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8.2.4. DallKirkham telescope
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8.2.6. Miscollimation, close focusing
► 8.2.5. Unusual twomirror systems: Loveday, Schwarzschild, Couder PAGE HIGHLIGHTS Twomirror systems can be modified so that the secondary reflects light back to the primary mirror, with the final focus forming after this last, third reflection. Best known system of this kind is LovedayCassegrain, using a pair of confocal paraboloids (Mersenne arrangement). After the third reflection (the second from the primary) the final focus is formed beyond the secondary. Coma is identical to that in a comparable Cassegrain, while the astigmatism is smaller by a factor of (m2+η)/(1+η)km2, resulting in lower field curvature as well. By aspherizing the mirrors somewhat more, systems corrected for either coma or both, coma and astigmatism can be obtained. In the Cassegrain configuration, however, design constraints impose severe limits to the useable field size, with the added drawback of relatively large effective central obstruction. In the Gregorian arrangement, while the central obstruction remains relatively large, much wider fields are possible, with the only remaining aberration being field curvature (FIG. 125). Such system was, to my knowledge  credit to Mr. Charles Rydel, President of the Commission des Instruments of the Societe Astronomique de France first described by Shaffer.
FIGURE 125: Twomirror 3reflection system in the Gregorian arrangement. Concave secondary mirror (S) reflects light back to the primary (P), which then forms the final focus through an opening on the secondary. Correction of all three primary pointimage aberrations, spherical, coma and astigmatism is possible with ellipsoidal primary and hyperboloidal secondary mirror. The only remaining aberration is relatively strong field curvature. The final system's focal ratio F is larger by nearly 1/3 than focal ratio of the primary. Originally, the arrangement was first published by Shaffer, but somewhat better corrected systems of this kind are achievable. These systems are effectively threemirror systems, and aberration coefficients are more complicated. It would suffice here to give a working prescription. Relative system parameters (units of the primary radius of curvature) of the Gregorian twomirror 3reflection anastigmatic aplanat are very simple:
S/R1 = 0.7252 S being the primarytosecondary separation, R1 and R2 the primary and secondary radius of curvature, respectively, K1 and K2 the primary and secondary conic, respectively, and Rp the Petzval (image) curvature which, in the absence of astigmatism, coincides with best image surface. These parameters are nearly optimized for an ƒ/3 system; they are scalable by either aperture, or primary's Fnumber. Scaling by the aperture doesn't require any changes, while slower system require slightly stronger secondary conic to optimally rebalance spherical aberration. All aberrations  except field curvature  are well corrected. Axial correction for 400mm ƒ/3 system is 0.041 wave RMS of balanced higherorder spherical, with the balanced higherorder coma limiting diffraction limited field to 0.63° radius. Higher order spherical aberration increases inversely to the 6th power of focal ratio, limiting the focal ratio at this aperture size to ~ƒ/2.7 for diffraction limited axial correction. Field curvature is strong, requiring either curved detector or field flattener. The simplest flattener form, a singlet positive planoconvex lens with front surface radius R=(11/n)Rp=0.15(11/n)R1, with small compensatory changes in the conics to optimize for coma and spherical, achieves good correction, except for lateral color. Its correction requires adding at least one more glass element, which can be as simple as a meniscus of equal radii in front of the field flattener. Plot below shows spots for such combination with the last glass surface 1.8 mm from the image, for 430700nm range. Correction in the green is not significantly worse than in the allreflecting arrangement, with the residual secondary spectrum being the primary source of chromatic error. Nearly eliminating chromatic error would require achromatizing one of the elements; also, somewhat more complex corrector is needed for larger correctortoimage separation. Correction level of this arrangement is somewhat better than in the original Shaffer arrangement (R2=S=0.75R1, K1=0.405, K2=6.04), which has similar correction level at 40% smaller aperture and ~ƒ/3.5. Another unusual astrographic system consists from two concave mirrors, with the secondary inside the focus of the primary FIG. 126). It was derived by Karl Schwarzschild as a solution for twomirror system with best correction of aberrations possible. Schwarzschild found that a twomirror system can correct only four Siedel aberrations: the remaining one is either field curvature, or astigmatism. The variant with astigmatism as the only remaining aberration is usually referred to as Schwarzschild telescope, and the alternative with no astigmatism but with curved image field is known as Couder telescope.
In the former, the minimum relative size of the secondary (in units of aperture diameter) k needs to be related to the secondary magnification m as k=(1mm2)/(1m2)=1m/(1m2). This implies that m has to be smaller than 1 for k<1, i.e. for the secondary smaller than primary. Unavoidably, the final focus falls in between two mirrors. Also, m<1 implies that the secondary is concave. For the maximum acceptable secondary size of k~0.5, the corresponding secondary magnification, from m=[(4k28k+5)0.51]/2(1k), is m~0.4. Larger secondary magnifications require smaller secondary, but secondary size reduction is limited by image accessibility. On the other hand, due to the wide primaryto secondary separation, needed secondary size to keep the outer field well illuminated becomes excessive at k~0.5; in the above system keeping 2degree field fully illuminated would require clear secondary mirror surface diameter of nearly 2/3 of the full aperture, and the effective obstruction almost certainly exceeding 70% of the aperture diameter. Thus the practical secondary magnifications value cannot deviate significantly from m~0.45 k~0.44 level. With k and m determined, mirror conics can be obtained from Eq. 8687. After substituting for k, the conic relations become: K1 = [2(1mm2)+m3]/m3 and K2 = (1m)(1m2)/(1m)3 For viable level of secondary magnifications m~0.45, the corresponding mirror conics are K1<8.6 and K2>2.1 for the secondary. As with all twomirror systems, the secondary mirror radius of curvature is given by R2=ρR1, with ρ=mk/(m1).
The field is flat, but the astigmatismcancelling field curvature limits
quality field size. Since the secondary contributes nearly as much of
astigmatism as the primary, the system astigmatism is approximated by
double that of the primary, thus as the PV wavefront error W~(αD)2/2R,
where α is
the field angle in radians, D the aperture diameter and R
the mirror radius of curvature (for lowerorder astigmatism, RMS
wavefront error is smaller than the PV by a factor 241/2).
In the Couder curvedfield anastigmatic aplanat, k=12m, hence the maximum secondary magnification is lower than in the flatfield aplanat. Needed mirror conics are: K1 = (m32m+1)/m3 and K2 = (m3+m2m)/(1m)3 For k~0.5 or smaller, the corresponding secondary magnification is m~0.25, or larger. Again, image accessibility requirements limit reduction of the secondary size to about a third of the primary mirror. Since at m~0.25 and k~0.5 needed secondary size to prevent vignetting of the outer field becomes excessive, the secondary magnification is confined to a narrow range around m~0.3 and k~0.4. The corresponding mirror conics are K1~16 and K2~0.5, respectively. In general, the primary is more strongly aspherised than in the Schwarzschild telescope, while the secondary becomes a mild prolate ellipsoid. From the fabrication point of view, the twice more strongly aspherised primary is mainly offset by switching to the prolate ellipsoid secondary. However, the drawback is significantly lower practical secondary magnification, potentiating the image accessibility problem. With zero astigmatism, the image curvature equals system's Petzval radius of curvature, given as: RP = mkR/2(m1mk), with R being, as before, primary's radius of curvature. For, say, m=0.3 and k=0.4, field curvature is R/13.7, remaining strong with the largest R values that meet practical requirements for system length. For the above 200mm ƒ/10/3 Couder telescope, the corresponding image curvature is Rc=RP=293mm. At 1° offaxis, it would induce as much as 4.8 waves PV (1.4 wave RMS) wavefront error of defocus. Clearly, curved detector surface matching the image curvature, or a field flattener lens, is a must. Similarly to the Schwarzschild system, residual higherorder spherical aberration  only about half as large  can be corrected by extending the secondary radius of curvature by about 1% (it also minimizes residual higherorder coma, but it was already negligible). With larger/faster systems, the higherorder residuals grow exponentially. While this type of twomirror system can achieve very good correction, it also has several potential drawbacks which, combined, probably prevented its more widespread use. FIG. 127 illustrates degree of field correction of allreflecting twomirror telescopes in their typical configurations, from classical Cassegrain and Gregorian, through their aplanatic arrangements, to DallKirkham, Schwarzschild/Couder and Loveday.
