Abstract

The two-point resolution problem has been discussed previously using the Sparrow criterion. However, this masks the real changes of the image structure as the degree of coherence between the two points changes. The only measurable quantity is, of course, the separation between the two object points. It is found that in all but the incoherent limit the ratio of the real to measured separation is not equal to unity except for specific values of the real separation. Both theoretical and experimental results are presented.

© 1967 Optical Society of America

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References

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  1. Rayleigh, Collected Papers (Cambridge University Press, Cambridge, England, 1902, Vol. 3, p. 84. See also Dover Publications, Inc., New York, 1964).
  2. R. Barakat, J. Opt. Soc. Am. 52, 276 (1962).
    [CrossRef]
  3. M. Born and K. Wolf, Principles of Optics3rd ed. (Pergamon Press Inc., New York, 1965), p. 524.
  4. H. H. Hopkins and P. M. Barham, Proc. Phys. Soc. (London) B63, 737 (1950).
    [CrossRef]
  5. G. Sparrow, Astrophys. J. 44, 76 (1916).
    [CrossRef]
  6. R. Barakat, J. Opt. Soc. Am. 53, 415 (1963).
    [CrossRef]
  7. F. Rojak, M. S. thesis, Lowell Technological Institute, Massachusetts (1961).
  8. M. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall Inc., Englewood Cliffs, New Jersey1964), p. 123.
  9. G. Toralda di Francia, J. Opt. Soc. Am. 45, 497 (1955).
    [CrossRef]

1963 (1)

1962 (1)

1955 (1)

1950 (1)

H. H. Hopkins and P. M. Barham, Proc. Phys. Soc. (London) B63, 737 (1950).
[CrossRef]

1916 (1)

G. Sparrow, Astrophys. J. 44, 76 (1916).
[CrossRef]

Barakat, R.

Barham, P. M.

H. H. Hopkins and P. M. Barham, Proc. Phys. Soc. (London) B63, 737 (1950).
[CrossRef]

Beran, M.

M. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall Inc., Englewood Cliffs, New Jersey1964), p. 123.

Born, M.

M. Born and K. Wolf, Principles of Optics3rd ed. (Pergamon Press Inc., New York, 1965), p. 524.

Hopkins, H. H.

H. H. Hopkins and P. M. Barham, Proc. Phys. Soc. (London) B63, 737 (1950).
[CrossRef]

Parrent, G. B.

M. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall Inc., Englewood Cliffs, New Jersey1964), p. 123.

Rayleigh,

Rayleigh, Collected Papers (Cambridge University Press, Cambridge, England, 1902, Vol. 3, p. 84. See also Dover Publications, Inc., New York, 1964).

Rojak, F.

F. Rojak, M. S. thesis, Lowell Technological Institute, Massachusetts (1961).

Sparrow, G.

G. Sparrow, Astrophys. J. 44, 76 (1916).
[CrossRef]

Toralda di Francia, G.

Wolf, K.

M. Born and K. Wolf, Principles of Optics3rd ed. (Pergamon Press Inc., New York, 1965), p. 524.

Astrophys. J. (1)

G. Sparrow, Astrophys. J. 44, 76 (1916).
[CrossRef]

J. Opt. Soc. Am. (3)

Proc. Phys. Soc. (London) (1)

H. H. Hopkins and P. M. Barham, Proc. Phys. Soc. (London) B63, 737 (1950).
[CrossRef]

Other (4)

M. Born and K. Wolf, Principles of Optics3rd ed. (Pergamon Press Inc., New York, 1965), p. 524.

Rayleigh, Collected Papers (Cambridge University Press, Cambridge, England, 1902, Vol. 3, p. 84. See also Dover Publications, Inc., New York, 1964).

F. Rojak, M. S. thesis, Lowell Technological Institute, Massachusetts (1961).

M. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall Inc., Englewood Cliffs, New Jersey1964), p. 123.

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Figures (8)

Fig. 1
Fig. 1

Two-point image formation.

Fig. 2
Fig. 2

Image illuminance distribution for various values of δ and a fixed a; γ = 0.80. The broken lines represent the separation of the gaussian-image points.

Fig. 3
Fig. 3

Image illuminance distribution for various values of γ from γ = 0 to γ = 1.0 in steps of 0.1 for δ = 3.2.

Fig. 4
Fig. 4

Image illuminance distribution for various values of γ from γ = 0 to γ = 1.0 in steps of 0.1 for δ = 4.0.

Fig. 5
Fig. 5

Image illuminance distribution for various values of γ from γ = 0 to γ = 1.0 in steps of 0.1 for δ = 4.8.

Fig. 6
Fig. 6

The ratio R, the ratio of the measured to the real separation of the two images as a function of the real separation δ for a diffraction-limited circularly symmetric system. A is the separation at the incoherent Sparrow criterion δ = 2.976; B the Rayleigh criterion δ = 3.832; and C the coherent Sparrow criterion δ = 4.600.

Fig. 7
Fig. 7

The ratio R as a function of the actual separation δ for a one-dimensional diffraction limited system. A is the separation at the incoherent Sparrow criterion δ = 2.606; B is the Rayleigh criterion δ = 3.142; and C is the coherent Sparrow criterion δ = 4.164.

Fig. 8
Fig. 8

Comparison between the theoretical (full curve) and the experimentally measured points (⊙) for a diffraction-limited circularly symmetric system; γ = 0.75.

Equations (12)

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incoherent 2 b s = 2.606 q / k a , coherent 2 b s = 4.164 q / k a ,
incoherent 2 b r = 3.142 q / k a . ( Rayleigh )
Sparrow incoherent 2 b s = 2.976 q / k a , coherent 2 b s = 4.600 q / k a , Rayleigh incoherent 2 b r = 3.832 q / k a ,
Γ ob ( x 1 , x 2 ) = I 0 γ ( x 1 , x 2 ) [ δ ( x 1 b ) + δ ( x 1 + b ) + δ ( x 2 b ) + δ ( x 2 + b ) ] ,
Γ im ( x 1 , x 2 ) = ob Γ ob ( x 1 , x 2 ) K ( x 1 q x 1 p ) × K * ( x 2 q x 1 p ) d x 1 d x 2 ,
I im ( x ) = Γ ob ( x 1 , x 2 ) K ( x q x 1 p ) × K * ( x q x 2 p ) d x 1 d x 2 .
I im ( x ) = I 0 ob γ ( x 1 , x 2 ) [ δ ( x 1 b ) + δ ( x 1 + b ) + δ ( x 1 b ) + δ ( x 1 b ) ] × K ( x q x 1 p ) K * ( x q x 2 p ) d x 1 d x 2 .
I im ( x ) = I 0 [ | K ( x b ) | 2 + | K ( x + b ) | 2 + 2 γ ( b , b ) × Re { K ( x b ) K * ( x + b ) } ] ,
K ( x ) = 2 a sinc ( k a x / q ) .
I im ( x ) = 4 a 2 I 0 [ sinc 2 ( ( k a / q ) ( x b ) ) + sinc 2 ( ( k a / q ) ( x + b ) ) + 2 γ ( b , b ) sinc ( ( k a / q ) × ( x b ) ) sinc ( ( k a / q ) ( x + b ) ) ] .
K ( x ¯ ) = 2 a [ 2 J 1 ( k a x / q ) k a x / q ] = 2 a Λ 1 ( k a x q ) ,
I im ( x ) = 4 a 2 I 0 [ Λ 1 2 ( ( k a / q ) ( x b ) ) + Λ 1 2 ( ( k a / q ) ( x + b ) ) + 2 γ ( b , b ) × Λ 1 ( ( k a / q ) ( x b ) ) Λ 1 ( ( k a / q ) ( x b ) ) ] .