Abstract

Images formed by aberration free optical systems with annular apertures are investigated in the whole range of central obstruction ratios. Annular apertures form images with central and surrounding ring groups, and the number of rings in each group is given by the outer diameter of the annular aperture divided by the width of the annulus. The theoretical energy fraction of 0.838 in the Airy disk of the unobstructed aperture increases to 0.903 in the central ring groups. Furthermore it is shown that the energy fractions for the central and surrounding ring groups are constant for all obstruction ratios and in each fractional energy level. The central image spot in the central ring group contains only a small energy fraction and therefore the central ring group as a whole is the effective image element.

© 1974 Optical Society of America

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References

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  1. G. C. Steward, The Symmetrical Optical System (University Press, Cambridge, 1928).
  2. B. L. Mehta, Appl. Opt. 13, 736 (1974).
    [CrossRef] [PubMed]
  3. H. F. A. Tschunko, Appl. Opt. 13, 22 (1974).
    [CrossRef]

1974 (2)

Mehta, B. L.

Steward, G. C.

G. C. Steward, The Symmetrical Optical System (University Press, Cambridge, 1928).

Tschunko, H. F. A.

Appl. Opt. (2)

Other (1)

G. C. Steward, The Symmetrical Optical System (University Press, Cambridge, 1928).

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Figures (4)

Fig. 1
Fig. 1

An unobstructed circular aperture and an annular aperture with obstruction ratio a = 0.8 and the images generated by these apertures.

Fig. 2
Fig. 2

Normalized irradiances I and integral energy functions E vs image radius x generated by an annular aperture with a radial obstruction ratio a of 0.8 and by an unobstructed aperture with its first minimum normalized to the radius of the central ring group for a = 0.8.

Fig. 3
Fig. 3

Normalized irradiances I in images vs image radius x generated by annular apertures with radial obstruction ratios a of 0 to 1. Envelopes of the maxima and group maxima are designated a = …; envelopes of the first, second, and further maxima of the surrounding ring groups are designated mI, mII, ….

Fig. 4
Fig. 4

Integral energy fractions E vs image radius x generated by annular apertures with radial obstruction ratios a of 0 to 0.99. The logarithmic ordinate values are (1 − Er) = E; Er is the fraction of energy outside radius r, thus giving ordinates of E [Eq. (2)].

Equations (9)

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I ( x ) = [ 1 / ( 1 a 2 ) ] 2 [ 2 J 1 ( x ) / x a 2 2 J 1 ( a x ) / ( a x ) ] 2 .
E ( x ) = [ 0.5 / ( 1 a 2 ) ] 0 x [ 2 J 1 ( x ) / x a 2 2 J 1 ( a x ) / ( a x ) ] 2 · x · d x .
E = E I + E II + E III ;
E I = 1 J 0 2 ( x ) J 1 2 ( x ) ;
E II = a 2 [ 1 J 0 2 ( a x ) J 1 2 ( a x ) ] ;
E III = 4 a 0 x [ J 1 ( x ) · J 1 ( a x ) / x ] d x .
I ( x ) = J 0 2 ( x ) .
n = 2 / ( 1 a ) .
x = π · 2 / ( 1 a ) = π · n ,

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