###
11.4. WAVEFRONT SPLIT ERROR IN THE AMICI PRISM

In order to restore the proper horizontal orientation to the image, Amici prism
uses configuration with its back side split into two surfaces coming together
in the plane containing optical axis, and at 45 degrees with respect to it.
As a result, converging wavefronts containing this line are split in two,
with each portion being reflected to the opposite side, and after reflection
on that side merging together in the point image. If the prism is less than
perfectly symmetrical, these two parts of the wavefront will have different
optical path lengths, with the phase differential will producing aberrated
diffraction images.

In addition, since a prism acts as a plane parallel plate, inducing longitudinal
chromatism, the color foci of the two wavefront portions won't coincide, which
can result in noticeable color infidelities. But this effect is generally smaller
and less important than the diffraction effect at the best focus.

Images below are OSLO simulations of these diffraction effects, for two
simple scenarios: (1) even phase error between the two wavefront portions, caused
by one side of the prism being slightly longer, and (2) the error gradually
increasing away from the dividing line, as a consequence of one back side being at
a slightly different angle. The two parts of the wavefront have a constant
path difference. In this case, the part of the wavefront left of the
central line is delayed, i.e. having the longer path with respect to the other one.
Converging beam has a relative aperture of f/5, with the prism front side 100mm
in front of the original focus, and about 10mm wavefront diameter at the splitting
line. About 1/9 wave of spherical aberration induced by 50mm in-glass path is
present in all simulations.

The in glass differential **δ** produces optical path differential
(n-1) δ, where **n** is the glass refractive index (in this case
the glass is Schott BK7, with n=1.517 for the 546nm wavelength).

The side length error is generally acceptable for δ~λ/4 and
smaller (corresponding to little over 1/8 wave of optical path differential).
It is still better than diffraction limited for twice as large error, but
doubling it again makes it unsuitable for higher magnifications (with the wavefront
diameter at the splitting line of 10mm, the width of the field affected
in the final image is nearly as much). At δ=1 and the wavefront split in two
halves, the resulting diffraction image is split in a double maxima (MTF graphs
below show the contrast consequence).

In the second scenario, the path difference, i.e. wavefront error gradually
increases away from the split. The prism side angle deviation is 1/4, 1/2 and 1
arc minute (the actual error is somewhat larger, due to the longer path to the
opposite side). Since the wavefront becomes folded, resulting aberration has similarities
with astigmatism, particularly when the two wavefront portions are comparable in size.

For smaller prism errors, resulting wavefront errors are the
smallest for the wavefronts split in two, since they are positioned over the area
of lower deviation (that changes with the largest prism error, because large
wavefront errors result in a different, less predictable phase combining).
MTF graphs on the bottom shows contrast loss for the three patterns with the
largest prism error. The simulations suggest that the acceptable prism error of this kind
should be below 10 arc seconds.

###
11.5. ABERRATIONS OF THE PRISM DIAGONAL

When in a converging light cone, prism diagonal generates aberrations, both
chromatic and monochromatic. Since it acts like plane parallel plate,
Eq.105.1 applies, with d/L becoming 1/2F, **F** being the focal ratio
(f/D, focal length by aperture diameter). Since d/L becomes a constant
for any given system, prism distance, i.e. beam diameter on its
front surface becomes non-factor, with the only remaining factors
being the focal ratio, in-glass path (thickness) and glass refractive
index. Taking for the index n~1.5, gives for the only two possibly significant
monochromatic aberrations the P-V wavefront error (mm) as W=T/1380F^{4}
(spherical aberration) and W=T/65F^{3} (coma). Graph below
shows how they change as a function of focal ratio (F).

Spherical aberration and coma affect all wavelengths neary equally,
which makes them a part of chromatic error as well. Purely chromatic
errors are longitudinal chromatism, caused by the change in refraction
with the wavelength, and lateral color, which is generally negligible.
Picture below illustrates these aberrations on a 32x32m prism (BK7) in
f/10 and f/5 cone. The objective is a "perfect lens", so all the
aberrations come from the prism.

At f/10, Zernike term for primary spherical aberration (8) is 0.002853,
which divided with 5^{0.5} gives
the RMS wavefront error as 0.001276. The corresponding P-V error is
larger by a factor 11.25^{0.5}, or 1/234 wave (both in units of 546nm
wavelength). It, expectedly, agrees with the equation, since no other
significant aberrations are present. The term for coma (4) is 0.003984,
which divided by 8^{0.5} gives the RMS error as 0.00141 (the
P-V error is larger by a factor 32^{0.5}). So, at
f/10 coma is somewhat larger than spherical aberration, but both are
entirely negligible.

Longitudinal chromatism has a form of reversed primary chromatism, with longer
wavelengths focusing shorter than shorter wavelengths (the consequence
of the refraction at the front surface being diverging).
It is a consequence of image displacement caused
by the cone angle narrowing inside the prism (looking at the raytrace
side view, it is causing the oblique line sections to become longer).
The displacement is given by (1-1/n)T, and the variation in **n** (δn)
with the wavelength produces longitudinal chromatism, given by (T/n^{2})δn.
It remains nominally unchanged with any focal ratio (of course, due to the
smaller Airy disc at the faster focal ratio, chromatic error increases
correspondingly). At f/5, the error in both, F and C line is over 0.4 wave
P-V. It will change relatively little in a fast achromat, unless a very small,
but it would introduce noticeable color error in the reflecting systems, or
any other fast systems with a very low level of chromatism.

Misaligned prism will induce all-field coma, astigmatism and lateral
color. Coma dominates astigmatism at fast focal ratios, while the latter
can be larger at ~f/10 and slower. At f/5, 1-degree prism tilt vs.
optical axis will induce 0.023 wave RMS of coma, and 0.0068 wave RMS
of astigmatism. Since coma changes with the 3rd power of focal ratio, and
astigmatism with the 2nd, at f/10 coma drops to 0.0028, and astigmatism
to 0.0017 waves RMS. But with the coma changing with the tilt angle,
and astigmatism with the square of it, at 2-degree tilt the latter will
be slightly larger. Since the magnitude of tilt -induced aberrations can be significant
only at fast focal ratios - except at insanely large tilt angles - it
is only coma that could be of concern.

Tilt-induced lateral color (*prism effect*) doesn't change nominally with the focal ratio, but its
magnitude vs. Airy disc does, in proportion to it. At f/5 and 1-degree
tilt, the mid-field separation of the F and C lines is 0.002mm, or about 30%
of the Airy disc diameter. For the field center, the separation shouldn't
be larger than half the Airy disc diameter. Since it increases with the
tilt anlgle, it should stay below 2 degrees. At f/10, 1-degree tilt will
induce only half the error at f/5, i.e. the F and C lines separation
will be about 15% of the Airy disc diameter.

11.6 DIFFRACTION EFFECT OF NON-SYMMETRICAL VANES ARRANGEMENT

The standard 4-vane spider is the simplest form of the kind, but
rotational stability is not its strongest point. To improve on that,
the vanes need to be rearranged, breaking the symmetry of a cross.
Simulations below show diffraction effect of two of such arrangements
vs. standard 4-vane form.

Due to the different converging angles from the vane sections,
the modified spider produces wider, complex spikes of similar length. While the
amount of energy transfered out of the Airy disc depends solely
on the vane face area, significantly wider spikes could appear either
more, or less pronounced, depending on the detector's filtering.
The eye could be biased toward wider, fainter spikes, or toward
narower, brighter ones (for identical vane area). Which it is, has
to be established experimentally; it is quite possible that it could
vary individually.

Doubling the vanes to increase spider rigidity will also alter their
diffraction effect. The reason is diffraction interference between the
vanes. As the simulation below shows, the spike of a doubled vane is

broken into bright segments and extended vs. single vane pattern.
The consequence is slightly more energy transferred to the outer areas,
even if the vane area is kept unchanged (graph on the right is effectively
magnified by showing 5 times smaller radius on the same frame size).
On the doubled area vanes' pattern
(right) can be detected presence of secondary side maximas, which would
imply less energy in the principal maxima, i.e. main spike. As with
the vane configurations above, whether the eye would be more sensitive to a
longer, segmented, and slightly less bright spike, only experimental
examination will answer.

###
11.7 RAYTRACING EYEPIECES: REVERSE vs. DIRECT

Eyepiece performance level is commonly determined by reverse raytracing,
i.e. in a setup when the eyepiece exit pupil becomes aperture,
and collimated light pencils passing through it travel through the
the eyepiece in reverse, to form an image in front of the field lens,
at the nearly same location where forms the image of the objective.
Image formed by raytracing eyepiece in this manner is real image,
but it is neither image that a perfect lens would form if placed
at the eyepiece exit pupil, nor image of the objective. Rather, it
is an image at the location of objective's image that would produce
perfectly collimated pencils at the exit pupil end. As such, this
image reflects aberrations of the eyepiece in their kind and magnitude,
some of them reversed in sign, some not. For example, if reversed raytracing
produces field curvature concave toward eyepiece, it implies that
such curvature would produce perfectly collimated exit pencils because
eyepiece itself generates curvature of opposite sign (it seems illogical
since the two curvatures seemingly coincide, but the image space of the
eyepiece is behind the eye lens, not in the objectve's image space; hence
with a flat objective's image the eyepiece would produce exit pencils becoming
converging toward outer field - because the field points in the image of the
objective would've been farther away than what it needs to form a collimated
beam - i.e. would form a curved best image surface
of the opposite sign). On the other hand, if
reverse raytracing produces overcorrection, this means that the eyepiece
would, with zero spherical aberration from the objective, form exit pencils
with rays becoming divergent toward pencil edge (since off-axis points on unaberrated
image surface are in this case closer to the eyepiece), focusing
farther than rays closer to the center, i.e. also generating overcorrection.

As long as the geometry of pencils passing through the exit pupil
is identical, so will be the aberrations generated. But this is
strictly valid only for points close to axis. The farther off
field point, the more likely that the perfect exit pencils that we
are starting with in reverse
raytracing won't exactly match those generated by a perfect input
from the opposite end, and that will
cause different aberration output as well, possibly significant.
One particular difference is that reverse raytracing can be done
from one fixed pupil location at a time. It is generally insignificant
with eyepieces having relatively small exit pupil shift with the change
of field angle (so called *spherical aberration of the exit pupil*,
with the pupil generally shifting closer toward eye lens with the
increase in field angle), but when it's significant, raytracing from
any fixed pupil location will cause gross distortion of the astigmatic
field for field zones with different exit pupil location. The only way
around it is to raytrace for several different exit pupil locations, and
piece up the actual field from that. This problem vanishes in direct raytracing,
where every point's cone simply goes to its actual exit pupil.

To make easier to detect the differences, a wide field eyepiece is
needed, and for the ease of accessing its optical process it should
be simple as possible. The perfect candidate is a modified 1+1+2
Bertele, which is for this purpose designed to produce 80°
apparent field of view (AFOV). While not in the league of the
(much) more complex designes for this field size, it is still
significantly better than other conventional designs. Below is
how it raytraces in reverse, and directly, the latter using
OSLO "perfect lens" as the objective and at the eye end. Eyepiece
focal length is 10mm, and exit pupil diameter is 1mm, hence it
processes f/10 beam.

Top half shows reverse raytracing. Surface #1 is the aperture stop,
and #9 the image. The section between two marginal cones
(6.4mm radius) is the actual image, and
the full length of the vertical dashed line (8.4mm radius) is Gaussian image,
i.e. image that would've been seen w/o distortion at the entry
field angle (in effect, the image seen at that angle from the distance
equal to the eyepiece focal length, as illustrated with the dotted
lines at left). Column of numbers at right are the heights of the marginal chief
ray (central ray) at all surfaces. Astigmatism plot shows reversal of
the tangential line (in the plane containing axis and chief ray) toward
field edge, preventing further increase in the outer 20%, or so, of the
field radius. At 40° off, the P-V wavefront error is 2 waves. Note that due
to the exit pupil shift (3.7mm eye relief for 40° to 5.5mm for
half-field), tangential line for 3.7mm pupil position slightly magnifies
the longitudinaludinal aberration for the upper half of the field;
the actual line is a bit flatter over that section. As the wavefront map shows,
on its way through the lenses the wavefront acquires vertical elongation,
despite it being cut into a horizontal ellipse (0.5x0.383mm) at the
aperture stop (OSLO Edu doesn't have pupil-that-tilts-with-field-angle feature).
Marginal cone focusing into the final image is noticeably wider than
the axial cone, which indicates negative (barrel) distortion. Since
wider cone in effect acts as one of the faster f-ratio, it produces
smaller diffraction pattern; with the effectively shorter focal length
image magnification drops and the entire image shrinks as a result
toward field edge (black square around red grid representing the image
area is the distortionless form of it, with its diagonal equal to the
field diameter). Nominally, distortion nears 25%. Illustration at left
shows how a beam of light passing through a lens surface generates
astigmatism: the two radii perpendicular in the centre of wavefront
footprint are different, creating astigmatic wavefront deformation.
It also shows how refracted beam changes its width, increasing
vertically when it travels from left to right (shown), or decreasing
when traveling in the opposite direction (for clarity, refraction
difference between the two rays is
neglected - they remain nearly parallel after refraction, as they
do when a section of lens surface is small enough to be nearly flat).

Longitudinal aberration plot shows entirely negligible spherical
aberration, and not entirely negligible axial chromatism. Defocus **δ** in the
blue F line of less than 0.1mm indicates ~0.2 waves P-V wavefront error
at f/10 (for 486nm wavelength, from δ/8F^{2}), and four times
as much at f/5. Error in the violet g-line is three times larger. lateral
color is well controlled accross the field, with the F-to-C separation
reaching half the Airy disc diameter at the field edge.

Bottom half shows direct raytrace of the same eyepiece with a "perfect
lens" as the 89.4mm f/10 objective (the focal length is determined from
the field-end angle of the marginal chief ray in reversed raytracing).
Focal length of the "perfect lens" at the image end is set to 10mm,
to produce a directly comparable f/10 system. Longitudinal chromatism
and spherical aberration are nearly identical to those in reversed raytracing.
But that is where the similarity ends.
Here, Gaussian image is smaller than the actual, apparent one, as a result of the
positive (pincushion) distortion, nominally the inverse of the
negative distortion in the reversed raytracing (1.32 vs. 0.76).
As a result of positive distortion, with the edge cone
visibly more elongated, the Airy disc at 40° is noticeably
larger than on axis, unlike the reversed tracing, where it is smaller.
Longitudinal astigmatism at 40° is about doubled, but
the P-V error is smaller: 1.8 wave. This is mainly the result of
the effective f-ratio for that cone being f/13.2, with the
transverse astigmatism smaller by a factor 1.75 vs. f/10,
and more so vs. effective focal ratio at this field point in the
reverse raytracing. Tangential curve now extends farther out, as a
result of the higher order astigmatism now being of the same sign
as the primary, adding to it instead of taking away as in reverse
raytracing. This is caused by the narrower cone of light passing
through the eyepiece for the outer field points, which much
more affects secondary astigmatism, changing with the 4th power of
cone width
(as opposed to the 2nd power with primary astigmatism),
with the 40° pencil not quite filling out the exit pupil circle
outlined by the axial cone, as it does in reverse raytracing.
As a result
of the different astigmatism plot, field curvature also changes:
instead of zero accommodation at the edge, and +1 diopter required
for 0.7 field radius, now edge requires somewhat over +1 diopter,
and the 0.7 field radius is flat, requiring zero accommodation
(small box bottom right, accounting for the effective 13.2mm focal
length of the perfect lens at this point). However, it should be noted
that if the two astigmatism plots would be corrected for the distortion
effect, they would become very similar, despite the difference
in the sign of secondary astigmatism.

Unlike the 40° wavefront in the reverse raytracing, elongated
vertically, here it's flattened, and noticeably more so. This is
caused by refraction at large angles, compressing or expanding wavefront
vertically, and shows the true extent of it
(the elongation in reverse raytracing is partly offset by the
horizontally elliptical wavefront outline determined at the entrance pupil).
This asymmetrical astigmatic shape will result in asymmetry of both,
ray spot plots and diffraction images along the extent of longitudinal
aberration. Here, the tangential line, laying in the sagittal plane, is
noticeably thicker, because it's formed by the (shorter) vertical
wavefront sections focusing into it (blue on the wavefront map is
its delayed area, hence it forms convex surface focusing farther away),
as well as about 50% longer, since
the wavefront extends that much more horizontally. As a result,
best focus is not in the middle between sagittal and tangential focus,
but closer to the sagittal line, which is
laying in the tangential plane (the one containing axis and the chief ray),
as indicated on the astigmatism plot.

Overall, the differences in aberrations magnitude are small, but one
needs to keep in mind that some of them do reverse in sign with the
change in light direction.
For instance, nominal lateral color error is reversed,
and roughly doubled, but the actual change is relatively small
due to the larger Airy disc (it is similar with astigmatism,
with the two plots appearing grossly different, but with the
actual P-V error diferential being near negligible).

###
11.8 POLYCHROMATIC STREHL: PHOTOPIC vs. MESOPIC vs. CCD

Polychromatic Strehl for telescopes with refracting elements
is commonly given for photopic (daylight)
eye sensitivity. Strictly talking, it is valid only for daytime
telescope use, but both sides of the market seem to be neglecting it,
or simply are unaware of it. Since the Strehl figure is used as a
qualifier of the level of optical quality - 0.80 for so called
"diffractin limited", and 0.95 for "sensibly perfect", it does
matter to know that it is limited to the sensibility mode used for
calculating the Strehl. In general, due to the higher overall sensitivity
in the mesopic mode - and particularly toward blue/violet -
and the error usually being greater in the blue/violet, the mesopic
(twilight level) Strehl will be lower than photopic (broad daylight).
It likely worsens somewhat toward scotopic (night conditions) mode,
but telescopic eye is most likely to be within the range of mesopic
sensitivity. How significant is the difference between photopic and
mesopic Strehl depends primarily on the magnitude of chromatic error
in the red and blue/violet, with the latter being more significant
since, unlike the blue/violet, sensitivity to the red generally declines toward mesopic and
scotopic mode. While these eye sensitivity modes are relevant for
visual observing, for CCD work it is the chip sensitivity that needs
to be used for obtaining the relevant, CCD Strehl. Here, CCD
sensitivity is a rough average of the range of sensitivities of
different chips.

As illustration of the difference between Strehl values for
photopic, mesopic and CCD Strehl, it will be calculated for a
highly corrected TOA-like triplet in two slightly different
arrangement: one with the standard Ohara crown, S-BSL7, and
the other with its low-melting-temperature form, L-BSL-7. Slight
difference in dispersion between the two is sufficient to
produce larger axial error, particularly in the red and violet,
which will show the difference in correction level between two
seemingly highly corrected "sensibly perfect" systems, when judged by the photopic
Strehl value alone. Strehl values are calculated using 9
wavelengths spanning the visual range, as shown below. Mesopic
sensitivity is approximation based on empirical results, somewhat
different from the official mesopic sensitivity, which is merely
a numerical midway between photopic and scotopic values.

From top down, first shown is a lens using S-BSL7, extremely
well corrected for axial chromatism. So much so, that the mesopic
Strehl, and even CCD Strehl are only slightly lower. This lens
practically has zero chromatism in the violet, a rarity indeed.

Replacing S-BSL7 with L-BSL7 (same prescription, except slightly
stronger R1 - 2380mm - to optimize red and blue) roughly doubles axial chromatism,
except for the violet, which is now at the same level as the
deeper red. Photopic Strehl is still excellent, suggesting there
is no noticeable difference in the chromatic correction between the two.
However, mesopic Strehl tells different story: this lens is not
"sensibly perfect", and its CCD Strehl sinks toward 0.80.

Mesopic Strehl gives different picture for achromats too. According
to its photopic Strehl, a 100mm
f/12 achromat is slightly better than the "diffraction limited"
at its best diffraction focus
(0.09mm from the e-line focus toward the red/blue; first column
shows Strehl values at the best green focus). But its mesopic
Strehl, also at the best diffraction focus, is only 0.63, and its
CCD Strehl dives down to 0.45. Knowing that the Strehl number reflects
average contrast loss over the range of MTF frequencies (for the
mesopic Strehl it is, for instance, 37%), implies that this achromat
is nowhere close to "diffraction limited" under average night-time
conditions. For that, it needs to be twice as slow, f/24.
In conclusion, it is hard to draw a precise line for where a
"sensibly perfect" photopic Strehl should be for telescopes used at night,
but it seems safe
to say that it does need
to be significantly better than 0.95; probably close to 0.99.

11.9 PLASTIC ACHROMATS

Optical plastics are widely used for production of small and not so
small lenses for all kinds of cameras, glasses and optical devices,
but rarely for telescopes, and when used, nearly without exception for those of
low quality. Most important optical plastics are acrylics, polycarbonates
and polystyrenes, but some other are also viable. Optically, they
can be as good as glass, but have several times higher thermal expansion,
and a 100-fold higher variation of the refractive index with temperature.
Also, they are more prone to static charges, and more difficult for coatings.
On the good side, they are ligther, safer, and cheaper. Technological advances
resulted in a wider number of optical-grade plastics available, which
makes their application for small telescope objectives easier. Follows
overview of the performance level of optical plastics - mainly those listed in OSLO Edu catalog -
as components of achromatic 100mm f/12 doublets and triplets. In general, they
have better color correction, occasionally approaching - even exceeding -
the minimum "true apo" requirement of 0.95 Strehl.

Doublets are of the Steinheil type, with the negative element in front, because
the "flint" element in most of objectives, polycarbonate, is more resistant to
impact and temperature (in general, order of elements does not significantly
change the output). Performance level is illustrated with a chromatic Focal
shift graph, against that for the standard glass achromat (BK7/F2, black plot).
Chromatic focal shift shows the paraxial focus deviation for other wavelengths
vs. optimized wavelength (546nm, e-line). In the absence of
significant spherochromatism - which is here generally the case -
it is a good indicator of the level of longitudinal chromatic correction.
The P-V error of defocus can be found from the graph for any wavelength,
using P-V=δ/8F^{2}, where **δ** is the focus
shift from the e-line focus (0 on the graph) and **F** is the focal number.
Graphs are accompanied with the corresponding photopic Strehl (25 wavelengths, 440-680nm),
except for the last two, whose rear element plastics are not listed in OSLO (direct
indexing for five wavelength was entered from ATMOS). The Strehl value
is for the diffraction focus, which for most of these objectives
does not coincide with the e-line focus (amount of defocus is given
as **z**, and can be positive or negative, depending on the
plot shape).

All but one plastic lens combination have a higher Strehl than the glass achromat
(0.81). Some combinations have near-apo correction in the blue/violet,
some in the red, but most important is how well corrected is the 0.5
to 0.6 micron section (approximately). Two doublets have Strehl value exceeding 0.9, as well as
two triplets, with one of them qualifying as a "true apo" by the
poly-Strehl criterion of 0.95 or better (#8).
It wouldn't satisfy the P-V apo criterion having 2.3 wave error
in the violet g-line (1/6 wave F-line, 1/5 wave C, and 1/2.5 wave r-line)
but due to the very low eye sensitivity to it in the photopic mode,
it has little effect on the photopic Strehl. The more appropriate for
night time use, mesopic Strehl, would be somewhat inferior to
objectives with a similar photopic Strehl, but better violet correction.

There are other plastics available, and more combinations possible (also,
the properties of any given plastic can vary somewhat depending on its
production process), but
these shown here suffice to conclude that optical-grade plastics
can be superior to the standard glasses in chromatic correction.
Some could even produce the "true apo" level in the range of mid
to moderately long focal ratios.

11.10 HOYA FCD1 vs. FCD100, TRIPLET OBJECTIVE

The older generation of extra-low dispersion glassess, with Abbe number
around 81, is commonly considered inferior in their performance limit to
the latest generation, with Abbe number around 95 (also called *
super-low dispersion*, or SD glasses). However, the larger
Abbe number gives one single advantage: with any given mating glass
the higher order spherical aberration residual is lower, allowing for
somewhat faster lens for a given design limit in the optimized
wavelength. But the difference is generally small. Let's illustrate this
with Hoya's FCD1 and FCD100 glasses in a 5-inch f/7.5 triplet objective.

Limiting the mating glass to Hoya's catalog, the best match for FCD1 is
BCD11, and for FCD100 BSC7 (Hoya's equivalent of Schott BK7). As image
below shows, a 5" f/7.5 triplet with FCD1 (top) has photopic polychromatic
Strehl rounding off to "sensibly perfect" 0.95 (mesopic value would be somewhat
lower, but not by much, considering relatively low errors across well
balanced spectrum). The FCD100 triplet (middle) does have better polychromatic
Strehl - rounding off to 0.98 - but about half of the differencial comes
from the optimized line correction. Since the limit in the optimized e-line
for the FCD1 triplet is at the level of 1/15 wave P-V of primary spherical
aberration, the actual units with a similar optimized line correction
would have no perceptible difference in color correction (granted,
any given optimized line correction level would be easier to achieves
in the FCD100 triplet, due to its more relaxed inner radii).

It is obvious on the LA graph that the FCD1 triplet has significantly
higher spherochromatism on the primary spherical level, most of it the
result of more strongly curved inner radii. But the sign of higher order
spherical residual - after optimally balancing the optimized wavelength -
significantly reduces the aberration in the blue/violet, while
increasing it only moderately on the red end. As a result, Strehl values
for non-optimized wavelengths are generally close to those of the FCD100
triplet.

One other possibility is using moldable glasses. Best match for FCD1
is M-BACD12. If one surface is aspherized, all four inner radii can
be equal, and the triplet is nearly as well corrected as the one with
FCD100 glass (bottom).

In all, how objective performs still depends more on the combination,
than any single glass. It is possible that the older ED glass objective
even performs better, although it is generally to expect small to
negligible advantage with the higher Abbe# varieties. The difference
is more pronounced in the doublets, because the higher order residual
increases exponentially with the lens curvature, and doublets require
them significantly stronger than triplets.

11.11 MICROSCOPE EYEPIECES IN TELESCOPES

While not common, use of microscope eyepieces in telescopes
does happen. How well these eyepieces can be expect to perform?
Is a good name on them implying they will be as good as those
made for telescopes, or even better? The answers are: "no one can
tell", and "no", respectively. There are two main differences
between the standard microscope and a telescope with respect to
the eyepiece performance: (1) due to the significantly shorter
objective-to-image distance - for a standard old-fashioned microscope
the main part of it the so called "*optical tube length*" (OTL), standardized
to 160mm - rays entering any given eyepiece field stop have
significantly larger divergence, and (2) due to the very small objective,
the effective focal ratio is very high (measured as a ratio of objective
diameter vs. objective-to-image separation; not to confuse with the
microscope numerical aperture, which is measured vs. objective-to-object
separation). The former generally increases off axis aberrations,
while the latter makes them smaller. In other words, looking only
at #1, eyepiece optimized for a microscope would have to be sub-optimized
for a telescope with respect to field correction. How much does #2
offset for this?

Since the microscope magnification can also be expressed as a product
of the objective and eyepiece magnifications - the former given by OTL/f_{o},
and the latter by 250/f_{e}, with **f**_{o} and
**f**_{e} being the objective and eyepiece focal length,
respectively - we'll illustrate the divergence vs. focal ratio offset
with an average objective of 10mm focal length, and a 20mm focal length
Huygenian eyepiece (from the above, they produce 160x250/10x20=200
magnification). Image below shows the optical scheme of a microscope
(top) and an actual raytraced system with the given parameters (bottom).
The objective and eye lens in the latter are "perfect lens", so neither contributes
aberrations (note that the correct magnification for perfect lens 1
should be -16.13, but makes no difference in the ray spot plot).

The eyepiece is upscaled 10mm Huygenian shown under "Individual
eyepieces" on eyepiece raytracing page, so its nominal
aberrations in a telescope are twice larger than those shown for the 10mm unit.
In this microscope setting, the 20mm unit (due to re-orienting its effective
focal length is around 22mm) shows entirely negligible
aberrations all the way up to its 10mm radius field stop. The effect
of the f/86 cone (the paraxial data given below objective is for the objective only)
makes the effect of significantly stronger divergence entirely
negligible over the strongly curved best image field (-4.5 diopters
of accommodation required at the edge). Even over flat field, it dwarfs
defocus effect to 1/12 wave P-V at the field edge. Note that the field
is given in terms of object height, with 0.618mm corresponding to
25° apparent FOV in the eyepiece, and 3.33° true field of
the objective, i.e. angular radius of the object (magnification is not, as with
a telescope, related to the angular size of the object in the system,
but to its angular size as seen from the standard least distance of distinct vision,
250mm; on the schematic microscope, that angle magnified by the objective is
**α**_{0}, and the final angular object radius is **α**).

In all, correction requirements for microscope eyepieces are much lower
than those used in telescopes. This applies to both, axial and off-axis
correction, and that is the main risk in using microscope eyepieces for
telescopes: those
performing just fine in a microscope, could become sub-standard in a
telescope. Another possible obstacle, not visible in this demonstration,
is that microscope eyepieces could be optimized to offset typical
aberrations of microscope objectives, while telescope eyepieces are
generally designed to produce best possible stand-alone image.

11.12 SCHMIDT vs. HOUGHTON TRIPLET CAMERA vs. STANDARD SCHMIDT

While looking for some modern, "extremely achromatic" camera prescription,
a drawing of a triplet catadioptric camera by Bernhard Schmidt caught my eye. It was
called an "alternative to the standard Schmidt", and made me curious:
just how close it is. Then, in an online PDF file which contained data
as close as possible to the prescription
(
Journal of Astronomical History and Heritage),
there was quite similar triplet camera patented by Houghton some 15-20
years latter (1944, US Pat.#2,350,112). Whether Houghton could know for Schmidt's work is
anyone's guess, but doesn't make less interesting finding out how do
these two cameras compare, and how close they come to the standard
(aspheric plate) Schmidt camera. For the Schmidt triplet, the original
handwritten prescription by Schmidt was used, scalled down to 100mm
aperture diameter, and optimized by very minor tweaks (there is also
a 1934. prototype of the same design which will be mentioned in
raytracing analysis). All three cameras are 100mm aperture f/1, to make them
directly comparable, and the field radius is 6 degrees.
Image below shows raytrace of the downscaled Schmidt 3-lens catadioptric
camera. The the outer two lenses are plano-convex and symmetrical with
respect to the biconcave mid element. A single glass, probably Schott's
old O15 crown (n_{d}=1.53, *v*_{d}=58.99) was
used; since it is not listed in OSLO Edu catalogs, the closest found
was used (during rescaling the lenses got somewhat squeezed up; increasing
the gaps to 6.9mm, needed to clear axial pencil, doesn't
appreciably change the
output).

Central obstruction size is not given. Image size sets the minimum size
at 20% linear, which would practically have to be somewhat larger. Since
the effect is near-negligible, both central obstruction and (possible) spider vanes
are omitted.

LA graph shows relatively significant higher-order spherical residual
on axis. The corresponding wavefront errors for five selected wavelengths
are given by the OPD (optical path difference) plot. Best image surface
doesn't fall midway between the tangential and sagittal surface due to
the presence of odd secondary (Schwarzschild) aberrations (in presence of
spherical aberration, the astigmatism plot, originating at the paraxial
focus, is shifted away from best focus, but best image surface is vertical
when its radius coincides with the one entered in raytrace). While all
five wavelengths have a common focus for the 75% zone ray, their best
foci - mainly due to spherochromatism - do not coincide, resulting in
a nominally significant chromatism. Still, the g-line error is only
about three times the error in the optimized wavelength. The ray spot
plots indicate relatively insignificant chromatism (Airy disc is a
tiny black dot; its e-line diameter is 0.00133mm, or 1/300 of the
0.4mm line). Polychromatic diffraction blur exceeds 0.02mm at 4.1°
and 0.04mm at 6° off
(the five wavelengths, even sensitivity). Should be mentioned that
other than the prescription, there is an actual unit, a prototype
of this camera type from 1934. According to the measurements taken
by the paper authors, it is nearly identical to the prescription,
except that the middle element has slightly weaker radii (perhaps
fabrication inaccuracy). When scaled to the comparable 100mm f/1 system
(originally 125mm f/1.1) the overall correction is somewhat worse.

Houghton's patented camera differs in that it has two biconvex lenses
framing in the biconcave central element. Also, it uses two different
glasses. Again, there was no near-exact match listed for a glass quoted for
the mid element, but the one used for raytracing is close enough not
to make the end result significantly different (the minor optimizing
tweaks are probably in better part due to the small differences in
glass properties).

While the LA graph looks better on the first sight,
due to considerably lower higher-order spherical residual, chromatic
correction is significantly suboptimal due to the five wavelengths
having a common focus too high, at the 90% zone. It is larger by a
factor of 2.6 than what it would be if the common focus was at the
70.7% zone. The ray spot plots indicate more chromatism than in the
Schmidt configuration, but diffraction blurring is significantly reduced.

However, when compared to the standard Schmidt below, even the Houghton
falls significantly behind.

There is not so much difference in the astigmatism plot - which shows
primary and standard secondary astigmatism - but significantly smaller
ray spot plots
and diffraction images indicate much lower odd secondary aberrations.
Nominal chromatism (spherochromatism) is also significantly smaller,
but not so relative to the optimized wavelength, which is much better
corrected than with the 3-lens correctors. Overall superiority
of aspheric plate is undisputable. It can be illustrated
with the magnitude of Zernike terms, and encircled energy, both 6°
off axis (below).

The standard Schmidt has only three significant terms, primary
astigmatism (#4), primary spherical (#8) and secondary astigmatism (#11).
The primary astigmatism term indicates 0.63 wave RMS (term divided by
√6), corresponding to
3.1 wave P-V, plus 0.43 wave RMS (2.7 wave P-V, term divided with √10) for secondary astigmatism.
That is more than ~4.5 waves P-V corresponding to ~0.02mm
longitudinal astigmatism on the plot, indicating the presence of
lateral astigmatism, of the same form as primary astigmatism, but
increasing with the 4th power of field angle, not included in the plot.
Similarly, the spherical aberration term indicates 0.53 wave RMS of primary
spherical aberration (term divided by √5), much more than what is present on axis. This is
due to the presence of lateral spherical aberration, of the same
form as the primary, but increasing with the square of field angle.

In the 3-lens Schmidt and Houghton, dominant term is primary astigmatism,
followed by primary spherical, secondary astigmatism, primary (#6) and
secondary coma (#13). Most of the terms are significantly higher than in the
standard Schmidt, and particularly for primary astigmatism. Simiarly
to the standard Schmidt, RMS error values indicated by the terms are
not in proportion to the graphical output, because it doesn't include
odd secondary Schwarzschild aberrations, lateral spherical, astigmatism
and coma. The ray spot plot in the paper is elongated vertically probably because
it is given for the plotted best astigmatic field not including odd secondary
aberrations; the actual best field, according to OSLO, is about 5% stronger
(also, spot structure is markedly different than for the one given by OSLO,
with the dense part of it for e-line more than twice larger than in OSLO:
over 0.05mm vs. 0.025mm; note that system in the paper has 25% larger
aperture, but also is somewhat slower, at f/1.13).
Polychromatic encircled energy plot (the 5 wavelengths, even
sensitivity) shows that the standard Schmidt has about three times
smaller 80% energy radius than its 3-lens corrector alternative, with the
Houghton midway between.

11.13 FLAT-FIELD QUADRUPLET APO

Flat-field quadruplets can come in various forms. The
particular arrangement given here is a contact air-spaced triplet
combined with a singlet meniscus at some distance behind
such as Ascar 130 PHQ from Sharpstar (its glasses are not
published, so that's where similarity ends). The triplet
does all corrections, central line and chromatic, but it is slightly
modified to compensate for the optical effect of the meniscus (it induces
corrective astigmatism and field curvature, but also significant amounts
of coma and spherical aberration). The
base triplet is NPN 130mm f/7.5 with Hoya's FCD1 and BCD11, given under
11.10 above. After adding the field-flattening meniscus, only
the 1st and last triplet radius were changed to correct for coma, and
one inner radius to correct for spherical aberration. However,
since the meniscus exerts negative power, the focal ratio went
from f/7.5 to f/8.6 (image below, top).

Field can be flattened with any meniscus form, but for minimized
lateral color a strongly curved meniscus is required. Off axis
monochromatic corretion is the best with the astigmatism cancelled,
and some slight residual field curvature remaining (flattening field
by indroducing a small amount of astigmatism roughly doubled the
edge field wavefront error). Meniscus location is
pretty flexible; cutting it in half only slightly worsens chromatic
correction. However, placing it right after the objective gives rise
to a significant higher-order spherical residual, due to correcting
for primary spherical requiring significantly larger inequality in
radius value between the three equal inner radii and the one correcting
for the spherical (trying to correct spherical by bending lenses
produces similar result).

One alternate way of correcting field curvature is by placing achromatized meniscus
significantly farther from the objective (bottom). The overall
chromatic correction is still good, but somewhat less than with
the above arrangement. The relative aperture also diminishes, to f/9.7.
The singlet meniscus extends the triplet's focal length by roughly 15%,
and the achromatized (more widely separaed) closer to 20%.
This means that these arrangements need to use triplets capable
of achieving good correction at f/6 to f/6.5 in order to produce
well corrected f/7 to f/7.5, or so, flat field systems.
Note that these are not necessarily the best glass
combinations, or separations: they illustrate general system
properties (however, with the singlet meniscus, as mentioned,
the differences are fairly small).

Does the triplet arrangement - NPN vs PNP - affects the outcame?
In general, like with the doublets, where reversing order of
positive and negative elements generaly has little effect on
chromatic correction, shouldn't be substantial, although it can be
significant in some respect. Ascar's 130 PHQ uses PNP triplet
which, since the positive element has to be ED glass, means it has
two ED glass elements. Similarly to the doublets, placing the
negative element in front requires significantly stronger inner
radii, because the glass used for it always have significantly
stronger index of refraction, requiring stronger radii to compensate
for the initial chromatic error by the weaker-index positive glass
(image below).

Significantly weaker inner radii of the PNP arrangement (bottom) seem
to be producing less spherochromatism and better overall correction. The exception
is the violet g-line, which is slightly worse, due to defocus, but the rest of lines
are significantly better. About twice smaller displacement of the
astigmatic field origin - which is by default at the paraxial focus -
indicates as much smaller spherical aberration in the e-line. However,
the NPN lens is set to produce the smallest error possible in the
g-line, which comes at a price of sub-optimal correction in the F and C.
By making the front radius 1-2mm stronger, F and C lines come to
their near-optimal correction, and the difference in F/C chromatic correction
becomes insignificant, while the g-line error becomes about 25% larger than in the
PNP. What remains unchanged is the twice larger
minimum error in the optimized wavelength.

11.14 ED doublets with lanthanum

ED doublets using lanthanum as mating element have become common these
days. While they leave something to be desired in the violet end
correction, their main advantage is their large Abbe# (dispersion)
differential vs. ED glasses, combined with sufficiently small relative
partial dispersion (RPD) differential to keep secondary spectrum
small to acceptable. Large Abbe# differential is a must for fast ED doublets, since the larger it is, the less
strongly curved inner radii required, and less higher-order spherical
aberration induced. Some other factors are also potentially significant -
like the refractive index ratio, actualy not favoring lanthanum in
general - but large enough Abbe# differential would compensate for
it too. The problem is that, due to the architecture of RPD diagram,
the larger Abbe# differential, the higher on it is lanthanum glass,
and the larger RPD differential vs. ED glass,
i.e. the larger becomes secondary spectrum (higher RPD is in general
offset by sufficently larger Abbe# differential, but the tendency is
secondary spectrum increase). Thus the choice of
lanthanum is always a compromise between a low higher-order spherical
residual, determining the central line correction level, and low secondary
spectrum. For the former the Abbe differential needs to be as high
as possible, for the latter - about as small as possible. The advantage
of high Abbe# ED glasses (~95) is that they can use lanthanums that are
lower on RPD diagram, i.e. with a smaller RPD differential, hence
smaller secondary spectrum as well. It will be illustrated how much
of a difference it makes vs. lower Abbe# ED glasses (~81), starting with
the latter.

Raytrace below shows three variation of an ED doublet using Chinese
(CDGM) glasses, FK61 ED and lanthanum (these are similar to the
Astro-Tech 4" f/7 AT102ED). Doublet of this type has R2 significantly stronger
than R3. From the LA (longitudinal aberration) and OPD (optical path
difference, i.e. wavefront error) plots it is immediately visible
that the soft spot is correction in the violet. Top doublet uses
lanthanum with smaller Abbe# differential than the other two, hence
has higher secondary spherical residual, and higher minimum error in the
central line. The middle doublet has the highest Abbe# differential,
and the bottom one is in between the two. Photopic polychromatic Strehl
(0.43-0.67 micron, shown boxed) slightly favors the latter (note that
the Strehl values are for the location of best e-line focus; due to
the presence of secondary spectrum, best poly-Strehl is shifted toward
F/C lines - nearly 0.02mm for all three - 0.890, 0.898 and 0.903,
top to bottom, respectively). OSLO quotes
the price of its lanthanum glass as 5 times the BK7, vs. 3.5 times for the other two, which
makes the top combination most likely.

F and C lines are nearly balanced in the second and third combination,
at ~0.155 and ~0.125 wave RMS, respectively (0.53 and 0.44 wave P-v
of defocus), which puts them at the level of a 100mm f/24 and f/27 achromat, respectively.
The top combination has correction somewhat biased toward blue/violet,
with 0.074 wave RMS (0.26 wave P-V of defocus) in the F line, and 0.15
(0.52) in the C. If made nearly equal (g-line in that case goes over 1.1 wave P-V), they come at ~0.39 wave P-V
of defocus, comparable to f/31 100mm achromat. The middle combination
has significantly larger error in the violet, but it has little effect
on the photopic polychromatic Strehl, due to the low eye sensitivity to violet
in this mode. However, in the more appropriate to night-time observing,
mesopic mode, eye sensitivity to violet is significantly larger, and
this combination would have more effect on contrast (0.72 vs. 0.76
mesopic poly-Strehl vs. top combination, in part due to the higher sensitivity
in the red vs. green/yellow as well) in addition to more violet fringing.
This magnitude of violet defocus (comparable to that in a 100mm f/18
achromat) would be visible on bright objects, unless a special lanthanum
doped coating, selectively absorbing in violet, is applied (which was
likely the case with the APM 140mm f/7 lanthanum doublet).

Using ED glass with higher Abbe# - also called "*super-dispersion*"
(SD) - makes possible to use lanthanums
with higher Abbe#, with less of RPD differentiial, i.e. inducing less of secondary spectrum, assuming
the ED glass is of similar RPD value (FCD100/FPL53 have somewhat
lower RPD than FK61/FCD1, thus the advantage is partly offset).
Taking Hoya's FCD100 and two possible Hoya lanthanum matching glasses,
show reduced secondary spectrum, and significantly better correction
in the violet (these are similar to Astro-Tech 102EDL). As a result, the poly-Strehl is significantly higher
than with the lower Abbe# ED glass.

Due to the presence of secondary spectrum, best polychromatic
focus is shifted from the best optimized line focus (defocus Z in mm).
Despite its lower F/C error (0.07 vs. 0.08 wave RMS, when the two lines are
equilized), the top
combination has lower polychromatic Strehl, because its lower
optimized-line Strehl weighs more in the poly-Strehl value. But its
poly-to-optimized-line Strehl ratio shows that it has better
chromatic correction (0.962 vs. 0.954).

Interestingly, going somewhat slower while using the lower Abbe#
lanthanum glass in order to eliminate higher-order spherical residual
is likely to result in a small drop in the chromatic correction,
not only due to a bit more of secondary spectrum, but also
due to the higher-orderspherical actually reducing error in the blue/violet.
For example, taking 125mm f/7.8 AT125EDL configuration with assumed
matching lanthanum CDGM's H-LAF50B (equivalent of Ohara S-LAH66, or
Hoya's TAF1), with no higher-order spherical residual, produces
photopic poly-Strehl at the diffraction polychromatic focus of 0.923
(shown is objective with the positive element in front, but the
reverse arrangement produces identical correction).

Even at f/7 there is no higher-order spherical, and chromatic correction
is only slightly worse (0.912 poly-Strehl) but it was probably made
slower to be geared toward visual observers, since its violet
correction leaves something to be desired on CCD level. Visually,
its violet g-line (0.436μ) is at the level of a 100mm f/28 achromat
(or 60mm f/17), i.e. unintrusive. Its F/C correction is at the level
of a 100mm f/33 achromat.

Non-lanthanum alternatives at this fast f-ratios do exist for SD
glasses, but only a few. Short flints, like Schott N-KZFS2, paired
with FCD100, Ohara FPL55/53 or LZOS OK4 would
produce better overall correction, with the poly-Strehl exceeding 0.95.
Schott N-ZK7 crown would produce better chromatic correction than
lanthanums, but because of the central wavelength limit to 0.96+ due
to more of the higher-order spherical residual its poly-Strehl is
lower, at ~0.91 (the inner radii of such objective would be also very
strongly curved, requiring very tight fabrication and assembly tollerances). The best match for lanthanums is fluorite, which
has the highest RPD value, hence the high Abbe differential
lanthanums matched with it would produce less of secondary spectrum.

When comparing secondary spectrum
in ED doublets with that in achromats, it should be kept in mind that
even at near-identical error levels the effect is not the same,
because in the former the aberration is a more or less balanced mix of
6th and 4th order spherical, with some amount of defocus, while in the
latter it is mainly defocus. They have different forms of intensity
distribution, and in so much different effect on contrast. Good indication
of this difference is given by their respective MTF plots (below).

Since at these (low to moderate) error levels spherical aberration
spreads energy wider (it is not fully apparent at the intensity
normalized to 0.1 shown, but would be more visible at lower normalization
values, or with logarithmic base), it causes more
of a contrast loss at low frequencies, but less at mid frequencies,
where is the approximate cutoff for bright low-contrast objects,
like planetary surfaces.
However, it is again more detrimental at high frequencies (lunar,
doubles, globulars). Diffraction simulations for 6th/4th order spherical
are based on the actual F-line wavefronts in the ED doublets, and defocus simulations are
pure defocus.

11.15 ED doublet with plastic

It was demonstated above that optical plastics can work well
replacing glass in an achromat. How well they can work as a
matching element to ED glass? Here's what it looks like with
some of plastic materials listed in OSLO Edu. Objective is 80mm
f/7, alike AT80ED, which is not using lanthanum (and probably
couldn't considering its low price), and cheap suitable crowns
have too small Abbe# differential to work well at f/7. It is not
to imply AT80ED uses plastic mating element, but it is a possibility.

With what appears to be a mix of styrene and acrylic (top), correction
in F and C is very good, just over 0.2 wave P-V of defocus. It is
comparable to a 100mm f/56 achromat, and satisfies the "tru apo"
requirement. At about 1 wave P-V, the red **r** line is at the
level of a 100mm f/22 achromat, while 4.1 wave P-V in the violet
**g** line puts it at the level of a 100mm f/14.5 achromat.

Using carbonate as mating element (bottom) produces markedly better
correction in the violet, but worse in the other three lines. With
just over 0.8 wave P-V in F and C, it is comparable to a 100mm f/15
achromat, with the red **r** line at the level of f/12, and
violet **g** line f/15. Other plastics did not produce good
correction levels, but it is very likely that better correction
than these two shown are possible. Of course, using plastics for triplets
widens the possibilities. These two combined, with STYAC in front
(reversed order is not as good) produce f/7 system with F/C lines
at the level of a 100mm f/12 achromat, and the violet g-line more than
2.5 times better, i.e. at the f/31 level. Still better is FK61/CARBO/STYAC
f/7 arrangement, with F/C at the level of a 100mm f/15 achromat, and
g-line at the level of f/25. These are unusual modes of
correction, illustrating that use of plastics could enhance both
correction level and correction choices in lens objectives
(note that optically there is no difference between the standard H-FK61 glass
and low-softening-temperature, moldable D-FK61, but the latter is nearly twice
more expensive).

Plastics have the advantage of being lighter, but their other physical
and chemical properties so should be at least close to those of
optical glass. As the production technology advances, it will be
becoming more viable as a glass substitute.

###
11.16 Can 52° AFOV fit 32mm 1.25" Plossl?

Different brands of this eyepiece come with anywhere from 44°
(Celestron) to 52° (Orion, Meade "Super Plossl", generic brands)
apparent fieldof view (AFOV) claimed. Taking that the apsolute
limit for the field stop
radius is the inner radius of the 1.25" barrel - around 14mm -
implying 23.6° zero-distortion angular field (~47° diameter),
the limit to the AFOV is imposed by field distortion. In most cases,
distortion is positive, enlarging image away from axis, in which
case the AFOV is bigger than zero-distortion FOV accoeding to the
extent of distortion. For Plossl eyepiece, it is about 10%,
implying nearly 52°. Raytrace exercise below tells somewhat
different story (it is illustrated using Plossl design, but in
general can be applied to any other).

Top design is downscaled Plossl from the Rutten/Vennrooij's book.
Reverse raytracing shows that the size of optical image is 14.2mm,
a bit over 14mm, implying 25.6° (51.2° diameter) as the
limit to AFOV. Design below, the upscaled patented Nagler Plossl
design, implies 26.5° (53° diameter) limit. The difference comes from different
distortion rates: the Nagler has somewhat larger distortion,
resulting in a larger AFOV transmitted (note that reverse
raytracing gives distortion of opposite sign to the actual
distortion, hence zero-distortion - or Gaussian - image is larger
than the optical image, showing so called *barrel distortion*).

However, the numbers come out differently with direct raytracing
- at list at first sight. The Nagler Plossl is plugged in with
two "perfect lenses": one for the 125mm f/8 objective, and
the other one at the eye end with a 17mm f.l.
Here, the true field angle just fitting into 14mm barrel radius
is 0.80°. With the objective focal length of 1000mm, and the
corresponding 31.25x magnification, the zero-distortion field
radius is 25°. That is 2.4° more than what the field stop
radius vs. eyepiece f.l. implies. The larger field produces
higher distortion (in proportion to the 3rd power of field radius),
now about 14%, with the coresponding 57° AFOV. Anyway, that
would be the usual way of calculating it. However, magnification
is not defined as magnification of the angle, rather as
magnification of its tangent. So the correct AFOV in this case
can be found from: (1) multiplying tan(0.8°) with 32mm f.l. to obtain
unmagnified height corresponding to the eyepiece focal length,
(2) having that height multiplied with 31.25x
magnification, and (3) from arctan of that height divided by 32mm
f.l. obtain the corresponding zero-distortion angle of view.
In this case, the unmagnified height is 0.447mm, the magnified
one is 13.96mm, and the corresponding zero-distortion angle is
23.6° (47.2° in diameter). With 14% positive (pincushion)
distortion, it gives 53.8° AFOV. It is less than a degree larger
than the AFOV obtained directly from the stop radius vs. eyepiece
f.l., i.e. the corresponding angle, multiplied with the distortion
ratio. This difference is most likely due to rounding off the nominal
distortion numbers.

The size of
transmitted AFOV doesn't depend on barrel length, or stop (i.e.
image) location within it - the last passing cone is vignetted
by over 50%, and the very next one higher is not making it trough at all.
Its change with the focal ratio (f/8 shown) is negligible.
There is little use of an oversized field lens transmission-wise, but
it is desirable in order to avoid light passing near the very edge
of a lens. Placing a 1mm wide stop into the barrel would reduce
the opening to 12mm radius, with the corresponding zero-distortion
field radius reduced to 20.25°, and the corresponding
AFOV to 22° (44° AFOV diameter). So, the answer is: it
is possible to pack 52° AFOV into 32mm 1.25" barrel eyepiece,
but only with no field stop in the barrel.

###
11.17 80° AFOV Plossl

What happens when the apparent field of a standard eyepiece, such as Plossl,
extends well beyond its usual 50 degrees? If, for instance, the same
Nagler Plossl from above is to expand to 80° AFOV? Image below
shows reverse raytrace of this design with the angle of divergence
from the aperture stop - in effect the exit pupil of the eyepiece -
equal to 40° (top). Obviously, the field lens had to be enlarged
in order to accept the wider field, but the astigmatic field looks
relatively good, with the astigmatism magnitude remaining nearly unchanged
over the last 30% of field radius. There is some field curvature, but
quite acceptable: edge of field best focus is less than 2mm away from
the field center focus. For 32mm f.l. eyepiece - from 32^{2}/1000 -
it translates to less than +2 diopters of accommodation
(infinity to over 0.5m distance). Distortion of about -30% means that
in the actual use the zero-distortion field - the
one determined by the eyepiece field stop - would be 56-57°.

However, coma - which increases
with the 3rd power of aperture (i.e. cone width) vs. astigmatism
increasing with the 2nd power - becomes obvious in
the outer field. Also, lateral chromatism is unacceptably large in
the mid 50%, or so, of the field radius. Coma can be diminished by
flattening R6 while strengthening R1 which, with the change of glasses, also lowers
astigmatism over most of the field, as well as lateral color error
(bottom). But longitudinal chromatism is significantly larger, and
field curvature is significntly more demanding: field edge requires
nearly +4.5 accommodation (infinity to ~0.24m distance).

Taking a compromise with some more astigmatism but flatter field
gives what is shown below (top). Note that
in reverse raytracing Gaussian image height is that of the apparent image (dashed line)
and the actual, zero-distortion image, determined by the converging marginal cone,
is the actual, "aberrated" image. Similarly, diverging cones entering
field lens are unequal in width - a consequence of the all field
pencils passing through aperture stop (i.e. eyepiece eit pupil) being
by default of equal width. The wider marginal converging cone
indicates lower magnification (by forming smaller Airy disc, i.e.
image scale), resulting in the negative (barrel) distortion.
Note that the half-diagonal of the square representing zero-distortion
(Gaussian) image equals the Gaussian image height (radius) in the image plane.

But
to find out how it is actually working in a telescope, it is
necessary to raytrace directly (bottom). The eyepiece is now working with the
field produced by a 100/1000mm (f/10) "perfect lens", and another
perfect lens is used on the opposite end to form the image. Field angle
needed for identical edge point height in front of the field lens is
1.09°, or 19.08mm. The
resulting astigmatic field is now significantly changed, with strong
higher-order astigmatism dominating the peripheral field. It is a
consequence of different ray geometry, with the diverging cones
coming from the objective being of the same width at the field lens,
while the marginal cone pencil exiting the eyepiece is, for that reason,
significantly more narrow than the axial pencil
(this is causing it reaching the retina as a narrower converging
cone, forming larger Airy disc i.e. generating positive image distortion).