12.4. telescope eyepieceS: COMPARATIVE RAYTRACING 

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 (3+2 variations, like Jean Cojan. 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 order astigmatism resulting in a practically astigmatism-free half-field.

According to its manufacturer, Edmund Scientific, RKE acronim should contain names "Rank, Kaspereit, Erfle", the first for its designer, David Rank, and the other two for no obvious reason. Its configuration is nearly identical to the Koenig 2+1, and seems the purpose of throwing in Kaspereit (2+2+2) and Erfle (2+1+2) was to divert from that fact. "Reversed Kellner", or the initial name for it by the ES, "Rank Kellner eyepiece" would be certainly more appropriate. It 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.

According to the Smith/Ceragioli/Berry (p499), the original design by Simon Plossl was actually achromatized Ramsden, i.e. a pair of widely separated positive achromats. It was Albert Koenig who brought the two achromats together and created the basic form of what we today call "Plossl eyepiece" (USP#2,217,281 Jan. 1939). Koenig himself called it "orthoscopic according to Plossl", and that is probably how it got to be called "Plossl". Koenig's "Plossl" was asymmetrical, using three different glasses (crown for the two inner elements, ordinary and heavy flint for the outer). It has good overall correction, but lateral color correction leaves something to be desired. 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, the above mentioned patented design of Albert Koenig (US#2,217,281), which was only produced under Clave's name. Smith/Ceragioli/Berry state that it is a post WW2 design by Maurice Paul. However, an article in French says it was Jean Texereau himself who modified the eyepiece, while Marcel Paul Clave and Serge-Rene Clave designed the Clave Barlow. Since no prescription for Clave Plossl was published, we'll show the Koenig's design, modified to two glasses mainly to minimize lateral color error (they are similar to some extent, with drawings on another French site suggesting that the true Clave had flat outside radii, with somewhat asymmetrical elements' radii and thicknesses). 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. Note that flattening the other outer radius would require use of different glasses - Schott SK2 and SF1 would do a good job - but the output would remain practically identical (small differences in thickness don't produce appreciable effect).

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. This design is no different in configuration than another patented Koenig design (same patent number as for his Plossl), which could be more advanced than Abbe's (will never know, since there is no known prescription, but Abbe's design was limited to 30° field). Unlike the Koenig's patent, the design shown uses only two different glasses, since it is all that's needed (the patent used barium dense flint - BASF64 or alike - for the eye lens, which is a high-index low-dispersion glass, close to some lanthanoids, now advertised by Zeiss Jena as being used in their Abbe orthoscopic eyepieces). 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. A very similar design, but using expensive, ultra-high-index glasses, is Janeczko et al. (from US 7,804,651B2, 2010), designed for night vision goggle systems and head mounted displays.

Yet another variation on the singlet+triplet form is Richter eyepiece, with one surface aspherized. The main difference with respect to Abbe is that the 3rd and last radius are considerably more strongly curved (similarly to the Abbe, the Richter is developed within the Zeiss company). This particular arrangement is a microscope variety, with the last surface concave and - to compensate for it - biconvex field lens. With no prescription found, it can be reconstructed from what appears to be accurate drawing of the design. One such reconstruction is raytraced below. Starting point for the glasses were those specified for the wide-angle inverse Richter eyepiece, which worked well, except that SF10 is replaced by SF2, as required to minimize lateral color error. The glasses are asigned to the elements based on best overall performance (different order tends to produce more curved field). Unit focal length is also 10mm, this time in direct raytracing, with OSLO "perfect lens" as objective and eyepiece. True angular field of 0.252° corresponds to 25° AFOV radius (the 2nd surface is a dummy surface placed at the front perfect lens focus, hence the "aperture radius" value for it represents Gaussian image radius at the focus of the objective).

With the radii closely conforming to those of the drawing, the astigmatic field is moderately curved, with the outer field becoming less curved due to the rapid increase in higher order astigmatism of the same sign (top). At -40mm field curvature radius, approximately midway between the edge and 70% zone foci, uneven astigmatic field makes required accommodation vary from negative at the 70% zone (-2.5 diopters), to positive (about +2.5 diopters at the edge). While neither alone is exessive, the combined differential of 5 diopters would be challenging for more than a few people. Color correction is good, particularly lateral color correction. Astigmatism can be reduced, with stronger field curvature, or increased to flatten the field. Magnitude wise, it is similar to that in the Abbe shown. Although cited aspherization calls for a parabolic surface 5, it would add far too much astigmatism and (positive) field curvature. As small as -0.02 conic flattens the field adding quite moderately to the eyepiece astigmatism; no other aberrations are significantly affected.

With relatively small changes in the radii, and the other parameters unchanged, better overall performance can be achieved (bottom). Best field is somewhat more curved, but accommodation changes evenly with the field height, and edge accommodation is still acceptable at -5.5 diopters. Needed aspheric to flatten the field is -0.09, still quite low, indicating that flat field could be also achieved with generally small changes in the design. The slight curl inside on the top of the tangential plot is due to the higher order astigmatism of opposite side. It leads to astigmatism decrease for higher field radii, where at some point T and S plots cross and astigmatism (of opposite sign) again starts to increase. This particular design would have good field quality at f/10 up to 65-70° AFOV, but required accommodation due to field curvature would become prohibitively high. Redesigning by allowing somewhat more of astigmatism for acceptably curved field is possible, giving to this configuration a wide-field potential.

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 the 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 considerably smaller size and a cleaner field. Although used for AFOV up to 50°, by design it is a widefield eyepiece and more comparable to designs from that group. Astigmatism is somewhat lower than in the designs usually employed for that field size, not very much, but enough to have smaller blur at 25° than Nagler Widefield at 23°. Laterall color error at the field edge is nearly non-existent in the blue/violet (according to SYNOPSYS; according to OSLO Edu it is roughly half of that in the red), 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. Note that the original Zeiss design had different mode of correction, with some negative (tip up) coma left in, somewhat lower field edge error but nearly by a third stronger field curvature, better lateral color correction except for the violet, which is worse. A simplified version, using SK2 for the three inner elements, and SF4 for the two outer elements, would have the best overall correction, but the differences remain small.

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, where the blur is half as large as at the edge. 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).

All previous designes were all-spherical, except the Richter microscope eyepiece above, which in the reconstruction attempt required very mild aspherization. An older aspheric design presented on Peter Smith's page (US Pat. #1,968,222 by Robert Richter, 1933) was intended for 65° AFOV. It is the same 1+3 arrangement as for the Richter microscope eyepiece in the previous group, but the field lens is now plano-concave, and the last surface is convex. No prescription is given, nor could be found online, so here is a reconstruction based on the raytrace drawing given on the page, with the starting glasses same as those for the microscope variety (change in the glasses is needed mainly for minimizing the lateral color error after the other aberrations are minimized by bending the radii and changing element's thickness). Unlike the group above, used is direct raytracing, with OSLO "perfect lens" as objective and eye lens. The 0.33° field angle corresponds to 65° AFOV (with ~8% positive distortion; design in the middle has as much as +20% distortion after aspherization, thus producing 72° AFOV with the same 0.33° field angle at the objective).

The original design (top) features a strongly curved 3rd surface, having as a result some low residual coma (coma contribution of the 3rd surface cannot be better offset by those of the other surfaces). Astigmatism-wise, it is midway between Zeiss Erfle and Nagler Widefield, but having color correction and field flatness nearly as good as the latter. Coma residual can be lowered by making the 3rd surface less strongly curved (middle). This configuration is made astigmatism-free when all-spherical, to illustrate the effect of aspherization for the purpose of field flattening. The first astigmatism plot shows astigmatic surface around -15mm field radius; below is the same vs. flat field, when the required edge accommodation is over 13 diopters. Putting -0.48 conic on the 5th surface (3rd surface of the eyepiece) flattens the field, by inducing needed amount of astigmatism. It results in a considerable increase in distortion, and have a relatively small effect on lateral color (increasing red vs. blue distortion). On the bottom are three variations of the arrangement illustrating that the conic can be varied from 0 to -1 without significant change in flat-field performance. The all-spherical configuration (right) has somewhat lower astigmatism, but more of longitudinal chromatism (it does not necessarily mean that it couldn't be improved in that respect, but if it could, it would require more involved re-designing). Note that these last three designs, while also 10mm f.l. at f/10, have smaller field and a different Airy disc scale.

Inverted configuration (1+3) was patented by Richter (Zeiss) as far back as 1932 as 82° 8.9% distortion eyepiece. There is no prescription available, but it can be assumed that this wide field coupled with low distortion was probably paid for with compromised astigmatism/field curvature and/or lateral color. This simple configuration does not allow for minimizing all aberrations at the same time; if it would, we wouldn't need the much more complex Smyth lens designs of the modern day.

An example of aspheric eyepiece with a nearly complete prescription given is the 60° AFOV design by Seymour Rosin and three other NASA employees (Wide angle long eye relief eyepiece, US Pat.#3,472,577 1969). It is a 3+1 arrangement with the strongly curved singlet surface aspherized. Patent states that the singlet does correction of monochromatic aberrations, with the zero-power triplet correcting for lateral color. It also points to the presence of coma, due to which it is not recommended to use exit pupils larger than 5mm (since the patent unit f.l. is 50.5mm, it implies f/10 cone, or slower; coma changes with the third power of focal ratio, hence it is eight times larger at f/5 than at f/10). As all other eyepieces on this page, the patent unit is scaled down to 10mm focal length. Surface conic in patent prescription is defined by the sagitta depth, but since no corresponding lens diameter is specified, it is somewhat speculative. However, raytrace shows that it is most likely approximately paraboloidal. Presented are both, reverse (top) and direct raytrace (bottom) of this design; the differences are not negligible.

The astigmatism plots with reversed raytracing show the effect of aspherization. With surface #3 left spherical, both astigmatism and field curvature are prohibitively strong. With -1 conic alone, both are greatly reduced, but field curvature is still far from acceptable (roughly 15 diopters accommodation for field edge). Given 6th order coefficient nearly flattens the field, at a price of significantly larger field edge astigmatism. Given 8th order coefficient is too low to have appreciable effect; it would significantly reduce edge astigmatism if nearly a dozen times higher. Overall correction would have been similar with -1.2 conic alone, but adding the 6th order term as the prescription calls for would produce unacceptably strong astigmatism and field curvature. Color correction is excellent, and distortion near negligible (note that the glasses in pattent application are given by the inde and Abbe#, nearly matched by closest contemporary glasses for raytracing purpose).

Direct raytracing uses "perfect lens" for the objective and the eye. It gives very similar output in the form of astigmatic field, but not in the magnitude. What is particularly puzzling, is that longitudinal astigmatism for field edge point is nearly 40% smaller, but the corresponding P-V error is nearly two and a half times larger. Taking a closer look reveals that the Airy disc presented by raytrace is significntly smaller for the edge point than on axis, and more so compared to the edge point Airy disc in the reverse raytracing, which is larger than the axial disc (bottom, boxed). Since distortion is negligible, most likely cause for this discrepancy is the way raytracing software processes width of the cone coming out of the eyepiece. Shown are wavefront, footprint shape on the singlet's flat surface, and Airy discs displayed (footprint shape in the reverse raytracing given by OSLO Edu has a peculiar shape of two elongated shapes overlaping perpendicularly, even on axis, indicating a software glitch). Apparently, cone width is measured vertically along the wavefront, hence the field edge Airy disc is smaller in direct raytracing, and larger in the reverse. In effect, it makes the former f/11.5, and the latter f/8.2 (approx.), with the P-V wavefront error differential for any given nominal longitudinal astigmatism being about two times. This explains a better part of the astigmatism contradiction; the rest is probably due to seemingly common differences in how astigmatism generates in reverse vs. direct raytracing. Main reason for the inequality is that reverse raytracing assumes identical pencil width at the pupil, which is in practice valid only for small field angles.

The wavefront elongation due to the cone width deformation at large field angles does affect diffraction image. In effect, it reshapes the aperture correspondingly, with the shorter diameter producing wider image than the longer one. It is not noticeabe in this case, due to the large magnitude of astigmatism, but SYNOPSYS offers as an option aberration-free diffraction image for any field point (below).

Other than differences in the spot size (left) and actual diffraction image (middle), it shows that the aberration-free diffraction images for the edge point (right) are indeed elongated, as expected (field angle in direct raytracing here is smaller than in OSLO because SYNOPSYS doesn't have "perfect lens" feature, and the aberration-free system used to simulate it has 1200mm f.l. vs. 1000mm for the "perfect lens in OSLO).

~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).

Another example is Nagler 1981b, which was omitted above, but is interesting because it's nearly 1/3 smaller than Nagler 1981a. It also has one extra element, being 2+2+2+2 configuration, which helps in better control of the lateral color error.

It has noticeably more astigmatism toward outer field, which could become noticeable at fast focal ratios. Longitudinal chromatism is very low, but at f/5 wouldn't satisfie the "true apo" criterion (not by much). Similarly to the 1981a, its true astigmatic field (botom leftmost, for e-line alone) is very different than that obtained for the fixed exit pupil location (the one for 40° field angle, next to it). Distortion exceeds -20% (since it is reverse raytracing, actual distortion is positive in sign, stretching the image bigger). All aberrations except lateral color are much lower at f/10 (bottom right; diffraction images are only for 40° angle, with e-line pattern shown for linear and logarithmic response).

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 has nearly identical configuration 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.

Another design can be interesting as a transitional form, even if it is relatively recent. It is a 80° AFOV 5-element Erfle with 2-element Smyth lens configuration by Imaizumi. In its patent application it is given as a part of the system with the objective and thick prisms, so it needed to be a bit modified for stand-alone performance. It should be somewhat better if optimized, but even without it performs significantly better than the comparable (flat-field) 5-element Erfle.

At f/5 its edge performance leaves something to be desired, as well as its longitudinal chromatic correction, but remain within acceptable. Overall, it is in the 80° AFOV performance category (more so considering likely improvement after optimization), and it was achieved just by adding Smyth lens to a ~65° AFOV configuration (the Airy disc at f/5 is midway between the two circles shown; scale is to small to draw it accurately with given screen resolution). Field is flat to beyond 80% radius, with the vey edge requiring about +2 diopters (infinity to 0.5m) accommodation (since edge point is closer to the eyepiece, when focused on the mid field it will cause the exit pencil come out not collimated, but slightly diverging, requiring natural, positive accommodation). Exit pupil shift is little over 1mm, much smaller than in the patented Naglers.

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.

As a curiosity, a design that shows how much can do Smyth configuration consisting of three singlets. It is T. Clarke design from 1983. that comes in several arrangements, from a single-glass to multi-glass combination. The author claims they can be used down to f/3 and 40° field diameter. Raytrace by OSLO does not seem to be confirming that, at least not for the field claimed. Below are bot reverse (top) and direct raytrace of the 1-inch focal length arrangement that does use some fancy glasses (and does significantly better than the BK7/SF2/BK7 combination). Both are with the same ray geometry, based on 1000mm f.l. f/10 objective (OSLO "perfect lens" in direct raytracing, with another "perfect lens" on the eye end). For comparison with 10mm f.l. units above, at 10mm f.l. the blur size and lateral color would be 2.54 times smaller, and distortion would remain unchanged.

The outputs at f/10 are similar, but not identical, given for axis, 14° and 20° field radius, for best focus locations (i.e. with full accommodation). Reverse raytracing gives distortion of opposite sign to that in direct raytracing, with the sign and magnitude of it indicated by the mismatch between the marginal (blue) focusing cone and end point of the Gaussian image (solid line, distortion graphs). Astigmatic fields are also different, although nearly equilizing in magnitude toward field edge (main reason is that the fixed exit pupil distance in reverse raytracing is correct for the field edge cone, but becomes increasingly inaccurate closer to the field center; for instance, correct exit pupil separation for 10° field radius is 32mm, which produces only a bit more astigmatism at that field height than direct raytracing, with proportionally smaller field curvature). The 20° ray spot plot for f/5 system, shown at left and up from the tri-color 20° spot, indicates insufficient edge correction at this focal ratio already (note that astigmatic field, determining longitudinal astigmatism, doesn't change with the focal ratio, but the aberration increases due to the combined effect of larger transferse aberration - i.e. wider cone - and smaller Airy disc).

Direct raytracing - which shows the actual eyepiece aberration - shows that astigmatism is negligible for about 60% of the field radius, partly supporting claim that the eyepiece could be used down to f/3 focal ratio. Hoever, at the 70% radius longitudinal astigmatism is already about 0.2mm, which at f/3 translates to 5 waves P-V wavefront error, i.e. blur of some eight Airy disc diameters. With the angular diameter of Airy disc in arc minutes given by 4.6F/f, for 25mm f.l. (f) eyepiece and F=3 it implies 4.3 arc minute blur, appearing as a spot to most people. But astigmatismm plot doesn't show coma, which increases inversely to the 3rd power of f-ratio, and becomes dominant at f/3, with about 1.5 times larger RMS wavefront error. Knowing that the coma blur is 2.5 times larger than astigmatism blur for the same magnitude of aberration, implies that the actual blur is roughly 15 arc minutes, half the diameter of full Moon (in addition, there's over half a wave of spherical aberration). At f/5, astigmatism is almost three, and coma almost five times smaller, making the blur roughly five times smaller, around 3 arc minutes, which can be deemed passable for the outer 30% of field radius.

Field edge accommodation requirement is little over 0.5mm, which for f=25.4mm f.l. eyepiece translates to less than 1 diopter (one diopter of accommodation being given by f²/1000). Direct raytrace shows that this point falls beyond retina, the consequence of a slightly diverging pencil coming out of the eyepiece. It requires contraction of eye lens, which is natural accommodation much easier on the eye than that of opposite sign.

◄ 12.3. Eyepiece aberrations II   ▐    13. THE EYE ►

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