Abstract

A resolution criterion due to Sparrow is studied. Unlike the well-known Rayleigh resolution criterion, the Sparrow resolution criterion is dependent upon the coherence properties of the incident light. The extreme limits of complete coherence and complete incoherence are investigated. An apodization scheme for increasing the coherent Sparrow resolution limit is formulated via the calculus of variations. The determination of the required pupil function (amplitude distribution over the exit pupil) necessitates the solution of an inhomogeneous Fredholm integral equation. Both slit and circular aperture are studied in detail.

© 1962 Optical Society of America

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References

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  1. Rayleigh, Collected Papers (Cambridge University Press, Cambridge, 1902), Vol. 3, p. 84.
  2. Reference 1, p. 85.
  3. C. Sparrow, Astrophys. J. 44, 76 (1916).
    [Crossref]
  4. G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).
    [Crossref]
  5. J. Arsac, Optica Acta 3, 114 (1956).
    [Crossref]
  6. R. Barakat (to be published).
  7. E. Wolf, Repts. Progr. Phys. 14, 95 (1951).
    [Crossref]
  8. H. Osterberg and J. Wilkins, J. Opt. Soc. Am. 39, 533 (1949).
  9. H. Osterberg and F. Wissler, J. Opt. Soc. Am. 39, 558 (1949).
    [Crossref]
  10. H. Osterberg, J. Opt. Soc. Am. 40, 295 (1950).
    [Crossref]
  11. R. K. Luneberg, Mathematical Theory of Optics (Brown University, Providence, Rhode Island, 1944), p. 390.
  12. R. Barakat, J. Opt. Soc. Am. 52, 264 (1962), this issue.
    [Crossref]
  13. See reference 11, p. 388.
  14. F. G. Tricomi, Integral Equations (Interscience Publishers, Inc., New York, 1957).

1962 (1)

1956 (1)

J. Arsac, Optica Acta 3, 114 (1956).
[Crossref]

1955 (1)

1951 (1)

E. Wolf, Repts. Progr. Phys. 14, 95 (1951).
[Crossref]

1950 (1)

1949 (2)

H. Osterberg and J. Wilkins, J. Opt. Soc. Am. 39, 533 (1949).

H. Osterberg and F. Wissler, J. Opt. Soc. Am. 39, 558 (1949).
[Crossref]

1916 (1)

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

Arsac, J.

J. Arsac, Optica Acta 3, 114 (1956).
[Crossref]

Barakat, R.

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (Brown University, Providence, Rhode Island, 1944), p. 390.

Osterberg, H.

Rayleigh,

Rayleigh, Collected Papers (Cambridge University Press, Cambridge, 1902), Vol. 3, p. 84.

Sparrow, C.

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

Toraldo di Francia, G.

Tricomi, F. G.

F. G. Tricomi, Integral Equations (Interscience Publishers, Inc., New York, 1957).

Wilkins, J.

H. Osterberg and J. Wilkins, J. Opt. Soc. Am. 39, 533 (1949).

Wissler, F.

Wolf, E.

E. Wolf, Repts. Progr. Phys. 14, 95 (1951).
[Crossref]

Astrophys. J. (1)

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

J. Opt. Soc. Am. (5)

Optica Acta (1)

J. Arsac, Optica Acta 3, 114 (1956).
[Crossref]

Repts. Progr. Phys. (1)

E. Wolf, Repts. Progr. Phys. 14, 95 (1951).
[Crossref]

Other (6)

See reference 11, p. 388.

F. G. Tricomi, Integral Equations (Interscience Publishers, Inc., New York, 1957).

Rayleigh, Collected Papers (Cambridge University Press, Cambridge, 1902), Vol. 3, p. 84.

Reference 1, p. 85.

R. Barakat (to be published).

R. K. Luneberg, Mathematical Theory of Optics (Brown University, Providence, Rhode Island, 1944), p. 390.

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Figures (7)

Fig. 1
Fig. 1

Total illuminance in the Fraunhofer plane of a slit aperture due to two coherent point sources for distances between the point sources corresponding to: 2β=4.4 (under the Sparrow coherent resolution limit); δ0=4.164 (at the Sparrow limit); 2β=4.0 (over the Sparrow limit).

Fig. 2
Fig. 2

Total illuminance in the Fraunhofer plane of a slit aperture due to two incoherent point sources for distances corresponding to: δ0=3.142 (Rayleigh resolution limit); 2β=2.8 (under the Sparrow incoherent resolution limit; δ0=2.606 (at the Sparrow limit); 2β=2.4 (over the Sparrow limit).

Fig. 3
Fig. 3

Normalized pupil function T(n)(δ0,r) for circular aperture.

Fig. 4
Fig. 4

Normalized pupil function T(n)(δ0,x) for slit aperture.

Fig. 5
Fig. 5

Normalized pupil function T(n)(δ0,x) for slit aperture.

Fig. 6
Fig. 6

Total illuminance curves for apodized and nonapodized slit apertures with coherent point sources at distance 2β=δ0=1.2.

Fig. 7
Fig. 7

Total illuminance curves for apodized and nonapodized slit apertures with coherent point sources at distance 2β=δ0≈0.

Tables (3)

Tables Icon

Table I Resolution limits for Airy type objectives.

Tables Icon

Table II Circular-aperture pupil-function parameters.

Tables Icon

Table III Slit-aperture pupil-function parameters.

Equations (58)

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D ( v ) = 0 1 T ( δ 0 , r ) J 0 ( v r ) r d r ,
D ( v ) = - 1 1 T ( δ 0 , x ) e i v x d x .
T = amplitude distribution over exit pupil amplitude distribution over entrance pupil .
I c ( v , 2 β ) = [ D ( v - β ) + D ( v + β ) 2 ] / D ( 0 ) 2
I i ( v , 2 β ) = | D ( v - β ) D ( 0 ) | 2 + | D ( v + β ) D ( 0 ) | 2
I c ( v , 2 β ) = 4 [ J 1 ( v - β ) ( v - β ) + J 1 ( v + β ) ( v + β ) ] 2
I i ( v , 2 β ) = [ 2 J 1 ( v - β ) ( v - β ) ] 2 + [ 2 J 1 ( v + β ) ( v + β ) ] 2 ,
I c ( v , 2 β ) = [ sinc ( v - β ) + sinc ( v + β ) ] 2
I i ( v , 2 β ) = [ sinc ( v - β ) ] 2 + [ sinc ( v + β ) ] 2 ,
sinc x = ( sin x ) / x .
( 2 / v 2 ) I ( v , δ 0 ) = 0             ( v = 0 ) ,
2 v 2 [ D ( v - δ 0 2 ) + D ( v + δ 0 2 ) ] 2 = 0.
2 v 2 I c ( v , δ 0 ) = [ D ( - δ 0 2 ) + D ( δ 0 2 ) ] 2 v 2 D ( δ 0 2 ) + [ D ( - δ 0 2 ) + D ( δ 0 2 ) ] 2 v 2 D ( - δ 0 2 ) = 0.
2 v 2 I c ( v , δ 0 ) = 2 v 2 D ( δ 0 2 ) = 0 ,             ( v = 0 ) .
( δ 0 / 2 ) J 3 ( δ 0 / 2 ) = J 2 ( δ 0 / 2 )
2 sinc ( δ 0 / 2 ) - 2 cos ( δ 0 / 2 ) - ( δ 0 / 2 ) sin ( δ 0 / 2 ) = 0
2 v 2 I i ( v , δ 0 ) = 2 v 2 D ( v , δ 0 ) 2 = 0             ( v = 0 ) .
( δ 0 / 2 ) [ J 2 ( δ 0 / 2 ) ] 2 = J 1 ( δ 0 / 2 ) [ J 2 ( δ 0 / 2 ) - ( δ 0 / 2 ) J 3 ( δ 0 / 2 ) ]
3 sin 2 ( δ 0 / 2 ) - δ 0 sin δ 0 + ( δ 0 / 2 ) 2 cos δ 0 = 0
0 1 [ T ( δ 0 , r ) ] 2 r d r = 1
2 v 2 D ( v , δ 0 ) = 0             ( v = 0 )
D ( 0 , δ 0 ) 2 = | 0 1 T ( δ 0 , r ) r d r | 2
0 1 r 3 T ( δ 0 , r ) J 0 ( δ 0 2 ) r d r = 0.
μ T ( δ 0 , r ) - λ r 2 J 0 ( δ 0 r / 2 ) + 0 1 T ( δ 0 , t ) t d t = 0 ,
T ( δ 0 , r ) = E 1 + E 2 r 2 J 0 ( δ 0 r / 2 ) ,
1 2 E 1 2 + E 2 2 C 2 + 2 E 1 E 2 C 1 = 1 , E 1 C 1 + E 2 E 2 = 0 ,
C 1 = 0 1 r 3 J 0 ( δ 0 r / 2 ) d r = 1 2 0 1 r 3 J 2 ( δ 0 r / 2 ) d r - 1 2 0 1 r 3 J 0 ( δ 0 r / 2 ) d r = 1 2 J 3 ( δ 0 / 2 ) ( δ 0 / 2 ) - 1 2 [ J 1 ( δ 0 / 2 ) ( δ 0 / 2 ) - 2 J 2 ( δ 0 / 2 ) ( δ 0 / 2 ) 2 ]
C 2 = 0 1 r 5 [ J 0 ( δ 0 r / 2 ) ] 2 d r .
E 1 = [ 1 2 - C 1 2 C 2 ] - 1 2 E 2 = - ( C 1 / C 2 ) E 1 .
J 2 ( x ) x 2 / 4 ,             J 1 ( x ) 1             ( x 1 )
T ( δ 0 , r ) E 1 - ( E 2 / 2 ) r 2 - 0 ( r 4 )             ( δ 0 1 ) .
- 1 1 [ T ( δ 0 , x ) ] 2 d x = 1
2 v 2 D ( v , δ 0 ) = 0             ( v = 0 )
D ( 0 , δ 0 ) 2 = | - 1 1 T ( δ 0 , x ) d x |
D ( v , δ 0 ) = 2 0 1 T ( δ 0 , x ) cos v x d x
2 v 2 D ( v , δ 0 ) = - 2 0 1 x 2 T ( δ 0 , x ) cos ( δ 0 2 x ) d x = 0.
0 1 T ( δ 0 , t ) d t + μ T ( δ 0 , x ) - λ x 2 cos ( δ 0 2 x ) = 0 ,
T ( δ 0 , x ) = E 1 + E 2 x 2 cos ( δ 0 / 2 ) x .
E 1 2 + 2 E 1 E 2 C 1 + E 2 2 C 2 = 1 2 E 1 C 1 + E 2 C 2 = 0 ,
C 1 = 0 1 x 2 cos ( δ 0 2 x ) d x C 2 = 0 1 x 4 cos 2 ( δ 0 2 x ) d x .
E 1 = 2 - 1 2 [ 1 - C 1 2 C 2 - 1 ] - 1 2 E 2 = - ( C 1 / C 2 ) E 1 .
I c ( v , δ 0 ) = [ D ( v - δ 0 / 2 ) + D ( v + δ 0 / 2 ) ] 2 [ D ( 0 ) ] 2 ,
D ( v ± δ 0 2 ) = 0 1 { E 1 + E 2 x 2 cos δ 0 2 x } × cos [ ( v ± δ 0 2 ) x ] d x
D ( 0 ) = 0 1 E 1 ( 4.164 ) d x = 1 2 .
I a ( v , δ 0 ) = 2 [ E 1 sinc ( v - δ 0 / 2 ) + E 1 sinc ( v + δ 0 / 2 ) + 2 E 2 S ( v , δ 0 / 2 ) ] 2 ,
S ( v , δ 0 2 ) = 0 1 x 2 cos 2 δ 0 2 x cos v x d x .
4 cos 2 δ 0 2 x cos v x = cos ( v - δ 0 ) + cos ( v + δ 0 ) + 2 cos v x
V = [ 0 1 T ( δ 0 , r ) r d r ] 2 + μ 2 0 1 [ T ( δ 0 , r ) ] 2 r d r - λ 0 1 r 3 T ( δ 0 , r ) J 0 ( δ 0 2 r ) d r = extremum .
T ( δ 0 , r ) + B ( r ) ,
d V / d = 0             ( at = 0 ) .
0 1 T ( δ 0 , t ) d t 0 1 B ( r ) r d r + μ 0 1 T ( δ 0 , r ) B ( r ) r d r - λ 0 1 r 3 B ( r ) J 0 ( δ 0 2 r ) d r = 0 ,
0 1 B ( r ) [ 0 1 T ( δ 0 , t ) t d t + μ T ( δ 0 , r ) - λ r 2 J 0 ( δ 0 2 r ) ] r d r = 0.
0 1 T ( δ 0 , t ) t d t + μ T ( δ 0 , r ) - λ r 2 J 0 ( δ 0 2 r ) = 0.
V = [ 0 1 T ( δ 0 , x ) d x ] 2 + μ 0 1 [ T ( δ 0 , x ) ] 2 d x - 2 λ 0 1 x 2 T ( δ 0 , x ) cos ( δ 0 2 x ) d x = extremum .
0 1 T ( δ 0 , t ) d t + μ T ( δ 0 , x ) - λ x 2 cos ( δ 0 2 x ) = 0.
0 1 J 0 ( β r ) J 0 ( v r ) r d r = β J 1 ( β ) J 0 ( v ) - v J 1 ( v ) J 0 ( β ) β 2 - v 2             ( β v )
C 2 = ( 3 v 4 + 23 v 2 ) J 0 2 ( v ) + 4 v ( 3 v 2 - 23 ) J 0 ( v ) J 1 ( v ) + ( 3 v 4 - 31 v 2 ) 30 v 4
C 2 = 20 J 0 2 ( v ) + 71 J 1 2 ( v ) - 152 J 0 ( v ) J 2 ( v ) + 46 J 1 ( v ) J 3 ( v ) - 124 J 2 2 ( v ) + 23 J 3 2 ( v ) 480