Abstract

Theoretical studies on image formation in a confocal scanning microscope with optical fibers as the transmission medium are reported. Theoretical analyses show that this new kind of microscope can be considered a coherent imaging system, even for finite fiber spot size. Based on these studies the coherent transfer functions in both in-focus and defocused cases are derived and calculated. The axial coherent transfer functions are also obtained, and, furthermore, the optical-sectioning property of the microscope system is investigated with the consideration of the image formation of a perfect-reflection planar object and a point object.

© 1991 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, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
    [CrossRef]
  3. T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).
  4. C. J. R. Sheppard, X. Q. Mao, “Confocal microscopes with slit aperture,” J. Mod. Opt. 35, 1169–1185 (1988).
    [CrossRef]
  5. C. J. R. Sheppard, C. J. Cogswell, “Confocal microscopy with detector arrays,” J. Mod. Opt. 37, 267–279 (1990).
    [CrossRef]
  6. C. J. R. Sheppard, T. Wilson, “Image formation in scanning microscopes with partially coherent source and detector,” Opt. Acta 25, 315–325 (1978).
    [CrossRef]
  7. I. J. Cox, C. J. R. Sheppard, T. Wilson, “Super-resolution by confocal fluorescence microscopy,” Optik 60, 391–396 (1982).
  8. C. J. R. Sheppard, T. Wilson, “Depth of field in the scanning microscope,” Opt. Lett. 3, 115–117 (1978).
    [CrossRef] [PubMed]
  9. C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik 72, 131–133 (1986).
  10. C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging II,” Optik 74, 128–129 (1986).
  11. C. J. R. Sheppard, X. Q. Mao, “Three-dimensional imaging in a microscope,” J. Opt. Soc. Am. A 6, 1260–1269 (1989).
    [CrossRef]
  12. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).
  13. J. W. Goodman, Introduction of Fourier Optics (McGraw-Hill, San Francisco, 1968).
  14. C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical microscope with extended depth field,” Proc. R. Soc. London Ser. A 387, 171–186 (1983).
    [CrossRef]
  15. T. Wilson, “Optical sectioning in confocal fluorescence microscopes,” J. Microsc. 154, 143–156 (1988).
    [CrossRef]
  16. X. Gan, M. Gu, C. J. R. Sheppard, “Fluorescent image formation in the fiber optical confocal scanning microscope,” J. Mod. Opt. (to be published).

1990 (1)

C. J. R. Sheppard, C. J. Cogswell, “Confocal microscopy with detector arrays,” J. Mod. Opt. 37, 267–279 (1990).
[CrossRef]

1989 (1)

1988 (2)

C. J. R. Sheppard, X. Q. Mao, “Confocal microscopes with slit aperture,” J. Mod. Opt. 35, 1169–1185 (1988).
[CrossRef]

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

1986 (2)

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).

1983 (1)

C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical microscope with extended depth field,” Proc. R. Soc. London Ser. A 387, 171–186 (1983).
[CrossRef]

1982 (1)

I. J. Cox, C. J. R. Sheppard, T. Wilson, “Super-resolution by confocal fluorescence microscopy,” Optik 60, 391–396 (1982).

1978 (2)

C. J. R. Sheppard, T. Wilson, “Image formation in scanning microscopes with partially coherent source and detector,” Opt. Acta 25, 315–325 (1978).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Depth of field in the scanning microscope,” Opt. Lett. 3, 115–117 (1978).
[CrossRef] [PubMed]

1977 (1)

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

Cogswell, C. J.

C. J. R. Sheppard, C. J. Cogswell, “Confocal microscopy with detector arrays,” J. Mod. Opt. 37, 267–279 (1990).
[CrossRef]

Cox, I. J.

C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical microscope with extended depth field,” Proc. R. Soc. London Ser. A 387, 171–186 (1983).
[CrossRef]

I. J. Cox, C. J. R. Sheppard, T. Wilson, “Super-resolution by confocal fluorescence microscopy,” Optik 60, 391–396 (1982).

Gan, X.

X. Gan, M. Gu, C. J. R. Sheppard, “Fluorescent image formation in the fiber optical confocal scanning microscope,” J. Mod. Opt. (to be published).

Goodman, J. W.

J. W. Goodman, Introduction of Fourier Optics (McGraw-Hill, San Francisco, 1968).

Gu, M.

X. Gan, M. Gu, C. J. R. Sheppard, “Fluorescent image formation in the fiber optical confocal scanning microscope,” J. Mod. Opt. (to be published).

Hamilton, D. K.

C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical microscope with extended depth field,” Proc. R. Soc. London Ser. A 387, 171–186 (1983).
[CrossRef]

Love, J. D.

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

Mao, X. Q.

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, X. Q. Mao, “Confocal microscopes with slit aperture,” J. Mod. Opt. 35, 1169–1185 (1988).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, C. J. Cogswell, “Confocal microscopy with detector arrays,” J. Mod. Opt. 37, 267–279 (1990).
[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, X. Q. Mao, “Confocal microscopes with slit aperture,” J. Mod. Opt. 35, 1169–1185 (1988).
[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, D. K. Hamilton, I. J. Cox, “Optical microscope with extended depth field,” Proc. R. Soc. London Ser. A 387, 171–186 (1983).
[CrossRef]

I. J. Cox, C. J. R. Sheppard, T. Wilson, “Super-resolution by confocal fluorescence microscopy,” Optik 60, 391–396 (1982).

C. J. R. Sheppard, T. Wilson, “Image formation in scanning microscopes with partially coherent source and detector,” Opt. Acta 25, 315–325 (1978).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Depth of field in the scanning microscope,” Opt. Lett. 3, 115–117 (1978).
[CrossRef] [PubMed]

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

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.

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

X. Gan, M. Gu, C. J. R. Sheppard, “Fluorescent image formation in the fiber optical confocal scanning microscope,” J. Mod. Opt. (to be published).

Snyder, A. W.

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

Wilson, T.

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

I. J. Cox, C. J. R. Sheppard, T. Wilson, “Super-resolution by confocal fluorescence microscopy,” Optik 60, 391–396 (1982).

C. J. R. Sheppard, T. Wilson, “Image formation in scanning microscopes with partially coherent source and detector,” Opt. Acta 25, 315–325 (1978).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Depth of field in the scanning microscope,” Opt. Lett. 3, 115–117 (1978).
[CrossRef] [PubMed]

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

J. Microsc. (1)

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

J. Mod. Opt. (2)

C. J. R. Sheppard, X. Q. Mao, “Confocal microscopes with slit aperture,” J. Mod. Opt. 35, 1169–1185 (1988).
[CrossRef]

C. J. R. Sheppard, C. J. Cogswell, “Confocal microscopy with detector arrays,” J. Mod. Opt. 37, 267–279 (1990).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Acta (2)

C. J. R. Sheppard, T. Wilson, “Image formation in scanning microscopes with partially coherent source and detector,” Opt. Acta 25, 315–325 (1978).
[CrossRef]

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

Opt. Lett. (1)

Optik (3)

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).

I. J. Cox, C. J. R. Sheppard, T. Wilson, “Super-resolution by confocal fluorescence microscopy,” Optik 60, 391–396 (1982).

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

C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical microscope with extended depth field,” Proc. R. Soc. London Ser. A 387, 171–186 (1983).
[CrossRef]

Other (5)

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.

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

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

J. W. Goodman, Introduction of Fourier Optics (McGraw-Hill, San Francisco, 1968).

X. Gan, M. Gu, C. J. R. Sheppard, “Fluorescent image formation in the fiber optical confocal scanning microscope,” J. Mod. Opt. (to be published).

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Figures (8)

Fig. 1
Fig. 1

Geometry of a reflection-mode fiber-optical CSM (L1, L2, lenses).

Fig. 2
Fig. 2

In-focus transverse coherent transfer functions c(l) for different dimensionless fiber spot sizes A.

Fig. 3
Fig. 3

Defocused transverse coherent transfer functions c(l, u) for A = 1: (a) real part, (b) imaginary part.

Fig. 4
Fig. 4

Defocused transverse coherent transfer functions c(l, u) for A = 5: (a) real part, (b) imaginary part.

Fig. 5
Fig. 5

Variations of c(0, u) with the axial optical coordinate u. The dashed curves represent the imaginary parts of c(0, u), while the solid curves represent the real parts of c(0, u).

Fig. 6
Fig. 6

Axial coherent transfer functions c(s) for different values of A.

Fig. 7
Fig. 7

(a) Variations of the detected intensity with the axial optical coordinate u in the case of a perfect reflection planar object. (b) Half-width of the curves of (a) as a function of the dimensionless fiber spot size A.

Fig. 8
Fig. 8

Variations of the detected intensity with the axial optical coordinate u in the case of a point object.

Equations (45)

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U 2 ( x s , x 2 ) = U 1 ( x 0 ) h 1 ( x 0 + x 1 M 1 ) r f ( x s x 1 ) × h 2 ( x 1 + x 2 M 2 ) d x 0 d x 1 ,
U ( x , z ) = j a j f j ( x ) exp ( i β j z ) ,
U ( x , z ) = a 1 f 1 ( x ) exp ( i β 1 z ) ,
a 1 = f 1 * ( x ) U ( x , z = 0 ) d x | f 1 ( x ) | 2 d x .
U 3 ( x s , x 3 ) = f 1 ( x 3 ) exp ( i β 1 z 0 ) f 1 * ( x 2 ) U 2 ( x s , x 2 ) d x 2 | f 1 ( x ) | 2 d x ,
I ( x s ) = | U 3 ( x s , x 3 ) | 2 d x 3 .
I ( x s ) = f 1 * ( x 2 ) U 1 ( x 0 ) h 1 ( x 0 + x 1 M 1 ) r f ( x s x 1 ) × h 2 ( x 1 + x 2 M 2 ) f 1 ( x 2 ) U 1 * ( x 0 ) h 1 * ( x 0 + x 1 M 1 ) r f * ( x s x 1 ) × h 2 * ( x 1 + x 2 M 2 ) d x 0 d x 0 d x 1 d x 1 d x 2 d x 2 ,
I ( x s ) = g 1 ( x 1 ) g 1 * ( x 1 ) r f ( x s x 1 ) r f * ( x s x 1 ) d x 1 d x 1 = | g 1 ( x s ) r f ( x s ) | 2 ,
g 1 ( x ) = U 1 ( x 0 ) h 1 ( x 0 + x 1 M 1 ) f 1 * ( x 2 ) h 2 ( x + x 2 M 2 ) d x 0 d x 2 = [ U 1 ( x / M 1 ) h 1 ( x / M 1 ) ] [ f 1 * ( x / M 1 ) h 2 ( x ) ] ,
R f ( m ) = r f ( x ) exp ( 2 π ixm ) d x ,
r f ( x s x 1 ) = R f ( m ) exp [ 2 π i ( x s x 1 ) m ] d m ,
r f * ( x s x 1 ) = R f * ( p ) exp [ 2 π i ( x s x 1 ) p ] d p .
I ( x s ) = C ( m ; p ) R f ( m ) R f * ( p ) exp [ 2 π i ( m p ) x s ] d m d p ,
C ( m ; p ) = g 1 ( x 1 ) g 1 * ( x 1 ) × exp ( 2 π i m x 1 + 2 π p x 1 ) d x 1 d x 1 ,
C ( m ; p ) = c ( m ) c * ( p ) ,
C ( m ) = g 1 ( x 1 ) exp ( 2 π i m x 1 ) d x 1 ,
c * ( p ) = g 1 * ( x 1 ) exp ( 2 π i p x 1 ) d x 1 ,
I ( x s ) = | c ( m ) R f ( m ) exp ( 2 π i m x s ) d m | 2 .
c ( m ) = [ Ũ 1 ( M 1 m ) P 1 ( λ f m ) ] [ f 1 ( M 1 m ) P 2 ( λ f m ) ] ,
c ( l ) = [ Ũ 1 ( M 1 l ) P 1 ( λ f l ) ] [ f 1 ( M 1 l ) P 2 ( λ f l ) ] ,
f 1 ( r ) = exp [ ( 1 / 2 ) ( r / r 0 ) 2 ] ,
f 1 ( l ) = 2 π r 0 2 exp [ ( 1 / 2 ) ( 2 π l r 0 ) 2 ] .
c ( l ) = 2 π [ 1 exp ( A ) ] exp ( A l 2 4 ) × 0 π / 2 [ 1 exp ( A ρ 0 2 ) ] d θ ,
ρ 0 = [ ( l cos θ ) / 2 ] + [ 1 + l 2 sin 2 ( θ / 4 ) ] 1 / 2 ,
A = ( 2 π a 0 r 0 / λ d ) 2 .
c ( l ) = ( 2 / π ) { cos 1 ( l / 2 ) ( l / 2 ) [ 1 ( l / 2 ) 2 ] 1 / 2 } .
P 1 ( r , u ) = exp [ ( i u / 2 ) ( r / a 0 ) 2 ] ( r < a 0 ) ,
P 1 ( r , u ) = 0 ( r > a 0 ) ,
u = ( 8 π / λ ) z sin 2 ( α / 2 ) ,
c ( l , u ) = [ f 1 ( M 1 l ) P 1 ( λ f l ) u ] [ f 1 ( M 1 l ) P 1 ( λ f l , u ) ] ,
c ( l , u ) = 2 A π [ 1 exp ( A ) ] ( A i u ) exp [ ( A i u ) l 2 4 ] × 0 π / 2 { 1 exp [ ( A i u ) ρ 0 2 ] } d θ ,
c ( 0 , u ) = A [ 1 exp ( A + i u ) ] ( A i u ) [ 1 exp ( A ) ] ,
c R ( 0 , u ) = { A [ 1 exp ( A ) cos u ] + exp ( A ) u sin u } A ( A 2 + u 2 ) [ 1 exp ( A ) ] ,
c I ( 0 , u ) = { u [ 1 exp ( A ) cos u ] exp ( A ) A sin u } A ( A 2 + u 2 ) [ 1 exp ( A ) ] .
c R ( 0 , u ) = sin u / u ,
c I ( 0 , u ) = ( 1 cos u ) / u .
s 0 = 1 / [ 2 sin 2 ( α / 2 ) ] .
I ( u ) = | c ( l = 0 , u ) | 2 = | A { 1 exp [ ( A i u ) ] } [ 1 exp ( A ) ] ( A i u ) | 2 = c r 2 + c i 2 ,
I ( u ) = sin 2 u / 2 ( u / 2 ) 2
I ( υ , u ) = | c ( l , u ) J 0 ( υ l ) l d l | 2 ,
υ = ( 2 π / λ ) r sin α .
I ( 0 , u ) = sin 4 ( u / 4 ) ( u / 4 ) 4
I ( x s ) = | U 2 ( x s , x 2 ) | 2 D 2 ( x ) d x 2 .
I ( x s ) = | U 2 ( x s , x 2 ) f 1 * ( x 2 ) | 2 d x 2 .
r 0 = ρ / ( 2 ln V ) 1 / 2 ,

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