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12. TELESCOPE EYEPIECE
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12.3. Eyepiece aberrations II
► 12.2. Eyepiece aberrations I : spherical, coma, astigmatism, field curvatureWhile the primary goal of the telescope objective is to transform flat incoming wavefronts into spherical, the eyepiece needs to accomplish exactly the opposite. If the wavefront entering the eye is not flat, the pointimage on the retina will suffer from wider diffraction energy spread, just as it does when the wavefront formed by the objective deviates from spherical. The extent to which it will affect perceived image quality depends on the detail size and properties, as well as image magnification. If the front eyepiece focal plane doesn't coincide with the image plane of the objective, the wavefronts originating from objectimage points will not exit the eyepiece flat, but curved (nearly spherical) and, as such, will be rejected by the eye in favor of the flat wavefronts coming from a point coinciding with the front focal plane of the eyepiece. The resulting error is defocus, easily correctable by moving the eyepiece to the proper position. If the objectimage and the front focal plane coincide, and wavefronts entering the eyepiece are spherical, it will emerge from it flat if ocular is aberrationfree. In the real world, these exiting wavefronts will be aberrated to some extent, and the aberrations are spherical, coma, astigmatism and field curvature, chromatism, image distortion and spherical aberration of the exit pupil. Due to a number of lens elements, aberration expressions for eyepieces are lengthier than for the objective. Also, eyepiece specs are commonly not known. For those reasons, eyepiece aberrations will be considered only in general terms. Before addressing specific aberrations, here's illustration of the main parameters of eyepiece aberrations using a simple Ramsdentype configuration.
While even a single lens eyepiece can produce a good axial correction  due to the generally small cone footprint on eyepiece surface  its field correction is very poor, with the offaxis aberrations increasing exponentially due to the larger footprint combined with stronger radii. For instance, the eyepiece shown is a 10mm f.l. Ramsden with a pair of BK7 PCX lenses w/8.5mm radius of curvature, 6mm lens separation and 5.7mm from field lens to objective's image, as well as from eye lens to exit pupil. At f/10, it is aplanatic, flat field with low astigmatism (0.11 wave RMS 15° off axis). If the eye lens is taken out, both, footprint and refraction (requiring r.o.c. reduction to 5.2mm) at the field lens  which already contributes nearly all off the astigmatism  have to increase, resulting in several times larger astigmatism severely limiting usable field. In addition, multiple elements may allow for partial offsetting of aberrations between them, further reducing their final level. For that reason, eyepieces utilize two or more lenses. In the above example, the field lens "sees" the image formed by the objective as its object. Its aperture stop, however, is at the objective, because it is where the chief incident off axis ray intersects the axis. The eye lens "sees" the virtual image, formed by reverse extension of rays refracted by the field lens, as its object, and its aperture stop is at the point of intersection of the extended chief raj converging from the field lens and the axis. Thus, to both apply the lens aberration relations for displaced aperture stop. Obviously, in order for the eye lens to produce a collimated pencil, its object  the image formed by the field lens  has to be at the distance equaling its focal length. In this particular example, specific aberrations are very low spherical aberration, less than 1/100 wave PV, most of it at the second lens due to the wider footprint (cone aperture) on it. Coma is practically zero on both lenses, due to the specific combination of object distance and stop separation for each lens. If, for instance, the focal length of lens changes, it changes the object and aperture stop locations relative to the lens' focus point, and the aberration is reintroduced, rather quickly. Nearly all of the astigmatism is on the field lens, and it is of opposite sign to that on the eye lens. The astigmatism is of the opposite sign to the Petzval curvature, offsetting it to produce flat, mildly astigmatic image surface. Eyepiece spherical aberrationThe wavefront error of spherical aberration in the eyepiece changes with the fourth power of the objective focal ratio. Thus, halving the objective Fnumber increases the wavefront error by a factor of 16. This makes eyepiece correction for spherical aberration critical. Even eyepieces superbly corrected at midtosmall focal ratios, may become noticeably affected at large relative apertures. Since spherical aberration in eyepieces is also in proportion with the eyepiece focal length, longer f.l. units are more affected. However, they normally don't produce sufficient magnification to show the effect of aberration. Degree of eyepiece correction for spherical aberration also varies with the eyepiece type and brand. Best corrected conventional 5mm f.l. eyepieces will have ~ 1/10 wave PV of spherical aberration (undercorrection) at f/4, while those on the opposite end are closer to 1/4 wave PV. This doesn't necessarily mean that the telescope performance will be noticeably affected. Most any objective contributes certain amount of spherical aberration of its own. If it is of the opposite sign to that of the eyepiece, the final result may still be admirable. But if the error is of the same sign for both, it will result in inferior performance. For instance, if the objective is 1/5 wave undercorrected, and the eyepiece 1/8 wave undercorrected, their cumulative error will be 1/3 wave PV of undercorrection. That is another example of the actual telescope aberration level being determined by the combined effect of both, objective and eyepiece. Moderntype eyepieces employing Smyth lens (negative field lens) behave differently. In general, they tend to induce overcorrection, which also may become significant at very fast focal ratios, especially when combined with the correction error of the objective that is of the same sign. The Naglertype eyepiece (Type 1) presented in the RuttenVenrooij's "Telescope Optics" raytraces to nearly 1 wave PV of overcorrection at f/5. Since it is a 100mm f.l. unit, the error would drop to nearly 1/20 wave PV with a 5mm f.l. unit at f/5. At f/4, it would be nearly 1/8 wave PV. Should the objective be 1/6 wave PV overcorrected at f/4, the combined error would come to ~1/3.5 wave PV of overcorrection. As an illustration of eyepiece performance at very fast focal ratios, here's raytrace of three actual designs: Nagler's patent from 1988(b), exceptionally well corrected for spherical aberration even among the modern, complex designs (top), Nagler's Plossl patent (bottom left), and a modern monocentric. The ray spot plots are given for field center (left) and outer field point given below focal length value (right).
Going from f/5 to f/3, this Nagler 2 does generate several times more of spherical aberration, but since the error at f/5 is so small  the Strehl indicates 0.003 wave RMS, or 1/100 wave PV  it only gets to 0.022 wave RMS at f/3. The Plossl, which is near the top correctionwise for the conventional eyepieces, already falls short at f/3. However, the modern monocentric still holds it's ground with 0.9+ Strehl. Astigmatism changes in inverse proportion to the square of fratio. Eyepiece comaOffaxis aberrations are more pronounced in eyepieces, due to their large viewing angles. Coma is, in general, not significant in eyepieces due to it being usually minimized by design, and much lower than astigmatism, which cannot be reduced nearly as efficiently. The amount of coma wavefront error in a positive lens changes in inverse proportion to the square of the focal length, and in proportion to the third power of the cone width. In effect, for given field angle, it changes in proportion to the eyepiece focal length. It also changes in proportion to the third power of telescope focal ratio. Given eyepiece design, the 30mm f.l. unit will have three times the coma of the 10mm f.l. unit. Either will have eight times more of the coma wavefront error at f/5, than at f/10 telescope focal ratio. As with the coma originating at the objective, eyepiece coma increases with the offaxis height in the image plane. Eyepiece astigmatismHowever, it is eyepiece astigmatism that usually dominates. It also depends on the lens focal length and the cone width, being inversely proportional to the eyepiece f.l. and in proportion to the square of the cone width. In effect, for given field angle it is, as coma, proportional to the eyepiece f.l. but, unlike coma, it changes in proportion to the square (not third power) of the telescope focal ratio (note that the geometric blur size changes in proportion to the focal ratio, but also with the square relative to the Airy disc). In other words, the wavefront error of astigmatism is four times larger at f/5 than at f/10. Unlike coma, eyepiece astigmatism increases with the square of apparent angle.
Also, unlike either spherical aberration or coma, lens astigmatism
 with the stop at the surface  is
independent of all three  refractive index, lens position factor and
shape factor; it only depends on the lens focal length. The means of controlling it with a set of positive lenses and lens
groups is generally limited, but the main reason why is it typically
strong in conventional eyepieces is that it is the third at the list of
priorities, after spherical aberration and coma. It is significantly
more complicated  or impossible, with simple designs  to have all three corrected. 0.01fe(ε/F)2<W<0.03 fe(ε/F)2, in units of 550nm wavelength, where fe is the eyepiece f.l., ε the eyepiece apparent field angle in degrees (radius), and F, as before, the telescope focal ratio. Plots at left show the range of astigmatism based on this approximation for three telescope ratios, fast, medium and slow. It scales with the eyepiece focal length.
Taking 20degree
nearedge AFOV for conventional eyepieces, it gives the following PV
values of astigmatism for selected focal ratios and eyepiece focal
lengths:
The error scales with the square of AFOV, which means it is four times smaller at 10° off axis, and 16 times smaller at 5°. The lower limit is not strict; the error can be still lower, but not significantly. On the other hand, poor designs can have significantly more astigmatism than the upper limit, up to twofold, or so. It can be assumed, however, that most of the conventional eyepieces made these days are closer to the lower limit shown in the table. For given magnification, eyepiece astigmatism scales inversely to the focal ratio. For instance, if an �/4 with 10mm f.l. eyepiece has 2.57.5 waves PV of astigmatism at 20 offaxis, an f/8 system with 20mm eyepiece of identical design will have 1.253.8 waves. An interesting example is Konig design given in "Telescope Optics" from Rutten and Venrooij. It has practically cancelled primary astigmatism (and well corrected higherorder term), at a price of somewhat stronger than usual coma (FIG. 213). In terms of the RMS wavefront error, at 10° offaxis it is  according to OSLO  superior to its lowcoma, strong astigmatism variant at both, f/10 and f/5, with 0.033 and 0.23 vs. 0.065 and 0.33 wave, respectively. At 20° field angle even more so: 0.16 and 0.7 vs. 0.94 and 3.7 wave. The enormous increase in the RMS error in the astigmatic Konig variant reflects the devastating effect of combined lower and higherorder astigmatism. Yet, it is the type more likely to be found on the market, because of the general notion that coma is less desirable aberration form. What actually puts the limit to usable field in a typical conventional eyepiece is the combination of lower and higherorder astigmatism. At relatively small field angles, usually below ~15°, the higherorder component is, in most properly designed eyepieces, negligible to nonexistent. But at once it creeps in at larger angles, it quickly explodes with the 4th power of field angle, adding to the already large lowerorder component. This puts an end to the acceptable field size. As mentioned, eyepiece astigmatism diminishes with the eyepiece focal length. A 10mm conventional eyepiece unit has ~3 times lower astigmatism than 30mm unit, resulting in about 1.7 times larger linear diffraction limited field. However, the difference is not that obvious in the eyepiece, due to shorter f.l. eyepieces having proportionally larger magnification, which mainly offsets the aberration decrease (FIG. 210). FIGURE 210: Offaxis aberration in a conventional eyepiece (from ~10" distance the view is similar to the actual one). The difference in the nominal level of aberration  from 16 waves of astigmatism combined with 2.4 waves of coma for the 30mm f.l. at f/5, to 0.65 wave of astigmatism and negligible coma for the 5mm at f/10, both at 20° offaxis, which is more or less the typical level  is much greater than what the actual view shows. Actual Airy discs are not of identical size at these three fratios (diameter being given by 4.6F/�e in arc minutes), for 550nm wavelength; it would be noticed visually, assuming sufficient magnification, but not with the given screen resolution. Note that the plot shows only aberrations originating in the eyepiece; the final spot shape and size vary with the amount of offaxis aberration induced by the objective, as well as with the combined field curvature of the objective and eyepiece. In general, eyepiece aberrations  particularly astigmatism  dominate in fast to medium telescope systems. For meaningful correction of eyepiece astigmatism it is necessary to introduce a negative (Smyth) lens, which induces astigmatism of opposite sign to that of the positive lens group. It also induces Petzval field curvature of the opposite sign, resulting in both astigmatism and field curvature minimized. Best known brand of this kind, the Nagler, has astigmatism reduced up to several times vs. comparable conventional eyepieces. Eyepiece field curvatureField curvature in the telescope eyepiece is directly related to its astigmatism. A hypothetical astigmatismfree eyepiece would form the image coinciding with the Petzval surface. Being formed by a positive power lens system, this surface is concave toward the eyepiece and, considering relatively short focal lengths, rather strongly curved (eyepiece's Petzval curvature, as well as other aberrations, are determined by reverse ray trace, with collimated light entering through the eyepiece's exit pupil). Real world eyepieces, however, produce strong astigmatism, particularly the conventional types. Usually, this astigmatism is of opposite sign to the Petzval, thus abaxial points form astigmatic surfaces less curved relative to the Petzval up to a certain level. At the point when the sagittal astigmatic surface is half as curved as Petzval curvature, best image surface is nearly flat (FIG. 211B). Further increase in the astigmatism causes best image surface to become increasingly curved to the opposite side (FIG. 211C). It is possible that astigmatism in the eyepiece is of the same sign as its Petzval (for instance, due to unbalanced higherorder astigmatism) in which case the astigmatic image surfaces are closer to the eyepiece than Petzval's, thus more strongly curved, and more so as the astigmatism increases.
FIGURE 211: Astigmatismfree eyepiece will form flat field if the image formed by the objective is also free from astigmatism and coincides with the eyepiece's Petzval surface P (A). Astigmatism modifies the field curvature depending on its sign and magnitude (B,C). If the sign is opposite to the Petzval's  usually the case  sagittal astigmatic surface S forms on the convex side of the Petzval's. Tangential surface T is always 3 times farther away from the Petzval than sagittal, with the best, or median surface M midway between the two. Note that the curves show primary astigmatism. Eyepieces with strongly curved lenses also generate secondary, higherorder astigmatism. Since it has significantly different rate of increase than the lowerorder form (4th and 2nd power of the field radius, respectively), once it reaches considerable level at larger field angles, it also strongly alters the initial field curvature. Depending on the design particulars, the effect with respect to field curvature can be either positive (field flattening) or negative; it can also be positive up to a certain image radius, then negative, or vice versa. Flat eyepiece field means that all offaxis pencils exiti the eye lens collimated, enabling the eye to focus on all points across the field simultaneously. When the eyepiece field is curved, off axis pencils are progressively more diverging, or converging, going farther off axis, hence they cannot be focused simultaneously with the central pencils on the retina (i.e. image points appear to be closer, for diverging pencils, or farther away, for converging ones). This can be either lessened or worsened when combined with the curvature of the image formed by the objective, as shown below. For simplicity, only best image surface is shown, i.e. field curvature is assumed to be either the Petzval with zero astigmatism, or best (median) astigmatic surface, when astigmatism is present. This is simplification in that it is not only the image curvatures, but also their respective astigmatism that interact, and since astigmatism directly influences field curvature it is a factor that shouldn't be entirely neglected. However, with the astigmatism of the objective being typically negligible in comparison, the astigmatism factor can be neglected in this respect for a general consideration, and assume that the interactions of field curvatures alone will give a good approximation of the combined field curvature. Most objectives generate curvature concave toward objective, and most eyepieces nowadays have near flat field, in which case the combined visual field has curvature similar to that of the objective (1a). The reason is that only light from those points of the objective's image laying in the front focal plane of the eyepiece will exit the eyepiece as collimated pencils. Those farther out will exit converging, hence eye will focus on the (defocused) point of the objective's image laying in the front focal plane of the eyepiece. In order to focus on the points curving out away from the eyepiece, the eyepiece has to be shifted toward them. In effect, the visual image field is curved away from the eye. Eyepiece itself can generate curved image surface, either convex or concave toward the eye. If it is convex and nearly coinciding with the image surface of the objective, light from all points will exit eye lens as collimated pencils, i.e. will focus onto the retina, and the visual field will be effectively flat (1b). If, however, eyepiece field curvature is opposite to that of the objective's, the effective visual field curvature will be a sum of the two, twice as strong (1c). Likewise, if the objective's image field is flat, the visual field will have the curvature of the eyepiece's field (2ac). With some objectives which generate field curvature convex toward them, like Gregorian twomirror telescope, the combined field curvature forms in the same manner, only the net sum with the same eyepieces is different (3ac). Astigmatism and field curvature of the eyepiece combine with those of the objective, to form astigmatism and field curvature of the final visual image. Just as the hypothetical astigmatismfree eyepiece would need objective's image to be astigmatismfree, with the two Petzval surfaces coinciding, in order to produce flat, astigmatismfree combined field, an astigmatic eyepiece would need objective whose astigmatic image would coincide with its own to result in astigmatismfree combined image. Eyepiece astigmatism is normally significantly stronger than that of the objective, especially for longer f.l. conventional eyepieces. Hence the astigmatism of the objective doesn't have much of effect on the final image: it is mainly determined by the eyepiece. Short focal length eyepieces have proportionally lower astigmatism (as the transverse aberration; lower to the square of it as wavefront error, due to the smaller Airy disc), and it can be more noticeably affected by the astigmatism of the objective. Astigmatic surface profile of most amateur telescopes (Newtonian, refractor, Cassegrain) is roughly similar in form to that shown on FIG. 211C, concave toward objective, with the tangential surface closer to it. Newtonian form is identical to it, while in refractors and Cassegrainlike systems  including SCT and Gregory MCT  sagittal surface is also concave toward converging light, and so is the system Petzval; in MaksutovCassegrain with separate secondary, system astigmatism can be nearly corrected, with the field curvature nearly coinciding with its Petzval. If the eyepiece has, say, form of astigmatism shown in (B), twice stronger than the astigmatism of the objective (i.e. double the sagittaltotangential surface separation) having the form shown in (C), the tangential surface of the combined image will be flat, with the sagittal unchanged, for the combined astigmatism half that in the eyepiece alone, and median image curvature of opposite sign, but of the same magnitude as that of the objective (Petzval surface becomes a factor only in astigmatismfree systems). If the eyepiece has the same amount of astigmatism, but with tangential surface flat and sagittal twice as strong as in the former example (which requires stronger eyepiece Petzval curvature), tangential surface in the combined image will have curvature equal to that in the objective, with the sagittal surface being as curved as in the eyepiece; the combined astigmatism is 50% greater than in the eyepiece, but with the combined median surface curvature somewhat weaker. Gregorianlike twomirror systems have, like Newtonian, Petzval surface curvature of the same sign as typical eyepiece's Petzval, but considerably stronger, with both sagittal and tangential astigmatic surface usually convex toward converging light. With both telescope types, this allows for the possibility of correcting both astigmatism and Petzval surface curvature in the final image with matching eyepiece (easier in the Gregorian, whose Petzval curvature is significantly closer in magnitude to that of the eyepiece).
Note that strong field curvature in eyepieces, if mainly result of
astigmatism as shown on FIG. 211C, does not implicate significant
defocus error, even without the ability of the eye to accommodate.
Sagittal surface is usually relatively weak, and the wavefront error
along it is identical to that along the tangential surface, with the
wavefront error along best (median) surface smaller by a factor of
1/√1.5.
In other words, it is the error of astigmatism that dominates
(but if the eye is unable to accommodate to the best image curvature,
the point images will be elongated). EYE ACCOMMODATIONNeeded eye accommodation to eyepiece field curvature (calculated based on the thin lens Gaussian equation, thus only approximation) depends on three factors, linear defocus of the off axis point in the visual focal plane(∆), eyepiece f.l. (f) end eye f.l. (E) as:
A=(∆E2)/[f2+∆(fE)]
A=∆ED/[f2+∆(fE)] = 1000∆/[f2+∆(f17)] = 1000∆/[(f+∆)f17∆]
for that defocus i.e. needed accommodation in diopters, where D is the eye focal length in diopters (D~59; while the optical f.l. of the eye, due to the image medium (n~1.33) is about 17mm, giving D=1000/17~59, its physical f.l. is about 23mm). Most often, point defocus ∆ is sufficiently small vs. eyepiece focal length, and the relation simplifies to A~1000∆/f^{2}. Taking A=1, for defocus corresponding to one diopter, gives ∆~f^{2}/1000. Or, described in plane terms, a point out of the centerpointplane by a given percent of the eyepiece f.l. in its image space, will form focus by as much percents of the eye f.l. off in its image space. For instance, field edge point 0.1mm off the central point plane in a 10mm f.l. eyepiece will be as much removed from the actual (objective's) image point in the flat image plane of the objective, thus the actual point it reimages will not form a parallel exit pencil, but slightly converging (actual field point farther away, illustration at left), or diverging (actual field point closer). Since change in the focal length f of xf approximately corresponds to bringing object from infinity to a distance of f/x (for x~0.1 and smaller), 1% difference in point distance vs. f.l. of the eyepiece means that the rays coming to the exit pupil won't be parallel, but either converging, as if coming from a point 100f=1m away, or diverging, as if coming from an imaginary point at the same distance but behind the objective (eyepiece). Since 1m vs. eye focal length numerically represents the eye focal length in diopters (59D), the defocus vs. infinity will be approximately 1/59 of the focal length, or 1 diopter. For the case shown, with the actual point farther away from the point that would form a parallel exit pencil, exiting rays will be converging, forming in the relaxed (infinity) eye focus point shorter than infinity focus, hence requiring eye lens relaxation (streaching) beyond that needed for infinity in order to bring it to the retina. Despite being numerically identical, only positive for extending, and negative for shortening the focus, the two forms of accommodation are very different to the eye. Shortening the focus, which requires compressing the eye lens to a stronger radii than in its infinity mode, is natural to the eye, and much easier than streching eye lens out to weaken the radii, which it can do in a much more limited way. Thus, assuming flat objective's image field, eyepiece field curvature convex toward the eye is much preferred to the opposite shape (note that field curvature resulting from reversed raytracing has opposite sign of its actual field curvature, which is the case with distortion as well; in systems w/o reversing reflection, light travels from left to right, so the proper eyepiece orientation is with exit pupil on the right side; in a Cassegrainlike system, eyepiece will have the orientation shown, but the objective's field curvature will be reversed due to reflection). Above formulae imply that scaling eyepiece up results in lower accommodation required, in inverse proportion to the focal length (all else equal). It is shown at left, for 10mm f.l. eyepiece, where 0.1mm of defocus due to field curvature closely approximates 1 diopter of accommodation required. Graph line is midely concave, which shows more clearly against straight line (dashed). Accommodation is negative for negative ∆, i.e. for the case shown on the previous picture. For comparison, shown is accomodation for eyepiece scaled up to 20mm f.l., keeping in mind that the nominal value of ∆ on its horizontal scale is twice larger, i.e. hence its graph has twice smaller angle. It doesn't feel intuitive that a scalled up eyepiece, covering twice larger linear field, requires less accommodation, but it is a consequance of two factors: its weaker best image curvature, and its longer focal length. The former makes the sagitta proportionally deeper, and that is compensated by its longer focal length. This means that the effective point distance, expressed in units of eyepiece focal length will be nominally identical to that for the shorter f.l. eyepiece, but the actual effective distance of the point for the eye, will be proportionaly larger, causing apropriatelly smaller degree of the exit pencil convergence or divergence, thus resulting in a lower accommodation requirement. This double effect of the focal length is why it shows squared in the formula. It is obvious that the same amount of accommodation will be required if the situation is reversed, i.e. with flatfield eyepiece, and objective having the same amount of image curvature. However, since the radius of curvature is constant, and field sagitta (s) for the latter is given by s=h2/2R, where h is the field height, and R the radius of field curvature, i.e. it changes with the square of field height, it will require near constant accommodation, regardless of the (flatfield) eyepiece focal length (all else equal). A simple way to approximate needed accommodation is to start with a 10mm f.l. eyepiece for which, approximately, every 0.1mm of defocus from the central point image plane corresponds to 1 diopter of accommodation. Knowing the radius of field curvature, sagitta s can be determined for any given field height h, and if it is expressed in mm, the corresponding accommodation is 10s in diopters. For instance, a standard 8inch f/10 SCT has best fild curvature of about 230mm. For the field height h=10mm, the coresponding sagitta is s=0.22mm, and the approximate accommodation vs. field center is 2.2 diopters (since it is ~1/27 of the eye's 59 diopters f.l., it is like accommodating from infinity to an object at a distance of 27 times eye's f.l. of 17mm, or about 0.46m).
