◄ 14.3. Observatory telescopes

With the Strehl of 0.35 and 0.65 in the violet g- and deep red r-line, 0.82 and 0.87 in the F and C, and 0.999+ in the e-line, it can be considered a "true apo" (just short in the g-line, but more than compensated for it with the excess of contrast-preserving energy in the red). It s confirmed by its 0.95+ photopic polychromatic Strehl. Expectedly, the Strehl is somewhat lower on both, CCD and mesopic sensitvity scale. The latter is approximated from experimental results, not the mere numerical average between photopic and scotopic that officially goes as "mesopic sensitivity". This vision mode is more representative for the night time observing, which implies that we need a higher than 0.95 photopic Strehl for the true "sensibly perfect" correction under such conditions.

This combination can't be used for oiled triplet without aspherizing, but it is not a plausible option since wouldn't result in a performance gain.

Longitudinal astigmatism of 0.37mm at 1-degree off axis implies 0.00094mm P-V wavefront error (from 0.37 divided by 8F², same as for defocus).

2 - 180mm f/7 fluorite apo refractor

Does fluorite guarantee superior performance? Another contender in the same aperture and focal ratio as the above lens is the TEC's fluorite triplet apo. While the only information available on this lens' design is that it uses fluorite as the middle element, all one can do is look at the available mating glasses and see what is the best that comes out.

After trying a good number of combinations with matching glasses according to both, PF,e (for the F/C correction) and Pg,F (for the violet), one of the best was a combination with two glasses that have similar, but opposite in sign deviation from the ideal match: BK7 and K10. Its design parameters are shown below (top).

Performance level of this fluorite triplet is practically identical to that of the above FPL55 triplet, with the latter having only a slightly higher polychromatic (photopic) Strehl. It is, of course, possible that fluorite triplet design could be better than this one, but such glass combination is not readily apparent - and the same applies to the FPL55 triplet. Another similar combination, this time with Chinese (CDGM) crowns, achieves practically identical level of performance (bottom), at least as long as the polychromatic Strehl is concerned. This combination has actually more than slightly better individual Strehl values, but for some reason - one of them being that the red r-line has quite low sensitivity figure - it is not reflected in the poly Strehl. Both objectives could be thrown just over 0.95 Strehl by minimizing the central line error, but that, of course, wouldn't produce noticeable improvement.

It comes to what is pretty much a common knowledge, and it is that there is nothing magical about fluorite itself vs. other extra low dispersion glasses. In the end, it is the design and fabrication quality what decides how well a lens will perform.

3 - Takahashi Mewlon 210

Takahashi's Dall Kirkham reflector comes in three sizes: 210mm, 250mm and the 300mm big gun. It is often speculated how much of off axis coma they generate. Luckily, Takahashi gives design specifications for all three, which makes it easy to create the exact optical design. The three are very similar, having ~f/3 primary and 4x secondary magnification. Since the primary is enlarged (+10mm on the diameter) it is not clear whether the focal ratios given were calculated vs. effective aperture or the actual diameter. It is assumed the latter, but it wouldn't make significant difference if it is the former.

The reason for enlarged primary is the displaced stop, placed at the primary's focus. It has no appreciable effect on coma, but reverses the sign of astigmatism, reducing field curvature nearly in half.

 Here's the raytrace for the smallest Mewlon, 210mm f/2.9/11.5 Dall-Kirkham system.  Diffraction limited field, limited by coma, is 0.055 degrees in radius if measured by the RMS wavefront error (0.0745 wave, for 546nm wavelength), and 0.63 degrees according to the P-V wavefront error (0.42 wave).

OSLO is inconsistent here, since the P-V/RMS ratio for primary coma is 32^0.5 (astigmatism is negligibly small in comparison this close to axis, and has no appreciable effect on the wavefront error). Taking 0.059 degrees, i.e. 2.5mm off axis, as the diffraction limited field radius, implies that its linear field is comparable to that of a f/6.1 paraboloid (from F=[h/48W]^1/3, where h is the off-axis height, and W is the P-V wavefront error - taken as 0.42 wave - both in the same units). Of course, the corresponding angular field is proportionally smaller in the Mewlon. Also, it has proportionally larger image scale, which means that the linear diffraction image of the aberration is nearly twice larger. No effect on visual observation, since the objective's magnification is a part of total magnification (objective x eyepiece), but results in better sampling with any given CCD chip.

The system has 32% linear central obstruction diameter.

4 - AP 10" f/14.6 Maksutov-Cassegrain

One of the cult telescopes in the amateurs' circles, the Astro-Physics' take on the Gregory-Maksutov design, never had its optics specs published, but since they are determined by the system parameters, it is possible to reconstruct them in the way that there is no appreciable difference in the performance level vs. the actual design.

The purpose of aspherizing the primary is to remove the residual coma, with the extra benefit of somewhat relaxing corrector's radii, hence reducing the higher-order spherical aberration residual as well. The limit to aspherization is set by the coma introduced by it. If it is chosen to eliminate coma, with the standard corrector thickness of 1/10 the aperture diameter, it results in a design below.

The central line is just short of 0.95 Strehl (0.947 without its 23.6% central obstruction, 0.958 with it dialed in, due to it taking out the central wavefront deformation), and it is the main reason why the polychromatic Strehl is also below 0.95 (around 0.94 when unobstructed, which implies that the Strehl degradation factor due to the chromatism is only ~0.99). There are only two practical ways to somewhat improve the central line correction: one is with a thicker meniscus, and the other with more strongly aspherized primary.

With 32mm thick meniscus, the central line's Strehl increases to 0.966 (0.971 with the obstruction), with a slightly larger (minimum size) secondary, and nearly 2" longer back focus.

With the primary conic increased to -0.23 (at which coma at 0.25 degree off axis reaches diffraction limit - shown below ray spot plots - and design still could be called "coma free" in practical sense) the Strehl increases to 0.956 (0.962 with the obstruction). With both, thickness and conic increase, the central line's Strehl goes to 0.975.

A 3% increase in the contrast averaged over MTF frequencies - which is what the Strehl value represents - is hardly noticeable at all. So the actual design may, and may not have these enhancements (with Roland's drive for perfection, it is more likely than not). Much more of a factor is seeing: even as good as 1 arcsec would lower the average Strehl of a perfect 10" aperture down to about 0.5 (D/r0~1.8). Yet, such 10" aperture still can deliver stunning views.

An interesting detail is the star test. These instruments with the balanced 4th and 6th order spherical show more asymmetry in out of focus images than the standard, 4th order spherical aberration alone. It was pointed out more than once by those manufacturing such telescopes (including Roland) that a highly corrected telescope of this type will still show noticeably different in- and out of focus images. Can we trust them, considering conflict of interest? The bottom row shows OSLO simulations for the design in question with 0.97 central line Strehl (the one with thicker meniscus).

5 - ARIES' original 10" Maksutov-Cassegrain (?)

Seems that quit a few amateurs believe that larger MC telescopes - and 10" would belong to that category - have to be aspherized in order to be well corrected. That, however, holds true only if the primary is kept at f/3, or so. Allowing for somewhat slower primary relaxes the required corrector radii, making possible to have significantly larger all-spherical systems well corrected.

More so, such systems can be also coma-free. Commercially-made MC that fits the requirement of a slower primary is the original Aries' 10" f/13.5 telescope from the 1990's. While I'm not aware of the official prescription, a design can be made to illustrate its performance level. It will be a f/4/14.5 system with separated secondary mirror.

Correction level is stunning: the central line Strehl is 0.99+, and so are the ratios in F and C (needless to say, the polychromatic Strehl too). Field curvature is mild, and the minimum obstruction size is 27% by diameter. Needed optical tube is longer, but not excessively long. Angular field that fits 2-inch barrel is nearly 3/4 of a degree in diameter.

The system could be easily rescaled to fit the original Aries, without appreciable change in the performance level.

6 - TAKAHASHI TSA 102 f/8 apo triplet

One in the series of exquisite Takahashi refractors, the TSA 102 had its optical design revealed during its marketing campaign. Here's how it raytraces.

On the design level, there is no chromatism to speak about. Polychromatic Strehl in the 430-670nm range comes to 0.992, while in the optimized wawelength it is 0.9998. The polychromatic MTF is practically coinciding with that of a perfect aperture.

There is some residual coma. While it is visually unnoticeable, it does make point image asymmetrical, and enlarges the 80% energy circle at 1-degree off axis by nearly 1/3. It is easily removed by adjusting the outer radii (R1=344mm, R6=-944mm, with R3=179.5mm and the second gap increased to 0.45mm to minimize spherical aberration), but the reason it's there, as graph on the Tak site shows, is that it gives a coma-free image with its dedicated focal reducer.

Just how good this triplet is illustrates the fact that its polychromatic Strehl is still as high as 0.962 even when scaled down to f/6, although the tolerances are much tighter. That is one of the Takahashi's secrets: they don't push the envelope, making it possible to execute design to near (practical) perfection.

7 - Cheap ED doublets: how good they are?

Cheap 4" ED doublets, like Svbony Sv503 and Astro-Tech AT102ED, use lower-grade ED glass - FPL51 and the optically near identical FK61, respectively. Both come at the fairly fast f/7 focal ratio. Mating material is not mentioned, which leaves a wide range of possibilities, from ordinary crowns, like Schott K7, K10 and alike, to somewhat obscured N-KZFS2, and to acrylic-like optical plastics.

Performance level is near-identical with either ED glass. Chromatic and overall correction is better when combined with K7 than K10 (not shown), despite the latter allowing for significantly more relaxed inner radii. Best correction level - near the "true apo" minimum - is with N-KZFS2, but it is unclear why it is not better known as a lens objective glass choice. Note that the Chinese (CDGM) equivalent to the Schott K7, H-K6, is quoted as being 4.5 times more expensive than their BK7 equivalent (H-K9L), with Schott K7 being only 2.10 times the N-BK7 price. Another possible Chinese match, H-K10 (equivalent to the Schott K3, now obsolete), quoted at 3.2 times the H-K9L price, produces similar output, somewhat tighter in the red and blue, but with larger error in the green and violet. There is no Chinese equivalent for the Schott N-KZFS2 short flint.

Finally, optical plastic going under ZC1 in OSLO catalog has similar properties to acrylic (which itself produces not negligible secondary spectrum with these ED glasses), is probably a kind of cyclic olefin polymer (COP), known as ZEONEX. It allows for a very similar correction level as K7 but - it should be safe to assume - at a lower cost.

With their 0.86 polychromatic Strehl, i.e. overall contrast level, the combinations with K7 and ZC1 would be comparable to a 100mm f/20 achromat. Factors that boost achromats poly-Strehl are its better design correction limit in the optimized wavelength, as well as the non-optimized wavelengths error increasing with the square of index differential vs. optimized wavelength, as opposed to the near-proportional increase with spherochromatism - according to the nominal error in F and C, the two ED lenses would be comparable to the f/25 achromat. However, with respect to the apparent chromatic error, they are significantly better, due to their color foci being much closer together, with most of the nominal chromatic error coming not from defocus, but from spherochromatism, which lowers contrast just the same, but is much less colorful.

Interesting detail is that despite its lower P-V and RMS wavefront error in the violet h-line, the objective with a plastic element has significantly lower Strehl value for this line than the K7 lens. At the same time, its radius of 80% encircled energy is significantly smaller - 0.0224mm vs. 0.0276mm for the lens with K7 element. It shows that the Strehl value at large wavefront errors - roughly over 0.15 wave RMS - becomes unreliable metric for the level of optical quality, as the central intensity becomes less representative of the overall energy distribution.

8 - Questar 3.5" Standard

No other commercial telescope had such an aura, and cult following as this small 89mm f/14.4 Maksutov-Cassegrain telescope in production since the 1950s, and in this form from 1960s. Going at 2-3 times the price of a quality apo triplet of the same aperture, how good it really is optically? There are all kinds of reports, most of them impressive, from 1/10 wave P-V, or better, tested, including an owner reporting as good as 0.987 Strehl - better than 1/16 wave P-V of primary spherical aberration - test result from "Company 7". Questar Corporation on its side states that the telescope has "no coma, astigmatism, or spherical aberrations". No wonder more than a few people believes that this little telescope actually performs above the limits set by the laws of physics.

Since there is no secrets on how the Maksutov-Cassegrain works, all needed for determining the performance level is a single number, the mirror focal length (the size of central obstruction is published). Knowing the final focal ratio determines mirror-to-corrector separation, since for a given corrector thickness there's only a single set of its radii that will correct both, spherical aberration and longitudinal chromatism. Since it is so-called "spot-Maksutov", with the secondary being an aliminized spot on the rear corrector radius (since 1960s), the constraints imposed in this configuration make possible deviations from the actual arrangement negligible. In absence of the mirror focal length, even the mirror-corrector separation will do, since the known final focal ratio determines the width of the axial cone on the rear corrector's side, and with that the effective focal length of the primary (for the light refracted by the corrector). Variations in the corrector thickness, within the standard range, have only minor effect.

This side cross section provides all information needed to reconstruct the Questar system. The dimensions correspond well to the actual O.T.A. (back focal length could be up to 20mm, or so, longer). The final system based on these values is shown below, with spherical (left) and aspherized primary.

It has practically perfect achromatism, but spherical aberration sets the correction limit to a nearly 0.92 Strehl in the all-spherical version, and ~0.96 in the one with aspherized primary (aspherization is determined by eliminating coma; further aspherization would make spherical aberration still lower, but would start inducing coma of opposite sign - for instance, putting -0.3 conic on the primary would produce a system with 25% less coma than with spherical primary, and the limiting axial design Strehl of 0.99). Astigmatism is negligible, but it is present: the 0.34mm longitudinal error at the field edge implies 0.37 wave P-V, or 0.80 Strehl level.

It is not known whether the primary is spherical or aspherized. According to the information available Questars were all-spherical in the 1954-57 period, and had the aluminized spot on the first radius up to 1978. Having aliminized spot on the first radius changes the performance level in this configuration only slightly. Corrector radii need to be somewhat stronger (-87.5/-92.6mm), and the secondary area on the front surface would need to have yet stronger radius (-82.1mm) in order to produce f/14.3 system. Back focal length would be insufficient, which means that either: (1) the mirror would have to be placed a bit closer to the meniscus (~1mm, in which case the system f/ratio would be ~f/15) with the secondary radius of ~-83mm, or (2) only the secondary radius would be made slightly stronger (-81mm) producing ~f/16 system.

The system can be coma-free only with the primary aspherized (all-spherical arrangement has linear coma-limited field equal to an f/7.4 parabola), which would imply aspherization, but how serious we can take that claim, standing next to "no astigmatism and spherical aberrations" too (Questar Standard 3.5" Telescope Specification Sheet)? Taking that "no" means "no perceptible", spherical aberration in this sistem can be brought down to that level only by putting higher order aspherics on the meniscus, or primary. Putting higher-order aspherics on the meniscus, or primary, doesn't affect the system coma, so the primary still needs to be aspherized to make it coma-free. Starting with the bare-bone, all-spherical version above, keeping the meniscus-to-mirror separation, central obstruction, and final focal ratio unchanged, will take a look of three possible scenarios of putting higher-order aspherics on: (A) front corrector surface, with primary, i.e. 4th order spherical reduced to near-zero by altering corrector radii, (B) also on the front corrector surface but with the 4th order spherical reduced by putting 4th order aspheric on the corrector, and (C) by putting higher-order aspherics on the primary (putting 4th order aspheric on it generates lower-order coma of the same sign as the system's higher-order coma, making it impossible to correct coma). Specifications, along with the performance level for all three is given below (secondary radius in the system B should be equal to the rear corrector radius, which causes imperceptible change in correction, but the primary r.o.c. needs to be increased to 399 to maintain the final focal ratio; for minimizing the aberration, AD needs to be changed to 2.02e-9 and AE to 3.33e-13, with the final output nearly identical to the one shown).

Arrangements A and C are very similar in their design limit, while B has more spherochromatism, as indicated with its deeper aspherics. The starting all-spherical system has about 1 wave of secondary spherical, about 0.2 wave of tertiary spherical (of the same sign), and little over 1/90 wave of 10th order spherical. The reason why A and C have less spherochromatism is that reducing 4th order spherical to near zero by manipulating corrector radii makes them more relaxed, hence all higher-order terms are smaller. How deep are these higher order curves, that need to be carved into the existing surface radius? Since OSLO calculates values for the correction at the paraxial focus, each higher-order term grows from zero on axis to the maximum at the edge, with the exponent equaling the order. So the secondary (6th order) spherical aberration term of 1.8e-13, needed to reduce it to near zero, gives the depth of glass to be taken of the edge when multiplied with the surface semi-diameter raised to 6th power - in this case 0.0014mm (this is about 2.5 times larger than the error at the best focus location, which is about 1 wave P-V; if correction was made at this focus, the needed curve depth would be 2.5 times smaller, maxing out at the 0.76 zone and falling back to zero at the edge). Since all terms are of the same sign, they add up (note that the 4th aberration term is only a small correction of about 1/20 wave that was little easier for me to put here, than to fiddle with corrector radius).

There is no way to know which - if any - of the three was used, but putting higher-order aspherics and conic onto the primary (arrangement C) seems to be the easiest, hence most likely. Omitting the 8th order term would put design limit to 1/14 wave, and would require adding the 6th order curve - 0.000222mm, or 0.4 wave at the edge, but since negative in sign would require that much deeper center - to the conic curve. Longitudinal astigmatism at 0.5 degree off axis is nearly 0.3mm, which gives 0.00018mm P-V wavefront error, or 0.33 wave for 0.00055mm wavelength (from 0.3/8F^2, same as for defocus). That is still below the "diffraction limited" 0.37 wave for primary astigmatism.

The MTF graph published by Questar Corporation (same source) should help, but it doesn't. As picture below shows, it doesn't match neither MTF graph for balanced primary spherical (middle), in case they used it for expressing the error, nor MTF for the actual (balanced secondary spherical, right) aberration of the all-spherical arrangement (MTF for the arrangement with aspherized primary is much tighter for the design limit, and would become similar only if the actual units would have significantly larger axial error), let alone the arrangements with higher-order aspherics (MTF not shown, practically coinciding with the limit). In the lower frequencies, the Questar's MTF plot is concave, while it should be mildly convex; in the high frequencies there is a very pronounced - and quite unusual - contrast drop. In fact, if we'd assume the customary 5% minimum contrast needed by the eye for resolving a line pair at the limit, the typical Questar would have its limiting field resolution about 5% lower than the limit imposed by the aberration-free MTF (given by Questar).

Worse yet, its limiting contrast for the 0.34 relative obstruction size is not acurate, and the measured "typical" MTF has an odd form, with the most of contrast loss toward high frequency range (top left, with the green line representing the proper MTF for 0.34 relative central obstruction). No known aberration produces such an MTF. It would require enlargement of the central maxima, without spreading energy beyond it. The closest that comes to it is astigmatism, and only in one orientation. Adding astigmatism to the all-spherical arrangement, to make the two MTFs roughly similar (bottom left, more contrast loss in the lower frequencies range, less in the high frequency range), results in the overall error at the level of 0.80 Strehl. In other words, the Questar's own graph implies this level of correction: 0.80, or so, Strehl wavefront on top of 0.34 relative central obstruction. The question is, if the telescope has "no spherical aberrations", what is causing this axial MTF degradation? Hmmmh...

9 - UniStellar eVscope

Would you pay $3000+ for a 4.5-inch Newtonian reflector? With a sophisticated goto mount, and 100,000X photon flux intensifier? Some people would and, judging on the published ratings, they are satisfied with what they've got for their money. The eVscope comes in two models, one w/o eyepiece, intended to use your smart phone display, and the other with a special eyepiece, connected to the photon-intensifying sensor placed in the mirror's image plane. Both are intended primarily for deep sky objects observations at a low (50X) optical magnification and up to 400x (up to 150x recommended) digital magnification. Perhaps for that reason, combined with the fairly small field covered by the sensor, the mirror image is left uncorrected for coma and field curvature. How does that affect optical quality?

Picture below illustrates main properties of the image supplied by a perfect 4.5-inch paraboloid, including the effect of a fairly massive obstruction by the spider vanes and sensor housing. The sensor comes in two sizes, 27'x37' in the model w/o eyepiece, and 34'x47' w/eyepiece, with the limiting resolution - determined by sensor properties - of 1.72 and 1.33 arc seconds, respectively (theoretical limit for the aperture is just about 1 arc second). Considering the intended purpose, the inferior limiting resolution is of no consequence since - as the MTF graph shows - cutoff frequencies for deep sky objects are several times the ultimate (high-contrast) cutoff at best.

The 0.3° field radius is just a tad smaller than the longer sensor half-size, in the model w/o eyepiece, and the 0.2° field radius a tad larger than the short half-side, which means that sensor corners have somewhat more coma, and so does the larger sensor, in proportion to its size. The spots are given for the best image surface (R=480mm) but it is practically unchanged on flat field, due to the small field size and the relative insensitivity of coma to defocus. At about 0.01mm longitudinally, astigmatism is negligible (P-V=0.01/8F²). However, as MTF graph shows, the loss in limiting resolution is significant even at the relatively low frequencies resolvable with deep sky objects, even if the contrast loss is quite moderate. At 50x magnification, sagittal coma at 0.3° is nearly 5 arc minutes, large enough for its shape to be recognized by the eye. On axis, encircled energy is ~0.64, comparable to the effect of a 0.45D (45% the aperture) central obstruction w/o vanes. While this may be acceptable considering the purpose, image quality would still benefit from a coma corrector - more so in the model with a larger sensor.

10 - APM 140mm f/7 SD Lanthanum confusion

On its site, APM states that this ED doublet uses Ohara FPL53 mated with the CDGM's H-LAF53 glass. However, this combination has too much of a residual secondary spectrum to fit into published Strehl-across-the-wavelengths graph for this refractor. As image below shows (top), the F (486nm) and C (656nm) lines are at, or near 1/2 wave P-V wavefront error, and their Strehl numbers fall significantly below 0.78 and 0.65 (approximately, since this line was cut out of the graph) respectively. The violet g-line is at 2.5 waves, comparable to a 100mm f/23 achromat.

Some time ago, on forum talk, Markus Ludes was cited as specifying for the mating element another Chinese Lanthanum glass, H-LAF7A. While coming closer to the published Strehl values, it is still falling short (middle). Glass that reproduces given F and C lines Strehl values, and also corresponds better to the published central line Strehl, is H-LAK51 (bottom). OSLO gives its cost as 3, in units of the BK7 cost, which also makes it more likely choice than H-LAF53 at 4.5 times the BK7 cost. The violet g-line error is still relatively large at 1.1 wave P-V, but this is comparable to a 100mm f/50 achromat, hence not very noticeable. Moreover, the scope description by some vendors uses the words "Lanthanum coatings" which, along with "new materials" indicates that a special kind of coating was used to cut out better part of the violet. Some materials, when doped with Lanthanum, can selectively absorb ultaviolet bands. For instance, LiNbO3 (Lithium niobate), doped with 5% Lanthanum, transmits only about 65% at 450nm, and less than 10% at 430nm, while transmitting over 90% of the longer wavelengths (undoped, it is near zero already at 450nm, but transmits only about 60% of the wavelengths over 500nm). With optimized transmission, light loss in the rest of wavelengths probably can be made negligible, with the absence of violet giving more abundance to the red hue - noticed in some reviews. The photopic polychromatic Strehl for the 430-670nm range comes to 0.89, which is about the level informally announced by Markus.

Note that polychromatic Strehl incorporates all colors, including the optimized, green-yellow. Hence it would have been better if the central line Strehl was better. If it was, say 0.99, the corresponding poly-Strehl would be 0.923, sensibly quite close to the "sensibly pefect" 0.95 (bear in mind that it is photopic Strehl, with the sensitivity in both blue/violet and red increasing significantly toward the more applicable field mode for night-time observing, mesopic, and the mesopic Strehl being correspondingly lower - this applies to all chromatically imbalanced systems, but affects more the blue/violet end, typically less well corected than the red). In other words, the farther the central line Strehl is from perfect, the more the off-color correction is better than the poly-Strehl indicates.

11 - 150mm f/5 ED doublet

Is the TS-Optics 152mm f/5 ED doublet well corrected for chromatism? Those somewhat familiar with lens design know that ED doublets require significantly more strongly curved inner radii, hence with such a fast system will generate much more of higher orders of spherical aberration - secondary (6th order), tertiary (8th), and even 10th order is not negligible. This, in fact, eliminates all mating glasses that would bring all colors close to each other, since Abbe differential required to bring higher order spherical aberrations to acceptable level leaves only glasses higher on the RPD diagram than Hoya's FCD1 - the ED glass cited for this refractor - thus inevitably producing some secondary spectrum.

Glasses suitable for the mating element are lanthnoids similar to the Schott's short flints with the lower Abbe number. Since the ED glass is from Hoya, it is more likely than not that the mating glass is from the same source. Hoya still lists some short-flint-like glasses, for instance E-ADF10 (Abnormal Dispersion Flint similar to Schott's KZFSN4). Somewhat better overall is another lanthanoid, NbFD11 (Niobium Flint, Dense, equivalent to Schott's N-LAF33, listed by OSLO at 4.4 times the BK7 cost). Image below shows performance level of this combination, compared to the standard BK7/F2 achromat.

The ED doublet does have significantly better correction, reflected in the photopic polychromatic Strehl (note different scales for both, longitudinal SA and OPD graphs). But how good it is? Well, the ~1 wave F/C average puts it at the level of a 100mm f/12 achromat, while the 5.3 waves in the violet g-line corresponds to f/10.6 achromat of the same aperture; the red r-line is at the level of f/10.8, and the poly-Strehl at the f/10.2 level. The violet can be tamed with lanthanum-doped coating, but it is not known whether or not such was applied.

12-Omegon AP110/660 ED doublet

Probably the fastest on record ED doublet for this aperture size raises question of how good its color correction really is. Since the two glasses are known, the answer is as simple as putting them together. The FPL53 positive element boosts confidence that it should be nearly as good as possible, but the mating element, S-NBM51, is what will determine the outcome. It is a near-equivalent of the Schott's N-KZFS4 "short flint", and the reason it's been selected is obvious: it is far right from FPL53 on the relative partial dispersion diagram (RPD), i.e. with a large Abbe number differential between them, which generally allows for more relaxed doublet's inner radii, hence also less of higher order spherical aberration. Also, short flints are generally better a match than lanthanum glasses, which is the only other group in this part of RPD diagram. Below is how this combination raytraces.

The LA plot shows some residual secondary spectrum, which was to expect considering that NBM51 to FPL53 RPD differential is not negligible. Most of the error in F and C lines comes from defocus (i.e. secondary spectrum): for instance, F line is 0.2 waves P-V at its best focus, and 0.75 waves at the best green focus. The design limit in the optimized line is 0.96 Strehl, which puts the best one could expect in the actual unit at ~0.95, or 1/8 wave P-V of spherical aberration equivalent. The weakest point is, as usual, the violet end: g-line error is over 3.5 waves P-V. This puts it at the level of a 100mm f/16 achromat, while the 0.8 wave averaged for the F and C lines correponds to a 100mm f/15 achromat. Considering it's 110mm aperture, it is roughly at the level of a f/17 achromat - except that the achromat should have a better central line correction.

Is this the best possible mating glass for this configuration? No. There are very few choices for such a fast doublet altogether, but Schott's N-KZFS2 would do considerably better. It would have no secondary spectrum residual to speak of, F/C average anywhere from 1/4 wave P-V to 0.4 wave (the latter with minimized g-line error), and g-line error from 0.7 wave (when minimized) to 1.1 wave (with minimized F and C). The only reasons not to use this glass is that Ohara probably doesn't have near equivalent, and that it would require somewhat stronger inner radii, with somewhat tighter tolerances. Note that there is no significant difference in performance between arrangement with FPL53 in front, and in the back, but the latter seems to be having a bit better correction.

13-The TS-Optics Photoline 125mm f/7.8 apo puzzle

This refractor is advertised as having a doublet objective employing Ohara FPL53 paired with unspecified lanthanum glass. Those that play with designing this kind of lens objectives learn quickly that lanthanum glasses are generally suboptimal match to extra-disperion (ED) glasses. One reason - readilly apparent from a glance at the relative partial dispersion diagram (RPD) - is that they are higher on the diagram, i.e. will produce a correspondingly high amount of secondary spectrum (fluorite, FPL51 and, to a lesser extent, FK61 being exceptions in that they wouldn't produce secondary spectrum with a group of lanthanum glasses with lower RPD values). The other is that due to their relatively high index-to-Abbe# ratio, they require stronger inner lens radii, i.e. produce more of the residual higher order spherical aberration. Yet another minus is their violet dispersion, which typically causes the violet end run farther off from the rest of the colors. Yet, user acconts describe the telescope as "sharp" and "colorless", and it is marketed as comparable to a triplet color correction wise. What could explain this discrepancy?

Choice of the lanthanum glass is determined by two factors: (1) its RPD differential, favoring those closest to the FPL53 level, for minimized secondary spectrum, and (2) relatively low violet end defocus. For a doublet, it is also necessary for it to have sufficient Abbe# differential, to minimize higher order spherical residual and spherochromatism. It also needs to satisfy physical and chemical requirements for fabrication and use. With respect to the first requirement, the best glass hand down is Ohara S-BSM81, producing zero secondary spectrum with FPL53 (there are also similar glasses from Schott and CDGM, but since their other properties are also similar, consideration will limit to Ohara glasses). However, it has insuficiently large Abbe# differential: at f/7.8 its design limit in the optimized wavelength is 0.22 wave P-V, due to the significant higher-order spherical residual. The best glass that can be found in the ATMOS base, with respect to the optical requirements, is S-LAL14. Image below shows what level of performance this lanthanum glass would produce when paired with FPL53 in both, doublet and triplet objective.

In a doublet (#1), correction at the given wavelengths level would be at the minimum required for the "true apo" if it wasn't for the large error in the violet g-line. Note that this "large error" compares to this line correction in a 100mm ~f/50 achromat, so it would be detectable only on bright targets. In a triplet (#2), there is no corrective action of the higher-order spherical residual on this wavelength anymore, hence the LA graph shows it more detached from the rest of the wavelengths (even if the triplet would be at f/7.8, the error would be still larger than in a doublet). This is perhaps why we don't see much of lanthanum in triplet objectives.

For a more complete picture, shown are triplets employing BSM81 (#3) and BSL7 (Ohara BK7 equivalent, #4). With the former, even at f/6.5 there is no noticeable higher-order spherical residual, and no secondary spectrum. The latter, due to its stronger front inner radii (i.e. its lower Abbe# differential) does generate more higher order spherical and spherochromatism, but closer to f/7.8, at f/7.5 - which is the actual Takahashi TSA 120 design, save for the residual coma left in to compensate for coma in the dedicated reducer - it is better corrected than the lanthanum doublet, and the FPL53/BSM81 triplet even more so.

In all, unless the TS Photoline uses some off-chart, custom lanthanum glass - which would be expensive - it is likely that it employs similar selective absorption coating as the one described under #10. The end result is a well performing telescope, and it is what matters.

14-CFF 200mm f/6.5 APO TRIPLET

What is known about this triplet is that uses Ohara FPL55 ED glass, and that has at least one aspherized surface. This means it can't be presented as the actual combination, but the choice of mating glasses is limited, due to FPL55 having lower relative partial dispersion (RPD) than other ED glasses, hence very few potential matching glasses for no appreciable secondary spectrum. The most likely choice is the cheapest, and also of the highest optical quality, S-BSL7, Ohara's equivalent of the Schott BK7. Image below shows BSL7/FPL55/BSL7 triplet with both, air and oil filled gaps (Schott K7 acts as oil substitute, having similar refractive index). Evidently, unlike the air-spaced lens, which has no noticeable higher-order spherical residual, in the oiled lens it is readily noticeable (the only difference in prescription, other than the gap medium, is the significantly stronger aspherization required with oiled triplet, mainly due to more 4th and 6th order spherical aberration at the 4th radius, as well as change of R1 to 644mm, to optimize color correction). Hoewer, it is a slight negative only in the optimized wavelength, while improving a bit the overall color correction - as the polychromatic Strehl (air-spaced graphs/spots are on top, oiled below).

Correction mode is, as usual, to have F and C lines nearly balanced, which is generally the best visual correction. A slight increase in the negative lens power pulls blue/violet and red away from the green (paraxial foci), in this case decreasing the error in violet and red, while increasing it in the blue, and would be better overall for CCD. The oiled lens qualifies as a "true apo" if judged by the errors in the five wavelengths - 0.8 Strehl or better in F and C, 0.4 or better in g and r (violet and deep red, respectively) - but still falls short of the "sensibly perfect" 0.95 polychromatic Strehl. Better part of it is caused by "only" 0.983 Strehl in the optimized wavelength; however, considering that it corresponds to a little more than 1/15 wave P-V of primary spherical aberration, no actual unit would be able to reach this Strehl value anyway). Doesn't seem to be possible to significantly reduce the higher-order residual in the optimized wavelength.

Possibility to combine two different mating glasses with similar but opposite in sign RPD deviation from FPL55 is also very limited, due to its low position on the RPD diagram. One viable combination is with S-BAL41 in front, and S-FSL5 in rear (bottom; the reverse combination is not as good). While the overall collor correction is slightly lower, it is higher in the optimized wavelength. Aspheric required is similar, also on R4, which should be the aspherized surface in the actual lens, since it generates most of the 6th order spherical residual, significantly lowered by aspherizing.

15-SkyWatcher 190mm f/5.3 MAKSUTOV-NEWTON

All that is needed to come up with Maksutov-Newton design of an actuall telescope is its aperture and focal ratio. Practical variations in the corrector thickness around the standard 1/10 of the aperture diameter cause negligible effects on the final output, and for any given corrector thickness there is a single pair of radii that optimally corrects for spherical and chromatic aberration. Central obstruction size is not specified, but from the photos it is about 52mm in diameter. Image below shows the system outline, prescription, and optical output. Outer field vignetting (top left) is illustrated with the converging cone (green) against diagonal's surface (projected axially as a circle). Half a degree off vignetting is acceptable both visually and CCD, and one degree off only visually.

Design limit, with perfect surfaces, is shown on top. Polychromatic photopic Strehl exceeds 0.999, which implies that all nine wavelengths used for calculation (0.43-0.67 micron, 0.03 increment) are better than 0.999. The Strehl value reflects quality of the wavefront, independent of the central obstruction effect. In this case, central obstruction diameter is 27-28% of the aperture diameter, hence it reduces the central diffraction maxima relative intensity by a factor (1-0.275²), while the encircled energy drops by approximately (1-0.275²)², due to the maxima's diameter reduction. This implies the PSF peak value of 0.923, and the encircled energy of ~0.854 for unit in a perfect aperture. The latter could be looked at as an indicator comparable to the Strehl for unobstructed aberrated wavefront, producing similar PSF encircled-energy wise, but keeping in mind that the obstructed PSF has smaller and brighter central maxima than that for the typical aberrations, and lower defocus sensitivity. The ray spot plots (left) show off-axis coma, a little over 1/2 wave P-V at 0.5° (8.8mm) off axis. That is nearly as good as f/9 paraboloid. Best field has 2200mm radius, convex toward mirror, resulting in a near negligible spot enlargement over flat field. Diffraction simulations (obstructed on top, unobstructed below) also show very little effect on image size for this field radius (0.5°).

No telescope comes with perfect surfaces, and of particular interest here is how a radius error on the corrector affects both, spherical aberration and chromatic correction. Effects of changing the front radius to -354mm, which is less than 0.1% radius error, are shown on the bottom. The central line error now goes to 0.0273 wave RMS, equivalent of 0.091 wave of primary spherical. That lowers the central line Strehl to 0.968 (from 0.999440), and polychromatic Strehl to 0.962. In other words, the Strehl degradation factor due to the chromatism is still only 0.994. Conclusion is that the increase in spherical aberration induced by a radius error on the corrector surface is always far greater than the accompanying increase in the chromatic error.

16-ORION Eon 110mm f/6 ED doublet

I said for the Omegon 110mm f/6.6 that it is probably the fastest ED doublet of that size, but it's not. Even faster is Orion's 110mm f/6 Eon doublet using Ohara FPL51 and NBM51. While the mating short flint is identical in both, the Omegon uses a higher-grade ED glass: FPL53. Since it is also significantly slower, it must be having significantly better overall correction, right? Wrong.

For one, FPL51 is higher on the relative partial dispersion diagram (RPD) than FPL53, hence having smaller differential vs. NBM51. That translates into a smaller secondary spectrum. More importantly, it matches the mating glass better in a way that its higher refractive index allows significantly more relaxed inner radii, resulting in markedly less of higher-order spherical aberration residual.

As a result, the e-line Strehl - design limit - is 0.997, vs. 0.961 with the Omegon. Photopic polychromatic Strehl is 0.832 vs. 0.81, in the same order (since the ratio for their central line Strehl is 1.037, assuming nearly identical central line correction, chromatic correction is slightly better in the latter; however, since both design limit and tolerance favor the Eon, it is reasonably to assume it will have a better central line correction average). Overall, chromatic correction vise, the Eon is not quite at the level of a 100mm f/15 achromat.

17-ES ED102 Essential Series Triplet

This Explore Scientific 102mm f/7 triplet OTA comes at a price lower than many a doublet. Sure, its finish leaves something to be desired, but what really matters is how good it is - or can be - optically, considering that it uses a "common" ED glass, Hoya's FCD1? The mating glass isn't specified, but looking at a RPD diagram, there is a very few Hoya candidates that would practically cancell secondary spectrum. The best one seems to be BACD11, ordinary crown equivalent of the Schott N-SK11, listed by OSLO at 1.4 times the BK7 price. Here's what level of correction these two glasses produce.

Turns out, it easily passes the "true apo" requirements with both, error magnitude for the five wavelengths, and polychromatic Strehl: 0.975. Sure, that is at the design limit and 0.999 e-line Strehl. With the more realistic and still excellent 0.95 (1/8 wave P-V of primary spherical level), the poly-Strehl would drop down to "only" 0.926. But this means that no apochromatic lens has a chance to pass the 0.95 poly-Strehl requirement unless it is better than 0.95 in the optimized wavelength. And, chances are, most of them aren't. Contrast-wise, it shouldn't matter if the central line is, for instance, 0.99, and the rest of wavelengths 0.96, or the other way around, as long as they give the same 0.95 poly-Strehl. But it would be useful to specify both components, for it would give better indication of what is correction level of non-optimized wavelengths (that may not matter a lot with systems where spherochromatism dominates, producing what Roland Christen calls white, or whithish, chromatism, as opposed to defocused colored fringes with secondary spectrum, but still would be good to know).

Back to the Explore Scientific "essential" triplet, if it uses this glass combination - and there is no good reason not to - optically it can be as good as any high-level apo. The unknown is its fabrication quality, which is what can make it or brake it in an actual unit.

18-Celestron C90 Maksutov-Cassegrain spotter

The all-spherical 90mm Maksutov-Cassegrain had always been popular among amateurs. It's highly portable, inexpensive, has sufficient aperture to show more than department store telescopes, and - hey - it's a Mak! Celestron's 90mm Mak is, as nearly all others, Gregory-Maksutov type, i.e. with a silvered spot on the rear meniscus surface serving as secondary. In the past few decades it went through different shapes and looks, and nowadays it sells as an f/13.9 spotter. Its prescription hasn't been published (to my knowledge), but due to limitations imposed of this type of a system, it is pretty straightforward to come close to its actual specs, through a few iterations. Based on the approximate 100mm of back focus, the primary is about f/2.3, and the system raytrace is shown below.

It does go as practically "color-free", but the design limit in the optimized wavelength - 0.90 Strehl - leaves something to be desired. It is a consequence of the fast primary, requiring strongly curved meniscus producing significant higher-order spherical residual. The actual units are probably closer to the "diffraction-limited" (0.80 Strehl) and, if the backfocus is a bit longer, as it could be, i.e. the primary little faster, even 0.80 Strehl may not be easy to get. This is probably why it goes as a "spotter" and not a telescope. For better correction, the primary - and the final focal ratio - would have to be slower. At 0.5° off axis, central diffraction maxima is still well defined, with some low level coma and astigmatism present. Even if the best image field seems to be strongly curved - 111mm radius - the flat field image is not much more distorted this far off: credits to the scope's slow focal ratio, making it relatively insensitive to defocus. In all, if well made, this little instrument can do more than little.

19-SkyWatcher EvoStar 120mm f/7.5 ED doublet apo confusion

This attractively priced refractor is quoted on the SkyWatcher US site - and probably from that source by some vendors as well - as being using Ohara FPL53 and Schott BK7 glasses (some vendors also hint at it having more than one ED element, and at least one quotes "ED BK7", but that's belonging to the marketing creativity domain). The problem is, these two glasses wouldn't produce acceptable performance level at this aperture size and focal ratio. They don't have large enough Abbe number differential, hence require very strongly curved inner radii for this focal ratio, resulting in unacceptably large higher-order spherical aberration residual. As shown below (top), their design limit in the optimized wavelength is slightly below 0.80 Strehl, and the polychromatic Strehl is yet lower (this is in the doublet with a positive element in front; with the reversed order, the required radii are even stronger). The RMS wavefront error implies 0.83 Strehl (with most wavefront deformation forms), but probably due to the high slope wavefront areas along the edge "wrinkle", showing on the 3-D wavefront map, the value is somewhat lower even at this moderate aberration level.

Assuming that the positive element is FPL53, and that its mating element is a Schott crown, the best - and practically only one - mating glass is N-ZK7 (it is also possible to mistype ZK7 to BK7). With it, the limit in optimized wavelength goes up to 0.977, and the overall chromatic correction is significantly better (bottom). In fact, it passes the "true apo" requirement setting the maximum allowed error in five widely separated wavelengths (less than λ/4 in F and C, less than λ/2 in the violet g and deep red r, with well corrected central wavelength; since originally it refers to primary spherical aberration, and today's objectives commonly have a mixture of primary and secondary spherical, with a significantly lower RMS to P-V wavefront error ratio, it is better to use the appropriate Strehl values: better than 0.80 for F/C and better than 0.40 for g/r - this objective has Strehl of 0.86, 0.84, 0.42 and 0.81 for F, C, g and r lines, respectively). OSLO qoutes ZK7 price at 1.90 times the BK7 price, making it viable in that respect. ZK7 has somewhat less favorable violet dispersion than BK7, but due to the presence of relatively significant secondary spherical aberration, which is minimized by a similar amount of primary spherical, in effect bending the LA plots to the left toward high zones, the error is reduced.

20-TS-Optics 6" f/5.9 RFT

Among many telescopes on the Telescope-Express site is this rich-field refractor. It is advertised as having better correction than an ordinary achromat, to the extent that it claims chromatic aberration "barely visible" even on bright objects. Yet the glasses specified - K9 and F4 - are a plain crown-flint combination no different than those of ordinary achromats (CDGM H-K9 and H-F4 are near-equivalents of the Schott BK7/F2). The LA (longitudinal aberration) graph shown may be giving a hint to what is behind this puzzling claim (below, top left; the 0.633 micron wavelength is omitted latter as unneeded for the analysis).

It shows that the optimized line is d-line (0.588 micron wavelength), although centered at its paraxial focus, which in the presence of spherical aberration never coincides with the best focus. Using these same glasses produces practically identical LA graph (the original only has a wider scale), either centered on the d-line paraxial focus (left) or its best focus (right). As a consequence of optimizing d-line, e-line (0.546 micron) is sub-optimized, with over 1/3 wave P-V wavefront error at the best d-line focus, and 1/5 wave at the best e-line focus. However, taking for the optimized wavelength d-line, which is closer to F- and C-line than e-line, nominally decreases the F/C error with respect to the optimized wavelength.

The OPD (optical path difference) graphs show that the P-V wavefront error of the averaged F and C lines (dashed line, top plot) is smaller than in the lens objective optimized for e-line (bottom plot; F and C lines are balanced here, as it in general should be, with the corresponding LA plot shown rightmost, in a box). The "gain" is, however, only superficial, since the error vs. yellow-green e-line, to which the day-light adapted eye is most sensitive (0.98 sensitivity, vs. 0.78 for d-line), is unchanged. The overall correction is actually worse, because of the sub-optimal e-line. The question which focus will eye prefer - the one with the lowest P-V error, or the one with the highest sensitivity - is easy to answer considering that the green light with 0.78 sensitivity sports 2/3 wave P-V wavefront error vs. e-line focus, and nearly 1/2 wave vs. d-line focus. There won't be preferred, or best focus, more so considering that in the more appropriate to night-time observing, mesopic mode, peak sensitivity shifts toward shorter green wavelengths.

This e-to-d optimized line shift has a very little (positive) effect on the violet end, and the only way to hide this widely defocused light is to apply a special coating - such as lanthanum doped coatings - selectively absorbing that portion of the spectrum. The fact that the objective vas designed to have correction biased to the red may be due to the blue also being partly absorbed, in addition to mainly absorbed violet. This would improve contrast with this type of lens objective, but at a price of compromised color fidelity. It could still be inferior to an ordinary achromat in overall correction, but it can't be decided without knowing specific absorption bands and rates.

The combined ray spot plot (bottom right) is for the best d-line focus, as well as the Strehl ratios shown (at the best e-line focus, the e-line Strehl is 0.87). The F line (blue) has better Strehl despite almost three times larger blur than the red C line, because the former is mainly defocused spherical aberration (somewhat beyond paraxial focus ad the location of best d-line focus), and and the latter is predominantly defocus, with defocus having the same P-V magnitude as primary spherical with eight times larger longitudinal aberration (nothing that Strehl ratio at this aberration magnitude level is not a reliable indicator of wavefront quality). This optimum line correction hoopla turns costly when it comes to (photopic) polychromatic Strehl: it is only 0.532 at its best diffraction focus, no better than a 100mm f/4 achromat. A 150mm f/5.9 achromat would have it 0.57.

21-Long Perng 90mm f/5.5 ED doublet refractor

Heart of this very compact and attratively priced refractor is, according to Long Perng, doublet consisting of Ohara FPL51 ED glass and NBM51 short flint (the closest Schott short flint is KZFSN4). Below is what this glass combination produces when raytraced.

Due to the short flint being higher on the RPD (relative partial dispersion) diagram, some secondary spectrum is present: the common best F (0.486μ) and C (0.656μ) focus is less than 0.1mm from the best e-line focus. Common to short flints in general (and similar to lanthanum glasses) the violet is running away. The P-V wavefront error in F and C is about 3/4 wave of defocus, which puts it at the level of a 100mm f/16 achromat (f/14.4 for 90mm aperture). The violet g-line (0.436 micron) is at the level of a 100mm f/14.5 achromat, and deep red r-line (0.707 microns) a 100mm f/19 achromat. In all, it is similar to a 100mm f/16 achromat with respect to the nominal color error - bearing in mind that the aberration forms are different: defocused spherical vs. defocus - contrary to the statement at a vendor's site that it has chromatic correction comparable to f/7 achromat (no aperture specified, but according to raytrace the corresponding aperture would've been 44mm). Due to the large Abbe# differential, limit to the correction in the optimized e-line is very high: over 0.99 Strehl. Best image surface is of 200mm radius, concave toward objective (this means that the outer field would be in a flat-field eyepiece farther from its front focal plane, resulting in converging existing pencils, harder to accommodate to). On flat field, defocused ray spot plot at 1° off axis is about 3x5 Airy discs in e-line (0.022x0.038mm; it has vertically elongated shape due to the presence of astigmatism).

This is defult secondary spectrum correction, with the errors in F and C nearly equal (considering that eye sensitivity is significantly higher to the blue line even in the daylight mode - 0.18 vs. 0.075 - practically optimal correction would have blue bias, but equaling F and C dates back from the time when the specific sensitivities haven't been measured). Error in the blue/violet can be reduced at the expense of increased error in the red. For instance, slightly weakening the inner radii (-187.3/-188.05) produces correction with 0.38 and 2.9 wave P-V in F and g-line, respectively, and 1.09 and 1.59 waves in the C and r-line (another way to control violet is to apply lanthanum doped coating for selective absorption). This may actually have been attempted, at least to some extent and at some time in the past (not necessarily properly), because some users reported excessive red end error in their units.

There is anoher version of this refractor marked with SD, instead of ED. The site that states the ED unit's correction is at the level of a f/7 achromat also states that the SD version uses the same ED glass. Since in the commonly used terminology ED and SD (extra-dispersion vs. super-dispersion) imply glasses with different dispersion properties - and price of the SD unit is 80% higher, with the only mechanical difference being in focuser type - they probably erred again. Assuming that the SD glass is Ohara FPL53, a combination with the same mating glass would produce near negligible improvement in the chromatic correction: 8% lower averaged F/C error, and 20% lower g-line error (but the optimized line design limit goes down to 0.976 Strehl). Better chromatic correction is possible with some other mating glasses. For instance, using Ohara's LAL59 short flint (near equal to Schott KZFN2) produces 0.39 wave P-V averaged F/C error, and 1.7 waves in the violet g-line. It is comparable to a 100mm f/30 achromat in the F/C, and f/33 in the g-line correction. Graph on FIG. 73 indicates as corresponding polychromatic Strehl 0.93-0.94, quite close to the "sensibly perfect" >0.95. It is significantly better than ~0.86 poly Strehl for the ED version. However, OSLO is a bit less optimistic. It gives 0.85 poly Strehl for the ED version (at 0.16mm from the best e-line focus toward F/C focus), and 0.90 for the SD (0.1mm off best e-line focus). Part of the difference comes from the long-focus achromats having practically design limit of 1 for the optimized wavelength; that would bring the two respective poly Strehl values to 0.855 and 0.922, respectively. Most of the rest of difference comes from defocus having about 3% lower RMS wavefront error than spherical aberration for given P-V error. This shows that fast systems like these two, with spherical aberration as the dominant error, are not fully comparable to long focus achromats based on the P-V wavefront error values. But the numbers obtained that way remain reasonably close.

22-Synta 180mm f/15 Maksutov-Cassegrain

This telescope comes in a few different brands - Celestron, SkyWatcher, Orion - but optically there is little or no difference between them. Synta hasn't published prescription for this system, but narrow constraints for this (Gregory) variety allow to come quite close to what the real system is - so close that there is no appreciable difference in the raytracing output. O.T.A. dimensions indicate ~f/2.7 primary which, with the back focus of about 0.6 times the mirror separation and known system focal ratio is all that's needed to reconstruct the system. The primary has to be aspherized, because it is impossible to have f/15 system at the given back focus: with spherical primary it is over f/17, and reducing it to f/15 by pulling the secondary a few mm farther away would shorten the back focus by nearly 100mm. Also, design limit in the optimized wavelength would have been little better than 0.05 wave RMS, which means that fabrication would require meeting very tight tolerances even for achievung λ/4 (0.0745 wave RMS) level. In order to have f/15 system with the given back focal length, the primary needs to be aspherised to about -0.25 conic (prolate ellipsoid). This also improves correction limit in the optimized wavelength, and cuts still very low coma in half (turning it from negative to positive, i.e. with the tail toward field edge).

Central obstruction by the secondary spot is 22% linear, but the actual one is determined by the baffle tube around secondary, and should be about 27%. Design limit in the optimized wavelength is less than 1/10 wave or, more accurately, 1/44 wave RMS, corresponding to less than 1/13 wave P-V of primary spherical aberration (the P-V error is superficially enlarged in presence of central obstruction, since it is measured from the reference sphere vertex, where there is no actual wavefront; also, secondary spherical has higher P-V to RMS error ratio than primary). Half an inch off axis point image is still "diffraction limited" (0.80 Strehl), which puts it at the level of f/10.3 paraboloid. Colors are so tightly wrapped up that the difference in Strehl ratio between optimized wavelength and 0.43-0.67μ spectral range is practically non-existent: 0.973. Taking f/2.6 primary results in the back focal length somewhat longer than 0.6 times the mirror separation (0.62) at f/15, with slightly more coma (also, with 0.965 Strehl in optimized wavelength, and about as high polychromatic Strehl). With small changes in the primary speed and/or conic and corrector thickness, the system can be entirely coma-free at the given back focus length.

23-APM/LZOS 130mm f/6 "Superapo"

APM/LZOS refractors use Russian glasses and fabrication. They are generaly based on the designs by Thomas Beck, but some may not be, and it is not known whether there were changes in the original designs. It is also unknown which glasses - except OK4 (for "osobiy kron", i.e. "special crown"), which is the only super-dispersion (SD) glass made by LZOS - are being used. What can be found online is that the matching glass is OK1 (UK vendor), or OF1 (for "special flint", customer review). The former is similar to the Shott N-PK51, with too small Abbe differential to produce viable output, and the later is the equivalent of the Schott KZFN2 short flint (obsolete), with more than half a wave P-V in F and C, and relatively poor overall chromatic correction. The two LZOS glasses that do fit as the possible matches are K8 (Schott BK7 near-equivalent) and BK8 (near-equivalent to the Schott N-PSK3). Their outputs with OK4 are shown below. Chromatic correction is modeled so that F and C line are nearly balanced in their error magnitude, with the violet and deep red determined by this requirement. It is considered generally optimal for visual use, although in low-light conditions typical for night time observing eye sensitivity in the blue-violet is significantly higher than in the red. This implies that the optimal mode here is toward photographic correction mode, in which correction of farther off wavelengths - particularly violet - is improved at a price of somewhat worsened F/C or even central line correction.

They are very similar, with the blue F and red C lines bordering the "true apo" requirement of no more than 1/4 wave P-V, i.e. 0.075 wave RMS error. So are their photopic polychromatic Strehl (9 wavelengths 430-670nm) values, only slightly higher for K8 (0.928 vs. 0.926 at the diffraction best focus, slightly defocused from the best e-line focus), with both remaining just below 0.93, hence falling short of the "sensibly perfect" (photopic sensitivity) 0.95. However, K8 has significantly lower error in violet: 0.61/0.182 vs. 0.96/0.264 P-V/RMS error, respectively. Note that the Strehl values for the violet g-line appear to be incorrect, not corresponding to the wavefront error values. It is a consequence of the Strehl - i.e. central diffraction intensity - becoming unreliable metric at large wavefront errors. Despite its lower Strehl, the K8 system encircles 80% of g-line energy in 0.0110mm radius circle, vs. 0.0124mm with BK8. Considering that K8 is also little cheaper, and of more consistent, high quality, it is very likely the mating glass in this case.

Bear in mind that the above is design limit, i.e. output with all surfaces and spacings are perfect. Actual units are always more or less imperfect, with a lower overall correction level. To illustrate how sensitive are these lens objectives, raytrace below shows the effect of very minor imperfections in one of the inner radii (R5) of the K8/OK4/K8 lens, lowering the central line Strehl to little above 0.95.

Surface radius error of +0.023% induces undercorrection lowering the e-line Strehl to 0.956, and the poly Strehl to 0.902 (left). That's despite the violet g-line improving to just over 1/2 wave P-V (photopic eye sensitivity to it is only 0.012, or over 80 times lower than for the central line) and the corresponding Strehl seems to be close to its correct value this time. Surface error of -0.032% induces overcorrection lowering the central-line Strehl to 0.955, and the poly-Strehl to 0.894 (right). The blue F-line falls down to 0.59 Strehl, and the violet g-line error increases to 0.73 wave P-V (the corresponding Strehl value given by OSLO is again lower than it should be), while the error on the red end is significantly decreased. This shows why two different units of the same model can have noticeably different mode of chromatic error, i.e. that they can significantly differ in the magnitude of error for specific segments of the visual spectrum, even when their central line correction is practically identical.

24-TS Photoline 96mm f/6 triplet apochromat

Telescope Express describes this objective as "color clean", which is more accurate than the more common "color-free" (no refracting objective - and we could say no objective - is color-free). While no mating glass to its FCD100 core element is specified, the very likely choice is BSC7 (Hoya's equivalent of the Schott BK7 or Ohara BSL7 borosilicate crown). Simply, there is no cheaper and better quality glass available and, as shown below, it provides performance level fitting the description (note that the specifications are not those of the actual model, but the color correction of this glass combination cannot be significantly different).

At its design limit, the objective is comfortably within the "true apo" requirements, better than "diffraction limited" (0.80 Strehl) in F and C, and with less than 1/2 wave P-V (more accurate measure is 0.15 wave RMS) in g (436nm) and r (706nm) lines. Its photopic polychromatic Strehl is also above the "sensibly perfect" 0.95. Blur distortion due to field curvature is obvious at 1°, hence it would require flattener for wider CCD fields. Visually, astigmatic blur at 1° off is roughly twice the Airy disc diameter, or nearly 6 arc seconds times telescope magnification. To the average eye, it will start appearing non-stellar at about 3 arc minutes apparent size, or 30x magnification. That would be the very edge of the standard 50-degree 20mm eyepiece.

The inward curving best image field would require accommodation which would depend on eyepiece astigmatism, field curvature and focal length. With 20mm Nagler Plossl (patented design), which is a near flat-field design with about 1 wave P-V of astigmatism at the field edge, required accommodation is little over -1 diopter, which should be easy enough for nearly everyone (comparable to accommodating from infinity to little less than 1m; negative accommodation results from the edge point being farther from the field stop plane than the center point, thus having its pencil slightly converging after passing through eyepiece, requiring unnatural beyond-infinity relaxation of the eye lens). With the 10mm Plossl, required accommodation would be four times as much, but that portion of the field wouldn't be within the field stop; the actual field edge of this eyepiece would require nearly identical accommodtion as that of the 20mm unit. However, with wide-field eyepieces accommodation would be generally more demanding.

25-140mm f/6.5 Super Triplet Apo

This refractor comes as iStar, SharpStar (which is also manufacturing it for the TS-Optics line) and possibly some other brends. What earned it the "Super Triplet" label is its nowadays fairly rare PNP (positive-negative-positive) configuration with two ED (some people call them SD) elements. The ED glass is specified as Hoya FCD100 super-low dispersion glass, but the mating glass is not revealed. A customer states that it is a lanthanoid, but without specifying source for that information. A look at the RPD diagram shows that there is a relatively few glasses that would make a good match: Schott BK7 and equivalents, and a few lanthanum glasses that are close to its RPD value (Schott N-ZK7 is very similar in the output to BK7, a bit sharper in the optimized wavelength, but since BK7 is cheaper and more reliable quality-wise, it is the more likely choice). The triplet cold be executed with equal pairs of radii, i.e. suitable for oiled interspaces, or with a single inner radius changed in order to balance spherical aberration. Turned out, the difference in output between the two is not negligible (image below).

Objective suitable for oiling (top) passes the "true apo" minimum requirement with respect to the wavefront error in non-optimized wavelengths. However, mainly due to the relatively low central line Strehl, it falls short of the "sensibly perfect" 0.95 polychromatic Strehl (photopic, nine wavelengths from 430nm to 670nm). If its central line Strehl was, for instance, 0.99, the corresponding polychromatic Strehl would've been higher by a factor 0.99/0.97, i.e. 0.957 (note that the polychromatic Strehl is the highest at a point defocused by -0.005mm from the best central line focus).

Objective below it, with three equal inner radii could be easier to fabricate. It has better color correction, but lower central line Strehl, resulting in a practically identical polychromatic Strehl. However, in night-time conditions, when vision shifts toward mesopic mode, with higher sensitivity to the blue and red, it would have an edge due to its better color correction. It may be a bit puzzling that an objective with the all color wavefronts comfortably within the "tru-apo" minimum requirements falls short of 0.95 Strehl, but it due to the fact that polychromatic Streh expresses overall correction level, not only chromatic correction. If the central line here would have 0.99 Strehl, polychromatic Strehl would increase to 0.965.

Lanthanum glass is rarely used for triplets, because its greatest advantage - high Abbe# differential vs. ED glass, allowing for fast f-ratios - is dwarfed due to triplet's inherent lower requirements for high Abbe# differential. And other than that, lanthanums in general are not that great as matching glasses: their high-index to (relatively) low dispersion property partly offsets their high Abbe# differential advantage, and they typically have less than great correction in violet - the higher their Abbe# vs. ED glass, the more so. It is illustrated with two LA graphs (bottom), for different lanthanum elements as the mating glass in this triplet. Lanthanums closest to FCD100 on the RPD graph are also those with the lowest RPD differential, i.e. producing secondary-spectrum-free output. However, even if they do have higher Abbe# number differential vs. FCD100 than BSC7 (which is Hoya's equivalent of the Schott N-BK7), the gain in the central line correction is small. And despite somewhat higher central line Strehl, their polychromatic Strehl is not better than that for the BSC7-element objective (note that the triplet configuration changes going toward higher Abbe# differential lanthanums with the 2nd and 3rd element needed to correct for chromatism taking meniscus form). The LA graphs illustrate the main disadvantage of lanthanum glasses: as Abbe# differential increases from Hoya's M-BACD5N to CDGM H-Lak12, the blue and red slide away from the green, and color defocus - particularly in the violet - becomes larger. The blue-red separation is what ultimately produces secondary spectrum, which is already detectable with H-Lak12. Considering that lanthanums close to FCD100 on the RPD diagram do not offer advantage in overall correction over BS7, the latter is a more likely choice, being cheaper and of highest optical quality.

Does PNP arrangement, with two ED elements necessarily make triplet "super", i.e. better corrected than the single-ED element NPN? Putting the same glasses into a NPN triplet with only a single inner radius changed and identical f-number, gives as an answer "no".

Mainly due to the significantly lower relative change in the radius needed, the higher order spherical residual is significantly lower, practically negligible, allowing for a significantly higher Strehl in the optimized wavelength. As a result, the polychromatic Strehl here is over 0.95, impying "sensibly perfect" correction at the design limit. The PNP arrangement could have as low - or even lower - higher order spherical residual only if employing wider interspace, like Takahashi TOA. Considering the low-bracket price of this refractor, it is unlikely. iStar objective cell does have its vertical adjustment screws somewhat more widely separated for the first two elements, but it is not more than a few mm, probably to accommodate spacing at the first two inner radii, with R3 significantly more strongly curved than R2 (objective cross-section above -2nd from top - has the two surfaces touching at the edge, possibly slighty overlapping). It indicates that the more likely arrangement would be the one with unequal R2/R3. Spacing increase to as much as 11mm results in only marginal improvement in color correction.

However, in this particular glass arrangement, with the most of higher order spherical residual coming from the third surface, even a relatively small gap between R2 and R3 can have it significantly reduced.

The e-line correction is now as good as 0.995 with the polychromatic Strehl exceeding 0.96. The gap is small enough not to impose significantly more stringent requirements with respect to cell fabrication and lens alignment, which should qualify this type of arrangement as the one most likely used for this objective.

26-Takahashi FC100DZ 100mm f/8 fluorite apo doublet

Sometimes described as the "best doublet ever", this refractor is marketed as consisting of a rear fluorite element and a high-index/low-dispersion front element. The latter generally describes lanthanoid glasses. Fluoride is "privileged" vs. other SD (super-low dispersion) glasses in being higher on the RPD (relative partial dispersion) diagram, hance being able to use low Abbe# glasses - for a large Abbe# differential - without introducing secondary spectrum. Lanthanoids in general tend to be more or less loose in the violet; those that perform the best in that respect with fluorite are having refractive index of ~1.697 and ~55.5 dispersion (e-line). They include Schott N-Lak14, Hoya Lac14, CDGM H-Lak12 and Ohara S-Lal14. While not the best - differences are small - it is the Ohara glass that produces longitudinal aberration graph nearly identical to the one published (below).

The LA graph differs from the published one only in having the violet-blue lines bending in somewhat more strongly toward edge (that reduces the aberration), which indicates somewhat smaller Abbe# differential than the actual glass. But the overall performance level is nearly identical. Note that the Takahashi graph may be a drawing of the actual raytrace output; it is shown on a large horizontal scale to leave a visual impression of the very tightly bound together color lines. Correction can be varied by slightly changing lens' positive or negative radii; for instance, violet g-line can have Strehl up to 0.72, with the C-line Strehl down to 0.94+ and the F-line Strehl practically unchanged. But those intricacies are beyond fabrication schedule. The polychromatic Strehl (photopic, 430-670m) is 0.975 at its limit. Needles to say, this particular match does qualify as a "true apo".

27-Agema 130mm f/8 fluorite apo refractor

The Strehl graph across visual spectrum displayed on Agema's web site shows this doublet to be superior to - among others - Zeiss APQ 130 f/7.7 and Takahashi 120mm f/7.5 TSA triplets. The level of correction is "impossible" for a doublet; there are glasses which in combination with fluorite would rate as a "true apo", but not even close to the Strehl above 0.95 from F to C, and beyond. Luckily, the answer to this puzzle comes in a form of patent application by Eduard Trygubov, Agema's owner (US2018/0149832 A1). It describes an ingenious doublet which uses a wide air gap to bring all colors tightly together, and then corrects unacceptably high secondary and tertiary spherical aberration by puting higher-order aspheric on the front surface. The actual design is, according to the published graph, not quite identical to the patent design, but the difference is very small.

The entire visual spectrum is comfortably within 1/5 wave P-V wavefront error, with the e-line Strehl rounding off to 0.9999 and the polychromatic Strehl (photopic, 430-670nm) reaching 0.987. The A4 term (AD in OSLO notation) is for lower-order spherical aberration which easily can be handled with small changes in one of the inner radii, but in this case it is of opposite sign to the other two terms, which means it decreases required depth of the aspheric (also, it is just another form of expressing conic, with the one needed to replace A4 being 0.31 - oblate ellipsoid). Since these are xⁿ curves, with n positive integer, the maximum depth is at the edge, given here by a product A_n65ⁿ; without the A4 term, A6 and A8 combined (AE and AF, respectively) would come to 0.00283mm, and with it the maximum depth is 0.002mm (absolute values), or nearly 4 waves.

This, however, is not the least amount of aspherization possible. Analogous to the Schmidt corrector, these xⁿ terms bring together paraxial foci of different wavelengths, where the error for non-optimized wavelengths is four times larger than at the best focus, which is for primary spherical the 0.707 zone focus. In order to bring color foci together at that zone, radius term is introduced which effectively induces needed defocus by altering the profile shape. In this case, that would amount to setting the A4 value to be equal to the sum of A6 and A8, which would result in zero aspherization at the edge. As illustrated below, the amount of glass to be removed is significantly smaller, not only due to the smaller maximum depth, but also due to it being at the inner zone.

Aspherization A is the one used in patent application, and B is the one that minimizes glass removal. Note that the curves show grossly exaggerated profiles, but the relative depth difference is, at least roughly, similar to the actual. Aspherization A is not as smooth as it appears, as it shows its form when applied to a concave surface (smaller graph below). Bottom graphs there is no difference in correction level between the two.

Similar reduction could be achieved with A6 and A8 alone. Since they are negative in sign, the curve they indicate is to the left of the actual (spherical) surface, i.e. glass should be grown on the sphere of that radius in order to form required asphere. That is easily done in raytracing, but the actual surface would require a slightly weaker sphere as the starting point. The best option is, again, zero aspherization at the edge. The sphere that fits that requirement has a radius of 538.77mm, and the required maximumm depth larger than in the previous case, but still requires significantly less of glass removal than the patent design (below). Note that the sphere required to match the aspheric turns concave due to enormously strong curves (f/0.25 base sphere), but the profile of aspheric vs. sphere deviation is similar to the actual deviation).

Performance remains practically unchanged, with the poly-Strehl rounding off to 0.99. Prescription shown uses the original radius - which means that the raytracing surface is suspended above the sphere - but there would be no difference if R1 was 538.77mm as required for actually fabricating the asphere. Of course, in order for the Strehl to be actually high as shown, all surfaces, centers and separations need to be extremely close to those of the design. If, for instance, the central wavelength measures "only" up to 0.96 Strehl (corresponding to less than 1/9 wave P-V of primary spherical aberration), one end of the spectrum would worsen, the other improve, but the poly-Strehl would be reduced by a 0.96 factor.

It is interesting that these two glasses would easily produce a "true apo" if aspherized as a contact doublet. But design limit in the violet g-line (436nm) would have been only around 0.45 Strehl, and about 0.85 in the red r-line (707nm). Since F and C would have it ~0.95 and ~0.93, respectively, the whole trouble of using wide gap was endured only to bring the two spectral ends - primarily the violet - above 0.9 Strehl as well. It brought its polychromatic Strehl (photopic) - as design limit - to 0.99, practically at the level of the special wide-gap(s) triplets like Takahashi TOA. A testament that only extrordinary drive can produce extraordinary achievement.

◄ 14.3. Observatory telescopes

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