Abstract

Images of circular phase objects in partially coherent light have been studied. It has been shown that recognizable contrast is obtained in the image of phase objects formed by an ordinary microscope even when the cone of the illuminating beam is sizable and the microscope is focused perfectly.

© 1963 Optical Society of America

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References

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  1. H. H. Hopkins, Rev. Optique 31, 142 (1952).
  2. M. De and S. C. Som, Opt. Acta 9, 17 (1962).
    [Crossref]
  3. See Ref. 2, p. 21.
  4. See Ref. 2, p. 29.
  5. See Ref. 2, p. 24.
  6. See Ref. 2, p. 25.
  7. H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 51, 709 (1961).
    [Crossref]

1962 (1)

M. De and S. C. Som, Opt. Acta 9, 17 (1962).
[Crossref]

1961 (1)

1952 (1)

H. H. Hopkins, Rev. Optique 31, 142 (1952).

De, M.

M. De and S. C. Som, Opt. Acta 9, 17 (1962).
[Crossref]

Hopkins, H. H.

H. H. Hopkins, Rev. Optique 31, 142 (1952).

Osterberg, H.

Smith, L. W.

Som, S. C.

M. De and S. C. Som, Opt. Acta 9, 17 (1962).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

M. De and S. C. Som, Opt. Acta 9, 17 (1962).
[Crossref]

Rev. Optique (1)

H. H. Hopkins, Rev. Optique 31, 142 (1952).

Other (4)

See Ref. 2, p. 21.

See Ref. 2, p. 29.

See Ref. 2, p. 24.

See Ref. 2, p. 25.

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Figures (5)

Fig. 1
Fig. 1

Schematic representation of the complete image forming system.

Fig. 2
Fig. 2

Intensity distribution in the image plane as a function of α = ω/σ. All curves normalized against the background. Object size: (a) σ = 1.0, (b) σ = 2.0, (c) σ = 3.0, and (d) σ = 4.0.

Fig. 3
Fig. 3

Variation of central intensity with the size ρ of the effective source. Normalized against the background. Object size: (a) σ = 2.0, (b) σ = 3.0, and (c) σ = 4.0.

Fig. 4
Fig. 4

Normalized central intensity as a function of optical path difference δ. Object size: (a) σ = 3.0 and (b) σ = 4.0.

Fig. 5
Fig. 5

Just-detectable optical path difference δ as a function of ρ. Limiting threshold contrast 20%.

Equations (61)

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u = ( 2 π / λ ) n sin α ξ , υ = ( 2 π / λ ) n sin α η ,
x = ξ / h , y = η / h ,
x c = x 0 / ρ , y c = y 0 / ρ ,
ρ = n c sin α c / n 0 sin α 0
B ( u , υ ) = | Φ ( x 0 , y 0 ; u , υ ) | 2 d x 0 d y 0 ,
Φ ( x 0 , y 0 ; u , υ ) = 1 2 π + a ( x x 0 , y y 0 ) f ( x , y ) e i ( u x + v y ) d x d y ,
a ( x , y ) = 1 2 π + A ( u , υ ) e i ( ux + vy ) d u d υ ,
A ( u , υ ) = g 2 + ( g 1 e i Δ g 2 ) g ( u , υ ) ,
g ( u , υ ) = 1 for u 2 + υ 2 σ 2 = 0 otherwise ,
a ( x , y ) = g 2 δ ( x , y ) + ( g 1 e i Δ g 2 ) σ 2 [ J 1 ( σ r 1 ) / σ r 1 ] ,
a ( x x 0 , y y 0 ) = g 2 δ ( x x 0 , y y 0 ) + ( g 1 e i Δ g 2 ) σ 2 [ J 1 ( σ r ¯ 1 ) / σ r ¯ 1 ] ,
r ¯ 1 = [ r 1 2 + r 2 2 r 1 r cos ( φ 1 φ ) ] 1 2 .
δ ( x x 0 , y y 0 ) = 2 π = 0 for x 0 2 + y 0 2 1 otherwise .
Φ ( x 0 , y 0 ; u , υ ) = g 2 e i ( u x 0 + υ y 0 ) + σ 2 ( g 1 e i Δ g 2 ) 2 π + J 1 ( σ r ¯ 1 ) σ r ¯ 1 e i ( u x + υ y ) d x d y ,
Φ ( r , φ ; ω , φ ) = g 2 Φ 1 ( r , φ ; ω , φ ) + 2 ( g 1 e i Δ g 2 ) Φ 2 ( r , φ ; ω , φ ) .
Φ 1 ( r , φ ; ω , φ ) = n = 0 i n n J n ( ω r ) cos n ( φ φ ) for r 1 = 0 otherwise ,
Φ 2 ( r , φ ; ω , φ ) = m = 0 p = 0 1 2 m i ( m 2 p ) m 2 p ( m + 1 ) J m + 1 ( σ r ) r I m + 1 , m 2 p ( σ , ω ) cos [ ( m 2 p ) ( φ φ ) ] ,
I m + 1 , m 2 p ( σ , ω ) = 0 1 J m + 1 ( σ r 1 ) J m 2 p ( ω r 1 ) d r 1 .
Φ ( r , φ ; ω , φ ) = [ g 2 ( Φ 1 R 2 Φ 2 R ) + 2 g 1 cos Δ Φ 2 R 2 g 1 sin Δ Φ 2 I ] + i [ g 2 ( Φ 1 I 2 Φ 2 I ) + 2 g 1 sin Δ Φ 2 R + 2 g 1 cos Δ Φ 2 I ] .
| Φ | 2 = g 2 2 | Φ 1 | 2 + 4 ( g 1 2 2 g 1 g 2 cos Δ + g 2 2 ) | Φ 2 | 2 + 4 ( g 1 g 2 cos Δ g 2 2 ) Re [ Φ 1 Φ 2 * ] + 4 g 1 g 2 sin Δ Im [ Φ 1 Φ 2 * ] .
Φ 1 R Φ 2 R + Φ 2 I Φ 1 I =  Re[ Φ 1 Φ 2 * ] , Φ 2 R Φ 1 I Φ 2 I Φ 1 R =  Im[ Φ 1 Φ 2 * ] .
B ( ω ) = g 2 2 0 ρ 0 2 π | Φ 1 | 2 r d r d φ + 4 ( g 1 2 2 g 1 g 2 cos Δ + g 2 2 ) 0 ρ 0 2 π | Φ 2 | 2 r d r d φ + 4 ( g 1 g 2 cos Δ g 2 2 ) 0 ρ 0 2 π Re [ Φ 1 Φ 2 * ] r d r d φ + 4 g 1 g 2 sin Δ 0 ρ 0 2 π Im [ Φ 1 Φ 2 * ] r d r d φ ,
0 ρ 0 2 π | Φ 1 | 2 r d r d φ = π ρ 2 for ρ < 1 , = π for ρ 1.
0 ρ 0 2 π | Φ 2 | 2 r d r d φ = 2 π m = 0 n = 0 ( m + 1 ) ( n + 1 ) Φ ¯ m + 1 , n + 1 ( σ , ω ) J ( m + 1 , n + 1 ; σ ρ ) ,
Φ ¯ m + 1 , n + 1 ( σ , ω ) = p = 0 1 2 l l 2 p I m + 1 , l 2 p ( σ , ω ) I n + 1 , l 2 p ( σ , ω ) , l = m or n , whichever is smaller ; J ( m + 1 , n + 1 ; σ ρ ) = 0 ρ J m + 1 ( σ r ) J n + 1 ( σ r ) r d r . }
0 ρ 0 2 π Re [ Φ 1 Φ 2 * ] r d r d φ = 2 π m = 0 ( m + 1 ) Φ ¯ m + 1 , m + 1 ( σ ρ , ρ ; ω ) for ρ < 1 , = 2 π m = 0 ( m + 1 ) Φ ¯ m + 1 , m + 1 ( σ , ω ) for ρ 1.
Φ 2 R Φ 1 I r d r d φ Φ 2 I Φ 1 R r d r d φ .
B ( ω ) = π g 2 2 ρ 2 + 8 π ( g 1 2 2 g 1 g 2 cos Δ + g 2 2 ) m = 0 n = 0 ( m + 1 ) ( n + 1 ) Φ ¯ m + 1 , n + 1 ( σ , ω ) J ( m + 1 , n + 1 ; σ ρ ) + 8 π ( g 1 g 2 cos Δ g 2 2 ) m = 0 ( m + 1 ) Φ ¯ m + 1 , m + 1 ( σ ρ , ρ ; ω ) for ρ < 1 ,
= π g 2 2 + 8 π ( g 1 2 2 g 1 g 2 cos Δ + g 2 2 ) m = 0 n = 0 ( m + 1 ) ( n + 1 ) Φ ¯ m + 1 , n + 1 ( σ , ω ) J ( m + 1 , n + 1 ; σ ρ ) + 8 π ( g 1 g 2 cos Δ g 2 2 ) m = 0 ( m + 1 ) Φ ¯ m + 1 , m + 1 ( σ , ω ) for ρ 1.
B ( ω ) = 2 π g 2 2 + 2 π ( g 1 2 2 g 1 g 2 cos Δ + g 2 2 ) σ 2 I 1 , 0 2 ( σ , ω ) + 4 π ( g 1 g 2 cos Δ g 2 2 ) σ I 1 , 0 ( σ , ω ) for ρ 0 ,
= π g 2 2 + 4 π ( g 1 2 2 g 1 g 2 cos Δ + g 2 2 ) m = 0 ( m + 1 ) Φ ¯ m + 1 , m + 1 ( σ , ω ) + 8 π ( g 1 g 2 cos Δ g 2 2 ) m = 0 ( m + 1 ) Φ ¯ m + 1 , m + 1 ( σ , ω ) for ρ .
B ( ω ) = π ρ 2 + 32 π sin 2 ( Δ 2 ) m = 0 n = 0 ( m + 1 ) ( n + 1 ) Φ ¯ m + 1 , n + 1 ( σ , ω ) J ( m + 1 , n + 1 ; σ ρ ) 16 π sin 2 ( Δ 2 ) m = 0 ( m + 1 ) Φ ¯ m + 1 , m + 1 ( σ ρ , ρ ; ω ) for ρ < 1 ,
= π + 32 π sin 2 ( Δ 2 ) m = 0 n = 0 ( m + 1 ) ( n + 1 ) Φ ¯ m + 1 , n + 1 ( σ , ω ) J ( m + 1 , n + 1 ; σ ρ ) 16 π sin 2 ( Δ 2 ) m = 0 ( m + 1 ) Φ ¯ m + 1 , m + 1 ( σ , ω ) for ρ 1 ,
= 2 π + 8 π sin 2 ( Δ 2 ) σ 2 I 1 , 0 2 ( σ , ω ) 8 π sin 2 ( Δ 2 ) σ I 1 , 0 ( σ , ω ) for ρ 0 ,
= π for ρ .
B ( ω ) = 2 π { 1 4 π sin 2 ( Δ / 2 ) σ I 1 , 0 ( σ , ω ) [ 1 σ I 1 , 0 ( σ , ω ) ] } .
B ( ω ) = 1 4 π sin 2 ( Δ / 2 ) u 0 ( 1 u 0 ) ,
B ( ω ) = 1 + 32 ρ 2 sin 2 ( Δ 2 ) p = 1 q = 1 p × q × Φ ¯ p , q ( σ , ω ) J ( p , q ; σ ρ ) 16 ρ 2 sin 2 ( Δ 2 ) p = 1 p × Φ ¯ p , p ( σ ρ , ρ ; ω ) for ρ < 1 ,
= 1 + 32 sin 2 ( Δ 2 ) p = 1 q = 1 p × q × Φ ¯ p , q ( σ , ω ) J ( p , q ; σ ρ ) 16 sin 2 ( Δ 2 ) p = 1 p × Φ ¯ p , p ( σ , ω ) for ρ 1 ,
= 1 + 4 sin 2 ( Δ 2 ) σ 2 I 1 , 0 2 ( σ , ω ) 4 sin 2 ( Δ 2 ) σ I 1 , 0 ( σ , ω ) for ρ 0 ,
= 1 for ρ .
Φ 1 = n = 0 i n n J n ( ω r ) cos n ( φ φ ) = e i ω r cos ( φ φ ) for r 1 = 0 otherwise .
0 ρ 0 2 π | Φ 1 | 2 r d r d φ = 0 ρ 0 2 π r d r d φ = π ρ 2 for ρ < 1 , = π for ρ 1.
Re [ Φ 1 Φ 2 * ] = Re { n = 0 m = 0 i n ( i ) ( m 2 p ) n ( m + 1 ) J n ( ω r ) J m + 1 ( σ r ) r × p = 0 1 2 m m 2 p cos n ( φ φ ) cos [ ( m 2 p ) ( φ φ ) ] I m + 1 , m 2 p ( σ , ω ) } .
Re [ Φ 1 Φ 2 * ] = n = 0 m = 0 ( 1 ) 1 2 n + 3 2 m 3 p n ( m + 1 ) J n ( ω r ) J m + 1 ( σ r ) r × p = 0 1 2 m m 2 p cos n ( φ φ ) cos [ ( m 2 p ) ( φ φ ) ] I m + 1 , m 2 p ( σ , ω ) .
0 ρ 0 2 π Re [ Φ 1 Φ 2 * ] r d r d φ = 2 π m = 0 ( m + 1 ) p = 0 1 2 m m 2 p I m + 1 , m 2 p ( σ , ω ) 0 ρ J m + 1 ( σ r ) J m 2 p ( ω r ) d r .
0 ρ 0 2 π Re [ Φ 1 Φ 2 * ] r d r d φ = 2 π m = 0 ( m + 1 ) Φ ¯ m + 1 , m + 1 ( σ , ω ) ,
Φ ¯ m + 1 , m + 1 ( σ , ω ) = p = 0 1 2 m m 2 p I m + 1 , m 2 p 2 ( σ , ω ) .
I m + 1 , m 2 p ( σ ρ , ρ ; ω ) = 0 ρ J m + 1 ( σ r ) J m 2 p ( ω r ) d r ,
0 ρ 0 2 π Re [ Φ 1 Φ 2 * ] r d r d φ = 2 π m = 0 ( m + 1 ) Φ ¯ m + 1 , m + 1 ( σ ρ , ρ ; ω ) ,
Φ ¯ m + 1 , m + 1 ( σ ρ , ρ ; ω ) = p = 0 1 2 m m 2 p I m + 1 , m 2 p ( σ ρ , ρ ; ω ) I m + 1 , m 2 p ( σ , ω ) .
I s , t ( σ ρ , ρ ; ω ) = 0 ρ J s ( σ r ) J t ( ω r ) d r .
I s , t ( σ ρ , ρ ; ω ) = ρ 0 1 J s ( σ ρ R ) J t ( ω ρ R ) d R .
I s , t ( σ ρ , ρ ; ω ) = I s , t ( σ ρ , ρ ; α ) = ρ α t m = 0 ( 1 α 2 ) m B ¯ s , t ( m , σ ρ ) ,
B ¯ s , t ( m , σ ρ ) = ( σ ρ / 2 ) m m ! n = 0 a s , t ( m , n ) A s , t ( m , n ) , A s , t ( m , n ) = J s + n ( σ ρ ) J t + m + n ( σ ρ ) + J s + n + 1 ( σ ρ ) J t + m + n + 1 ( σ ρ ) ,
a s , t ( m , n ) = ( s + t + 1 ) ( s + t + 2 n 1 ) ( s + t + 2 m + 1 ) ( s + t + 2 m + 2 n + 1 ) ,
a s , t ( m , 0 ) = 1 s + t + 2 m + 1 .
I s , t ( σ ρ , ρ ; 0 ) = lim α 0 I s , t ( σ ρ , ρ ; α ) = ρ m = 0 B ¯ s , 0 ( m , σ ρ ) = ρ I s , 0 ( σ ρ , ρ ; 0 )
I s , t ( σ ρ , ρ ; 1 ) = lim α 1 I s , t ( σ ρ , ρ ; α ) = ρ B ¯ s , t ( 0 , σ ρ ) .
I s , t ( σ ρ , ρ ; ω ) = I s , t ( σ ρ , ρ ; β ) = ρ β s m = 0 ( 1 β 2 ) m B ¯ s , t ( m , ω ρ ) ,
B ¯ s , t ( m , ω ρ ) = ( ω ρ / 2 ) m m ! n = 0 a s , t ( m , n ) A s , t ( m , n ) , A s , t ( m , n ) = J t + n ( ω ρ ) J s + m + n ( ω ρ ) + J t + n + 1 ( ω ρ ) J s + m + n + 1 ( ω ρ ) ,