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8.2.4. Dall-Kirkham telescope   ▐    8.2.6. Miscollimation, close focusing

8.2.5. Unusual two-mirror systems: Loveday, Eisenberg-Pearson, Schwarzschild, Couder

Two-mirror 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 Loveday-Cassegrain, 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: Two-mirror 3-reflection 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 point-image aberrations, spherical, coma and astigmatism is possible with ellipsoidal primary and hyperboloidal secondary mirror. The only remaining aberration is a 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 three-mirror 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 two-mirror 3-reflection anastigmatic aplanat are very simple:

S/R1 = 0.7248
R2/R1 = -0.7084
K1 = -0.428
K2 = -6.58      and
Rp = 0.147R1,

S being the primary-to-secondary 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 f/3 system; they are scalable by either aperture, or primary's F-number. Scaling by the aperture doesn't require any changes, while slower system require slightly stronger secondary conic to optimally re-balance spherical aberration.

All aberrations - except field curvature - are well corrected. Axial correction for 400mm f/3 system is 0.04 wave RMS of balanced higher-order spherical, with the balanced higher-order coma limiting diffraction limited field to 0.6° radius (with 50% linear central obstruction, the field size limit is about 1.5 degree in diameter. Higher order spherical aberration increases inversely to the 6th power of focal ratio, limiting the focal ratio at this aperture size to ~f/2.7 for diffraction limited axial correction. With a more realistic system, such as f/4.9 with f/3 primary, the overall correction is exceptional (0.0036 wave RMS on axis, and 0.022 wave at 0.75 degrees off), and field size is limited only by acceptable central obstruction. With about 50% linear central obstruction, the unvignetted field is 1.5 degree in diameter.

Field curvature is strong, requiring either curved detector or field flattener. The simplest flattener form, a singlet positive plano-convex lens with the front surface radius

R=(1-1/n)Rp =0.15(1-1/n)R1 ,

(the actual radius should be about 10% stronger, since the flat side induces roughly 10% as much astigmatism of opposite sign; better correction, however, gives biconvex lens slightly weaker in power than what is indicated by the equation) 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 430-700nm range. Correction in the green is not significantly worse than in the all-reflecting 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 corrector-to-image separation.

Correction level of this arrangement is somewhat better than in the original Shaffer arrangement (R2 =S=0.75 R1, K1=-0.405, K2=-6.04), which has similar correction level at 40% smaller aperture and ~f//3.5.

Somewhat different take on the two-mirror 3-reflection configuration is the Eisenberg-Pearson system. The secondary is convex, resulting in a significantly weaker field curvature. A 300mm f/4.3 system shown below has a minimum obstruction by secondary of 43% linear, but the practical minimum is closer to 50%.

In this simple system, it is not possible to flatten field by changing mirror parameters. A quick attempt showed that a simple zero-power flattener consisting of a pair of singlets, planoconcave followed by a planoconvex (both BK7, 222mm radius, placed at 440mm from the primary, facing it with their flat side) flattens the field with no significnt aberrations induced. Better correction without a field corrector is attainable if the third reflection comes of a surface that has different conic from that of the primary. The field is perfectly flat, and there is no aberrations to speak about (bottom). It does require stronger aspherics, and the central portion of the primary is a significantly stronger hyperboloid than the outer primary area. Since the stronger hyperboloid is flatter, with a smaller sagita, having these two hyperboloids share the same vertex - as in this prescription - means that the edges of the inner hyperboloid would be carved into the outer one (the depth is roughly 1/100 of a mm).

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 two-mirror system with best correction of aberrations possible. Schwarzschild found that a two-mirror 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.

FIGURE 126: For a given system focal ratio, the Schwarzschild flat-field aplanat (
SPECS) is significantly more compact than Couder (SPECS). The latter is about as long as a comparable Schmidt camera, with zero chromatism, but no other advantages. Its image radius is about twice more strongly curved than in the Schmidt, it does not have efficient control of higher order aberrations at focal ratios significantly faster than f/3, and its image is less accessible, being located in the converging cone. Couder arrangement can be made more compact by using faster primary, but the limit to it is imposed by its high surface conic. With an f/8 primary, the system would be only moderately shorter, but it would be at the limit of acceptable correction due to the higher-order residuals, and the mirror would become very difficult for fabrication (since the aspherizing glass volume for given mirror diameter changes inversely to the cube of focal ratio, a -16 conic f/8 hyperboloid is about as difficult to make as f/3.2 paraboloid). On the other hand, the more compact Schwarzschild has quality field limited by astigmatism. Both, Schwarzschild and Couder could have the final focus made accessible at the side, by use of a diagonal flat, but it may not be sufficient for broader range of accessories, especially with the latter.

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=(1-m-m2)/(1-m2)=1-m/(1-m2). 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=[(4k2-8k+5)0.5-1]/2(1-k), 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 primary-to secondary separation, needed secondary size to keep the outer field well illuminated becomes excessive at k~0.5; in the above system keeping 2-degree 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. 86-87. After substituting for k, the conic relations become:

K1 = -[2(1-m-m2)+m3]/m3      and     K2 = (1-m)(1-m2)/(1-m)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 two-mirror systems, the secondary mirror radius of curvature is given by R2=ρR1, with ρ=mk/(m-1).

The field is flat, but the astigmatism-cancelling 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 P-V 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 lower-order astigmatism, RMS wavefront error is smaller than the P-V by a factor 241/2).

     For the above 200mm f/6.7/3 Schwarzschild telescope, astigmatism wavefront error at 1° off-axis should be nearly 4 wave P-V, or 0.8 wave RMS (raytrace gives 3.7 wave P-V). Note that the above relations are for lower-order aberrations. Due to the strong mirror conics, higher order spherical aberration and coma are not entirely negligible. In this particular system, the residual higher order spherical aberration was somewhat over 1/7 wave P-V, and the residual coma was making blurs somewhat elongated. Both were corrected by widening the mirror separation by about 0.5%, and reducing the secondary's conic by about 5%.

In the Couder curved-field anastigmatic aplanat, k=1-2m, hence the maximum secondary magnification is lower than in the flat-field aplanat. Needed mirror conics are:

K1 = -(m3-2m+1)/m3      and     K2 = (m3+m2-m)/(1-m)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(m-1-mk),

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 f/10/3 Couder telescope, the corresponding image curvature is Rc=RP=-293mm. At 1° off-axis, it would induce as much as 4.8 waves P-V (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 higher-order spherical aberration - only about half as large - can be corrected by extending the secondary radius of curvature by about 1% (it also minimizes residual higher-order coma, but it was already negligible).

With larger/faster systems, the higher-order residuals grow exponentially.

    While this type of two-mirror 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 all-reflecting two-mirror telescopes in their typical configurations, from classical Cassegrain and Gregorian, through their aplanatic arrangements, to Dall-Kirkham, Schwarzschild/Couder and Loveday.

: Best image surface ray spot plots for (left to right) classical Cassegrain (CC) and Gregorian (CG), aplanatic Cassegrain (AC) and Gregorian (AG - aplanatic Gregorian - with twice as fast primary as the AC for similar secondary size, and AG* with the same f/3 primary, but over 0.7D minimum secondary size required), Dall-Kirkham (DK), with the spot size reduced three times in order to fit in, and the Gregorian 3-reflection anastigmatic aplanat (AA). Aperture diameter D=400mm for all. In order to minimize DK coma, practical systems use slow primary (~f/4) and low secondary magnification (~2.5). An f/12 system based on these parameters would have the coma over angular field nearly twice lower, and over two and a half times lower over the corresponding linear field, compared to the above f/8 system. The circle outlines the system e-line Airy disc diameter.  

8.2.4. Dall-Kirkham telescope   ▐    8.2.6. Miscollimation, close focusing

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