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▪ ** **CONTENTS
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8.2.4. Dall-Kirkham telescope
▐
8.2.6. Miscollimation, close focusing
►
**
8.2.5. Unusual two-mirror
systems: Loveday, Schwarzschild, Couder**
Two-mirror systems can be modified so
that the secondary reflects light back to the primary mirror, with the
final focus forming after this last, third reflection. Best known system
of this kind is **Loveday-Cassegrain**,
using a pair of confocal paraboloids (Mersenne arrangement). After the
third reflection (the second from the primary) the final focus is formed
beyond the secondary. Coma is identical to that in a comparable
Cassegrain, while the astigmatism is smaller by a factor of (m2+η)/(1+η)km2,
resulting in lower field curvature as well. By aspherizing the mirrors
somewhat more, systems corrected for either coma or both, coma and
astigmatism can be obtained. In the Cassegrain configuration, however,
design constraints impose severe limits to the useable field size, with
the added drawback of relatively large effective central obstruction. In
the Gregorian arrangement, while the central obstruction remains
relatively large, much wider fields are possible, with the only
remaining aberration being field curvature (**FIG. 125**). Such system
was, to my knowledge - credit to Mr. Charles Rydel,
President of the Commission des
Instruments of the
Societe Astronomique de France
-first described by Shaffer.
**FIGURE 125**: Two-mirror
3-reflection system in the Gregorian arrangement. Concave secondary
mirror (**S**) reflects light back to the primary (**P**),
which then forms the final focus through an opening on the
secondary. Correction of all three primary point-image aberrations,
spherical, coma and astigmatism is possible with ellipsoidal primary
and hyperboloidal secondary mirror. The only remaining aberration is
a relatively strong field curvature. The final system's focal ratio F is
larger by nearly 1/3 than focal ratio of the primary. Originally, the arrangement
was first published by Shaffer, but somewhat better
corrected systems of this kind are achievable.
These systems are
effectively three-mirror systems, and aberration coefficients are
more complicated. It would suffice here to give a working
prescription. Relative system parameters (units of the primary
radius of curvature) of the Gregorian two-mirror 3-reflection
anastigmatic aplanat are very simple:
S/R**1 **= 0.7248
R**2**/R**1 **= -0.7084**
**
K**1** = -0.428**
**
K**2**
= -6.58
**
**
and **
**
R**p **= 0.147R**1**,
**S** being the primary-to-secondary separation, **R****1**
and **R****2**
the primary and secondary radius of curvature, respectively, **K****1**
and **K****2**
the primary and secondary conic, respectively, and **R****p**
the Petzval (image) curvature which, in the absence of astigmatism,
coincides with best image surface.
These parameters are nearly optimized for an f/3
system; they are scalable by either aperture, or primary's F-number.
Scaling by the aperture doesn't require any changes, while slower system
require slightly stronger secondary conic to optimally re-balance
spherical aberration.
All aberrations - except field curvature - are well corrected.
Axial correction for 400mm f/3 system is 0.04 wave RMS of balanced
higher-order spherical, with the balanced higher-order coma limiting
diffraction limited field to 0.6°
radius (with 50% linear central obstruction, the field size limit is about 1.5 degree in diameter. Higher order spherical aberration increases inversely to the 6th
power of focal ratio, limiting the focal ratio at this aperture size to
~f/2.7 for diffraction limited axial correction.
With a more realistic system, such as f/4.9 with f/3 primary,
the overall correction is exceptional (0.0036 wave RMS on axis, and 0.022 wave at 0.75 degrees off), and field size is limited only by acceptable
central obstruction. With about 50% linear central obstruction,
the unvignetted field is 1.5 degree in diameter.
Field curvature is strong, requiring
either curved detector or field flattener. The simplest flattener form, a
singlet positive plano-convex lens with the front surface radius
R=(1-1/n)R**p**=0.15(1-1/n)R**1**,
(the actual radius should be about 10% stronger, since the flat side
induces roughly 10% as much astigmatism of opposite sign; better correction, however,
gives biconvex lens slightly weaker in power than what is indicated by the equation) with small compensatory changes in the conics
to optimize for coma and spherical, achieves good correction, except for
lateral color. Its correction requires adding at least one more glass
element, which can be as simple as a meniscus of equal radii in front of
the field flattener. Plot below shows spots for such combination with
the last glass surface 1.8 mm from the image, for 430-700nm range.
Correction in the green is not significantly worse than in the
all-reflecting arrangement, with the residual secondary spectrum being the
primary source of chromatic error. Nearly eliminating chromatic error
would require achromatizing one of the elements; also, somewhat more
complex corrector is needed for larger corrector-to-image separation.
Correction level of this arrangement is somewhat better than in the original
Shaffer arrangement (**R2**=**S**=0.75**R1**, **
K1**=-0.405, **
K2**=-6.04),
which has similar correction level at 40% smaller aperture
and ~f/3.5.
Another unusual astrographic system consists from two concave mirrors,
with the secondary inside the focus of the primary **FIG. 126**). It
was derived by Karl Schwarzschild as a solution for two-mirror system
with best correction of aberrations possible. Schwarzschild found that a
two-mirror system can correct only four Siedel
aberrations: the remaining one is either field curvature, or
astigmatism. The variant with astigmatism as the only remaining
aberration
is usually referred to as *Schwarzschild telescope*, and the
alternative with no astigmatism but with curved image field is known
as *Couder telescope*.
**FIGURE 126**: For given system focal ratio, the Schwarzschild
flat-field aplanat (SPECS)
is significantly more compact than Couder (SPECS).
The latter is about as long as a comparable Schmidt camera, with zero
chromatism, but no other advantages. Its image radius is about twice
more strongly curved than in the Schmidt, it does not have efficient
control of higher order aberrations at focal ratios significantly faster
than f/3, and its image is less accessible, being located in the
converging cone. Couder arrangement can be made more compact by using
faster primary, but the limit to it is imposed by its high surface
conic. With an f/8 primary, the system would be only moderately shorter,
but it would be at the limit of acceptable correction due to
higher-order residuals, and the mirror would become nearly difficult for
fabrication (since the aspherizing glass volume for given mirror
diameter changes inversely to the cube of
focal ratio, a -16 conic f/8 hyperboloid is about as difficult to make
as f/3.2 paraboloid). On the other hand, the more compact Schwarzschild
has quality field limited by astigmatism. Both, Schwarzschild and Couder
could have the final focus made accessible at the side, by use of a
diagonal flat, but it may not be sufficient for broader range of
accessories, especially with the latter.
In the former, the minimum relative size of the secondary (in units of
aperture diameter) **k** needs to be related
to the secondary magnification **m** as k=(1-m-m2)/(1-m2)=1-m/(1-m2).
This implies that **m** has to be smaller than 1 for k<1, i.e. for
the secondary smaller than primary. Unavoidably, the final focus falls
in between two mirrors. Also, m<1 implies that the secondary is concave.
For the maximum acceptable secondary size of k~0.5, the corresponding
secondary magnification, from m=[(4k2-8k+5)0.5-1]/2(1-k),
is m~0.4. Larger secondary magnifications require smaller secondary, but
secondary size reduction is limited by image accessibility.
On the other hand, due to the wide primary-to secondary separation,
needed secondary size to keep the outer field well illuminated becomes
excessive at k~0.5; in the above system keeping 2-degree field fully
illuminated would require clear secondary mirror surface diameter of
nearly 2/3 of the full aperture, and the effective obstruction almost
certainly exceeding 70% of the aperture diameter. Thus the practical
secondary magnifications value cannot deviate significantly from m~0.45
k~0.44 level.
With **k** and **m** determined, mirror conics can be obtained from
Eq. 86-87. After
substituting for **k**, the conic relations become:
K**1 **
= -[2(1-m-m2)+m3]/m3
and
K**2**
= (1-m)(1-m2)/(1-m)3
For viable level of secondary magnifications m~0.45,
the corresponding mirror conics are K**1**<-8.6
and K**2**>2.1
for the secondary.
As with all two-mirror systems, the secondary mirror
radius of curvature is given by R**2**=ρR**1**,
with ρ=mk/(m-1).
The field is flat, but the astigmatism-cancelling field curvature limits
quality field size. Since the secondary contributes nearly as much of
astigmatism as the primary, the system astigmatism is approximated by
double that of the primary, thus as the P-V wavefront error W~(αD)2/2R,
where **α** is
the field angle in radians, **D** the aperture diameter and **R**
the mirror radius of curvature (for lower-order astigmatism, RMS
wavefront error is smaller than the P-V by a factor 241/2).
For the above 200mm f/6.7/3
Schwarzschild telescope, astigmatism
wavefront error at 1° off-axis should be
nearly 4 wave P-V, or 0.8
wave RMS (raytrace gives 3.7 wave P-V). Note that the above relations
are for lower-order aberrations. Due to the strong mirror conics, higher
order spherical aberration and coma are not entirely negligible. In this
particular system, the residual higher order spherical aberration was
somewhat over 1/7 wave P-V, and the residual coma was making blurs
somewhat elongated. Both were corrected by widening the mirror
separation by about 0.5%, and reducing the secondary's conic by about
5%.
In the Couder curved-field anastigmatic aplanat, k=1-2m, hence the
maximum secondary magnification is lower than in the flat-field aplanat.
Needed mirror conics are:
K**1**
= -(m3-2m+1)/m3
and
K**2**
= (m3+m2-m)/(1-m)3
For k~0.5 or smaller, the corresponding secondary magnification is
m~0.25, or larger. Again, image accessibility requirements limit
reduction of the secondary size to about a third of the primary mirror.
Since at m~0.25 and k~0.5 needed secondary size to prevent vignetting of
the outer field becomes excessive, the secondary magnification is
confined to a narrow range around m~0.3 and k~0.4. The corresponding
mirror conics are K**1**~-16
and K**2**~-0.5,
respectively. In general, the primary is more strongly aspherised than
in the Schwarzschild telescope, while the secondary becomes a mild
prolate ellipsoid. From the fabrication point of view, the twice more
strongly aspherised primary is mainly offset by switching to the prolate
ellipsoid secondary. However, the drawback is significantly lower
practical secondary magnification, potentiating the image accessibility
problem.
With zero astigmatism, the image curvature equals system's
Petzval radius of curvature, given as:
R**P** = mkR/2(m-1-mk),
with **R** being, as before, primary's radius of curvature. For, say,
m=0.3 and k=0.4, field curvature is R/-13.7, remaining strong with the largest **R** values that meet practical requirements for system
length. For the above 200mm f/10/3
Couder telescope, the corresponding image curvature is R**c**=R**P**=-293mm.
At 1° off-axis, it would induce as much as
4.8 waves P-V (1.4 wave RMS) wavefront error of defocus. Clearly, curved
detector surface matching the image curvature, or a field flattener
lens, is a must.
Similarly to the Schwarzschild system, residual higher-order spherical
aberration - only about half as large - can be corrected by extending the
secondary radius of curvature by about 1% (it also minimizes residual
higher-order coma, but it was already negligible).
With larger/faster systems, the higher-order residuals grow
exponentially.
While this type of two-mirror system can achieve
very good correction, it also has several potential drawbacks which,
combined, probably prevented
its more widespread use.
**FIG. 127** illustrates degree of field correction of all-reflecting two-mirror telescopes
in their typical configurations, from classical Cassegrain and Gregorian,
through their aplanatic arrangements, to Dall-Kirkham, Schwarzschild/Couder
and Loveday.
**
**
FIGURE
127: Best image surface ray spot plots for (left to right)
classical Cassegrain (**CC**) and Gregorian
(**CG**), aplanatic Cassegrain (**AC**)
and Gregorian (**AG** -
aplanatic Gregorian - with twice as fast primary as the AC for
similar secondary size, and
**AG*** with
the same f/3 primary, but over 0.7D minimum secondary size required), Dall-Kirkham
(**DK**),
with the spot size reduced three times in order to fit in,
and the Gregorian
3-reflection anastigmatic aplanat (**AA**). Aperture diameter
D=400mm for all. In order to minimize DK coma, practical systems
use slow primary (~f/4) and low secondary magnification (~2.5).
An f/12 system based on these parameters would have the coma
over angular field nearly twice lower,
and over two and a half times lower over the corresponding linear field,
compared to the above f/8
system. The circle outlines the system e-line Airy disc diameter.
SPEC'S
◄
8.2.4. Dall-Kirkham telescope
▐
8.2.6. Miscollimation, close focusing
►
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