + l / 2 ) 2 + n 2 ] 1 / 2 .

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