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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. The coma residual could have been left in on purpose, to compensate for the positive coma in some TAL telescopes.

¹ http://www.svenwienstein.de/HTML/es_14mm_ethos_13mm_English.html
SPECS.


Individual eyepieces

Due to the large number of eyepiece designes shown, image above could not go into more detailed account of each one. That will be attempted below, with the eyepieces organized in four different groups: (1) narrow-field, 30°-40°, (2) standard field, 40°-50°, (3) standard widefields, 55°-65°, and (4) Smyth-lens widefields (the field criteria is based on the usual practice; the possible field size is somewhat flexible category). In general, the same order will be followed, with some interesting designs added to those already presented. Also, some less interesting designes will be omitted. As before, all units are of 10mm focal length, but in addition to f/5, ray spot plots will be shown for f/10 and f/15 systems. Spectrum is also widened, to include the violet g and red r lines. Since the chromatic errors at fast focal ratios are dwarfed by spherical aberration and astigmatism, the f/5 ray spot column shows only green e-line.

25°-40° FIELD EYEPIECES

The first group includes one and two-group lens designs with field diameter up to 40° (in their original designs). The Huygenian consists of two plano-convex lenses facing objective with their convex side. It achieves best overall correction with the focal length of the field lens about twice that of the eye lens, and the lens separation nearly 40% of their focal length sum. At the canonical equal lens separated by half their focal lengths for corrected lateral color, aberrations are too strong, dominated by coma, and including crippling field curvature due to the strong negative astigmatism being of the same sign as the Petzval curvature (on top of that, objectve's image falls into the field lens, and exit pupil forms between the lenses). With the increase of the field lens f.l. relative to the eye lens, both coma and astigmatism diminish, coming to their minimum at ~2:1 ratio. Coma can be cancelled, and weak positive astigmatism begins to countereffect the strong Petzval. Further ratio increase leads into coma turning negative and larger, as well as positive astigmatism, softening image curvature at the price of larger wavefront error. Lens thickness, particularly of the eye lens, is also important aberration factor; also, thinner lenses, in general, as well as a reduction in lens separation, allow for a larger eye relief. All these factors matter, and have to be optimized for the best possible correction. For instance, design posted in "comparative raytracing" has some significant residual negative coma and lateral color. The coma can be reduced to non-significant by reducing the eye lens thicknes from 1.25mm to 0.7mm; eye relief gets shorter by ~0.1mm, and best field curvature goes from -10mm to -11mm. Increasing lens separation from 10mm to 11mm takes care of the lateral color, at a price of eye relief reduced to ~2.2mm. Field curvature, due to the stronger positive astigmatism, goes to -12mm. Reducing front lens thickness from 1.5mm to 1mm extends the exit pupil to 2.9mm at 15° field angle (at 25° it is only 2.4mm). Such optimization leads to a Huygenian with clean ray spot plots, as shown here, even with the spectrum extended on both sides.

The Huygenian is usually limited to ~30° field diameter, not because of point-image aberrations, but by its strong field curvature. Nominal field curvature can be misleading: here, radius of a sphere containing best astigmatic focus for the 15° field radius (on top of the shortest vertical dashed line) is significantly shorter than the one containing best focus at 25°, despite the latter being more than twice more defocused with respect to the central spot. The reason is that what matters is the sagitta of field curvature - i.e. point coordinate determined by its height and the radius - not the radius itself. Thus, it is better to express field curvature in diopters of accommodation. Good approximation for diopter of accommodation is given by D~f2/1000, with f being the eyepiece focal length. For a 10mm f.l. every 0.1mm of defocus requires one diopter of accommodation (in terms of object distance, one diopter represents accommodation from infinity to 1m, and three diopters accommodation from infinity to 0.33m). As the astigmatism graph shows, this 10mm Huygenian requires over -8 diopters accommodation at 25° field, nerly -6 diopters at 20°, and -3.5 diopters at 15°. Even -3.5 diopters, or accommodation from infinity to -29cm can be challenging for some people. The minus sign means that eye has to deal with diverging pencil of light exiting the eyepiece - since the field point is closer to the eyepiece than the center point, forming parallel pencil of light - which is natural to the eye, and much easier than positive accommodation, when eye needs to relax beyond its infinity mode. Axial spot at f/10 shows best focus primary spherical nearly filling up the Airy disc, corresponding to 1/13 wave P-V wavefront error. Since it changes with the 4th power of focal ratio, at f/5 it would be 16 times greater - or over 1 wave P-V - making the eyepiece useless. Huygenian also has a significant negative ("barrel") distortion at 25° field radius, but since it changes with the 3rd power of field angle, it is negligible at its usual field size.

Note that the reverse raytracing image is not the image seen by the eye. Raytracing follows a perfect parallel pencil entering eye lens, converges toward the field lens, and exits it converging further toward objective's image (with most eyepieces; with the Huygenian, objective's image is behind the front lens, so the rays are diverging after the front lens, and raytrace forms imaginary focus by extending them backwards, toward eye lens). Eyepiece image is, in fact, image of the objective which would produce perfectly collimated exiting pencils and, as such, accurately reflects the eyepiece aberrations. Image seen by the eye is formed on the retina from near-collimated pencils entering the eye, with each point projected toward infinity in the direction of the entering pencil.

The Ramsden, similar to the Huygenian in that consists of two planoconvex lenses, but facing each other with their convex side, is plagued with similar problems: at the lens separation equaling lens focal length (assuming two identical lenses, as proposed by the original design) needed for corrected lateral color, objective's image is inside the field lens, exit pupil is on the inner side of the eye lens for all but very central field points, for which is at the lens, and field curvature severely limits usable field (with 10mm f.l. unit, needed accommodation at 10° is over 11 diopters). Reducing lens separation achieves some positive, even still short, eye relief and lessens field curvature, but at a price of introducing lateral color. With a pair of two identical lenses at about 2/3 of the focal length separation, eye relief at 10° off axis is 2.3mm (only 1.3mm at 15° yes, these simple eyepieces have that much of the spherical aberration of exit pupil), and lateral color is acceptable - with acceptable taken as C and F lines still within the Airy disc - up to about 10° field radius. Better performance is achieved with the field lens of somewhat longer f.l. than eye lens, as in the design shown (-5.66/8.08mm radii, 0.66/0.96mm lens thickness, 9.6mm lens separation - 72.5% of one half of the focal lengths sum). Lateral color is acceptable up to nearly 15° field radius. Required accommodation at 15° off axis is -2.5 diopters (as refocusing from infinity to 0.4m) and eye relief for the same field height is 2.4mm. At f/10, and more so at f/15, astigmatism is very low, and distortion is negligible within this field. Best focus ray spot plot for primary spherical aberration at f/5 is about 4 times the Airy disc diameter, indicating 0.3 waves P-V wavefront error.

Note that this design differs from the one in "comparative raytracing", despite both having a longer f.l. field lens and identical f.l. sum. Even with the other design having smaller lens separation, its eye relief at 15° is only 0.2mm, due to the weaker eye lens and relatively stronger field lens. In other words, the stronger eye lens in this design spreads the converging cones wider, and they are bent down less with the weaker field lens, so that the aperture stop - i.e. exit pupil - has to move farther out in order for the final cone direction on the other end of the eyepiece to become slightly converging toward optical axis.

The Kellner, also known as achromatized Ramsden, in its basic form keeps the same configuration, but the eye lens is now a cemented achromat. Although the additional, cemented surface, allows for better correction of lateral color, it wasn't utilized in its early versions. This is illustrated by the design given by Rutten and Venrooij, who did modify those old designs only to the point where they still retain the basic similarity (very similar in that respect is design given in the "Handbook of Optical Sytems" by Herbert Gross, as well as some other quality sources). In adition to the significant lateral color and short eye relief, the early Kellners had much more of the positive astigmatism, enough to overpower the strong negative Petzval so much that the resulting best image field had a positive curvature - i.e. curving toward the objective - almost as strong as the negative curvature of the Ramsden (as mentioned before, this form of curvature, requiring accommodation to the converging beams, is much harder on the eye). Smith/Ceragioli/Berry show in their book that better correction within the basic design is possible, including flat field, but something is left to be desired: the objective's image is almost touching the field lens surface, and the eye relief is only ~18% of the focal length (the book puts it at 22%, but it would require a very short focal length objective, binocular-alike); also, lens size limits the field at 20° radius even with f/10 beams. Design shown here under #3 (0/5/-5/10/0mm radii, 0.5/1.9/1.4 lens thicknesses, 8.9mm separation with F2/BK7/BK7 glasses from left to right) has about 50% larger lateral color error, requires little over 2.5 diopters field-edge accommodation, and has about a third more of astigmatism, but offers round spot plots, lens surface safely away from the objective's image, and 3mm of eye relief at 20° off. It also can accomodate field up to 45° in diameter. It could also be made flat-field, with nearly identical level of aberrations to that in the S/C/B, and 4mm eye relief, by weakening the field lens to 12mm, and increasing thickness of the 2nd element of the eye lens to 2.4mm (most of the reduction in positive astigmatism). Small changes to the basic configuration allow yet better correction, as shown with the "modified Kellner" under #4. The lenses are significantly closer, allowing for longer eye relief, astigmatism further reduced, and required accommodation at 20° is less than -2.5 diopters. The astigmatism reduction is achieved by bending the rear eye lens surface, and making the rear surface of the field lens concave took care of the lateral color. Another way of achieving lateral color correction is to make the cemented surface somewhat more strongly curved. Other modification are also possible, and such modified Kellner can have significantly better performance than its old fashioned cousins.

The true Monocentric is eyepiece design dating back to the 19th century, when it was invented by Adolf Steinheil (1883). As its name implies, its surfaces are concentric about a common center of curvature. In the form, it is a cemented triplet with the crown element sendwiched between two flint elements. The original version by Steinheil wasn't symmetrical, and some others, like that by Philip Taylor (US Pat. 2,604,012, 1952) were. Taylor's version had very good correction over ~25° field diameter at f/10, with acceptable edge accommodation of about -2.5 diopters (10mm f.l. unit). The monocentric shown here (5.85/2.75/-2.75/-5.85 radii, 3.1/5.5/3.1mm thicknesses, F2/BK7/F2 from left to right) is symmetrical, with 4.2mm eye relief, exceptional correction over most of the field and somewhat demanding edge accommodation of over -3 diopters required. Its high field correction is based on a delicate balance between lower and higher order astigmatism (generated by the two inner radii), which can hold within a relatively small field. The curvature can be significantly lessened by a small strengthening of the inner radii, but at a price of more astigmatism in the wider edge area. Also, correction can be somewhat improved by making design asymmetrical. This design is only of historical significance, since similar and better performance now can be achieved with less glass (modern "monocentrics").

The modern "monocentric" shown here - same as the one in "comparative raytracing" - is a cemented triplet made of three different glasses. It achieves better correction than the symmetrical true monocentric, and has significantly longer eye relief of 7.2mm. Required edge accommodation is also lower at -2.3 diopters. It should be noted that the 15° ray spot plots for these eyepieces are given for their optimum location, different than that for the 0.7 field spots; by default, best astigmatic image is midway between the sagittal and tangential surface. In general, since their astigmatic field curls back due to the higher-order astigmatism, edge points require less accommodation. When comparing the commercially available Steinheil and Hastings triplets, they have somewhat smaller quality field of about 25° in diameter, exception being the Edmund Scientific Hastings, whose 12.5mm unit has somewhat worse edge correction with 30° field than this triplet, but significantly lower edge accomodation of little over +3 diopters required (when downscaled to 10mm, as opposed to +10 and more required by ES Steinheil, and Thor Labs units). It is not that the other three are poor designs, it's about different design priorities. The three narrower field eyepieces have zero-accommodation flat field up to 10-12°, and are paying for it with more astigmatism beyond that point. The ES Hastings has a curved field with the accommodation required reaching maximum of -2.6 diopters at about 0.7-0.8 of the field radius. It's a common trade off in accepting some field curvature for better edge performance.

The last design in this group (#7), that could be called simplecentric, is a cemented doublet (10/-3.8/-7.9mm radii, 3.1/1.3mm thicknesses, N-SK11, SF1 from left to right), with its correction level approaching that of the triplet. In fact, by changing the second radius to -3.85mm, correction becomes nearly identical. With the required edge accomodation of less than -2 diopters, and only two glass elements, it is a viable option to the cemented triplets.

40°-55° field eyepieces

These are 2+1, 2+2 and 3+1 designs. They can be looked at as more advanced vs. previous group, in that better correction of aberrations allowed for wider corrected fields. The simplest Koenig, 2+1, in its original patented form had planoconvex eye lens, more of a lens separation (3.4mm at 10mm f.l.), no more than 45° usable field (with +5 diopters accommodation required at that field radius), over 2.5 waves P-V of astigmatism at the field edge, and lateral color unacceptable by today's standards. Rutten and Venrooij improved somewhat on this original design, but according to what seemed to be their policy, not to extent that its performance would radically change; it amounted to slightly less field curvature and (significantly) better lateral color correction. More radical versions, such is the one shown below, have significantly better overall correction, allowing fields in excess of 50° (for comparison, at 25° radius astigmatism is only 1 wave P-V). This particular design requires much less of a much easier for the eye to handle, negative accommodation at 25° field. It also has a peculiar astigmatic field, with the balance between lower and higher astigmatism resulting in a practically astigmatism-free half-field.

RKE, or "reversed Kellner" (also "Rank Kellner Eyepiece", for its designer, David Rank), was designed back in the 1970s for Edmund's Scientific 105mm f/4.2 Astroscan. Shown is the 28mm unit scaled down to 10mm f.l. Similarity in the lens arrangement to the 2+1 Koenig extends to the similarity in performance level.

The Plossl shown here is based on the patent prescription of Albert Nagler. Correction level is similar to the previous two designs, with the advantage being less of field edge accommodation required: -1.25 diopters, low enough to make it practically flat-field visually.

The symmetrical eyepiece is just a form of the symmetrical Plossl. Astigmatism could be reduced to the level of the Nagler Plossl, if traded for acceptable field curvature (-2.5 diopters).

Clave Plossl is, according to Chris Lord, another patented design of Albert Koenig (US#2,217,281), which was only produced under Clave's name. It is asymmetrical Plossl, a flat-field design with good edge correction and some residual coma. Since coma increases with the 3rd power of focal ratio, and astigmatism with the 2nd, it is more noticeable at f/5. It has lower distortion than other designs in this group.

The Brandon eyepiece is, according to Chris Lord, reversed Clave Plossl. The similarity is there, in both, lens arrangement and output. Exact prescription is not known, only that it uses four different glasses (this arrangement uses SF1/PSK3-SK5/SF10, from left to right). There is no effect from making the field lens significantly thicker, other than a small (10-15%) reduction in the astigmatism/field curvature. Residual coma is probably what makes these two designs less suitable for fast focal ratios than designs without it (as the diffraction images show, coma transforms roundish astigmatic image into a triangular shape with a bright, curved base).

Well regarded for its correction, Abbe orthoscopic eyepiece is usually tought of as confined to somewhat smaller fields. As raytrace shows, its field correction can be just as good as that of other designs in this group, with its nearly flat field as added bonus. If desired, astigmatism can be traded for field curvature, and if the ultimate in correction is pursued, field can be somewhat reduced, to stay within acceptable field edge accommodation requirements. Such a design, H-ortho (from "highly orthoscopic") is shown under #8. Its correction remains unbelieveably good even at f/5, with the central diffraction maxima nearly intact at 20° off axis. The price to pay for it is -3.5 diopters edge accommodation required, which would be too much for some people.

55°-65° designs

The simplest Bertele is 1+1+2 arrangement. Its original patent design from 1925. (US pat. 1,699,682) claims up to 70° field diameter, but by today's standard is nowhere close to it. When scaled down to 10mm f.l., at 33° off axis it generates 3.4 waves P-V wavefront error of astigmatism (5.5 times the Airy disc), on its best image surface which have 21mm radius, convex toward the eye. This means that the eye has to relax nearly 9 diopters beyond its infinity mode to focus onto it, after being focused on the field center (it is like accommodating from infinity to a 12cm distant object). In other words, field center and perifery can't be in focus at the same time, with the latter requiring action by the focuser. On top of that, lateral color error limits sufficiently corrected field to little more than 10° radius. It illustrates well the limitations imposed by the lack of available glasses and by today's standard primitive designing means (lateral color error can be reduced to negligible if the original SK7/SK7/FK5/SF2 sequence is replaced by FK5/FK5/FK5/SF10). Nearly identical lens-shape wise arrangement could indeed be stretched out to a 75-degree field, or so, with the worst error of 1.5 waves of astigmatism at the 0.7 zone, and better than "diffraction limited" at the edge - really accomplishment, considering its simplicity - but still with more than 3 diopters of beyond-infinity accommodation between the center/edge and 0.7 zone required. The modified Bertele shown here still maintans nearly identical original design, but with somewhat different glasses and lens surface radii, resulting in a more favorable form of astigmatic field. The ray spot plots are for the best focus location, by definition midway between the tangential and sagittal surface (it may vary somewhat in the presence of significant higher-order astigmatism, or other aberrations), so that the approximate accommodation required can be determined from the graph. In this particular design, the outer ~15% of the field radius require divergent (natural) accommodation, maxing out at nearly -2.5 diopters at the edge (vertical dashed line) and most of the rest of the field require convergent (beyond infinity) accommodation up to 1.5 diopter. Not eye-friendly, but significantly less so than the original design. Using other common glasses the beyond-infinity accommodation can be entirely eliminated. Lateral color is well controlled, but due to design simplicity, longitudinal chromatism can't be at the same time (unlike the original design, which has entirely negligible axial color, at the expense of unacceptable lateral). The error is larger in blue/violet, 0.25 wave in the F-line, and 0.7 waves in g-line (436nm) at f/10. It can be somewhat reduced - down to 0.17 waves in F, and 0.45 in g, possibly more - but for a complete correction the field lens needs too be achromatized as well. Negative (barrel) distortion is near 18%.

Similarly to the Bertele, the original Van Hofe design from 1924. (US pat.1,759,529) had unacceptably large lateral color error, although somewhat lower astigmatism and less curved convex-to-eye field (the original design, which proposes somewhat longer eye relief, that would correspond to a very short-focal-length objective, had a flat field). Modification shown here uses the same glasses, in the same order, except that the eye lens is PSK2 instead of SK6, biconvex (planoconvex in the original), and with the central element made thinner, but with more strongly curved radii. Lateral color is corrected, except the field edge, edge astigmatism about 10% larger, and the inner field astigmatism about 20% lower. Best image field is practically flat. Created with the idea of an expanded-view monocentric, the Hofe eyepiece achieved wider field, but there is no comparison correction wise.

For decades, the Erfle was the synonym for a widefield astronomical eyepiece. The original 2+1+2 design from 1921. (US pat. 1,478,704) like nearly all old designs was poorly corrected by today's standards. Somewhat modified Erfle presented by Rutten and Venrooij had significantly less of lateral color and astigmatism/field curvature than the original, but not so much as to lose a connection to it. At f/10, the edge astigmatism (33° radius) is 3 waves P-V, required field-edge accommodation is still over 3.5 diopters. Distortion is relatively low, nearly 10%. By further weakening of the two radii contributing most of astigmatism, and changing glasses as necessary for color correction, the field can be flattened, with the astigmatism reduced to less than 2.4 waves P-V. Continuing in the same direction, reduction in astigmatism is traded for the increasing field curvature, and the near-optimum balance is given by the "modern Erfle" from Smith/Ceragioli/Berry. Its lateral color peaks at 0.7 field radius, where is nominally acceptable (F and C lines separated by less than Airy disc diameter), but with the blue inside Airy disc, and both red lines outside of it. Distortion is significantly greater, at 20%. Maximum accommodation required is about -1.5 diopter (diverging), much easier to the eye than up to a few times higher converging accommodation of the old-style designs. Similar to Erfle, 2+2+2 (three achromats) arrangement, known as Kaspereit eyepiece (1923), allows for better lateral color correction; this was more significant back in the day when glass selection was limited.

Zeiss Astroplan was apparently an attempt to adopt Erfle design to a smaller field. It allowed for the considerably smaller size and a cleaner field. Astigmatism is somewhat lower than in the designs usually employed for this field size, but not by much. Laterall color error at the field edge is non-existent in the blue/violet, while the two red lines are somewhat overcorrected (as illustrated on the diffraction image, with even sensitivity). Maximum accommodation required, at the edge, is -1.6 diopters, much friendlier to the eye than the full-blown Erfles from that era.

Zeiss Erfle from 1959, as presented by the Gross' Handbook (most prescriptions there should be accurate, but there are some errors; that for the 1+1+2 Bertele is unusable) is quite massive eyepiece for that time, with the outer field correction leaving something to be desired. Lateral color is worse on the red end (overcorrected), nominally nearly constant thru the outer field, hence the worst around 0.7 zone. It requires +4 diopters edge accommodation, which means the only way to focus on it is to refocus mechanically. It is unsuitable for fast focal ratios: at f/5, its astigmatic blur at the field edge has angular size of the full Moon. It is possible it was designed for some partly compensating arrangement.

Coming in the same 2+1+1+2 configuration as the Zeiss Erfle, the Nagler Widefield brings significant improvements in the overall correction. Its astigmatism at 33° is only marginally higher than that of the Zeiss at 23°, hence nearly cut in half. Its lateral color correction is very good, with the partial exception of the violet toward field edge. Its field edge requires -2 diopters of accommodation - i.e. as from infinity to 0.5m - easy enough for most people. It, however, has relatively high distortion and some coma, negligible at slower focal ratios, but becoming visible at the fast ones, making diffraction blur asymmetric. Since astigmatism between aprox. 0.5-1 wave P-V wavefront error (about 1-2 Airy discs in diameter) at sufficient magnifications shows as a little cross, it would in the Widefield have that kind of deformation at about 10°-15° off axis, quite close to the midfield. I read somewhere that it was these little crosses that prompted Al Nagler to come up with something better: the Panoptic.

There is no published prescription for the Panoptic, but the design from S/C/B should illustrate well its level of correction. Astigmatism is practically non-exsistent, except at the field edge, with the longitudinal aberration of ~0.06mm at the very edge implying ~0.5 wave P-V (divided by 8F2, same as for defocus), or ~0.1 wave RMS. Dominant aberration is trefoil (which doesn't show on the astigmatism plot), but even at the field edge the central diffraction maxima is still intact, with 0.45 Strehl (0.56 at the 0.7 radius). The only part of the image that deviates from excellent is axial and lateral color error un the violet, but it has generally little importance (axial blur at f/5 is twice the blur at f/10, or little over 1.5 wave P-V wavefront error).

~70° and over

There is no official field size at which the extra-wide fields begin, but considering that traditional wide-angle eyepieces - or at least those that were in practical use - generally remained below 70° field diameter, that could be viewed as the dividing level. The first design that boldly made a step over this line, into new generation of exceptionally well corrected extra wide-fields was the 1981 Nagler, known as Nagler 1. Below is the Nagler 1 design "a" in Al Nagler's patent application. It will be singled out here to illustrate some specifics related to this class of eyepieces. The #1 is so called Smyth lens, a negative lens acting as tele-extender (Barlow) some distance in front of the objective's focal plane. It changes geometry of rays entering the following it, positive lens group in that they are now spreading wider, and more so toward the field edge. This requires larger, more strongly curved lenses. It makes possible to achieve lens combination which will have compensatory action on field astigmatism over a significantly larger radius, up to a point when higher order aberratios spring out of control. The Smyth lens alone is not enough, as the Koehler 110° eyepiece illustrates. But this design acrobatics comes at a price: light cones traveling through different lens areas, of very different curvatures, tend to exit the positive group differently - the parallel pencils exiting the eye lens from the outer cones, passing through more strongly curved part of lens surfaces, tend to intersect optical axis closer to the eye lens than the pencils from inner-field cones. That is known as spherical aberration of the exit pupil, with the exiting parallel pencils not intersecting the axis nearly at the same location, forming a tight exit pupil, but rather extending pupil erea longitudinally. If excessive, it can cause darkening of the outer field areas, the "kidney bean effect". Conventional eyepieces also have spherical aberration of the exit pupil, but generally smaller, inconsequential. It is more likely to be noticed in the extra-wide field designs, and it makes it their #2 specific.

In raytracing, this exit pupil disarray makes impossible to accurately raytrace such eyepiece from a fixed exit pupil location. Picture below (bottom) shows how different, grossly inaccurate astigmatic field becomes in field areas with different exit pupil locations. This, of course, means just as inaccurate ray spot plots. For accurate output, every narrow zone needs to be raytraced from its appropriate exit pupil location, so that the entance beam on the other side is only midly converging toward optical axis. For that reason, a standard raytracing software can't produce a single accurate astigmatic field of such eyepiece: it has to be pieced together from separately raytraced different zones. So the first plot from left, showing astigmatic field generated when raytracing from the exit pupil for the field edge, has that pieced up real astigmatic field as the redish area around vertical axis (approximately). The rest of plots, from left to right, show astigmatic plots for exit pupil of the 0.8, 0.6, 0.4 and 0.2 zone, with the corresponding exit pupil locations (ER is eye relief, the exit pupil separation from the eye lens). The rightmost plot shows astigmatic field for the entire 40-degree radius when raytraced from the midfield exit pupil: the more distant entrance pushes the outer light cones farther toward lens perifery, and the non-existing astigmatism jumps of the chart for the outer zones. Nagler 1981a is a 10mm f.l. unit, and it is shown for f/10 entry cone.

The field is for all practical purposes flat, with the required accommodation remaining below 1 diopter. The actual ray spot plots (left) show that at f/10 even the very edge still preserves the central diffraction maxima (the e-line blur is about 0.7 the Airy disc diameter, implying about 0.085 wave RMS wavefront error). Lateral color is well corrected for the blue/violet in the inner field, with the violet flying off toward field edge, while the red end splits off higher in both, midfield and edge. Field edge shows significant lateral color, looking incospicious in the photopic mode, but becomes fairly obvious in the more likely for night time observing, mesopic mode (as a reminder, lateral color doesn't change nominally with the F-ratio, which means it is twice as bad at f/5, due to the twice smaller Airy disc).

Since the Koehler 110° spot plots are large enough, this time will start with another interesting design from the pre-Nagler era: the Ludewig 90° eyepiece from 1953 (US Pat.#2,637,245). It could be looked at as a transitional form between conventional and new Smyth-lens designs, in that it does employ a negative field lens, but it is placed right next to the positive lens group (in this arrangement, separating the negative lens worsens the lateral color, this is why it is "glued" to the next lens). Although Ludewig claimed fields of up to 90°, it is shown only for 80° field, since astigmatism already becomes objectionable. At 80° the mainly astigmatic blur (with some residual coma) is over 5 Airy disc diameters, which translates to more than 3 waves P-V wavefront error (over 0.6 waves RMS). Since the green f/10 Airy disc in a 10mm f.l eyepiece is 4.6 arc minutes wide, it corresponds to some 25' apparent diameter. It is still better than the traditional Erfle, which would, even in the improved version given by Rutten and Venrooij, have about 50% larger blur. However, required field edge accommodation to the best focus is over +11 diopters, which means that most people would focus closer to the elongated, line-like sagittal focus (more so considering the difficulty of focusing beyond-infinity). Lateral color error has a reversal at about 70% zone, diminishing somewhat toward the edge, but remaining unacceptably large in the outer field. Distortion is surprisingly small for this field size, and the exit pupil shift is negligible. With today's raytracing software, the design can be improved to a nearly flat-field, with about 50% lower astigmatism and significantly less lateral color (below; the defocused field shows about -1 diopter required for the field edge, and about half as much for the 0.7 zone). Similar configuration has been patented in 1998 as a 65° design (Koizumi/Watanabe, Fujinon Corp. #6,069,750; this design is nearly identical to the William Optics' SWAN, and likely was the basis for that series). Note that unlike the rest of designs in this group, this one is given for f/10 and f/15 beams, with the raytrace illustration for f/10 (1mm exit pupil); it means that it is shown twice larger relative to the others.

Nagler 2 (1988a) in its patented design has significantly reduced eye relief vs. Nagler 1, and exit pupil shift of ~3.6mm, only slightly less relative to the central field pupil, but significantly smaller nominally. The field is flat to -0.06mm, or -0.6 diopters. Axial color correction not quite as good, but still more than satisfactory, and lateral color correction is similar, except for the violet end, which is significantly worse. An interesting coparison with the Smyth lens design from the far 1923. (Taylor, #1,468,762, filed 1920) shows that Smyth lens alone is not enough. Patented as a binocular eyepiece coupled with a small 0.85-inch achromat at 10x magnification (somewhat more, with the eyepiece f.l. of 6.2mm and f/3.5 objective), it focused on obtaining flat field and corrected astigmatism. For some reason, that was achieved only near field edge (with some residual negative coma to the tune of 0.4 wave P-V at f/5), while the inner field suffered from significant astigmatism - roughly three times the Airy disc diameter at f/5 - which at f/3.5 leaves something to be desired even at 10x magnification. Lateral color correction is not quite acceptable at f/10, but eye relief is comfortable, and the exit pupil shift negligibly small.

The 1988b Nagler has everything significantly smaller, including eye relief, which makes it more suitable for the longer f.l. units. It has somewhat more astigmatism - enough to bloat Airy pattern into unrecognizable shape - and similar color correction, with somewhat less of the violet lateral color error. The Speers-Waller configuration here illustrates the gain from allowing for field curvature. Astigmatism is low enough to leave Airy pattern preserved over the entire 40° field radius, but the required field edge accommodation of -5.8 diopters is more than what most people could handle (for 10mm f.l.; since the diopter equivalent in mm changes with f2/1000, the 20mm f.l. unit, with twice larger longitudinal astigmatism, would require half as strong edge accommodation). This same configuration could be slightly modified to one with acceptable field curvature, and still a very good correcction of astigmatism; if, for instance, R15 is made 88mm and rescaled to 10mm f.l., edge accommodtion drops to -3.7 with the Airy pattern mainly preserved across the field. This arrangement has a good eye relief, and somewhat reduced exit pupil shift. Ethos-like design by Smith/Ceragioli/Berry, shows nothing short of amazing correction, with one exception: lateral color. It, of course, doesn't imply that the actual Ethos has it in similar magnitude, but this design would have it noticeable, as the simulations for the mesopic eye mode suggest. The exit pupil shift is further reduced. The Fukumoto Nikon patent from 2013. has excellent field correction, except the last 20% of the radius, or so. Lateral color error is, again, less than satisfactory. It is interesting that the edge astigmatism can be nearly eliminated by a correction of a single radius (R15, from 18.8mm to 18mm, with a small correction in exit pupil location, to keep the ligh cone in a proper orientation), with even the lateral color somewhat improved. The only purpose of the edge astigmatism could be to keep the field flatter, with the edge accommodation kept below -1 diopter, at -0.8. But twice stronger accommodation required with the astigmatism greatly reduced is still quite comfortable for most people (infinity to 0.63m). The smaller Fukumoto has so much of edge astigmatism that the spot had to be reduced to half its size. Unlike its larger cousin, it affects only the edge area; even at 47.5° the spot is significantly smaller, as shown in the smaller box at left. The edge astigmatism can also be significantly reduced by a small change in two radii, with the dominant aberration remaining at 50° being trefoil. Lateral color error is still too large. Finally, the miniature in comparison Ageev-style design has also the shortest eye relief (which according to the patent description should be somewhat longer with the actual unit, 80% of the focal length), has good correction over the practically flat field, but the lateral color error is again the weakest point. Note that the patent document (RU2548745C1) gives illustration of the lens assembly, but describes the glasses only in general terms, without specified radii/separations. Hence this particular design is not to reflect the actual unit's performance, rather to show that such configuration can indeed achieve a high level of correction. The astigmatic plot given in the patent document is unusual in that tangential and sagittal surfaces cross twice before beginning to spread wider at the 50° mark. Maximum accommodation requred, according to the plot, is between -0.05mm and 0.07mm for 12.5mm f.l. unit which, with one diopter of defocus given by 12.52/1000=0.156mm, translates to -1/3 and nearly +1/2 diopters, respectively.


12.3. Eyepiece aberrations II   ▐    13. THE EYE

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