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.......................................................................................... CONTENTS
11.6.1. Approximating Maksutov corrector radiiDue to very strongly curved surfaces, amount and type of spherical aberration generated by achromatic meniscus corrector in the Maksutov camera is very sensitive to both, surface radii differential and variation in corrector thickness. Also, significant higher-order spherical aberration it generates makes lower-order expression alone insufficient for determining corrector properties needed to minimize spherical aberration of the mirror. Thus, there is no simple way to find out specific corrector radii and thickness for a given mirror. However, it is possible to arrive at a good approximation, either for informative purposes, or to be used as a starting point in designing corrector. Expressing meniscus thickness and first radius in terms of the mirror radius of curvature R as t=τR, and R1=ŕR allows to set a relation between these two parameters based on Eq. 125, as [ŕ-(1-1/n2)τ]ŕ3=-(n+2)(n-1)2τ/2n3. With t«ŕ, neglecting (1-1/n2)τ leads to the approximation ŕ4~(n+2)(n-1)2τ/2n3 and the first iteration for needed value of the first radius for corrected third-order spherical: R''1~[(n+2)(n-1)2τ/2n3]1/4R (126) where n is, as before, the corrector glass refractive index, and τ its center thickness. Once the first approximation of R1 is known, a correction factor compensating for the neglected value can be applied, giving closer approximation of the needed first radius as: R'1~[1-(1-1/n2)τ/ŕ]R''1 (126.1) The original corrector thickness, chosen by Maksutov, was t=D/10. Thicker corrector reduces somewhat the residual spherical aberration, but at the price of increased lateral color. Optimum thickness is, as noted by Rutten and Venrooij (p101), probably some 50% greater, thus t~D/7 (either can be easily expressed in terms of mirror radius of curvature R for any specific mirror). As mentioned, the corrector with these parameters nearly corrects third-order spherical aberration of the mirror of radius of curvature R. However, due to strong lens curvatures, there is always residual fifth-order aberration present. It is negligible compared to a total third-order aberration of the mirror, but once it is near corrected, the higher-order aberration becomes a factor. For a minimum total aberration, the two need to be optimally balanced, by combining their near-equal amounts of opposite signs. Due to this, as well as due to the approximations made for simplicity, near-optimum first radius R1 will be somewhat smaller than what is directly indicated by Eq. 126-126.1. Typically, the best first radius is closely approximated by R1~0.94R'1. Nearly as good final approximation is given by R1~0.9R''1, which indicates that the first iteration is a sufficiently good basis for estimating the needed first radius. Once R1 is known, the second radius is obtained from: R2=R1-(1-1/n2)t (127) Keep in mind that the corrector is convex toward the mirror, thus with the radii nominally negative. This means that R2 is weaker than R1, and the meniscus is a weak negative lens, thicker at the edge than at the center. With these parameters, a system is roughly corrected; however, the approximate character of the procedure, combined with high system sensitivity to changes in optimum meniscus parameters, require verification and likely correction. Further detailed optimization improves quality level, but the main benefit is achieved by balancing the two forms of spherical aberration. Final optimization is best done with the help of ray tracing software, as illustrated on FIG. 93.
It should be noted that the above approximation is for the corrector-to-mirror separation of ~2R/3, needed for zero system coma. With the change in the position of the corrector, its radii needed for zero system spherical aberration also change. This is due to Maksutov corrector having relatively significant power (much more so than the Schmidt corrector), which effectively changes for the mirror with their separation. In general, decrease in separation increases the effective corrector power, requiring weaker radii, and vice versa. Also, with focal ratios (~ƒ/3 and larger) the higher-order coma of the corrector increases sufficiently to cause the zero-coma corrector separation be somewhat smaller. In practice, there is no need to calculate for best paraxial correction, knowing that the optimum overall correction is different. The correction factor for the optimum overall system correction can be applied directly to R"1 given by Eq. 126, in an empirical relation taking into account slight variations due to the changes in relative thickness τ and mirror focal ratio F as: R1~(1-3τ-F/100)R"1 (128) and R2=R1-(1-1/n2)t/0.97 (128.1) Evidently, needed first radius of the corrector becomes slightly stronger as the relative corrector thickness and mirror F-number increase. The 0.97 factor is not strictly fixed; it determines the level of best chromatic correction, not necessarily the very point of numerical maximum. Slight deviation from it may offer correction modes more appropriate in some instances, but the difference - or benefit, if any - is typically very small.
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