◄
8.3. Three-mirror telescopes
▐
8.4. Off-axis and tilted element telescopes
►

**
8.3.1. Paul-Baker and other three-mirror anastigmatic aplanats**

**
Paul-Baker
three-mirror system** combines concept of the Mersenne
telescope with the unique property of a sphere to be free from primary coma and astigmatism
with the aperture stop at the radius of curvature. The basic, *curved
field PB concept*, uses a pair
of confocal mirrors - concave paraboloidal primary and convex spherical secondary -
with the third, concave spherical mirror placed so that the secondary vertex coincides with its center
of curvature. In effect, the secondary acts as an aperture stop for the
tertiary (**FIG. 129**). As long as radii of curvature of the tertiary
and secondary are identical, their spherical aberration contributions
are also identical and of opposite sign, and the system is free from primary spherical
aberration, as well as from coma and astigmatism.

**
**
**
FIGURE 129**:
Paul-Baker three-mirror anastigmatic aplanat. The basic arrangement
(left) consists from a concave paraboloidal primary (**P**), convex spherical
secondary (**S**) and concave spherical tertiary (**T**). The only remaining
third-order aberration is field curvature, which can be corrected by
extending secondary-to-tertiary separation and aspherizing the
secondary (right). Secondary in either arrangement is at the
center of curvature of the tertiary mirror.

The parameters are
exceptionally simple for a three-mirror telescope: the system is fully
specified by the above description for the primary of ~f/3 and slower.
Faster PB systems require higher-order aspherics added to one or more surfaces for optimum
performance. The PB image curvature is** R****p**=R1/2,
**R****1**
being the radius of curvature of the primary, with the curvatures of the secondary and tertiary cancelling each other.

The *PB flat-field variant* also
uses
paraboloidal primary and spherical tertiary. Primary-to-secondary
separation is given by **S****1****=(1-k)R****1****/2**,
with the secondary conic **K****2=-1+(1-k)3**
and the radius of curvature **R****2****=kR****1**.
Distance to the tertiary - and tertiary radius of curvature - is **S****2=R3=kR1****/(1-k)**.
The **k** parameter is the height of the marginal ray at the
secondary (and tertiary), in units of the aperture radius. It is also the
secondary-to-primary-focus separation in units of the primary's focal
length. In
either case, it is determined by the secondary location, which is
arbitrary (except for the small/large extremes). The tertiary is behind primary
for k>0.268; for k=0.268 the two mirror centers coincide, and for smaller **k** values
tertiary is located between primary and secondary.

The
**
FAA1** and
**FAA2** are also relatively simple systems, rivaling the
Paul-Baker in image quality. The FAA1 consists of the
near-paraboloidal (ellipsoidal) primary and a pair of spherical mirrors, as shown on
**FIG. 130a**. A pair of spherical mirrors in this arrangement can also
work with paraboloidal primary, but can't correct all four 3rd

**FIGURE 130**: **(a)** **
FAA1**, flat-field anastigmatic
aplanat consisting from: (1) concave ellipsoidal primary, (2) small convex
spherical secondary outside the primary focus, and (3) concave spherical tertiary. The field is
somewhat more limited in size than in either Paul-Baker or FAA2, due
to vignetting caused by restricted tertiary size. Configuration
can't vary significantly, but the system
f/ratio is very flexible.
Excellent performance extends to ~f/3, without
a need to add higher-order surface aspherics (in the range of
amateur apertures).

**(b)** **FAA2**, a flat-field anastigmatic aplanat
consisting from: (1) concave ellipsoidal primary, (2) small convex
spherical secondary placed inside the primary focus, and (3) concave ellipsoidal tertiary. Effective
central obstruction achievable with this design is significantly
smaller than with either Paul-Baker of FAA1. This makes it viable
for visual observing, after a diagonal flat is added to make the
final image accessible. Not quite as well corrected as the FAA1 at
very large relative apertures, but the difference is
of no practical significance. This system was first published by
Korsch in a very
similar form; the secondary in that arrangement is hyperboloidal,
but it doesn't seem it significantly changes correction level.

order aberrations (leaving distortion out).
Very low residual coma
remains present. Required system parameters are more complex than those
for the PB. For that reason, only actual prescriptions are given for the
main FAA1 arrangement, the anastigmatic aplanat, and the version with
paraboloidal primary, with slight residual coma (the parameters scale with the aperture, or
with the primary radius of curvature).

- **FAA1**, flat-field anastigmatic aplanat:

**σ1=0.54515,
σ2=-0.11423,
ρ2=0.14433,
ρ3=0.16907,
K1=-0.95,
K2,3=0**

- **FA**, flat-field anastigmat: **
**

σ1**=0.5429,
****σ2****=-0.1115,
****ρ2****=0.1429,
****ρ3****=0.1666,
K****1****=-1,
****K2,3****=0**

**- AP**, aplanat:

**σ1****=0.544,
****σ2****=-0.113,
****ρ2****=0.12,
****ρ3****=0.167,
K****1****=-1,
****K2,3****=0**

where
**σ1** and
**σ2**
are primary-to-secondary and secondary-to-tertiary separation,
respectively, and **ρ2** and
**ρ3**
are secondary and tertiary radius of curvature, respectively, all expressed in units of the primary radius of curvature.
The effective system focal length is typically somewhat (up to ~10%)
smaller than primary's f.l. The parameters should give near-optimum
performance; it can be maintained within a small degree of compensatory
changes in parameters (for instance, the aplanat gives nearly identical
performance with the secondary-to-tertiary distance changed to -0.11R,
and tertiary radius to 0.164R)

The **FAA2** uses concave ellipsoidal
primary, convex spherical secondary forming the focus between the two
mirrors, and concave ellipsoidal tertiary placed behind the primary
(**FIG.** 130**b**). It is very similar to Korsch's
three-mirror flat-field anastigmatic aplanat, in which the secondary is
hyperboloidal (there is no need for aspherizing in my version). Mirror positioning is similar to the flat-field PB, but mirrors
are somewhat easier to make, baffling is better and central
obstruction required for a given field is smaller. Correction of primary
aberrations is even slightly better than in the PB, but the difference
has no practical consequences. An actual system prescription, with the
above notation, is as follows:

**
σ1****=0.41896,
****σ2****=-0.70833,
****ρ2****=0.25,
****ρ3****=1/3,
K****1****=-0.7196,
K**2**=0
and K****3****=-0.3496**

The effective system
focal length is typically ~50% longer than that of the primary. In
setting up the design, a near flat-field condition requires near-zero
Petzval, which is achieved by selecting needed radii of curvature of the
secondary and tertiary for given primary. Spherical aberration is
controlled with the primary conic, coma by the primary-to-secondary
separation, and astigmatism by the secondary-to-tertiary separation (the
former influences coma much more than astigmatism, the latter the other
way around, so it takes a few steps to have them optimally balanced).

**FIG.
131** illustrates performance of the
above systems with the ray spot plot over 1 degree field diameter.

**
**

FIGURE
131: 1o
field ray
spot plots for (left to right) curved and flat-field Paul-Baker (PB),
anastigmatic aplanats FAA1 and FAA2 (FA is FAA1 versions with
parabolic primary), and folded Cassegrain-Gregorians,
3A (reduced
by a factor of 4 to fit in) and
3AA. Aperture diameter is
300mm for all except 3AA (D=400mm). At f/3, higher order spherical becomes noticeable, in
particular in the flat-field PB variant (requires higher-order aspheric
term surface correction). Note that the angular field for 3AA
is different than for the rest of systems, 0°,
0.23° and 0.33°
(for 25mm field radius); also,
spots for the PB (curved field), 3A and 3AA are on best curved image
surface, with the radii -900mm, 333mm and -240mm, respectively. SPEC'S

As mentioned, visual field quality in
the Cassegrain-Gregorian can be expected to be significantly better, due
to its astigmatism partly cancelling that of the eyepiece. For instance,
the RMS error of the above CG at 0.5o
off-axis is some 2.5 times the error of a comparable Newtonian in the
image produced by the objective. With a typical 30mm Kellner - which
still has a few times stronger astigmatism, opposite in sign to that in
the CG - the resulting CG RMS error is about 25% smaller than that in
the Newtonian. With modern eyepiece types better corrected for
astigmatism, nearly full field correction at a certain eyepiece f.l. is possible.

In principle, similar field correction
effects should be expected in combination with a focal reducer lens.

Another system with internal focal plane that produces excellent flat field
performance is in the line of systems pioneered by Korsch and Robb.
This particular one is given with a detailed account of the designing
process (*Analytic design procedure of three-mirror telescope corrected
for spherical aberration, coma, astigmatism and Petzval curvature*,
J.U. Lee, 2009). All three mirrors are aspherized - near
paraboloidal primary, hyperboloidal secondary and ellipsoidal
(prolate ellipsoid) tertiary, but the conics
are relatively mild. The original design (top) is a 0.8m f/22.3 system,
which can have the final image located either in front of the primary,
or with a reflecting diagonal added diverted out to the side and fully
accessible. The system is scalable both, in aperture size and focal
ratio, and can be easily converted into a much more compact 0.4m f/11.1
(below), which is much more compact. Shown is arrangement with a diagonal,
in order to address field vignetting issues arising from its use.

Field quality at the design limit is still exquisite, even with the
primary almost as fast as f/1. However, due to the hole in the
diagonal necessary to transmit the folded light beams, illuminated
field is annular, with a bit less than the inner half radius lost.
The hole is placed around diagonal's center, and circular in form
in the plane of surface. This means that positioned at 45° has
elliptical shape (as shown in the rear view at right; the diagonal
has elliptical shape because it was enetered as a circular surface,
with the diameter equal to the needed major axis).
Extending top and bottom side to make it circular would
negatively affect illumination; in general, the hole should be nearly as
small as needed to pass the field edge beams. Beam footprints on the
bottom show vignetting at the diagonal, i.e. at the hole,
with this size, position and shape of the diagonal hole (vignetting
left and right from the hole, i.e. along X-axis, are identical, and
only shown on both sides for 0.33° field angle).
Red areas of the footprint are vignetted at the hole after reflection
from the tertiary mirror; first vignetting at the hole is not
shown - that part of beam is simply missing - but it is shown
more clearly as the red area on the secondary (botto corner left).
In general, only one side of the field is
vignetted, and that could be addressed by decentering/reshaping
the hole. Note that the minimum central obstruction here is about 0.25D
by the secondary (omitted because of the small scale of footprints),
and that the diagonal hole does not act as central obstruction.

◄
8.3. Three-mirror telescopes
▐
8.4. Off-axis and tilted element telescopes
►

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