|
telescopeѲptics.net
▪
▪
▪
▪
▪▪▪▪
▪
▪
▪
▪
▪
▪
▪
▪
▪ CONTENTS
10.2.6.4.
Maksutov-Cassegrain aberrations: axial
Aberrations of a 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 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 spherical aberration 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 sL,
sP
and sS
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,
n being the glass refractive index, R1
and R2
the front and back surface radius and t the lens thickness. Since
the lens focal length
ƒL
is numerically negative, the
meniscus aberration coefficient is numerically positive, expressing
over-correction.
For the primary, the
lower-order aberration coefficient is given by sP=(1+K1)a'4/4R3,
with K1
being the primary conic and σ the corrector-to-primary separation
in units of the primary's radius of curvature RP.
The fourth power factor a'=1+(σRP/ƒL)+[(n-1)t/nR1],
with σ being the primary-to-corrector separation in units of the
primary's radius of curvature - shows dependence of the primary
aberration contribution on the corrector location vs. primary. The
farther, and/or the thicker corrector, and/or the stronger corrector's
radii, the larger primary
diameter needed to grasp diverging cone of light, and the larger its
aberration contribution (FIG. 120). This 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 included in the following
consideration.
It should be noted that 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, this can't be done by calculating the third-order
aberration alone.
For spherical primary
K1=0
and the aberration coefficient is approximated by
sP~a'4/4R3.
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 r.o.c. 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. Due to the primary focal length being effectively
reduced by a factor of ~[1-(σR/ƒL)],
ρ is also larger, as well as the
effective secondary magnification m. Denoting the changed
parameters as k', r' and m', the system aberration
coefficient for spherical aberration can be approximated as:
with the first,
second and third bracket quantity being the aberration contribution of
the meniscus, primary and secondary mirror, respectively, and R1,
RP
and RS
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 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 needed first meniscus radius - after substituting r'RP
for RS
- 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)RP3/2t]1/2/(n-1)a'2.
For n~1.5 and an average value for a'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.
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 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 increase on both
mirrors after the introduction of meniscus lens, but the proportion will
not change significantly). The resulting ratio of the secondary to
primary aberration contribution, given by s'~sS/sP
(Eq. 154) is the ratio of
reduction in the corrector's contribution. Since the aberration
contribution of the corrector is (approximately) inversely proportional
to the forth power of R1,
the above value for the front lens radius needs to be increased by a
factor of [1/(1+s')]1/4.
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 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 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: a',
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 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 need
only minor optimization can be determined using empirical relation given
for Maksutov camera. 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 than
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~R(1-2τ-0.01F)[τ(n+2)(n-1)2/2n3(1+s')]1/4,
and R2=R1-{[1-(1/n2)]t/0.97}
As shown in the section on off-axis aberrations, coma
and astigmatism are inherently low in a typical MCT system (k~0.25, m~4)
with separated spherical secondary.
In the arrangement with an aluminized
spot on the back of corrector as the secondary (Gregory-style), R2=RS
and R1=R2-[1-(1/n2)]/0.97.
Thus, changes in the radius affect similarly both, corrector's and
contribution of the secondary's: they either raise or fall in a similar proportion. The system
geometric properties are also maintained by manipulating the
corrector/secondary-primary spacing.
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. 121),
the balanced form
is considerably less detrimental to
image quality.
FIGURE 121: Balanced lower/higher spherical aberration (LHSA) 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 alone (corresponding to 1/3 wave
P-V). The difference between the two, at similar error levels, is that
the LHSA affects contrast at lower MTF frequencies somewhat more, and
those closer to the mid-range somewhat less than LSA.
This is due to LHSA's more steeply curved wavefront edges, causing
energy transfer farther away from the Airy disc. LHSA is still noticeable in the star test at as low as
0.02λ
RMS level, the error level well below the threshold of detectable
in the actual performance. 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 nearly the same as LSA.
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 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
►
Home
| Comments |