Download Telescopes

Observing Infinity
Tim D. Gerke
1.
Introduction
The uses of telescopes may seem quite broad; from navigating a ship from Portugal to the
Americas (inadvertently) in the 15th century, to birdwatching, stargazing, exploring the
cosmos, and spatial filtering images. The underlying concept in all of the aforementioned
uses for telescopes is simply viewing things from afar. At first glance this seems simple, but
of course, there is a lot of work necessary to make a very useful telescope, especially for
viewing stars, planets, etc. due to limited light, extreme distances and multiple wavelengths.
In the following sections I will analyze different types of telescopes in their functionality,
resolvability, chromaticity, etc. and discuss the pros and cons of different types.
2.
Which Telescope and Why?
The two main types of telescopes are refractive and reflective, within each category being
a multitude of designs. The why portion of the question hinges on a few factors, among
which are chromaticity, intensity of object viewed, size, and actual type of use (portable,
long range, short range etc.) to name a few.
3.
Refractive
The simplest types of refracting telescopes are the Keplerian and Galilean, shown in Fig. 1.
As you can see, the way all these telescopes are set up is with a couple of lenses separated
by the sum of their focal lengths. Therefore, any rays coming from far off (nearly parallel
to the axis), would pass through the coinciding focal points between the two lenses, and
exit the system again parallel to the axis with some modification to the off-axis placement
(transverse magnification). For the three element Keplerian telescope seen in the figure,
there is an additional lens placed in a 2F configuration between the focal points of the
outer lenses to right the otherwise inverted image.
The system matrices for any optical system is often quite useful, and the matrix for
two-lens simple configurations shown above is,
M = L(f2 ) T(f1 + f2 ) L(f1 )
1
(1)
L1
f1
L2
f2
(a)
L1
f1
L2
f2
L3
f3
(b)
L1
f1
L2
f2
(c)
Fig. 1. (a)Keplerian, (b)Keplerian (3 element) (c) Galilean.
2


f2
 − f1
= 
f1 + f2 
− ff21
0

where L and T are the thin lens and transmission matrices respectively. The corresponding
matrix for the three lens system is,
M = L(f2 ) T(f2 + 2fr ) L(fr ) T(f1 + 2fr ) L(f1 )


= 
f2
f1
0
(2)

f1 + f2 

f1
f2
which can be seen to be nearly the same as Eq. 2 besides a minus sign. This is intuitively
obvious since the second, “relay” lens simply inverts the rays entering the final lens...meaning
the image is simply erect upon exiting the system relative to the inverted image of the
Keplerian setup. The Galilean setup has a negative f2 , so the image here is also erect.
The transverse magnification of these systems is,
MT = ±
f2
.
f1
(3)
It is interesting that typically, f1 >> f2 . This says that a telescope actually demagnifies
an image, which may be counterintuitive. However, the correct way to think about this
is to imagine the object being viewed, say, a planet. When thought about in this way,
demagnification makes sense. Rather than magnification of the object, what is desired is to
make it appear closer, which would be a large angular magnification. Angular magnification
is related to transverse magnification as,
Mang =
f1
1
=±
MT
f2
(4)
and since f1 >> f2 , it can be seen that Mang is typically quite large. This is how a telescope
takes a huge object that is far away, and makes it appear very small, but much much closer.
For a simple example with numbers for the two-element Keplerian system, let’s assume
f1 = 100 cm and f2 = 1 cm. For these components, the paraxial layout would be a lens
1
, an angular magnification
seperation d = 101 cm, transverse magnification MT = − 100
h
Mang = −100, an image height h0 = − 100
, and an output angle α0 = −100α. For these
numbers, objects would appear 100 times smaller, but 100 times closer. These sorts of
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numbers might be close to what you would want if you were interested in seeing some
features on the moon, but it is probably insufficient for seeing features the size of a person
on the moon for example.
4.
Reflective
Reflective telescopes are also widely used, mainly due to their good chromatic performance
and the ability to scale them to very large sizes (huge lenses tend to sag under their large
weight). The one main negative aspect that comes to mind directly is the finite reflectivity
of metal (broadband) coatings which will lower the percentage of optical power transmitted
through the system. They are, despite this one minor downside, definitely worth an analysis.
Three major types of simple reflective telescopes are the Newtonian, Cassegrain, and
Gregorian, which are all pictured in Fig. 2. As you may see immediately, the Newtonian is
the same system as the Keplerian, except the objective lens is a positive mirror. If we were
to unfold the system and ignore the flat mirror which is used to deflect the image to an
eyepiece mounted on the side of the apparatus, it would be the exact same as the Keplerian
refractive telescope. For this reason, I will not include an analysis on it here. Any quantities
of interest can be found directly from the Keplerian analysis, where f =
R
2
for the mirror.
The Cassegrain and Gregorian can be unfolded to be as shown in Fig. 3.
The system matrices for these systems are extremely messy if computed directly as,
M = L(f3 ) T(d2 ) L(
R2
R1
) T(d1 ) L( ).
2
2
(5)
Rather, it is appropriate to note that the first two mirrors together cause the incoming rays
parallel to the optical axis to focus to a spot located at the objective mirror which has a
small hole in it to allow them to pass through to the eyepiece. Therefore, to simplify the
analysis, we can combine the two mirrors into a single equivalent lens in the unfolded system,
which will then make the analysis similar to the refractive systems already discussed. This
lens would have an effective focal length of,
1
1
1
d1
=
+
−
.
F
f1 f2 f1 f2
(6)
and the principal plane, P 0 , would be located,
A
−F
C
A−1
C
P0 =
=
4
(7)
M1
f1
L2
f2
(a)
M2
f2
L3
f3
M1
f1
(b)
M2
f2
L3
f3
M1
f1
(c)
Fig. 2. (a) Newtonian, (b) Cassegrain, and (c) Gregorian.
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M1
f1
M2
f2
L3
f3
(a)
M1
f1
M2
f2
L3
f3
(b)
Fig. 3. Unfolded versions of the (a) Cassegrain and (b) Gregorian telescopes.
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relative to the second lens, where A and C are from the ABCD matrix for the two lens
subsystem.
Again, the angular magnification is one of the main quantities of interest when dealing
with telescopes of any kind, so it would be appropriate to discuss this here. This is the
reason for combining the first two lenses into a single equivalent thick lens. As you can see
from Fig. 3 the first two lenses combine to effectively do the job of the first lens of the two
lens systems. Therefore, the angular magnification can be directly and simply stated to be,
Mang = −
F
f3
(8)
where F is the effective focal length of the first two lenses (mirrors), and f3 is the focal
length of the eyepiece, or third lens.
To do a simple example of a Newtonian telescope with standard optics from Newport,
I would choose to use a broadband metallic concave mirror with a large focal length. A
possible good choice would be 20DC2000 mirror with f = 100 cm and f # = 19.7 which
is 5.08 cm in diameter. This would make for a nice portable hobbyists telescope, but not
for any serious viewing applications. For an eyepiece, I would use a precision achromatic
doublet with the smallest focal length Newport offers (PAC010), 1.27 cm. These numbers
would provide magnifications of MT = 0.0127 and Mang = 78.74.
On a final note to the design, the axis-folding mirror should be placed at the coincident
focal lengths of the mirror and the eyepiece lens, and should be large enough to pass the
desired angular field of view. The constraints of this angular field of view would depend on
application and needs, but a general overview is warranted here. If we desire to view an
object that has an approximate size of D, and is z away, we must allow for an angular field
D
z.
The size of the mirror is then dictated by it’s necessity of catching
√
the entire angular field of view, which would require it to be Dmirror = 2f1 AF oV (the
√
2 comes from the 45◦ incidence) and the eyepiece would then need to be Deyepiece =
of view of AF oV =
(f1 + f2 ) AF oV . For example, the moon is 384, 403 km away and 3, 476 km wide, so to view
it with the telescope designed above, Dmirror ≥ 12.788 mm and Dlens ≥ 9.16 mm, which is
unfortunately a pretty fast lens. We could relax this to closer to an
F
2
lens if we would allow
for the mirror focal length to be closer to 70 cm with some loss to the angular magnification.
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5.
Telescopes with Power
Telescopes are also often used in imaging systems not focused at infinity. The most common
system would be the afocal Keplerian telescope with the object and image located at the
front and back focal planes respectively. This system is most often referred to as a 4f
system if the two lenses are identical. The main use of this setup is spatial filtering. The
important concept is the existence of the Fourier plane at the coincident focal points of
the two lenses (centered between them for equal f ’s). A mask can be placed at the Fourier
plane to spatially filter the object plane, and if the lenses do not have the same focal length,
a magnification (or minification) MT = − ff12 can also be added by using different lenses.
Another key element to these systems is their resolvability. This is discussed in Section 8.
6.
Gaussian Design
To follow up with the paraxial example given at the end of Section 3, I will now provide a
similar example with actual lens parameters as one would find in a Newport catalog. Actual
distances between lens surfaces are most likely useful quantities to know in manufacturing
the telescopes, so such quantities will be provided. Remember, for the example previously
done, f1 = 100 cm and f2 = 1 cm. Readily available in the Newport catalog for precision achromatic doublets are f1 = 75 cm (PAC094) or 50 cm (PAC091) and f2 = 1.27 cm
(PAC010), so we will do a quick design with these two. The back focal distances for the
first lenses are 74.18 and 49.46 cm and the front focal distance for the second lens is 1.07 cm
so the distance between surfaces of the two lenses would be 75.25 and 50.53 for the 75 and
50 cm focal length lenses respectively.
The other quantity that would likely be of interest is the location of the principle planes
relative to the surfaces. These quantities would be the effective focal length minus the back
or front focal lengths, P A = F − F F L and P 0 B = F − BF L, which are P A = 3 mm
and P 0 B = 8.2 mm for the first lens and P A = 0.1 mm and P 0 B = 2 mm for the second
lens. A final quantity of interest that would be useful in determining the placements of the
object and image/eye is the lens thickness. With this, as can be seen in Fig. 4, we have all
information necessary to set up a real telescopic system with standard lenses from Newport.
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A
B
BFL
F1
P
P
P‘
A
B
FFL
F2
F2
P‘
Fig. 4. Gaussian layout for a two-element Keplerian telescope.
7.
Radiometry
Having the systems worked out in details of the quantities of interest, namely the focal
lengths, effective and actual, and the magnification factors, Mang , the next useful bit to
analyze would be the aperture stops, field stops, etc. To start, it is interesting to get an
idea of how much optical power typical telescopes collect. To make a quick estimate, I will
assume a 100 mm objective size, which might be a bit large for a typical hobbyists system,
but would definitely be small relative to a telescope used to astronomical research.
Two numbers of interest would be the percentage of power emitted by the sun or
our “nearest” star (it’s actually not the nearest, rather the nearest “bright” star), Alpha
Centauri, collected by this telescope. The percent optical power collected by our 100 mm
objective lens is the ratio of the solid angle from the source terminating on the objective
lens edges to the total solid angle emitted,
Γ=
2
πDobj
W
=
4π
16πL2
(9)
where L is the distance from the earth to the source object. For the sun (L = 150x109 m),
this gives a percentage, Γ ∼ 3x10−26 and for Alpha Centauri (L = 3.8x1016 m), Γ ∼ 5x10−37 ,
which puts the astronomical optical output of the sun into perspective and also explains
why we can’t see stars during the day since they are being washed out by the scattering of
the 10+ orders of magnitude greater collected power (for any optical system). If we were to
increase the size of our objective lens, the collecting power would increase by the square, so
it is apparent why high quality telescopes have very large objective lenses.
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This is also the first reason why a reflective telescope is often a better bet than refractive. Large lenses tend to sag under their own weight, and consequently increase the
abberations of the system whereas reflective elements can be made quite large since they
are usually some type of coating on a substrate which can be designed to support it’s weight
The eyepiece for the simple systems discussed so far is the field stop. If we trace the
PPR of a system (the aperture stop of the simple telescopes discussed is obviously the
objective lens), then we can define the angular field of view of the system as,
F OVang =
Deyepiece
,
2(f1 + f2 )
(10)
where f1 is either the focal length of the first lens for the two element systems, or the
effective focal length of the first two elements for the three element systems. From this
relation it is apparent that as the eyepiece size goes up, so does the field of view. But in
any real system, one’s eye will be the limiting factor in the field of view. Therefore the
eyepiece need not be any larger than the eye, which is good since the eyepiece typically has
a much shorter focal length and is consequently limited in size. Also, since f1 >> f2 , we can
make the approximation, f1 + f2 ∼ f1 which then tells us that to make the angular field of
view maximal f1 would have to be minimal, but this would lower the angular magnification
unless we also scaled f2 back. All in all, one can see there is a design problem here involving
angular magnification, system size, angular field of view, component cost, and resolvability,
as well as other possible constraints.
8.
Resolvability
In any imaging system resolvability is an important aspect, and telescopes are no exception.
The main difference though is that telescopes are not imaging systems in that they do not
take an object and relay it to a finite image plane. Instead, they take rays from large objects
far away coming in at small angles and magnify them such that they appear to be coming
from a much smaller object that is located much closer. We therefore define our system
resolvability in terms of incident angles. The minimally resolvable angle of incidence is
defined as the minimal angular separation between to resolvable bodies and is expressed as,
∆φ =
1.22λ
.
Dobj
(11)
To put some typical numbers on this, we can assume a 0.5 inch objective lens and 0.5 µm
light to give ∆φ = 49 µrad, which would correspond to 3.5x1012 m at the Alpha Centauri
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star system. This is approximately twice the distance from Alpha Centauri A to Alpha
Centauri B, which was discovered to be a double star in 1752 by Nicholas Louis de La Caille
using a refractive telescope with a 0.5 inch objective, a perfect example of resolvability in
action.
For the 4f system setups, we can discuss a spatial resolvability as opposed to angular.
Similar to the calculation above, the spatial resolvability is expressed as
∆x =
1.22λf1
.
Dobj
(12)
So, suppose we have an object with spot sizes of ∆xspot = 1µm. To image these spots would
require F # =
f1
Dobj
≥ 1.67 (0.5 µm illumination). The other main constraint, though, is
vignetting. To avoid vignetting, the lenses must be at least the size of the field, X, plus the
height of the P M R at the first lens (which was the objective size calculated above),
Dobj min = 2X + Dobj = 2X +
f1
.
1.667
(13)
For the simple example above, assuming a field size of 1 cm and a focal length of f1 = 50 mm,
Dobj min = 5 cm. If a shorter system is desired, we can crank the focal length down to 10 mm,
making Dobj min = 2.6 cm, which is conveniently close to the common size of 1 inch optics.
9.
Pupil Matching
For telescopes used in viewing stars or other stellar objects, a final point of interest is the
exit pupil. If the exit pupil is placed at some place in front of the eye, it could make for
an odd image to the viewer. What is desirable is to place the exit pupil at the eye, a short
distance beyond the eyepiece. This can easily be seen in any binoculars or telescopes simply
by looking through them from a distance.
The exit pupil is the image of the aperture stop as seen from the image space. If the eye
is assumed to be 2 cm behind the eyepiece, which has a focal length of f2 , then an aperture
stop should be placed,
dA.S. =
20f2
20 − f2
(14)
where dA.S. is relative to the eyepiece and f2 is in mm. The other important factor is that
the pupil of a human eye is as large as 7 mm, so the exit pupil should be equal to or
greater than this to get the best transfer of light from the object into the eye. For this, the
aperture stop would have to be MT 7 where MT =
10
dA.S. .
As an example, if f2 = 10 mm,
then dA.S. = 20 mm and MT = 1, so the aperture stop would need to be 7 mm or larger.
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10.
Chromaticity
Telescopes are most often used to observe far away objects that are illuminated with, or
emitting white light, so imaging these objects well requires optics that have little or no
chromatic dispersion. Dispersion will cause the different colors/frequencies contained in the
white light to focus through lenses at different spots. The reflective elements do not have
this problem if they are made from simple metals coatings, but, as I have mentioned, they
are typically less efficient. If the reflective elements are made from high reflective, multilayer coatings, they will have some wavelength dependence and will consequently have a
wavelength-dependent efficiency. One possible solution to this problem for the refractive
optics would be to use achromats. System design using these would be equivalent to a
typical Gaussian design for regular lenses. For the afocal systems discussed thus far, the
focal points would be coincident between the lenses. The way to do this is to measure the
distance between the principle planes of the objective and eyepiece lenses and set it to be
the sum of the effective focal distances. Of course, achromats are not perfectly achromatic,
so the refractive systems will still have slightly worse performance than reflective systems
for imaging multicolored objects.
11.
Conclusions
Given the information I have purveyed thus far, one could begin to make an informed
decision on what kind of telescope to buy. For viewing astral objects, one must first decide
what sort of resolution is desired, and find a system with the appropriate objective lens size.
One must also consider light collecting ability, which is also a function of objective size as well
as of loss. If a small system is desired for portability, one must give up resolution and light
collection power to utilize smaller components. Reflective systems are more appropriate for
imaging systems containing wavelengths all across the visible spectrum due to the chromatic
dependence of the focal length of a refractive lens although this can be corrected to second
order using an achromatic doublet, or even better for more complex achromats. A final
noteworthy fact is that reflective systems are exclusively used in large, extremely high
quality systems because reflectors can be made to be quite large, whereas lenses typically
cannot.
For the imaging/spatial filtering telescopes, the main constraints were discussed in
Section 8. These systems are typically simple Keplerian systems with identical lenses, unless
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a magnification is desired in which case the appropriate focal lengths are chosen. If the image
was desired to be erect rather than inverted, another lens can be added after the telescope
at a distance of 2f from the image plane so that an erected image will exist 2f behind
that lens. Finally, the lens sizes would have to be designed correctly to achieve the desired
resolution.
In conclusion, I have presented an analysis for a few types of afocal telescopes. I discussed resolution, pupil matching, chromatic performance (in a qualitative manner), and
spatial filtering systems. With these tools one should be capable of selecting or designing a
basic telescope for whatever purpose one has in mind.
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