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▪ ** **CONTENTS
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8.1.3. Newtonian reflector diagonal flat
▐
8.2.1. General aberrations
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#
**
8.2. all-reflecting Two-mirror
telescopes**
Aberrations -
Classical and aplanatic - Dall-Kirkham
- Loveday
- Miscollimation and close focusing
As the title implies, this section as about mirror-only telescope
systems. Mirror/lens systems are presented in section
10. Catadioptric telescopes.
While the Newtonian does use a pair of
mirrors, one of them is flat and, therefore, a passive imaging element.
Systems with both mirrors active in shaping up the wavefront - in other
words, with both mirrors curved - are termed two-mirror telescopes.
Classical two-mirror arrangements are **Gregorian** -
the very first reflective
telescope design, conceived by *James Gregory* in 1663 - and
**Cassegrain** (**FIG. 121**),
probably introduced in 1672 by *Laurent Cassegrain*. In the former,
concave secondary mirror is placed outside the focal point of the
primary mirror, and in the latter the secondary is convex, placed inside
the primary mirror focus. In either case, the final focus is made
accessible either by focusing through an opening on the primary, or by
inserting small diagonal flat in front of it, to reflect converging cone out to
the side (Nasmyth arrangement).
**
FIGURE **
**121**:
TOP:** Left:** **Gregorian reflector** consists from a pair of
concave mirrors. The secondary, placed axially outside of the primary's
focus
**F****1**,
forms properly oriented final focus **F** by re-focusing
diverging light cone coming from the primary.** **In the
classical arrangement, primary is paraboloid, hence for the
secondary to maintain zero spherical aberration it has to be an
ellipsoid with its near focus coinciding with the primary's focus.
Right:
**Cassegrain reflector** also
consists from a pair of mirrors, but the secondary is convex, and
placed inside the primary's focus. In the classical arrangement with
paraboloidal primary, secondary has to be hyperboloidal, with its
near (virtual) focus coinciding with the primary's. The
image is reversed, left-to-right and up-side-down. Effective focal length is determined
by extending marginal
ray converging from the secondary to the final focus
backwards to the point of
intersection with the incident marginal ray, or
**
**=s/k, **k** being the relative (in units of aperture radius)
height of marginal ray on the secondary.
Since the primary is paraboloid in
the classical arrangement, its focus in either Gregorian or
Cassegrain has to coincide with the near (conic) focus of the
secondary (not to confuse with secondary's Gaussian focus **F****2**,
for object at infinity),
in order for the latter to form an aberration-free axial image.
BOTTOM: Projected image of the primary is, effectively,
the object for the secondary. It is located at a distance **
ℓ** from
secondary's surface, which re-images it to the final focus.
The final image is constructed
by a reverse projection of the
reflections from the concave side of the secondary of two imaginary rays, one parallel with the axis
(**blue**) and
the other extended to secondary's vertex (**red**).
The final top point image forms at the point of their intersection. The primary
is the exit pupil for the secondary (i.e. chief, of central ray **CR** for
the off-axis image
point is coming from the
primary's center), and it is also the
aperture stop. The
stop-to-secondary separation **σ**, in units of the secondary
radius of curvature **R**2
(secondary-to-primary separation **s**=σR2) is, according to the
sign convention,
negative; it is a factor in calculating
off-axis aberrations at the secondary (due to the chief
ray - and with it the entire converging wavefront - being shifted off
the center of the secondary, as a result of displaced stop). Optical
image of the primary formed by the secondary is the system exit
pupil (**ExP**2);
it is a factor - aperture stop - in calculating aberrations of the
tertiary mirror in three-mirror systems, due to the chief ray of
off-axis points appearing as if coming (onto the tertiary) from its
center, and the marginal rays (**1** and **2**) appearing as
if coming from its boundary.
The final system focal length,
is determined by the focal lengths and separation of the two mirrors. It
determines secondary magnification as
where **
1**
and **
2**
are the focal length of the primary and secondary mirror, respectively, and
**s** is the mirror separation. According to the above relation, for
light moving from left to right the final system
focal length is numerically positive in the Cassegrain, and negative in
the Gregorian. Primary mirror focal length (**1**)
is always negative, while the secondary mirror focal length **
2**
is positive in the Gregorian arrangement, and negative in the Cassegrain; mirror separation **s** is
negative for both.
In order to correct for spherical
aberration, the two mirror must have specific
conic forms. Although Gregory
established - based on conic properties established by Renι Descartes (*La
G*ι*ometri*ι, 1637) - that in his arrangement primary and secondary need to be
paraboloid and ellipsoid, neither him, nor anyone else at the time knew
how to make aspheric surfaces. For some time, it was difficult to
produce even spherical mirrors accurately. As a result, early
instruments were suffering from all of the three primary "image quality"
aberrations - spherical, coma and astigmatism. It took about half a century for mirror making to advance
to that next level, when *John
Hadley, *in 1720, built first two-mirror system close to its
theoretical specifications. It eliminated most of spherical aberration.
Not long after, *James Short* was the first to make a true
Gregorian system, with paraboloidal primary and ellipsoidal secondary.
He went on to produce well over a thousand Gregorian telescopes.
Due to the added difficulty of making
strongly aspheric convex surface (hyperboloid) for the secondary, the Cassegrain was the last to come into
efficient use,
but soon became the most popular two-mirror arrangement. First
significant telescope of this type was probably *James Nasmyth'*s
20-inch in 1845.
Two-mirror systems using paraboloidal primary are termed
classical Cassegrain, or Gregorian.
Nowadays, unlike the telescope pioneers, we can determine lower-order
aberrations of such a system at the best focus location with a few simple
relations. The system aberration is a sum of aberrations at the primary
and secondary mirror. For the former, the P-V wavefront error at the best
focus for object at infinity is, from
Eq. 7,
12 and
18, given by:
for spherical aberration, coma and astigmatism, respectively, with **K1**
being the primary conic, **D1**
the aperture diameter, **F1** the
primary's focal ratio, **α**
the field angle, **h** the height in the focal plane and **
1**
the primary's focal length.
For the secondary, the effect of close object distance and displaced
aperture stop make the aberration relations more complex. Change in stop
position doesn't affect the coefficient for lower-order spherical, but
the object distance does; thus
Eq. 9.1. applies
(note that 2ψ=Ω). With coma, both
stop separation and object distance affect the aberration coefficient,
thus the appropriate expression is Eq.
15.3. And with astigmatism, stop separation affects the
coefficient, but object distance doesn't, as given with
Eq. 22.
Substituting these coefficients in their respective wavefront aberration
functions which are, same as for the primary, given by
Eq. 7,
12 and
18, results in the
following P-V wavefront error at the best focus for spherical aberration,
coma and astigmatism, respectively:
** **
with **K2**
being the secondary conic, **D2**
the secondary minimum aperture (the cone width at the secondary), **R2**
the secondary radius of curvature, **Ω** the inverse of the
secondary's surface-to-object distance **ℓ** in units of its radius of curvature
(**Ω=R2**/**ℓ**,
see **FIG. 121**, bottom), and **
σ** the secondary to the aperture
stop (i.e. primary) separation in units of secondary's radius of
curvature (according to the sign
convention, **
σ** is numerically negative in
Cassegrain, while positive in the Gregorian, and **Ω** is positive in
both).
Some useful identities are (Ω-1)=(m+1)/(m-1)=(2ρ/k)-1, with **k**
being the relative height of marginal ray at the secondary in units of
the aperture radius, and **ρ**=R**2**/R**1**,
the secondary radius of curvature in units of the primary's (both
parameters extensively used on the next page).
The system P-V wavefront error at the best focus for each aberration is
given by a sum W=W**1**+W**2**.
It is obvious from **Eq. 78-78.1** that with paraboloidal primary,
secondary conic for corrected spherical aberration is K**2**=-(1-Ω)2.
Spherical secondary has coma of opposite sign to that of the primary in
both, Cassegrain and Gregorian, but significantly smaller; aspherizing
it to correct for spherical aberration adds additional offset and lowers
the system coma to the level of a comparable paraboloid.
Astigmatism of spherical secondary is, on the other hand, of opposite
sign to that of the primary in the Cassegrain, and of the same sign in
Gregorian; aspherizing it for corrected spherical adds astigmatism of
the opposite sign in both, resulting in the two ending at a similar
system astigmatism levels. As the astigmatism relation in **Eq. 78.1**
shows, it is independent of object (i.e. primary's image) distance.
According to **Eq. 78-78.1**, needed secondary mirror conic for zero
coma is
σK**2**=-(R**2**2D**1**3/R**1**2D**2**3)-(1-Ω)(1-σ),
and the corresponding conic for the primary (to compensate for spherical
aberration induced by changing the secondary conic), K**1**=-1-[K**2**+(1-Ω)2]R**1**3D**2**4/R**2**3D**1**4.
Alternately, needed mirror conics in terms of the primary and system
focal lengths, and mirror separation are given as K**1**=-1,
K**2**=-[(-**1**)/(+**1**)]2,
and K**1**=-1-2(**1**-s)**1**2/s2,
K**2**=-[(-**1**)/(+**1**)]2-2**1**3/s(+**1**)3
for the classical and aplanatic arrangement, respectively. Note that the
primary focal length** 1**
and mirror separation **s** are always numerically negative, while
the system focal length ****
is numerically positive for the Cassegrain, and negative for the
Gregorian.
The next step in two-mirror system
evolution was modifying mirror
conics
for the correction of coma, in order to obtain
wider well corrected photographic field. Coma-free two-mirror systems are termed
aplanatic
Cassegrain (Ritchey-Chrιtien) and Gregorian. A two-mirror system
interesting from the practical point of view is
Dall-Kirkham, consisting of
ellipsoidal primary and spherical secondary.
Two-mirror arrangement
worth mentioning is the Gregorian with three reflections, which can be made anastigmatic aplanat (corrected for primary
spherical aberration, coma and astigmatism).
Front baffle tube
Although formally not a part of their
optical system, axially configured two-mirror telescopes regularly use
baffle tube in front of the primary, as well as baffle tube around
secondary, to protect the final image from
stray light.
Front
baffle tube should be extended forward from the primary as much as
possible, not to intrude into the light converging from the primary. As
illustration at left shows, the limit is set by the plane where diameter
of the cone converging from the secondary equals diameter of the hollow
core (caused by obscuration by the secondary) in the cone converging
from the primary. The common diameter, in units of the aperture diameter
**P**, is given as:
T=(1+η)/[m+(1/o)]
where **η** is the back focal length **B** in units of primary's
f.l. and **o** is the relative size of central obstruction in units
of the aperture (o=S/P).
Obviously, the distance from this plane to
the final focus is a sum of its separation from the primary **L** and
back focal length **B**. Since L+B=FT, with **F** being the final
system focal ratio, L=FT-B. Substituting in this relation
T=(1+η)P/(m+1/o) and dividing with **1**
- the primary focal length - gives the
limit to the front baffle extension in dimensionless parameters, measured from the primary, in units of
the primary mirror f.l. as:
where ε=L/**1**.
As mentioned, the width of the cone converging from the secondary at this point equals the
hollow core diameter in the axial cone converging from the primary,
caused by obstruction,
given by
in units of the primary mirror diameter. At this baffle position, field
of full illumination is limited to the axial spot. Moreover, baffle tube
won't protrude into the light converging from primary only if with zero
wall thickness (assuming its inner diameter equaling **T**, in order
to allow all axial light from the secondary to pass through). Thus, the
front opening of an actual front baffle tube will have to be pulled
somewhat toward primary, to avoid light intrusion. Since the inner
hollow cone converging from the primary has the effective focal ratio F**c**=F**1**/o,
with **F****1**
being the primary's focal ratio, every mm of shortening the front baffle
tube clears 1/2F**c**
mm for the baffle wall thickness. So, if baffle wall at the opening is
1mm thick, baffle tube extension should be a bit more than 2F**c**
mm shorter than what **Eq. 79** implies.
This will also create some "breathing room" between
baffle opening and cone converging from the secondary. For instance, a
system with /3 primary
(F**1**=3)
and P/3 central obstruction diameter (o=0.33), will have F**c**=9
and need some 10mm shorter front baffle tube. If the secondary
magnification is 4 (so F=12), that will create 10/2F=0.4mm gap between
the converging cone and inner edge of the baffle opening. As
SCT systems show, such tight front
baffle opening causes negligible vignetting (of more concern is the rear
baffle tube opening).
This baffle gap **g** will also create a
small 100% illuminated image field radius **i100**,
which is well approximated with i**100**~(**1**+B)g/(**1**-L)
or, in dimensionless units, i**100**~(1+η)g/(1-ε).
Pulling the front opening closer to the primary would create wider gap,
and increase 100% illuminated field, but would likely require increase
in the secondary baffle diameter, i.e. central obstruction. The two
baffles, around secondary and in front of primary have to work in
concert, i.e. have to be tightened so that no ray passing next to the
secondary baffle pass the baffle tube and fall directly onto the image.
For that to happen, a ray projected from the bottom of baffle tube rear
opening, that touches top of its front opening, is not allowed to clear
the secondary baffle. For that to occur, opening of the secondary baffle
has to be at a distance **d** from the rear baffle tube opening
not exceeding:
**d=H/[2F1b+1]**,
preferably a bit smaller, where H=**1**+E,
with **E** being the length of baffle tube behind the primary, and **
b** is the ratio of baffle tube opening vs. length.
Since the position of this plane determines the width
of cone converging from primary, it also determines the minimum needed
central obstruction. If the secondary is already larger, the two baffles
will be well matched. If, however, the secondary is smaller than the
width of this cone cross section, secondary baffle and corresponding
central obstruction need to be larger.
If the secondary - i.e. its cell - is larger than
this cross section, the baffle should extend toward primary by
2(s-sm)F**1**,
with **s** being the actual secondary's radius (or size of its cell),
and **s**m
the radius of its minimum size. If it is smaller, its cell needs
to be made larger, to fit the cross section of the primary's cone at
distance **d** from the rear end of the baffle tube. Similarly to the
front baffle tube opening, that needs to be corrected for the actual
baffle wall thickness, with the actual baffle extension being shorter by
a bit more than 2F**1**t,
**t** being the wall thickness
(slight vignetting induced to off-axis image points as a consequence of
the tight secondary baffles
negligible even in photographic applications).
◄
8.1.3. Newtonian reflector diagonal flat
▐
8.2.1. General aberrations
►
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