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◄ 10. CATADIOPTRIC TELESCOPES   ▐    10.1.2. Single-lens field flattener ►

10.1.2. Sub-aperture corrector examples: Single-mirror systems

Systems with sub-aperture correctors are relatively infrequent and, in the commercial telescope arena, often come with second-grade products. This doesn't mean that high-quality catadioptric systems with sub-aperture lens elements can't be built. One example is hyperbolic astrograph, consisting of a hyperbolic primary and either two (Rosin), three (Wynne) or four element (a pair of doublets) sub-aperture lens corrector. Sub-aperture Schmidt corrector can also be used to enhance off-axis performance of a telescope (for instance, to reduce or cancel astigmatism in the Ritchey-Chrιtien), but it is seldom used for this purpose in amateur telescopes.

Due to simplicity and popularity of Newtonian reflectors, among the most frequently encountered sub-aperture corrector forms are coma corrector for paraboloidal mirror, and spherical aberration corrector for a sphere. The goal of such correctors is to minimize or eliminate the dominant aberration without introducing significant other significant aberrations. In the case of spherical mirror, it is desirable to correct both, axial aberration and coma. For visual use, these tasks often can be accomplished with relatively simple sub-aperture correctors.

Even a single-element lens corrector in the form of equal or near-equal radii meniscus, can significantly improve field performance of a fast paraboloidal mirror (FIG. 97, left), or even make possible to use fast spherical mirror (FIG. 97, right). Since paraboloid forms good near-axis image on its own, the former is a hybrid catadioptric. The latter is a true catadioptric, since fast spherical mirrors alone are pretty much useless for astronomical use. In both instances the corrector is placed in front of the diagonal flat.

FIGURE 97: (A) Ray spot plot for a fast paraboloid with meniscus-type coma corrector placed in front of the diagonal flat. With the coma nearly corrected, dominant aberration is astigmatism, limiting diffraction-limited field to 0.18° radius, over three times that of the paraboloid alone; since the astigmatism offsets with that of the eyepiece, actual visual field is larger, limited by eyepiece astigmatism. (B) Ray spot plot for a fast sphere with sub-aperture meniscus corrector in front of the diagonal. This one has slightly different radii, which is needed to generate sufficient corrective spherical aberration at a reasonable lens thickness (mirror alone has 1.4 wave P-V of spherical aberration). Both correctors offer similar diffraction-limited field (angular), and near perfect longitudinal chromatic correction. Lateral chromatism is somewhat greater in the latter, due to the thicker meniscus, but still within tolerable. Neither corrector is fully optimized, but should illustrate most of their potential. While appearing quite simple to make, they require very high surface radius accuracy for given meniscus thickness, which has nearly as tight tolerance itself. This is partly offset by greater simplicity of mounting and collimating a single element. The black circle represents e-line (546nm) Airy disc.   SPEC'S

Main priority with the meniscus corrector in combination with spherical mirror is correction of spherical aberration, without introducing significant aberrations of other types and, then, to reduce mirror's coma. Meniscus orientation is concave toward primary; the reverse orientation wouldn't work due to significantly less spherical aberration generated, and unfavorable aberration distribution between the two surfaces (the corrective front surface generates much less of the aberration than the rear surface, unless it is made much more strongly curved, which would make chromatism unacceptable). This is also viable orientation for coma-corrector in a paraboloid, since a meniscus of equal radii in this form doesn't introduce appreciable amount of spherical aberration but, if properly designed, can reduce or nearly cancel mirror's coma.

Sub-aperture meniscus corrector for a single mirror can be also placed closer to the focal plane, at the bottom of the focuser. Off-axis correction is nearly as good as with the corrector placed in front of the diagonal flat, while offering advantages of smaller meniscus and lighter, smaller diagonal flat assembly. It is described in more detail toward page bottom.

A simple doublet corrector of Jones-Bird type can also achieve good overall correction level in combination with spherical mirror (FIG. 98). Even simpler version, a split meniscus concave toward mirror, with the front surface somewhat stronger (intermediate form toward the above meniscus corrector) , also corrects for spherical aberration and coma, while introducing strong astigmatism and field curvature. It offers better control of secondary spectrum, but has some lateral color.

 Another more recent sub-aperture corrector type for spherical mirror, incorporated in the Cape Newise telescope, consists of a pair of widely separated doublets (one in front of the diagonal, the other at the bottom of the focuser tube). It seems to be capable of very good performance. However, its specifications are not available.

FIGURE 98: LEFT: Ray spot diagram for 200mm ƒ/5.9 system with ƒ/4 spherical primary and Jones-Bird type corrector (Telescope Optics, Rutten/Venrooij), placed in front of the flat. Black circle is the e-line Airy disc. The corrector is a single-glass lens doublet: bi-concave front lens and positive meniscus. Its strong astigmatism can be advantageous for visual use, to partly offset strong astigmatism in conventional eyepieces (which also relaxes the effective field curvature, which is as strong as 70mm for the objective's image). Chromatic correction of the Jones-Bird is approximately at 4" ƒ/30 achromat level - but with excessive chromatism in the violet. As the LA graph shows, it is mostly due to chromatic defocus (secondary spectrum). Two-glass J-B corrector gives better results; replacing F2 with SSK3 reduces the h/r error six fold, tenfold using PFCB19-61 and FN11).
RIGHT: Same corrector type, slightly modified (the front lens becomes negative meniscus) with paraboloid. Level of correction is similar, except for traces of spherical aberration and coma. Chromatic correction is noticeably improved due to the use of two different glasses, BK7 and SSK2. It also resulted in significantly lower corrector magnification (f/4.6 effective system). Correction of all aberrations can be further improved by use of low dispersion glasses (orange frame).      SPEC'S
 

For imaging applications, strong field curvature is undesirable. Thus correctors intended for imaging (or all-purpose) have one more requirement to fulfill, nearly flat image field which, in turn, requires good correction of astigmatism as well. With simple two-lens corrector, it either results in relatively significant residual spherical aberration, which can only be remedied by hyperbolizing primary. 

As a result, flat-field two-element coma correctors for paraboloid induce spherical aberration; the larger, faster mirror, the more so. The basic corrector form here is the Ross corrector, originally a negative meniscus followed by biconvex lens. According to Bratislav Čurćić, the MPCC coma corrector is of this type, while the former Lumicon coma corrector is modified by replacing biconvex lens with positive meniscus. Examples of these two corrector types, as well as of the split meniscus form are given in FIG. 99.


FIGURE 99: Three types of a flat field coma corrector with BK7 lenses. Modified Ross, which is actually Rosin-type corrector, has best color correction, although not significantly in practical terms. All three correctors induce spherical undercorrection, particularly split meniscus, which would require slower parabola for acceptable performance. The P-V wavefront errors with 200mm f/4 paraboloid are 0.73, 0.76 and 1.07 wave, respectively (note that the axial blurs for paraboloid are in reduced proportion). Since the error scales with DF3, a quarter-wave 200mm paraboloid for the three would be f/5.7, f/5.8 and f/6.5, respectively (for given aperture, needed mirror conic for cancelled spherical remains unchanged, due to the offset between the increased corrector's and mirror's aberrations).      SPEC'S

For complete correction of a paraboloid, more complex correctors are required. In general, they need to have three (or more) single lenses, or two or more achromatized doublets. The lenses are separated, so that combined aberrations can be brought to a negligible minimum. Examples of this advanced corrector type are TeleVue's Paracorr and Wynne triplet.

Wynne-type corrector offers excellent correction level with either paraboloidal or hyperboloidal mirrors (it is also used in two-mirror systems), both in regard to monochromatic aberrations and chromatism. However, while one of the three lens elements is a simple near plano-convex or PCX, the other two are strongly curved, thin menisci, very demanding in both, fabrication and positioning/centering. Still, the Wynne corrector is not out of reach for advanced amateur telescope makers and designers. Large, fast amateur mirrors in particular benefit from a corrector of this type (FIG. 100).


FIGURE 100: Three-element Wynne-type corrector for fast paraboloid designed by Richard Snashall (optical design pages). It uses a single common crown glass (BK7) to achieve near perfect correction over 0.5° field radius (raytrace by OSLO EDU). Black circle represents the e-line (546nm) Airy disc. Corrector performance further improves with reducing the aperture, and more so with the relative aperture reduction. This level of correction probably reaches near-limit with a 3-element Wynne corrector made of a single common glass. While it is safe to assume that even better correction level is possible if using different glasses, that wouldn't result in practical benefit to an average user, since the size of corrected field is already exceeding standard requirements, and the aberration level is well below detectable (even without considering much greater magnitude of the seeing error). Note that this type of corrector can also be used to correct hyperboloidal mirror, with similar results (it is used to correct prime focus image of large observatory Ritchey-Chrιtien telescopes). While the correction level achievable with Wynne corrector is impressive, these lenses pack large amounts of aberrations - astigmatism, coma and chromatism - thus require tight fabrication and mounting standards. The raytrace indicates that they are well worth the effort and time.                     SPEC'S
 

With smaller mirrors, it is easier to achieve high level of correction by combining sub-aperture corrector and hyperboloidal mirror. This is due to simple coma correctors generating under-correction, thus introducing one aberration while correcting for the other. Having the mirror hyperboloidal practically takes spherical aberration out of the equation, allowing corrector design to be optimized for correcting other aberrations. For that reason, hyperbolic astrographs can be designed to a high level of correction with quite simple doublets in place of the correcting element (FIG. 101).


FIGURE 101: Ray spot plot for 10" ƒ/4 hyperbolic astrograph with Rosin-type corrector designed by Mike I. Jones (black circles represent the e-line Airy disc). The corrector consists of a positive and negative meniscus, placed at the bottom of the focuser base. The primary is a hyperboloid with the conic K=-1.408. Evidently, image quality is excellent across the flat 1o field. The LA graph shows most colors focusing tightly together. Only the violet end departs somewhat, resulting in the violet h-line blur of approximately 12 microns in diameter (~0.2 wave RMS error), also exhibiting some lateral color error. However, it is still quite small, and can easily be remedied by using slightly different glasses, for instance LAF9 for the front element. Low level of aberrations allows upscaling to significantly larger apertures while preserving high correction level. The two corrector lens elements are relatively easy to fabricate; main fabrication difficulty is aspherizing the primary. Needed in-focus distance for the corrector is relatively small, allowing it to be mounted below the focuser base. The final focus to corrector separation, on the other hand, is large enough to accommodate use of various accessories, if desired. A very good example of how successful can be combining hyperboloidal mirror and relatively simple two-element sub-aperture corrector in creating near-perfect photo-visual telescope/astrograph.     SPEC'S
 

Follow a few examples of less sophisticated, but potentially useful sub-aperture correctors for a single mirror:

EXAMPLE 1: Close meniscus corrector - As illustrated on the top, meniscus corrector can be used for both, spherical and paraboloidal mirror. While single meniscus can always correct for coma, the amount of residual astigmatism depends on its location. In general, the closer it is to the focal plane, the higher residual astigmatism. Still, meniscus corrector closer to the image plane can significantly improve field definition, and may be preferred for being smaller and easier to mount and dismount. Follows an example of such a corrector. 

The starting point for either type of meniscus corrector is a form with the front radius transforming the incident converging cone into near-collimated pencil. This requires front surface radius R1~(n-1)L, with n being the glass refractive index and L being the surface-to-mirror-focus separation. Having ray heights at the two meniscus surfaces similar roughly minimizes balances 4th and 6th order spherical aberration, so that they can be combined to optimize for a minimum total aberration. The 4th order aberration total is given by a sum of the three 4th order peak aberration coefficients (equaling the P-V wavefront error at paraxial focus) - Sm=(K+1)D4/64R3, S1=(n-1)n2h14/8R13 and S2=-(n-1)n2h24/8R23 - for the mirror, front and rear meniscus surface, respectively (D is the mirror diameter, and h1,2 is the marginal ray height at the corrector front and rear surface, respectively).

With R being numerically negative, and both R1 and R2 positive, spherical aberration of the meniscus is cancelled in the first approximation for R1=R2. Chromatic correction is also at the optimization level. For paraboloid, all that is needed is to find out the appropriate thickness that will correct for mirror's coma. Since it typically involves balancing lower- and higher-order coma, it is best done with ray-tracing software, such as OSLO (it is also needed to optimize higher- and lower-order spherical aberrations, as well as chromatic correction). First approximation of needed thickness is ~1/14 of the corrector-to-original-focus separation. For instance, BK7 coma corrector for 200mm ƒ/4 paraboloid located (front  surface) at 100mm in front of the mirror focus has, in the first approximation R1,2=52mm and 7mm center thickness.

With these parameters, the optimum location is found at 95mm in front of the original mirror focus (shown to the left). Diffraction limited field is 0.27 degrees in diameter, set by astigmatism, about four times stronger than mirror's own. That makes corrected field nearly 5 times larger, linearly, than the original coma-limited field, and expectedly larger in visual use, due to the offset with (stronger to much stronger) eyepiece astigmatism of opposite sign. Chromatic correction is nearly perfect with respect to secondary spectrum, with the RMS error 1/130 wave at the blue F-line, and 1/116 wave at the red C-line. Center-field correction is 1/160 wave RMS. Lateral color is present, but low, approximately at a level found in eyepieces (significantly lower than in the typical Kellner). Since meniscus generates coma according to the f-ratio, it works for any aperture size, as long as it is at the same distance from mirror's focus (slight axial adjustments may be necessary to optimize color correction). Tolerances are fairly forgiving: up to 2-3% deviation in axial displacement, thickness deviation, or radius (it needs to be near equal on both sides) will not result in appreciable error. If suitable lenses are available, corrector can be made out of a PCX/PCV pair.

Correcting spherical aberration of a spherical mirror requires slightly weakening rear corrector radius relative to the front. Since the aberration contribution of either surface of the corrector is several times larger than that of the mirror, and the aberration contribution changes with the 3rd power of the radius, the rear radius is typically ~5% weaker. Due to the change in the powers, chromatic correction is compromised, and typically requires reduction in both radii by 20-30% to arrive at a near-minimum level. Astigmatism and field curvature increase significantly. In order to minimize coma, the meniscus needs to be 2-3 times thicker (roughly) than with a paraboloid.

EXAMPLE 2: Sphere with sub-aperture doublet lens corrector - Simple doublet corrector for spherical mirror, made of two equal-radius plano-convex and plano-concave K11 lenses (ne=1.5) in near contact. The simplest way to keep them achromatic is to have the two powers near equal and of opposite signs. To simplify the calculation, only correction for spherical aberration will be sought. The mirror is 200mm with ƒm=1000mm (ƒ/5), with the following lower-order peak aberration coefficients: Sm=0.003125, Cm=0.001091, and Am=-0.000381 for spherical aberration, coma and astigmatism, respectively, with the last two for 0.5° field angle.

With the two lenses facing the mirror with their curved side, positive lens in front, the shape factors are q1=q2=1. With the front lens to mirror image distance O1, the lens position factor p1= (2ƒ1/O1)-1. For the second lens, the object is the image formed by the front lens, which is at half its focal length to the right behind it (obtained from lensmaker's formula, which also can be used to calculate the rear lens' stop separation). That determines position factor for the rear lens as p2=(4ƒ2/O1)-1 (keep in mind that with the light coming to the mirror from left, ƒ1 is numerically negative, while ƒ2 and object-to-lens distance O1 are positive).

Substituting n, q1 and q2 in Eq. 97 gives simplified expressions for the aberration coefficients for the front and rear lens that can be used for calculations (when obtaining peak aberration coefficient, D is the marginal ray height at the lens surface). With the front corrector lens placed 100mm inside the mirror focus, spherical aberration of the mirror is corrected with |f1,2|~240mm; however, the system coma is prohibitive, being over four times that of the mirror. At the other viable corrector position, about 250mm inside the mirror focus (in front of the diagonal), with |f1,2|=1000mm, color correction is still perfect, but the coma is reduced to double that of the mirror (the wavefront error is effectively appropriate to that of a mirror with the F number smaller by a factor of 0.8, or ƒ/4 in this case). Astigmatism is comparably negligible, effectively flattening the field. Note that both lenses use BK7 crown glass (SPEC'S).

Performance of the corrector could be improved by bending the lenses and using two different glasses. However, in this simple form, any significant gain in correcting one aberration - in this case coma - can only be achieved by allowing significant increase in one or more of other aberrations. For instance, the alternative Jones-Bird corrector does correct for coma, but at a price of introducing strong astigmatism/field curvature. It also requires more complicated lens shapes, with two different glasses to control chromatism (which remains compromised in the violet). Nevertheless, it is still advantageous field-wise for visual observing, not only because it offers some 50% wider diffraction limited field (linear) in the image of the objective, but also because its astigmatism partly cancels that of the eyepiece.

Note that aberration calculations using 3rd order thin lens formulas can be very approximate, due to possibly significant higher order aberrations, and lens' thickness factor. Final design optimization requires ray-tracing.

EXAMPLE 3: Corrective tele-extender lens - A simple doublet of negative power placed at the bottom of a focuser in fast Newtonians would, by extending converging cone, make possible reduction in the minimum size of the diagonal mirror. It can also be designed to reduce coma, while inducing low spherical aberration and acceptable astigmatism. Since even this simple sub-aperture corrector requires quite involved procedure, the calculation will be only outlined, and the effect illustrated with a slightly modified (SPECS) tele-extender from Telescope Optics, Rutten and Venrooij.

A few words about Barlow Lens. It is designed to extend the focal length, without introducing significant aberrations to a telescope. Any Barlow with a single lens group at the bottom of the barrel has magnification factor M=1-L/ƒB, with L being the length between the lens and the eyepiece field stop, and ƒB the Barlow focal length, numerically negative (this means that, for instance, inserting a diagonal into Barlow appropriately increases its magnification factor). The focal length can be approximated if the magnification factor is known from ƒB~-L/(M-1) with L being the distance between the lens group and the top of Barlow's barrel. Also, approximate axial separation of the original focus from Barlow lens is given by ~L/M.

The first step in designing corrective tele-extender is determining the doublet focal length ƒD. From Eq. 100, it is approximated from ƒD=(ƒ1-t)ƒ/(ƒ1-ƒ), where ƒ1 is the mirror focal length, t the mirror-to-doublet separation and ƒ the desired final focal length (the value for ƒD is approximate mainly due to it being relatively close to the image for lenses to fit tightly into a thin lens definition). Once the doublet focal length ƒD is known, the needed focal lengths of its two elements needed to achromatize are, from Eq. 43, fD1=(V1-V2)ƒ/V1 and ƒD2=(V2-V1)ƒ/V1, with V1 and V2 being the respective glass Abbe numbers.

The next step is determining needed lens shape for corrected spherical aberration. With the glasses and lens location known, after substituting index of refraction and position factor for the two lenses in Eq. 97, it can be reduced to quadratic equation; then, deciding for the initial shape factor of one of the lens elements, the other can be solved for its shape factor q. This gives the frame within which the lenses can be bent in order to optimize for off-axis aberrations.

Off-axis aberrations can be calculated either surface by surface, as in the previous example (field flattener), or using lens relations, Eq. 98-99.1. First two or three outcomes usually outline direction and limitations within given set of parameters.

Such procedure would have lead into designing corrective tele-extender for a fast Newtonian. Although originally designed as a Barlow for Schmidt-Cassegrain telescope, tele-extender from the Rutten/Venrooij's book happened to fit for this purpose due to significant negative coma, needed to offset mirror's coma. The results are quite acceptable, as can be seen on the ray spot plot (for flat field, best field Rc=-300) for a catadioptric 200mm ƒ/4/8.4 catadioptric Newtonian with paraboloidal mirror (SPECS). Other than correcting for the coma, the extender lens, if permanently mounted, allows for some 20% smaller diagonal mirror.

The doublet consists of a positive flint (F2) front element, and negative crown (BK7) element, with very small spacing (lenses are nearly touching). Center field correction is 0.025 wave RMS, with diffraction-limited (0.80 Strehl) flat field diameter, set by the  lens' astigmatism, of ~0.3°. Being opposite in sign to that of most conventional eyepieces, astigmatism induced by this doublet is likely to have small positive effect on the visual field quality. At 0.074 wave RMS, the system is at the diffraction-limited level in the blue F-line (486nm), and still better in the red C line, at 0.039 wave RMS. This is roughly the chromatism level of a 100mm ƒ/70 doublet achromat, and nearly as good as an apochromat.

Sub-aperture correctors can also be used with two-mirror telescopes, usually with the goal of improving field quality. As with the above examples of the Newtonian telescopes corrector, they can be either integral part of a system, or an optional add-on. Typical sub-aperture correctors in two-mirror systems are coma-corrector in Dall-Kirkham, or astigmatism/field curvature corrector in aplanatic Cassegrain (Ritchey-Chrιtien) telescope. But they also can be used in systems with full-aperture corrector, either to maximize performance, or to allow for easier fabrication of the full-corrector, or both. One such example is aplanatic Houghton-Cassegrain with both, full- and sub-aperture being a plano-convex/concave lens pair (FIG. 138).

In general, aberrations induced by a sub-aperture corrector are determined by its effective diameter, as well as the element shape and power. Ideally, its monochromatic aberrations would nearly balance out with those of the mirror (or mirrors), while the chromatism it induces should be negligible. It is not always possible; in principle, axial monochromatic correction is a priority, followed by acceptable chromatic correction and correction of off-axis aberrations. While sub-aperture correctors can be very complex, a simple single-lens doublet, as illustrated with the above examples, can be very effective. A brief overview of the aberration properties of a thin-lens-element sub-aperture corrector follows.


◄ 10. CATADIOPTRIC TELESCOPES   ▐    10.1.2. Single-lens field flattener ►

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