Abstract

Three-dimensional imaging in transmission and incident light microscopes is considered. Two-point resolution and images of lines and planes are discussed. The transfer functions for partially coherent systems are developed.

© 1989 Optical Society of America

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References

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  1. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,”J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  2. E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  3. B. R. Frieden, “Optical transfer of a three-dimensional object,”J. Opt. Soc. Am. 57, 56–66 (1967).
    [CrossRef]
  4. N. Streibl, “Three-dimensional imaging in a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
    [CrossRef]
  5. C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik 72, 131–133 (1986).
  6. C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging, II,” Optik 74, 128–129 (1986).
  7. C. J. R. Sheppard, H. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
    [CrossRef]
  8. T. K. Gaylord, M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–4 (1982).
    [CrossRef]

1987 (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)

1982 (1)

T. K. Gaylord, M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–4 (1982).
[CrossRef]

1969 (1)

E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

1967 (1)

1964 (1)

Frieden, B. R.

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–4 (1982).
[CrossRef]

Matthews, H.

McCutchen, C. W.

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–4 (1982).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, H. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[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).

Streibl, N.

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Appl. Phys. B (1)

T. K. Gaylord, M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–4 (1982).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Commun. (1)

E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[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

Geometry of the optical system in (a) transmission and (b) incident light.

Fig. 2
Fig. 2

Image of an axial line in a transmission system. s ≥ 1.

Fig. 3
Fig. 3

Image of an axial line in an incident light system, s = cot(α/2).

Fig. 4
Fig. 4

Coherent transfer function for a transmission system with a small condenser pupil: normalized spatial frequencies.

Fig. 5
Fig. 5

Coherent transfer function for a transmission system with small condenser pupil: absolute spatial frequencies.

Fig. 6
Fig. 6

Transfer function H(mm′, 0, rr′) for a transmission system with a large condenser.

Fig. 7
Fig. 7

Cutoff for the transmission weak-object transfer function. Curves (a), (b), (c), and (d) correspond to Eqs. (45a), (45b), (45c), and (45d), respectively.

Fig. 8
Fig. 8

Transfer function C(m, n, r; 0, 0, r0) for an incident-light system.

Equations (87)

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v 0 = k x 0 sin α 1 ,             v = k x sin α , w 0 = k y 0 sin α 1 ,             w = k y sin α , u 0 = 4 k z 0 sin 2 ( α 1 / 2 ) ,             u = 4 k z sin 2 ( α / 2 ) ,
h 1 ( v 0 ) = P 1 ( ξ 1 , η 1 ) exp { j u [ 1 4 sin 2 ( α / 2 ) - 1 2 ( ξ 1 2 + η 1 2 ) ] } × exp [ - j ( ξ 1 v 0 + η 1 w 0 ) ] d ξ 1 d η 1 ,
h 2 ( v ) = P 2 ( ξ 2 , η 2 ) exp { j u [ 1 4 sin 2 ( α / 2 ) - 1 2 ( ξ 2 2 + η 2 2 ) ] } × exp [ - j ( ξ 2 v + η 2 w ) ] d ξ 1 d η 1 ,
t ( v 0 ) = j k 2 [ 1 - n 2 ( v 0 ) ] .
I ( v ) = S ( v 1 ) | h 1 ( v 0 - v 1 ) t ( v 0 ) h 2 ( v - v 0 ) d v 0 | 2 d x 1 d y 1 = g ( v 0 - v 0 ) t ( v 0 ) t * ( v 0 ) h 2 ( v - v 0 ) h 2 * ( v - v 0 ) d v 0 d v 0 ,
g ( v 0 - v 0 ) = h 1 ( v 0 - v 1 ) h 1 * ( v 0 - v 1 ) S ( v 1 ) d x 1 d y 1 .
g ( v 0 - v 0 ) = P 1 ( ξ 1 , η 1 ) 2 exp { - j [ ξ 1 ( v - v 0 ) + η 1 ( w 0 - w 0 ) ] } exp { j ( u 0 - u 0 ) [ 1 4 sin 2 ( α 1 / 2 ) - 1 2 ( ξ 1 2 + η 1 2 ) ] } d ξ 1 d η 1 .
I ( u ) = h 2 ( u - a ) 2 + h 2 ( u + a ) 2 + 2 Re [ g ( 2 a ) h 2 ( u + a ) h 2 * ( u - a ) ] .
g ( u ) = exp { j u 4 [ cot 2 α 2 + ( 1 - s 2 ) ] } [ sin ( s 2 u / 4 ) s u 2 / 4 ] ,
h 2 ( u ) = exp ( j u 4 cot 2 α 2 ) [ sin ( u / 4 ) u / 4 ] ,
s = sin ( α 1 / 2 ) / sin ( α / 2 ) ,
I ( u ) = [ sin ( u - a ) / 4 ( u - a ) / 4 ] 2 + [ sin ( u + a ) / 4 ( u + a ) / 4 ] 2 + 2 cos [ a ( s 2 - 1 ) 2 ] × sin ( s 2 a / 2 ) s 2 a / 2 { sin [ ( u + a ) / 4 ] ( u + a ) / 4 } { sin [ ( u - a ) / 4 ] ( u - a ) / 4 } ,
a = 4 Δ k sin 2 ( α / 2 ) ,
h 2 ( u ) = exp ( - j u 4 cot 2 α 2 ) [ sin ( u / 4 ) u / 4 ] ,
I ( u ) = { sin [ ( u - a ) / 4 ] ( u - a ) / 4 } 2 + { sin [ ( u + a ) / 4 ] ( u + a ) / 4 } 2 + 2 cos { [ cot 2 α 2 ( 1 - s 2 ) 2 ] a } sin ( s 2 a / 2 ) s 2 a / 2 × { sin [ ( u - a ) / 4 ] ( u - a ) / 4 } { sin [ ( u + a ) / 4 ] ( u + a ) / 4 } .
t ( v 0 ) = δ ( v 0 ) δ ( w 0 ) .
I ( v ) = { P 2 ( ξ 2 , η 2 ) exp [ - j ( ξ 2 v + η 2 w ) - 1 2 j u ( ξ 2 2 + η 2 2 ) ] δ ( ξ 1 2 + n 1 2 - ξ 2 2 - η 2 2 ) d ξ 2 d η 2 } × { P 2 * ( ξ 2 , η 2 ) exp [ - j ( ξ 2 v + η 2 w ) - 1 2 j u ( ξ 2 2 + η 2 2 ) ] δ ( ξ 1 2 + η 1 2 - ξ 1 2 - η 1 2 ) d ξ 2 d η 2 } × P 1 ( ξ 1 , η 1 ) 2 d ξ 1 d η 1 .
I ( ρ ) = P 1 ( ξ ) 2 P 2 ( ξ ) 2 J 0 2 ( ξ ρ ) ξ d ξ ,
I ( ρ ) = 0 1 J 0 2 ( ξ ρ ) ξ d ξ = 1 2 [ J 0 2 ( ρ ) + J 2 2 ( ρ ) ] ,
I ( ρ ) = 0 | P 1 [ ( 1 sin 2 ( α / 2 ) - ξ 2 ) 1 / 2 ] | 2 P 2 ( ξ ) 2 J 0 2 ( ξ ρ ) ξ d ξ .
I ( ρ ) = J 0 2 ( ρ ) ,
t ( v 0 ) = T ( m ) exp ( j m · v 0 ) d m ,
I ( v ) = C ( m ; m ) T ( m ) T * ( m ) exp [ j ( m - m ) · v ] d m d m ,
C ( m ; m ) = P 1 ( ξ , η ) 2 P 2 ( ξ - m , η - n ) P 2 * ( ξ - m , η - n ) × δ [ r - m ξ - n η + ½ ( m 2 + n 2 ) ] δ [ r - m ξ - n η + ½ ( m 2 + n 2 ) ] d ξ d η ,
m n = n m
ξ = ξ - ½ ( m + m ) ,
C ( m , 0 , r ; m , 0 , r ) = | P 1 [ ξ + 1 2 ( m + m ) , η ] | 2 × P 2 [ ξ - ½ ( m - m ) , η ] P 2 * [ ξ + ½ ( m - m ) , η ] × δ ( r - ξ , m - ½ m m ) δ ( r - m ξ - ½ m m ) d ξ d η ,
ξ = r m + m 2 = r m + m 2 .
C ( m , 0 , r ; m , 0 , r ) = 1 m m δ [ r m - r m - 1 2 ( m - m ) ] × | P 1 [ ξ + 1 2 ( m + m ) , η ] | 2 P 2 [ ξ - 1 2 ( m - m ) , η ] × P 2 * [ ξ + 1 2 ( m - m ) , η ] δ [ ξ - ( r - r m - m ) ] d ξ d η .
C ( 0 , 0 , r ; 0 , 0 , r ) = π s 2 δ ( r ) δ ( r ) ,             s 1 ,
C ( m , 0 , 0 ; m , 0 , 0 ) = { 2 m m δ ( m - m ) Re [ s 2 - ( m 2 ) 2 ] 1 / 2 s 1 2 m m δ ( m - m ) Re [ 1 - ( m 2 ) 2 ] 1 / 2 s > 1
C ( m , 0 , 0 ; 0 , 0 , 0 ) = { 2 m Re ( s 2 - m 2 4 ) 1 / 2 s 1 2 m Re ( 1 - m 2 4 ) 1 / 2 s > 1 .
C ( m ; m ) = c ( m ) c ( m )
c ( m ) = P 2 ( - m , - n ) δ [ r + ½ ( m 2 + n 2 ) ] ,
Λ x = ( sin α ) / λ ,             Λ z = 2 sin 2 ( α / 2 ) / λ ,
C ( m , 0 , r ; m , 0 , r ) = 1 m m δ [ r m - r m + 1 2 ( m - m ) ] P 2 [ ξ - 1 2 ( m - m ) , η ] × P 2 * [ ξ + 1 2 ( m - m ) , η ] δ [ ξ - ( r - r m - m ) ] d ξ d η
= 1 m m δ [ r m - r m - 1 2 ( m - m ) ] H ( m - m , 0 , r - r ) ,
H ( m , 0 , r ) = Re { [ 1 - ( r m + m 2 ) 2 ] 1 / 2 } ,
t ( v 0 ) = δ ( u 0 ) + t 1 ( v 0 ) ,
I ( v ) = C ( 0 , 0 , r ; 0 , 0 , r ) d r d r + C ( m ; 0 , 0 , r ) T 1 ( m ) exp ( j m · v ) d m d r + C ( 0 , 0 , r ; m ) T 1 * ( m ) exp ( - j m · v ) d m d r ,
I ( v ) = C D ( 0 ; 0 ) + C W ( m ; 0 ) T 1 ( m ) exp ( j m · v ) d m + C W ( 0 , m ) T 1 * ( m ) exp ( - j m · v ) d m .
C ( m ; m ) = C * ( m ; m ) .
T 1 ( m ) = T 1 r ( m ) + i T 1 i ( m ) .
I ( v ) = C D ( 0 ; 0 ) + [ C A ( m ; 0 ) T 1 r ( m ) + C P ( m ; 0 ) T 1 i ( m ) ] exp ( j m · v ) d m
C A ( m ; 0 ) = C W ( m ; 0 ) + C W * ( - m ; 0 ) , C P ( m ; 0 ) = C W ( m ; 0 ) - C W * ( - m ; 0 )
C W ( m ; 0 ) = P 1 ( ξ , η ) 2 ( ξ - m , η - n ) P 2 * ( ξ , η ) × δ [ r - m ξ - n η + ½ ( m 2 + n 2 ) ] d ξ d η = 2 ρ Re { [ 1 2 ( 1 + s 2 ) - ρ 2 4 - r 2 ρ 2 - | r - 1 2 ( 1 - s 2 ) | ] 1 / 2 } ,
r ½ ( ρ - 1 ) 2 - ½ .
r ½ ( ρ + 1 ) 2 - ½ ;
r - 1 2 ( ρ + s ) 2 + s 2 2 ,
r - 1 2 ( ρ + s ) 2 + s 2 2 .
C W ( m , n , 0 ; 0 ) = 2 ρ Re ( s 2 - ρ 2 / 4 ) 1 / 2 ,
C D ( 0 ; 0 ) = P 1 ( ξ , η ) 2 P 2 ( ξ , η ) 2 d ξ d η ,
C ( m ; m ) = P 1 ( ξ , η ) 2 P 2 ( ξ - m , η - n ) × P 2 * ( ξ - m , η - n ) δ [ r + 1 2 sin 2 ( α / 2 ) - ( ξ 2 + η 2 ) + m ξ + n η - m 2 + n 2 2 ] δ [ r + 1 2 sin 2 ( α / 2 ) - ( ξ 2 + η 2 ) + m ξ + n η - m 2 + n 2 2 ] d ξ d η ,
C ( 0 , 0 , r ; 0 , 0 , r ) = π δ ( r α - r α ) ,
r α = r + 1 2 sin 2 ( α / 2 )
r α { s 2 s 1 1 s > 1 .
C ( m ; m ) = c ( m ) c ( m ) ,
c ( m ) = P ( - m , - n ) δ [ r + 1 2 sin 2 ( α / 2 ) - m 2 + n 2 2 ] ,
- 1 2 cot 2 α 2 r - 1 2 sin 2 ( α / 2 ) .
C ( m ; m ) = P 2 ( ξ - m , η - n ) P 2 * ( ξ - m , η - n ) × δ [ r + 1 2 sin 2 ( α / 2 ) - ( ξ 2 + η 2 ) + m ξ + n η - m 2 + n 2 2 ] × δ [ r + 1 2 sin 2 ( α / 2 ) - ( ξ 2 + η 2 ) + m ξ + n η - m 2 + n 2 2 ] d ξ d η = P 2 ( ξ - m , η - n ) P 2 * ( ξ - m , η - n ) × δ [ r α - ( ξ 2 + η 2 ) + m ξ + n η - m 2 + n 2 2 ] × δ [ r α - ( ξ 2 + η 2 ) + m ξ + n η - m 2 + n 2 2 ] d ξ d η .
ξ = ξ - m
η = η - n ,
C ( m ; m ) = P 2 ( ξ , η ) P 2 * [ ξ + ( m - m ) , η + ( n - n ) ] × δ { r α - [ ( ξ + m 2 ) 2 + ( η + n 2 ) 2 ] - m 2 + n 2 4 } × δ [ ( r - r ) + ( m - m ) ξ + ( n - n ) η + ( m - m ) 2 + ( n - n ) 2 2 ] d ξ d η .
C ( m , 0 , r ; m , 0 , r ) = 1 m - m Re { [ m r α - m r α m - m - ( r - r m - m ) 2 - m 2 + m 2 4 ] - 1 / 2 } ,
r α + m ( r - r ) m - m - m m 2 1
r α + m ( r - r ) m - m - m m 2 1
C ( m , n , r ; 0 , 0 , r ) = P 1 ( ξ , η ) 2 P 2 ( ξ - m , η - n ) P 2 * ( ξ , η ) × δ [ r + 1 2 sin 2 ( α / 2 ) - ( ξ 2 + η 2 ) ] × δ [ r + 1 2 sin 2 ( α / 2 ) - ( ξ 2 + η 2 ) + m ξ + n η - m 2 + n 2 2 ] d ξ d η .
- 1 2 sin 2 ( α / 2 ) r - 1 2 sin 2 ( α / 2 ) + 1 2 ( 1 + s 2 - s 2 - 1 ) ,
r = r 0 = - 1 2 sin 2 ( α / 2 ) + 1 2 ( 1 + s 2 - s 2 - 1 ) .
C ( m , n , r ; 0 , 0 , [ s b : r : 0 ] ) = 1 ρ Re { [ 1 2 ( 1 + s 2 ) - ρ 2 4 - r β 2 ρ 2 - | r β - 1 2 ( 1 - s 2 ) | ] 1 / 2 } ,             r β 1 - s 2 2 ,
I ( v ) = C ( m , n , r ; 0 , 0 , r 0 ) T ( m ) × T * ( 0 , 0 , r 0 ) exp [ j ( m v + n w + r β u ) ] d m + C ( 0 , 0 , r 0 ; m , n , r ) T ( 0 , 0 , r 0 ) × T * ( m ) e - j ( m v + n w + r β ( u ) d m .
I ( v ) = C A ( m , n , r ; 0 , 0 , r 0 ) [ T r ( m ) T i ( 0 , 0 , r 0 ) + T r ( 0 , 0 , r 0 ) T i ( m ) ] exp [ - j ( m v + n w + r β u ) d m + C P ( m , n , r ; 0 , 0 , r 0 ) [ T i ( m ) T r ( 0 , 0 , r 0 ) + T i ( 0 , 0 , r 0 ) T r ( m ) ] exp [ - j ( m v + n w + r β u ) ] d m ,
C A ( m , n , r ; 0 , 0 , r 0 ) = C ( m , n , r ; 0 , 0 , r 0 ) + C * ( - m , - n , - r ; 0 , 0 , - r 0 )
C P ( m , n , r ; 0 , 0 , r 0 ) = C ( m , n , r ; 0 , 0 , r 0 ) - C * ( - m , - n , - r ; 0 , 0 , - r 0 ) .
t ( v , w , u ) = δ ( v + γ u ) ,
T ( m , n , r ) = δ ( n ) δ ( r - m γ ) .
I ( v ) = C ( m , 0 , m γ ; m , 0 , m γ ) × exp [ j ( m - n ) ( v + γ u ) ] d m d m
I ( v ) = P 1 ( ξ , η ) 2 P 2 ( 2 γ - ξ , η ) 2 4 ( γ - ξ ) 2 d ξ d η .
m 0 - 1 2 γ sin 2 ( α / 2 ) .
I ( v ) = C ( m 0 , 0 , r ; m 0 , 0 , r ) exp [ j ( r - r ) u ] d r d r .
I ( v ) = P 1 ( ξ , η ) 2 P 2 ( ξ - m 0 , η ) 2 d ξ d η .
π 2 ( ξ ) = h 2 ( v ) exp ( - j ξ · v ) d ξ ,
π 1 ( ξ ) = g ( v ) exp ( - j ξ · v ) d ξ .
π 1 ( ξ ) = P 1 ( ξ , η ) 2 δ [ ζ - 1 4 sin 2 ( α / 2 ) + 1 2 ( ξ 2 + η 2 ) ] ,
π 2 ( ξ ) = P 2 ( ξ , η ) δ [ ζ - 1 4 sin 2 ( α / 2 ) + 1 2 ( ξ 2 + η 2 ) ] ,
C ( m ; m ) = π 1 ( ξ ) π 2 ( ξ + m ) π 2 * ( ξ + m ) d ξ .
π 2 ( ξ ) = P 2 ( ξ , η ) δ [ ξ + 1 4 sin 2 ( α / 2 ) - 1 2 ( ξ 2 + η 2 ) ] ,

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