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10.2.3. Maksutov camera
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10.2.3.2. Maksutov corrector - telescopes
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#
**10.2.3.1. Maksutov corrector surface radii**

Due to enormous
amount of spherical aberration generated by strongly curved surfaces of achromatic meniscus
corrector in the Maksutov camera, it is very sensitive to both, surface
radii differential and variation in corrector thickness. Also,
significant higher-order spherical aberration it generates makes
lower-order expression alone insufficient for determining corrector
properties needed to minimize spherical aberration of the mirror. Thus,
there is no simple way to find out specific corrector radii and
thickness for a given mirror. However, it is possible to arrive at a
good approximation, either for informative purposes, or to be used as a
starting point in designing corrector.

Expressing meniscus center thickness **t**
and front surface radius of curvature **R**1
in terms of the mirror
radius of curvature **R** as t=-τR (which determines parameter
**τ** as numerically positive), and R1=κR
allows for establishing relation between these two parameters based on
Eq. 125, as
[κ-(1-1/n2)τ]κ3=-(n+2)(n-1)2τ/2n3.
With **τ** being much smaller than **
κ**,
neglecting [1-(1/n2)]τ
leads to the approximation
κ4~(n+2)(n-1)2τ/2n3
and the **first iteration** for needed value of the first radius for
corrected lower-order spherical aberration:

where **n** is, as before, the
corrector glass refractive index, and **
τ** its relative center thickness t/R.

Once the initial approximation of **R****1**
is known, a correction factor compensating for the neglected value can
be applied, giving **closer approximation** of
the needed first radius as:

The original corrector thickness, chosen by Maksutov, was t=D/10.
Thicker corrector reduces somewhat residual spherical aberration,
but at a price of increased lateral color. Optimum thickness is, as
noted by Rutten and Venrooij (p101), probably some 50% greater, thus t~D/7
(either can be easily expressed in terms of mirror radius of curvature **R** for any specific
mirror).

As already mentioned, meniscus corrector with these parameters nearly
corrects for third-order spherical aberration of a mirror of the radius of curvature **R**.
However, due to strongly curved surfaces, there is always residual
fifth-order aberration present. It is negligible compared to
the lower-order aberration of the mirror, but once it is near corrected, the
higher-order aberration becomes a factor. To minimize the aberration, the two need to be optimally balanced, by combining their
near-equal amounts of opposite signs. Due to this, as well as due to the
approximations made for simplicity, near-optimum first radius **R****1**
will be nominally somewhat smaller than what is indicated by **Eq. 126-126.1**.

Typically, **best radius of curvature for the
front meniscus surface** is closely approximated by R1~0.94R'1.
Nearly as good final approximation is given by R1~0.9R''1,
which indicates that the first iteration is a sufficiently good basis
for estimating the needed first radius.

Once **R****1**
is known, the second radius is obtained from:

which in turn determines the thickness **t** in terms of the radii
and index of refraction, as given with
Eq. 124.

Keep in mind that the corrector is convex
toward the mirror, thus with the radii nominally negative. Since the
thickness is positive, this means
that
**R****2**
is weaker than **R****1**,
and the meniscus is a weak negative lens, thicker at the edge than at
the center.

With these parameters,
a single-mirror Maksutov system is roughly corrected; however, the approximate character of the
procedure, combined with the high system sensitivity to changes in
optimum meniscus parameters, require verification and likely correction. Further detailed optimization improves quality level, but the main benefit is achieved by
balancing the two forms of spherical aberration. Final optimization is
best done with the help of ray tracing software,
as illustrated on **FIG.
183**.

**FIGURE 183**:
TOP: Illustration of the
process of finding best meniscus parameters. For D=200mm
f/3
sphere with t=25mm and BK7 glass, 0.9R''1=-249mm
which, taken as **R****1**,
with **R****2**
obtained from **Eq. 128**, gives an
f/2.9 system ready for
optimization (**a**). Strengthening the radii to R1/R2=-246.25/-260.4
results in near-optimum correction for the central line (**b**).
However, while the paraxial foci nearly coincide, the outer zonal foci
are more widely separated, resulting in less than optimum sphero-chromatic
correction. Optimizing is accomplished by changing the radii relation
to R2=R1-[1-(1/n2)]t/0.97.
This results in the outer zonal foci coming close together (**c**).
BOTTOM: The other example is for a D=200mm f/5 sphere with t=20mm,
which is actually a telescope in the Newtonian configuration.
First iteration (0.9R"1)
already results in a very useable system (**a**). However, it can and
should be optimized through the same procedure. Bringing all colors as close as possible would require slightly stronger radii (-352.5/-364 vs. -357.5/-369.2mm) but the gain wouldn't be detectable in field use).
SPEC'S
(optimized only)

It should be noted that the above approximation is for the
corrector-to-mirror separation of ~2R/3, needed for zero system coma.
With the change in the position of the corrector, its radii needed for
zero system spherical aberration also change. This is due to Maksutov
corrector having relatively significant power (much more so than the
Schmidt corrector), which effectively changes for the mirror with their
separation. In general, decrease in separation increases the effective
corrector power, requiring weaker radii, and vice versa. Also, with
large mirror focal ratios (~f/3 and larger),
higher-order coma of the corrector increases sufficiently to require
somewhat smaller zero-coma corrector-to-mirror
separation.

In practice, there is
no need to calculate for best paraxial correction, knowing that the
optimum overall correction is different. The correction factor for the
**optimum overall system correction**
can be applied directly to **R"****1**
given by **Eq. 126**,
in an empirical relation taking into account slight variations due to
the changes in relative thickness **τ**=t/R, **R** being the
mirror radius of curvature, and mirror focal ratio **F**.
This gives the meniscus radii approximation as:

with **n** being the glass refractive index, **F** the mirror
focal ratio and **t**, as before,
the nominal meniscus center thickness.

This first
radii approximation should bring the wavelengths quite close together,
with relatively small residual spherical aberration present. If the blue
is farther away than the red, while spherical aberration is relatively
low, both radii need to be strengthened, while keeping the differential
nearly unchanged. Opposite if the red focuses farther away. When the
wavelengths are close together, spherical aberration is minimized by
small adjustment in one of the corrector radii (if the residual
spherical is significant, then it needs to be minimized first, by
finding needed radius differential; after that the radii are changed as
needed, with relatively small changes in their differential, if needed,
until both chromatic and spherical aberration are minimized).

Evidently, needed first
radius of the corrector becomes slightly stronger as the relative
corrector thickness and mirror F-number increase. The 0.97 factor is not
strictly fixed; it determines the level of best chromatic correction,
not necessarily the very point of numerical maximum. Slight deviations may offer correction modes more appropriate in some instances,
but the difference - or benefit, if any - is typically very small.

Maksutov himself gave empirical formulas for determining corrector
radii for the camera (i.e. aplanatic) configuration (*New catadioptric
meniscus systems*, p273). They are as follows:

R1=-0.612DF^{0.66}

R2=-(0.612F^{0.66}-0.0565-0.007/F)D

R3=-2.107DF^{0.983}, and

s=1.11DF^{1.14},

where R1, R2, R3 and **s** are the front and rear meniscus radii,
mirror radius of curvature and corrector-to-mirror separation,
respectively (for aperture diameter **D** and focal ratio number F=f/D,
**f** being the mirror focal length).
Relations are for corrector thickness 0.1D and BK7 glass,
or equivalents, but small differences in the refractive
index are inconsequential. Relations are good approximation for
the Newtonian configuration as well, only adding a small amount
of overcorrection (for the ultimate tightening of the colors
seems that corrector radii need to be reduced by a factor of 0.99).

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10.2.3. Maksutov camera
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10.2.3.2. Maksutov corrector - telescopes
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