with the system
coma aberration
coefficient **c****s**
being the sum of the individual aberration coefficients
for the two mirrors, cs=c1+c2,
and **α**
the field angle.
Not surprisingly, there is a direct similarity with all-reflecting
two-mirror systems, the only difference being in the position of
aperture stop. The individual coma coefficients for SCT primary and secondary,
respectively - with the secondary's coefficient adjusted for the
difference in ray height, so that the two can be directly added - are given by:

with
**σ**1
being the primary mirror stop separation in units of the primary radius of
curvature (with the stop at the corrector, σ1=-sc/R1,
**s**c being the corrector-to-primary separation, positive
in sign), **m** the secondary magnification, and

is
the secondary mirror stop separation (i.e. exit pupil,
or image of the
aperture stop formed by the primary to secondary separation), in units
of the secondary's radius of curvature (pupil-to-secondary separation is
numerically negative for the aperture stop located inside primary's
focus); as before, ρ=R2/R1.
In the typical f/2/10
commercial SCT (i.e. with secondary magnification m~5), k~0.25,
σ1~0.4, ρ~0.31
and
σ2~-7.6.
Note that the negative value of **σ**_{2} gives negative
value to the secondary's coma coeficient, which means it is of opposite
sign to that of the primary. Using these approximate values gives
c_{1}=0.6/R_{1}^{2}, and c_{2}=-0.4125/R_{1}^{2},
with the system coma aberration coefficient c_{s}=0.1875/R_{1}.
For R_{1}=-800mm, i.e. an 8-inch SCT, the corresponding P-V error is
W=0.195α, with **α** being the field angle in radians
(for small angles equaling the angle tangens). Putting W=0.00023mm,
for the value equaling
0.42 wave P-V (550nm wavelength), or 0.80 Strehl, gives α=0.00118,
i.e. the corresponding field angle 0.068° and field height 2.37mm.
With "diffraction-limited" field radius, with repect to coma, h=F^{3}/90
in mm, this implies that an 8-inch standard SCT has coma-limited linear field
comparable to an f/6 paraboloid (from F^{3}=90h).

Taking
σ1~0.4
(it does not change significantly with the change in secondary size
within practical range), and substituting for **k** and **ρ** in
terms of secondary magnification **m**, gives a good approximation in
terms of secondary magnification as
σ2~(η-5m-4)(m-1)/2m(1+η),
where **η** is the back focal length in units of primary's focal
length.
For the standard η=0.5, the secondary mirror stop separation is
σ2=0.5-(5m/3)+(3.5/3m).

Setting c1=-c2 gives the secondary conic needed for zero coma as:

This conic only
corrects the lower-order coma. The higher-order coma in the SCT is low,
about 5% of the lower-order coma magnitude, and of the same sign. This
means that correcting for this residual higher-order coma - actually,
minimizing it by balancing with the lower order coma of opposite sign -
requires about 5% stronger conic, which will induce lower order coma
needed to minimize the higher-order term.

Graph at left shows the
secondary conic needed to correct coma as a function of secondary
magnification, based on **Eq. 116.2**), for the back focal length
η=0.5. Since the change in magnification also changes the secondary
size, as k=(1+η)/(m+1), radius of curvature, as ρ=mk/(m-1) and
secondary stop separation **
σ**2,
these are substituted in the equation to obtain accurate conic value
(corrector location is assumed constant, with σ1=0.4,
since the change in its value are near negligible). The coma correcting
secondary is an prolate ellipsoid for secondary magnifications of about
4.5 and larger, paraboloid for magnification just over 4, and a
hyperboloid for lower magnifications. The required nominal conic
increases sharply for secondary magnifications lower than 3, as the
secondary increases in size, and its surface becomes less strongly
curved.

Alternative expression for needed
secondary conic for zero coma using aggregate parameters can be written
as K2=[(BA+1)A-(2-2σ1)C)]/(1-B),
where A=(m+1)/(m-1), B=(1-2σ1)k
and C=[m/(m-1)]3.

Needed **σ**1
value for zero coma in the arrangement with two spherical mirrors is σ1=[2C-(kA+1)A]/2(C-kA2). Alternately, it can be
expressed in terms of **ρ** and **k** as σ1=[2ρ3-k2(2ρ-k)(2ρ+1-k)]/2[ρ3-k2(2ρ-k)2].

The standard commercial SCT has both mirrors
spherical, thus K1=K2=0.
Obviously, coma in such arrangement is not corrected; for the linear
field, the standard 8-inch f/2/10 SCT is approximately at the level
of an f/6.1
paraboloid, with the coma increasing to the diffraction-limited level
(0.80 Strehl) at about 2.5mm off-axis. Larger SCT apertures have
somewhat more coma, since they use a smaller secondary (due to the
relatively smaller back focal length); for instance, in an 11-inch
standard SCT coma is about 15% larger.

SCT system satisfying **Eq. 115.2** is so called *aplanatic*
SCT (Fig. 175b). It recently became commercially available advertised as the "improved
Ritchey-Chretien".
While admittedly with less astigmatism than comparable RC, it is - needless to
say - still "only" an aplanatic SCT.

Another option for correcting the coma
is to keep the mirrors spherical, but move the corrector (i.e. stop) farther away
from the primary. Taking the common ~f/2/10 system with k~0.25,
ρ~0.31 and m~5
gives secondary stop separation σ2~0.4-[1.6/(1-2σ1)].
Substituting these values in the relation
for secondary's
aberration coefficient (**Eq. 116**)
gives c2=0.288σ1-0.528
(omitting common denominator ** R**1) which, after setting c2=-c1
thus 0.288σ1-0.528=-(1-σ1),
gives needed corrector separation for corrected coma as σ1=0.663, or
nearly 2/3 of the mirror
radius of curvature.

● **
lower-order astigmatism** P-V wavefront error
for object at infinity is:

with the system
astigmatism aberration
coefficient **a****s**
being the sum of the two individual mirror coefficients for the primary
and secondary, as=a1+a2,
with:

While generally low in the typical
compact commercial SCT, astigmatism still requires attention. Relatively small design
changes can result in significantly increased astigmatism level. For
instance, typical f/2/10 SCT with spherical primary, σ1=0.4 and k=0.25
would, in an aplanatic arrangement, require secondary conic
K2=-0.77 for
cancelled coma; its astigmatic P-V error would have been W=-0.000155(αd)2.
Change of the secondary magnification to m=4, corrector separation **σ**
to 0.375 and relative marginal ray height on the secondary **k** to 0.3 (for
identical back focus distance), would require zero-coma secondary conic
K2=-1.176 and would have the P-V wavefront error of astigmatism
W=-0.000374(αd)2
- greater by a factor of 2.4 (despite that, best image surface
curvature would slightly improve, due to the lower Petzval curvature).

Graph
at left shows the P-V wavefront error of coma and astigmatism in the
typical 8-inch f/2/10 commercial
SCT with spherical mirrors (both change in proportion to aperture
diameter). Obviously, coma dominates, having nearly 4.5 times large P-V
wavefront error at 1° off axis, and increasingly more toward inner
field. For given eyepiece, the field size also changes in proportion to
the aperture.

For comparison, coma in an f/10
paraboloid (dashed blue) is nearly five times smaller. Since the linear
coma of a paraboloid changes with the 3rd power of focal ratio, the SCT
linear field is, coma-wise, comparable to an f/6
paraboloid. Angularly, coma in a paraboloid changes with the square of
focal ratio, thus the SCT field compares to that of an f/4.7
paraboloid. Specifically, the diffraction limited field radius in the
SCT (only considering coma, but astigmatism is negligibly small at those
field angles) is 2.5mm vs. 11.2mm in an f/10
paraboloid. Angular diffraction limited field radius in the SCT (4.3 arc
minutes) is as many times smaller than that in the paraboloid, and equal
to the diffraction limited field of an f/4.7
paraboloid.

Astigmatism in the SCT is
only slightly stronger than in the comparable paraboloid, 1.6 vs. 1.4
wave P-V at 1° off axis. While this far off axis it equals coma in the
paraboloid (as the P-V wavefront error), in the SCT is still only a
small fraction of the coma wavefront error. It does not matter much in
practical terms, since it is normally eyepiece astigmatism that
dominates the outer field; it merely gets more distorted by coma in the
SCT.

● **field curvature**; of interest are
Petzval field curvature, which is not affected by the corrector and,
thus, remains as for any two-mirror system, 1/RP=(2/R2)-(2/R1)=2(1-ρ)/ρR1,
and best image surface curvature
(or *median astigmatic curvature*) which is, in the presence of
astigmatism, given by 1/Rm=(1/RP)+4as** **or:

which for spherical mirrors (K1=K2=0)
reduces to:

Substituting for **
ρ** in terms of **m** and **k**, with k=(1+η)/(m+1),
the Petzval surface curvature can be also written as
RP=(1+η)mR1/2[m2-1-(1+η)m]
with the back focal length in units of primary's f.l. being η~0.5
for systems with f/2 primary.

Graph below shows field curvatures for the
typical 8-inch f/2/10 SCT with
η=0.5, based on **Eq. 118**. Due
to the difference in their astigmatism, a system with spherical
mirrors has somewhat weaker, and aplanatic (w/aspherized secondary)
somewhat stronger best image curvature than their Petzval. If the Petzval and astigmatism are of the same
sign, they add up arithmetically to a stronger best image curvature,
and they offset each other when of opposite sign, resulting in theee
weaker median image curvature. Since the median surface is midway
between the sagittal and tangential surface, with the tangential
surface always 3 times farther away from
the Petzval than sagittal (by 6a vs. 2a, **a** being the system
astigmatism coefficient), median surface is flat when the
astigmatism coefficient is one fourth of the Petzval value, and of
opposite sign.

While Petzval surface becomes flat with secondary
magnification m~2, neither spherical-mirrors nor aplanatic (aspherized
secondary) can have flat best image surface. It is weaker with lower
secondary magnifications (i.e. with larger secondary), and is
generally more curved in the aplanatic arrangement.

Graph at right is a blown out shot of the graph
at left, showing wider area with field curvatures for secondary
magnifications below 2. Note that the green - aplanatic - plot
apparently dives down toward flat surface due to the needed conic
for coma correction raising to infinity for m=1 (flat secondary),
and taking the curvature figure with it. In the actual system, the
curvature bows back toward that for the primary with the stop at the
corrector (similarly to that for the arrangement with spherical
mirrors, only shifted toward lower secondary magnification). This is
of little practical importance, since any practical secondary
magnification is larger than 1.5.

In regard to
**misalignment**** sensitivity**,
the SCT, expectedly, has elements of both, two-mirror telescope and a
Schmidt camera. In a typical SCT arrangement secondary mirror is mounted
on the corrector. If they are centered with respect to each other, their
decenter (i.e. lateral shift) with respect to the optical axis produces
error of similar magnitude but of opposite sign on their two respective
surfaces (Eq. 109
and Eq. 91.2 for the corrector and
secondary, respectively), making Schmidt-Cassegrain with spherical
mirrors less sensitive to small decenter errors. The effect is
a relatively small shift of the best focus from the field center; some
negligible residual astigmatism is also present. It can be illustrated
on a raytracing example (below).

From left to right: (1) perfectly collimated system vs. one with (2)decentered corrector
(that amounts to center misalignment of the corrector and secondary),
(3) decentered corrector and secondary by the same amount, (4) tilt of the
corrector alone, and (5) tilt of corrector and secondary together.
It shows that decenter generates somewhat more of opposite in sign coma (primarily)
on the secondary than on
the corrector, with the best focus shifted from the physical field
center - marked with ON-AXIS - by about 0.13° opposite
to the decenter direction (note that physical field center deviates
from the optical field center determined by the axial ray entering aperture;
in this case optical center is little over 0.1° below the physical).
Corrector tilt alone has no appreciable effect, but when it includes
the secondary too, the effect is, for the same 1° tilt nothing
short of disasterous (rightmost), with mainly coma, negative for
counterclockwise tilt (shown, with light entering telescope from left to
right), and positive for the clockwise tilt. As enourmously large as
it is, this collimation aberration can be nearly entirely offset by
decentering secondary by 4.26mm (bottom). Even the image tilt is
corrected. It shows that even grossly
misaligned SCT can act as well collimated. This, of course, implies that
decenter error also can be offset by tilt.

Element tilt error always induces
optical field center shift away from the physical center, as well as image tilt.
In the case of mirrors, the shift angle is twice larger than
the tilt angle, but it is not directly comparable to the field angle
at the final image. This is
due to the secondary field angle being
magnified vs. primary field angle, here by about 3.4 times. Hence,
if the original field angle at the final image is, say 0.5°, the
actual angle of reflection for the central ray is 3.4 times larger,
1.7°. So the secondary tilt of 1° will induce angular
optical field center shift corresponding to a little more than
0.5° in the final image field. But the coma generated at the
secondary is commensurate to its physical tilt, i.e. for 1° tilt it
will be at the level of coma at 1° off axis in a perfectly
aligned system (over 7 waves P-V at the optical field center in this case;
at the phisical field center, nearly 1/2 degree below the optical
center, coma is over 10 waves). Coma at the optical field center is
proportional to the tilt angle, which means that 0.1° secondary
tilt would induce over 0.7 waves of coma at that field point. At the
physical field center, however, the final magnitude is a sum of
preexisting coma, astigmatism and the all-field coma induced at the secondary
by its tilt; in this case, it is 3.7 waves P-V.

In an aplanatic SCT (K~-0.8 for
conventional f/2/10 system)
secondary mirror decenter induces more of
coma, making it about three times more sensitive to decenter error.
Also,
there is no possibility to compensate for decenter error by adjusting
secondary tilt; these systems have to be well centered.

Tilt of the corrector/secondary,
due to a negligible
error at the corrector, results in center field coma induced by the
secondary, as given with Eq. 91.1. Again,
in an SCT with spherical mirrors, due to the presence of system coma, at
some point off-axis, in the direction of tilt, the two coma
contributions will nearly cancel out, and the original field quality can
be restored by making plane of the secondary parallel to that of the
primary by tilting the former. Same applies to an aplanatic SCT.

Since coma due to tilt or lateral surface shift originates from surface
deviation, it doesn't change with the field angle. It is added equally
to the entire field, which leaves the field center with the amount of
coma given by corresponding equations, while subtracting as much coma from one
side of the field, and adding it to the other, in the orientation
coinciding with that of the corrector's shift. As a result, best focus
point shifts off center, in the direction where the misalignment coma is subtracted.
The amount of shift depends on the level of
SCT coma: in a typical 8" SCT with spherical mirrors, coma induced by
as little as 0.2° tilt of the corrector/secondary approximately equals system's coma at
0.25° off-axis.

For larger SCTs with spherical mirrors, coma increases in proportion to
the aperture diameter, and the shift angle per mm decreases accordingly.
In an aplanatic SCT, there is no system coma, and both decentered and
tilted corrector/secondary result in coma evenly distributed across the field.
However, astigmatism is also present and it may be significant. In the
Edge 8 model,
0.1° secondary tilt counterclockwise (top away from the primary)
shifts optical field center about 1/20 degree above the physical
center, with 0.9 waves P-V of coma in the former (slightly more at
the physical center). At 0.4° off (0.45° from the physical center)
coma and astigmatism are of similar magnitude, producing 1.2 waves P-V
wavefront error miscollimation aberration.

Decentered secondary
also induces image tilt, which can be significant in an aplanatic SCT
(less so in the standard model, due to its field already compromised by
strong coma), and can't be corrected
by tilt-collimation.

Despace
sensitivity of the SCT corrector is practically zero; since in practice
it supports the secondary mirror, its axial shift would cause error
appropriate to mirror decenter, as detailed in
SCT focusing errors. Of course, any combination in misalignment of the two
mirrors and corrector is possible. When compared
to miscollimation aberrations in the
classical and aplanatic Cassegrain there is an overall similarity.

**
EXAMPLE**: Taking
the typical commercial
8" f/2/10 SCT configuration, with D=8, d=4, R1=-32,
k=0.25, m=5 and σ1=0.4,
both mirrors spherical, **Eq. 112** gives the **
spherical
aberration coefficients** s1=-0.000007629
and s2=0.000002197
for the primary and secondary, respectively. This determines the needed
relative corrector power as
P=(s1+s2)/s1=0.712.

With the relative
exit pupil separation for the secondary mirror σ2=-7.6,
the mirror coma coefficients are **c****1**=0.00058594
and **c****2**=-0.0004031
for the primary and secondary, respectively,
resulting in the system **coma aberration coefficient** **c****s**=c1+c2=0.00018282
and the P-V wavefront error of coma **W**c=0.0078α,
with the RMS wavefront error given by
**ω**c=Wc/**√**32=0.00138α.
With the RMS wavefront error for 0.80 Strehl ωc=λ/**√**180=
0.0000016136 for λ=550nm=0.00002165",
the field angle
**α**
at which coma reaches this level is,
α=0.0000016136/0.00138=0.00117
radians, or 4 arc minutes. This is somewhat smaller angular field
than that of an f/5 parabola. However, the linear field is doubled
at f/10, thus having the field appearance of an ~f/6 paraboloid.
Secondary conic needed to cancel lower-order coma is K2=-0.77.
Ray tracing gives slightly stronger coma, the result of added
low-level higher-order coma of the same sign (consequently, needed
conic to cancel coma is slightly higher).

The
**astigmatism aberration
coefficients** are **a****1**=-0.01125
and **a****2**=0.01849
for the primary and secondary, respectively, giving the system
aberration coefficient **a****s**=a1+a2=0.00724,
for the P-V wavefront error Wa=0.11584α2.
For α=0.00117
radians, it gives **W****a**=0.000000158,
or 1/137 wave for λ=550nm=0.00002165".
Entirely negligible at the field height where the coma reaches 0.80
Strehl level and, for all practical purposes, for the rest of
useable field as well.

Petzval
**field curvature** of
the system is **1/R**P=(2/R2)-(2/R1)=-0.14,
or -7.16", and the best image surface curvature is

**1/R**m=1/RP+4as=-0.11
or -9".

Finally, the **
spherochromatism**
P-V wavefront error is smaller vs. unit power corrector by a factor
of 0.712. From
Eq. 106, taking n=1.518, s=0.25 (neutral zone
at 0.707 radius), the index differential
ι=0.0036 and 0.0044 for the F- and
C-line respectively, gives 1/8.1 wave (F) and 1/8.9 waves (C) of
spherochromatism at the best focus. In terms of the blur size, it is
~1.8 and ~1.6 times their respective
Airy disc diameter for the F- and C-line, respectively. The usual
neutral zone placement at 0.866 radius results in twice smaller
blur, and double the wavefront error.

A few words about **Schmidt-Gregorian
arrangement**. In the arrangement with two spherical
mirrors it can have coma cancelled with the corrector placed in
the proximity of primary's focus (with the secondary protruding
in front of it). That would make it even somewhat more compact
that coma-corrected SCT, but the drawbacks of significantly
stronger astigmatism and nearly doubled spherochromatism (due to
contributions from the two mirror being of the same sign) make
such a system less attractive.

◄
10.2.2.4. Schmidt-Cassegrain telescope
▐
10.2.2.4.2. SCT focusing errors
►

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