Abstract

In viewing systems that employ critical, Kohler, or collimated illumination, the illumination may frequently be characterized by the complex degree of coherence in the form of a first-order Bessel function divided by its argument. The anomalies that occur in the image of a tri-bar target viewed in partially coherent illumination of this form are discussed for a two-dimensional circular diffraction-limited imaging system. The Sparrow criterion, which in this case correlates with experimental measurements of resolving power, is applied to determine the resolution limit. The computed resolution-limit curve for a tri-bar target is shown and compared to the previously published two-point resolution-limit curve for a one-dimensional system and to the limiting values obtained in a two-dimensional system for coherent and incoherent illumination. In the latter case, the full curve is shown as a special case of the application of the Sparrow criterion to the partially coherent imaging equations for a circular two-dimensional system. Experimental confirmation of the calculations is given.

© 1971 Optical Society of America

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References

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  1. D. H. Kelly, Appl. Opt. 4, 435 (1965).
    [CrossRef]
  2. W. N. Charman, Phot. Sci. Eng. 8, 253 (1964).
  3. R. Becherer and G. B. Parrent, J. Opt. Soc. Am. 57, 1479 (1967).
    [CrossRef]
  4. R. Swing and J. Clay, J. Opt. Soc. Am. 57, 1180 (1967).
    [CrossRef]
  5. R. Barakat, Opt. Acta 17, 337 (1970).
    [CrossRef]
  6. A. Offner and J. Meiron, Appl. Opt. 8, 183 (1969).
    [CrossRef] [PubMed]
  7. Rayleigh Lord, Collected Papers (Cambridge U. P., Cambridge, England, 1902), Vol. 3 (also Dover, New York, 1964).
  8. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 524.
  9. H. H. Hopkins and P. M. Barham, Proc. Phys. Soc. (London) B63, 737 (1950).
  10. C. Sparrow, Astrophys. J. 44, 76 (1916).
    [CrossRef]
  11. F. Rojak, M. S. thesis, Lowell Technological Institute, Lowell, Massachusetts, 1961.
  12. M. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N. J., 1964), p. 123.
  13. R. Barakat, J. Opt. Soc. Am. 52, 276 (1962).
    [CrossRef]
  14. R. Barakat, J. Opt. Soc. Am. 53, 415 (1963).
    [CrossRef]
  15. D. N. Grimes and B. J. Thompson, J. Opt. Soc. Am. 57, 1330 (1967).
    [CrossRef] [PubMed]
  16. B. J. Thompson and E. Wolf, J. Opt. Soc. Am. 47, 895 (1957).
    [CrossRef]
  17. On the surface, these values may not seem realistic. However, the ANSI preliminary draft of the standard on MTF of photographic emulsions specifies these values.

1970 (1)

R. Barakat, Opt. Acta 17, 337 (1970).
[CrossRef]

1969 (1)

1967 (3)

1965 (1)

1964 (1)

W. N. Charman, Phot. Sci. Eng. 8, 253 (1964).

1963 (1)

1962 (1)

1957 (1)

1950 (1)

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

1916 (1)

C. 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).

Becherer, R.

Beran, M.

M. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N. J., 1964), p. 123.

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 524.

Charman, W. N.

W. N. Charman, Phot. Sci. Eng. 8, 253 (1964).

Clay, J.

Grimes, D. N.

Hopkins, H. H.

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

Kelly, D. H.

Lord, Rayleigh

Rayleigh Lord, Collected Papers (Cambridge U. P., Cambridge, England, 1902), Vol. 3 (also Dover, New York, 1964).

Meiron, J.

Offner, A.

Parrent, G. B.

R. Becherer and G. B. Parrent, J. Opt. Soc. Am. 57, 1479 (1967).
[CrossRef]

M. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N. J., 1964), p. 123.

Rojak, F.

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

Sparrow, C.

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

Swing, R.

Thompson, B. J.

Wolf, E.

B. J. Thompson and E. Wolf, J. Opt. Soc. Am. 47, 895 (1957).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 524.

Appl. Opt. (2)

Astrophys. J. (1)

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

J. Opt. Soc. Am. (6)

Opt. Acta (1)

R. Barakat, Opt. Acta 17, 337 (1970).
[CrossRef]

Phot. Sci. Eng. (1)

W. N. Charman, Phot. Sci. Eng. 8, 253 (1964).

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

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

Other (5)

On the surface, these values may not seem realistic. However, the ANSI preliminary draft of the standard on MTF of photographic emulsions specifies these values.

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

M. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N. J., 1964), p. 123.

Rayleigh Lord, Collected Papers (Cambridge U. P., Cambridge, England, 1902), Vol. 3 (also Dover, New York, 1964).

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 524.

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Figures (13)

F. 1
F. 1

Coordinates and constants for the image of a tri-bar object.

F. 2
F. 2

Image illuminance distribution for δ equal to 6.0 and varying values of γ(δ).

F. 3
F. 3

Image illuminance distribution for δ equal to 5.4 and varying values of γ(δ).

F. 4
F. 4

Image illuminance distribution for δ equal to 4.6 and varying values of γ(δ).

F. 5
F. 5

Image illuminance distribution for δ equal to 3.8 and varying values of γ(δ).

F. 6
F. 6

Ratio, R, of the apparent to real period of a tri-bar object as a function of the real normalized period for varying values of γ(δ).

F. 7
F. 7

(a) Two-point resolution limit for a one-dimensional imaging system, (b) two-point resolution limit for a two-dimensional circular imaging system, and (c) tri-bar resolution limit for a two-dimensional circular imaging system.

F. 8
F. 8

Illustrating the method of finding the roots of the second derivative of the image illuminance to determine the resolution limit of a tri-bar target using the Sparrow criterion.

F. 9
F. 9

(a) Two-point resolution limit and (b) tri-bar resolution limit as a function of the argument of the coherence function Λ1(αδ).

F. 10
F. 10

The resolution limit of a tri-bar object as a function of the ratio of numerical apertures of condenser and objective for λ ¯ = 550 nm. The numerical aperture of the condenser is shown as the parameter, on each curve.

F. 11
F. 11

Visibility curves for the image of a tri-bar target for indicated values of γ(δ). The modulation-transfer curve of an ideal lens is shown by the heavy solid curve. The incoherent limit, according to Charman,2 is shown by the lighter and partially broken curve.

F. 12
F. 12

Experimental system for tri-bar investigation. Elements are (a) mercury arc, (b) interference filter (λ = 5461 Å), (c) diffuser, (d) pinhole, (e) collimator (f=105 mm), (f) tri-bar target (8.98 cycles/mm), (g) imaging lens, (h) apertures (4–6 mm), and (i) film plane.

F. 13
F. 13

Experimentally measured values of R for γ(δ)=0.72 (●), and nearest calculated curve for γ(δ) = 0.70 from Fig. 6. The solid point indicates an unresolved image.

Equations (13)

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t ( x ) = { 1 when | x | < P / 4 0 P / 4 < | x | < 3 P / 4 1 3 P / 4 < | x | < 5 P / 4 0 elsewhere ,
Γ im ( y 1 , y 2 ) = object Γ ob ( x 1 , x 2 ) K ( y 1 q x 1 p ) × K * ( y 2 q x 2 p ) d x 1 d x 2 ,
γ ( x 1 , x 2 ) = { 2 J 1 [ α ( x 1 x 2 ) ] } / α ( x 1 x 2 ) = Λ 1 [ α ( x 1 x 2 ) ]
Γ ob ( x 1 , x 2 ) = Λ 1 [ α ( x 1 x 2 ) ] t ( x 1 ) t * ( x 2 ) ,
K ( y / q ) = 2 a [ 2 J 1 ( k ¯ a y / q ) ] / ( k ¯ a y / q ) = 2 a Λ 1 ( k ¯ a y / q ) ,
I ( y ) = 4 a 2 Λ 1 [ α ( x 1 x 2 ) ] t ( x 1 ) t * ( x 2 ) Λ 1 [ k ¯ a ( y q x 1 p ) ] × Λ 1 [ k ¯ a ( y q x 2 p ) ] d x 1 d x 2 .
( d 2 / d y 2 ) [ I ( y ) ] y = 0 = 0 ,
d 2 d y 2 [ I ( y ) ] = 4 a 2 Λ 1 [ α ( x 1 x 2 ) ] t ( x 1 ) t * ( x 2 ) d 2 d y 2 × { Λ 1 [ k ¯ a ( y q x 1 p ) ] · Λ 1 [ k ¯ a ( y q x 2 p ) ] } d x 1 d x 2 .
( d / d z ) { [ J 1 ( z ) ] / z } = [ J 2 ( z ) ] / z ( d 2 / d z 2 ) { [ J 1 ( z ) ] / z } = [ J 3 ( z ) ] / z [ J 2 ( z ) ] / z 2 ,
J 1 [ α ( x 1 x 2 ) ] [ α ( x 1 x 2 ) ] t ( x 1 ) t * ( x 2 ) ( J 1 ( k ¯ a x 1 / p ) ( k ¯ a x 1 / p ) · J 3 ( k ¯ a x 2 / p ) ( k ¯ a x 2 / p ) + J 3 ( k ¯ a x 1 / p ) ( k ¯ a x 1 / p ) · J 1 ( k ¯ a x 2 / p ) ( k ¯ a x 2 / p ) J 1 ( k ¯ a x 1 / p ) ( k ¯ a x 1 / p ) · J 2 ( k ¯ a x 2 / p ) ( k ¯ a x 2 / p ) 2 J 1 ( k ¯ a x 2 / p ) ( k ¯ a x 2 / p ) · J 2 ( k ¯ a x 1 / p ) ( k ¯ a x 1 / p ) 2 + 2 J 2 ( k ¯ a x 1 / p ) ( k ¯ a x 1 / p ) · J 2 ( k ¯ a x 2 / p ) ( k a x 2 / p ) ) · d x 1 d x 2 = 0 .
t ( x ) = δ ( x + b ) + δ ( x b )
( 1 + γ ) J 1 ( k ¯ a b / p ) J 3 ( k ¯ a b / p ) ( 1 + γ ) J 1 ( k ¯ a b / p ) J 2 ( k ¯ a b / p ) k ¯ a b / p + ( 1 γ ) [ J 2 ( k ¯ a b / p ) ] 2 = 0 ,
1 γ 1 + γ = [ J 1 ( k ¯ a b / p ) / ( k ¯ a b / p ) ] · J 2 ( k ¯ a b / p ) J 1 ( k ¯ a b / p ) J 3 ( k ¯ a b / p ) [ J 2 ( k ¯ a b / p ) ] 2