**telescope**Ѳ**ptics.net **
▪** **
** **
▪
▪**
** ▪
▪▪▪▪
▪
▪
▪
▪
▪
▪
▪
▪
▪ ** **CONTENTS
◄
8.4.1. Herschelian
▐
8.4.3. TCT 2
►
#
**
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
**Schiefspiegler **and **Yolo**.
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 by 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
~f/20 and smaller.
**FIGURE 134**: The two-mirror TCT final field forms as a combination
of the effective field at each mirror. Since the combining fields are
each a circular off axis area of mirror's field, they are not symmetric
about their respective center points. Since they are likely to have not
only different magnitude of astigmatism, but also different ratio of
coma vs. astigmatism - each changing differently with the field angle -
once the aberration of their combined centerfield point is minimized
with a given combination of tilt angles, aberration distribution over
the field is more or less asymmetrical.
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
**τ**1,
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:
W**a1**=-D1(**τ**1+**α**)2/8F** **
and ** **
W**c1**=(**τ**1+**α**)D1/48F**2**
respectively, with **D**1 being the aperture
diameter, **α**
the angular differential, in radians, between the tilt angle **τ**1
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 **τ**2
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 the best focus location, are given by:
W**a2** = -**τ22**D**22**/4R**2
**
and ** **
W**c2** =** τ2**(1-**Ω**)D**23**/12R**22**
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 the 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
f/24.7 schiefspiegler, with concave
f/14.7 spherical primary and spherical convex secondary,
with the primary-to-secondary separation of 965mm. Radii
of curvature **R**1=**R**2=-3240mm, secondary to primary's
projected focus distance * ***l**=-655mm,
the inverse of relative object distance for the secondary
**Ω**=**R**2/**l**=4.95, secondary mirror effective diameter **D**2=44.5mm,
and the tilt angles **τ**1=2.65°
and **τ**2=6.65°
for the primary and secondary mirror, respectively. Best image field is
astigmatic but flat.
In the field center (α=0), primary
mirror astigmatism and coma contribution is
**W**a1=-0.00193mm
(3.5 waves P-V, in 550nm wavelength units), and **
W**c1=0.00048mm
(0.9 waves P-V), respectively. With the secondary mirror contributions of
**W**a2=0.00198mm
and **W**c2=-0.00032mm,
the system aberrations are
**W**aS=**W**a1+**W**c1=0.00005mm,
or 0.09 wave P-V of Seidel astigmatism (higher-order is entirely
insignificant), and **
W**cS=**W**c1+**W**c2=0.00016mm,
or 0.29 wave P-V of coma. This is close to 0.27 wave P-V output by the
exact ray trace (OSLO) for the final center field aberration (relatively
low astigmatic deformation has little effect on the P-V error of coma).
This system is titled "anastigmatic" due to the astigmatism being practically
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. That could be remedied by giving to the
secondary toroidal shape: as the astigmatism plot shows, either
tangential (along vertical diameter) focus would have to be extended,
or sagittal (horizontal) brought closer. Resulting system (below) is
practically free of coma and astigmatism.
The vertical secondary radius is now -3,186mm, and image
tilt is now 6.5°. Obviously, this arrangement would allow for
significantly faster systems.
Coma could be also diminished with the reduction in
the primary (and secondary) tilt angles, but that would cause secondary
intruding into the light falling onto the primary (at 1.6° primary
tilt, the center field RMS drops to 0.032 wave, but the
secondary is halfway into the axial pencil of light falling onto the
primary). Better option is to increase secondary tilt to correct
coma, and add a third mirror to take care of astigmatism, which
leads a system like Terry Platt's Buchroeder.
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° counterclockwise to the axial ray, as shown in inset top
left).
Since the aberrations of two tilted mirrors are a sum of their
respective partly offsetting aberrations, their center-field residue
will increase in proportion with aperture, but exponentially in inverse
proportion to the system focal ratio. For instance, at a twice as fast
focal ratio, coma contribution will be four times larger by each mirror
due to the radius cut in half, and nearly doubled due to both tilt
angles doubled (nearly, because it is in proportion to the sum of tilt
angle, and comparably small field angle). As a result, center field coma
in f/12.3 system of this aperture would have been nearly eight
times larger.
Also, since the field astigmatism here is a residual to the opposite in
sign field astigmatism of each mirror, roughly comparable in magnitude,
it is not increasing with the square of field angle, as in
centered systems, but approximately in proportion to it (the rate varies
somewhat with the field radius).
Note that the raytrace indicates that the primary tilt is not quite
sufficient to place the secondary out of incoming rays for 0.5°
field angle. Required tilt for that is 2.9°,
which would increase the center-field aberration by about 10%, and also
slightly
(about 5%) the off axis astigmatism. Even t 0.25°
field radius the full size secondary would be touching marginal ray of the
incoming light.
To answer a question whether it is better to have eyepiece aligned with
the axis, or with image plane (i.e. tilted with respect to axis), we'll
raytrace a somewhat more relaxed 108mm f/27 Kutter design by Dave Groski. The
eyepiece is 32.5mm f.l. Koenig with apparent field of view of ~52°.
Astigmatic field for mirrors alone (top right) is given without adjustment
for tilt, and the ray spot plots are for the best focus (since coma
originates from mirror tilt, it is all-field coma with the same
orientation and magnitude across the field).
Axis-aligned eyepiece (bottom left) produces curved field with required eye
accommodation between zero (bottom edge) and +2.4 diopters (top edge,
with one diopter of accommodation given by f^{2}/1000, **f**
being the eyepiece f.l.). Aligning eyepiece with the field plane does
reduce eyepiece astigmatism one one side of the field, but at a price
of increasing it somewhat on the other, also inducing asymmetrical
lateral color error and distortion (the latter indicated by the mismatch
between the tip of converging cone and image plane, representing
undistorted, Gaussian image). Eyepiece-alone spots (bottom corner right)
indicates that most of the lateral color error is inherent to the
eyepiece. However, except the field edge, most of astigmatism is
contributed by the mirrors. The relative insensitivity of the eyepiece
is due to the very slow focal ratio and moderate image tilt.
Effects of misalingment are significantly more
pronounced with faster systems, like Stevick-Paul
and off-axis Newtonian, which tend to have
more of image tilt as well (also, larger eyepice AFOV for given f.l.
shows exponentially larger aberrations toward field edge).
Center-field performance in this configuration can be improved making a
system catadioptric, by placing a strongly
tilted plano-convex lens in the path of light converging from the
secondary. The gain is in the center field correction, with the outer
field remaining of similar quality for given focal ratio. Longitudinal
chromatism is practically non-existent, but lateral color error may not
be in larger apertures (~150mm and larger at ~f/20).
As shown below, lateral color is asymmetric, diminishing toward one,
while increasing toward the opposite direction. It can be corrected by
making the lens slightly wedged. In this case, the wedge angle is 0.049°
(2.94 arc minutes); in order to compensate for the change in
astigmatism, lens' tilt is slightly reduced.
Another option is to make the secondary
toroid. It allows for
better overall correction of aberrations. The secondary can be either
convex, or concave. The former will, all else equal, result in a slower
system, but it is the system configuration that can also have
significant impact in this respect. In the conventional configuration
(top below), the design minimum in the field center is less than a half
of the minimum error in the comparable catadioptric design, while the
off axis error is roughly cut in half. Making secondary smaller requires
it to have more of a tilt (with the radii kept identical), but will
result in somewhat faster, more compact configuration (bottom below).
The aberration level, both on and off axis, is only slightly higher for
given system focal ratio (note that the toroid is specified under
"surface tag data" as the reverse of the radius value; for instance, in
the 100mm f/24.7 system the secondary radius in the tangential plane -
the one containing axial and chief ray - is -3,240mm, while it is
-3182.3mm in the plane orthogonal to it, with the inverse value of
-0.00031424).
In the arrangement with concave secondary, the
secondary tilt has to have sign opposite to that in the system with
convex secondary, with light reflected from it crossing the incoming
light to focus out on the opposite side of the incoming beam. The first
system of this type, and the best known, is
*Yolo* by Arthur
Leonard (top below). In it, toroidal form of the secondary is achieved
by placing it into a specially made cell ("warping harness") which folds
it by the action of a tightening nut (top below). The system shown
is Leonard's 12-inch f/15 (slightly tweaked) scaled down to 110mm
aperture. Originally, it has hyperboloidal primary, which also
compensates for spherical aberration of the secondary (which indicates
that secondary here induces roughly three times more spherical
aberration, since it would have been zero if both mirrors were
paraboloids). Raytrace shows that aspherizing does not produce
appreciable gains. Making system faster in this configuration would
increase aberrations approximately in inverse proportion to the focal
ratio. An alternative is to use smaller, more
strongly tilted toroidal secondary (mid/bottom).
The level of aberrations remains similar to the
slower Yolo, mainly due to the less of primary tilt required. The
primary is considerably faster, thus parabolizing it has more of an effect
(secondary's contribution is negligible due to its small size),
but it still doesn't make for appreciable difference: axial blur
nearly the size of the Airy disc implies less than 1/12 wave P-V
wavefront error. There is little difference in correction between
arrangement w/tilted focusing cone (middle) and Newtonian-like
configuration (bottom). While astigmatism can be always corrected, for
any given secondary tilt angle there is only one secondary r.o.c. value
at which this toroidal surface will also correct for axial coma. Off
axis, there is some residual coma but, as wavefront maps show,
astigmatism dominates. Trefoil is still significantly smaller than coma,
thus only noticeable on axis (note that CVX in OSLO prescription stands
for toric curvature, as the reciprocal of the radius value, not
"convex").
These are the traditional two-mirror TCTs.
Some of the most notable more recent developments in this field are
described on the next page.
◄
8.4.1. Herschelian
▐
8.4.3. TCT 2
►
Home
| Comments |