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▪ CONTENTS
10.2.3.5.
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 at 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 also 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 previously, approximated by cL~-1.2q/ƒL2,
with q being the shape factor (R1+R2)/(R1-R2).
With q expressed in terms of the meniscus focal length ƒL
and first radius 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:
also for
spherical surfaces, with the values in the brackets, from left to right, being the aberration
contributions of the meniscus lens
cL, primary,
cP, and secondary,
cS, respectively.
As before, σ1 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
σ'2 the relative exit pupil separation
for the secondary, obtained from
Eq. 115.1 with σ1, k' and m'
substituted for σ, k and m, respectively (m' and k'
represent the final values of these parameters determined by corrector's
power, as explained with
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,
t being, as before, the corrector center thickness.
In terms of the coma of the primary mirror, for the typical relative
separation σ1~0.4, 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. 154), 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, σ1~0.4), 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.
190). While the
FIGURE
190: 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
the form:
with the system P-V wavefront error W=aα2D2/4.
For n~1.5, the meniscus' coefficient can be approximated by aL~t/4R1R2,
and taking R1~R2
gives aL~t/4R12.
With R12~-|tRP3|1/2/3,
the coefficient can be roughly approximated by
aL~-3|t/RP|1/2/4RP
which, for the 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 the typical stop separation ~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 value needs to be adjusted for the weaker
corrector due to the presence of the secondary mirror. The adjusted value is
approximated by aL/aP~-2[|t/RP|(1-1.2k)]1/2.
This puts corrector's astigmatism
contribution to a system with the above parameters at nearly -1/4 that of the primary's.
For the average values of k~0.25, m~4 and σ1~0.4
(giving σ2'~-7.1),
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
σ2' (Eq. 132/131). Roughly,
astigmatic contribution in the typical Maksutov-Gregory is double that in
usually somewhat faster MCT with a larger, separated secondary. Thus, the secondary
contribution can be roughly approximated by
aS/aP~-1/12k2,
in units of the primary mirror astigmatic contribution.
Hence, the MCT system coefficient of astigmatism, as
a sum of its elements' contributions, is approximated by a~{1-2[|t/RP|(1-1.2k)]1/2-(1/12k2)}(1-σ1)2/RP.
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:
Note that the meniscus doesn't affect appreciably system's
Petzval curvature, which is
for all practical purposes determined by mirror radii.
EXAMPLE: Actual vs. approximated
level of coma and astigmatism in the 6" ƒ/3/17
Gregory-Maksutov and 6" ƒ/3/10 Maksutov-Cassegrain two-mirror system with a
separate secondary from
FIG. 119 (down-scaled to 6"). With t/RP
being 0.021 for both, and k=0.18 and 0.27,
respectively, the coma coefficient ratio 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 0.03 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
relative (in units of the primary's)
astigmatism aberration contribution for the Maksutov-Gregory and the
separate secondary MCT are, from
aL/aP~-2[|t/RP|(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. The 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 -0.18, 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.
From the above approximation for
the system coefficient of astigmatism, it is a~0.0007 for the
Maksutov-Gregory (slightly more than given with Eq. 132).
With the system Petzval surface obtained from
Eq. 30, best image surface
curvature is, from Eq. 133, approximated as Rm~-190mm. The
actual value for the system is -210mm, 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~0.00015. With that value, best (median) image curvature approximation
for this system is Rm~-345mm,
with the actual best field curvature Rm~-360mm.
The discrepancy is mainly
caused by somewhat greater actual values for k, m and
ρ than those obtained from two-mirror relations, the result
of the power of meniscus corrector. For a quicker, and fairly
accurate estimate of the median image curvature in an MCT, in a
typical system it can be expected to be numerically quite close to
the secondary mirror radius of curvature.
◄
10.2.3.4. MCT aberrations: spherical
▐
10.2.4. Full-aperture Houghton corrector
►
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