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

Abstract

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.

Full text

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.
Received 19 December 1994; revised manuscript received 13
March 1995.
0003-6935@95@286337-13$06.00@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
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1 October 1995 @Vol. 34, No. 28 @APPLIED OPTICS 6349