Aperture Configuration and Imaging Performance H. F. A. Tschunko and P. J. Sheehan It is generally accepted for optical apertures that obstructions of small proportions are relatively harm- less for good imaging performance and that larger aperture obstructions should be avoided. This work presents some quantitative data on this subject. Optical simulation was used to measure parameters of images created by different families of aperture configurations. Twenty-four apertures were selected, and the following data given: aperture shape, energy portion within an area equivalent to the Airy disk and to a circle including the third bright ring of the ideal diffraction image, modulation transfer func- tion, and a photograph of the diffraction image. Design criteria are discussed for aperture configurations of large telescope systems. Introduction The imaging performance of a perfect optical system is fixed by the shape of its aperture. Deviations from a circular clear aperture-like an obstruction inserted in the light paths-cause changes in the images. Aperture obstructions may include intrusion of structural parts, the gaps between the segments of a larger mirror, or interspaces between the aperture elements of a syn- thesized aperture. Qualitative experimental investigations of modified images diffracted by different apertures were published in 18941 and more recently. 2 3 It has been assumed generally that obstructions should be avoided. The purpose of the work presented in this paper is to estab- lish some numerical data on the image effects of different aperture configurations. Numerical data on images diffracted by different apertures may be gained by theoretical numerical analysis. This has been done for simple aperture shapes. For more complicated aperture shapes the numerical digital handling by computers becomes less economical. For this reason an optical analog com- puter, an optical experimental simulation, was chosen to determine selected image parameters. Apertures were considered that are not only theoretically inter- esting but also technically significant for the design of modern large telescopes, for instance, apertures with This work was done while the authors were with NASA Elec- tronics Research Center. H. F. A. Tschunko is now with NASA Goddard Space Flight Center, Greenbelt, Maryland 20771, and P. J. Sheehan is at Babson College, Wellesley, Massachusetts 02157. Received 21 October 1970. larger obstructions, apertures composed for several mirror segments, and apertures synthesized by separate smaller apertures. This paper gives first a short review of the image analysis for the simplest cases, then it describes the instrumentation used for the optical simulation, further it presents the resulting data, and finally it discusses the results and derivable design criteria for large telescopes. Image Analysis The image performance of an otherwise ideal optical system is determined by the configuration of the aper- ture. Its effects are recognizable in the image by its energy distribution. To investigate image properties, images of point sources on the optical axes in the focal planes are used. A first indication of the quality of an image is given by the fraction of the whole image energy concentrated in its central image portion. The energy values within different image radii yield further information, i.e., the energy integral curve. For concentrically obstructed apertures, these integral values are given by E = c [2,qi(z')z' - a22,j1(az')/az']2-z'dz' as textbooks show, for instance Ref. 3. The term c is a constant, a designates the ratio of the diameter of the obstructing element to the over-all diameter of the aperture, and z' is the image radius in optical units. Another presentation of the image quality can be given by the modulation transfer function (MTF) presenting the contrast in the images of sinusoidal patterns. For apertures symmetric to axes in preferred directions, the MTF's will be different for different orientations. The discussion of the results will show the significance of different image parameters. 1432 APPLIED OPTICS / Vol. 10, No. 6 / June 1971
Transcript
Aperture Configuration and Imaging Performance
H. F. A. Tschunko and P. J. Sheehan
It is generally accepted for optical apertures that obstructions of small proportions are relatively harm-less for good imaging performance and that larger aperture obstructions should be avoided. This workpresents some quantitative data on this subject. Optical simulation was used to measure parametersof images created by different families of aperture configurations. Twenty-four apertures were selected,and the following data given: aperture shape, energy portion within an area equivalent to the Airydisk and to a circle including the third bright ring of the ideal diffraction image, modulation transfer func-tion, and a photograph of the diffraction image. Design criteria are discussed for aperture configurationsof large telescope systems.
Introduction
The imaging performance of a perfect optical systemis fixed by the shape of its aperture. Deviations from acircular clear aperture-like an obstruction inserted inthe light paths-cause changes in the images. Apertureobstructions may include intrusion of structural parts,the gaps between the segments of a larger mirror, orinterspaces between the aperture elements of a syn-thesized aperture.
Qualitative experimental investigations of modifiedimages diffracted by different apertures were publishedin 18941 and more recently. 2 3 It has been assumedgenerally that obstructions should be avoided. Thepurpose of the work presented in this paper is to estab-lish some numerical data on the image effects of differentaperture configurations.
Numerical data on images diffracted by differentapertures may be gained by theoretical numericalanalysis. This has been done for simple apertureshapes. For more complicated aperture shapes thenumerical digital handling by computers becomes lesseconomical. For this reason an optical analog com-puter, an optical experimental simulation, was chosento determine selected image parameters. Apertureswere considered that are not only theoretically inter-esting but also technically significant for the design ofmodern large telescopes, for instance, apertures with
This work was done while the authors were with NASA Elec-tronics Research Center. H. F. A. Tschunko is now with NASAGoddard Space Flight Center, Greenbelt, Maryland 20771, andP. J. Sheehan is at Babson College, Wellesley, Massachusetts02157.
Received 21 October 1970.
larger obstructions, apertures composed for severalmirror segments, and apertures synthesized by separatesmaller apertures. This paper gives first a shortreview of the image analysis for the simplest cases,then it describes the instrumentation used for theoptical simulation, further it presents the resultingdata, and finally it discusses the results and derivabledesign criteria for large telescopes.
Image Analysis
The image performance of an otherwise ideal opticalsystem is determined by the configuration of the aper-ture. Its effects are recognizable in the image by itsenergy distribution. To investigate image properties,images of point sources on the optical axes in the focalplanes are used. A first indication of the quality of animage is given by the fraction of the whole imageenergy concentrated in its central image portion. Theenergy values within different image radii yield furtherinformation, i.e., the energy integral curve. Forconcentrically obstructed apertures, these integralvalues are given by
E = c [2,qi(z')z' - a22,j1(az')/az']2-z'dz'
as textbooks show, for instance Ref. 3. The term c is aconstant, a designates the ratio of the diameter of theobstructing element to the over-all diameter of theaperture, and z' is the image radius in optical units.Another presentation of the image quality can be givenby the modulation transfer function (MTF) presentingthe contrast in the images of sinusoidal patterns. Forapertures symmetric to axes in preferred directions, theMTF's will be different for different orientations.
The discussion of the results will show the significanceof different image parameters.
1432 APPLIED OPTICS / Vol. 10, No. 6 / June 1971
Experimental Simulation
To investigate such aperture effects on the image, anoptical system of zero aberration has to be simulated inorder to isolate these aberrations caused by the apertureconfiguration. In such a system, a collimating sectionprovides a coaxial monochromatic, coherent, andparallel incoming light bundle with a plane wavefront.This simulates radiation arriving from an infinitely fardistant point source to the imaging portion of the setup.The effective apertures are inserted between the col-limating part and the imaging part. The wave opticaleffects created by the aperture shapes appear in theresulting point images. With this setup, the energyfractions contained within circles of different imageradii can be measured. The modulation transferfunctions can be determined by convolving two identicalapertures in selected directions. The instrumentationused provides a plane wavefront through the apertureplane. The minimum diameter of the aperture masksto be used determines the physical length of the setupshown in Fig. 1.
P.
Fig. 1. Scheme of the optical system for simulation of apertureeffects and energy measurements on diffracted point images.
From the pinhole P to the primary focus F,, the setuprepresents the classical arrangement by Abbe,' 6' which isa modified Tpler setup. The scheme shows the laserbeam L fed through the illuminating optics 0 to thepinhole P. The collimating optics C is coherentlyilluminated by the resulting wavefront. Optic Ccreates the coaxial parallel bundle to the imagingoptics I. The aperture masks are inserted betweenC and I in the aperture plane AP. The primary focalplane of I is in F1 ; 0, is the reimaging optics to magnifythe primary diffracted image of F. into F1 1, the, second-ary focal plane. A pellicle is inserted as a beamsplitter BS. A photometer Po is used to monitor theconstancy of the integral light flux. The other pho-tometer PI serves to measure the flux in different imageareas. The light path between the pinhole P and thereimaging optics should not be folded, to avoid distor-tions in the diffracted image when taking quantitativemeasurements. Pinholes of the smallest availablediameter were used to achieve a constant illuminationacross the aperture plane. The best available lenssystems for C, I and for 0, 0I were chosen to producean optimal energy distribution in the primary and inthe secondary images at F and F, respectively.Optical elements which are not of extreme high quality
yield qualitative images but render the setup uselessfor quantitative measurements of image parameters.
Several methods for manufacturing aperture maskswere considered. Masks produced by chemical millingwith a precision of better than 0.1 mm and a diameterof about 50 mm were found well suited. However, thisis an expensive and especially time-consuming acquisi-tion. To facilitate the mask making, a larger aperturediameter of about 100 mm is advantageous to achievethe needed mask precision. Using flat black kraftpaper offered a less complex and much faster method.The larger diameter proved to be most significant forthe handling of the masks for the MTF measurements.Photographic masks had to be excluded because thetransparent mask carrier in the light beam destroys theintended measurement precision of <0.01 by effects ofreflection, dispersion, and distortion.
For C and I in Fig. 1, the collimating and the imagingunits, respectively, telescope lenses with three ele-ments were chosen. Their aperture diameter of 150mm was stopped down to 100 mm, thus increasing thefocal number FN from 15 to 22.5. The smallest avail-able pinholes with nearly circular shape of 0.005-mmdiam rendered a flux constancy across the apertureplane of about -4 0.005.
The available phototubes and photometers yielded ameasurement precision of not less than zt0.005. Thelaser stability was monitored by the photometer P0, andthe alignment of the pinhole unit 0 + P was monitoredby photometric crosstracks in the aperture plane.Errors in the alignment were checked regularly bycalibrations, using standard apertures with knownimage parameters. Each measurement was repeatedseveral times with several alignments and laser cali-brations. A ratiometer for the signal ratio PI/PO wasfirst used but finally eliminated to exclude additionalinstrumental errors. The achieved precision the MTFand E values is estimated to be equal to or better than0.01.
Aperture Configurations
More than a hundred aperture configurations werescreened for testing, and a number of apertures andaperture families were selected for more detailed in-vestigations. Data on twenty-four apertures arepresented in Tables I to XXIV. Measured were thefollowing image parameters:
(a) the energy fraction Eo within an area equivalentto the Airy disk;
(b) the energy fraction E3 equivalent to an areaincluding the third bright ring of the ideal diffractionimage by aperture 1;
(c) the modulation transfer function values as theflux through a pair of shifted identical aperture maskswith the overlapping aperture portion centered on theoptical axis of the setup.
In this way the mathematical convolution wasphysically realized in different orientations of the aper-tures. These data are presented with the apertureconfiguration and the photographic recording to theimage.
June 1971 / Vol. 10, No. 6 / APPLIED OPTICS 1433
Tables I-XXIV represent the data for apertures 1-24. sD = width of spiders or gaps in aperture divided by aperture diameter.
- n - _
E8 . 0.838
I
E. 0.69
E, 0.87
IV
E,- 0.89
II III
.935
.920
.887
S E E,
008 .79 .89
016 0.57 .86
V VI
I
VII VIII
E0 E.
0.004 0.823 0.942
0.008 0.808 0.932
0.016 0.780 0.912
Ix
1434 APPLIED OPTICS / Vol. 10, No. 6 / June 1971
Tables I-XXIV represent the data for apertures 1-24. s/D = width of spiders or gaps in aperture divided by aperture diameter.
U~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
- E, E,
.000 .81 E, Et~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.0 \ ° .8°° 25 D
11 .004 .81 .92 \ o E,, 073\ E, *.85.008 .78 .87 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~E8 .5
_ 016 72 .83 <t E3 086 \ ET .85
0 If
XV fVI VI
June 1971 / Vol. 10, No. 6 / APPLIED OPTICS 1435
Tables I-XXIV represent the data for apertures 1-24. s/D = width of spiders or gaps in aperture divided by aperture diameter.
XIX
I -- -- ~
0 00 I
XXII
We may distinguish different groups of apertureshapes in the tables. One group, 1 and 2 togetherwith some concentrically obstructed apertures of a =0.5, serves as reference apertures for measurementchecks. A second group, apertures 3 and 4, shows theentrance pupils of a real telescope construction and of aproposed telescope design. The third group, twelveapertures 5-16, shows larger aperture configurationscomposed of smaller segments or mirror elements.The fourth group of apertures, 17-23, of larger over-alldiameter are formed by a number of smaller circularapertures as applied in aperture synthesis. Aperture24 shows the configuration of a photo shutter in an in-termediate position of operation.
The third group (apertures 5-16) was investigatedas aperture families, with different gap widths betweenthe mirror segments as the variable parameter of theseconfigurations. Other families of apertures appear
if other parameters are varied or different parametersare compared in the evaluation: for instance, apertures5 and 6 or 13 and 14, without and with central ob-struction, such as 7 and 8, which show different shapesof the center segment; the same appears in 5 and 9;or apertures with a different number of segments, suchas 5, 13, and 15; 10 and 11; 9 and 12; or again 12 and16, both with a circular outline but with rather differentsubdivision of the aperture area.
Similarly in group four, apertures 17 and 18 or 19and 20 show a comparable type of central obstructionand a varied number of mirror segments as in the pair22 and 23. The following discussion gives more detailabout their optical image effects.
Discussion of ResultsThe data for apertures 1-24 are presented in Tables
I-XXIV. For apertures 6 and 12-15 only small dif-
1436 APPLIED OPTICS / Vol. 10, No. 6 / June 1971
xx XXI
E. *0.10E. 0.18
ER 071
XXIII XIV
I
0.95
Fig. 2. The energy integral values Eo as a function of the ratiogap width to the aperture diameter s/D for selected apertures.
ferences in the MTF values appear and average MTFvalues are given.
A look at the data for apertures 5-16 shows MTFvalues rather close to comparable apertures, that is,to apertures with the same outlines and comparablecentral obstruction. The E0 numbers for the fractionof the energy content in the central portion of the dif-fraction image, however, show lower values. Thesituation with the E3 values is less obvious.
If we arrange the E values as a function of the ratios/D, the width of the gap to the aperture diameter,then as a first approximation, straight lines (Figs. 2and 3) result. If the s values go to zero, for a circularoutline the E values approach the maxima E =0.8378 and E3 = 0.9525, the values for the circularclear open aperture. For s/D larger than 0.005, theapertures with broken outlines show higher E values.
The influence of the shape of the aperture outlines isobvious if we consider the E values of apertures 10 and11, with seven and nineteen hexagons. Here the higherE values belong to 11 with more obstructed area butwith a larger number of differently positioned stretchesin the outline and within the aperture. If these com-posite apertures are bounded by circular outlines,apertures 9 and 12, then again for smallest gap widths,the Eo and E3 values approach the ideal values 0.8378and 0.9525. However, now the circular boundedaperture with the smaller number of elements and withless obstruction at the same s/D, aperture 9, is superiorto 12; the influence of the differences in the outlines iscancelled and the obstructions determine the energyconcentrations. Moreover, the MTF values of thecircular bounded apertures are much smoother andhigher than those with broken outlines.
Another interesting feature is recognizable: aper-tures 5, 13, and 15 with one, two, and three rings of
0.90 \
0.86
Fig. 3. The energy integral values E3 as function of the ratio gapwidth to aperture diameter s/D for selected apertures.
arranged segments show decreasing E values. Toexplain this, we have to remember that in practice theabsolute size of the individual mirror elements will bethe same in all three cases with resulting over-alldiameter in the proportions of 3, 5, and 7, respectively.We further assume that the absolute gap width s is thesame too, then the relative gap width sD with dif-ferent absolute diameters decreases. As a result, thecomparable E values increase in a similar way. InFigs. 2 and 3 the respective lines move together orcoincide. The energy concentrations in the images donot decrease but are the same. Similar geometricratios are valid for apertures 10 and 11, with sevenand nineteen hexagons within a circular outline.
Apertures 5 and 9 or 12 and 13 show circular outlinesbut different subdivisions in annular concentric ar-rangements of segments or in hexagons. The createdimages show different energy concentrations withlower E values for apertures 5 and 13. The differencesare small, less than 0.01 for s/D = 0 to 0.01. However,it is interesting to understand why this occurs. Aper-tures 9 and 12 with the hexagons show higher energyconcentrations E and E than apertures 5 and 13with the concentric annular gaps. The image energy isshifted from larger to smaller image radii and is moreconcentrated in radial directions. Unfortunately theenergy is not evenly distributed within zones of thesame radii but simultaneously shifted and concentratedwithin preferred azimuths. This azimuth concentra-tion produces different and/or less smooth MTF curvesfor different orientations in azimuth.
The group of apertures 17-23 represents larger aper-tures formed by six to nineteen smaller circular aper-tures, as it is used in aperture synthesis. Apertures17 and 18 are related to apertures 5 and 6 or 7 as areapertures 19 and 20 related to 13 and 14. Apertures
June 1971 / Vol. 10, No. 6 / APPLIED OPTICS 1437
s(S
22 and 23 can be considered to lead to the extreme of acentrally obstructed aperture with a radial obstructionratio of 5/7. The MITF curves are rather irregular,representing the effects of the gaps between the smallerapertures synthesizing the over-all aperture. Aperture21 represents one third of the sector of aperture 20and shows a necessarily rather irregular MTF curve.To synthesize aperture 20 by overlapping three 20apertures in proper orientations cannot produce acomparable coherent image.
Irregularly outlined apertures such as 24, formed bydifferent operational positions of a three-sectoredphotographic lens shutter, form a variety of deteri-orated images, represented by widely spread E valuesand MTF values. A more detailed treatment of thiscase is not included in this report.
The energy distributed in the outer portion of thediffracted photographic images can be rendered byproper exposure to appear outstanding. We rememberthat not only the relative energy fraction in the outerimage parts matters, but its distribution, the arrange-ment of its irradiance, the ratio of the irradiancemaxima to the irradiance of an ideal image. Theenergy concentration to smaller image radii means ingeneral the simultaneous formation of bright rays,spots, and halos, too. When we are dealing with areal mirror system, effects due to the limited precisionof the optical imaging surfaces will cause deteriorationsindependent in cause but compounded with the apertureeffects on the image performance. These precision-limited image effects are not covered in this investiga-tion.
This investigation is of significance mainly for largerapertures when improvements in the image performanceof an optical system can be achieved either by increasingthe aperture diameter or by improving the opticalaperture parameters-or both.
Still more detailed and more precise measurement ofdiffracted imaging may result in numerical values forimage qualities for which the presented EO, E3, and theMTF values are elements. Complete E curves overthe image radii in different image azimuths will increasethe values of the derivable numerical figures of theachievable image performances.
Summary
The investigations give numerical data emphasizingqualitative experiences related to imaging effects ofdifferent aperture configurations:
(a) aperture obstructions should be kept to a min-imum to prevent the spreading of more energy intoouter image portions;
(b) obstruction edges should be arranged to lie inno preferred azimuth direction and thus not formingradial energy concentration (rays);
(c) the application of circular outlines representsthe first approach to the ideal image;
(d) for the evaluation of the optical performance,the values of the energy concentration should not beconsidered alone, but together with the modulationtransfer functions;
(e) the modulation transfer functions should besmooth continuous functions with no ups and downsand should have nearly the same function values for allazimuths orientations, as the images and the E datashow;
(f) synthesized apertures are better suited for in-coherent than for coherent imaging, as data in TableXXI show;
(g) large apertures with circular outlines, subdividedby narrow gaps between component mirrors, will per-form optically near perfection.
References1. J. Scheiner and S. Hirayama, Photographische Aufnahmen
