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12.3. Eyepiece aberrations II   ▐    13. THE EYE

12.4. telescope eyepieceS: COMPARATIVE RAYTRACING 

In order for any two eyepieces to be comparable with respect to their ray trace output, they have to be of the same focal length, and used at the same focal ratio. In general, the larger focal length at a given focal ratio, the larger geometric blur; however, since magnification drops in the same proportion, its angular size, as well as the corresponding diffraction image remains nearly unchanged (this is valid for relatively large aberrations, when the ray spot is multiple times larger than Airy disc). Likewise, the larger relative aperture, exponentially larger the aberration: primary spherical aberration with the 4th, primary coma with the 3rd, and primary astigmatism with the 2nd power of the change (that is because the relative aperture effectively determines width of the cone at the field lens, i.e. eyepiece "aperture").

Eyepieces shown below are all 10mm f.l. at f/5; they are also comparable in size, being approximately given in 1:1 scale (FIG. 216). It expands on FIG. 213 with respect to the number and types of eyepieces shown, also making the eyepieces comparable in size. Due to that latter requirement, the simplest eyepiece types are shown smaller, and some others bigger than the optimum scale. Likewise, the Airy disc and 5 arc-minute circle are quite small, so that the large aberrated spots of poorly corrected eyepieces, which are much larger than those in highly corrected eyepieces, can fit in.

Selection is far from all-inclusive, but attempts to follow the evolutionary path of telescope eyepiece, from its simplest, narrow-field forms, to the complex, highly corrected ultrawide-field examples of our day. Of course, it is also a subject of data availability. While ray trace output for any given design can vary in the level of specific aberrations - common trade offs are less astigmatism for more field curvature and/or coma, or the other way around - most of the examples shown are within their typical, or usual mode or according to a specific prescription (e.g. patented eyepieces).

Ray spot plots are given for the axial, midfield and near-edge point (except for the Huygenian and Ramsden, which are shown for their historical role), in the optimized wavelength (e-line), as well as the combined e/F/C spot. Given are the axial Strehl in the e-line, distortion % for field edge, as well as the approximate maximum eye accommodation required with respect to the central field point (for 10mm f.l. every diopter of accommodation approximates 0.1mm defocus relative to the vertical plane containing central point; xD of accommodation approximates one required to refocus from infinity to a 60/x times 17mm distant object). Eye relief, measured from the eye lens, is given for the midfield and near-edge point (omitted where negligible); it indicates most of the so called exit pupil spherical aberration, since the effect becomes generally negligible for field angles below 20 degrees.

FIGURE 216: 1 - The Huygenian is pretty much useless at f/5 due to spherical aberration (0.24sqrt[-logS] gives 0.24 waves RMS, corresponding to 0.8 wave P-V, which means it requires f/7 do drop below quarter wave p-v), but the spots reveal good color correction, and low off axis aberrations at 15 degrees; field is limited mainly by its strong Petzval curvature. Lens separation can be reduced for longer eye relief w/o significantly altering performance level
2 - Ramsden has significantly better monochromatic axial correction and mild field curvature, but more of off axis aberrations. Reducing lens separation vs. classical prescription increases eye relief, while reducing off axis aberrations, except lateral color.
3 - Kellner has insufficient axial correction at f/5, with field limited by astigmatism.
4 - Monocentric - shown is the modern form of this eyepiece, but performance level is similar to the original - maintains excellent correction even at f/5, but field is limited by secondary astigmatism, exploding as field approaches 15 degrees; since it has a significant higher order aberration component, as well as primary coma, its aberration load diminishes more quickly with slower focal ratios than in eyepiece types limited mainly by primary astigmatism.
5 - Koenig has good overall correction, with the field limited by astigmatism
6 - Plossl - shown is Nagler's 1985 patent - is comparable to Koenig (somewhat more of astigmatism is mainly the consequence of the flatter field)
7 - Symmetrical is similar to Plossl, which may tend to have marginally better correction, just as the Koenig may have with respect to Plossl
8 - Classical orthoscopic eyepiece output is very similar to Plossl's. With comparably thick lenses, it has somewhat longer eye relief.
9 - The "highly orthoscopic", or H-Ortho has somewhat different triplet configuration, resulting in a highly corrected field, similar to that of the monocentric eyepiece, only significantly wider. In part, this exceptional of field astigmatism correction is a trade off for the significantly curved best image surface. However, while the standard orthoscopic would also have astigmatism reduced with more strongly curved best field, it doesn't seem it could achieve the same level of correction.
10 -
Simple 4-element Bertele eyepiece from 1925. is an interesting design, in that with some change in the radii values it can produce astigmatism which at a certain point reverses from expanding to contracting. So the ray spot plot is smaller at 30 than 20 degrees off axis and, with larger lenses, would come to near-zero point closer to 35 degrees off (in this particular case). If modified this way, its astigmatism 30 degrees off axis is 3-4 times smaller than with the original design. Its drawback is relatively short eye relief (astigmatism in the more complex Bertele designs, with 80 and 90 degree AFOV, falls between the R/V and Zeiss Erfle for comparable field angles but, similarly to the original Erfle, suffer from significant lateral color error). 
11 - Van Hofe is a well corrected old-type widefield with a nearly flat field and relatively low distortion.
12 - Rutten and Venrooij's, or R/V Erfle has similar configuration to the original patent from 1921. but with reduced astigmatism (20-25%), field curvature and (originally unacceptable) lateral color.
13 - S/C/B Erfle (Smith/Ceragioli/Berry) is a modern take on this configuration, with very much reduced astigmatism, but somewhat reduced eye relief and relatively large distortion.
14,15 - Despite being intended for sub-50° field, Zeiss Astroplan - another Erfle variation - is, within its field limits imposed by the lens size, better corrected for astigmatism than the more complex Zeiss Erfle from 1959.
16,17 - Scidmore 1965 6-element design is better than the original Erfle, but not from Van Hofe, or Zeiss Erfle. Another Scidmore design, superficially resembling one with a Smyth lens (it is a near zero-power lens) may have been an attempt to reduce lateral color error with still limited choice of glasses.
18, 19 - Nagler's 1985 Widefield further, and significantly reduces astigmatism in 2+1+1+2 configuration, with impeccable color correction (Meade's 4000 Series Super Wide was somewhat rearranged design of this type, with one of the two singlets used as eye lens, i.e. in 2+1+2+1 configuration; performance level remains comparable). The next, probably near-ultimate improvement for this configuration came with the Nagler Panoptic, whose level of correction should be well represented with the S/C/B "New WF". The coma-like edge field aberration is actually predominantly trefoil, as shows on the diffraction image. It is the first widefield eyepiece that came meaningfully close to having preserved Airy pattern near the field edge (see #21).
20 - Koehler 110° eyepiece from 1959 is often seen as one of a kind with respect to the field size at the time, but there were similar attempts before (for instance, Slyusarev 100° eyepiece from 1947). Its significance is in being the first example of the new, ultra-widefield breed of eyepieces using the Smyth lens. Due to such configuration, Koehler was able to reduce astigmatism somewhat within the standard 65-70° widefield, but more so - roughly, nearly twice (compared to the Slyusarev) - at the 100+ degree mark. To understand this eyepiece, one needs to know that it was designed as an integral part of 15x75 binoculars, with
f/5.2 objective lenses. At these field sizes, these lenses generate significant amount of astigmatism, of opposite sign to that of the eyepiece, which significantly reduces astigmatism in the final image. Possibly for the same reason, its axial correction at f/5 (atypical overcorrection) leaves something to be desired, as it does its axial chromatic correction, it has short eye relief, and a massive astigmatism/field curvature when on its own.
21 - Ray spot plots speak clearly why the 1981 Nagler's "ultrawide" (type 1) marks the beginning of new era, in which highly corrected eyepieces with 80+ degree AFOV become the standard against which all others are measured. It has unprecedented field correction, with the astigmatism reduced to a small fraction of that in other ultrawide-field eyepieces.
22,23 - The 1988 Nagler (type 2) reaches the level of astigmatism correction beyond which further improvement becomes of little practical importance. As the diffraction image for the Nagler's 1988(a) patent prescription shows 40° off axis, astigmatism is so low that the Airy pattern is easily recognizable, with the most obvious aberration being lateral color error (F and C lines are given increased weight, 0.35 and 0.25 respectively, to show more clearly the color shift). Correction-wise, further improvement is limited toward making the residual lateral color negligible; also, its axial chromatism is larger than in most other eyepiece types, which is common to all patented Nagler's ultrawides (for F and C lines, it is in the 0.6-0.8 Strehl range, which compares to, say, 0.85-0.95 Strehl for the top corrected designs, like Dilworth's). Other possible improvements, such is reduction in size, are not related to aberrations.
24 - Patent application for Dilworth 90° eyepiece comes before the Nagler type 2, has wider field and even better astigmatism correction, but for some reason had never made it onto the marketplace.
25 - Shown configuration appeared on the Telescope-Express site as an illustration of the Speers-Waller eyepiece, thus this represents merely an attempt to see if such configuration can work. It can. It has obvious similarity to the Nagler 1988(a), only with the singlet and meniscus switching places, with the latter turned around to keep the same orientation vs. diverging ray pencils. In fact, just making the initial design starting with the Nagler and changing its configuration, roughly trying to preserve each element's power, resulted in eyepiece design close to the optimization level. Overall correction achievable is very good, with the only aberration of some significance being field curvature, at little over 4 diopters of accommodation - i.e. infinity to about 35cm - required between field center an edge. It nearly certainly can be reduced w/o significant compromises; appears it could be easier with somewhat wider Smyth lens separation. Similar configuration was used for Meade's 4000 Series Ultra Wide, difference being that both doublets had a positive meniscus shape, convex side facing the Smyth lens.

26 - Nagler-type eyepieces can come with different number of elements and configurations; one example given as "modern Nagler" by Smith/Ceragioli/Berry has correction comparable to the type 2 Nagler, in a somewhat longer train.
27 - Further expansion in the field size led to breaking into the 100°-120° AFOV range. One example is Ethos-like 100° eyepiece given by Smith/Ceragioli/Berry, in configuration nearly identical to the Nagler Ethos. The only residual aberration worth mentioning is lateral color, comparable in magnitude to that in the Dilworth's eyepiece. The configuration can be made more compact by reducing the two widest interspaces, with the lateral color also significantly reduced. This type of configuration, more or less modified, is likely used in some other brands (online source¹ supplies evidence that 13mm Ethos and 14mm 100° ES are essentially identical configuration).
28 - Fukumoto 100° eyepiece (the 1st of 5 examples in the patent application) is a basis of the
Nikon NAV-HW ultrawide (7/10 arrangement). There is obvious similarity in the general configuration with the Ethos, but it is also a design with its own characteristics, such as more complex Smyth lens, and near non-existent spherical aberration of exit pupil.
29, 30 - Another Fukumoto-Nikon design from the same time frame is 82° AFOV design with a very complex Smyth lens and relatively simple positive counterpart. Five of the six examples in the patent application do not feature the front two singlets; most have similar edge performance as the one shown here, but somewhat more astigmatism in the inner field. Reduction in the overall size is very obvious. Subsequent 2015 Fukumoto-Nikon patent expands the field of this eyepiece type to 100°. Patented version shown (the preferred one, out of five) shows very good correction up to 45°, or so, and the astigmatism flare up toward field edge doesn't show on the astigmatism graph in the patent application. Since a small change in one of the radii (R8, box to the right) nearly brings it to where it should be, there is probably a typo in the patent prescription.

31 - Design based on the published configuration for APM "Ultraflat" 84° eyepiece (originally 12.5mm f.l.) with no other data. It also turned out to be a viable configuration, with a very good correction over the entire, flat field, and long eye relief.
32 - Ageev-Parko 100° eyepiece (Shvabe, Moscow) is one of the latest newcomers to this arena. Its patent application only describes configuration and glasses in general terms, so the design shown above only illustrates configuration and performance, using Schott glasses (original glasses are LZOS). The astigmatism plot supplied with the patent application implies the highest level of correction (noting that astigmatism plot says nothing about coma or other aberrations possible in this kind of eyepiece, such is trefoil). Its major advantage is small size, nearly half the length of the "small" 100° Fukumoto (noting that longer eye relief claimed in the patent would require larger lenses); eye relief in the patent description is nearly twice longer.
33 - TAL widefield is a recent modification of the Rusinov design from the 1970's by Klevtsov. Its field is not as wide, and its correction level is not as high as in the modern Smyth-type designs, but it is significantly better than in the conventional widefields.


12.3. Eyepiece aberrations II   ▐    13. THE EYE

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