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10.2.4.6. Plano-symmetrical HCT
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10.2.4.8. HCT comparison
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#
**10.2.4.7. Houghton-Cassegrain telescope: designing**

In designing a two-mirror full aperture Houghton
catadioptric telescope, the most
important additional factor with respect to a single-mirror system is that of the secondary mirror. It induces
both spherical aberration and coma of the opposite sign to that of the
primary. In effect, it changes the amount of aberration to correct. The new corrector requirement in this respect is obtainable by
correcting the system aberration coefficient for spherical aberration
and coma for the effect of the secondary.

For the conventional, symmetrical
Houghton corrector,
the secondary-to-primary mirror aberration contribution for the
lower-order spherical is given by:

with **s1**
and
**s****2**
being the aberration coefficients of spherical aberration for the
primary and secondary, respectively and, as before, **k** is the
relative height of the marginal ray at the secondary in units of the
primary's semi-diameter and **m** is the secondary magnification.
Since k=(1+η)/(m+1), the relation s'=(1+η)(m2-1)/m3 can be more
practical for systems with the fixed value of back focal length in units
of primary's f.l., **η**.

For the lower-order coma, the secondary contribution relative to
that of the primary mirror is:

with **c1**
and **c2**
being the coma aberration coefficients for the primary and secondary,
respectively, and **σ**
the stop (corrector) to primary separation in units of the primary
radius of curvature (this means that 2σ-1 is numerically negative for
σ<0.5, which is required for compact systems).

Both, **s'**
and **c'** are negative in the Cassegrain, due to the two contributions being of the
opposite sign. In the Gregorian, the two concave spherical
mirrors induce spherical aberration of the same sign, thus **s'** is
positive, and significantly greater than in a comparable Cassegrain.
Since it is proportional to the total of system's spherical aberration,
it indicates that Houghton-Gregorian systems, in general, would require
significantly stronger corrector, with correspondingly higher
spherochromatism and higher-order spherical.

The two Gregorian mirrors induce coma of opposite signs, making **c'**
negative; however, due to the stop placed farther away from the
spherical primary, the secondary mirror coma contribution is likely to
be significantly greater than that of the primary, i.e. **c'** smaller than -1.
This, in turn, makes the required shape factor for zero coma negative
(requires curve reversal, the weaker radius first). In other words, coma
and spherical aberration cannot be both corrected in the
Houghton-Gregorian system with symmetrical corrector (with plano-symmetrical
type, higher-order spherical and spherochromatism are considerably
lower, but coma is stronger than with symmetrical type).

The two ratios give the effective amount of aberrations to be cancelled
by the corrector. Thus, for spherical aberration, the aberration
coefficient in the form (1+s')/4R3
replaces 1/4R3
for a single-mirror system, and for the coma it is now (1+c')(1-σ)/R2
instead of
(1-σ)/R2,
**R** being the primary mirror radius of curvature. Both
coefficients are for the spherical mirror surface; for aspheric surfaces,
appropriate values for **s'** and **c'** can be obtained from
Eq. 115/116.

With these substitutions, the zero spherical aberration shape factor is
qs=n(n-1)(1+s')ƒ3/(n+1)R3,
and for the coma qc=2n(n-1)(1-σ)(1+c')ƒ2/(n+1)R2.
Setting qs=qc
closely approximates the **lens element focal length** (absolute value) for the corrector
cancelling spherical aberration and coma as:

while the appropriate shape factor **q** (obtained from the **q**
factor for zero spherical aberration) is:

Again, **s'** and **
c'** are numerically negative in the Cassegrain. Needed lens radii are then obtained
from
Eq. 142-143, or from the following summary. A small secondary mirror location adjustment and slight
correction of one of the two curves are usually sufficient to bring a
system obtained through this procedure to a near-optimum level. Note
that this calculation is for lower-order spherical aberration; as the
primary relative aperture exceeds ~ƒ/2.5,
higher-order spherical becomes significant with the standard, aplanatic
Houghton corrector, and will result in a discrepancy between
results obtained for the lower-order aberration alone and the actual
system error.

**
EXAMPLE**: 200mm ƒ/2.8/10.5 symmetrical-type
BK7 corrector
Houghton-Cassegrain (SPECS) with the primary-to-secondary separation 0.72ƒ1,
thus **k**=0.28, secondary magnification **m**=4 (determining
secondary radius of curvature in units of the primary's as
ρ=0.373, and
back focal distance in units of the primary focal length
η=0.4) and
the corrector-to-primary separation in units of the primary's radius
of curvature **σ**=0.38.
From **Eq. 154**, secondary-to-primary mirror spherical
aberration ratio is **s'**=-0.328 (opposite in sign) and, from **
Eq. 155**, the coma ratio is **c'**=-0.63. Substituting these
two values into **Eq. 156** gives the approximation of needed
**lens element focal length** (absolute value) as
**
ƒL**~0.683R,
**R** being the primary mirror radius of curvature. Needed
**lens
shape factor** for BK7 glass (n=1.519) is, from **Eq. 157**, **q**=0.067.
It determines needed **lens radii** as
**R****2,4**~ 2(n-1)ƒL/(1-q)~0.76R
and
**R****1,3**
= (1-q)R2,4/(1+q)
= 0.66R.

Ray
trace shows that coma is corrected, but the system suffers from 0.15
wave RMS (546nm unit) of spherical aberration when the two lenses
are in contact (1mm center separation). It is mainly the consequence
of the higher-order aberration term not accounted for. Increasing the separation to
1.6mm cancels the lower-order contribution, reducing the error to
0.12 wave RMS of
higher-order spherical aberration. Further increase in lens
separation to 4mm induces as much of lower-order spherical needed to
balance the combined aberration to 0.025 wave RMS. Similar effect is
achieved keeping the lenses in contact by either by relaxing all corrector's
radii by 0.7% (larger by a 1.007 factor), or to relax radii 1/3 to
746mm (absolute value), with no appreciable change
in other aberrations. However, the increase in lens separation
worsens longitudinal chromatism, which drops roughly at the level of a
4" ƒ/8-9 doublet achromat.
In comparison, relaxing 1/3 radii while keeping lens separation at
1mm maintains near minimum chromatism, at the level of a 100mm ƒ/18
achromat.

**Chromatism
**of the Houghton
corrector can be reduced, while keeping it symmetrical radii-wise,
either by opting for two-glass
corrector, or by significantly relaxing the radii, at a price of
re-introducing some coma. If, for instance, we opt for **
ƒL**=|Re|,
with the effective radius to correct **R****e**=R/(1-s')1/3=-1279mm,
due to to the primary's aberration being diminished by the secondary
contribution (spherical aberration is inversely proportional to R3),
the needed shape factor
** q**=n(n-1)ƒ3/(n+1)Re3=0.313.
This determines corrector radii (absolute values) as **R****2,4**~ 2(n-1)ƒL/(1-q)~1932,
and **R****1,3**
= (1-q)R2,4/(1+q)
=1011. The chromatism is more than twice smaller than the minimum in
the standard arrangement, and can be
considered negligible (the system is also somewhat slower, at ƒ/11).
The coma increases to the diffraction-limited level (0.80 Strehl) at
5mm off-axis (0.3 degrees in diameter, or about 50% larger than in a
comparable SCT with spherical mirrors).

Better
redesigning option is to achromatize corrector. Just by switching
the rear element glass from BK7 to BK8, slightly relaxing 3rd radius
(from -739 to -745mm) and increasing lens separation to 1.7mm, chromatism is reduced
nearly four times vs. minimum in the standard arrangement,
without further optimization.

Larger
relative secondary size generally reduces both, chromatism (due to
greater offset of primary's spherical aberration and, therefore, weaker lens required) and coma (due to greater offset of the
primary's coma).

Two-mirror Houghton system can also be arranged with a pair of
plano-convex and plano-concave lenses (plano-symmetrical
Houghton corrector) with identical curvatures (as
illustrated on
Fig. 108, right) of
opposite signs. In this case, needed
**corrector lens radius**
(absolute value) for corrected 3rd order spherical aberration is, from
Eq. 142, closely approximated by:

Since for the corrector consisting of plano-lenses **q** equals 1 for
both elements, and needed lens element focal length, absolute value, is approximated
by
ƒ~[(n+1)/n(n-1)(1+s')]1/3|R|, its coma coefficient is, from Eq. 138,
approximated by:

Evidently, the **s'**
parameter affects both, spherical aberration and coma of the corrector.
With the primary mirror coma coefficient given by
cp=(1-σ)/R2,
the system coma aberration coefficient can be approximated by cs~ccr+(1+c')cp,
and the resulting P-V wavefront error of coma at the best focus is:

with **α**
being the field angle in radians.

**
EXAMPLE**: 200mm
ƒ/2.8/11 HCT
with the identical optical configuration as above, only with plano-symmetrical BK7
Houghton corrector (n=1.519) instead of symmetrical, needs the **lens element focal length**,
absolute value,
|**ƒ**|~[(n+1)/n(n-1)(1+s*)]1/3|R|=1.68|R|,
with the corresponding **surface radii** value |**R****1,3**|=(n-1)|ƒ|=0.872|R|=977.
With the two lenses in near-contact (5.5mm center separation), ray trace gives less than 1/50
wave P-V of spherical aberration, and nearly six times lower
chromatism than the symmetrical corrector type aplanatic HCT
telescope.

With
**s'**=-0.33, the coma aberration coefficient for the corrector
is
**c****cr**~-0.84/R2
which, with the mirror pair coma coefficient **c****m**=(1+c')(1-σ)/R2=0.23/R2,
gives the system coma coefficient as **c****s**~ccr+(1+c')cp~-0.61/R2.
At 0.1 degree off-axis, it results in the P-V wavefront error of
**coma**
W=0.00028mm, or 0.51 wave P-V for the e-line (546nm) - nearly 15%
less than in the ƒ/2/10 commercial SCT. That nearly
coincides with ray-trace results.

In general, the plano-symmetrical two-mirror Houghton system has coma
comparable to that of an all-spherical (mirrors) SCT, which also can be
cancelled or reduced by aspherizing the secondary (oblate ellipsoid), or
by reducing stop separation.

Houghton-Cassegrain coma, as already mentioned, can be cancelled for any
corrector location with the symmetrical corrector type, as
well as asymmetrical aplanatic varieties. It is only a matter of
generating exactly as much of the coma of opposite sign by the corrector
to cancel that of the two mirrors (which is identical to that in the
Schmidt-Cassegrain, given with
Eq. 115). Calculating needed properties of the asymmetrical
aplanatic Houghton corrector for two-mirror systems is based on the same
principles as for the symmetrical types. It is matched against the
combined spherical aberration and coma of the two mirrors, with the lens
focal lengths and radii for zero coefficients obtained using
Eq. 134 and
Eq. 137.

Since Houghton corrector
induces only negligible astigmatism, SCT relations
for **astigmatism**
and
**field curvature**,
Eq. 116-117, are applicable to the Houghton-Cassegrain as
well.

**
Misalignment sensitivity**
of the Houghton-Cassegrain secondary is given by the general two-mirror
system relations (Eq.
91.1-91.2). For the corrector itself, raytrace suggests sensitivity
to decenter similar to that of the Schmidt corrector, and sensitivity to
tilt about tenfold greater (similar to the tilt sensitivity of a
full-aperture meniscus corrector). Sensitivity to despace is practically
non-existent for all three corrector types. Sensitivity to despace
between the two lens elements of the Houghton corrector is very
forgiving, and so is the lens thickness tolerance.

◄
10.2.4.6. Plano-symmetrical HCT
▐
10.2.4.8. HCT comparison
►

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