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▪ CONTENTS
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8.4.1. Herschelian
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8.4.3. Off-axis Newtonian
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8.4.2.
Two-mirror tilted
component telescopes
By replacing the flat in a Herschelian with a toroid,
in the same configuration, or with a second curved mirror directing
light back toward primary, needed mirror separation can be reduced
roughly two two three times, for a similar level of correction. Field
asymmetry is also reduced. Best
known systems of this type of tilted component telescopes (TCT) are the variants of Yolo and
Schiefspiegler.
The former is introduced by S.A. Leonard in the early 1960s. It uses two concave mirrors, one
of them toroidal (a sphere deformed into a toroid in a specially
designed cell). In general, it achieves better - and exceptionally good
- performance than the Schiefspiegler (Schief
for short), and the systems can be considerably faster. On the other
hand, it is also more complex. The Schief is created by Anton Cutter,
and originally uses a pair of spherical mirrors, the secondary being
convex. The aberration here is present in the field center, and has to
be kept acceptably low by limiting a system to quite small relative apertures,
typically
~/20 and smaller.
Aberrations of
a tilted-mirror system are a sum of the aberrations at each mirror.
The principles of aberration compensation can be outlined using the
basic two-mirror arrangement. Here, aberrations at the primary are simply off-axis aberrations for the field angle equal to a mirror tilt angle
τ1,
typically ~3°, and possibly residual spherical aberration
to compensate for that induced by the secondary (not needed in
traditional TCT arrangements, due to mild surface curvatures).
For the secondary, the
object is the image projected by the primary. Since the
secondary is centered around the reflected axis of the primary (i.e.
around the chief ray of the axial cone), its
aperture stop coincides with its surface for the center-field
aberrations, not only for spherical aberration, but also for
center-field astigmatism and coma. For other field points, the chief ray
height at the secondary differs from zero, and the aperture stop is
shifted to the primary.
These features make two-mirror TCT alike
axial two-mirror systems in one, and different in another respect. The
similarity extends to secondary magnification and spherical aberration,
for which the same general two-mirror system relations (Eq. 78 and
Eq. 80,
respectively) apply to TCT's as well, including center-field astigmatism
and coma, formally abaxial aberrations. For TCT's abaxial aberrations
outside the field center, regular stop-shift relations for abaxial
aberrations of the two-mirror system secondary apply.
This means that for astigmatism wavefront
error, which doesn't change with object distance,
and for coma,
which changes with object distance, the aberration relations that
apply for the secondary mirror contribution in the field center are
Eq. 19, and
Eq. 15, respectively. For all
other field points, astigmatism and coma at the secondary are calculated
either from general two-mirror system relations (Eq.
82-82.1), or from Eq. 15.2-15.3
and Eq. 19.1 and
Eq. 22 for coma and
astigmatism, respectively.
The consequence of TCT mirror tilt is
that it places
off-axis aberrations in the field center by effectively moving the field
far off-axis - with individual aberration
contributions of the primary and secondary being determined by their
tilt angles.
The primary mirror field is a circle far off from the
optical axis (left on the above illustration),
with strong astigmatism and coma at the same level along the field
cross-radius centered at
the optical axis (the field center for axially oriented mirror). Along the
central field
meridian orthogonal to this radius, coma and astigmatism increase
according to the angle measured from the optical axis. In between, there
is a gradual transition from one form of aberrated field to another.
Actual TCT field radius is
typically only a fraction of the tilt angle at the primary.
TCT's primary mirror center
field astigmatism and coma, commonly at the level of 10 and 2-3 waves
P-V, or higher (respectively) are, for best focus location, given by:
Wa1=-D1(τ1+α)2/8F
and
Wc1=(τ1+α)D1/48F2
respectively, with D1 being the aperture
diameter, α
the angular differential, in radians, between the tilt angle τ1
and the angle determined by the point distance from the optical axis
(for the tangential - vertical - field meridian, it corresponds to field angle relative to the field center), and
F the focal ratio of the primary.
The main goal of the two-mirror TCT's
secondary is to minimize
this center-field aberration induced by tilted primary. This is
accomplished by tilting it to induce offsetting aberrations,
with the secondary mirror tilt angle τ2
measured with respect to the axial ray reflected from the primary (i.e.
the chief ray for axial cone).
Center-field astigmatism and coma at the secondary, also as the P-V wavefront error
at best focus location, are given by:
Wa2=-τ22D22/4R2
respectively,
with R2,
D2,
τ2
being the secondary radius of curvature, effective aperture diameter
(i.e. the minimum secondary size) and tilt angle, respectively,
α being the
field angle as defined earlier, and Ω=R2/l
the inverse of relative object distance for the secondary
l (secondary to projected focus of
the primary separation) in
units of the secondary radius of curvature. Since both, R2
and l are, according to the
sign convention,
numerically negative, the sign of Ω is positive.
Note that
only a convex secondary, with numerically negative radius of curvature, offsets astigmatism
and coma generated by tilted primary.
For the field points outside the center, astigmatism and coma at the secondary,
as the P-V error at best focus location, are given
by:
with
σ being the secondary to the
aperture stop (i.e. primary) separation in units of secondary's radius
of curvature, numerically negative for convex secondary (note that these relations are valid
for two-mirror system secondary in general, with the tilt angle
τ=0 for axial systems). The field angle
α, as mentioned
before, is the angular differential between the tilt angle of the
secondary to the reflected optical axis of the primary and field point
angle measured from the point to the reflected primary's axis.
Numerically, it can be either positive or negative (zero for the field
center).
With coma and astigmatism being in different proportions throughout the
field of the two mirrors, due to their different tilt angles (that of
the secondary is commonly in excess of double the primary's tilt angle), the final image in
a simple TCT, after the center-field aberration is minimized, has only partially corrected field - unless of
very small relative aperture - with uneven distribution of off-axis
aberrations. Uneven compensation of astigmatism can, and most often does induce image tilt. Even
the very field center is a compromise: since it is impossible to
generate with the secondary's tilt the exact proportion of astigmatism and
coma that are induced by primary's tilt, some level of residual aberration
remains present.
These consequences of the uneven aberration match of two different far off-axis field
segments limit simple two-mirror TCTs - and most more complex tilted
systems as well -
to small relative apertures and smallish aperture diameters.
EXAMPLE: Anton Kutter's 110mm
/24.7 schiefspiegler, with concave
/14.7 spherical primary and spherical convex secondary,
with the primary-to-secondary separation of 965mm. Radii
of curvature R1=R2=-3240mm, secondary to primary's
projected focus distance l=-655mm,
the inverse of relative object distance for the secondary
Ω=R2/l=4.95, secondary mirror effective diameter D2=44.5mm,
and the tilt angles τ1=2.6°
and τ2=6.5°
for the primary and secondary mirror, respectively.
In the field center
(α=0), primary
mirror astigmatism and coma contribution is
This system is titled "anastigmatic" due to the astigmatism being nearly
cancelled in the field center; however, there is still strong
astigmatism present in the outer field. Center-field coma could be
cancelled by tilting the secondary more, to 9.7°, but astigmatism would
become unacceptably large.
In the
course of years, arrangements with three and four mirrors are added, in
various combinations of surfaces. A few of those systems can achieve
satisfactory performance at focal ratios closer to /10, but most of
them are in the /15 to /25 range. Further reduction of aberrations,
allowing for faster focal ratios and more compact instruments, can
be achieved with some type of lens corrector. Among the best examples of
this type of tilted-element telescopes are
catadioptric schiefspieglers
designed by Ed Jones, using a pair of tilted PCX and PCV lenses to completely
correct the astigmatism of a tilted 12-inch /7 mirror. Other
arrangements are possible, with very good field correction despite
unusually fast focal ratios for a TCT. Chromatic error is also reduced
to near-negligible. While this type of systems can have high degree of
correction, especially in the mid field - and for visual use the entire
field of view - they commonly require more strict tolerances in spacing,
centering and tilt angles than a comparable axial Newtonian telescope.
This is the only negative of this type of system, since imperfect
collimation can more easily make a difference between high and
mediocre optical quality in actual use. On the other hand, both the
primary and the lens pair in the Jones catadioptric schiefspiegler are
significantly less sensitive to miscollimation than secondary mirror in
the Ritchey-Chrιtien of comparable aperture and -ratio.
Reducing the
aperture size allows for larger relative apertures, but there is not much
room in that direction. An interesting compact solution is given by
Herrig's two-mirror
4-reflection system. And a design that surpasses all other tilted mirror
telescopes in regard to image quality is the
Stevick-Paul three-mirror
system (FIG. 88). It is the only TCT anastigmatic
aplanat in the /10-/12 relative aperture range, with only a mild field curvature
remaining.
Despite
some of these systems being very well corrected, obstruction-free and
relatively insensitive to miscollimation, they never became really
popular, even in the small-aperture range. Telescopes of this kind are
usually
built by amateur enthusiasts. Among the reasons are probably
their odd appearance, relatively
complicated element positioning and bulkiness of the tube assembly. A
system somewhat less affected by these drawbacks is the off-axis Newtonian
(other off-axis configurations are possible, but even less price-competitive).
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FIGURE 87: The two-mirror TCT final field forms as a combination
of the effective field at each mirror.
The ray spot plots illustrate the field match in this two-mirror TCT
system. Secondary's field is farther away from its axis, thus its ray
spots have larger proportion of astigmatism, lower degree of
variation in the size of aberrations across the field's vertical
diameter, and less of the angular spot inclination toward the ends
of field along the diameter orthogonal to it. These differences between
the two combining field result in the final seemingly
disarrayed, asymmetrical field (the plot is for the best image surface,
tilted at ~4.5° to the axial ray).
FIGURE 88:
Stevick-Paul 3-mirror TCT bases its exceptional performance
on the freedom from off-axis aberrations of a sphere with the stop at
the center of curvature. The convex secondary sphere (2) is confocal with
the paraboloidal primary (1), producing collimated beam and acting as an
aperture stop for the concave spherical tertiary (3). Having identical radii of
curvature, secondary and tertiary cancel each other's spherical
aberration. A small flat (4) makes the final image accessible. The ray
spot plot is for an 8" /11.7 system. The
circle within the square shows Airy disc size relative to the
spot size (unrelated to the field angle): image quality is
practically perfect across the field (design scheme and field
generated by