Abstract

The tedious numerical computations associated with the calculation of partially coherent imagery are alleviated by a method which uses dimensionless coordinates and takes advantage of the properties of the Fourier transform. A 1-D periodic object function can model many objects of practical interest, including nonperiodic objects. The properties of a given optical system are described in terms of the transmission cross coefficient. For aberration-free systems with circular pupils, including annular sources (dark-field illumination), the cross coefficient can be calculated analytically. For aberrated or apodized systems, a 1-D approximation can be used. The effect of a convolving slit in the image plane of a scanning microscope can also be included.

© 1978 Optical Society of America

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References

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  1. D. Nyyssonen, Appl. Opt. 16, 2223 (1977).
    [CrossRef] [PubMed]
  2. E. C. Kintner, “Investigations Relating to Optical Imaging on Partially Coherent Light,” Ph.D. Thesis, U. Edinburgh (1975).
  3. E. C. Kintner, R. M. Sillitto, Opt. Acta 24, 591 (1977).
    [CrossRef]
  4. H. H. Hopkins, Proc. Roy. Soc. A 217, 408 (1953); J. Opt. Soc. Am. 47, 508 (1957).
    [CrossRef]
  5. Y. Ichioka, T. Suzuki, J. Opt. Soc. Am. 66, 921 (1976).
    [CrossRef]
  6. D. Nyyssonen, in preparation (1978).
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).
  8. F. S. Acton, Numerical Methods that Work (Harper and Row, New York, 1970), p. 11.
  9. H. H. Hopkins, Proc. Phys. Soc. 79, 889 (1962).
    [CrossRef]

1977 (2)

D. Nyyssonen, Appl. Opt. 16, 2223 (1977).
[CrossRef] [PubMed]

E. C. Kintner, R. M. Sillitto, Opt. Acta 24, 591 (1977).
[CrossRef]

1976 (1)

1962 (1)

H. H. Hopkins, Proc. Phys. Soc. 79, 889 (1962).
[CrossRef]

1953 (1)

H. H. Hopkins, Proc. Roy. Soc. A 217, 408 (1953); J. Opt. Soc. Am. 47, 508 (1957).
[CrossRef]

Acton, F. S.

F. S. Acton, Numerical Methods that Work (Harper and Row, New York, 1970), p. 11.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Hopkins, H. H.

H. H. Hopkins, Proc. Phys. Soc. 79, 889 (1962).
[CrossRef]

H. H. Hopkins, Proc. Roy. Soc. A 217, 408 (1953); J. Opt. Soc. Am. 47, 508 (1957).
[CrossRef]

Ichioka, Y.

Kintner, E. C.

E. C. Kintner, R. M. Sillitto, Opt. Acta 24, 591 (1977).
[CrossRef]

E. C. Kintner, “Investigations Relating to Optical Imaging on Partially Coherent Light,” Ph.D. Thesis, U. Edinburgh (1975).

Nyyssonen, D.

Sillitto, R. M.

E. C. Kintner, R. M. Sillitto, Opt. Acta 24, 591 (1977).
[CrossRef]

Suzuki, T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Opt. Acta (1)

E. C. Kintner, R. M. Sillitto, Opt. Acta 24, 591 (1977).
[CrossRef]

Proc. Phys. Soc. (1)

H. H. Hopkins, Proc. Phys. Soc. 79, 889 (1962).
[CrossRef]

Proc. Roy. Soc. A (1)

H. H. Hopkins, Proc. Roy. Soc. A 217, 408 (1953); J. Opt. Soc. Am. 47, 508 (1957).
[CrossRef]

Other (4)

E. C. Kintner, “Investigations Relating to Optical Imaging on Partially Coherent Light,” Ph.D. Thesis, U. Edinburgh (1975).

D. Nyyssonen, in preparation (1978).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

F. S. Acton, Numerical Methods that Work (Harper and Row, New York, 1970), p. 11.

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Figures (9)

Fig. 1
Fig. 1

(a) Periodic chart, (b) Periodic tri-bar chart.

Fig. 2
Fig. 2

Computation of the cross-coefficient function T12).

Fig. 3
Fig. 3

Computation of the cross coefficient for an annular source.

Fig. 4
Fig. 4

Computing the area of intersection of two circles.

Fig. 5
Fig. 5

Large source envelopes intersection of unit pupils.

Fig. 6
Fig. 6

Area determined by both unit pupils and large source.

Fig. 7
Fig. 7

Area bounded by one unit pupil and large source.

Fig. 8
Fig. 8

Small source overlaps one unit pupil.

Fig. 9
Fig. 9

Small source overlaps intersection of both units pupils.

Equations (52)

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x = [ ( ρ 0 ) / λ ] x ¯ ,
y = [ ( ρ 0 ) / λ ] y ¯ .
ξ = ξ ¯ / ρ 0 ,
η = η ¯ / ρ 0 .
K ( ξ , η ) = K ( x , y ) exp [ 2 π i ( x ξ + y η ) ] d x d y .
I ( ξ , η ) = I ( x , y ) exp [ 2 π i ( x ξ + y η ) ] d x d y ,
I ( x , y ) = I ( ξ , η ) exp [ 2 π i ( x ξ + y η ) ] d ξ d η .
F ( ξ , η ) = F ( x , y ) exp [ 2 π i ( x ξ + y η ) ] d x d y .
J ( ξ , η ) = J ( x , y ) exp [ 2 π i ( x ξ + y η ) ] d x d y
I ( ξ , η ) = T ( ξ + ξ , η + η ; ξ , η ) × F ( ξ + ξ , η + η ) F * ( ξ , η ) d ξ d η ,
T ( ξ 1 , η 1 ; ξ 2 , η 2 ) = J ( ξ , η ) K ( ξ 1 + ξ , η 1 + η ) · K * ( ξ 2 + ξ , η 2 + η ) d ξ d η .
I ( ξ ) = T ( ξ + ξ , ξ ) F ( ξ + ξ ) F * ( ξ ) d ξ ,
T ( ξ 1 , ξ 2 ) = J ( ξ , η ) K ( ξ 1 + ξ , η ) × K * ( ξ 2 + ξ , η ) d ξ d η .
F ( x ) = n = a n cos ( 2 π n ξ p x ) a n = a n ,
F ( ξ ) = n = a n δ ( n ξ p ξ ) .
I ( ξ ) = n = ( a n a o * T ( n ξ p ; 0 ) + n = 1 { a n + n a n * T [ ( n + n ) ξ p ; n ξ p ] + a n n a n * T [ ( n n ) ξ p ; n ξ p ] } ) δ ( n ξ p ξ ) = n = c n δ ( n ξ p ξ ) ( c n = c n ) .
I ( x ) = n = c n cos ( 2 π n ξ p x ) .
cos n x = 2 cos x cos ( n 1 ) x cos ( n 2 ) x . 8
I s ( ξ ) = S ( ξ ) · I ( ξ ) = sin ( π w ξ ) π w ξ · I ( ξ ) ,
I s ( x ) = c 0 + 2 n = 1 c n s n cos ( 2 π n ξ p x ) ,
c n = a n a o * T ( n ξ p ; 0 ) + n = 1 { a n + n a n * T [ ( n + n ) ξ p ; n ξ p ] + a n + n a n * T [ ( n n ) ξ p ; n ξ p ] } ,
s n = sin ( n π w ξ p ) n π w ξ p .
a n = 2 P 0 P / 2 F ( x ) cos ( 2 π n x / P ) d x ,
a o = t b + ( 2 w t / P ) · ( t t t b ) ,
a n = sin ( 2 π n w t / P ) / π n .
a o = t b + ( 6 w t / P ) · ( t t t b ) ,
a n = [ sin ( 2 π n w t / P ) + sin ( 2 π n ( 3 w t + w b ) / P ) sin ( 2 π n ( w t + w b ) / P ) ] · ( t t t b ) / π n .
T ( ξ 1 ; ξ 2 ) = J ( ξ ) K ( ξ 1 + ξ ) K * ( ξ 2 + ξ ) d ξ ,
K ( ξ ) = { exp [ 2 π i Φ ( ξ ) ] | ξ | 1 0 | ξ | > 1 ,
A ( r 1 , r 2 , d ) = A 1 + A 2 = ( r 1 2 θ 1 c d 1 ) + ( r 2 2 θ 2 c d 2 ) = ( r 1 2 θ 1 + r 2 2 θ 2 ) c d .
θ 1 = arccos ( d 2 + r 1 2 r 2 2 ) / 2 d r 1 ; θ 2 = arccos ( d 2 + r 2 2 r 1 2 ) / 2 d r 2 ;
c = r 1 sin θ 1 or c = r 2 sin θ 2 .
ρ s < 0.01 ?
( ξ 1 ξ 2 ) 2 ?
ξ max = ξ 2 + 1 ,
ξ min = ξ 1 1.
ξ min ρ s and ξ max ρ s ?
ρ s < 1 ?
ρ s ξ max and ρ s ξ min ?
T ( ξ 1 ; ξ 2 ) = A [ 1,1 , ( ξ 1 ξ 2 ) ] ( finish ) .
ρ s 2 ( ξ 1 ξ 2 + 1 ) ?
T ( ξ 1 ; ξ 2 ) = A [ 1,1 , ( ξ 1 ξ 2 ) ] + A [ ξ s , 1 , min ( | ξ 1 | , | ξ 2 | ) ] π ( finish ) .
T ( ξ 1 ; ξ 2 ) = A ( ξ s , 1 , max ( | ξ 1 | , | ξ 2 | ) ) ( finish ) .
Condition ( a ) ξ min > ρ s or Condition ( b ) ξ max < ρ s ?
T ( ξ 1 ; ξ 2 ) = π ρ s 2 ( finish ) .
T ( ξ 1 ; ξ 2 ) = A ( 1 , ρ s , | ξ 1 | ) ( finish ) .
T ( ξ 1 ; ξ 2 ) = A ( 1 , ρ s , | ξ 2 | ) ( finish ) .
( ξ 1 ξ 2 + 1 ) ρ s 2 ?
T ( ξ 1 ; ξ 2 ) = A ( 1 , ρ s , | ξ 1 | ) + A ( 1 , ρ s , | ξ 2 | ) π ρ s 2 ( finish ) .
T ( ξ 1 ; ξ 2 ) = A [ 1,1 , ( ξ 1 ξ 2 ) ] ( finish ) .
J ( ξ , η ) δ ( ξ ) δ ( η ) .
T ( ξ 1 , ξ 2 ) = { 1 | ξ 1 | 1 and | ξ 1 | 1 , 0 elsewhere , ( finish ) .

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