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12.2. Eyepiece aberrations I   ▐    12.4. Telescope eyepiece: comparative raytracing

12.3. Eyepiece aberrations II 

 Eyepiece distortion

Due to the eyepiece being working with much larger field angles than the objective, image distortion becomes significant aberration. As explained in 2.6. Distortion, it is caused by wavefront tilt error, shifting Gaussian image point away from its ideal position. There is no effect on point-image quality, only its position in the image space. Distortion has two different forms, linear and angular (FIG. 212).

FIGURE 212: Image distortion in the eyepiece is a result of large apparent angles it produces. The difference between relatively large angle (in radians) and its tangent becomes significant. As a result, if the projected apparent heights h'1 and h'2 are proportional to the appropriate point-source P1 and P2 heights in the object image plane (IP), then the ocular-produced angles α'1 and α'2 can't be proportional to the appropriate field angles α1 and α2 at which the point-source heights are seen in the object image plane - and the other way around. When the heights are not proportionally imaged, the eyepiece suffers from linear (or rectilinear) distortion, which can be negative (barrel distortion, with the length of a given linear section decreasing with field angle) or positive (pincushion distortion, with the length of a given linear section increasing with field angle). Likewise, when the angles are not proportionally re-imaged, eyepiece suffers from angular distortion - also either positive or negative - which manifests itself by elongating objects, vertically or horizontally, farther out in the field. Obviously, the two forms of field distortion can't be both cancelled at the same time. In amateur astronomy, the preference is, in general, minimized linear distortion, for esthetic reasons. In professional astronomy, angular precision is more important.  

     Discrepancy between (recti)linear and angular distortion is a consequence of large viewing angles typically associated with eyepieces. Illustration below shows the relation between apparent angles and equal linear sections for small and large (magnified) image, where the magnified image has zero rectilinear distortion, i.e. every linear segment of field radius is equally magnified. In such case, due to large viewing angles, upper radius of a circular shape near field edge (or anywhere in the field where distortion is large enough) will appear smaller than the lower one,

and the circular form - despite being perfectly re-imaged in geometrical sense - will appear as if having flattened top. In order for it to appear without such distortion, the top radius has to be longer as needed to make its apparent viewing angle equal to that of the lower radius. In other words, it requires a specific amount of (positive) rectilinear distortion (if it exceeds needed amount, the shape will appear elongated toward edge). Which, in turn, will produce field distortion shown on the distortion page.

Most eyepieces exhibit both forms of distortion, although one is usually dominant. In astronomy, angular attributes are more important than linear, although in amateur astronomy the aesthetic quality of low linear distortion is usually more of a priority. Manufacturers seldom give the specs on eyepiece distortion; linear distortion, expressed as a percent, %ld=100(1-tanα'/tanα'), with α' being the actual angle, and α the ideal (zero distortion) angle, is considered negligible if up to ~5% for conventional ~45-degree apparent field of view (AFOV) eyepieces. However, since it increases with the third power of field angle, it is hard to control with wide AFOVs. Consequently, wide-field eyepieces can have it exceeding 20%.

Likewise, the percent of angular distortion can be expressed as %a.d.=100(1-α'/α).

Eyepiece chromatism

Eyepiece chromatism can be expected to be cancelled longitudinally. However, lateral chromatism is hard to correct for entirely. In general, it will be more noticeable in wide-field eyepieces, but ordinary oculars can have noticeable amounts of it as well (FIG. 213).

FIGURE 213: Basic forms of conventional eyepiece types in amateur astronomy, with ray spot diagrams illustrating their typical aberrations in e (green), F (blue) and C (red) spectral lines. The spots are given for f/10 and f/5 focal ratio (except for those that can't handle satisfactorily f/5 cone), for the field center, 7° and 10° off-axis (black circles are the e-line Airy disc, and RC is the best image curvature). 
Of the older types - Huygens, Ramsden, and monocentric - the latter has better axial correction (with somewhat limited field, due to higher-order astigmatism). More recent types, like Kellner, Abbe (orthoscopic), Konig, Erfle and Plossl have similar level of correction, with the Kellner being somewhat inferior.
The last two "types" are a pair of either singlet or achromatic lens. They can have good correction, except for the excessive lateral color (as the Ramsden arrangement shows, increasing the separation does not reduce it significantly).
Axial color correction is good for all (where it appears to be excessive, it is due to the residual spherical). Lateral color - which increases with the field angle and doesn't change with f-ratio - is acceptable, in general, although may not be at the fast focal ratioe, when it is more pronounced due to the smaller Airy disc.
Axial image quality is limited by spherical aberration at fast f-ratios (>f/5). Off-axis field quality is limited by the astigmatism, although coma can also be relatively strong (the two version of Konig differ significantly only in two of the radii values). 

It should be noted that the above eyepiece designs are not fully optimized, but can be considered within their standard quality range (most of them are down-scaled designs from Rutten and Venrooij, with some tweaking added in the process). Significant variations in the amount of specific aberrations is possible within the same eyepiece type, depending on the design goal and design/fabrication quality.

In general, eyepieces have longitudinal chromatism as some form of primary chromatism: shorter and longer wavelengths focus on the opposite sides of the mid (green-yellow) wavelengths. Figure below illustrates some of the variations, as well as a range of magnitude (black circle is the e-line Airy disc). The standard Plossl has entirely negligible longitudinal and lateral chromatism, the former roughly ten times lower than the modified 10mm Bertele, interesting in that with only 2+1+1 lens combination produces astigmatic field similar to the sophisticated modern eyepieces employing Smyth lens (its chromatism can be significantly reduced with different glasses/radii, but it is possible that a commercial eyepiece has this level of chromatism). And even the latter, in its biggest offender, the violet g-line, has the error at the level of a 100mm f/90 achromat, and the overall logitudinal chromatism is at the "true apo" entry level.

Unlike the blue-violet, the red wavelengths in the Plossl have near-zero focus deviation from the mid wavelengths. Such asymmetries are possible, and probably more likely at a very low error level. For example, cemented triplets at the bottom - also 10mm f.l. - are of a similar configuration, using identical glasses (F2/BK7/F2), yet they have mutually inversed forms of primary spectrum.

Above graphs give good idea about the magnitude of eyepiece longitudinal chromatism, and how negligible is its chromatic effect. For comparison, longitudinal spectrum in the same wavelength range stretches over nearly 2.5mm in a 100mm f/12 achromat, considered generally acceptable in that respect. The effect of the eyepiece can be directly judged based on their respective LA graphs. Since the eyepiece LA graph is a product of reverse raytracing, with the collimated light entering through exit pupil and exiting through field lens, it means that the aberration in the objective focal plane will produce perfect collimated pencils exiting eyepiece if it is of the exact shape, magnitude, and orientation as the aberration produced by reverse raytracing. This means that if we look at the LA graphs of the Plossl above, and a 100mm f/12 achromat below, eyepiece chromatism will reduce the achromat's g-line (436nm) error by less than 0.05mm, F-line (486nm) error by 0.02mm, and C (656nm) and r-line (707nm) by less than 0.01mm.

Even if we'd try to design eyepiece so that its form of longitudinal chromatism is the same, with the opposite sign to that of the achromat, it would be very hard to generate that much of aberration on such a small scale. The simplest form of it would have been a negative reversed doublet (BK7 for the negative lens and F2 for the positive) placed at its focal length in front of the objective's focal plane (vertical line crossing the rays behind the eyepiece is the "perfect lens", focusing collimated beams exiting the eyepiece, playing the role of a perfect eye). Top shows the "eyepiece" and longitudinal chromatism it produces (C and F lines are coinciding with their paraxial foci, so that it doesn't change their mutual position in the achromat). Even the -53mm f.l. eyepiece of this kind shown - and the error generated scales with focal length - would change very little in the achromats longitudinal error (the biggest gain would be reduction in the violet line from 4.94 to 4.7 waves P-V). To accumulate the error of needed magnitude, such eyepiece - or rather corrector - would need to employ multiple elements with special glasses, and it may not be practical, or marketable. A challenge of its own is controlling astigmatism/field curvature of such a lens close to the focal plane.

Spherical aberration of the exit pupil

The last eyepiece aberration to address is spherical aberration of the exit pupil. Ideally, telescope ocular will re-image the aperture into a single plane; in other words, all the bundles of parallel rays exiting the ocular would merge into a common circle symmetrical around the optical axis (FIG. 209). In reality, in re-imaging the entrance pupil (aperture opening), an eyepiece acts as any positive lens does - in other words, it suffers from under-correction. As a result, object-image points toward the edge will cross closer to the eye lens (FIG. 214). In effect, exit pupils for the outer zones are shifted toward the eye lens, away from the eye, and - depending on the extent of shift - it may result in the outer field either being not visible from the same eye location as the mid-field (central field area is always visible), or it being vignetted. If the eye needs to be moved back and forth in order to view the entire field, it may become decentered , having parts of the field vignetted or lost from the

FIGURE 214: Exaggerated illustration of the spherical aberration of exit pupil. The eyepiece (EP) transforms diverging light cones emerging from the image points into parallel pencils. Those coming from the points higher in the image plane are intersecting the axis closer to the eye lens than those from the lower image points. As a result, exit pupils for the former are shifted closer to the eyepiece. Large enough pupil separation  will make it difficult or impossible to hold the entire field in view for any single eye position. The effect can be noticeable in wide-field and long-focus eyepieces, both having, in general, greater longitudinal exit pupil aberration. Point image quality is unaffected, only the axial position of its exiting pencil.

view (the "kidney bean effect"). Other than that, spherical aberration of the exit pupil doesn't affect image quality (it only occurs when the light within the pencil exiting the eyepiece is not collimated). Also, it can be minimized in any particular design, and not all wide-field or long-focus eyepieces necessarily suffer from it to a significant degree.

There is a number of eyepiece designs, but vast majority of those used in astronomy are variations of just a few basic configurations (FIG. 215). In general, more elements allow better overall correction, but at the price of increased scatter and absorption of light. The exit pupil range in the figure below indicates the extent of the spherical aberration of exit pupil for the few basic eyepiece types.

FIGURE 215: Basic forms of astronomical eyepiece, with its front focal point coinciding with the focal plane FP of the objective. Ex indicates the extent of spherical aberration of the exit pupil, normally not troublesome in conventional eyepiece types. The simplest of them - Kellner - features 45-50 deg. apparent field and satisfactory correction for spherical aberration at ~f/5 and slower f-ratios. Pl�ssl has better overall correction, setting quality standards for conventional eyepieces. Orthoscopic (Abbe) usually has somewhat smaller field than Plossl, not due to inherently  inferior field definition, but for its high standards and customary use for planetary observing. Conventional wide-fields, Konig and Erfle, have better spherical correction than Kellner, but more intrusive edge astigmatism, due to their larger, 60-70 deg. fields. Most of today's standard wide-fields are variations of these two basic concepts. Nagler (Type 1), whose negative front lens allows for better correction of astigmatism, offers well corrected fields exceeding 80°. Its wide exit pupil range indicates more intrusive spherical aberration of the exit pupil ("kidney bean effect"), better corrected with later types. Darker color indicates glass of higher refractive indici. (data from Telescope Optics, Rutten/Venrooij).

12.2. Eyepiece aberrations I   ▐    12.4. Telescope eyepiece: comparative raytracing

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