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▪ ** **CONTENTS
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10.2.4.2. Houghton camera 2
▐
10.2.4.4. Secondary spectrum reduction
►
#
**10.2.4.3. Houghton-Newtonian**
The only difference between Houghton camera and a telescope in Newtonian
configuration is in the focal length of the mirror, and position of the
corrector. In general, corrector separation in Houghton-Newtonian is
somewhat larger, to accommodate for a smaller obstruction by diagonal
mirror. The two corrector types most interesting to amateurs are
symmetrical aplanatic and plano-symmetrical. The only remaining
aberration with the former is mild field curvature (**FIG.
198b**). The latter retains low residual coma of little or no
consequence for general observing, with similar mild field curvature (**FIG.
198a**). Follows its description in more details.
For n=1.52, the corrector position for zero coma would be at 0.264R (Eq.
145).
Moving it to a more practical location in the Newtonian arrangement,
say, 0.43R, wouldn't change coma coefficient of the corrector (Eq. 138), but would change that of the mirror.
Consequently, sum of the two coefficients is not zero anymore and, for
spherical primary, it determines the system coma aberration coefficient
as cs=(σ*-σ)/R2,
where **σ**
and **σ***
are the zero-coma and the actual corrector-to-mirror separation,
respectively. In this case, cs=0.166/R2,
which is 1/6 of the mirror coma aberration coefficient, given by 1/R2.
This means that the system will have 1/6 of the mirror coma. For an
ƒ/4.5 primary, that would result in the coma-free linear field at the
level of an ƒ/8.2 mirror. It is about half the coma of
the comparable
Maksutov-Newtonian, and ~1/3 the coma of the
Schmidt-Newtonian.
**
FIGURE
198: **
Ray spot plots for Houghton-corrector systems in a Newtonian arrangement.
**(a) **Plano-symmetrical corrector type consists of a plano-convex and plano-concave element, thus does not
cancel the system coma for every corrector location. However, this has little practical
significance.** **At the location supporting diagonal flat in a
regular Newtonian configuration (~0.85f in front of the mirror), some residual coma is visible in the
spots, but hardly any in actual observing.
**(b)** Symmetrical
aplanatic corrector type, with two radii on four surfaces, also corrects
for coma. Coma with plano-symmetrical corrector can be cancelled
if the corrector is moved half-way between the mirror and the focal
plane **(c)**, which would result in larger central obstruction, still acceptable for photographic purposes. The
small circle represents the size of the e-line (446nm) Airy disc.
SPEC'S
#
Chromatic correction is excellent, as long as
the mirror is not significantly faster than ~ƒ/4.
#
10.2.4.4. Two-mirror
Houghton-cassegrain telescope - aplanatic
Full-aperture Houghton corrector can
also be used as a part of two-mirror catadioptric system (**FIG. 199**). The only difference vs.
single-mirror arrangement on the above is that the aberrations of the
secondary are to be added. They are identical to those given for
Schmidt-Cassegrain telescope. The
Houghton-Cassegrain has the advantage of allowing coma correction in an
all-spherical arrangement for any location of the corrector. However,
for that it requires putting a curve on at least three out of four
surfaces.
**FIGURE 199**: Symmetrical aplanatic
Houghton-Cassegrain (**a**) has all four corrector surfaces curved,
with the two lens elements having identical radii of curvature. It can cancel coma for any corrector location. The
plano-symmetrical corrector (**b**) can only cancel coma for a single
corrector location, where the coma of the corrector cancels equal
opposite coma of the primary, determined by the stop location. In practice, due to the coma of the plano-symmetrical corrector
being typically about 3/4 of that of the primary, only of the
opposite sign, and coma of the secondary being typically about 1/2
of the primary's and of the same sign as the corrector's coma, no
corrector location can result in cancelled system coma. In general,
coma is lower in systems with larger secondary, allowing to place
corrector closer to the mirror, but the difference is not
significant. Typically, a two mirror HCT with plano-type corrector
will have ~1/4 of the primary's coma.
For
lower-order spherical
aberration of the primary cancelled, the sum of the three aberration coefficients - for the
corrector (Eq. 134 in general,
Eq. 135 for q1=q2),
primary and secondary mirror (Eq. 113) -
has to be zero.
The coefficient sum
sH+s**1**+s2=sS,
respectively,
will result in the system P-V wavefront error at the best focus of:
W**s**=
-s**s**D**4**/64**
**
(146)
with **D** being the aperture diameter.
Presence of the glass refractive index **n** in **Eq. 130-131**
indicates that spherical aberration can be cancelled only for a single
wavelength. The non-optimized wavelengths are affected by spherical
aberration, resulting in **spherochromatism**. In combination with
spherical mirror, needed corrector's aberration coefficient for
zero spherical aberration at the optimized wavelength with the
refractive index **n** is **
s=-n(n-1)(n+1+**ι**)/4(n+1)(n+**ι**)(n-1+**ι**)R3****=-1/****4R3**
for the index differential
ι=0 (from
Eq.135, with **q** substituted by
Eq. 136). Thus, the aberration coefficient for spherochromatism of the Houghton corrector is given by:
with
ι
being the index differential
ι=n-ni, and **R**
the mirror radius of curvature. Hence, the P-V wavefront error of
spherochromatism at the best focus for non-optimized
wavelengths (i.e.
ι≠0) is:
Ws=s'D**4**/64 (148)
A
good empirical approximation of the spherochromatism coefficient,
yielding ~5% greater value, is given by s'~-0.8ι/R3,
or s'~ι/10ƒ3
in terms of the mirror radius of curvature, or focal length,
respectively. Substituted in **Eq. 148** it
gives the P-V
wavefront error of spherochromatism at the best focus as:
Ws~ Dι/640F**3** (149)
with
**F** being the mirror focal ratio. Negative index differential for shorter wavelengths makes them
over-corrected at the best focus, while longer wavelengths are
under-corrected. Compared to the P-V wavefront error of spherochromatism at best
focus of the Schmidt corrector (Eq.
106),
that of the Houghton is greater by a factor of ~1.6 for the Schmidt with
the neutral zone at 0.707 the radius, and lower by a factor ~0.8 for the
Schmidt with the neutral zone at 0.866 the radius. However, Schmidt
corrector with 0.707 radius neutral zone has the advantage of best foci of all the wavelengths coinciding, ensuring
virtually zero secondary spectrum. This is
not the case with this form of the Houghton corrector (symmetrical aplanat
type), which can have significant
secondary spectrum. While there is no significant difference in
spherochromatism level with other Houghton corrector types, their
secondary spectrum is generally significantly smaller.
Also, the comparison is for
the lower-order spherochromatism alone; at some point higher-order spherical
becomes significant aberration contribution, again, in particular with
the symmetrical aplanat type, and needs to be taken into
account.
These are comparisons
for a single-mirror system, but they remain nearly unchanged for
two-mirror systems as well. The only difference is that the power of the
corrector - and its chromatism - are somewhat lower, due to the
aberration of the primary being partly offset by the secondary.
While spherochromatism does contribute to the level of chromatism of the
Houghton corrector, more of a limiting factor can be its secondary spectrum.
This is definitely the case with the symmetrical aplanat type, while
much less with the plano-symmetrical and asymmetrical type, the later
being limited by spherochromatism (**FIG.****
200**).
In comparison, Schmidt corrector has secondary spectrum practically
cancelled, with the only source of chromatism being spherochromatism,
while the Maksutov has nearly cancelled secondary spectrum, but suffers from
strong higher order spherical at all
wavelengths.
**
**
FIGURE 200:
Longitudinal aberration curves for various Houghton two-mirror systems and a camera.
Stronger lens surface curvature needed to correct faster mirrors
induce exponentially higher fifth order spherical aberration,
increasing defocus for the outer zones, significantly worsening chromatism for the
symmetrical aplanatic corrector type
(**a**). It sets the limit
of acceptable chromatism for this corrector type at ~ƒ/3 mirror
focal ratio. Plano-symmetrical Houghton corrector doesn't correct for
coma, but suffers significantly less from chromatism, both, in a
single- or
two-mirror arrangement (**b**). In comparison, Maksutov corrector
produces much tighter longitudinal focusing (secondary spectrum),
especially for the outer zones, where the best foci are
(**e**).
However, it suffers from unacceptable residual spherical aberration,
nearly evenly at all wavelengths. The SCT has by far the best
overall correction below ƒ/3
(**d**), approached only by that of
optimized (asymmetrical) Houghton corrector
(**c**). SPECS:
HCT/SCT/MCT
While not a factor with
ordinary Newtonian-style systems, chromatism level of two-mirror
Houghton systems sets the limit to how fast the primary can be at ~ƒ/3
with the aplanatic single-glass Houghton corrector (**FIG. 201**).
Reduction in the chromatism can be obtained by allowing for a relatively
small amounts of residual coma, allowing for more weakly curved lenses,
as illustrated on **FIG. 201c**.
**
**
FIGURE 201: Aplanatic two-mirror
Houghton-Cassegrain with symmetrical corrector type begins to show excessive chromatism as the
relative aperture of primary mirror exceeds ~ƒ/3. It results from the combined effect of
increased secondary spectrum and higher order spherical aberration (**FIG**.** 95**). A system with an
ƒ/2.5 primary
**(a) **has more than doubled chromatic blur diameter compared to a system with
ƒ/3
primary mirror **(c)**. Reduction in the amount of system
chromatism in Houghton-Cassegrain two-mirror systems is possible with
some residual
coma allowed **(d)**. This
level of chromatism is unnoticeable visually, and so is the amount
of coma in the system, which is, approximately, at the level of a
200mm ƒ/9 paraboloid. However, best performance is achieved
with the asymmetrical corrector type, having as little as 0.075 wave
RMS error in the violet h-line, which compares to 1.3 wave RMS
with the symmetrical type. SPEC'S
Fortunately, Houghton
corrector as an optical arrangement offers various possibilities for
significant reduction in the level of chromatism. Some that are most
appropriate are presented in more details in the following section.
◄
10.2.4.2. Houghton camera 2
▐
10.2.4.4. Secondary spectrum reduction
►
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