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**4.1.3. Higher-order spherical aberration**

Considering the
importance of center field correction,
what is commonly called *higher-order* - and more specifically *secondary
spherical aberration*, being
a result of the second term in the
conic surface equation - does have some significance in certain
types of amateur telescopes with typically more strongly curved optical
surfaces. In particular, in most
apochromatic refractors,
as well as Maksutov-Cassegrain
(MCT)
and Schmidt-Cassegrain (SCT) telescopes. Since the
Schmidt corrector can directly compensate for higher-order terms as
well, they are not a factor in the properly made SCT.

Putting Schmidt-type aspheric on one or
more surfaces of an apochromat or MCT is not a practical option; they
also generate significantly more of higher-order spherical aberration
than a comparable SCT. Higher-order spherical in the MCT can be very
effectively lowered by moderately aspherizing primary mirror, which
allows to relax corrector's radii. However, all-spherical MCT, as well
as apochromats, may have significant level of residual
higher-order spherical aberration - which is referred to either as 5th order transverse (ray), or 6th
order on the wavefront. It has to be minimized by balancing it with a similar amount of lower-order spherical aberration and defocus.
This is usually accomplished by adjusting surface curvature radii, so
that they induce needed amount of the lower order aberration. Near-optimum
reduction of the 6th order aberration P-V wavefront error by balancing is by a factor of
~0.23, with the PV-to-RMS wavefront error ratio also reduced, by a
factor 0.64. In all, the RMS error is reduced
approximately by a factor of 0.15 (**FIG. 37**,** **B).

Higher-order spherical aberration is
merely a consequence of using (close) approximation of the conic surface,
instead of the actual surface, in calculating the aberration. General
conic surface is described by an expansion series, determining its
sagitta (depth at the center) as

z=(d2/2R)+[(1+K)d4/8R3]+[(1+K)2d6/16R5]+...
(Eq. 4.6),

with **d** being the surface half-diameter, **R** the radius of
curvature and **K** the conic
constant. The first term alone accurately describes paraboloid
(K=-1) which, consequently, doesn't have higher-order aberration
components. The second term describes surface approximations for K≠-1,
producing lower-order (3rd order transverse ray, 4th order on the wavefront) spherical
aberration for object at infinity (and any object location except that
coinciding with the conic surface's
geometric focus); the third term in the series adds a correction to that surface
approximation, and with it an additional, relatively small amount of the
aberration, the already mentioned 5th order transverse, or 6th order on
the wavefront.

The order designation is based
on the sum of exponents in surface half-diameter **d** and field
angle **α**.
Since spherical aberration doesn't change with the field angle, the
exponent in **α**
is zero, and the second term in the expansion series for the surface -
and thus for the wavefront as well - is of 4th order; since the
exponent in **d** for the transverse geometric aberration resulting
from the 4th order surface approximation is 3, it is a 3rd order
transverse aberration.

Terms beyond third in the expansion series are
insignificant in amateur-size telescopes.

**FIGURE 37 **
TOP:
(**A**) - Higher-order spherical
aberrations of reflecting (shown) or refracting conic surface result from corrections to the lower-order surface
approximation. The higher-order term that can be significant in amateur
telescopes is following the lower-order term in the expansion series.
According to the power to the zonal height in its expression, it is referred
to either as 5th order transverse ray aberration term, or 6th order
on the wavefront for the transverse and wavefront aberration form,
respectively (by the same token, longitudinal aberration is 2nd
order in its lower-order form, and 4th order in the next higher-form). In the presence of the lower-order aberration
on a single surface, it adds to
both, its ray (**LA**=longitudinal aberration, and **TA**=transverse aberration,
shown for the paraxial focus plane)
and wavefront error (**W**). With the lower-order aberration
cancelled, it remains as a residual surface aberration.

(**B**)
- The form of wavefront deviation for pure secondary spherical at the best
focus is very similar to that for the primary aberration (the wavefront
graph is rotated 90° clockwise to put it in the same
orientation as the LA graph). Its function
is ρ6-0.9ρ2,
which for its first derivative 6ρ5-1.8ρ
(i.e. 6ρ4-1.8
after reduced by **ρ**)
set to zero gives the value of **ρ**
with the maximum deviation as ρ=0.30.25=0.74. When significant,
this higher order residual
needs to be minimized by correcting the surfaces in their lower-order
approximation so that they better approximate the actual
surface profile. This is typically the case in telescopes using strongly
curved surfaces, such as meniscus corrector in
Maksutov-Cassegrain, or lens objective in
apochromatic refractors.
Secondary spherical can be either removed by applying aspheric term on a
surface, or minimized - i.e. "balanced" - with primary spherical
aberration, keeping surfaces spherical (note that the relevant form in
balancing
for
both aberrations is the one balanced with defocus, i.e. best focus
aberration). Balancing can be described as "adding" similar amount of
lower-order aberration of opposite sign to that of the higher-order.
Specifically, the P-V wavefront error of the balancing primary spherical
is 0.885 of the P-V wavefront error of secondary spherical, both at
their respective best focus location. Balancing is usually accomplished
by an appropriate change in the radius of one or more surfaces. Since
the two forms of deviations are only similar, not identical, the
aberration cannot be cancelled out, but is significantly reduced, by a
factor of ~0.15 in the RMS wavefront error. Graph at left shows
minimizing 6th order spherical residual in a wider context, starting
with the error at paraxial focus. The first two terms in the relation
for the balanced 6th/4th order aberration are a sum of the respective
errors at the paraxial focus, i.e WFE at that location. The third,
defocus term, represents the shift to the best focus location.

(**C**) - The
minimized wavefront
deviation form (expanded in the width for clarity) has an additional curl toward the edge.
Compared to primary spherical at best focus, it has greater deviation
above 91% of the radius, and below 64%, but the overall deviation is
smaller due to its smaller deviation over the 64-91% radius section. Since the P-V/RMS ratio of an
aberration is given by a sum of the (absolute) values of its
Zernike aberration term
for the two values of **ρ**
corresponding to the maximum wavefront deviation from zero mean and unit
coefficient (**z**nm=1)
value, it is here given by a sum of absolute values of
√7(20ρ6-30ρ4+12ρ2-1)z60
for ρ=0 and
ρ=1
(as can be seen above, under **C**), i.e. 2√7. Consequently, for given P-V wavefront error, the RMS wavefront error for balanced 4th and 6th
order spherical aberration, given by RMSB=P-VB/**√**28,
i.e. the RMS error is smaller by a factor of ~0.63 than for primary
spherical (and slightly larger for pure secondary spherical) aberration). Note that
this balanced form of spherical aberration is often called simply
"higher (5th or 6th) order" spherical, which may be confused with the
higher-order spherical directly related to the surface term. For this
reason, it is
better to use the term "balanced 6th/4th-order spherical" for the
former.

BOTTOM: As with spherochromatism in general, only a single wavelength
can be fully optimized, or in the case of balancing 6th and 4th order
aberration optimally balanced, i.e. fully minimized. Other wavelengths
will be either balanced only partially, or will have the two aberrations
add up (being similar in form, when adding up, the sum is roughly
arithmetic). Below are illustrated stages of partially balanced 6th order
spherical for both, longitudinal aberration (**LA**) and the
corresponding wavefront.

The top shows five stages of balancing, starting with
pure secondary spherical aberration and adding one quarter of the full
primary spherical needed for optimum balancing at the time. Bottom inset
starts with the fully balanced form and deducts one quarter of the full
amount of 6th order spherical aberration in it, ending with the
full balancing 4th order alone (generated by OSLO). Thus, top shows the
full amount of balancing higher order with various amounts of primary
spherical, and the inset shows the full amount of balancing spherical
with various amounts of higher-order spherical. The number columns show
the corresponding P-V and RMS wavefront error (**WFE**) for the P-V
error of the fully balanced aberration normalized to unity, for best (**B**),
paraxial (**P**), marginal (**M**) and the point of maximum
defocus (**MD**, with respect to paraxial focus, when different from
the marginal focus), from top to bottom. Since the two forms are
generally similar, but somewhat different, the LA graphs and wavefront
shapes for partial balancing are also somewhat different when the 6th
order aberration is dominant (top) and when the 4th order is dominant
(bottom inset). However for the same numerical proportion of one and the
other one, the best focus error and the extent of LA remain at a similar
level (comparable mixed aberrations are starting from either full order
toward the optimally balanced form, e.g. second from left on top to the
forth in the inset). The only significant difference is that the error
is the largest at the marginal focus for the former, and at the paraxial
focus for the latter. Since the reduction rate at the optimum balance is
by a factor of 4.6, 6 and 3.9 for the P-V WFE, RMS WFE and LA extent,
the relation between the LA and P-V WFE, given by LA/64F2
(exact for the 4th order and nearly exact for the 6th order), gradually
transforms to approx. LA/76F2
at the optimum balance, with the RMS error smaller by a factor of 4.44.
The rate of change is somewhat slower up to the level when one of the
aberrations is less than a half of the other one, increasing after that.
For the points outside the LA range, the P-V WFE error increases roughly
by the amount of defocus **l**
from the paraxial or marginal focus, given by W=l/8F2.
Note that the specific full magnitude of 6th order aberration here is
0.28 waves P-V, hence with the optimally balanced 6th+4th order of 0.06
wave P-V. Note that OSLO somewhat deviates from the values implied by
the theory - 4.3 times smaller P-V and 6.7 times smaller RMS for the
minimized vs. pure 6th order spherical - giving somewhat larger
reduction in the P-V, but smaller in the RMS error; the difference
probably comes from the best focus criterion used by the software.

For object at infinity, reflecting surface
forming a perfect wavefront is the paraboloid. Since surface error
doubles in the wavefront for paraxial focus, the lower-order **
P-V
wavefront error** with respect to the reference sphere centered at
paraxial focus is given by double the second term, or W4=(1+K)(ρd)4/4R3.
Likewise, the remaining 6th order P-V wavefront aberration is W6=(1+K)2(ρd)6/8R5,
with **ρ** being the height in pupil normalized to 1 and **d** the
pupil (aperture) radius. At the best focus, as mentioned, the P-V wavefront
error for the 4th order spherical aberration is smaller by a factor of
0.25, as given by the respective aberration functions, ρ4
for ρ=1 and ρ4-ρ2
for ρ=0.51/2,
for paraxial and best focus, respectively. Due to somewhat different form of deviation, best focus
P-V error for
the 6th order aberration (balanced with defocus) is reduced somewhat less, by a factor of ~0.4,
corresponding to its aberration function for paraxial (ρ6
for ρ=1) and best focus location (ρ6-0.9ρ2
for ρ=0.74);
location of the best focus is also shifted from midway between the
marginal and paraxial foci to somewhat closer to paraxial focus.

When 6th order spherical aberration is minimized by combining with the
lower-order form, the resulting aberration - the 6th/4th order spherical -
is also balanced by a specific amount of defocus needed for the shift from
paraxial focus to the location with the highest peak diffraction
intensity. In other words, it is balanced by both, the lower-order form
and defocus (this balancing should not be confused by balancing of the 6th
order spherical alone, which is only done by defocusing). As a result,
the aberration function changes from

WS6=S6ρ6
and WS6=S6(ρ6-0.9ρ2)

for the P-V wavefront error of pure 6th order spherical aberration at paraxial
and best focus, respectively, where **S**6
is the peak aberration coefficient for 6th order (secondary) spherical
aberration S6=(1+K)2d6/8R5, to

WS(6/4)=S6(ρ6-1.5ρ4+0.6ρ2-0.05)

for the peak wavefront error (1/2 the P-V error) of balanced 6th/4th order spherical
aberration. The term in **ρ**4
is for balancing with lower-order spherical, and the term in **
ρ**2
for balancing with defocus. The constant term quantifies the relative
peak aberration for ρ=1 or
ρ=0, which are identical in
magnitude, but of opposite sign (as illustrated on **FIG. 37**, C).
Compared to the relative P-V wavefront error of pure (unbalanced, i.e.
at the paraxial focus) 6th
order spherical (1 for ρ=1),
the fully balanced form's P-V error is smaller by a factor of 10; and it
is smaller than the pure 6th order balanced with defocus by a factor of
4.3.

The **longitudinal**
5th order spherical - which is, strictly, 4th order, being proportional
to the 4th power of pupil radius - is given by LA5=(K+1)(K+3)(ρd)4/16R3,
and adds to the 3rd order aberration, LA3=(K+1)(ρd)2/4R.
For K≠1, it is smaller than the lower-order aberration by a factor of
(K+3)(ρd)2/4R2.

The **transverse** 5th order spherical
aberration for reflecting conic surface, as the difference between height of marginal ray reflected from 3rd and 5th order surface
approximation in the paraxial focus plane (**FIG. 37**, A) is given by
TA5=(K+1)(K+3)(ρd)5/8R4.
Hence, it is smaller than 3rd order transverse aberration, whose blur
radius is TA3=(K+1)(ρd)3/2R2,
by a factor of (3+K)/64F2
(note that this applies for K≠-1). The total transverse aberration is a
sum of the two terms, which are of identical sign.
Looking at the wavefront-error-to-transverse-aberration ratio, we see
that the one for the lower-order aberration is independent of the conic,
with the paraxial focus transverse error being larger than the wavefront
error by a factor of 16F.

For the higher-order aberration, the wavefront
error and transverse aberration doesn't scale evenly with the conic.
This means that the aberration changes its form somewhat with the conic,
with the paraxial transverse aberration being greater than the wavefront
error approximately (due to the best focus wavefront error varying
slightly with respect to the error at paraxial focus) by a factor of
16(K+3)F/3(K+1), again, for K≠-1. Given P-V wavefront error, the 5th order transverse
paraxial blur for spherical reflecting surface (K=0) is, as expected due
to the similar form of wavefront deformation, nearly identical to the
lower-order aberration paraxial blur; the corresponding RMS wavefront
error is only slightly lower in the former. Best focus blur, being
located closer to the paraxial focus, is somewhat larger for the
5th
order aberration.

The best focus transverse blur for given P-V
wavefront error of balanced 4th and 6th order spherical aberration is
larger than the 3rd order blur, mainly due to the strong deviation at the edge of
the wavefront. It, however, affects relatively small wavefront area. The RMS wavefront error for the balanced
6th order
aberration is actually smaller by a factor of ~0.75, due to relatively
large portion of the P-V error being caused by the central deformation,
relatively small in area (**FIG. 37**, C). Contrary to the 3rd order
aberration, the paraxial blur for balanced 3rd and 5th order aberration
is comparable in size to the best focus
blur (**FIG. 38**). This is due to the paraxial and marginal foci nearly coinciding
for optimally balanced 5th order aberration (**FIG. 37**, B, bottom).

**
**

FIGURE 38:
Characteristics of pure secondary (5th order transverse, 6th order
wavefront) spherical aberration (top) and its
minimized form when balanced with primary spherical (bottom), the latter
being the one usually referred to as 3th order (transverse), or 4th
order (wavefront) spherical aberration. Wavefront errors are normalized
to 1 at the pure secondary spherical aberration (**PSSA**) error at paraxial
focus (actual error 0.4 wave P-V), and the balanced longitudinal
aberration (**LA**) graph is scaled 2.5 times the arbitrary unit on the PSSA
LA graph (note that the location of paraxial focus remains nearly
unchanged after balancing, but longitudinal defocus shifts to the
opposite side of it). Longitudinal defocus **Λ**
is normalized to 2, with Λ=0
being the location of paraxial focus, and Λ=2
location of the farthest defocus point (it coincides with the marginal
focus point for the unbalanced form, and with 0.72 zone focus with the
balanced form of aberration). Note that normalizing LA to 2 for pure and
balanced secondary spherical is only for the convenience of making it
comparable to primary spherical. For given LA, defocus P-V wavefront
error is 3 times larger than the s.a. P-V wavefront error at paraxial
focus for pure spherical (thus actual
Λ=3), and 1.33 times for the balanced
form (Λ=1.33).

PSSA wavefront data show overall
similarity with primary spherical, main exception being that the error
at the paraxial focus is somewhat smaller, and at the marginal focus
somewhat larger relative to the best focus error, than with primary
spherical (note that the P-V WF error ratio at the paraxial vs. best
focus for pure secondary spherical is 2.33, not 2.56 as implied by OSLO,
probably due to less than perfect accuracy, or different criterion in
determining the best focus; related to it, it is to expect that the RMS
error reduction at the best focus is somewhat greater than implied and,
indeed, it is about 3.6 times smaller, rather than nearly 3.5, as
implied by raytrace). This extends to the
size of corresponding geometrical blurs. The smaller error at paraxial
focus, relative to that at the best focus location, means that less of
defocus error is needed to balance PSSA itself (analogous to balancing
primary spherical with defocus, not to confuse with balancing PSSA with
primary spherical and defocus in order to have it minimized). However,
since both are proportionally smaller, the aberration function for best
focus location is proportional to ρ6-ρ2
(it gives the smallest P-V error, with the shape similar to primary
spherical, with zero deviation at the center and the edge, with the
maximum deviation at ~0.7 zone; however, the smallest RMS error is
achieved with slightly less defocus, ρ6-0.9ρ2, and edge deviation of nearly 30% of the maximum peak deviation at 0.74
zone, opposite in sign). The PSSA LA is nearly identical to that for
primary spherical for given P-V wavefront error at the best focus. Since the secondary aberration function ρ6-0.9ρ2
generates larger maximum deviation than
the primary function ρ4-ρ2,
the latter is multiplied by a 1.5 factor of 1.5, i.e.
1.5(ρ4-ρ2)
to bring its P-V to the needed proportion vs. secondary aberration P-V
to result in the minimum RMS error (specifically, the primary aberration
P-V wavefront error is smaller by a factor of 0.885).
Thus the wavefront aberration function for balanced secondary spherical aberration
is

ρ6-0.9ρ2-1.5(ρ4-ρ2)=ρ6-1.5ρ4+0.6ρ2.

As the plot implies, its maximum is for ρ=1. Setting
first function derivative, 6ρ5-6ρ3+1.2ρ,
to zero does not give direct solution for the values of **ρ** for
which the tangent of the function is vertical (points of inflection),
but solving it by dividing the derivative by 6ρ and reducing it to
quadratic function by dividing it with ρ2
gives the solutions as ρ2=0.5± 0.051/2,
i.e,

ρ1=(0.5-0.051/2)1/2=0.526
and ρ2=(0.5+0.051/2)1/2=0.851.

However, the balanced 5th/3rd (i.e. 4th/6th)
order spherical is a distinctly different form of spherical aberration.
It is already
indicated by the shape of its LA graph, where the paraxial and marginal
foci nearly coincide. The wavefront error is nearly six times smaller at
the best vs. paraxial focus, the latter having twice the error of the
marginal focus. As the numbers show, fully minimizing pure higher order
spherical aberration by balancing it with primary spherical, results in
4.6 times smaller P-V, and six times smaller RMS wavefront error. The
ray spots are much more even in size, but the best focus spot is
significantly larger for given RMS wavefront error: with this specific
error (0.0077 wave RMS) best focus spot diameter with primary spherical
would have been less than 1/3 of the Airy disc's, which makes it nearly
three times smaller. The reason is that the dense central portion of the
balanced aberration blur, where most of the rays are (bottom corner,
right), is of similar size in both aberration forms, and how wide out
are spread remaining 20%, or so, of the rays matters little. The
significance of uneven rate of aberration change toward the opposite
sides of best focus, indicates that the conventional wave of star
testing at nearly equal amounts of defocus either side from best focus will be less
reliable than with primary spherical aberration.

While the primary and pure secondary spherical aberration at best focus
location have very similar effect with respect to image quality,
balanced secondary spherical differs more than negligibly even at
relatively small error levels. Plots at left show OSLO output for 0.0745
wave RMS wavefront error of these three forms of spherical aberration.
Due to its double bright ring, balanced secondary spherical encircles
about as much energy at 3λF radius as the other two do at 2λF. It also
has somewhat more energy spread out in the wider area beyond this point.
The corresponding Strehl is also slightly lower, with encircled energy closer to 0.79. The
contrast transfer (shown only for balanced primary and secondary, for
clarity) is similar, on the average, except the frequencies close to the
resolution threshold, which is lower for the balanced form, the
consequence of its steep edge wavefront curl being similar to that of
turned edge.

An alternative, generally more effective
way of correcting higher order spherical aberration is deforming a
smooth conic - usually sphere - or flat surface, into a needed irregular
radially symmetrical profile, often referred to as *higher-order
aspheric*. Unlike the balancing, which only minimizes present
higher-order term, it can completely cancel out not only secondary (5th
order transverse, or 6th order wavefront) spherical aberration, but also
yet higher order terms. Example of such higher order aspheric is the Schmidt corrector;
it could also be used for higher-order aberration balancing (for
instance, overcorrecting it with
the appropriate amount of 4th order aberration alone would significantly reduce the
6th order aberration
error; in, say, 200mm
f/2 Schmidt camera, from 0.83 wave P-V/0.227
wave RMS of 6th order spherical to ~0.19 wave P-V and less than 0.05 wave RMS error
of balanced aberration), but it is, of course, inferior to the complete
correction.

Higher-order spherical is more of a problem in larger apochromatic
refractors and, particularly, in instruments using strongly curved
meniscus correctors. As the wavefront error of 6th order aberration
increases inversely to the 5th power of surface radius, meniscus
corrector can generate enormous amount of higher spherical aberration,
that cannot be minimized to an acceptable level. This limits relative
apertures of mirrors used in these instruments to ~f/3, or slower, even
in moderate to small apertures. An effective - although
adding to the expense - solution to this problem is aspherizing the
primary, which then requires weaker corrector, with significantly lower
higher-order spherical aberration contribution.

◄
4.1.2. Lower-order spherical:
aberration function
▐
4.2. Coma
►

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