12.2. Eyepiece aberrations I : spherical, coma, astigmatism, field curvature

However, 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.
 
Roughly, for an ordinary 30mm f.l. eyepiece, diffraction limited field set by the astigmatism is ~7 degrees in radius at f/10, or roughly 1/4 of the entire field. This is only a nominal limit, since at this focal length the magnification produced normally is not sufficiently high to show the effect of this level of aberration. Actual limit to a quality field size is somewhere in the second half of the field radius. Approximately, typical P-V wavefront error of primary astigmatism in a conventional eyepiece is, based on the parameters for a single lens relation (Eq. 22, with the aperture D effectively proportional to the inverse of the telescope focal ratio F), within the range given by:

  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 20-degree near-edge AFOV for conventional eyepieces, it gives the following P-V 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 2.5-7.5 waves P-V of astigmatism at 20 off-axis, an f/8 system with 20mm eyepiece of identical design will have 1.25-3.8 waves.

An interesting example is Konig design given in "Telescope Optics" from Rutten and Venrooij. It has practically cancelled primary astigmatism (and well corrected higher-order term), at a price of somewhat stronger than usual coma (FIG. 213). In terms of the RMS wavefront error, at 10° off-axis it is - according to OSLO - superior to its low-coma, 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 higher-order 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 higher-order astigmatism. At relatively small field angles, usually below ~15°, the higher-order component is, in most properly designed eyepieces, negligible to non-existent. But at once it creeps in at larger angles, it quickly explodes with the 4th power of field angle, adding to the already large lower-order 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).

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 curvature

Field curvature in the telescope eyepiece is directly related to its astigmatism. A hypothetical astigmatism-free 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 higher-order 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.

Note that the curves show primary astigmatism. Eyepieces with strongly curved lenses also generate secondary, higher-order astigmatism. Since it has significantly different rate of increase than the lower-order 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 off-axis 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 (2a-c). With some objectives which generate field curvature convex toward them, like Gregorian two-mirror telescope, the combined field curvature forms in the same manner, only the net sum with the same eyepieces is different (3a-c).

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 astigmatism-free eyepiece would need objective's image to be astigmatism-free, with the two Petzval surfaces coinciding, in order to produce flat, astigmatism-free combined field, an astigmatic eyepiece would need objective whose astigmatic image would coincide with its own to result in astigmatism-free 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 Cassegrain-like systems - including SCT and Gregory MCT - sagittal surface is also concave toward converging light, and so is the system Petzval; in Maksutov-Cassegrain 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 sagittal-to-tangential 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 astigmatism-free 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.

Gregorian-like two-mirror 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 ACCOMMODATION

A=(∆E2)/[f2+∆(f-E)]

for the actual defocus inside the eye it needs to compensate for, or

A=∆ED/[f2+∆(f-E)] = 1000∆/[f2+∆(f-17)] = 1000∆/[(f+∆)f-17∆]

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∆/f². Taking A=1, for defocus corresponding to one diopter, gives ∆~f²/1000.

Or, put simply, light from a point displaced from the point-image defined surface in its image space (which is in the actual setup the objective's image space, and object space for the eyepiece), will not form a parallel exit pencil, but either slightly converging one - for a point farther away - or diverging, if closer than the distance producing collimated exit pencil. For instance, a field edge point 0.1mm off that imaginary point-surface 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 Cassegrain-like 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 (not in inverse proportion to the square of it, as it appears, because the value of Δ also changes, in proportion to the focal length). Graph below allows to extract the values for point displacement and accommodation requirement for eyepieces of different focal lengths. On the horizontal scale it shows relative point displacement values, accounting for the change in Δ with eyepiece focal length f, and on the vertical scale the relative value of corresponding required accommodation, given as the actual accommodation in diopters multiplied with f/10 factor (these are a consequence of using Δ that changes with eyepiece focal length in the above equation for accommodation).

For instance, the actual point displacement for 0.1mm point displacement in the 10mm f.l. unit, given by (10/f)Δ, f being the eyepiece focal length, will be a bit less than 0.05mm in the 5mm unit, and 0.2mm in the 20mm unit. The corresponding accommodation requirements will be 2 and 0.5 diopters, respectively (from the value on the vertical scale divided with f/10). Value of Δ alone represents point displacement corresponding to 1 diopter accommodation; it is obtained by dividing the scale value - in this case 1 - with 10/f. For the 5mm unit, it is slightly less than 0.025. In other words, value of Δ corresponds to the scale value only for the 10mm unit, as well as the scale value for accommodation requirement. For different focal lengths, both change as indicated. The plots are not straight lines due to the form of the equation, with those for f<17mm being concave, and for f>17mm convex in the point of origin, 0;0 (the larger differential vs. 17mm f.l. the more so).

It is obvious that the same amount of accommodation will be required if the situation is reversed, i.e. with a flat-field 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 (flat-field) 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 8-inch 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).  

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