Abstract

Encircled energy calculations show the effects of focus errors, spherical aberration, and a central stop on image contrast. The complementary quantity (excluded energy) gives the illumination at the center of a dark spot on an incoherent background and is useful in some photometric situations. A simple formula is given for the approximate calculation of the excluded energy at large distances from the center of the diffraction pattern.

© 1970 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. H. Linfoot, E. Wolf, Proc. Phys. Soc. B66, 145 (1953).
  2. H. F. A. Tschunko, J. Opt. Soc. Amer. 55, 1 (1965).
    [CrossRef]
  3. E. Wolf, Proc. Roy. Soc. A204, 533 (1951).
  4. R. Barakat, A. Newman, J. Opt. Soc. Amer. 53, 1365 (1963).
    [CrossRef]
  5. R. Barakat, M. V. Morello, J. Opt. Soc. Amer. 54, 235 (1964).
    [CrossRef]
  6. I. L. Goldberg, A. W. McCullough, Appl. Opt. 8, 1451 (1969).
    [CrossRef] [PubMed]
  7. A. T. Young, Appl. Opt. 6, 1063 (1967).
    [CrossRef] [PubMed]
  8. G. Lansraux, G. Boivin, Can. J. Phys. 36, 1696 (1958).
    [CrossRef]
  9. R. Barakat, A. Houston, J. Opt. Soc. Amer. 53, 1244 (1963).
    [CrossRef]
  10. R. Barakat, A. Houston, J. Opt. Soc. Amer. 55, 538 (1965).
    [CrossRef]
  11. K. Serkowski, Observatory 87, 259 (1967).
  12. D. L. Fried, J. Opt. Soc. Amer. 56, 1372 (1966).
    [CrossRef]
  13. R. F. Hufnagel, N. R. Stanley, J. Opt. Soc. Amer. 54, 52 (1964).
    [CrossRef]

1969

1967

A. T. Young, Appl. Opt. 6, 1063 (1967).
[CrossRef] [PubMed]

K. Serkowski, Observatory 87, 259 (1967).

1966

D. L. Fried, J. Opt. Soc. Amer. 56, 1372 (1966).
[CrossRef]

1965

H. F. A. Tschunko, J. Opt. Soc. Amer. 55, 1 (1965).
[CrossRef]

R. Barakat, A. Houston, J. Opt. Soc. Amer. 55, 538 (1965).
[CrossRef]

1964

R. Barakat, M. V. Morello, J. Opt. Soc. Amer. 54, 235 (1964).
[CrossRef]

R. F. Hufnagel, N. R. Stanley, J. Opt. Soc. Amer. 54, 52 (1964).
[CrossRef]

1963

R. Barakat, A. Houston, J. Opt. Soc. Amer. 53, 1244 (1963).
[CrossRef]

R. Barakat, A. Newman, J. Opt. Soc. Amer. 53, 1365 (1963).
[CrossRef]

1958

G. Lansraux, G. Boivin, Can. J. Phys. 36, 1696 (1958).
[CrossRef]

1953

E. H. Linfoot, E. Wolf, Proc. Phys. Soc. B66, 145 (1953).

1951

E. Wolf, Proc. Roy. Soc. A204, 533 (1951).

Barakat, R.

R. Barakat, A. Houston, J. Opt. Soc. Amer. 55, 538 (1965).
[CrossRef]

R. Barakat, M. V. Morello, J. Opt. Soc. Amer. 54, 235 (1964).
[CrossRef]

R. Barakat, A. Newman, J. Opt. Soc. Amer. 53, 1365 (1963).
[CrossRef]

R. Barakat, A. Houston, J. Opt. Soc. Amer. 53, 1244 (1963).
[CrossRef]

Boivin, G.

G. Lansraux, G. Boivin, Can. J. Phys. 36, 1696 (1958).
[CrossRef]

Fried, D. L.

D. L. Fried, J. Opt. Soc. Amer. 56, 1372 (1966).
[CrossRef]

Goldberg, I. L.

Houston, A.

R. Barakat, A. Houston, J. Opt. Soc. Amer. 55, 538 (1965).
[CrossRef]

R. Barakat, A. Houston, J. Opt. Soc. Amer. 53, 1244 (1963).
[CrossRef]

Hufnagel, R. F.

R. F. Hufnagel, N. R. Stanley, J. Opt. Soc. Amer. 54, 52 (1964).
[CrossRef]

Lansraux, G.

G. Lansraux, G. Boivin, Can. J. Phys. 36, 1696 (1958).
[CrossRef]

Linfoot, E. H.

E. H. Linfoot, E. Wolf, Proc. Phys. Soc. B66, 145 (1953).

McCullough, A. W.

Morello, M. V.

R. Barakat, M. V. Morello, J. Opt. Soc. Amer. 54, 235 (1964).
[CrossRef]

Newman, A.

R. Barakat, A. Newman, J. Opt. Soc. Amer. 53, 1365 (1963).
[CrossRef]

Serkowski, K.

K. Serkowski, Observatory 87, 259 (1967).

Stanley, N. R.

R. F. Hufnagel, N. R. Stanley, J. Opt. Soc. Amer. 54, 52 (1964).
[CrossRef]

Tschunko, H. F. A.

H. F. A. Tschunko, J. Opt. Soc. Amer. 55, 1 (1965).
[CrossRef]

Wolf, E.

E. H. Linfoot, E. Wolf, Proc. Phys. Soc. B66, 145 (1953).

E. Wolf, Proc. Roy. Soc. A204, 533 (1951).

Young, A. T.

Appl. Opt.

Can. J. Phys.

G. Lansraux, G. Boivin, Can. J. Phys. 36, 1696 (1958).
[CrossRef]

J. Opt. Soc. Amer.

R. Barakat, A. Houston, J. Opt. Soc. Amer. 53, 1244 (1963).
[CrossRef]

R. Barakat, A. Houston, J. Opt. Soc. Amer. 55, 538 (1965).
[CrossRef]

H. F. A. Tschunko, J. Opt. Soc. Amer. 55, 1 (1965).
[CrossRef]

R. Barakat, A. Newman, J. Opt. Soc. Amer. 53, 1365 (1963).
[CrossRef]

R. Barakat, M. V. Morello, J. Opt. Soc. Amer. 54, 235 (1964).
[CrossRef]

D. L. Fried, J. Opt. Soc. Amer. 56, 1372 (1966).
[CrossRef]

R. F. Hufnagel, N. R. Stanley, J. Opt. Soc. Amer. 54, 52 (1964).
[CrossRef]

Observatory

K. Serkowski, Observatory 87, 259 (1967).

Proc. Phys. Soc.

E. H. Linfoot, E. Wolf, Proc. Phys. Soc. B66, 145 (1953).

Proc. Roy. Soc.

E. Wolf, Proc. Roy. Soc. A204, 533 (1951).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Contours of equal excluded energy in the (x,r) plane for perfect circular apertures with obscuration ratio .

Fig. 2
Fig. 2

See caption to Fig. 1.

Fig. 3
Fig. 3

See caption to Fig. 1.

Fig. 4
Fig. 4

Contours of equal excluded energy in the (r,) plane for perfect optical systems and x = 0, Note the rapid growth of excluded energy with ; at a given r, X(r, = 0.5) is twice as large as X(r, = 0). The kinks in the contours indicate the crossings of minima (dark rings) in the focal-plane diffraction pattern. The dependence of the first two minima on is shown by the dashed lines.

Fig. 5
Fig. 5

The universal outer-contour curve for aberration-free systems.

Fig. 6
Fig. 6

Contours of equal excluded energy in the (x,r) plane for a clear aperture with β = 2 rad of spherical aberration.

Fig. 7
Fig. 7

Same as Fig. 6, but for an obscured aperture ( = 0.4).

Fig. 8
Fig. 8

Same as Fig. 6, but with β = 4 rad of spherical aberration.

Fig. 9
Fig. 9

Contours of equal excluded energy in the (x,r) plane for an obscured ( = 0.4) aperture with 4 rad of spherical aberration. Compare with Figs. 68.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

X ( r ) = 1 E ( r ) ,
X ( r ) = 2 / π 2 r 1 / 5 r ,
X ( r ) [ 5 r ( 1 ) ] 1
I ( r / π ) = 4 [ J 1 ( r ) / r 2 J 1 ( r ) / ( r ) ] 2 / ( 1 2 ) 2 ,
Δ x = ( π D 2 / 4 λ F 2 ) Δ F = ( π / 4 λ f 2 ) Δ F ,
E ( W ) = C s = 0 s = 1 d s t = 0 t = s d t T ( s ) T ( t ) cos [ ψ ( s ) ψ ( t ) ] D ( W s , W t ) ,
D ( s , t ) = 4 s t [ s J 1 ( s ) J 0 ( t ) t J 1 ( t ) J 0 ( s ) ] s 2 t 2 ;
ψ ( y ) = α y 2 + β y 4 ,
C = 1 / ( 1 2 ) .

Metrics