as the P-V wavefront error at the best focus, where **m** is the secondary
magnification and **F**1
the primary focal ratio ƒ1/D.

The relation implies that, for
nominal primary mirror shift **∆**, error induced by extending back
focus is independent of aperture. It only depends on secondary
magnification and primary's focal ratio. For
F1=2,
m=5, and K2=0,
every mm of reduction in mirror separation (∆=1), or nearly for every inch of
focus extension, induces ~1/23 wave
P-V wavefront error of
over-correction, and as much of under-correction for
widening the separation.

Aspherizing the secondary would result in the
error of similar magnitude, only the sign of aberration would reverse:
focus extension (i.e. reducing mirror separation) would induce
under-correction. Focus extension also induces undercorrection to
Celestron's Edge, about 1/20 wave P-V for every inch of extension. In
addition, its off axis astigmatism nearly triples, transforming the
field from practically flat into a very curved one (for 8-inch aperture,
190mm curvature radius, convex toward the secondary, with the flat-field
wavefront error 0.5° off axis nearly four times larger).

This level of error remains nearly unchanged with the change in
aperture. This is due to any given mirror shift being producing
identical nominal change of the height of marginal ray at the secondary.
In other words, the relative ray height change is proportionally smaller
for larger aperture. Since the aberration contribution of the secondary
is in proportion to the fourth power of its effective aperture (Eq.
78.1), the relative change in its aberration contribution is
approximated by 1+(4∆'/D), **
∆'** being the change in marginal ray height,
∆'=∆/2F1.
Hence, the relative increase in secondary mirror contribution is
proportionally smaller for larger aperture, but since its contribution
is also proportionally greater, the final nominal change in its
aberration contribution remains at the nearly same level.

Spacing error also induces coma. For reduced mirror separation, it adds
to the system coma, but not significantly (less than 10% for 5 inch
focus extension).

**Close object focusing **

However,
this only applies to the change in mirror separation to accommodate the
use of various accessories, while the object of observing remains at a
large distance. In addition to the change in mirror separation, focusing on relatively
**close objects** has to deal with
optical consequences of the change in shape of the incident wavefront (which is now convex spherical)
as well. Follows more detailed consideration of
how it affects performance level of a typical
commercial SCT.

Observing
close terrestrial objects is not what telescopes are made for. Any optical system can
give optimum performance only within a limited object distance range or,
in terms of geometric properties of incident light, for a certain degree
of curvature
of the incident wavefront, determining geometry of rays projected from
it. Telescopes are optimized for distant objects observing, hence for a
near-flat incident wavefront, or bundles of near-collimated (parallel) rays. The
smaller object distance, the more convex-spherical shape of the
wavefront, and the more diverging the rays. This change in light
geometry, without an adequate change in properties of the optical
surfaces of telescope objective, results in the wavefront it
forms being different from spherical - the closer object, the more so.

In an SCT
optimized for infinity, near-zero level of spherical aberration is
achieved by balancing aberration contributions of its three optical
elements, Schmidt corrector, primary
and secondary mirror, for near-flat incoming wavefront. As the shape of
the incident wavefront changes, so do the individual aberration contributions of
these three elements. The problem is that they do not change in
proportion that would preserve the near-zero balance. Instead, certain
amount of aberration is generated, increasing exponentially with the
reduction in object distance.

Typically,
the most significant aberration generated while observing close object
with telescopes is spherical. The same applies to the SCT. Change in its
off-axis aberrations is insignificant.

Spherical
incoming wavefront affects the three SCT optical components differently.
Due to its low power, changes to the wavefront at the corrector are
small enough to be neglected. Not so with the primary mirror. New
wavefront/ray geometry result in larger effective mirror diameter and
extension of its converging cone. These, in turn, increase the width of
converging cone at the secondary, extending its converging cone
significantly more. Most importantly, the aberration contributions to
the wavefront at the primary and secondary change in different
proportions, resulting in the aberrated final wavefront (**FIG. 177**).

**
**

FIGURE
177: RIGHT: Wavefronts from close objects are convex spherical - i.e. their
rays (red) are diverging. With displaced stop, it causes the effective diameter of the
primary to increase, and the converging cone reflected from its surface
to extend, focusing farther away than its infinity focus (blue; the
effective separation of focal length from mirror is also grossly
exaggerated).
Consequently, height of the marginal ray at the secondary and its magnification both increase, with the final focus location shifting from **F**
to **F'** (from another angle, the image formed by the primary -
which is object of imaging for the secondary - is shifted closer to the
secondary's focus **F****2**,
resulting in the secondary forming the image at a greater distance). To move the final focus back to its original location, the
mirror separation is increased, reducing marginal ray height on the secondary until its magnification is
lowered nearly to its original level. However, due to different
form of the incident wavefront, the final wavefront is no more spherical.
LEFT:
Exaggerated illustration of how spherical incident wavefront **W****S **
generates different
amount of spherical aberration at the corrector, compared to the flat
(infinity) wavefront **W****F**.
Spherical wavefronts from closer objects flatten out as they enter the
glass; as a result, the rays, projected orthogonally to the wavefront, change directions (refract).
The refracted wavefront is
non-spherical, but the amount of aberration generated is
negligible (Eq. 105). More interesting is the Schmidt surface
side, which
generates huge amounts of spherical aberration as it selectively delays
portions of the wavefront by (n-1)z. As the illustration
shows, diverging rays of the spherical incident wavefront exit the rear
(Schmidt)
surface at a slightly higher point than collimated rays of the flat
incident wavefront (**ρ'**
and **ρ**,
respectively). This changes the optical path length (the extra path
generated over **T****min**
due to the cosine factor is negligible in comparison) and the effective
surface profile, which now acts thicker toward the edge (since the
object ray incidence angle **β** increases with the zonal
height).
With the neutral zone placed at 0.707 radius, the wavefront points above
it retard (i.e. refract) more
with respect to the chief ray. However, the effect is similar to the
change of the Schmidt profile, which is now slightly shifted toward the
profile that would bring all rays to paraxial focus (i.e. with neutral
zone at the center). Thus most of the change induced to the wavefront is
cancelled out by refocusing, with the residual error remaining entirely negligible in practical use.

In order to establish more
specifically the amount of spherical aberration generated in an SCT used
for close object observing, we need to track down changes in the
aberration contribution for each of its three optical elements. While
the change in contribution of the corrector and primary mirror is
independent of the type of focusing, the aberration at the secondary
mirror will be different in SCT systems that use mirror focusing (the
common commercial type) from that generated by systems that use conventional
focusing via focuser tube.

If the two mirrors are stationary, the amount of aberration induced by close
object observing is practically limited by the available focuser travel.
Since closer objects are imaged farther back, at a certain point the
image simply runs out of a focuser reach. The limit to object distance
set by given focuser travel length can be approximated by treating the
SCT as a two mirror system with the aperture stop at the corrector
location. The effect of the corrector plate on ray geometry is minor, as
described in 11.5. Schmidt-Cassegrain telescope.

For given
mirror separation **s**, back focus
distance - the separation from mirror surface to the final focus - in a two-mirror system is given by
B=s+(ƒ1-s)ƒ2/(ƒ1-s-ƒ2),
with **ƒ1**
and **ƒ2**
being the primary and secondary mirror focal length, respectively (note
that, according to the sign convention, all three parameters are
negative). With a close object, it is given by B=ρ'k'ƒ1'/(ρ'-k'),
with the

▪ new primary's effective aperture
radius d'=(1-k)ψd/[1-(1-k)ψ]

▪ new (effective) primary's
focal length ƒ1'=ƒ1/(1-ψ),

▪ new secondary-to-primary
radius ratio ρ'=(1-ψ)ρ,

▪ new marginal ray height at the
secondary k'=1-(1-ψ)(1-k)=(1-ψ)k+ψ,
and

▪ primary focus extension ψf1/(1-ψ)

with **ρ** and **ρ'** being the R2/R1
ratio for object at infinity and close object, respectively (with the
secondary radius shrinking relative to the effective primary's radius),
**k** and **k'** the relative secondary size vs. the effective
primary diameter, also for infinity and close object, respectively, and
**ƒ**1
and **ƒ**1'
the primary's focus distance for object at infinity (focal length) and
close object, respectively.

For the typical commercial SCT configuration, with s~0.75ƒ1
and (ƒ/ƒ1)~5,
or ρ~0.3125 and k~0.25, the back focal distance is well approximated by

B~0.23(0.33+ψ)ƒ1/(0.06-ψ),

with **ψ**=ƒ1/O
being the inverse object distance **O** in units of the
primary's focal length. For an 8" ƒ/10 SCT with ƒ1~400mm, the focus
shift between ψ=0
(object at infinity) and ψ=0.01
(object at 100ƒ1,
or 40m) is 0.3ƒ1,
or 120mm - most likely already out of the focuser range.

For ψ~0.06,
back focus distance approaches infinity (the secondary forms collimated
beam of light), and for ψ>0.06
back focus distance has negative value (secondary forms diverging beam
of light).

To determine
the level of spherical aberration induced to a commercial SCT at this
and other object distances, it is necessary to determine the changes in
aberration contributions of its three optical elements. For
the typical commercial SCT configuration with spherical mirrors, the system wavefront
error, as the P-V error of lower-order spherical aberration at the best
focus, can be approximated from Eq. 113.2,
by

with
**P**, **1**
and **S** being the proportional contributions of the corrector, primary
and secondary mirror, respectively, and aperture diameter **D** in mm. The sum of contributions -
under-correction by the primary, over-correction by the corrector and
secondary - is near zero in the well corrected system. Relatively small deviation from perfect figure on any of
the three optical elements can be cancelled out by matching deviations
on other two elements, but it is probably as hard to achieve high
correction level doing this as it is to produce three near-perfect
matching surfaces. For an
ƒ/2/10 200mm
system, a sum in the brackets of 0.011 would result in ~0.25 wave of spherical
aberration, over-correction if positive, under-correction if
negative.

Follows simplified analysis of the change in these
contributions for close objects. Its primary goal is to illustrate the
mechanism of change, but it should also give useful quantitative
approximation of the effect.

As already mentioned, change at the corrector is negligible. Spherical
aberration added to it due to close object distance is merely that of a
plane parallel plate of the same thickness (Eq.
105.1).

For the mirrors, close object distance itself is a factor that changes
the aberration contribution - for the primary directly, and for the
secondary by changing the separation between it and primary's image
(which is the object for the secondary) - and the compounding factor is
the change in the effective aperture of both, primary and secondary.

At the primary, according to
Eq. 9.1, the
aberration contribution changes with
(1-2ψ)2,
with **ψ** being the
inverse of the object distance **O** in units of primary's focal
length **ƒ**1,
i.e. the inverse of the relative object distance o=O/ƒ1,
hence ψ=ƒ1/O=1/o.
Since the effective aperture of the primary, due to the displaced stop,
changes in proportion to (1+2σψ),
**σ**
being the corrector-to-primary separation in units of the primary's
radius of curvature, the combined change in its contribution due to
these two factors is in proportion to (1-2ψ)2(1+2σψ)4.

Unlike the primary, for which there is only that one single scenario,
the change of aberration contribution at the secondary depends on the
focusing modality: it is either with a standard focuser and fixed
secondary, or focusing by moving the secondary.

Fixed secondary

With fixed secondary, the extension of primary's cone due to object
distance reduction has two effects: one is the wider cone at the
secondary, resulting in magnification increase, and the other is the
change in the cone geometry, resulting in the primary's image forming
farther away from the secondary.

The cone width at the secondary due to focus extension alone (i.e. with
fixed primary's aperture) changes in proportion to 1-ψ+ψ/k.
However, since the marginal ray at the primary is higher by a factor of 1+[2σψ/(1-2σψ)]
or, slightly rounded, 1+2σψ,
the final height is proportional to [1-ψ+(ψ/k)](1+2σψ).

The aberration contribution due to the change in its object
(primary's image) distance, according to
Eq. 78.1, is a function of the inverse of the relative image distance **Ω**, in
units of secondary's radius of curvature, in proportion to (Ω-1)2.
With the focus extension of the primary due to close object distance
being
ψ/(1-ψ) in units
of primary's focal length,
the change in **Ω** is proportional to (1-ψ)k/[(1-ψ)k+ψ],
where **k** is the relative height of marginal ray at the secondary
for object at infinity (and also the relative separation of primary's
image from the secondary in units of primary's focal length). Thus, the
secondary aberration contribution as a function of object distance is
proportional to {(1-ψ)kΩ/[(1-ψ)k+ψ]-1}2,
and its change vs. (Ω-1)2, with Ω~2.5 and k~0.25 for the typical ƒ/2/10
commercial SCT, comes to [(1-3.66ψ)/(1+3ψ)]2.

With the ratio of change in the effective secondary diameter (determined
by the width of the axial cone at its surface) given by [1-ψ+{ψ/k)](1+2σψ),
with **k** being the relative height of marginal ray at the secondary
for object at infinity, its aberration contribution to the lower-order
spherical aberration of the system is now changed in proportion to [(1-3.66ψ)/(1+3ψ)]2[(1+3ψ)(1+2σψ)]4,
leading to [(1-3.66ψ)(1+3ψ)]2(1+2σψ)4,
or (1-0.66ψ-11ψ2)2(1+2σψ)4.
With σ~0.4 in the typical ƒ/2/10
commercial SCT, it is (1-0.66ψ-11ψ2)2(1+0.8ψ)4.

Alternately, since object distance is directly related to image
magnification, so that (Ω-1)=(m+1)/m-1), the secondary contribution can
be expressed in terms of secondary magnification **m**2.
Starting with the general relation for
secondary magnification in two-mirror systems, m2=ρ/ρ-k
for object at infinity,
and substituting for **ρ'** and **k'** in m2'=ρ'/(ρ'-k')
for object at a relatively close distance,
gives the change of (m2'+1)/(m2'-1)
vs. (m2+1)/(m2-1)
as [2ρ-k-ψ/(1-ψ)]k/[k+ψ/(1-ψ)](2ρ-k),
with the secondary contribution changing with the square of it. With ρ~0.3125
and k~0.25 of the typical ƒ/2/10
commercial SCT, it comes, again, to [(1-3.66ψ)/(1+3ψ)]2.

Here too, change in the aberration contribution due to the magnification
factor gets multiplied with the change in the secondary aperture factor
to yield the combined change in the aberration contribution at the
secondary, for the same final contribution ratio [(1-3.66ψ)(1+3ψ)]2(1+2σψ)4.

With the effect of changes in **D** and **F** being
accounted for within the brackets, the P-V error of lower-order
spherical aberration at the best focus as a function of the object distance
in an SCT with **fixed mirrors** (i.e.
standard focuser) can be expressed as a lengthy but simple
approximation:

**
**

with the aberration contributions from the **corrector**,
**primary** and **
secondary** mirrors given in green, blue and red,
respectively. For ψ=0.01,
or object at 100 primary mirror's focal lengths away, D=200mm, F1=2,
σ=0.4R1
and k=0.25, it gives the error as W~(0.71-0.9915+0.2948)D/2048F13~0.000171mm,
or 0.3 wave of over-correction in units of 550nm wavelength. This is
slightly less from what OSLO gives for such a system (0.34 waves); the
discrepancy of 10-15% persists for other values of ** ψ**,
mainly due to the OSLO system having slightly smaller **σ**
value (0.375)
and including a small higher-order aberration
contribution, accounting for about 3% of the total aberration.

Moving secondary

In the SCT with mirror focusing, the mirror separation **s**
is increased - usually by moving the primary - in order to lower
secondary magnification back to its infinity level, and bring focus for close objects back to
its infinity location. For this to occur, the effective decrease in the
R2/R1
ratio due to the extended primary's cone has to be offset by a lower
secondary magnification in order to produce the same final focus
location. Since the secondary-to-final-focus distance is mkƒ1,
with object at infinity, and m'k'ƒ1'
for close object (where **ƒ**1**'**
is the "acting" primary's focal length equaling the length of extended
cone), the final focus will nearly coincide when m'k'ƒ1'
equals mkƒ1.
This neglect small additional shift of the secondary forward needed
to produce slightly higher magnification to compensate for the backward
shift of the secondary (relative to primary) from its infinity location,
which has insignificant effect.

The needed ray height on the secondary for m'k'ƒ1'=mkƒ1
is k"=(1-ψ), which in turn
determines secondary magnification at that location as m2"=ρ'/(ρ'-k")
which, with ρ'=(1-ψ)ρ, gives m2"=ρ/(ρ-k).
In other words, m2"=m2,
with secondary magnification nominally unchanged vs. that with object at
infinity. The ray height at the secondary is also nearly unchanged,
since the slightly lower nominal **k** value is mainly offset by the
slightly larger effective aperture of the primary. The actual ray height
is slightly less than for object at infinity, but it is small enough to
be neglected, more so considering that the secondary does need to be
moved slightly forward in order to compensate for its shift away from
the primary and reach the same final focus location.

So, we can assume that the secondary mirror contribution is nearly the
same as for object at infinity. The change in primary's contribution is
the same as with fixed mirror, and there is no appreciable change in the
corrector's contribution. Thus the P-V error of lower-order
spherical aberration at the best focus induced by reduced object distance
O/ƒ1=1/ψ in an SCT with **mirror focusing** is:

For the same object distance **O** of 100 primary's focal
length (ψ=0.01),
the error for the above system is now W~0.18 wave of over-correction, practically identical to the 0.18 wave P-V error given by OSLO.
That is 40% smaller error than with fixed mirrors. Since the error is, as in the fixed mirror system, nearly entirely
lower-order spherical, the corresponding RMS wavefront error is smaller
by a factor of ~0.3.

Thus, correction error induced by close focusing is
considerably smaller in SCT systems with mirror focusing (**FIG. 178**).

**FIGURE 178**: Best focus P-V error of
spherical aberration induced by close object observing in 8-inch
*f*/2/10
SCT. The error is generally greater in the fixed mirror SCT system,
exceeding 0.25 wave already for 100 primary's focal length object
distance. However, since that is at the limit of reach for
conventional focusers - or a bit beyond it - this is also the limit
to the error (inset top left shows how back focal length increases
with **ψ**). For star testing, the error shouldn't exceed 0.05 wave,
which is induced at ~500 and ~270 primary's focal length object
distance for fixed mirror and mirror focusing SCT systems,
respectively (under assumption that a system is near perfectly
corrected for infinity; if not, system under-correction at infinity is
deducted, and over-correction added to the error induced by close
object observing). Due to the wider range of object distance it can
handle (fixed mirror generate 5 inch focus extension already at
object distance of 100*f***1**), mirror focusing generates larger error toward its close
focus limit, exceeding 1 wave P-V for 8-inch SCT. For given relative
object distance, the error increases in proportion to the aperture,
but the close focus limit diminishes generally more, making the
error toward the limit somewhat smaller in larger SCTs (close focus
limit based on data from Meade catalog).

Inset bottom right shows
the big picture, with the aberration contributions of the corrector,
primary and secondary combining to produce the final error. The
error magnitude is very similar for aplanatic SCT systems, as well
as the Edge-type, the latter having the error about midway between
that with fixed and moving secondary (note that plot colors are not related to the color
"code" used for the aberration contributions of SCT elements in the
above formulas).

Simple empirical formulas roughly approximating close
focusing P-V wavefront error in units of 550nm wavelength for mirror focusing and
fixed mirror SCT systems are: