10.2. Sub-aperture corrector: Examples

EXAMPLE 1, single mirror with sub-aperture lens corrector: simple doublet corrector for spherical mirror, made of two 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 calculation for spherical aberration, the lenses are plano-convex/concave, with the curved side facing the mirror. The mirror is 200mm with ƒm=1000mm (ƒ/5), with the following 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 the 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 (wavefront error, effectively appropriate to a 0.8F mirror, or f/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, shown on the page top, corrects 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 - especially spherical aberration - using the above formulas are only approximate. It is due to both, possibly significant higher order aberrations and lens' thickness/separation. Final design optimization requires raytracing.

EXAMPLE 2, two-mirror system with sub-aperture corrector: 250mm f/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 secondary 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 the chosen location, it is needed to calculate the 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.

With proper choice of lens radii (thus focal length and position factors) both, lower-order coma and astigmatism can be corrected by slightly adjusting corrector location. However, the lenses are positioned differently, 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, especially astigmatism, which required some more of 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.

The chromatism, being very low, is insignificant disadvantage. However, seems that the corrector is limited to a relatively narrow secondary magnification range between 2 and 3 (with accessible focus position); lowering secondary magnifications lowers higher-order astigmatism, but increases lateral color, while lowering it results in a quickly rising residual coma and higher-order astigmatism. These can be suppressed by abandoning the simplicity of the corrector design, but it does have 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, a system employing the lens corrector can use faster primary, for the system focal ratios as fast as ~f/7.