Since the systems are scaled, they may not be fully optimized, but they
should be close enough that further improvement wouldn't produce
appreciable practical benefit. The medial triplet astrograph has nearly
identical configuration and correction level to Honders-Riccardi. All
elements are made of a single crown glass, yet the systems practically cancel
all five Siedel aberrations and chromatism, producing in addition flat image field.
The single-reflection configuration is quite forgiving to the increase in
relative aperture; correction level doesn't change significantly at f/3, or even somewhat faster ratios.

In fact, the CE is a core around which this system is built. It can be
said that both, front lens and field corrector are accessory elements
added to minimize its aberrations. As for the CE aberrations, they are
nearly reduced to those of the refracting surface in the light
converging from the front lens (thus, with its exit pupil - i.e. stop -
at the front lens, and its object the image formed by the lens), and
aberrations of the reflecting surface of the radius of curvature **R**4
in the light refracted by **R**3
(thus, with its exit pupil at the image of the front lens formed by **R**3,
and the virtual image of the front lens' image formed by **R**3
as its object). The distances are found using
paraxial approximation, and the
aberrations (monochromatic) from corresponding
aberration coefficients.

The refracting CE surface also induces chromatism. Since most of the
system's optical power is produced by the reflective CE surface, the refracting
CE surface - as well as two positive lenses - are of weak power and
relatively low chromatism. Secondary spectrum can be corrected by
balancing optical powers of the positive front lens and negative CE
refracting surface **R**3
with their separation. However, it wouldn't affect lateral color created
at **R**3
due to its displaced stop and, considering large stop separation, it would be
significant. Solution to the lateral color problem requires an
additional optical element; a positive field lens can bend deviant chief
rays of different wavelengths dispersed by a concave refracting surface
of the CE, but doing so it induces other aberrations, including
secondary spectrum. To offset for this secondary spectrum, the
front-lens/CE combo needs to be left appropriately imbalanced in this
respect.

Likewise, monochromatic aberrations of the three elements need to be
balanced in order to nearly cancel out. In general, chromatic
aberrations need to be minimized first, because they are determined by
elements power and separation. Subsequent bending of the surfaces,
while keeping elements' powers nearly unchanged, doesn't affect appreciably
chromatic correction, but does monochromatic aberrations, which can be
minimized in this manner.

**EXAMPLE**: A sort of re-construction of
an actual Honders design - this particular one created by Massimo Riccardi,
thus could be referred to as Honders-Riccardi - a 300mm f/3.7
catadioptric dialyte with a positive front lens, catadioptric rear
element, and positive field lens, all three elements of BK7
crown (n=1.52, V=64.4). Based on the final system already known - which
makes it easier - it illustrates basic design principles and its
optical architecture.

Starting with the front refractive surface of the
catadioptric element (CE), its radius **R**3
is to be somewhat longer than the desired final focal length,
considering the need for a positive field lens to correct for lateral
color. The separation **S** between the front lens and CE should be
**S**~900mm, to make image accessible using a diagonal flat mounted
onto the front lens. If ~1100mm is desired for the system, **R**3~-1200mm
is as good a starting value as any. Standard center thickness of glass
element with reflective surface is 1/6 (so called "full thickness") to
1/10, so the CE's thickness **t**2
should be in that range. These values are the basis for
approximating needed radius of curvature for the reflective surface **R**4
as somewhat smaller than 2R3,
which would be needed for the **R**4
value if the light falling at it would be collimated. Since we can
expect the light falling at it to be somewhat diverging, in order for
the final secondary spectrum to be balanced at the positive field lens,
the first approximation of **R**4
should be somewhat smaller, say, 10-20%, for R4~1.7R3~-2000mm. Note that both radii are, according to the
sign convention,
numerically negative, and the thickness is positive for light traveling
from left to right.

With the power of **R**3
approximately known, we can also approximate needed power of the small
field lens to cancel its lateral color. Approximately, lateral color is
proportional to hd/fV, where **h** is the height of the
chief ray on the
surface, **d** is the effective distance past the refracting element in the
direction of light travel, **f** is its focal length and **V** its
Abbe number.

For the field lens, the chief ray height **h**F
is determined by the position of its
exit pupil **Ex**F
which, with the initial assumption that light has little or no
interaction with **R**3
on its way to the field lens, is the image of the exit pupil **Ex**3
for **R**3
formed by the reflecting surface. The chief ray appears as if coming
from the center of the exit pupil, thus determining the chief ray angle
at its origin on the reflecting surface. With the exit pupil **Ex**3
for **R**3
being at the front lens, its image of it, whose location is determined
from Eq. 1, is the exit pupil
**Ex**4
for the reflecting surface **R**4,
which re-mages it into the field lens' exit pupil **Ex**F.

Using the approximate values of S~900mm, R3~-1200
and R4~2000
gives Ex3=-984mm
and Ex4=72,400mm
(for the pupil distance for the reflecting surface -984-t2).
The positive **Ex**3
value implies that the chief ray refracts toward axis after passing **R**3.
The **Ex**4
value implies that it reflects from **R**4
nearly parallel with the optical axis, slightly diverging. For the **R**3
surface on its return, it will appear as if coming from **Ex**4.
For the field lens, it will appear as if coming from the exit pupil **
Ex**F,
formed by **R**3
imaging of **Ex**4
(unlike the axial rays, **R**3
surface does affect off-axis rays of the light reflected from **R**4).
With -1.52 for **n**, and -1 for **n'** (due to light traveling
from right to left), **Eq. 1** gives the exit pupil distance for the
field lens ExF=2240mm,
to the right of the CE. Since the chief ray appears as if coming from
the pupil center, it is diverging toward the field lens, and will hit
its surface approximately at the height h~[1-1000/2240]h3~1.45h3.

Also, by diverging light, this second refraction at **R**3
is opposite in effect to the first one, and stronger. Since the height
of chief rays is nearly identical for both, the combined effect with
identical dispersion power is determined by the respective surfaces'
optical power, i.e. focal lengths. For air-to-glass surface (first
refraction at **R**3)
the focal length is fA=nR/(n-1)=-3510mm,
and for glass-to-air surface (second refraction) it is given by fG=R/(n-1)=-2310,
with the combined (diverging) power of fC=-6760mm.

As a result of the negative refractive power after
second refraction at **R**3,
the chief rays of opposing (vs. optimized green) wavelengths will
effectively reverse their divergence, turning toward each other, red
(long wavelengths) moving lower and blue (short wavelengths) upward.
Since the new negative divergence is nearly half as strong as the
positive divergence of the first refraction at **R**3,
the red and blue will cross after they travel about double the CE
thickness (**t**2).
The point of this divergence is now the relevant distance **d** past
refractive surface (or element) for **R**3.
Since the combined power of both refraction is nearly three times weaker
than that of the first **R**3
refraction, this point of convergence will lie nearly 3 times the double
center thickness of the CE, or ~200mm from **R**3,
toward the field lens. Thus the effective distance **d**3
past the imaginary combined refractive surface can be approximated as d3~S-200~700mm
(note that the reflection on **R**4
does change the geometry of color divergence, but since it results from
its magnification factor and doesn't actually influence the chromatism,
it can be neglected, for simplicity).

Lastly, will take a distance **d**F
from the field lens to the final image to be **d**F~100mm~.
With these results, the condition for near-zero lateral color, (hFdF/fFVF)~-(h3d3/fCV3),
with VF=V3
and hF~1.45h3
reduced to (1.45dF/fF)~(700/fC),
gives fF~1.45dFfC/700~fC/4.8.
With fC=~-6760mm,
needed power of the field lens to neutralize lateral color resulting
from two refractions at **R**3 is fF~1400mm.
Since neutralizing divergence still doesn't correct lateral color,
unless the colors nearly cross at the neutralizing element, the field
lens needs to be twice as strong, thus with the focal length f/font>F~700mm
(analogy to the optical power: a lens of identical opposite power will
collimate diverging beam; for reversing it into conversion, it needs
to be twice as strong). With BK7 glass and plano-convex lens, this
translates into RF=(n-1)fF/n~240mm
(the sign depending on lens orientation).

Taking one more hint from the actual design to save
approximating best lens shape for balancing monochromatic aberration,
will approximate the radii as R1/R2~-0.5
for the front lens, and near plano-convex field lens. With the front
lens' focal length put at ~6 times its separation from the CE, or f1~5400mm,
by substituting R2~-2R1
into Eq. 1.2, this
gives **R**1~4200mm
and **R**2~-8400mm.
Similarly, **R**F~-(n-1)fF/n~-240mm
(convex to the right). The respective lens thicknesses are **t**1~30mm
and **t**F~10mm.
With the parameters already approximated, **S**~900mm, **R**3~-1200mm,
**R**4~-2000mm,
**t**2~30mm,
this completes the first contour of the final system. Since the lens
power, so far, has been adjusted for minimizing lateral color,
additional adjustments for minimizing secondary spectrum, through
separations/radii adjustments, are likely to
be needed.

Plot
to the left shows the result after the above parameters ("first
approximation") were input in OSLO . Mainly due to relying on an
established main frame, the resulting 300mm f/3.4 (somewhat faster than
planned) system is ready for optimization. Spherical aberration is only
1/4 wave P-V, lateral color roughly minimized, coma, and to a smaller
degree astigmatism significant farther out in the field, with the
secondary spectrum being by far the worst of the aberrations. Nearly
optimized 300mm f/3.7 Honders system to the left designed by Massimo Riccardi (SPECS)
has all of the aberrations minimized
to negligible, or near-negligible by relatively small adjustments in
radii and separations. Secondary spectrum is at the level comparable to
4" f/70 standard achromat, and can be further reduced by a factor of 3
with a few minor tweaks (framed inset on the right).

Expectedly,
correction of aberrations gets even better with slower systems of this
type. As the
plot shows, there is no aberrations to speak of with a 300mm f/7.4 Honders-Riccardi variant (SPECS).
The field is limited to that usable with 2" eyepiece barrel, but even at
2f (about 3 inches) off-axis, the
combined aberrations are still well below "diffraction-limited" level.
This particular system is not optimized for the minimum central
obstruction size, which is here about 0.25D. As small as 0.15D central
obstruction is possible with re-arranged systems, but at a price of
sacrificing sufficient illumination in the outer field. In that respect,
longer-focus Honders variants such as this one do not differ appreciably
from other catadioptric systems with full aperture lens corrector (Schmidt-Newtonian, Maksutov- and Houghton-Newtonian). With mildly curved
spherical surfaces, they are relatively easy to fabricate, and somewhat
more compact than other comparable systems.

What makes Honders telescope really stand
out is its unsurpassed correction level; Houghton-Newtonian is closer
to it than the other two types, but still suffers from appreciably
stronger astigmatism and field curvature (not a factor in visual
observing, but a minus in astrophotography). Visual chromatism defined
by F/C (blue and red line, respectively) is entirely negligible; Toward
the ends of visual spectrum, correction is still excellent, although
unbalanced: at the best focus, h-line (405nm) and r-line (707nm) RMS
wavefront errors are 0.1 and 0.012, respectively (it can be balanced by
using another ordinary glass, PK2, for the field lens, to 0.04 and 0.026
wave RMS, respectively).

This Honders-Riccardi
system is also fairly tolerant to miscollimation: the most sensitive
element in this respect, the CE, doesn't induce appreciable aberrations
if despaced or decentered up to a couple of mm. A 0.1 degree tilt only
induces lateral color with F/C line separation of about the Airy disc
diameter, due to effectively misaligned field lens (in comparison, a
300mm f/5 paraboloid with the same amount of tilt would have 2/3 wave
P-V of coma in the field center). All these qualities make
this type of catadioptric dialyte a very viable alternative for
high-quality instruments.