10.1.2. Sub-aperture corrector examples (3)

EXAMPLE 2: Meniscus corrector - As illustrated on previous page, meniscus corrector can be used for both, spherical and paraboloidal mirror. Main priority with the former is correction of spherical aberration, without introducing significant aberrations of other types and, if possible, 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.

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 3: 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 4: 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.

EXAMPLE 5: Two-mirror system with sub-aperture corrector - 250mm ƒ/3.6/10 Dall-Kirkham telescope with a simple two-element corrector consisting of a pair of plano-convex and plano-concave lens with equal radii of curvature. This is the same corrector type as in the first example (also single glass, BK7), but the result should be better if the corrector is designed as an integral part of the system; that is, if the primary conic can be adjusted to compensate for spherical aberration induced by a coma-inducing doublet corrector. Since the calculation is considerably more complex than for a single mirror, it will be only verbally outlined.

Starting with the corrector location nearly coinciding with that of the primary, the attention can be concentrated on coma, since spherical aberration of the corrector can always be offset by adjusting the primary conic, and hoping that astigmatism will also be corrected, or remain low. To find out if the coma of the two mirrors - given by Eq. 82 as coma aberration coefficient, which determines the peak aberration coefficient as given by Eq. 13 - can be corrected at a chosen location, it is needed to calculate coma coefficient for the two lenses, from Eq. 98.1, 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. However, the lenses are positioned differently than in Newtonian configuration, facing the secondary with their flat side, the negative lens in front. 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. The result is a modified Dall-Kirkham telescope (SPEC'S) with integrated sub-aperture lens corrector with significantly improved field quality. This is not a fully optimized unit, but does show most of the benefit obtainable with this type of sub-aperture lens corrector.

Diffraction limited field in green light is about 0.9° in diameter along best image surface, some 7 times larger, linearly, than in a comparable all-reflecting Dall-Kirkham, 2 times larger than in a comparable all-reflecting classical Cassegrain, and about 50% greater than in a comparable all-reflecting Ritchey-Chrétien. Farther off-axis image quickly deteriorates - more so than in all-reflecting Dall-Kirkham, classical Cassegrain and Ritchey-Chrétien - as a result of higher-order astigmatism. It, however, has little importance since the field size is typically limited to about 1°, or less.

Diffraction-limited flat field is nearly 0.7° in diameter, five times greater than in a comparable all-reflecting DK, nearly double the diffraction-limited flat field of a comparable classical Cassegrain, and 1.8 times greater than in a comparable RC. If desired, nearly flat field is attainable as well, but only with secondary size approaching D/2. However, quality over flat field remains good with smaller secondary sizes as well, making Dall-Kirkham with integrated sub-aperture corrector preferred choice among other two-mirror systems, either all-reflecting or comparable catadioptric arrangements.

The chromatism, being very low, is insignificant disadvantage. However, seems that the corrector's best performance is limited to a relatively narrow secondary magnification range between 2 and 3 (with accessible focus position); lowering secondary magnification (i.e. increasing its relative size) lowers higher-order astigmatism, but increases corrector-to-focus distance, thus also needed lens diameter, while systems with higher secondary magnifications (i.e. smaller secondary size) have reduced corrector-to-focus separation and smaller lenses, but stronger lens radii and, consequently, higher-order astigmatism. Overall performance can be further improved by abandoning the simplicity of the corrector design, but it has insomuch limited appeal. This sets the minimum secondary size in 0.3D to 0.4D range, not significantly different from the all-reflecting arrangement. However, DK system employing the lens corrector can use faster primary, for the system focal ratios as fast as ~ƒ/7, or even faster.