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▪ CONTENTS ◄ 10. CATADIOPTRIC TELESCOPES ▐ 10.1.2. Field flattener ► 10.1.2. Subaperture corrector examples: Singlemirror systems
PAGE HIGHLIGHTS Systems with subaperture correctors are relatively infrequent and, in the commercial telescope arena, often come with secondgrade products. This doesn't mean that highquality catadioptric systems with subaperture 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) subaperture lens corrector. Subaperture Schmidt corrector can also be used to enhance offaxis performance of a telescope (for instance, to reduce or cancel astigmatism in the RitcheyChrιtien), but it is seldom used for this purpose in amateur telescopes. Due to simplicity and popularity of Newtonian reflectors, among the most frequently encountered subaperture 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 subaperture correctors. Even a singleelement lens corrector in the form of equal or nearequal radii meniscus, can significantly improve field performance of a fast paraboloidal mirror (FIG. 153, left), or even make possible to use fast spherical mirror (FIG. 153, right). Since paraboloid forms good nearaxis 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 153: (A) Ray spot plot for a fast paraboloid with meniscustype coma corrector placed in front of the diagonal flat. With the coma nearly corrected, dominant aberration is astigmatism, limiting diffractionlimited 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 subaperture 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 PV of spherical aberration). Both correctors offer similar diffractionlimited 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 eline (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 comacorrector 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. Subaperture meniscus corrector for a single mirror can be also placed closer to the focal plane, at the bottom of the focuser. Offaxis 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 JonesBird type can also achieve good overall correction level in combination with spherical mirror (FIG. 154). 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 subaperture 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
154: LEFT: Ray spot diagram for 200mm /5.9 system with
/4 spherical
primary and JonesBird type corrector (Telescope Optics, Rutten/Venrooij),
placed in front of the flat. Black circle is the eline Airy disc. The
corrector is a singleglass lens doublet: biconcave 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 JonesBird
is approximately at 4"
/30 achromat level 
but with excessive chromatism in the violet. As the LA graph shows, it is
mostly due to chromatic defocus (secondary spectrum).
Twoglass JB corrector
gives better
results; replacing F2 with SSK3 reduces the h/r error six
fold, tenfold using PFCB1961 and FN11). For imaging applications, strong field curvature is undesirable. Thus correctors intended for imaging (or allpurpose) have one more requirement to fulfill, nearly flat image field which, in turn, requires good correction of astigmatism as well. With simple twolens corrector, it either results in relatively significant residual spherical aberration, which can only be remedied by hyperbolizing primary. As a result, flatfield twoelement 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. 155.
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 more widely separated, creating more degrees of freedom, so that combined aberrations can be brought to a negligible minimum (for instance, the Paracorrlike corrector corrects coma with the front achromat without inducing spherical aberration, but it does come at a cost of inducing enormous astigmatism, then corrected with the rear achromat  something that cannot be done without widening lens separation). Examples of this advanced corrector type are TeleVue's Paracorr and Wynne triplet. Wynnetype corrector offers excellent correction level with either paraboloidal or hyperboloidal mirrors (it is also used in twomirror systems), both in regard to monochromatic aberrations and chromatism. However, while one of the three lens elements is a simple near planoconvex 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. 156).
FIGURE 156:
TOP  Paracorrlike coma
corrector for paraboloid  which also flattens the field  consists of a
pair of achromats, negative in front. It has a negative net power,
extending focal length for 1020% (approx.). At left is design published
in Smith, Ceragioli and Berry (Telescopes, eyepieces, astrographs),
scaled to 250mm f/4.5 mirror. It uses four different nonED glasses. In
the middle is a similar arrangement using only two simple glasses
(BK7/F2), and at right is arrangement using perhaps best glass
combinations possible  or close to  including ED glass, designed by
Mike I. Jones (also scaled to 250mm f/4.5 mirror). Expectedly, the
latter has the best correction, except near edge, where higher order
astigmatism starts exploding (it is still of little consequence
visually, not only for the corrector's astigmatism being likely
dominated and offset by that of the eyepiece, but also for being still
angularly small for the eye  even in a f=24mm 80°
AFOV eyepiece, which would have field stop radius about 20mm, the Airy
disc is 4.6F/f=0.96 arc minutes, so at F=5 the blur size where the eye
just begins to recognize shape is about five times the Airy disc
diameter; in a 50mm 40° AFOV even the edge
blur would still appear as a point). Visually, the 2glass corrector is
not significantly inferior to the 4glass at left, and photographically
is only marginally inferior. In practical terms, the two are not
significantly inferior to the better corrected ED arrangement. Note that
the lenses are limited to 48mm in diameter, resulting in a significant
vignetting of the edge beams at the rear element. As mentioned, this wide corrected fields do not result in practical benefit to the visual observer (even without considering much greater magnitude of the seeing error), but can be advantageous for imaging. With smaller mirrors, it is easier to achieve high level of correction by combining subaperture corrector and hyperboloidal mirror. This is due to simple coma correctors generating undercorrection, 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. 157).
Follow a few examples of less sophisticated, but potentially useful subaperture 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 nearcollimated pencil. This requires front surface radius R1~(n1)L, with n being the glass refractive index and L being the surfacetomirrorfocus separation. Having ray heights at the two meniscus surfaces similar roughly minimize/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 PV wavefront error at paraxial focus). For the mirror, object is at infinity, stop at the surface, and, from TABLE 2, peak aberration coefficient for primary spherical alone (equaling the PV wavefront error at paraxial focus) is Sm=WPV=(K+1)D4/64R3 (for K, D, R the mirror conic, aperture diameter and radius of curvature, respectively). For the meniscus, the object is relatively close (virtual image formed by mirror for the first surface, and the image of the mirror's image for the second), and appropriate stopshift relations apply (note that all relations are for the aberration at the Gaussian focus, thus for any nonzero sum the actual PV wavefront error for primary spherical is four times smaller). Despite this being a singlelens corrector, it is not simple to formulate required specifications. One reason is that it is not a thin lens, and relatively simple thin lens aberration relations do not apply. Instead, aberrations are to be calculated for each of the two surfaces, with the object for the first lens surface being the (virtual) image formed by the mirror, and for the second lens surface the (virtual) image formed by the first surface. Both lens surfaces have displaced stop: for the first, it is at the mirror, and for the second it is the image of the stop (mirror) formed by the first surface. All these elements need to be calculated in order to obtain parameter values for the stopshift relation applicable to the surface, which is fairly complex in its form. In addition, strongly curved surfaces tend to generate significant higherorder aberration terms, requiring additional, more complex calculations for determining design specifications accurately. Keeping it simple, details of parameter calculation for subaperture correctors will be only outlined in general terms. 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, if the ray height differential on its two surfaces is negligible. 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 higherorder coma, it is best done with raytracing software, such as OSLO (it is also needed to optimize higher and lowerorder spherical aberrations, as well as chromatic correction). First approximation for the needed thickness is ~1/14 of the correctortooriginalfocus separation. For instance, BK7 coma corrector for 200mm /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 comalimited 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 Fline, and 1/116 wave at the red Cline. Centerfield 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 fratio, 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 23% 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 2030% to arrive at a nearminimum level. Astigmatism and field curvature increase significantly. In order to minimize coma, the meniscus needs to be 23 times thicker (roughly) than with a paraboloid. EXAMPLE 2: Sphere with subaperture doublet lens corrector  Simple doublet corrector for spherical mirror, made of two equalradius planoconvex and planoconcave 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 m=1000mm (/5), with the following lowerorder 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, from q=(R2+R1)/(R2R1), are q1=q2=1. With the front lens to mirror image distance O1, the lens position factor p1= (21/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=(42/O1)1 (keep in mind that with the light coming to the mirror from left, 1 is numerically negative, while 2 and objecttolens 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 /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 JonesBird 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 fieldwise 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 raytracing. EXAMPLE 3: Corrective teleextender 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 subaperture corrector requires quite involved procedure, the calculation will be only outlined, and the effect illustrated with a slightly modified (SPECS) teleextender 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=1L/B, with L being the length between the lens and the eyepiece field stop, and B 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 B~L/(M1) 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 teleextender is determining the doublet focal length D. From Eq. 100, it is approximated from D=(1t)/(1), where 1 is the mirror focal length, t the mirrortodoublet separation and the desired final focal length (the value for D 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 D is known, the needed focal lengths of its two elements needed to achromatize are, from Eq. 43, fD1=(V1V2)/V1 and D2=(V2V1)/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 offaxis aberrations. Offaxis aberrations can be calculated either surface by surface, as the example of a singlelens field flattener, or using lens relations, Eq. 9899.1. First two or three outcomes usually outline direction and limitations within given set of parameters. Such procedure would have lead into designing corrective teleextender for a fast Newtonian. Although originally designed as a Barlow for SchmidtCassegrain telescope, teleextender 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 /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 diffractionlimited (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 diffractionlimited level in the blue Fline (486nm), and still better in the red C line, at 0.039 wave RMS. This is roughly the chromatism level of a 100mm /70 doublet achromat, and nearly as good as an apochromat. Subaperture correctors can also be used with twomirror 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 addon. Typical subaperture correctors in twomirror systems are comacorrector in DallKirkham, or astigmatism/field curvature corrector in aplanatic Cassegrain (RitcheyChrιtien) telescope. But they also can be used in systems with fullaperture corrector, either to maximize performance, or to allow for easier fabrication of the fullcorrector, or both. One such example is aplanatic HoughtonCassegrain with both, full and subaperture being a planoconvex/concave lens pair (FIG. 138). In general, aberrations induced by a subaperture 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 offaxis aberrations. While subaperture correctors can be very complex, a simple singlelens doublet, as illustrated with the above examples, can be very effective. A brief overview of the aberration properties of a thinlenselement subaperture corrector follows.
