Imaging Performance of Annular Apertures

Hubert F. A. Tschunko

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

Introduction

The irradiance distribution in images formed byaberration free optical systems with annular aper-tures was first given by Steward' who showed the ir-radiance distribution over the image radius for aper-tures with a central obstruction ratio of 0.5 and 1.0.Steward presented further the expressions for theradii of minima-dark rings-and of the maxima-bright rings. Mehta2 gave the analytic integrationof the irradiance distribution and showed curves forthe total energy over the image radius for central ob-struction ratios of a = 0.25, 0.5, and 0.75. The au-thor3 gave a short aspect of the images formed byapertures of high obstruction ratio and showed in theexample of a = 0.8 that significant modificationsoccur in energy distribution as compared with thefull aperture a = 0.

In this paper images are investigated that areformed by aberration free optical systems with annu-lar apertures in the whole range of central obstruc-tions ratios approaching a = 1.0. The significanttransformation of these images will be shown anddiscussed. Images formed by annular apertures areevaluated by determining the energy distributionand the energy content as a function of the image ra-dius.

Theoretical Analysis

The irradiance distribution of an aberration freeimage formed by an annular aperture with a ratio ofcentral obstruction a is given by the well known ex-pression'

The author is with NASA Goddard Space Flight Center,Greenbelt, Maryland 20771.

Received 2 January 1974.

I(x) = [1/(1 - a2)]2[2J,(x)/x - a22J,(ax)/(ax)]2. (1)

The energy integral is given by the normalized ex-pression

E(x) =

[0.5/(1 - a2)]F [2J,(x)/x - a2 2Ji(ax)/(ax)]2 x dx. (2)

The integration in Eq. (2) yields three main terms:

E = El + El, + El,,;

El = 1 - J 2(x) - J12(x);

El = a2[1 - Jo2(ax) - J 12(ax)];

El,, = - 4a f [Jl(x) J(ax)/x]dx.

(3)

(4)

(5)

(6)

These three expressions, required to determine thetotal energy function, are seldom used in numericalevaluations because four Bessel functions and an in-tegral of the product of two Bessel functions have tobe calculated. If Eq. (2) is used to calculate E, con-siderable time is saved in lieu of using Eqs. (4), (5),and (6).

However, El, El,, and E111 have an obvious mean-ing. Equation (4) represents the integral energy forthe unobstructed aperture of ratio a = 0; Eq. (5)gives the integral energy for an obstructed apertureof radius a; Eq. (6) is equivalent to the differencebetween the square of the amplitude sum and thesum of the amplitude squares. The integral energyfunction indicates which energy fraction is comprisedby an area within a circle of radius x.

For the infinitely thin circular line aperture, a = 1,the image irradiance of Eq. (1) is transmuted' to

I(x) = J02(x). (7)

1820 APPLIED OPTICS / Vol. 13, No. 8 / August 1974

minus third power of the image radius. These enve-lope curves emerge out of the limiting envelope forthe obstruction ratio a = 1.0, i.e., the J 0

2 (x) func-tion.

In Fig. 4, the energy integral (total energy) is plot-ted as expressed by Eq. (2) for the same obstructionratios as in Fig. 3. Here the abcissa scale is thesame as in Fig. 3. For a = 0, Eq. (2) changes to Eq.(4). The ordinate scale shows the energy values.The energy fractions Er outside of radius x are de-creasing with the minus first power of the increasingradius. The ordinate axis shows the values of (1 -Er) in logarithmic scale. This graphical arrange-ment has the advantage of approaching straight lineplots in the upper part of this energy integral func-tion and thereby increases sensitivity to limited pre-cision in the integrations of irradiances and the sub-sequent normalization of total energy content tounity.

Discussion of Results

(1) In Figs. 1, 2, and 3, it is recognizable how thecenter spot, the Airy disk, and its surrounding ringsare transmuting with increasing obstruction ratiointo central ring groups with increasing number ofrings in each group and with increasing radii. Thenumber of rings n in each group is given by

Fig. 1. An unobstructed circular aperture and an annular aper-ture with obstruction ratio a = 0.8 and the images generated by

these apertures.

The corresponding energy expression is meaning-less in this limiting case.

Results

In Fig. 1, an aperture with no obstruction and anaperture with a central obstruction ratio a = 0.8 areshown with the corresponding image photographs.

In Fig. 2, the irradiance distribution [Eq. (1)] foran aperture with an obstruction ratio of a = 0.8 isshown. The irradiances are normalized to be 1.0 atthe center of the image and are plotted in logarith-mic scale over a linear radius scale. The annularaperture forms a central ring group and surroundingring groups, as shown in Fig. 1. In the same Fig. 2,the irradiance distribution for a = 0 is superimposedwith a radius scale so chosen that the first minimum,the radius for the Airy disk, coincides with the radi-us of the central ring group for a = 0.8. Further, theenergy integrals [Eq. (2)] for both parameters arealso shown. The abscissa scale for the energy is lin-ear.

In Fig. 3, the normalized irradiance distributionsfor annular apertures [Eq. (1)] are shown for the fullrange of obstruction ratios, for a = 0, 0.5, 0.8, 0.9,0.95, 0.98, 0.99, and 1.0. Here a logarithmic presen-tation for both axes is used. For a = 0, Eq. (1)simplifies to [2J1 (x)/x]2 . This is the first curve setplotted in the foreground of Fig. 3. Note that theenvelopes of the functions of irradiance for the ringmaxima are straight lines that are inclined with the

0 10 50 X 100

Fig. 2. Normalized irradiances I and integral energy functions Evs image radius x generated by an annular aperture with a radialobstruction ratio a of 0.8 and by an unobstructed aperture withits first minimum normalized to the radius of the central ring

group for a = 0.8.

August 1974 / Vol. 13, No. 8 / APPLIED OPTICS 1821

200 X

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 surround-

ing ring groups are designated mlI, mI1 , ....

n = 2/(1 - a). (8)

This has an obvious physical relationship to theaperture geometry. If we divide the diameter of theaperture by the width of the annulus (the differencebetween outer and inner aperture radius), we obtainthe number of rings n in the ring groups. The ener-gy fraction within the center spot for a = 0 is equalto 0.8378. This fraction grows to 0.9028, the energyfraction in all central ring groups for all higher ob-struction ratios. The very central spot that is com-parable in absolute size to the Airy disk now onlycontains, for the example of a = 0.8, an energy frac-tion equal to 0.18 (Fig. 2). Therefore, the centralring group replaces the Airy disk as the dominantimage element.

In Fig. 3, the radii for the first minima decreasefrom 3.83 for a = 0 to 2.4 for a = 1.0. This is thewell known increased resolution for obstructed aper-tures, which is of some significance only in very lowobstruction ratios a. For higher a, the very centerspot similar to the Airy disk is replaced by the dis-cussed central ring groups.

In Fig. 3, the superimposed normalized irradiancecurves show higher maxima for higher obstructionratios a. However, in the case of the Airy disk, a =

0, the distribution of the irradiance of the centerspot is not the lowest, but the highest. The curveset for each central obstruction a shows for the maxi-ma of the corresponding ring groups decreasing irra-diance values. The envelope curves, m for themaxima of the first surrounding ring groups, are forhigher obstruction ratios a parallel to the J2(x), the

envelope curve of the central ring groups. The sameis valid for the other ring groups with curves M2 ,

M 3 .... This means that as the obscuration ratio in-creases, the maxima of the surrounding ring groupsare decreasing, and the central group becomes rela-tively brighter. For a = 0, the first ring maximum is0.0175; for a = 0.5, the maximum of the first sur-rounding ring group is 0.0022, about 1 order of mag-nitude lower, and for a = 0.95, this maximum of thefirst ring group is already 2 orders of magnitude(0.000175) lower than that for the full aperture a =0.

This must have a favorable effect on the distribu-tion of the total energy over the radius within theimage. This we consider now.

(2) The energy integral curves show oscillations,which are due to the contribution of single rings inthe ring groups. The level stretches in these curvesindicate the minima between the ring groups.

It is significant that for all obstruction ratios, theE values in the level zones are the same; beginningwith Eo = 0.9028, E = 0.9499, E 2 = 0.9664, and soforth. One exception is the E curve for a = 0. HereEo = 0.9028 is not the energy fraction of the centerspot, the Airy disk, but approximately the centerspot plus the first surrounding ring. For the lowestobstruction ratios, the layout of rings and ring radiiis not yet fully regular (i.e., for a = 0 the radius ofthe first dark ring is 3.83 and not r). This is causedby a distribution of the roots in the Bessel functionsthat are approximately r apart for larger arguments.This is already recognizable in Fig. 2, where theminima in the irradiance for a = 0 are not regularity

1822 APPLIED OPTICS / Vol. 13, No. 8 / August 1974

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)].

distributed, while the minima between the ringgroups for a = 0.8 in Fig. 1 and for all higher valuesof a in Fig. 2 and Fig. 3 are regularity spaced. Theseradii are equal to the number of rings multiplied bylr, or

x = 7r 2/(1-a) = 7r n, (9

where n is given by Eq. (8), the number of rings inthe ring groups. Equations (8) and (9) also hold fora = 0, where n = 2 in correspondence with the afore-mentioned exception for E, which comprises thecenter spot plus the first bright ring.

The size of the larger images of the annular aper-tures is partially compensated by the qualitative im-provement in the concentration of the image energyin the central ring group.

(3) As stated in the Introduction, this paper in-vestigates images formed by aberration free opticalsystems.

For nonobstructed apertures, it is well known thatimperfections in the optical elements and operationin unfavorable environments (seeing) creates harm-ful light scattering. This diminishes the image con-trast, reduces the clear appearance of dark rings sur-rounding the Airy disk, and lowers the image qualitybelow the optimally achievable theoretical limit.

This is valid for all aperture shapes including ob-structed annular apertures.

Conclusions

The investigation of the imaging performance ofannular apertures showed that

(1) annular apertures form central and surround-ing ring groups;

(2) the energy fraction in the center spot of0.8378 increases to 0.9028 in the central ring groups;

(3) the very central spot in the central ring groupis no longer decisive, but the central ring group as awhole is the new image element;

(4) the energy fractions in the central and sur-rounding ring groups do not decrease with increasingobstruction ratio, but stay constant for all obstruc-tion ratios and for all fractional energy levels.

(5) images formed by unobstructed apertures arequantitatively superior because they are smallerthan those formed by annular apertures. Imagesformed by annular apertures are qualitatively superi-or because their energy content is more concentratedin the central ring group.

The author is grateful to J. D. Mangus for manystimulating discussions and valuable suggestions.

References

1. G. C. Steward, The Symmetrical Optical System (UniversityPress, Cambridge, 1928).

2. B. L. Mehta, Appl. Opt. 13, 736 (1974).3. H. F. A. Tschunko, Appl. Opt. 13, 22 (1974).

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