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8.2.5. Loveday two-mirror telescope

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. 79). 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.

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=0.7252
R2/R1=-0.7083
K1=-0.428
K2=-6.55      and
Rp=0.15R1,

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. Field curvature is strong, requiring either curved detector o field flattener. The simplest flattener form, a singlet positive plano-convex lens with front surface radius R=(1-1/n)Rp=0.15(1-1/n)R1, gives good results when combined with somewhat lowered secondary conic, in order to compensate for the lens' coma. The only remaining aberration is astigmatism; at 1mm lens-to-image separation, the wavefront error at 0.5° off-axis in a 400mm ƒ/4 system is 1/3 wave P-V of astigmatism (~5 micron blur diameter), and at 5mm separation the error at 0.5° off-axis is 1.3 wave P-V of astigmatism (blur diameter below 15 microns).

To compensate for the additional astigmatism-induced curvature, the lens radius needs to be somewhat stronger. In the above example, about 5% stronger for every mm of lens-to-image separation than what the formula for flattening the Petzval surface indicates.

These parameters are nearly optimized for an ƒ/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. For 400mm aperture at ƒ/3, correction level is 1/25 wave RMS in the field center, mostly residual higher-order spherical, and 1/16 wave RMS at 0.5° off-axis (best surface), mostly residual higher-order coma. From there, it changes nearly in proportion with the aperture; spherical changes with the 4th power of the ƒ-ratio, which limits useable ƒ-ratio to ƒ/2.5-ƒ/3, depending on the aperture size. Higher-order coma and astigmatism are also hard to control close to these levels. Relatively small compensatory variations in the conics and radii/separation are possible, but have little effect.

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.

FIG. 80 illustrates degree of field correction for typical two-mirror telescopes.