Abstract

Power contained in a square area of an image formed by a diffraction-limited imaging system with a centrally obscured circular pupil is calculated and compared with the power contained in a circular area. It is shown that, regardless of the amount of obscuration, the difference between the corresponding ensquared and encircled powers is less than 9% of the total image power. Approximate expressions are obtained for the power lying outside a large square or a circular area.

© 1978 Optical Society of America

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References

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  1. Lord Rayleigh, Philos. Mag. 11, 5 (1881);Scientific Papers (Dover, New York, 1964), Vol. 1, p. 513.
  2. W. T. Welford, J. Opt. Soc. Am. 50, 749 (1960).
    [CrossRef]
  3. A. T. Young, Appl. Opt. 9, 1874 (1970).
    [PubMed]
  4. B. L. Mehta, Appl. Opt. 13, 736 (1974).
    [CrossRef] [PubMed]
  5. H. F. A. Tschunko, Appl. Opt. 13, 1820 (1974).
    [CrossRef] [PubMed]
  6. N. M. Weiner, Opt. Eng. 13, 87 (1974).
    [CrossRef]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 416.
  8. In dimensional units (as opposed to dimensionless units used in the text), if I0 is the irradiance at the pupil, the total power in the image is Pt = πD2 (1 − ∊2)I0/4, and the central irradiance is I(0;∊) = π(1 − ∊2)Pt/4λ2F2 = [πD(1 − ∊2)/4λF]2I0.
  9. This was suggested by the late Ted Edelbaum.
  10. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., New York, 1966), p. 195.

1974 (3)

1970 (1)

1960 (1)

1881 (1)

Lord Rayleigh, Philos. Mag. 11, 5 (1881);Scientific Papers (Dover, New York, 1964), Vol. 1, p. 513.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 416.

Mehta, B. L.

Rayleigh, Lord

Lord Rayleigh, Philos. Mag. 11, 5 (1881);Scientific Papers (Dover, New York, 1964), Vol. 1, p. 513.

Tschunko, H. F. A.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., New York, 1966), p. 195.

Weiner, N. M.

N. M. Weiner, Opt. Eng. 13, 87 (1974).
[CrossRef]

Welford, W. T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 416.

Young, A. T.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

N. M. Weiner, Opt. Eng. 13, 87 (1974).
[CrossRef]

Philos. Mag. (1)

Lord Rayleigh, Philos. Mag. 11, 5 (1881);Scientific Papers (Dover, New York, 1964), Vol. 1, p. 513.

Other (4)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 416.

In dimensional units (as opposed to dimensionless units used in the text), if I0 is the irradiance at the pupil, the total power in the image is Pt = πD2 (1 − ∊2)I0/4, and the central irradiance is I(0;∊) = π(1 − ∊2)Pt/4λ2F2 = [πD(1 − ∊2)/4λF]2I0.

This was suggested by the late Ted Edelbaum.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., New York, 1966), p. 195.

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Figures (5)

Fig. 1
Fig. 1

Angular function θ(r) for a square detector of halfwidth α. θ(r)/4 as shown represents the angle subtended by those detector points in the first quadrant which lie at a distance r from its center.

Fig. 2
Fig. 2

Normalized point-spread function In, ensquared power Ps, encircled power Pc, and 10(PsPc) when = 0. The distance from image center represents r in the case of point-spread function and α in the case of included power, in units of λF.

Fig. 3
Fig. 3

Same as Fig. 2 except that = 0.25.

Fig. 4
Fig. 4

Same as Fig. 2 except that = 0.50.

Fig. 5
Fig. 5

Same as Fig. 2 except that = 0.75.

Tables (1)

Tables Icon

Table I Encircled and Ensquared Powers for a Centrally Obscured Circular Pupil with a Linear Obscuration ratio ot

Equations (19)

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I ( r ; ) = 1 ( 1 2 ) 2 [ 2 J 1 ( π r ) ( π r ) 2 2 J 1 ( π r ) π r ] 2 I ( 0 ; ) ,
I ( 0 ; ) = ( π / 4 ) ( 1 2 ) .
P c ( α ; ) = 2 π 0 α I ( r ; ) r d r = 1 1 2 [ P c ( α ) + 2 P c ( α ) 4 0 1 J 1 ( π α u ) J 1 ( π α u ) d u u ] ,
P c ( α ) = P c ( α ; = 0 ) = 1 J 0 2 ( π α ) J 1 2 ( π α ) .
P s ( α ; ) = det I ( r ; ) r d r d θ ,
P s ( α ; ) = 0 2 α I ( r ; ) θ ( r ) r d r ,
θ ( r ) = 2 π , 0 r α = 2 π 8 cos 1 ( α / r ) , α < r 2 α .
P s ( α ; ) = P c ( 2 α ; ) 8 0 2 α I ( r ; ) cos 1 ( α / r ) r d r = P c ( 2 α ; ) 8 π ( 1 2 ) 1 2 × [ J 1 ( π α u ) J 1 ( π α u ) ] 2 cos 1 ( 1 / u ) d u u .
X = 1 P ,
J 1 ( z ) ( 2 / π z ) 1 / 2 sin ( z π / 4 ) .
I ( r ; ) 8 π 4 r 3 ( 1 2 ) 2 [ sin ( π r π / 4 ) sin ( π r π / 4 ) ] 2 I ( 0 ; ) .
I ¯ ( r ; ) 4 I ( 0 ; ) π 4 r 3 ( 1 2 ) ( 1 ) .
X c ( α ; ) 2 π α I ¯ ( r , ) r d r = 2 π 2 α ( 1 ) .
X s ( α ; ) | x | > α d x | y | > α d y I ¯ ( r , ) = 4 2 π 3 α ( 1 ) ,
X s ( α ; ) = 0.900 X c ( α ; ) .
I n ( r ; ) = I ( r ; ) / I ( 0 ; )
X s ( α ; ; x 0 , y 0 ) | x | > α d x | y | > α d y I ¯ { [ ( x x 0 ) 2 + ( y y 0 ) 1 ] 1 / 2 } = 2 π 3 ( 1 ) { [ ( α x 0 ) 2 + ( α + y 0 ) 2 ] 1 / 2 ( α x 0 ) ( α + y 0 ) + [ ( α + x 0 ) 2 + ( α + y 0 ) 2 ] 1 / 2 ( α + x 0 ) ( α + y 0 ) + [ ( α x 0 ) 2 + ( α y 0 ) 2 ] 1 / 2 ( α x 0 ) ( α y 0 ) + ( α + x 0 ) 2 + ( α y 0 ) 2 ] 1 / 2 ( α + x 0 ) ( α y 0 ) } .
X s ( α ; ; x 0 = y 0 = b ) = 2 2 π 3 ( α 2 b 2 ) ( 1 ) [ α + ( α 2 + b 2 ) 1 / 2 ] .
X s ( α ; ; x 0 = y 0 = 0 ) = 2 π 3 α ( 1 ) × { [ 1 + ( α α b ) 2 ] 1 / 2 + [ 1 + ( α α + b ) 2 ] 1 / 2 } .

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