Abstract

A confocal scanning fluorescent microscope is suitable for 3-dimensional (3-D) imaging. The 3-D optical transfer functions (OTF’s) for such a microscope are calculated to show their dependence on the wavelength of the fluorescence. These calculations reveal that when the wavelength of the fluorescence is equivalent to that of the excitation light, the 3-D OTF has no missing-cone region. However, as the wavelength becomes longer, the 3-D OTF approaches that of an incoherent conventional microscope at the wavelength of the excitation light.

© 1989 Optical Society of America

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References

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  1. T. Wilson, C. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984).
  2. C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 5, 1051–1073 (1977).
    [Crossref]
  3. C. J. R. Sheppard, T. Wilson, “Depth of field in the scanning microscope,” Opt. Lett. 3, 115–117 (1978).
    [Crossref] [PubMed]
  4. I. J. Cox, C. J. R. Sheppard, T. Wilson, “Super-resolution by confocal fluorescent microscopy,” Optik 60, 391–396 (1982).
  5. M. Y. Chiu, H. H. Barrett, R. G. Simpson, C. Chou, J. W. Arendt, G. R. Gindi, “Three-dimensional radiographic imaging with a restricted view angle,”J. Opt. Soc. Am. 69, 1323–1333 (1979).
    [Crossref]
  6. C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik 72, 131–133 (1986).
  7. C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging II,” Optik 74, 128–129 (1986).
  8. K. Carlsson, N. Åslund, “Confocal imaging for 3-D digital microscopy,” Appl. Opt. 26, 3232–3238 (1987).
    [Crossref] [PubMed]
  9. J. S. Ploem, “Laser scanning fluorescence microscopy,” Appl. Opt. 26, 3226–3231 (1987).
    [Crossref] [PubMed]
  10. R. W. Wijnaends van Resandt, H. J. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Stelzer, R. Stricker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. 138, 29–34 (1985).
    [Crossref]
  11. I. J. Cox, “Scanning optical fluorescence microscopy,” J. Microsc. 133, 149–154 (1984).
    [Crossref] [PubMed]
  12. B. R. Frieden, “Optical transfer of the three dimensional object,”J. Opt. Soc. Am. 57, 56–67 (1967).
    [Crossref]
  13. H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
    [Crossref]
  14. P. A. Stokseth, “Properties of a defocused optical system,”J. Opt. Soc. Am. 59, 1314–1321 (1969).
    [Crossref]
  15. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).
  16. N. Streibl, “Fundamental restrictions for 3-D light distributions,” Optik 66, 341–354 (1984).

1987 (2)

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).

1985 (1)

R. W. Wijnaends van Resandt, H. J. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Stelzer, R. Stricker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. 138, 29–34 (1985).
[Crossref]

1984 (2)

I. J. Cox, “Scanning optical fluorescence microscopy,” J. Microsc. 133, 149–154 (1984).
[Crossref] [PubMed]

N. Streibl, “Fundamental restrictions for 3-D light distributions,” Optik 66, 341–354 (1984).

1982 (1)

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

1979 (1)

1978 (1)

1977 (1)

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 5, 1051–1073 (1977).
[Crossref]

1969 (1)

1967 (1)

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[Crossref]

Arendt, J. W.

Åslund, N.

Barrett, H. H.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

Carlsson, K.

Chiu, M. Y.

Chou, C.

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 5, 1051–1073 (1977).
[Crossref]

Cox, I. J.

I. J. Cox, “Scanning optical fluorescence microscopy,” J. Microsc. 133, 149–154 (1984).
[Crossref] [PubMed]

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

Davoust, J.

R. W. Wijnaends van Resandt, H. J. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Stelzer, R. Stricker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. 138, 29–34 (1985).
[Crossref]

Frieden, B. R.

Gindi, G. R.

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[Crossref]

Kaplan, R.

R. W. Wijnaends van Resandt, H. J. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Stelzer, R. Stricker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. 138, 29–34 (1985).
[Crossref]

Marsman, H. J. B.

R. W. Wijnaends van Resandt, H. J. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Stelzer, R. Stricker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. 138, 29–34 (1985).
[Crossref]

Ploem, J. S.

Sheppard, C.

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

Sheppard, C. J. R.

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 fluorescent microscopy,” Optik 60, 391–396 (1982).

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 5, 1051–1073 (1977).
[Crossref]

Simpson, R. G.

Stelzer, E. H. K.

R. W. Wijnaends van Resandt, H. J. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Stelzer, R. Stricker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. 138, 29–34 (1985).
[Crossref]

Stokseth, P. A.

Streibl, N.

N. Streibl, “Fundamental restrictions for 3-D light distributions,” Optik 66, 341–354 (1984).

Stricker, R.

R. W. Wijnaends van Resandt, H. J. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Stelzer, R. Stricker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. 138, 29–34 (1985).
[Crossref]

Wijnaends van Resandt, R. W.

R. W. Wijnaends van Resandt, H. J. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Stelzer, R. Stricker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. 138, 29–34 (1985).
[Crossref]

Wilson, T.

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

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

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

Appl. Opt. (2)

J. Microsc. (2)

R. W. Wijnaends van Resandt, H. J. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Stelzer, R. Stricker, “Optical fluorescence microscopy in three dimensions: microtomoscopy,” J. Microsc. 138, 29–34 (1985).
[Crossref]

I. J. Cox, “Scanning optical fluorescence microscopy,” J. Microsc. 133, 149–154 (1984).
[Crossref] [PubMed]

J. Opt. Soc. Am. (3)

Opt. Acta (1)

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 5, 1051–1073 (1977).
[Crossref]

Opt. Lett. (1)

Optik (4)

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

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).

N. Streibl, “Fundamental restrictions for 3-D light distributions,” Optik 66, 341–354 (1984).

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

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[Crossref]

Other (2)

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

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

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Figures (3)

Fig. 1
Fig. 1

Diagram of the confocal fluorescent microscope.

Fig. 2
Fig. 2

3-D OTF’s for the confocal fluorescent microscope. The altitude represents the 3-D OTF, Cn(s, 0, w). Fluorescence Wavelengths: (a) λ, (b) 2λ, (c) 3λ, (d) 6λ, (e) 10λ.

Fig. 3
Fig. 3

3-D OTF for the incoherent conventional microscope at the excitation wavelength. The altitude represents ln[Cn1[(s, 0, w) + 1].

Equations (23)

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i f ( x , y , z ) = h 1 ( x , y , z ) 2 o ( x - x s , y - y s , z - z s ) .
I ( x D , y D , z D ; x s , y s , z s ) = - | h 2 ( x D M - x , y D M - y , z D M - z ) | 2 h 1 ( x , y , z ) 2 × o ( x - x s , y - y s , z - z s ) d x d y d z ,
I ( 0 , 0 , 0 ; x s , y s , z s ) = - h 2 ( x , y , z ) 2 h 1 ( x , y , z ) 2 × o ( x - x s , y - y s , z - z s ) d x d y d z .
psf ( x , y , z ) = h 2 ( x , y , z ) h 1 ( x , y , z ) 2 .
C ( f x , f y , f z ) = - h 2 ( x , y , z ) 2 h 1 ( x , y , z ) 2 × exp [ - 2 π i ( x f x + y f y + z f z ) ] d x d y d z = - [ T 1 ( f x , f y ; z ) T 2 ( f x , f y ; z ) ] × exp ( - 2 π i z f z ) d z ,
T 1 ( f x , f y ; z ) = - h 1 ( x , y , z ) 2 exp [ - 2 π i ( x f x + y f y ) ] d x d y , T 2 ( f x , f y ; z ) = - h 2 ( x , y , z ) 2 exp [ - 2 π i ( x f x + y f y ) ] d x d y .
v x = 2 π λ x sin α 2 π λ a d x , v y = 2 π λ y sin α 2 π λ a d y , u = 2 π λ z sin 2 α 2 π λ ( a d ) 2 z ,
s = λ sin α f x ,             t = λ sin α f y ,             w = λ sin 2 α f z .
T n 1 ( s , t ; u ) = - h n 1 ( v x , v y , u ) 2 exp [ - i ( v x s + v y t ) ] d v x d v y ,
T n 1 ( s , 0 ; u ) = g 1 ( s ) { J 1 [ u g 2 ( s ) ] u g 2 ( s ) } 0 < s < 2 = 0 2 s ,
g 1 ( s ) = 2 ( 1 - 0.69 s + 0.0076 s 2 + 0.043 s 3 ) , g 2 ( s ) = s - 0.5 s 2 ,
h n 2 ( v w , v y , u ) = h n 1 ( v x β , v y β , u β ) ,
T n 2 ( s , t ; u ) = T n 1 ( β s , β t ; u β ) ,
T n 1 ( s , t ; u ) = T n 1 [ ( s 2 + t 2 ) 1 / 2 , 0 ; u ] .
T n 1 ( s , t ; u ) T n 2 ( s , t ; u ) = - T n 1 { [ ( s - s ) 2 + ( t - t ) 2 ] 1 / 2 , 0 ; u } T n 1 [ β ( s 2 + t 2 ) 1 / 2 , 0 ; u β ] d s d t
C n ( s , t , w ) = F [ h n 1 ( v x , v y , u ) 2 ] F [ h n 2 ( v x , v y , u ) 2 ] ,
f ( x L , y L , u ) = p ( x L , y L ) exp [ ½ i u ( x L 2 + y L 2 ) ] ,
p ( x L , y L ) = { 1 x L 2 + y L 2 1 0 x L 2 + y L 2 > 1 .
C n 1 ( s , t , w ) = F [ h n 1 ( v x , v y , u ) 2 ] = - f ( s , t , u ) f * ( s - s , t - t , u ) × exp ( - i w u ) d s d t d u .
C n 1 ( s , t , w ) = - p ( s + ½ s , t + ½ t ) p * ( s - ½ s , t - ½ t ) × exp [ i u ( s s + t t - w ) ] d s d t d u = - p ( s + ½ s , t + ½ t ) p * ( s - ½ s , t - ½ t ) × δ ( s s + t t - w ) d s d t ,
C n 1 ( s , t , w ) = { 1 ρ [ 1 - ( w ρ + ρ 2 ) 2 ] 1 / 2 Π [ w 2 g ( ρ ) ] ρ < 2 0 otherwise ,
Π ( x ) = { 1 x < ½ 0 x > ½
g ( ρ ) = ρ - ½ ρ 2 , ρ = ( s 2 + t 2 ) 1 / 2 .

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