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# Diffraction effects of telescope secondary mirror spiders on various image-quality criteria

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Diffraction from secondary mirror spiders can significantly affect the image quality of optical telescopes; however, these effects vary drastically with the chosen image-quality criterion. Rigorous analytical calculations of these diffraction effects are often unwieldy, and virtually all commercially available optical design and analysis codes that have a diffraction-analysis capability are based on numerical Fourier-transform algorithms that frequently lack an adequate sampling density to model narrow spiders. The effects of spider diffraction on the Strehl ratio (or peak intensity of the diffraction image), full width at half-maximum of the point-spread function, the fractional encircled energy, and the modulation transfer function are discussed in detail. A simple empirical equation is developed that permits accurate engineering calculations of fractional encircled energy for an arbitrary obscuration ratio and spider configuration. Performance predictions are presented parametrically in an attempt to provide insight into this sometimes subtle phenomenon.
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Diffraction effects of telescope secondary
mirror spiders on various image-quality criteria
James E. Harvey and Christ Ftaclas
Diffraction from secondary mirror spiders can signiﬁcantly affect the image quality of optical telescopes;
however, these effects vary drastically with the chosen image-quality criterion. Rigorous analytical
calculations of these diffraction effects are often unwieldy, and virtually all commercially available optical
design and analysis codes that have a diffraction-analysis capability are based on numerical Fourier-
transform algorithms that frequently lack an adequate sampling density to model narrow spiders. The
effects of spider diffraction on the Strehl ratio 1or peak intensity of the diffraction image2, full width at
half-maximum of the point-spread function, the fractional encircled energy, and the modulation transfer
function are discussed in detail. A simple empirical equation is developed that permits accurate
engineering calculations of fractional encircled energy for an arbitrary obscuration ratio and spider
conﬁguration. Performance predictions are presented parametrically in an attempt to provide insight
into this sometimes subtle phenomenon.
Key words: Spider diffraction, telescope diffraction effects, image-quality criteria.
1. Introduction
The qualitative effect of secondary-mirror spiders on
the image-intensity distribution or point-spread func-
tion 1PSF2of an optical telescope is well known to
every amateur astronomer who has observed the
familiar diffraction spikes accompanying star images
as shown in Figs. 11a2and 11c2. These effects can be
important in certain scientiﬁc applications, depend-
ing on the image-quality criterion. However, more
attention seems to have been devoted to this problem
by amateur astronomers than by scientists and engi-
neers. Several discussions concerning the use of
curved spiders to reduce or eliminate these objection-
able diffraction spikes have been reported by amateur
astronomers.1–5 These include a diffractionless
mount that is achieved by merely attaching the
secondary mirror to a section of pipe that is then
attached to the telescope structure. Indeed the re-
sulting star images are devoid of the objectionable
diffraction spikes, as illustrated in Fig. 11b2. Another
amateur astronomer reported on an antidiffraction
mask for a telescope that effectively eliminated the
diffraction spikes at a considerable reduction in collect-
ing area.6Curved spider conﬁgurations similar to
those shown in Fig. 11d2that do not produce prominent
diffraction spikes are also available commercially.7
Clearly diffraction effects that can degrade the image
still exist; however, the azimuthal variations in the
image-intensity distribution have been eliminated.
Whether this results in a superior image depends on
the application and the appropriate image-quality
criterion for that application. Only four references
have been found in the professional technical litera-
ture that deal speciﬁcally with spider diffraction.8–11
Several additional papers were found that deal with
the more general subject of aperture conﬁgurations
but include a peripheral discussion of spider diffrac-
tion effects.12–15 Incidentally, the images shown in
Fig. 1 were actual photographs taken in a laboratory
setup where different masks were illuminated with
laser light. Note the detailed structure in the images.
This structure is also a diffraction 1or interference2
effect produced by the aperture and is discussed
below.
2. Image-Quality Criteria
If visual aesthetics is the chosen image-quality crite-
rion, or if one is trying to observe bright binary stars
visually, the curved spiders that eliminate diffraction
spikes are probably desirable. However, the diffrac-
J. E. Harvey is with the Center for Research and Education in
Optics and Lasers, The University of Central Florida, 12424
Research Parkway, Orlando, Florida 32826. Christ Ftaclas is with
Hughes Danbury Optical Systems, Inc., 100 Wooster Heights Road,
M@S 813, Danbury, Connecticut 06810.
March 1995.
0003-6935@95@286337-1306.00@0. r1995 Optical Society of America. 1 October 1995 @Vol. 34, No. 28 @APPLIED OPTICS 6337 tion spikes caused by conventional 1narrow2spiders do not signiﬁcantly broaden the image core. The full width at half-maximum 1FWHM2of the PSF, an appropriate image-quality criterion when bright point sources are observed, is therefore not degraded by diffraction effects from secondary mirror spiders. This FWHM has been the astronomers’ classical deﬁnition of resolution. Perhaps this explains the apparent lack of concern about this subject as evi- denced by the small number of technical papers in the literature. The complex pupil function describes the amplitude 1aperture shape including obscurations and spider conﬁguration2and phase 1wave-front aberrations2 variations in the exit pupil of the telescope that determine image quality. Wave-front aberrations 1which are neglected in this discussion of diffraction effects2are rendered observable and measured by interferometric techniques. Single-number merit functions derivable from interferometric data include the rms wave-front error and the peak-to-valley wave- front error. The PSF is the squared modulus of the Fourier transform of the complex pupil function as illustrated in Fig. 2.16 The intermediate quantity called the amplitude spread function is not an observable quan- tity with ordinary sensors. Frequently used single- number merit functions 1or image-quality criteria2 obtained from the PSF are the resolution 1FWHM2, the Strehl ratio, and the fractional encircled energy. Fractional encircled energy, or the closely related half-power radius, of the PSF have become common image-quality requirements imposed on telescope manufacturers in recent years. These image-quality criteria are particularly relevant if the telescope is to be used to collect light and place the image on the slit of a spectrographic instrument. The autocorrelation theorem of Fourier-transform theory permits us to deﬁne the optical transfer func- tion 1OTF2as the normalized autocorrelation of the complex pupil function. Various properties of the OTF, or its modulus, the modulation transfer func- tion, may provide more appropriate image-quality criteria if the application involves studying ﬁne detail in extended objects. Limiting resolution and the transfer factor at a speciﬁc spatial frequency are single-number merit functions derivable from the OTF. In the remainder of this paper we deal quantita- tively with diffraction effects of secondary mirror spiders on several different image-quality criteria including the Strehl ratio, the fractional encircled energy, and the modulation transfer function. 3. Strehl Ratio The Strehl ratio, deﬁned as the ratio of the peak irradiance of an aberrated PSF to the peak irradiance of the diffraction-limited PSF, is a commonly used image-quality criterion. A slight modiﬁcation of this deﬁnition 1the diffraction-limited peak irradiance with spiders divided by the diffraction-limited peak irradi- Fig. 1. Diffraction effects of secondary mirror spiders on telescope image quality. Fig. 2. Relationship among the complex pupil function, the PSF, and the OTF. Frequently used image-quality criteria associated with each function are indicated. 6338 APPLIED OPTICS @Vol. 34, No. 28 @1 October 1995 ance without spiders2is appropriate for this study. From the central ordinate theorem of the Fourier- transform theory 1which states that the area of a function is equal to the central ordinate of its Fourier transform217 and the relationships shown in Fig. 2, it is clear that the Strehl ratio can be expressed as Strehl ratio ;S5 1 Aannulus 2Aspiders Aannulus 2 2,112 where Aannulus is the area of the annular aperture without spiders and Aspiders is the total area of all spiders. For any number Nof straight radial spiders of width dDin an annular aperture of linear obscura- tion ratio E, we can write Aannulus 5pD 2 112E 2 2@4, 122 Aspiders 5ND2d112E2@2, 132 S5 3 122Nd p111E2 4 2.142 This expression, and indeed the discussion through- out this paper, requires that the spiders be thin compared with the central obscuration 1d9E2. Figure 3 is a parametric plot of the Strehl ratio for the case of four spiders as a function of the spider width for a variety of obscuration ratios. For an obscuration ratio E50.20 and four spiders of fractional width d50.03, we thus obtain a Strehl ratio of ,0.88 as highlighted in Fig. 3. 4. Fractional Encircled Energy The complex amplitude distribution in the focal plane of a telescope is given by the product of some complex exponentials with the Fourier transform of the com- plex pupil function evaluated at spatial frequencies j5x 2 @lfand h5y 2 @lf. 16 U1x2,y225exp1ikf 2 ilfexp 3 ip1x221y222 lf 4 3F5U11x1,y1260j5x2@lf,h5y2@lf.152 Here k512p2@l,fis the focal length of the telescope, F denotes the Fourier-transform operation, and the complex pupil function is given by U11x1,y125B1x1,y12T11x1@D,y1@D2exp3ikW1x1,y124, 162 where B1x1,y12is the incident amplitude distribution 1ﬁeld strength2,T11x1@D,y1@D2is the aperture function of outer diameter D1including any obscurations of spiders2, and W1x1,y12is the wave-front aberration function18 describing any phase variations in the exit pupil of the telescope. For a uniform amplitude, normally incident plane wave 1no aberrations2, the pupil function is just the constant Btimes the aperture function, U11x1,y125BT11x1@D,y1@D2,172 and the irradiance distribution in the image plane illustrated in Fig. 4 is given by I1x2,y2250U1x2,y2202 5B2 l2f2 0 F 3 T1 1 x1 D,y1 D 240 j5x2@lf,h5y2@lf 0 2.182 By applying the central ordinate theorem of the Fourier transform theory to Eq. 182, we see that the on-axis irradiance in the image plane is given by I10, 025B2 l2f2Aapeture2.192 The normalized irradiance distribution 1normalized to unity at the origin2is thus expressed in dimensionless coordinates x5x2D@lfand y5y2D@lfas In1x,y25I1x2,y22 I10, 02 51 Aaperture2 0 F 5 T1 1 x1 d,y1 D 260 j5x2@lf,h5y2@lf 0 2,1102 where Aaperture is just the area of the aperture. The fractional encircled energy, a commonly used image-quality criterion, is deﬁned as the radiant energy contained in a circle of radius r2divided by the Fig. 3. Parametric plot of the ratio of the peak irradiance in the diffraction-limited PSF produced by an annular aperture of obscu- ration ratio Eand four spiders of width dDdivided by that produced by an annular aperture without spiders. Fig. 4. Diffraction-limited irradiance distribution in the focal plane of a telescope depending on the dimensions of the pupil function in the exit pupil, the focal length of the telescope, the wavelength, and the incident ﬁeld strength. 1 October 1995 @Vol. 34, No. 28 @APPLIED OPTICS 6339 total radiant energy reaching the focal plane: EE1r225 e f50 2p e r50 r I1x2,y22r2dr2df e x250  e y250  I1x2,y22dx2dy2 .1112 Substituting Eq. 182, we see that the denominator of Eq. 1112can be written as e x250  e y250  I1x2,y22dx2dy2 5B2 l2f2 e x250  e y250  0 F 5 T1 1 x1 D,y1 D 260 j5x2@lf,h5y2@lf 0 2 3l 2 f2 djdh,1122 and when Rayleigh’s theorem is applied,17,19 which states that the integral of the squared modulus of a function is equal to the integral of the squared modulus of its Fourier transform 1which corresponds to Parseval’s theorem for the Fourier series2, Eq. 1122 is equal to e x250  e y250  I1x2,y22dx2dy25B2 e x150  e y150  0 T1 1 x1 D,y1 D 20 2 3dx1dy1.1132 But for a binary amplitude aperture that has a transmittance of either unity or zero, 0 T1 1 x1 D,y1 D 20 2 5T1 1 x1 D,y1 D 2 .1142 Hence the denominator of Eq. 1112can be written as e x250  e y250  I1x2,y22dx2dy25B2Aaperture.1152 Now, substituting I1x2,y225B2Aaperture2 l2f2In1x,y21162 into the numerator of Eq. 1112and noting that r5 r2D@lfand dr5dr2D@lfin dimensionless coordi- nates, we obtain the fractional encircled energy in terms of the normalized irradiance distribution: EE1r25Aaperture D2 e f50 2p e r50 r In1x,y2rdrdf.1172 The difficulty comes in obtaining the Fourier trans- form of the aperture function in Eq. 1102for arbitrary spider conﬁgurations. There are two choices. The Fourier transform of the aperture function can be obtained analytically or numerically. In this day of fast and inexpensive computers and many commer- cially available optical-analysis codes, the numerical approach seems to be the obvious choice. However, virtually all the commercially available optical-design Fig. 5. Diffraction-limited PSF for an annular aperture with a narrow opaque strut consisting of two parts: an image core and a diffraction ﬂare perpendicular to the strut. Fig. 6. Encircled energy caused by the diffraction-limited annular apertures. This ﬁgure can be used as a set of characteristic curves from which to obtain values of EEannulus1r2, which are necessary when the empirical equation is applied to various aperture conﬁgurations. 6340 APPLIED OPTICS @Vol. 34, No. 28 @1 October 1995 and -analysis codes that do have a diffraction- analysis capability are based on numerical Fourier- transform algorithms that lack adequate sampling density to model narrow spiders. There must be several numerical samples across the width of the spider if the code is to predict In1x,y2 accurately. Suppose we require three samples across a spider whose width is 1% of the aperture diameter. This implies 300 samples across the aperture. A 50% guard band to prevent aliasing would then require at least a 600 3600 numerical array. This is larger than can be conveniently handled by many computers. Also, the normalization must be performed properly. Fractional encircled energy is just that, a fraction. It has a numerator and a denominator. It is not sufficient to calculate the numerator accurately. One must then divide by the proper denominator. The denominator is merely the total energy reaching the focal plane. One commercially available code merely adds up the energy in the ﬁrst seven Airy rings and divides by that value.20 This technique is ad- equate for clear annular apertures without a high- spatial-frequency structure; however, if a small per- cent of the energy is diffracted into wide angles by narrow struts or spiders, erroneously encircled en- ergy predictions result. The problems above can lead to one of several unfortunate situations: 1a2The user may ignore spi- der diffraction altogether and thus impose image- quality requirements that are physically unobtainable. 1b2The user may think the diffraction effects of spiders are being accurately modeled when in fact they are not. 1c2The user may be aware of the sampling problem and thus try to model the effects of spiders by merely increasing the central obscuration to account for the additional obscuration caused by the spiders. It will presently become clear why this latter situa- tion is not a good approximation to actually modeling the diffraction behavior of narrow spiders. These problems led us to develop a code that employs a hybrid technique for making encircled energy predictions in the presence of narrow second- ary mirror spiders. For any number of straight spiders lying along the radius of the aperture, we obtain the Fourier transform of the pupil function analytically to obtain the normalized irradiance distri- bution. We then perform a numerical integration Table 1. Fractional Encircled Energy from Diffraction-Limited Annular Apertures Radius 1r2D@lf2 Fractional Encircled Energy E50.0 E50.1 E50.2 E50.3 E50.4 E50.5 0.00 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.20 0.1031 0.1020 0.0988 0.0934 0.0858 0.0763 0.40 0.3381 0.3340 0.3218 0.3018 0.2746 0.2406 0.60 0.5587 0.5801 0.5546 0.5136 0.4588 0.3930 0.80 0.7592 0.7455 0.7055 0.6424 0.5612 0.4677 1.00 0.8281 0.8103 0.7592 0.6809 0.5843 0.4787 1.20 0.8379 0.8185 0.7639 0.6835 0.5891 0.4913 1.40 0.8424 0.8245 0.7755 0.7069 0.6309 0.5544 1.60 0.8638 0.8497 0.8129 0.7650 0.7146 0.6594 1.80 0.8909 0.8815 0.8587 0.8324 0.8037 0.7586 2.00 0.9069 0.9011 0.8894 0.8789 0.8620 0.8143 2.20 0.9100 0.9058 0.8996 0.8971 0.8831 0.8283 2.40 0.9112 0.9065 0.9004 0.8995 0.8853 0.8296 2.60 0.9186 0.9116 0.9022 0.9002 0.8875 0.8417 2.80 0.9293 0.9195 0.9058 0.9027 0.8944 0.8654 3.00 0.9362 0.9242 0.9075 0.9041 0.9006 0.8876 3.20 0.9377 0.9249 0.9078 0.9043 0.9027 0.8990 3.40 0.9383 0.9263 0.9116 0.9072 0.9028 0.9015 3.60 0.9240 0.9322 0.9215 0.9146 0.9041 0.9016 3.80 0.9476 0.9406 0.9342 0.9233 0.9060 0.9025 4.00 0.9515 0.9468 0.9435 0.9284 0.9067 0.9032 4.20 0.9524 0.9489 0.9471 0.9294 0.9071 0.9033 4.40 0.9527 0.9490 0.9474 0.9301 0.9109 0.9042 4.60 0.9549 0.9500 0.9480 0.9342 0.9196 0.9067 4.80 0.9584 0.9518 0.9496 0.9408 0.9300 0.9092 5.00 0.9609 0.9530 0.9507 0.9465 0.9371 0.9100 Fig. 7. Fractional encircled energy resulting from diffraction-limited narrow rectangular apertures. This ﬁgure can be used as a set of characteristic curves from which to obtain values of EErect1r2, which are necessary when the empirical equation is applied to various aperture conﬁgurations. 1 October 1995 @Vol. 34, No. 28 @APPLIED OPTICS 6341 over the desired circle to obtain the fractional en- circled energy. Particular attention was given to the problem of normalization. This hybrid approach pro- vides accurate results for arbitrarily narrow spiders. We also developed an intuitive empirical equation for the encircled energy based on Babinet’s principle and Rayleigh’s theorem. This empirical equation was then validated by comparison with rigorous solutions and used to study the parametric behavior of encircled energy for arbitrary spider conﬁgura- tions. A. Empirical Equation Describing Spider-Diffraction Effects The empirical equation is extremely simple. Given an annular aperture with a narrow opaque strut of rectangular cross section as shown in Fig. 5, the image consists of two parts. Part Ais the image core that is just the diffraction-limited PSF caused by the annual aperture 1diminished somewhat by the pres- ence of the strut2, and Part B is the diffraction ﬂare caused by the strut. We ignore, for the moment, the interference effects that occur in the region in which these two functions overlap. A consequence of Babinet’s principle is that the Fraunhofer diffraction patterns of complementary apertures are identical except in the neighborhood of the center of the diffraction pattern.21–23 Hence we know that the functional form of the diffraction pattern caused by the narrow strut 1referred to as the diffraction ﬂare2is identical to the Fraunhofer diffrac- tion pattern caused by a thin rectangular aperture of width band length a, i.e., sinc21x2b@lf2sinc21y2a@lf2. Furthermore the total energy contained in the ﬂare 1Part B2is proportional to area ab of the strut 1Ray- leigh’s theorem applied to the complementary rectan- gular aperture2. Clearly the presence of the strut also decreases the total energy passing through the aperture 1by an amount proportional to its area, ab2, and again, by Rayleigh’s theorem, the total energy in the composite diffraction pattern is reduced by an equal amount. If we ignore the interference effects that occur in the region in which these two functions overlap Table 2. Fractional Encircled Energy from Narrow Rectangular Apertures Radius 1r2D@lf2 Fractional Encircled Energy d50.01 d50.02 d50.05 d50.10 d50.20 d50.50 0.00 0.0003 0.0006 0.0015 0.0029 0.0052 0.0050 0.20 0.0043 0.0086 0.0215 0.0429 0.0851 0.2027 0.40 0.0083 0.0166 0.0415 0.0828 0.1641 0.3874 0.60 0.0123 0.0246 0.0614 0.1225 0.2414 0.5485 0.80 0.0163 0.0326 0.0813 0.01618 0.3163 0.6789 1.00 0.0203 0.0406 0.1012 0.2007 0.3880 0.7757 1.20 0.0243 0.0485 0.1210 0.2391 0.4559 0.8405 1.40 0.0283 0.0566 0.1407 0.2769 0.5195 0.8783 1.60 0.0323 0.0645 0.1604 0.3140 0.5783 0.8963 1.80 0.0363 0.0725 0.1799 0.3504 0.6320 0.9021 2.00 0.0403 0.0805 0.1993 0.3858 0.6804 0.9026 2.20 0.0443 0.0884 0.2186 0.4203 0.7234 0.9034 2.40 0.0483 0.0964 0.2377 0.4538 0.7609 0.9067 2.60 0.0523 0.1043 0.2567 0.4862 0.7933 0.9132 2.80 0.0563 0.1122 0.2755 0.5174 0.8205 0.9220 3.00 0.0602 0.1201 0.2942 0.5475 0.8430 0.9313 3.20 0.0642 0.1280 0.3126 0.5763 0.8612 0.9395 3.40 0.0682 0.1359 0.3309 0.6038 0.8754 0.9453 3.60 0.0722 0.1438 0.3490 0.6301 0.8861 0.9486 3.80 0.0762 0.1516 0.3668 0.6550 0.8940 0.9498 4.00 0.0802 0.1595 0.3844 0.6785 0.8990 0.9499 4.20 0.0841 0.1673 0.4018 0.7007 0.9028 0.9501 4.40 0.0881 0.1751 0.4189 0.7216 0.9047 0.9510 4.60 0.0921 0.1829 0.4358 0.7411 0.9057 0.9531 4.80 0.8096 0.1190 0.4524 0.7592 0.9060 0.9560 5.00 0.1000 0.1984 0.4688 0.7760 0.9061 0.9592 Fig. 8. 1a2Annular aperture with a 0.7 obscuration ratio and four spiders whose width is 3% of the aperture diameter. 1b2Correspond- ing PSF clearly showing complex interference effects. 1c2Three- dimensional isometric plot of In1x,y2on a log scale when deter- mined by the ﬁrst step of our hybrid approach to making rigorous calculations. Fig. 9. Comparison of predictions from an empirical equation 1continuous curve2with rigorous calculations 1bold dots2indicating a less than 0.5% error. Table 3. Validation of Empirical Equation prExact Approx. % Error 5 0.4593 0.4595 0.04 10 0.7306 0.7345 0.53 15 0.8450 0.8470 0.24 20 0.8747 0.8776 0.33 25 0.8820 0.8849 0.33 30 0.9013 0.9043 0.33 35 0.9258 0.9286 0.30 40 0.9333 0.9359 0.28 45 0.9365 0.9390 0.27 50 0.9446 0.9470 0.25 55 0.9522 0.9544 0.23 6342 APPLIED OPTICS @Vol. 34, No. 28 @1 October 1995 1intuition tells us that the integration performed in calculating the encircled energy renders it insensitive to the exact nature of the interference pattern2, the fractional encircled energy resulting from such an aperture can be written as EE1r2 5 e f50 2p e r50 r Part A rdrdf1 e f50 2p e r50 r Part B rdrdf e x250  e y250  1Part A 1Part B2dxdy . 1182 Because the presence of the strut both blocks energy from passing through the aperture and dif- fracts an equal amount out of the image core, we subtract an amount out of the core that is equal to twice the area of the strut, then add back in that small portion of the ﬂare that lies inside the circle of interest and divide this numerator by the total energy that has been diminished by the area of the strut. When additional spiders are present, Part B becomes the sum of several diffraction ﬂares; however, its fractional encircled energy does not change provided the spiders all have the same width. Because we have deﬁned Aannulus as the area of the annulus and Aspiders as the area of all the spiders, we can merely write EE1r2 51Aannulus 22Aspiders2EEannulus1r21AspidersEErect1r2 Aannulus 2Aspiders , 1192 where EEannulus1r2and EErect1r2are the fractional en- circled energy caused by an annular aperture and a narrow rectangular aperture, respectively. When the annular aperture function shown in Fig. 61a2is substituted into Eq. 1102, the normalized irradi- ance distribution in the focal plane is shown to have the well-known analytic solution24 In1x,y251 112E222 3 2J11pr2 pr2E22J11Epr2 Epr 4 ,1202 where Eis the linear obscuration ratio. The frac- tional encircled energy EEannulus1r2is thus easily calcu- lated. Figure 61b2illustrates this quantity graphi- cally for a variety of obscuration ratios as a function of radius 1in units of lf@D2. Table 1 provides the tabu- lated data making up the graph. Values of EEannulus1r2 for any obscuration ratio can be obtained from this graph or table 1directly or by interpolation2when the empirical equation is used to make encircled energy predictions in the presence of secondary mirror spi- ders. Similarly, when the rectangular aperture function shown in Fig. 71a2is substituted into Eq. 1102, the normalized irradiance distribution in the focal plane has the analytic solution In1x,y25sinc21dx2sinc21by2,1212 Fig. 11. Parametric plot of fractional encircled energy versus circle radius for different spider widths. Table 4. Encircled Energy for Different Wavelengths and Spider Widths l1µm2 Encircled Energy in 25-µm Radius d50.0000 d50.0082 d50.0204 0.55 0.9350 0.9219 0.9069 0.65 0.9115 0.8994 0.8842 0.75 0.9083 0.8960 0.8803 Fig. 10. Aperture conﬁguration with small obscuration and three narrow spiders producing a relatively mild scalloping of the ﬁrst few rings of the PSF. 1 October 1995 @Vol. 34, No. 28 @APPLIED OPTICS 6343 where x5x2D@lf,y5y2D@lf, and the fractional encircled energy from a narrow rectangular aperture 1or opaque spider2is given by EErect1r25bd e f50 2p e r50 r sinc21dx2sinc21by2rdrdf,1222 where a5bDand b5dDare the length and width of the rectangle, respectively. When d9b, this calcula- tion can be approximated by the one-dimensional integral EErect1r2<bd e x52r r sinc21dx2sinc21by2dx.1232 This quantity is plotted in Fig. 71b2for a variety of spider widths as a function of radius 1in units of lf@D2. Table 2 provides the corresponding tabulated data. Figures 6 and 7 can now be used as characteristic design or analysis curves for predicting encircled energy in the presence of secondary-mirror spiders by picking values of EEannulus1r2and EErect1r2from these curves and plugging them into the empirical equation discussed above. Alternatively, Tables 1 and 2 can be used as look-up tables for the quantities EEannulus1r2 and EErect1r2when the empirical equation to make encircled energy predictions in the presence of second- ary mirror spiders is used. B. Validation of the Empirical Equation Because our empirical equation for encircled energy ignores the interference between the image core and the diffraction ﬂares, we chose an example where these effects are quite pronounced to compare the resulting predictions with rigorous calculations. Figure 8 shows a rather highly obscured 1E50.72 annular aperture with four spiders whose width is 3% of the aperture diameter 1d50.032and a photograph of the resulting PSF that clearly shows the rather complex interference effects produced by these four subapertures.25 Also illustrated in Fig. 8 is a three- dimensional isometric plot of In1x,y2on a log scale as determined by the ﬁrst step of our hybrid approach to making rigorous calculations. A careful comparison of this isometric plot and the photograph of the PSF indicates excellent qualitative agreement in detail. The continuous curve in Fig. 9 is the fractional encircled energy prediction from our simple empirical equation given by Eq. 1192. The superimposed bold dots represent the rigorous calculations. Note from the tabulated data in Table 3 that the difference between the two methods is less than 0.5%. This is remarkably accurate for a method that ignores the prominent interference effects present in this example. However, the nature of the interference is such that it redistributes energy azimuthally but not radially. Hence, although integration of the diffraction pattern over a circle averages the azimuthal structure in the intensity distribution, it does not change its radially integrated value. C. Parametric Encircled Energy Plots For a much smaller obscuration ratio and three narrow spiders, the interference is less pronounced and the effects of spider diffraction show up as a relatively mild scalloping of the ﬁrst few rings of the irradiance distribution in the focal plane as illus- trated in Fig. 10. For a 12.25- in.- 131-cm-2diam aperture with an obscuration ratio of 0.1735 and a 150-in. 1381-cm2 focal length, fractional encircled energy was predicted with the empirical equation and illustrated parametri- cally in Fig. 11 as a function of circle radius 1in units of lf@D2for spiders of three different widths. Note that for narrow spiders and modest circle sizes the curves run essentially parallel to one another. This parametric representation is particularly useful be- cause one family of curves includes data for all wavelengths and circle sizes. For example, if one had a requirement for encircled energy in a given sized circle 1i.e., 25-µm radius2for several different Fig. 12. Variations in the nature of the diffraction ﬂares with spider width. Fig. 13. Corresponding fractional encircled energy curves provid- ing insight into the image-degradation effects of secondary mirror spiders of varying widths. 6344 APPLIED OPTICS @Vol. 34, No. 28 @1 October 1995 wavelengths, the appropriate data points can be extracted from the graph and tabulated as shown in Table 4. Figure 12 shows variations in the nature of the diffraction ﬂares with spider width. Note the very broad central lobe of the sinc2function for very narrow spiders. This central lobe becomes narrower as the spider width increases, and considerable struc- ture 1many lobes2becomes evident for very wide spiders. Because more light is diffracted into smaller angles by the wide spiders, the corresponding encircled energy curves shown in Fig. 13 do not remain parallel and can in fact cross one another. For example, if we compare the encircled energy of an annular aperture 1E50.32with four spiders that are narrow 1d50.012, wide 1d50.082, and very wide 1d50.152, we see that the curve for the very wide spiders crosses that of the wide spiders at a radius of ,2.6 lf@Dand crosses the curve for the narrow spiders at a radius of ,11 lf@D. There are also other interference effects apparent on close inspection of the diffraction patterns in Fig. 12. The double lines making up each lobe of the diffraction ﬂare are real interference effects in the diffraction-limited PSF 1not artifacts of the optics, detector, or ﬁlm used to record the images2. They are in fact Young’s interference fringes produced by the two halves of the aperture separated by the spiders parallel to that diffraction ﬂare. D. Arbitrary Spider Shapes Because our empirical equation for fractional en- circled energy depends only on the spider width and the total area of the spiders, it can also be used with conﬁdence for arbitrary spider shapes. Richter10 has discussed in some detail how a curved spider produces a searchlight effect 1subtended angle 5angle of arc2 rather than the narrow diffraction ﬂare produced by a straight spider. This is intuitive if one thinks merely of the curved spider as a set of straight segments, each of which produces a narrow diffraction ﬂare perpendicular to the segment as illustrated in Fig. 14. As discussed above, the length of the ﬂare depends only on the width of the spider, and because the fractional encircled energy of the individual ﬂares are the same regardless of their orientation, the shape of a spider segment does not affect the encircled energy. Changing the orientation of a spider segment redistrib- utes the diffracted energy only azimuthally, not radi- ally. However, as discussed above, the total dif- fracted energy is proportional to the spider area and therefore its length. If several curved spiders are designed to produce adjacent searchlight beams that neither have gaps nor overlap, presumably there would be no azimuthal variations in the resulting telescope PSF. Either three or four curved spiders produce this result if they can be combined to yield a semicircle and their relative orientation is correct. The single spider made from a section of pipe referred to above as a diffractionless mount and illustrated in Fig. 1 has a similar effect; however, the searchlight fans overlap by exactly 360 deg. Thermal, structural, or dynamic considerations may determine the spider design or conﬁguration for a given application; however, if fractional encircled energy is the image-quality requirement, the empiri- Fig. 14. Curved spider producing two searchlight beams emanat- ing in opposite directions from the image core. This is intuitive if one approximates the curve as a set of straight segments. Fig. 15. Fractional encircled energy curves compared for a variety of spider conﬁgurations. The spider width is held constant. 1 October 1995 @Vol. 34, No. 28 @APPLIED OPTICS 6345 cal equation presented in this paper can be used with conﬁdence to compare the performance of candidate designs. The family of curves in Fig. 15 shows the encircled energy produced by the seven spider conﬁgu- rations illustrated. 1Spider conﬁguration 6 has no structural or practical merit and is included here only to emphasize that the image-quality degradation of truly arbitrary spider shapes can be evaluated by our empirical equation.2The width of the spiders and the central obscurations have been held constant for all conﬁgurations; hence the fractional encircled en- ergy decreases with increasing total spider length. Clark et al.12 have presented an asymptotic approxi- mation to the encircled energy for arbitrary aperture shapes that agrees well with rigorous calculations for the larger circle sizes. Figure 16 graphically shows a few discrete data points providing a detailed compari- son between this asymptotic approximation, our em- pirical formula, and a rigorous calculation for an obscuration ratio E50.5 and four spiders of fractional width d50.04. The inset shows a comparison of the asymptotic approximation and an exact calculation over a wider range of the circle radii. Our empirical equation is considerably more accurate for the smaller circle sizes. E. Other Obscurations and Particulate Contamination Clearly the diffraction effects of obscurations other than secondary-mirror spiders can be predicted by a modiﬁed version of the simple empirical equation discussed above. The Hubble Space Telescope 1HST2 has three small circular pads that obscure the beam in addition to the central obscuration and four spiders. Equation 1192can be generalized to include the effect of these pads by merely adding a term to account for the pads: EEHST1r2531Aann 22Aspid 22Apad2EEann1r2 1AspidEErect1r21ApadEEcirc1r24@1Aaperture2, 1242 where Apad 5Mpg2D2@4, M5number of pads 1252 is the area of the aperture covered by the pads and EEcirc1r2is the fractional encircled energy of an unob- scured circular aperture of diameter gD. The aper- ture conﬁguration is shown in Fig. 17 along with the fractional encircled energy curves for A, an unob- scured circular aperture; B, an annular aperture with an obscuration ratio of 0.33; C, an annular aperture with four secondary-mirror spiders of fractional width 0.0107; and ﬁnally D, the HST aperture complete with three small circular pads with fractional diam- eters of 0.06365. Even the effects of particulate contamination 1dust2 on a telescope mirror can be modeled in this way. Parametric predictions of image degradation caused by particulate contamination of the HST primary mirror was calculated by generalizing Eq. 1242to include two extra terms: EEdust1r2531Aann 22Aspid 22Apad 22Adust2EEann1r2 1AspidEErect1r21ApadEEcirc1r2 1AductEEdust1r24@1Aaperture2,1262 where Adust is the area of the aperture covered by dust. EEdust1r2is the fractional encircled energy of the scattered light distribution caused by the dust. The value of EEdust1r2is unknown, but certainly negligible, because the dust is so small that its Fig. 16. Comparison of an asymptotic approximation, an empiri- cal formula, and a rigorous calculation for a few discrete data points. The inset shows a comparison of the asymptotic approxi- mation and an exact calculation over a wider range of circle radii. Fig. 17. HST aperture and diffraction-limited performance includ- ing the effects of central obscuration, spiders, and pads. Table 5. HST Image Degradation Areal Dust Coverage Fractional Encircled Energy Case 1 Case 2 0.00 0.730 0.730 0.01 0.723 0.716 0.02 0.715 0.700 0.03 0.707 0.658 0.04 0.698 0.670 0.05 0.689 0.655 6346 APPLIED OPTICS @Vol. 34, No. 28 @1 October 1995 diffracted@scattered light distribution is very broad compared with lf@D. We must be careful now because encircled energy is fractional quantity whose denominator can be propor- tional to the total energy reaching the focal plane: Aaperture 5Aann 2Aspid 2Apad 2Adust, Case 1, 1272 or it can be proportional to the total energy collected by a clean mirror: Aaperture 5Aann 2Aspid 2Apad, Case 2. 1282 Equation 1262was used to calculate the fractional encircled energy for both cases. A Strehl ratio of 0.906 was used to approximate the additional image- quality degradation caused by the maximum allow- able rms wave-front error of 0.05 waves. We ob- tained the encircled energy predictions tabulated in Table 5. The denominator for Case 1 is reduced by dust and hence degrades the encircled energy substan- tially less than for Case 2. Case 2 permits only about half as much contamination as Case 1. The HST image-quality requirement stated that 70% of the total energy reaching the focal plane from a stellar image must be contained within a radius of 0.10 arcsec 1Case 12. Our predictions therefore indi- cate that the HST image-quality requirement can be satisﬁed with 3.8% areal dust coverage on the pri- mary mirror. 5. Modulation Transfer Function The modulation transfer function 1MTF2is deﬁned as the normalized autocorrelation of the pupil function as illustrated in Fig. 2. Closed-form analytical solu- tions for the MTF of annular pupil functions with narrow spiders are rather cumbersome for many spider conﬁgurations. However, we encounter sam- pling problems when performing the alternative calcu- lation with fast-Fourier-transform 1FFT2routines: MTF 5 e 2 e  f1a,b2f1a2x,b2y2dadb e 2 e  0f1a,b202dadb .1292 For a binary amplitude 1opaque@transparent2dif- fracting aperture the numerator of the above equa- tion is the area of overlap as a function of the shift parameter, and the denominator is just the total area ATof the aperture26: MTF 5Rff AT ,Rff 5 e 2 e  f1a,b2f1a2x,b2y2dadb. 1302 For a circular aperture of diameter D, it is straightfor- ward to show that17 MTF 51D2@225cos211r@D221r@D23121r@D2241@26 pD2@4. 1312 A. MTF of AnnularApertures The closed-form analytical solution for the MTF of an annular aperture of outer diameter D1and inner diameter D2is now obtained by MTF 5Rff p1D122D222@4,1322 where Rff 5Rf1f122Rf1f21Rf2f2, where Rf1f1is the autocorrelation of the circle function deﬁning the outer radius of the annulus, Rf2f2is the autocorrelation of the circle function deﬁning the inner radius of the annulus, and Rf1f2is the cross-correlation of the two circles deﬁning the inner and outer radii of the annulus. Figure 18 illustrates the MTF of an annular aper- ture for several different obscuration ratios. B. Analytic Solution of MTF with Spiders Closed-form analytical solutions for the MTF of annular pupil functions with four narrow spiders Fig. 18. MTF for an annular aperture with obscuration ratio E. Fig. 19. Analytical solution for the MTF proﬁle in the xdirection 1u502. 1 October 1995 @Vol. 34, No. 28 @APPLIED OPTICS 6347 have been obtained by breaking the autocorrelation integral of Eq. 1302into the sum of four different integrals: Rff 5 e a50 2E f1a2f1a2x2da1 e a52E 12E f1a2f1a2x2da 1 e a512E 11E f1a2f1a2x2da1 e a511E 2 f1a2f1a2x2da, 1332 where the limits of integration are for an annulus with an outer radius of unity as illustrated in Fig. 19. A similar closed-form integral can be solved for the MTF proﬁle in the u545° direction. For an annular aperture with obscuration ratio E50.30 and frac- tional spider width d50.05 in Fig. 20 we compare these two MTF proﬁles with that for no spiders at all. Note that the presence of secondary-mirror spiders actually improves the MTF over certain spatial- frequency regimes. For example, it is readily shown that for spatial frequencies greater than Œ2@2times the cutoff spatial frequency, the 45° proﬁle is en- hanced by the factor Aann@AT5pD2112E22@4 pD2112E22@422D2d112E251.108. 1342 The above relationship is restricted to the case of four narrow spiders in an annular aperture with obscura- tion E#Œ221. C. MTF Sensitivity Curves Figures 21 and 22 show the sensitivity of the MTF to variations in the obscuration ratio and the spider width, respectively. In Fig. 21 we hold the spider width ﬁxed at 5% of the aperture diameter and vary the obscuration ratio from 0.2 to 0.3 to 0.4. In Fig. 22 we hold the obscuration ratio ﬁxed at E50.3 and vary the spider width from 2% to 5% to 8% of the aperture diameter. Secondary mirror spiders thus have a signiﬁcant but modest effect on the MT, with an initial abrupt drop in the modulation at a very low spatial frequency. At high spatial frequencies, there is actually an increase in the MTF resulting from the normalization by the reduced area of the aperture. 6. Summary and Conclusions We have shown that diffraction effects from second- ary mirror spiders can signiﬁcantly degrade telescope image quality; however, these effects vary drastically with the particular image-quality criterion. Rigor- ous analytical diffraction calculations are often un- wieldy for complicated aperture shapes, and virtually all commercially available optical-design and analy- sis codes that have a diffraction analysis capability are based on numerical Fourier-transform algorithms that frequently lack adequate sampling density to model very narrow spiders. Scalar diffraction theory and Fourier techniques have been applied to model the effects of spider diffraction parametrically on the Strehl ratio 1or the peak intensity of the diffraction image2, the fractional encircled energy, and the MTF. Parametric performance predictions are presented as a function of the central obscuration ratio, the particu- lar spider conﬁguration, and the width of the spiders. In particular, when fractional encircled energy is the image-quality criterion of choice, a simple empirical equation is presented in this paper and validated to be remarkably accurate for arbitrary obscuration ratio and spider conﬁgurations. Fig. 21. Sensitivity of the MTF to the central obscuration ratio. Fig. 22. Sensitivity of the MTF to the spider width. Fig. 20. Effect of telescope secondary mirror spiders on MTF. Note the improvement at high spatial frequencies. 6348 APPLIED OPTICS @Vol. 34, No. 28 @1 October 1995 The authors are grateful for the many useful comments and suggestions of the reviewers of the original manuscript of this paper. It is a much improved paper because of their conscientious efforts. References and Notes 1. A. Couder, ‘‘Dealing with spider diffraction,’’ in Amateur Telescope Making, Advanced 1Book Two2,A. G. Ingalls, ed. 1Scientiﬁc American, New York, 19462, pp. 260–262. 2. R. E. Cox, ‘‘Spider diffraction in moderate-size telescopes,’’ Sky Telesc. 166–171 1Sept. 19602. 3. R. C. Ludden, ‘‘A 10-inch reﬂector fashioned in wood,’’ Sky Telesc. 112–114 1Feb. 19692. 4. C. H. Werenskiold, ‘‘A note on curved spiders,’’ Sky Telesc. 262–263 1Oct. 19692. 5. R. E. Cox, ‘‘Secondary mirrors and spiders,’’ Telesc. Making 7, 4–7 1Spring 19802. 6. W. A. Rhodes, ‘‘An antidiffraction mask for reﬂectors,’’ Sky Telesc. 289–290 1Apr. 19572. 7. Kenneth Novak & Co., Catalog, Box 69, Ladysmith, Wisc. 54848. 8. C. H. Werenskiold, ‘‘Improved telescope spider design,’’ J. R. Astron. Soc. Can. 35, 268–273 119412. 9. E. Everhart and J. Kantorski, ‘‘Diffraction effects produced by obscurations in reﬂecting telescopes of modest size,’’Astron. J. 64, 455–463 119592. 10. J. L. Richter, ‘‘Spider diffraction: a comparison of curved and straight legs,’’Appl. Opt. 23, 1907–1913 119842. 11. L. M. Beyer and L. C. Clune, ‘‘Intensity and encircled energy for circular pupils obscured by strut supported central obscura- tions,’’Appl. Opt. 27, 5067–5071 119882. 12. P. P. Clark, J. W. Howard, and E. R. Freniere, ‘‘Asymptotic approximation to the encircled energy function for arbitrary aperture shapes,’’Appl. Opt. 23, 353–357 119842. 13. P. J. Peters, ‘‘Aperture shaping—a technique for the control of the spatial distribution of diffracted energy,’’ in Stray Light Problems in Optical Systems, J. D. Lytle and H. E. Morrow, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 107, 63–69 119772. 14. H. F. A. Tschunko and P. J. Sheehan, ‘‘Aperture conﬁguration and imaging performance,’’Appl. Opt. 10, 1432–1438 119712. 15. A. B. Meinel, M. P. Meinel, and N. J. Woolf, ‘‘Multiple aperture telescope diffraction images,’’ in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. 1Academic, New York, 19832, Vol. 9, pp. 149–201. 16. J. W. Goodman, Introduction to Fourier Optics 1McGraw-Hill, New York, 19682, pp. 61, 113. 17. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics 1Wiley, New York, 19782, pp. 194, 216. 18. H. H. Hopkins, Wave Theory of Aberrations 1Oxford U. Press, New York, 19502, Chap. 2, p. 21. 19. R. Bracewell, The Fourier Transform and its Applications 1McGraw-Hill, New York, 19652,p.112. 20. A. A. Dantzler, ‘‘Encircled energy correction method for ray- trace programs,’’Appl. Opt. 27, 5001–5002, 119882. 21. M. Born and E. Wolf, Principles of Optics 1Pergamon, Oxford, 19802, Chap. 8, p. 381. 22. E. Hecht, Optics 1Addison-Wesley, Reading, Mass., 19872, Chap. 10, p. 458. 23. M. V. Klein, Optics 1Wiley, New York, 19702, Chap. 7, p. 298. 24. H. F. A. Tschunko, ‘‘Imaging performance of annular aper- tures,’’Appl. Opt. 13, 1820–1823 119742. 25. These photographs were provided by H. F. A. Tschunko, NASA@GSFC. 26. R. R. Butts and C. B. Hogge, ‘‘The modulation transfer func- tion of an annular aperture with supporting struts,’’Air Force Weapons Laboratory reportAFWL-TR-75-311 1Feb. 19762. 1 October 1995 @Vol. 34, No. 28 @APPLIED OPTICS 6349 ... As early as 1941 43 , Werenskiold, an amateur astronomer and telescope builder has recorded that curved spiders on secondary mirror could greatly improve the quality of the planetary observation, followed by similar experimental results from Couder (1952) 44 and Everhart and J. Kantorski (1959) 45 . 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