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Astronomy 203/403, Fall 1999
15 Lecture, 21 October 1999
15.1 Diffraction-limited telescopes
We have considered radiating, or transmitting, apertures in the diffraction calculations given above. The
case of a focussing or receiving aperture is similar, since it involves subsequent interference of light that
encounters the aperture as a plane wave. This is obviously the case for telescopes, if we consider what
happens to light from a very distant point source, like a star. Suppose a star lies on the telescope’s axis.
The subsequently focussed image consists of electric fields added up only from that part of the plane
wave lying on the telescope aperture, of course, and this is exactly what we calculated above in Lecture
13, except that the waves are now propagating in rather than out. One can reverse the direction of a wave
simply by changing the sign of the wavevector κ, so that the stellar intensity as a function of the off-axis
angle is given by the final result, Equation 13.15, replacing κ by -κ. Because Equation 13.15 only contains κ
in the form of squares and even functions, the answer doesn’t change:
af
Iθ =
a f2 LM 2J 1 a−κaθ f OP2 = cEN2 A2 κ 2 LM 2J 1 aκaθ f OP2
a2π f2 N −κaθ Q 8πr 2 a2π f2 N κaθ Q
2 2
cEN
A −κ
8πr 2
.
(15.1)
This is an example of reciprocity: an optical system is called reciprocal if the fields and intensities it
receives as a function of angle are the same as it would broadcast if used as a transmitter.
The telescope therefore forms in its focal plane an image of an Airy pattern if diffraction is the dominant
blurring effect and the telescope is used to observe a distant point source such as a star. Distance in the
focal plane is related to angle on the sky by the plate scale:
PS ≡
1
f eff
=
1 f1
fp f2
(Cassegrain telescope),
(15.2)
and as a function of the distance ρ from the center of the focal plane, the intensity of the image of a point
source is
LM 2J 1 eκaρ / f eff j OP2
I b ρ g = I ( 0)
MN eκaρ / f eff j PQ
.
(15.3)
The radius of the first dark ring, and the diameter of the half-peak-intensity level (the full width at half
maximum, or FWHM), are obtained from application of the plate scale to Equation 13.16 and Figure 13.3:
ρ 1 = θ 1 f eff = 1.22
d FWHM ≅ 1.2
λf eff
D
λf eff
D
= 1.22 Fλ
= 1.2 Fλ
,
(15.4)
,
where D is the telescope primary diameter and F = f eff / D is the final focal ratio (or f-number) of the
telescope.
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Most of the power in the starlight focussed by the telescope winds up in the central maximum of the Airy
pattern, as is easy to set up and integrate numerically. For instance, the fraction of the total power that lies
inside the first dark ring is
zbg
z
2π 1.22 Fλ
z LMN a f OPQ
z LMN a f OPQ
3.832
I ρ ρdρdφ
Pcentral maximum
= 02π ∞0
=
Ptotal
I ρ ρdρdφ
zzb g
0
∞
0
0 0
2J 1 u
udu
u
2J 1 u
udu
u
= 0.838 ;
(15.5)
that is, 83.8% of the total power lies within the central peak. Similarly, the power in the bright rings can be
computed by changing the bounds of the integral in the numerator of Equation 15.5 to the various other
zeroes of J 1 u (see Table 13.1). This results in 7.2% of the total power between the first and second dark
rings, and 2.8% between the second and third. The central maximum dominates to the extent that the
other bright rings can be ignored to first approximation.
af
15.2 Single-mode propagation.
Under the assumption that the Airy pattern subtends a small angle on the sky, the solid angle it occupies
is
FG
H
Ω 1 ≅ πθ 21 = π 1.22
λ
2a
IJ 2 ≈ 4 λ2
K A
,
(15.6)
and that of the full width at half maximum (FWHM) is
Ω FWHM ≅ π∆θ 2FWHM ≅
λ2
A
.
(15.7)
This latter relation between the effective solid angle of the central part of the central maximum and the
area of the telescope is a general and important result that can be derived quickly from the Kirchhoff
diffraction integral, Equation 12.24. If the field amplitude on the aperture (area A) is constant and equal to
E A , the electric field at a large distance r away, but near the optical axis, is
EF =
e i(κr −ωt )
EA
λr
zz
e
e
i κ x x ′+κ y y ′
jdx ′dy ′
.
(15.8)
aperture
To zeroth order in κ x x′ and κ y y′ , the exponential inside the integral is unity:
e
iκ x x ′
2
iκ x x ′ g
b
= 1 + iκ x x ′ +
+ ≈ 1
2!
,
(15.9)
so to this order the integral simply gives the aperture’s area:
EF =
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e i(κr −ωt )
EAA ,
λr
2
(15.10)
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Astronomy 203/403, Fall 1999
EA A
EF =
or
.
λr
(15.11)
If the far-field diffraction pattern extends over an effective solid angle Ω, then the total power in the far
field is
PF = I F r 2 Ω =
c
2
EF r 2Ω ,
8π
(15.12)
while the total power passing through the aperture is
PN =
c
2
EN A .
8π
(15.13)
Energy is conserved ( PF = PN ), so the last three expressions can be combined to produce
2
2
E A A = EF r 2Ω =
or
AΩ = λ2
2
E A A2 2
r Ω ,
λ2 r 2
;
(15.14)
(15.15)
the area-solid angle product of a diffraction-limited beam is constant and equal to λ2 as the beam
propagates.
It is easy to show that Equation 15.15 applies to any reciprocal receiver or transmitter of electromagnetic
radiation – anything from a telescope to an electric dipole antenna. Suppose we have two receivertransmitters, separated by some large distance R as in Figure 15.1. Device 1 has effective area A1 and
emits into an effective solid angle Ω 1 , with corresponding parameters A2 and Ω 2 for device 2. If device
1 transmits a total power PT at wavelength λ, and device 2 lies well within the solid angle into which
device 1 radiates, then the power received by device 2, PR , is
PR = PT
4π A2
Ω 1 4πR 2
,
(15.16)
since the power emitted by device 1 is concentrated within Ω 1 rather than 4π steradians, and the
receiving area of device 2 is just A2 rather than the sphere with radius R surrounding device 1. Now if
one reverses the roles, letting device 1 receive and device 2 transmit, one obtains
PR = PT
4π A1
Ω 2 4 πR 2
.
(15.17)
If both antennas are reciprocal and energy is conserved, the same values of PR and PT would have to be
obtained in these two experiments. Equations 15.16 and 15.17 therefore combine to produce
A1 Ω 1 = A2 Ω 2
.
(15.18)
Thus all devices that can emit and receive electromagnetic waves reciprocally – antennas and telescopes –
involve beams that have the same area-solid angle product. But we already know that this product is
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simply equal to λ2 for one such device – the circular diffracting aperture – so Equation 15.15 holds for all
reciprocal receivers/transmitters of electromagnetic waves.
In the argot of antenna engineers a diffraction-limited beam is one example of a single mode of
electromagnetic radiation. Beams with AΩ = λ2 are thus often called single-mode beams.
Ω1
A2
R
Figure 15.1: transmission of a single-mode beam with solid angle Ω 1 from device 1 (left)
to device 2 (right), which receives it with total collecting area A2 .
15.3 Radio telescopes
At short wavelengths – near-infrared, visible, and ultraviolet – atmospheric turbulence reduces the
coherence of wavefronts from celestial objects, adding an additional blurring mechanism that turns out to
dominate over the effects of diffraction for all but very small telescopes. Light from point sources is
spread over a focal-plane area called the seeing disk. At the best astronomical observing sites, like Mauna
Kea or the Chilean Andes, the seeing disk is routinely 0.5-1 arcsecond in diameter, compared to an Airy
disk FWHM size of 0.017 arcsec for wavelength 0.55 µm on an 8 m diameter telescope. Some of the
blurring effects of turbulence can be overcome by use of active optics (see below, Lecture 17), but the only
way to eliminate all of the effects of seeing is to remove the atmosphere. This of course is the reason for
having the Hubble Space Telescope, which is diffraction-limited at visible and near-infrared wavelengths.
The size of the seeing disk varies approximately as λ−1/5 ; seeing gradually improves as the observed
wavelength increases. At radio wavelengths ( λ > 1 mm ) seeing is a minor effect, and ground based
telescopes are therefore diffraction limited in general.
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Most, but not all, radio telescopes “look like telescopes, 1” in that they have large conic-section primary
mirrors. Since so many astronomical objects have structure on very small angular scales, radio telescopes
that are designed to be used alone (single dish observing) are generally made as large as is mechanically
possible, for the smallest possible Airy disk. At the longest wavelengths the surfaces are often made of
metal mesh, rather than a solid reflective surface, in order to keep the dish lightweight; as long as the
holes in the mesh are much smaller than the wavelength, metal mesh still reflects light perfectly well. The
largest single-dish radio telescopes are the 300 m diameter telescope in Arecibo, Puerto Rico, built and
operated by Cornell University, and the 100 m diameter telescope in Effelsberg, Germany, which belongs
to the Max Planck Institute for Radio Astronomy in Bonn.
We have discussed the Arecibo telescope briefly above (§10.2), and highlighted its spherical primary and
its world-record case of spherical aberration. This telescope and its two spherical-aberration
countermeasures are shown in Figure 15.2 and Figure 15.3. The original line feed is still available, as seen
in Figure 15.3, but nowadays the new Gregorian corrector is normally used. Extensive metal screens have
also been added around the edge, to prevent the detectors from seeing the ground even when the beam
spills off the primary. These upgrades lead to a larger viewable field and greatly improved sensitivity.
For three decades the Effelsberg 100 m, illustrated in Figure 15.4 and Figure 15.5, has been the world’s
largest fully steerable telescope. It has an altitude-azimuth mount, and its paraboloidal surface is good
enough to use the whole aperture at frequencies as high as 24 GHz (λ = 1.3 cm). The central 60 m is
composed of solid metal panels that can be used at somewhat shorter wavelengths (below 7 mm, these
days), but the outer parts are made of metal mesh that is fairly transparent at such short wavelengths.
Under construction at present is a slightly larger fully steerable single dish radio telescope at the US
National Radio Astronomy Observatory’s Green Bank, WV site. The Green Bank Telescope (GBT) will
have a 110 m by 100 m offset paraboloid primary mirror, for which the area projected to incident light will
be 100 m in diameter and suffer no obscuration or diffraction from the secondary mirror or its supports
(Figure 15.6, Figure 15.7).
The highest frequency at which a telescope can be used is determined by the size of random
displacements from the designed, ideal conic-section surface: to reach the diffraction limit the telescope
surface accuracy must be about λ/10 or smaller. As one might imagine, this is a difficult task, even at long
wavelengths, when telescope diameters in the 100 m class are contemplated. The Effelsberg 100 m is
paraboloidal within an RMS surface accuracy of about 0.5 mm over the whole dish, an amazing feat when
one considers that the structure is likely to bend by more than that whenever the telescope pointing is
changed significantly. The secret is in the space-frame “backup” structure, shown well in Figure 15.5: it is
designed so that the surface always is paraboloidal, though the apex curvature of the paraboloid changes
in different telescope orientations owing to the different aspect of its weight. Such a surface change can be
accounted for simply by refocusing: moving the secondary mirror or the prime-focus detector slightly
along the optical axis to accommodate the new primary focal length. The Green Bank Telescope adds to
such “homologous” structural design a system of remotely-actuated positioners and encoders for three
points on each panel of the primary mirror. These are designed to maintain 0.36 mm RMS surface
accuracy, and enable observations with the full aperture at wavelengths down to about 3.8 mm.
Many pictures of radio telescopes, including some unusual and imaginative antenna configurations,
may be found in the book by John Kraus: Radio astronomy, second edition (Powell, OH: Cygnus-Quasar
Books, 1986), pp. 6.49 – 6.97.
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Figure 15.2: aerial view of the spherical-primary, 305 m diameter Arecibo radio telescope
in Puerto Rico. (NAIC and Cornell University)
Figure 15.3: closeup of the prime focus region at Arecibo. The arc-shaped rail lies along
the primary’s spherical focal surface. On the left, enclosed by the geodesic dome, is the
new two-mirror Gregorian corrector. Hanging from the rail on the right, along a radius of
the primary, is the line feed. Each of these SA compensators has its own detector. The
telescope points in elevation by moving the Gregorian corrector or the line feed along the
arcuate rail, and in azimuth by rotation of the circular ring from which the rail hangs.
(NAIC and Cornell University)
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Figure 15.4: view from the front of the 100-m radio telescope of the Max Planck Institute
for Radio Astronomy, in Effelsberg, Germany. (MPIfR, Bonn.)
Figure 15.5: side view of the MPIfR 100-m radio telescope. (MPIfR, Bonn.)
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Figure 15.6: diagram of the 100 m Green Bank Telescope. (NRAO)
Figure 15.7: the Green Bank Telescope, under construction in summer 1999. (NRAO)
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