10.1.2. Sub-aperture corrector examples: Two-mirror systems
The most common two-mirror candidates for sub-aperture corrector are classical Cassegrain, Dall-Kirkham and Ritchey-Chretien. In addition, the simplest two-mirror arrangement, consisting of a pair of spherical mirrors, also can be made (very) useable with the addition of sub-aperture corrector. The two examples are Field's two-mirror telescope, using a single meniscus corrector, and Klevtsov, using two-element corrector to offset the aberrations of a pair of spherical mirrors.
Sub-aperture corrector with two-mirror systems usually fills one of these two roles: it either serves as field corrector in systems that have good axial correction, or comes as an integral part of a system that cannot function without it. The former is usually referred to as field corrector, and the latter, somewhat informally, will be referred to as integrated (sub-aperture) corrector.
The advantage of sub-aperture vs. full-aperture corrector is in much less glass required, also resulting in less added weight. The disadvantage is that they generally have more stringent tolerances for both, fabrication and alignment. The chromatism of a corrector is typically very low, and usually not a factor.
Sub-aperture correctors for classical Cassegrain
In general, correcting field aberrations in classical Cassegrain, often with field curvature included, requires more or less simple two-element sub-aperture corrector. Depending on corrector's design, it can be either add-on field corrector, or integrated system corrector. Follow examples of both types. All four systems below are fast, or very fast for a Cassegrain, but have flat, well corrected fields. System A, published by Charles Rydel (SPECS), features two BK7 lenses, one plano-concave and the other bi-convex. System B, designed by Mike I. Jones (SPECS), uses two meniscus elements (FK5/BK7). Both these systems are classical Cassegrains that can function without field corrector. Unlike them, the last two systems with paraboloidal primary and spherical secondary, are useless without their integrated correctors. System C (SPECS) uses a Harmer-Wynne type corrector. System D (SPECS) uses three simple plano-convex/concave lenses in a very fast, well corrected arrangement.
For comparison, at 0.5° off-axis, coma blur in an ƒ/10 classical Cassegrain w/o corrector would be three times the Airy disc diameter, and in an f/5.9 as much as 14 times larger than the Airy disc (0.82 and 3.7 waves P-V, respectively).
Sub-aperture correctors for Dall-Kirkham telescope
Due to its spherical secondary and less aspherised primary, Dall-Kirkham (DK) telescope is easier to fabricate than any other all-reflecting two-mirror system. However, its relatively strong coma severely limits quality field already at medium focal ratios. A simple two-element sub-aperture corrector can effectively eliminate both, coma and field curvature. However, since such corrector induce under-correction, the primary is somewhat more strongly aspherised (still prolate ellipsoid). Thus these are integrated correctors. Also, DK with sub-aperture corrector is significantly more sensitive to misalignment.
The DK correctors can come in various forms, but the three general configurations are: (1) a pair of plano-concave/convex lenses of identical reversed radii, (2) two meniscus lenses in () configuration, and (3) two meniscus lenses in )( configuration. Degree of correction is similar for all (meniscus correctors are likely to show smaller ray spots, but at that high level of correction it has little or no practical consequences).
Design A (SPECS) uses corrector made of two thin PCV/PCX BK7 lenses, for near-perfect overall correction. Corrector in the design B (SPECS) has advantage of being closer to the focal plane, thus using smaller lenses. On the other hand, lenses are more difficult to make. Designs C (SPECS) and D (SPECS) illustrate the influence of primary's focal ratio on corrector's performance. In general, the faster primary, the more aberrations generated at the secondary, and the harder it is to achieve high level of correction. The last design uses 3 easy to make lenses, with the role of the separated negative lens being to cancel lateral color.
The A sub-correctors for classical Cassegrain and DK are the simplest type, only requiring two plane surfaces and two identical reversed radii. Despite their apparent simplicity, needed calculation is quite complex. The focus is on coma, which requires establishing coma mirrors' aberration coefficient for the two mirrors - given by Eq. 82 as coma aberration coefficient, which determines the peak aberration coefficient as given by Eq. 13 - and find at which corrector location it can be can be cancelled by the opposite coma induced by corrector, calculated from Eq. 98, with the effective aperture stop separation T for the corrector's front lens being the distance to the exit pupil formed by the secondary (measured from the pupil to the surface).
With proper choice of lens radii and corrector location both, lower-order coma and astigmatism can be corrected. As mentioned, primary conic is made somewhat stronger, to correct for under-correction induced by the lens corrector. The raytrace showed the presence of higher-order aberrations, which required additional radii/location adjustment, in order to have it minimized by balancing it with the lower-order form. Corrector's final parameters are the result of a lengthy series of iterations.
The result is a modified Dall-Kirkham telescope with integrated sub-aperture lens corrector, and significantly improved field quality. Diffraction limited field in the green light typically approaches 1° in diameter on flat field, up to a dozen times, or more, greater than in a comparable all-reflecting Dall-Kirkham. Quality field is also up to several times greater than in a comparable all-reflecting classical Cassegrain, and significantly greater than in a comparable all-reflecting Ritchey-Chrétien. Beyond 1° off-axis, or so, image often quickly deteriorates - more so than in all-reflecting Dall-Kirkham, classical Cassegrain and Ritchey-Chrétien - as a result of higher-order astigmatism and coma. That, however, has little importance since the field size is typically limited to about 1°, or less.
Sub-aperture corrector for the Ritchey-Chretien
The two remaining aberration in the Ritchey-Chretien (RC) are astigmatism and field curvature. Astigmatism alone significantly limits quality field even over best (curved) image surface. Over flat field, it is further reduced by defocus resulting from field curvature. Neither aberration is significant in visual use. However, considering that the usual purpose of RC systems is imaging, some form of corrective action is necessary to improve quality of the outer field. Bending detector's surface to conform to best image curvature cancels out field curvature, but astigmatism still remains. Fortunately, correcting both aberrations is possible with relatively simple sub-aperture correctors. Since generating astigmatism by a lens corrector does not necessarily generate spherical aberration as well - such as the case with coma correctors - RC systems can use add-on field correctors for that purpose. Two examples are shown below.
System at left (SPECS) uses a pair of plano-concave and plano-convex lens of the same glass (BK7), designed by Richard from U.K. Lens elements have somewhat different surface radii, which is compensated for by widening the separation. Field correction is excellent, with the error in e-line being still only 0.068 wave RMS at the edge of 0.4-degree field radius. Color error is literally non-existent: as little as 0.01 wave RMS axially at 404nm and nearly half as much at 830nm. System at right (SPECS) uses two meniscus lenses, also BK7, for just as good, near-perfect correction: the e-line RMS wavefront error ranges from 0.017 on axis to 0.031 at 0.4° off-axis; axial color error is 0.068 wave RMS at 404nm, and 0.026 at 830nm, not changing appreciably over the entire field.
Other sub-aperture correctors for two-mirror system: Field-Maksutov, Klevtsov, Celestron Edge
A simple corrector for all-spherical two-mirror system would be the ultimate in convenience, since it would make aspherizing the mirrors unnecessary. Such correctors exist, but are less in use than full-aperture correctors for all-spherical systems. The reason is that, in general, they don't offer as good overall level of correction, and/or that their fabrication and alignment tolerances are significantly more stringent. It is particularly pronounced here, due to this particular corrector type having to correct an enormous amount of aberrations. Two examples of all-spherical two-mirror systems with integrated sub-aperture corrector are shown below (obviously, the corrector here must by integrated, since spherical mirrors alone would be entirely useless as a telescope).
Field-Maksutov (SPECS) mainly corrects for spherical aberration. The corrector offsets only a small fraction of primary's coma; with the secondary offsetting nearly half of it, the system as a whole has approximately half of the primary's coma, as angular wavefront aberration. Since the final linear field for given field angle is larger than that of the primary by the factor of secondary magnification m, the final linear diffraction-limited field radius is, from Eq. 70.2, approximated by mF13/45, or FF12/45 in mm, with F1 and F being the focal ratio of the primary and system, respectively. In this particular system, with F1=3, m=3.3 and F=10, diffraction limited field radius is little over 2mm. In that, an ƒ/10 Field-Maksutov is nearly identical to a commercial ƒ/2/10 SCT (SCT with an ƒ/3 primary would, of course, have significantly less coma).
Field-Maksutov coma is somewhat dependant on the meniscus thickness. In general, thicker meniscus offsets more of primary's coma, and vice versa, but the difference is not significant. In the above system, reducing meniscus thickness by 25% increases coma by about 10%. Placing aperture stop at the tube opening (as indicated above) nearly halves the astigmatism; however, since the coma is by far the dominant aberration, and not affected by stop position (the change in coma contribution by the spherical primary is nearly entirely offset by the opposite change at the corrector and secondary) the effect is relatively small: only about 4% reduction in the RMS wavefront error at 0.5° off-axis).
Addition of the second lens in the Klevtsov design (SPECS) enables correction of both, coma and spherical aberration (ƒ/3/10 system is rescaled version from Busack's PointSpread). However, corrector adds astigmatism, rising the total system astigmatism near to double that of the primary mirror. Still, field correction is significantly better than Field's, with the RMS wavefront error of astigmatism at 0.5° off axis being about 1/3 smaller (considering that astigmatism changes with the square of field radius, vs. coma changing with field radius, Klevtsov's field quality advantage is considerably greater than what the edge error implies).
In addition, visual field in the Klevtsov should be further improved due to partial offset of its astigmatism with eyepiece astigmatism (in other words, visual field quality in the Klevtsov is limited by somewhat reduced astigmatism of the eyepiece).
Relatively recent use for sub-aperture corrector has been found in the commercial SCT telescope as well. While coma of this compact arrangement can be corrected by aspherizing the secondary, field curvature still remains, and can only be corrected by adding a field flattener. Since a pair of lenses can do both, correct for coma and field curvature, aspherizing the secondary becomes unnecessary. Two examples below are based on the Celestron's published data (EdgeHD White Paper - Final). Since no specific prescription has been given in the paper, the systems shown are not the exact replica, but the performance level should be practically identical.
Coma and field curvature can be corrected with a similar corrector consisting of a pair of BK7 lenses, but the astigmatism is nearly twice greater. It implies that better correction might be possible with some other, more expensive glass combination, As mentioned, off axis error of these systems is already practically negligible, and added expense for the further improvement - if possible - probably wouldn't be justified.
Unusual sub-aperture corrector...
interesting idea from amateurs' circles can be used for larger and
faster systems. It places sub-aperture Schmidt corrector at the focus of
the primary, in the converging cone from the secondary. Therefore, it is
Gregorian arrangement with Schmidt corrector between two concave
mirrors. It allows for
spherical primary, but corrected coma still requires aspherizing the
secondary, although considerably less than in a comparable SCT. The corrector is
significantly stronger than the full-aperture version, but it is also
significantly smaller. Corrector's depth and amount of spherochromatism
induced are similar for both. The spherochromatism is nearly doubled
versus comparable SCT; since it changes inversely to the third power of
the primary's F-number, an ~ƒ/2.4 primary would be necessary to reduce
spherochromatism in this system to the level of a commercial SCT of
similar aperture. Another option is making achromatic corrector, which
practically eliminates spherochromatism.
As shown to the left, chromatism is not intrusive in 8" aperture SGT with single-glass sub-aperture corrector (SPECS). It is at the level of 4" ƒ/70 doublet achromat. Achromatizing it with a crown/flint combination with three identical Schmidt surfaces (the one on the flint element is inverse Schmidt surface) reduces spherochromatism several times, and may be desirable for larger apertures (SPECS). Further reduction of spherochromatism is possible with separated, much more strongly aspherised Schmidt elements. Since more strongly curved Schmidt surface also induces more of coma, a compact aplanatic Gregorian with both mirrors spherical can be achieved. As this advanced design by Mike I. Jones shows, the chromatism also has nearly vanished. The downside is over 50% greater astigmatism, and fabrication difficulty of very strong double Schmidt corrector: in that respect, its stronger element is comparable to a corrector for ~ƒ/0.5 standard Schmidt system.
Obviously, the aberration calculation is, in general, considerably more complex for sub-aperture lens corrector, as opposed to a full-aperture lens corrector, mainly as a result of displaced aperture stop, especially when more than a single element (e.g. single sub-aperture meniscus lens) is required.
In principle, sub-aperture corrector is capable of correcting any single aberration. The difficulty arises from it tending to generate significant multiple aberrations, which may be difficult to match with the aberrations of the rest of the system. Often times, strongly curved surfaces are required, generating significant higher-order aberrations, and/or setting very tight fabrication/collimation tolerances. This is generally less of a problem with more relaxed radii-wise full-aperture correctors, which also can be used to manipulate main mirror's off-axis aberrations by varying the stop location. For that reason, most of quality catadioptric telescopes are made with full-aperture correctors, despite their often higher cost. To those most common in amateur telescopes will be given consideration in the following pages.