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10.2.4.5. Secondary spectrum reduction
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10.2.4.7. Houghton-Cassegrain: designing
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#
**10.2.4.6. Two-mirror Houghton
telescope with plano-symmetrical corrector**

As mentioned, a
single-glass plano-symmetrical Houghton corrector type,
using a pair of plano-convex and plano-concave lenses of equal surface
radii, can cancel mirror spherical aberration inducing chromatism considerably
lower than comparable symmetrical aplanatic version (**FIG.
204**). The price to pay in a two-mirror
system, however,
is introduction of coma somewhat greater than
that in a comparable SCT with spherical mirrors (**FIG. 204a**), only of the
opposite sign (which indicates that coma contribution of the corrector
and secondary is stronger than that of the primary). Consequently, the
coma can be reduced by moving corrector closer to the the primary, rather than
farther away, as it is the case with SCT systems.

**
**

FIGURE 204:
Chromatism in a
Houghton-Cassegrain using single-glass plano-lenses is significantly
reduced compared to a single-glass aplanatic corrector (FIG.
201a), but the field coma is is
uncorrected and approximately by 1/3 stronger than in a comparable SCT
(**a**).
It is possible, however, to reduce and even cancel the coma by moving the
corrector closer to the primary, increasing the coma of the primary
until it balances out with corrector's coma. This, of course, requires larger minimum secondary mirror.
At the minimum secondary size of 0.5D[1] - thus
with the corrector at about half the focal length from the primary - and
magnification m=2, the system becomes
ƒ/5
(**b**), all four curved surfaces (two
at
the corrector and the two mirrors) have identical radius of curvature and
the coma is effectively cancelled (the system **a** has the minimum
secondary size of 0.3D). This works due to the coma of
the primary mirror increasing as the stop (corrector)
separation decreases, thus cancelling more of the corrector's coma.
Obviously, this wouldn't work with the SCT, since the Schmidt corrector
has practically zero coma. Therefore, reduction of the coma here requires increase in
the stop separation. The coma level in the two systems nearly equalizes
when the back focal distance is reduced from 9" at
ƒ/10 to 5" (for an
ƒ/8 system); the linear aberration is reduced in the HCT, while nearly
unchanged in the SCT (**c**).
However, HCT chromatism in the violet is about four times larger (for
0.707 SCT neutral zone placement), which puts it at the level of a 4"
~ƒ/200 achromat. Since the chromatism of a system changes with the
fourth power of the primary's relative aperture, a single glass
corrector HCT with an ƒ/3 primary has the chromatism level
of an SCT with an ƒ/2 primary. As
Eq. 148 implies, the
chromatism also changes in proportion to the aperture.
SPEC'S

[1]**For the BK7 glass index.
Higher refractive index lowers coma of the corrector, thus allowing for
greater corrector-to-primary separation and, for n~1.65-1.7,
significantly smaller secondary. Higher refractive index
also results in increased chromatism, particularly
in the violet, making
a two-glass corrector desirable option.**

**
**

FIGURE 205:
Ray spot plot for the
chromatism and field coma in the comparable
ƒ/10 SCT and MCT. The blur size can be poor indicator
of the level of chromatism, as it is the case for aberrations in
general. While the geometric blur size indicates similar levels of
chromatism in both, the ƒ/2.5/10 plano-Houghton and
ƒ/2.5/10 SCT, the actual chromatism measured by
the wavefront error* in the violet h-line
(405nm) is some four times greater in
the HCT (0.24 wave RMS of the combined secondary spectrum and
spherochromatism) than in the SCT (0.06 wave
RMS of spherochromatism). The Houghton chromatism could be
reduced by the use of a slightly different glass type for the
second lens element, as described in the previous section (for instance, by replacing
BK7 glass in the rear lens by BK8 and the increase in lens spacing to
15mm, the h-line error reduces to 0.16 wave RMS, and the r-line from
0.07 to below 0.05). The gain is, in general, smaller, since most of the chromatic error
here comes
from the spherochromatism, not secondary spectrum. In the Maksutov-Cassegrain,
chromatism is even lower than in the SCT, but the image quality suffers from higher order spherical aberration, not correctable without adding
higher-order surface term. The system shown has little better than 1/4 P-V wavefront error of mostly higher order spherical, resulting
in slightly over 1/20 wave RMS error (comparable to 1/6 wave PV of
lower-order spherical) in the optimized e-line. It probably can be
reduced somewhat by further optimizing, but not significantly. This underlying
correction error increases toward non-optimized wavelengths, also
combining with the very small defocus (secondary spectrum) element. As a
result, the h-line chromatism for the MCT falls between the other two at
~0.1 wave RMS. The combined h-line/r-line RMS error at ~0.15 wave
RMS is still somewhat smaller than in the Houghton (~0.18 wave RMS). SPEC'S:
SCT
MCT

***It
should be noted that the energy lost to the Airy disc grows
exponentially with the RMS wavefront error.**

While the coma in the
plano-lens HCT is not as severe as to make it unusable, it does
compromise its field quality to a significant degree. In the SCT, it can
be corrected either by aspherizing the secondary, or by moving the
corrector significantly farther from the primary, as mentioned. The
latter is not an option for the HCT, due to its coma originating from
the corrector and secondary, thus requiring more (opposite) coma from
the primary; in other words, it requires corrector moved significantly
closer to the primary. Secondary conic for a coma-corrected HCT needs to
be oblate spheroid - not an easy figure to make, especially with smaller
convex mirrors. Fortunately, there is another, easier way of correcting
the HCT coma, and that is by employing a sub-aperture
corrector placed at the front opening of the baffle tube (**FIG.
206**). This simple doublet fully corrects

**FIGURE 206**: 200mm diameter
ƒ/9.75 aplanatic all-spherical Houghton-Cassegrain with a plano-lens
full-aperture corrector and integrated sub-aperture corrector, also
consisting of a pair of plano-lenses. The sub-aperture corrector is
very easy to fabricate - it can be cut out of the full-aperture
corrector - and at a favorable location, the least sensitive to
misalignment and not
interfering with the accessory end of the telescope. It only
requires minor adjustment (weakening) in the power of the
full-aperture corrector. Ray spot plots show the only remaining system
point-source aberration, low astigmatism. Field curvature is weaker than in
the single corrector
system. Chromatism is significantly reduced in comparison to the single (full-aperture) corrector version
(compare** FIG. 113a**): the h-line (405nm) RMS wavefront error
is reduced to 0.08, or 1/3 the error of the single-corrector system,
with the r-line (707nm) remaining nearly unchanged at and
practically negligible. All four lens elements are of a single
common glass; achromatizing one or both correctors would likely
result in a further significant reduction of chromatism which, as
the ray spot diagram to the left shows - already surpasses required
chromatism level of a true apo refractor (ray spot diagram generated by
OpTaliX-LT
raytracing software).
SPEC'S

the system coma, at the price of somewhat stronger, but still low
astigmatism. Stronger astigmatism has a positive effect of lessening the
field curvature. Chromatism is also reduced, and the overall performance
level is high, even without detailed system optimization (**FIG. 207**).
As with the camera, aberration coefficients of the sub-aperture
corrector are different than those given for the full-aperture
corrector, due to it being placed in a converging cone of light.
Consequently, the position factor **p** for both lens elements of the
sub-corrector has its value changed according to
the relation given for
Eq. 97. In a two-mirror
system, however, it is the image formed by the secondary (without the
sub-corrector in place) that is the object for the first lens element of
the sub-corrector.

◄
10.2.4.5. Secondary spectrum reduction
▐
10.2.4.7. Houghton-Cassegrain: designing
►

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