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10.2.6.4. Maksutov-Cassegrain aberrations: axial**
Aberrations of the Maksutov-Cassegrain system are a sum of the
aberrations of the meniscus corrector and the two mirrors. In that
sense, they are no different than system aberrations of a
Schmidt-Cassegrain, given with Eq. 113-119,
with the only difference coming from the aberrations induced by a
corrector. System aberration coefficients for two mirror systems with
meniscus corrector are found from the sum of the coefficients for the
three system elements. The main difference, as already mentioned, is
that
higher-order spherical
aberration
(HSA) is significantly greater with the meniscus-type corrector, making
the third-order expressions for spherical aberration more approximate
even with relatively slow mirrors. Designing a two-mirror Maksutov-Cassegrain
is considerably more calculation extensive also due to the relatively
significant corrector power.
**Lower-order s****pherical
aberration**
(LSA) of a two-mirror system with full aperture Maksutov-type corrector
is a sum of spherical aberration contributions of the corrector and two
mirrors. Thus, the system aberration coefficient can be written as s=sL+sP+sS,
with **s**L,
**s****P**
and **s****S**
being the aberration coefficients for the meniscus, primary and
secondary mirror, respectively. The system wavefront error is given as
W=sD4/64
as the best focus P-V wavefront error (**D** being the aperture
diameter).
Lower-order spherical
aberration aberration coefficient for the
meniscus is, according to
Eq. 121.1, approximated by:
sL~
-(n+2)/8nƒLR12,
with the meniscus focal length ƒL=-n2R1R2/(n-1)2t,
and the lens center thickness t=n2(R1-R2)/(n2-1),
**n** being the glass refractive index, **R**1
and **R**2
the front and back surface radius, respectively. Since the lens focal
length **ƒ**L
is numerically negative, the meniscus aberration coefficient is
numerically positive, expressing over-correction. As already mentioned
on the camera page, this approximation gives a few percentage points
larger than the actual value for the third order term alone, but for ~ƒ/3
primary mirror typical for the MCT configuration, it is near exact for
the first two terms (lower- and higher-order) combined. Also, due to its
greater center thickness, it acts more like a thick lens, with more of
the lower- and higher-order aberration generated at the rear surface
than what the thin lens concept implies. As a consequence, the actual
aberration of the meniscus is some 10-15% lower than what the above
approximation gives.
For the primary, the lower-order aberration coefficient is given by
sP
= (1+KL)p/4R**P**3,
with **K**1
being the primary conic and **σ** the corrector-to-primary separation
in units of the primary's radius of curvature **R****P**.
The **p** factor p={1+[(n-1)t/nR1]+(σR**P**/ƒL)}4[1-(R**P**/ƒL)]2,
with **σ** being the primary-to-corrector separation in units of the
primary's radius of curvature, shows dependence of the primary's
aberration contribution on the corrector power and location vs. primary,
which determine both, ray height at the primary (the fourth power term)
and the effective object distance for the primary due to the ray
divergence by the meniscus (the quadratic term). The second factor in
the fourth power term expresses the increase in ray height at the rear
corrector surface, due to refraction at the front surface, which is
compounded by the ray height increase due to the corrector-to-primary
separation, expressed by the third term. As mentioned, the quadratic
term expresses the drop in the aberration level at the primary due to
the ray divergence caused by the corrector, effectively placing object
for the mirror at the distance approximated by the corrector's focal
length.
This **p** factor is
neglected in calculations for the single-mirror system (Eq.
122-125), because it partly offsets with the factors neglected in
Eq. 121.1, thus making the
approximation more accurate. It will be, however, included in the
two-mirror system consideration, because the ray height factor is
roughly cut in half due to reduced corrector-to-primary separation
which, with the increase in the effective object distance for the
primary due to the weaker corrector, sufficiently reduces offsetting
between these factors to make them significant.
The purpose of going through the Maksutov
two-mirror system aberration coefficients is primarily to show how it
generates the aberrations, not to have them precisely calculated; as
already mentioned, due to the likely significant higher order terms
generated (mainly) by the corrector, this cannot be done by calculating
the third-order aberrations alone.
For spherical primary
K1=0
and the aberration coefficient is approximated by
sP~p/4R3.
It is numerically negative (under-corrected), opposite in sign to the
lens aberration.
For the secondary, the aberration coefficient
expression is identical to that for the SCT secondary (Eq.
113.1). However, due to the relatively significant negative power of
Maksutov corrector, all three parameters, height of the marginal ray at
the secondary **k** (in units of the aperture radius), secondary
radius of curvature in units of the (effective) primary's curvature
radius **ρ**
and secondary magnification **m**, have changed with respect to those
for the two mirrors alone. The increased height of the marginal ray at
the primary, coupled with the slight reduction in its effective relative
aperture, results in the increased height of the marginal ray at the
secondary as well, and larger **k**. While to the primary's nominal
focal length is effectively reduced by a factor of ~[1-(σR/ƒL)],
the actual cone length is larger by a 1/(1-RP/ƒL)
factor (FIG. 188),
hence the effective
**ρ'** is slightly smaller than
**ρ**, and the effective secondary
magnification **m' **is larger than that for the mirrors alone. So
the system aberration coefficient for spherical aberration can be
approximated as:
with the
first, second and third term being the aberration contribution of the
meniscus, primary and secondary mirror, respectively, and **
R**1,
**R**P
and **R**S
the radii of curvature of the first meniscus surface, primary and
secondary, respectively. The relation is for spherical surfaces;
conics can be added, as given
for the SCT primary and secondary.
Contributions of the meniscus and the secondary are of the same sign
(positive), and opposite to that of the primary. For given primary
mirror, the system needs to be configured so that its aberration
contribution is cancelled by those of the meniscus and secondary. With
R1/R2~1,
the meniscus' focal length is approximated
by ƒL~(nR1)2/(n-1)2t,
and the needed first meniscus radius - after substituting **ρ'R**P
for **R**S
- can be approximated as:
The extraction is not
straightforward, as all three of the secondary's parameters are affected
by the corrector's power, an unknown beforehand. Probably the simplest
initial configuration can be arrived at by starting out with the primary
and meniscus alone. Taking again R1/R2~1
and ƒL~(nR1)2/(n-1)2t,
the meniscus lens focal length **ƒL**
is approximated in units of the primary radius of curvature as:
ƒL~
[n(n+2)RP3/2**t**]1/2/(n-1)p1/2.
For n~1.5
and the average value for **p**1/2
of ~1.1, it gives the meniscus focal
length ƒL~3(RP3/t)1/2,
which in turn gives the appropriate front lens surface radius as
R12~3(n-1)2(tRP3)1/2/n2
or, for n~1.5,
R12~(tRP3)1/2/3.
The rear corrector radius for best chromatic correction is, as before,
R2=R1-[1-(1/n2)]t/0.97,
**t** being the corrector center thickness.
This gives the first
approximation of the meniscus radii needed to correct primary's
spherical aberration. Adding the secondary offsets more of the
aberration of the primary, requiring somewhat weaker corrector for the
system. The degree of weakening can be approximated, most simply, by
neglecting the corrector for a moment, and figuring out the aberration
coefficient for secondary mirror that would be produced by a system
similar to the intended MCT in regard to the relative aperture,
obstruction ratio etc. (the level of aberration will change on both
mirrors after the introduction of meniscus lens, but the proportion will
not change by much). The resulting ratio of the secondary to primary
aberration contribution, given by s'~sS/sP
(Eq. 154), approximates the
ratio of reduction in the needed corrector's contribution vs. that
needed for primary mirror alone. Since the aberration contribution of
the corrector is (approximately) inversely proportional to the 3rd power
of **R1**,
(4th power before factoring in the **R1**
term in the corrector thickness **t**) the above value for the front
lens radius needs to be increased by a factor of [1/(1+s')]1/3.
The above procedure would
result in the first outline of a system, most likely not yet at the
optimization level. The main reason is that the meniscus radii
approximation would nearly cancel the lower-order spherical, which
leaves higher-order spherical aberration - significant with faster
mirrors - unbalanced. Thus, radii adjustments are needed to produce
usable system. Since it is normally the corrector's properties that need
to be adjusted in the first place, the calculation is complicated due to
the changes at the corrector causing changes in all other parameters
down the optical train: **p**,
**k**, **ρ** and **m**, hence in
the aberration contribution of the two mirrors as well. In order to
maintain needed geometric properties of the system, effects induced by
the adjustments of the corrector's power need to be compensated for with
the appropriate changes in mirror separation. Obviously, the two-mirror
Maksutov system is more complicated than the single-mirror system, and
warrants even more the use of ray tracing software for system
optimization and final verification.
Still, MCT system
approximation that usually needs only minor optimization can be
determined using empirical relation given for the Maksutov camera.
The needed front corrector radius for the primary mirror alone is determined
from
Eq. 126 and Eq.
128 (slightly modified **Eq. 128**, as R1~[1-2τ-0.01F)R1"
should be better suited for faster MCT primaries). This value is then
corrected for the aberration contribution of the secondary mirror, by
multiplying it with [1/(1+s')]1/4
- noting that **s'** is according to
Eq. 154
numerically negative - and the rear radius value is then obtained from
Eq. 128.1. It
can be all summarized in an initial approximation for the corrector
surface radii given as:
R1
~ RP(1-2τ-0.01F)[τ(n+2)(n-1)2/2n3(1+s')]1/4,
and R2
= R1-{[1-(1/n2)]**t**/0.97}
with
**
τ**=t/R, as before,
and **F** the system focal ratio. Taking
τ~1/60, F~16,
n~1.5 and s'~-0.2, gives R1~0.22RP.
As shown in the section on
off-axis aberrations, coma and astigmatism are inherently low in the
typical MCT system (k~0.25, m~4) with separated spherical secondary.
In the
**arrangement with an
aluminized spot** on the back of the corrector that acts as the secondary
(Gregory-style), R2=RS
and R1=R2-[1-(1/n2)]/0.97.
As mentioned, achieving needed level of the compensatory spherical
aberration at the corrector here requires more strongly curved - hence
smaller for given back focal length - secondary, producing more of coma
and astigmatism. The common radius constraint of this arrangement can be
illustrated by substituting n~1.5, R1~0.95R2,
t~R2/14,
σ~0.4, R2=ρRP,
ρ=mk/(m-1)=(1+η)m/(m2-1),
and setting RP=1
and back focal length η=0.3 in **Eq. 129**. Relative aberration
contributions of the three elements (in units of **R**P)
now can be approximated as:
**1/400ρ3**-**0.28**+**1.34/2(m+1)4ρ****3**
= **(m2-1)3/880m3**-**0.28**+**(m****2****-1)3/1.5(m+1)m3**
for the
meniscus (blue), primary (red) and secondary mirror (green). Plots at
left illustrate how the primary spherical aberration contributions of
the three Gregory-Maksutov optical elements determine secondary
magnification at which the sum will be zero (about 6-inch aperture with
the standard ~ƒ/3 primary and ~0.1D meniscus thickness).
The
primary mirror undercorrection contribution is nearly constant (it does
change but only slightly with the meniscus' radius/power). Secondary's
overcorrection is fairly constant for secondary magnifications larger
than 3, due to the effect of smaller secondary being mainly offset by
the effect of stronger surface radius.
It is overcorrection of the meniscus that changes rapidly with its radii
strength - as the relation at left indicates, approximately with the third power
of it. At low secondary magnifications (i.e. large secondary with
weak radius of curvature, imposing it onto rear meniscus surface) it is
near negligible. Even at m~5, it is still not sufficient to result in
near-zero system coefficient, with the system having about 2 waves P-V
of undercorrection.
The zero sum occurs at the secondary magnification of ~5.8, i.e. system
focal ratio ~ƒ/17.4. The zero-sum secondary magnification doesn't change
significantly with faster primaries; at ƒ/2.5 it is still about 5.8 (for
the same nominal BFL, nearly 140mm, which is about the practical minimum
for this aperture; thus it does decrease somewhat for the same relative
BFL). Increasing corrector thickness beyond the standard ~0.1D
(D=aperture diameter) lowers the minimum secondary magnification only
slightly within the practical thickness range.
Note that the above consideration is for chromatically corrected
meniscus that is corrected for both, spherical aberration (as minimized
secondary spherical)
and chromatism. Spherical correction - in the form of minimized
spherical aberration - can be achieved for any rear meniscus
radius - i.e. back focus - value with the appropriate front radius, but
for given meniscus thickness there is only one
radii combination that will correct both, spherical and chromatism. At
secondary magnifications lower than the above minimum, meniscus corrected for both does not
have enough of rear side curvature to create sufficient back focus
distance.
Extending BFL by a slight reduction in the
corrector-to-mirror separation will also increase the secondary
magnification and system focal ratio. The effect on spherical correction
is minimal, and can be compensated for by a slight change in the front
meniscus radius value. In practice, this means that the range of
secondary magnification for the optimum correction extends to up to 10%,
or so, above the secondary magnification for near-minimum BFL.
Similarly to spherical aberration, chromatism can be corrected for any rear meniscus radius
value, but with a weaker than needed rear radius (i.e. lower secondary magnification) spherical aberration will
not be. Solution to it is to aspheize the primary, which would
also allow use of somewhat faster primary (thus more compact design),
since correcting residual spherical even at the secondary magnification of
about 4 requires significant aspheric (up to -0.4, or so, conic) on the
primary, and that in turn allows for a weaker corrector and less of the
higher-order spherical residual.
**
Maksutov-Cassegrain star test**
An interesting aspect of the
commercial Maksutov-Cassegrain is the question of its **
star test**. There is a notion that its
optics has special properties, making it sort of exception in that its
intra and extra focal pattern are not supposed to be identical, even
when it is near perfectly corrected. Or, put somewhat differently, that
it doesn't need to have near-perfect star test for near-perfect
performance.
The answer to this special status is in its higher order spherical
aberration. Due to its steeply curved optical surfaces, especially those
of the meniscus corrector, Maksutov-Cassegrain systems generate
6th-order spherical aberration that can't be cancelled (w/o aspheric
surface terms), only minimized by balancing it with the 4th-order
aberration. While roughly as much noticeable in the star test as the
lower-order spherical aberration for given P-V wavefront error (**FIG.
189**), the balanced form is considerably less detrimental to image
quality.
**FIGURE 189**: Simulation of the balanced higher order spherical
aberration in a star test
(size of defocused patterns reduced roughly 10 and 20 times for 4 and 8
waves defocus, respectively). The difference in appearance between
extrafocal patterns is noticeable at 1/10 wave P-V error (0.017 wave
RMS, comparable to 1/17.5 wave P-V of lower-order spherical). As the
correction error increases, the intrafocal pattern becomes noticeably
dimmer than extrafocal pattern at 4 waves of defocus, with the latter
becoming brighter, more contrasty in its inner portion. At the doubled
defocus error, the main difference between the extrafocal patterns is
that one side shows contrasty concentric pattern, while the other is
increasingly dimmer and diffuse in its outer portion, with noticeably
larger and darker central hole (note that for the reversed sign of
aberration the patterns would switch the sides). At the diffraction
limit, the difference in extrafocal patterns is glaring (generated by
APERTURE, Suiter). Note that the actual pattern appearance vary, with
the actual eye brightness response.
In an MCT, the higher-order
aberration originating mostly at the corrector, needs to be minimized by
balancing it with a similar amount of lower-order aberration. Ideally,
they are near-optimally balanced one against another, in which case the
error reduction factor is about 0.2 (1/5 of the HSA aberration RMS error
alone, balanced with nearly identical amount of the opposite in sign LSA).
Thus, if an MCT system is, say, 0.95 Strehl, it has ~1/28 waves RMS of
combined higher-and lower-order aberrations. In a near perfectly
balanced system, each form alone would be over 1/6 wave RMS. Of course,
it is possible that the two are less than perfectly balanced, which
makes quantifying the star test more difficult, unless the higher-order
component is clearly dominant, when it tests similarly as LSA.
The difference between
balanced higher-order (BHSA) and lower-order (primary) spherical
aberration (LSA), at a similar error level, is that the former affects
contrast at lower MTF frequencies somewhat more, and those closer to the
mid-range somewhat less than LSA. This is due to BHSA's more steeply
curved wavefront edges, causing energy transfer farther away from the
Airy disc.
This mysterious property of the seemingly compromised star test combined
with an excellent performance level, is also characteristic of other
systems with steeply curved optical surfaces, like
apo refractors.
Schmidt-Cassegrain telescopes also can have the
higher-order component present, if the higher-order term is not
accurately put on the corrector.
◄
10.2.3.3. Maksutov-Cassegrain
▐
10.2.3.5. MCT off-axis aberrations
►
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