Abstract

The scalar three-dimensional optical transfer function is derived without using the paraxial approximation. The weak-object transfer function for a partially coherent system with equal condenser and objective apertures and the coherent transfer function for a confocal transmission system are of identical form. The coherent transfer function of a confocal reflection system is also derived. Both uniform angular illumination and systems obeying the sine condition are considered. In all cases the transfer functions can be expressed analytically.

© 1994 Optical Society of America

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References

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  1. B. R. Frieden, “Optical transfer of a three-dimensional object,”J. Opt. Soc. Am. 57, 56–66 (1967).
    [CrossRef]
  2. N. Streibl, “Three-dimensional image in a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
    [CrossRef]
  3. C. J. R. Sheppard, X. Q. Mao, “Three-dimensional imaging in a microscope,” J. Opt. Soc. Am. A 6, 1260–1269 (1989).
    [CrossRef]
  4. C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 164, 337–390 (1991).
  5. C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode scanning microscope,” Opt. Commun. 81, 281–284 (1991).
    [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. T. Noda, S. Kawata, S. Minami, “Three-dimensional phase contrast imaging by an annular illumination microscope,” Appl. Opt. 29, 3810–3815 (1990).
    [CrossRef] [PubMed]
  9. C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
    [CrossRef]
  10. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,”J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  11. T. Tsuruta, “Imaging of 3-D object,” O + E (Japan). 156, 112–123 (1992).
  12. R. Barakat, D. Lev, “Transfer functions and total illuminance of high numerical aperture systems obeying the sine condition,”J. Opt. Soc. Am. 53, 324–332 (1963).
    [CrossRef]
  13. C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
    [CrossRef]

1993 (1)

C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

1992 (1)

T. Tsuruta, “Imaging of 3-D object,” O + E (Japan). 156, 112–123 (1992).

1991 (2)

C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 164, 337–390 (1991).

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

1990 (2)

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
[CrossRef]

T. Noda, S. Kawata, S. Minami, “Three-dimensional phase contrast imaging by an annular illumination microscope,” Appl. Opt. 29, 3810–3815 (1990).
[CrossRef] [PubMed]

1989 (1)

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)

1967 (1)

1964 (1)

1963 (1)

Barakat, R.

Cogswell, C. J.

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
[CrossRef]

Frieden, B. R.

Gu, M.

C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 164, 337–390 (1991).

Kawata, S.

Lev, D.

Mao, X. Q.

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode scanning microscope,” Opt. Commun. 81, 281–284 (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]

McCutchen, C. W.

Minami, S.

Noda, T.

Sheppard, C. J. R.

C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 164, 337–390 (1991).

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (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, “The spatial frequency cut-off in three-dimensional imaging II,” Optik 74, 128–129 (1986).

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

Streibl, N.

Tsuruta, T.

T. Tsuruta, “Imaging of 3-D object,” O + E (Japan). 156, 112–123 (1992).

Appl. Opt. (1)

J. Microsc. (2)

C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 164, 337–390 (1991).

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
[CrossRef]

J. Mod. Opt. (1)

C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

J. Opt. Soc. Am. (3)

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

O + E (Japan). (1)

T. Tsuruta, “Imaging of 3-D object,” O + E (Japan). 156, 112–123 (1992).

Opt. Commun. (1)

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

Optik (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).

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Figures (8)

Fig. 1
Fig. 1

Autocorrelation of a spherical shell, calculated as the volume of overlap. The figure shows a cross section through the shells. P is a general point somewhere on the region of overlap, so that OP is a radius of a circle with center O perpendicular to the plane of the figure.

Fig. 2
Fig. 2

OTF for an incoherent system with constant angular illumination: (a) α = 60°, (b) α = 9°.

Fig. 3
Fig. 3

OTF for an incoherent system that obeys the sine condiion: (a) α = 60°, (b) α = 9°.

Fig. 4
Fig. 4

Cross sections C(l,0) through the 3-D OTF, which apply for transverse imaging of a thick structure with no axial variations: (a) constant angular illumination, (b) sine condition.

Fig. 5
Fig. 5

Autoconvolution of a spherical shell, giving the CTF for confocal reflection.

Fig. 6
Fig. 6

CTF for a confocal reflection system with constant angular illumination: (a) α = 60°, (b) α = 90°.

Fig. 7
Fig. 7

CTF for a confocal reflection system obeying the sine condition: (a) α = 60°, (b) α = 90°.

Fig. 8
Fig. 8

2-D transfer function C(l) for imaging of a thin object: (a) constant angular illumination, (b) sine condition.

Equations (31)

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l = ( m 2 + n 2 ) 1 / 2
K = ( l 2 + s 2 ) 1 / 2 .
C ( l , s ) = 2 / K ,             0 K 2.
s = - l K cos β ( 1 - K 2 4 ) 1 / 2 ,
C ( l , s ) = 4 π K 0 β 1 P 1 ( θ 1 ) P 2 ( θ 2 ) d β ,
cos θ 1 , 2 = l K cos β ( 1 - K 2 4 ) 1 / 2 s 2 .
p = 2 l K s ( 1 - K 2 4 ) 1 / 2 ,
cos θ 1 , 2 = s 2 ( p cos β 1 ) .
β 1 = arccos [ s / 2 + cos α l K ( 1 - K 2 4 ) 1 / 2 ]
= arccos [ 1 p ( 2 cos α s + 1 ) ] ,
P ( θ ) = 1 ,
C ( l , s ) = 4 π K arccos [ 1 p ( 2 cos α s + 1 ) ] .
2 ( l sin α - s cos α ) = K 2 ,
P ( θ ) = cos 1 / 2 θ ,
C ( l , s ) = 0 β 1 2 s π K ( p 2 cos 2 β - 1 ) 1 / 2 d β
= 2 s π K ( p 2 - 1 ) 1 / 2 E [ β 1 , p ( p 2 - 1 ) 1 / 2 ] ,
C ( l , 0 ) = 4 π l arccos [ cos α ( 1 - l 2 4 ) 1 / 2 ]
C ( l , 0 ) = 4 π l ( sin 2 α - l 2 4 ) 1 / 2
cos θ 1 , 2 = s 2 ( 1 p cos β ) .
c ( l , s ) = 4 π K 0 β 2 d β ,
β 2 = arcsin [ 1 p ( 1 - 2 cos α s ) ]
s 2 cos α .
c ( l , s ) = 4 π K arcsin [ 1 p ( 1 - 2 cos α s ) ] ,
c ( l , s ) = 0 β 2 2 s π K ( 1 - p 2 sin 2 β ) 1 / 2 d β
= 2 s π K E ( β 2 , p ) .
2 ( l sin α - s cos α ) K 2 ,
c ( l , s ) = 2 s π K E ( p ) .
c ( 0 , s ) = 1 ,             2 cos α s 2.
c ( 0 , s ) = 2 / s ,             2 cos α s 2.
C ( 0 ) = 2 ln sec α
C ( 0 ) = 4 sin 2 ( α / 2 )

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