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10.2.2.3. Schmidt-Newton telescope
▐
10.2.2.4.1. SCT off-axis
aberrations
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#
**10.2.2.4. Schmidt-Cassegrain telescope (SCT)**

Among the most popular commercial
designs, the *Schmidt-Cassegrain telescope* (or SCT, **FIG. 174**)
owes its success mostly to the possibility of relatively inexpensive
commercial production of a
well-performing Schmidt corrector, particularly in combination with two
spherical mirrors. The road for the SCT commercial production was paved by Celestron's Tom Johnson who, back in
the '60-ties, pioneered the method for
corrector's mass-production.

**FIGURE 174**: Schmidt-Cassegrain
telescope is a Cassegrain-like two-mirror system combined with a
full-aperture Schmidt corrector. Various combinations of
corrector separation and mirror
conics are possible, with
somewhat different image field properties. Prevailing commercial
arrangement is a compact design with fast spherical primary and
usually also spherical secondary mirror, resulting in ~ƒ/10
system. All-spherical SCT is
corrected only for spherical aberration, with low astigmatism, as well as relatively strong field
curvature and coma remaining. The corrector also induces low
level spherochromatism, undetectable visually and negligible for most
photographic
applications.

Optical
effect of the corrector on the system parameters is small, but not
negligible. It slightly increases the marginal ray height on the
primary, as determined by its
refraction angle
**
δ**
(Eq. 107), with
this ray being then directed toward a different focus focus point, as a
result of corrector's interference.

With, say, 200mm
ƒ/2 spherical primary, combined with a 0.707 neutral
zone corrector at σ1=0.4R**1**
(primary's radius of curvature) in front of it, marginal ray on the primary will
be only ~0.23mm higher (for ~0.71 corrector power in an all-spherical
arrangement). The two, corrector and primary, "act" as a prolate ellipsoid (K~-0.71),
nearly 200.5 mm in diameter, with only slightly extended marginal focus.
Since it still retains ~29% of the original D(mm)/32F
longitudinal spherical aberration, in order to find out secondary
magnification we need to trace the 0.707 zone ray, the only one whose
height and orientation after passing the corrector and primary didn't
change, and to whose focus the rest of rays will be directed after
reflection from the secondary.

Taking the 0.707 ray as marginal, the
primary becomes 141.4mm diameter ƒ/2.79 mirror (the 0.707 ray focuses at
the mid point of the original longitudinal defocus, 1.56mm inside the primary's paraxial focus).
Slightly shorter focal length - and the corresponding radius of curvature -
increase the effective secondary-to-primary radius of curvature ratio ** ρ** from 0.3125 (in an ƒ/2/10 mirrors-only system
with paraboloidal primary) to 0.3137, with the
relative ray height at the secondary in units of the aperture radius
**k** reduced from 0.25 to 0.2471, and the resulting secondary
magnification **m** reduced to ~4.7.

Applying
this magnification value to the effective 200mm ƒ/1.992 primary results
in a final ƒ/9.38 system. So, if the two mirrors without
corrector would form an ƒ/10 system, optical effect of the corrector
changes it into ~ƒ/9.4. In order to have an ƒ/10 system with an ƒ/2
primary, the secondary needs to be slightly more (~1.5%) strongly curved, thus with the
R2/R1
radii ratio ρ~0.308, for the secondary
magnification m=~5.05. The relative back focal distance in units of the
primary focal length is only slightly reduced, from 0.5 to η~0.49.

Aberration-wise, there are two
significant differences between the SCT and all-reflecting Cassegrain
varieties. One is that the SCT can be made free from both, coma and
astigmatism, while an all-reflecting arrangement can only correct for
one. On the other hand, the Schmidt corrector induces some sphero-chromatism.
Follows overview of SCT axial aberrations, spherical (as a sum of
spherical aberration contributions
of its three elements) and spherochromatism.

**SCT spherical aberration**

The only significant monochromatic
aberration introduced by the Schmidt corrector in collimated light is
spherical. Its purpose is to offset
spherical aberration on the two mirrors, resulting in a
spherical-aberration-free system. Thus, the P-V wavefront errors for
lower-order spherical aberration at the best focus for an SCT system can be
written as:

with
**s**cr
being the corrector spherical aberration coefficient, **d**
the pupil (aperture) radius and the mirror aberration coefficient
**s**M
being the sum of the individual mirror
spherical aberration coefficients, sM=s1+s2.
The individual mirror coefficients for object at infinity are same as for
a two-mirror system alone (Eq.
9.2 and Eq. 9
for the primary and secondary, respectively), given by:

for the secondary, with
**K**1
being the primary conic, **R**1
the primary radius of curvature, **k** the height of marginal ray at
the secondary in units of the aperture radius, **m** the secondary magnification
and
** ρ**=R2/R1,
the secondary radius of curvature in units of **R**1.

Note that in order to be able to directly add the two coefficients, the
secondary aberration coefficient had to be corrected for the difference
in apertures by multiplying it with **k**4 factor, the relative height of
marginal ray at the secondary in units of the ray height at the primary.
The two forms for the secondary aberration coefficient have parameters
interchangeable through k/ρ=kR1/R2=(m-1)/m=1-(1/m).

Thus, the system P-V wavefront error at
the best focus can be written as:

with **P**
being the needed corrector *power* to cancel system spherical
aberration, and ss={}/4R13 being the system
aberration coefficient.

Of course, for zero system spherical
aberration, the aberration coefficient for the corrector **s**cr,
which is related to the corrector *power* **P** as scr=-P/4R13,
needs to be equal to the sum of mirror aberration coefficients
**s**M,
and of the opposite sign. If the primary is spherical, to cancel its
aberration alone, the corrector aberration coefficient needs to be scr=-*b*/8
(with the aspheric term *b*=2/R13,
it comes to scr=-1/4R13,
same as the spherical primary, but of the opposite sign). The corrector
with this value of the aspheric term is said to have a *unit *power*
*
**P**.
In such arrangement, for cancelled spherical aberration of the system,
the secondary mirror conic needs to be K2=-[(m+1)/(m-1)]2,
same as in the classical Cassegrain. With both, primary and secondary
spherical (K1=K2=0),
needed power **P** of the corrector for zero system spherical aberration is
(from **Eq. 113.1**) P=1-k(m-1)(m+1)2/m3.
For the typical ƒ/2/10 SCT with
k~0.25 and m~5, P~0.71, i.e. needed corrector strength, or depth, is
about 71% of that needed to correct the primary alone. In terms of the
F-ratio, this corrector has the strength needed to correct some 12%
slower primary.

In general, for any combination of conics, needed
corrector power to cancel spherical aberration of two mirrors is
determined by a value of the sum of opposite aberrations contributions
of the primary and secondary relative to spherical aberration of the
primary alone. Thus, it can be written as P=1-(s'2/s'1),
with the aberration contribution of the secondary **s'**2
in proportion to k[K2+(m+1)2/(m-1)2](1-1/m)3,
and aberration contribution of the primary **s'**1
proportional to (K1+1). The prime notation is to differentiate the
proportionate relative contributions from the corresponding actual aberration
coefficients s1=s'1/4R13
and s2=s'2k4/4R23.

Thus, the lower-order aspheric parameter **A**1
for the corrector in a Schmidt-Cassegrain system can be, analogously to
the Schmidt camera, written as A1=*b*/8(n'-n),
but with the corresponding aspheric coefficient * ***b** changed in
proportion to the needed corrector's power, as *b*=2P/R13.
Reduction of the 5th order aspheric parameter of the SCT corrector, **A**2,
relative to the value for primary alone, is typically greater than that of **A**1,
due to significant higher-order spherical aberration of opposite sign
generated at the secondary as a result of relatively close object (i.e.
image of the primary) distance, as well as due to reduction in the
higher-order aberration resulting from reduced corrector separation
(i.e. height of marginal ray at mirror surface). For spherical secondary
and typical 8" ƒ/2/10 SCT configuration with σ~0.4, the parameter,
given by **A**2=*b*'/16(n'-n),
with the higher-order aspheric coefficient *b*' approximately 1/6
of that needed for primary alone (with the stop at the center of
curvature), or *b*'~[1-(k6/ρ5)]/R15.

Thus, the higher-order corrector's power in the typical
commercial unit is only ~0.16 of that needed to correct higher-order
spherical aberration of a comparable ƒ/2 Schmidt system. The parameter
changes for different values of **σ**1
approximately in proportion to σ1/0.4,
thus the generalized approximation for * ***b**' can be written as
*b*'~2.5σ1[1-(k6/ρ5)]/R15.

For closer objects, spherical
aberration coefficients for all three, corrector, primary and secondary change
(it is negligible for the corrector),
disturbing presumed near-zero balance for distant objects, and resulting in spherical aberration.
The chage of aberration contribution on the two SCT mirrors is similar
to that in all reflecting two-mirror systems (Eq. 92).
Main difference is with SCT systems that focus by moving the primary.
Here, the error
induced by a relatively small object distance is in part offset by
under-correction induced by refocusing, which requires an increase in
mirror separation.

More specifically, reduced object distance lowers
under-correction of the primary, and increases over-correction of the
secondary. That makes the system over-corrected; the increased
mirror separation needed to bring the focus point to its fixed location
diminishes the
effective diameter of the secondary, reducing over-correction induced by
it, and by that the overall system over-correction as well. This
makes a typical commercial SCT better suited for terrestrial
observations than a similar system with fixed-mirror focusing. More
details on this subject are given in
11.5.2. SCT focusing errors.

**
SCT spherochromatism**

Spherochromatism in the SCT
originates at the corrector, whose corrective power is optimized for one
- usually green/yellow - wavelength. Since shorter wavelengths refract
more strongly, and the longer ones refract more weakly, the effective
corrector power increases toward the former, and decreases toward the
later. With the combined spherical aberration of the two mirrors being
undercorrection, this means that shorter wavelengths (blue/violet) will
be overcorrected, and the longer ones (red) undercorrected. This
wavelength-dependant spherical aberration increasing with the refractive
index differential vs. optimized wavelength is the only source of
chromatism in a SCT.

Given aperture and F#, SCT
spherochromatism is proportional
to the relative power of the Schmidt corrector **P**. In other words, to the relative
value of the aspheric term * ***b** needed to cancel spherical aberration of the
system. It can be written as:

with * ***b****s**
being the aspheric factor - or the corrector's spherical aberration
coefficient - needed to correct for spherical aberration of
the primary alone. As mentioned, *b*=2/R13,
from the general form of the aspheric coefficient *b*=2n[(m+1)/(m-1)]2/R13,
for mirror magnification m=0 (object at infinity) and index of
incidence n=1. Aspheric coefficient cancelling the aberration of a
spherical mirror is *b*s=-b/8=-1/4R13,
while ** **that for an SCT system, equal to its aberration
coefficient **s**cr,
is *b*SCT=-P/4R13,
where **P** is the corrector *power*, positive in sign.

The P-V wavefront error of
spherochromatism for a particular SCT arrangement is obtained by
multiplying relative power of
its corrector (**Eq. 118**) with the wavefront error for the unit power
corrector (Eq. 106). For the transverse aberration (ray spot diameter), the relative
power is to be multiplied with the transverse aberration for the unit
power corrector (Eq.
107.1) and the SCT secondary mirror magnification.

In general,
SCT
spherochromatism is low. For a typical commercial ~ƒ/2/10 version, with both mirrors
spherical, k~0.25, σ~0.4
and m~5, relative corrector power ~0.72, and 0.866 neutral zone
placement, the red (C-line) and blue (F-line) geometric blurs are still within the
Airy disc. For the 0.707 radius neutral zone placement (**FIG. 175**),
the blurs are doubled, but the wavefront error halves for the lowest
chromatism level achievable with the Schmidt corrector.

FIGURE
175:
Ray spot plot of the typical commercial 8" SCT with spherical mirrors
(**a**) and the more recent aplanatic variety
(**b**).
The 0.5° field radius is at the limit of the usable field (due to vignetting by the baffle tube),
visible only in 30mm to 50mm f.l. eyepieces. Thus, the ~1/20 mm sagittal coma of
the 8" is 5-6 arc minutes in the eyepiece, which is
acceptable for the field edge. Its color error is only significant in deep violet: the blur
size of 5-6 Airy disc diameters at the best focus location corresponds to
~0.12 wave RMS of spherical aberration. The red and blue blurs are
approximately 1.5 times the Airy disc diameter, or ~0.04 wave RMS level.
A 4" ƒ/12 doublet achromat has blue/red blurs ~3.2 times the Airy disc
diameter, but the error is defocus, thus worth ~0.3 wave RMS. The SCT
RMS error is roughly 10 times lower, making its visual chromatism all but
invisible. The 12" (aplanatic SCT) has much smaller off-axis blurs
due to the absence of coma and more symmetrical (astigmatic) images with
nearly four times smaller wavefront error 0.5° off-axis (0.17 vs.
over 0.6 wave RMS). Most of the increase in chromatism (~40%)
is due to its larger aperture (it is roughly at the level of a 4"
ƒ/70
achromat). SPEC'S

Residual spherical aberration in the system will
alter spherochromatic error due to the increased error in the optimized
wavelength (**FIG. 176**).

FIGURE 176: Residual
spherical aberration in the optimized wavelength of an SCT
affects correction in all other wavelengths as well. Non-optimized wavelengths with spherical aberration
at near-perfect correction of
identical sign to the

e-line residual will have the aberration
increased by similar amount (scaled according to the wavelength), while those with the base spherical
aberration of opposite sign will have it reduced by nearly as
much (generally, the error will be somewhat greater for shorter
wavelength, and somewhat smaller for the longer ones).

The overall correction level is nearly certain to be worse, not
only due to the error in the optimized wavelength, to which (and
those relatively close to it ) the eye is most sensitive.
As the plot at right shows, the wavelengths with initially the
opposite error sign (red), may only switch the sign, without
major error decrease. This particular SCT would show noticeably
more color in the blue/violet than when optimally corrected.

Since an SCT near-perfectly corrected for spherical aberration
have blue wavelengths overcorrected, and red wavelengths
undercorrected, a residual of, say, 1/4 wave P-V wavefront error
of overcorrection in the optimized wavelength will increase
spherochromatic error in the blue, and reduce it in the red (the
plots are for a typical ƒ/2/10
8-inch SCT with spherical mirrors). If the residual spherical is
relatively significant, spherical correction in the red is
likely to be actually better than in the optimized wavelength.

Consequently, the instrument will have more or less pronounced
chromatic imbalance, with the negatively affected portion of the
spectrum (blue/violet wavelengths in this example) possibly
showing noticeably inferior chromatic correction in both, visual
observing and imaging.

Follows an overview of the SCT off-axis aberrations, after a quick look at the properties of the standard SCT with an off axis mask.

EFFECT OF OFF AXIS MASK ON 11-INCH STANDARD SCT

Off axis mask can be used with large-aperture SCTs in conditions of strong atmospheric turbulence, in order to significantly reduce its negative effect on image quality. The obvious negative is the reduction in aperture, which is assumed to be at least partly compensated for by significantly improved wavefront quality. There are, however, some additional, relatively minor negatives.

Since the mask allows light to use only the upper portion of the optical train, colors are unwrapped and widely separated. Due to the very slow focal ratio, it is of little consequence if observing plane nearly coincides with the best image surface. The problem is, this surface is tilted, in this case by 9 degrees, which in the image plane perpendicular to optical axis (i.e. for unaccommodated eye) causes not only lateral deformation in the optimized color, but also lateral chromatic dispersion.

Bottom spots and images show this effect at the edge of the widest field possible with a 2-inch eyepiece barrel (best image surface tilt is slightly smaller, at 8 degrees).

◄
10.2.2.3. Schmidt-Newton telescope
▐
10.2.2.4.1. SCT off-axis
aberrations
►

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