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11.8.2.
Maksutov-Cassegrain off-axis aberrations
Off-axis aberrations
of the meniscus, unlike
those of the Schmidt corrector, are not negligible. Both coma and
astigmatism of the meniscus are of the opposite sign to those of the
primary mirror, and of the same sign as those of the secondary. Due to
the secondary spherical aberration contribution, also opposite to that
of the primary (for spherical mirrors), the meniscus in two-mirror
systems is somewhat weaker, thus with somewhat lower contribution of
(opposite) coma and astigmatism relative to those of the primary.
The
coma
lower-order aberration coefficient
for the meniscus is, as
mentioned in the previous section, approximated by cL~-1.2q/ƒL2,
with q being the shape factor (R1+R2)/(R1-R2).
With q expressed in terms of
ƒL
and R1,
it can also be written as cL=
-1.2/R1ƒL.
The system coma coefficient is a sum of the contributions from the lens
and the two mirrors:
c=cL+cP+cS
= [-1.2/R1ƒL]
+ [(1-σ)/RP2]
+ [k'(2σ-1)(1-u')(m'2-1)/m'2RP2]
(131)
also for
spherical surfaces, with the values in the brackets being the aberration
contributions of the meniscus lens, primary and secondary, respectively.
As before, σ is the stop (corrector) separation from the primary
in units of the primary's r.o.c., k' is the relative height of
marginal ray at the secondary, m' is secondary magnification and
u' the relative exit pupil separation obtained from
Eq. 116.1 with σ, k' and m'
substituted (m' and k' represent the final values
determined by corrector's power, as explained under
FIG. 98 and
Eq. 129).
The system P-V wavefront error is given by
Wc=cαD3/12,
with α
being the field angle in radians. Expressing R1
and
ƒL
in terms of the primary's radius of curvature RP,
the meniscus coma coefficient is approximated by cL~1.2t1/4/1.7RP9/4.
In terms of the coma of the primary mirror, for the typical relative
separation σ~0.4, that gives cL/cP~|t/RP|1/4.
It shows that coma contribution of the corrector slowly increases with
its relative thickness. However, this doesn't take into account weaker
corrector due to the presence of secondary mirror. With the secondary
spherical aberration contribution approximated with -1.2k (from
Eq. 153), in terms of the primary mirror
contribution, and the aberration contribution of the corrector changing
approximately in proportion to R14,
the front meniscus radius needed to correct the two mirror's spherical
aberration is weaker approximately by a factor of (1-1.2k)1/4.
Since the meniscus' focal length
ƒL
changes nearly in proportion with R1,
it gives better approximation of the corrector coma contribution as cL/cP~1.2|t/RP|1/4(1-1.2k)1/2.
For the average |t/RP|~0.02
ratio and k~0.25, corrector coma contribution is nearly -0.4 of the primary's,
increasing only about 5% for k~0.2. For the average k~0.25 and m~4 (u=-7.1), secondary's coma
contribution is, from Eq. 131, about -0.63 of the primary's, going to little over -0.7
for k~0.2, m~6 and u~-10. So for a typical MCT telescope, the secondary
coma contribution can be roughly approximated by cS/cP~-0.32/√k.
The above numbers indicate that the lower-order coma level
in the MCT is generally low, as long as usual designing freedoms are available.
Higher-order
coma in a typical MCT is not negligible, and that needs to be kept in
mind. It is mainly produced by the
corrector and, for good off-axis correction, it needs to be balanced with a
similar amount of the lower-order coma of opposite sign.
Lower-order astigmatism in the MCT is also generally
low, but both less predictable and potentially greater than either in
the SCT or Houghton-Cassegrain. The reason is that neither Schmidt nor
Houghton correctors have significant power, thus their astigmatic
contribution is negligible. On the other hand, Maksutov corrector has
relatively significant power and thickness, both resulting in a
potentially significant amount of astigmatism, opposite in sign to that
of the primary (hence of the same sign with the astigmatism of the
secondary). So while the effect of the corrector's position on primary's
astigmatism in a two-mirror system is
generally similar to that in a Newtonian-style Maksutov or Schmidt, due
to the stop (corrector) position relative to the primary being similar
(typically, σ~0.4), the system error is different. If astigmatism of the two
mirrors nearly balances out, it leaves the meniscus' contribution as the
dominant in the final system error. While it is relatively low, even low-level astigmatism can significantly change
best field curvature, and also may become less than tolerable farther
off-axis. If the secondary astigmatism overpowers that of the primary,
combined with the meniscus' contribution it would likely result in a
more than insignificant system error. And vice versa, if astigmatic
contribution of the secondary is lower than that of the primary, astigmatism of the
corrector will play balancing role.
As mentioned in the previous section, aberration coefficient
of astigmatism for the Maksutov corrector is disproportionately greater
than its power, due to it being a strongly curved thick meniscus (FIG.
99). While the
FIGURE
99: Exaggeration of the effect of meniscus' thickness on its
astigmatism[1].
While the effective focal length is relatively weak, both surfaces
are strongly curved, of high powers and, consequently, astigmatism.
As long as the meniscus is thin, their aberration nearly
offset due to similar opposite powers. As the lens thickness (t)
increases, the chief ray (CR) arriving at the front surface
at an angle α, arrives at the
second surface at an increasingly smaller angle
α',
due to an effective rotation of the second surface around its center
of curvature C. The effective inclination angle
α' at the second surface is given by
α'=α-β.
As a result, astigmatism at the second surface diminishes, which in
turn increases the total lens' (meniscus) aberration. Depending on
the system configuration, it may or may not be desirable.
[1] Oversimplified; actual ray path involves meniscus' principal points,
typically at a significant separation from the meniscus itself.
aberration coefficient of astigmatism
for a thin lens is a=-1/2ƒ,
ƒ
being the focal length, Maksutov
corrector generates significantly more of the aberration, due to the
imbalance in contributions of the two surfaces caused by meniscus'
thickness (it also affects meniscus' coma, but in a lesser degree, due
to it changing with the angle, not the square of it). Astigmatism added due to
the meniscus' thickness, in a form of the aberration
coefficient, is a'=t/2ƒ1R2,
with
ƒ1
being the first surface focal length. System coefficient of astigmatism is
approximated by a~(n2+n-1)(n-1)t/2n2R1R2.
Most of the numerical value is the portion caused by lens thickness,
which can be written as a'=(n-1)t/2R1R2.
Thus the system aberration coefficient of astigmatism takes form:
a=aL+aP+aS=
[(n-1)t/2R1R2]
+ [(1-s)2/RP]
+ [-k'(m'-1)(2s-1)2(1-u')2/m'RP] (132)
with the system P-V wavefront error W=aα2D2/4.
For n~1.5, the meniscus' coefficient can be approximated by a~t/4R1R2,
and taking R1~R2
gives a~t/4R12.
With R12~-|tRP3|1/2/3,
the coefficient can be roughly approximated by
aL~-3|t/RP|1/2/4RP
which, for a typical average t/RP~0.02, gives the meniscus' astigmatism
as about -1/10 that of the primary's, with the stop at the surface. For a
typical stop separation of ~0.4RP,
it gives corrector's astigmatism as aL/aP~-2|t/RP|1/2,
in units of the primary mirror contribution.
Again, as for the coma, this needs to be adjusted for the weaker corrector due to the
presence of secondary mirror, which comes to aL/aP~-2|t/RP|1/2(1-1.2k)1/2.
This puts corrector's system
astigmatism with the above parameters at about -1/4 that of the primary's.
For average values of k~0.25 and m~4
(giving u=-7.1), and σ~0.4,
secondary's astigmatic contribution is nearly -0.5/RP,
or nearly 40% greater than that of the primary (~0.36/RP),
and of the opposite sign.
For smaller k and higher magnification m, characteristic
of Gregory-Maksutov in general, the relative secondary contribution can
be considerably higher, mostly due to increase in u' (Eq. 132). Roughly,
astigmatic contribution in a typical Maksutov-Gregory double that in
a faster MCT with larger, separated secondary. Thus, the secondary
contribution can be roughly approximated by
aS/aP~-1/12k2,
in units of the primary mirror astigmatic contribution.
Image field
curvature in the MCT is, similarly to other two-mirror
arrangements, given in terms of the system's Petzval curvature R0
and aberration coefficient of astigmatism, as best, or median field
curvature:
1/Rm=(1/R0)+4a
(133)
EXAMPLE: Actual vs. approximated
level of coma and astigmatism in the typical 6" ƒ/3/17
Gregory-Maksutov and 6" ƒ/3/10 Maksutov two-mirror system with a
separate secondary. With t/RP
being 0.021 in both, and k=0.18 and 0.27,
respectively, the approximation cL/cP~1.2|t/RP|1/4(1-1.2k)1/2
gives the corrector lower-order coma
contribution of
about -0.4 and -0.37 in units of the primary's coma.
Adding the secondary contribution of nearly -3/4 and about -0.6 of
the primary's coma, respectively, according to
cS/cP~-0.32/√k,
gives the approximated system
lower-order coma sum of about -0.15 and 0.03 in units of the
primary's coma for the Gregory-Maksutov and system with separate secondary,
respectively.
The actual sums for the two
systems are -0.19 and 0.05 of lower-order coma, respectively, with the higher
order coma being nominally nearly 40% of the lower-order coma (thus
adding up) in the former, and nearly identical, but of opposite sign
(thus practically cancelling out) in the later. Approximations for
lower-order coma are close enough to be useful.
Approximate values for the
astigmatism aberration contribution for the Maksutov-Gregory and the
separate secondary system are, from
aL/aP~-2|t/RP|1/2(1-1.2k)1/2,
-0.26 and -0.24 for the
two correctors and, from
aS/aP~-1/12k2, -2.5 and -1 for the secondary, respectively, in units
of the primary's astigmatic contribution. Actual system
contributions are -0.27/-0.22 and -2.3/-0.85, respectively. This
puts the system astigmatism sum approximation at about -1.8 and -0.24,
versus the actual -1.6 and -0.07, respectively. Again, not really
accurate, but sufficiently so to reflect gross proportions of the
individual elements' contribution and system error level.
Higher-order astigmatism in these
systems is generally negligible.
With the corrector astigmatism
aberration coefficient approximated by
aL~-3|t/RP|1/2(1-1.2k)1/2/4RP
or, for the Gregory-Maksutov,
aL~-0.1/RP,
the system coefficient gets multiplied by 1.8, giving a~0.00015.
With the system Petzval surface obtained from
Eq.30, best image surface
curvature is, from Eq.130, approximated as Rm~-250mm. The
actual value for the system is -290mm, which is close enough for the
initial assessment. For the system with separated secondary, the
value of aL
is only slightly ~10% smaller, but the approximated astigmatism is
nearly 1/4 of that of the primary, so that the system astigmatism
coefficient (approximation) comes to a~0.00001. Using the same
equations, it gives best (median) surface curvature approximation
for this system as Rm~-515mm,
with the actual best field curvature being Rm~-560mm.
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. Due to its steeply
curved optical surfaces, especially those of the meniscus corrector,
Maksutov-Cassegrain systems generate hard-to-impossible to correct (w/o
aspheric surface terms) higher-order spherical aberration (HSA). While
roughly as much noticeable in the star test as the lower-order
aberration (FIG. 100), it is considerably less detrimental to the
image quality.
FIGURE 100: Higher-order spherical aberration in as it would
appear in a star test
(about 4 waves of defocus, the in-focus patterns scaled up ~2x, for
clarity). At
0.1λ
RMS wavefront error level (corresponding to ~1/2 wave P-V), it is as
clearly noticeable - if not more so - than 0.1λ
RMS of the lower-order spherical aberration (corresponding to 1/3 wave
P-V). The difference between the two, at similar error levels, is that
the HSA affects contrast at lower MTF frequencies somewhat more, and
those closer to the mid-range somewhat less than its third-order cousin.
This is due to HSA, having more steeply curved
wavefront edges, causes energy transfer farther away from the
Airy disc. The HSA is still noticeable in the star test at as low as
0.02λ
RMS level, the error level well bellow the threshold of detectable
in the actual performance. Since this form of spherical aberration is
generally low in Maksutov-Cassegrain systems (it is highly unlikely it
will be more than a small fraction of 0.1λ wave RMS wavefront
error), detecting it in its characteristic form indicates that the
system is well corrected for the (more important) lower-order
aberration. However, that doesn't warrant plainly accepting imperfect
star test. Quantifying the aberration level, even approximately,
requires establishing with certainty which of the two aberrations is
clearly dominant - if there is such - or what is the combined
aberration, which can be a
difficult task when they are balanced one against another (as it is
usually the case), despite their respective patterns do noticeably differ in
appearance.
This mysterious
property of the 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 low levels of the HST, if the higher-order term
is not accurately put onto the corrector.
◄
11.8.1. MCT aberrations: spherical
▐
11.9. Full-aperture Houghton corrector
►
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