◄ 8.2.1. Classical and aplanatic   ▐    8.2.3. Miscollimation, close focusing ►
 

8.2.2. Dall-Kirkham and Loveday two-mirror systems

An interesting two-mirror variant is one with the secondary spherical. It is known as Dall-Kirkham and, although with stronger coma than its classical counterparts, its mirrors are easier to make, and its miscollimation sensitivity is lower, which makes it a viable practical alternative. With the secondary spherical, the needed primary conic for corrected lower-order spherical aberration, from Eq. 80, is

                           K1= -1+(m2-1)(1+η)/m3 = -1+k(m-1)(m+1)2/m3              (90)

In the configuration with convex secondary, the primary is ellipsoidal (it is hyperboloidal with concave secondary,  which makes such arrangement impractical). Coma wavefront error in Dall-Kirkham configuration is larger by a factor 1+(m2-1)(m-η)/2m, than in a comparable classical Cassegrain. The astigmatism, however, is smaller by a factor [(m2-1)(m-η)2/4(m2+η)]-1. As a result, the DK median field surface is less curved, as given by:

                  1/Rm= 2[(m2-2)(m-h)+(m+1)m - (m2-1)(m-h)2/2m]/(1+h)R1m2         (90.1)

In addition to be less curved, the DK image is fairly insensitive to defocus error caused by image field curvature, due to the present off-axis coma being significantly larger than the defocus error caused by field curvature (to a smaller extent, also due to the coma itself being relatively insensitive to defocus). As a result, Dall-Kirkham flat-field performance is very close to its best field performance.

Note that since the coma error in a classical Cassegrain is inversely proportional to m2, it actually diminishes in the DK with the increase in secondary magnification m.

While it is evident that both, secondary magnification and final system focal ratio are factors determining the level of the DK system coma, it is good to clarify the specifics of it. Secondary magnification is only the factor with a given system focal ratio. In other words, an ƒ/15 DK system will have more coma if the secondary magnification is 5 (with an ƒ/3 primary) than when it is 3 - with an ƒ/5 primary. From the above relation for the DK system coma, good approximation of the ratio of increase of the coma wavefront error in the DK vs. classical Cassegrain (or paraboloid) of the same focal ratio is obtained by setting h to zero: w~(m2+1)/2. For the above example, that gives the coma error @5x secondary magnification some 13 times larger than in a comparable Cassegrain, and larger by a factor of 3.6 than @3x secondary magnification.

Knowing that for a w factor of the change in coma wavefront error, the corresponding effective focal ratio number is found from F'=F/√w, the focal ratio number of a comparable paraboloid with respect to the size of Dall-Kirkham's angular coma-free field is approximated by

                                        F'~1.4F/(m2+1)1/2                                         (90.2)

F being the DK system focal ratio number. Also, knowing that the effective focal ratio number F' corresponding to the size of linear diffraction-limited field is, for the factor of coma wavefront error change w, found from F'=F/w1/3, the focal ratio number of a comparable paraboloid in this respect is approximated by

                                        F'~1.26F/(m2+1)1/3                                        (90.3)

Note that simplified forms of either of the two approximations - F'~1.4F1 and F'~1.26F1m1/3, respectively,  is still quite useable as an approximation (F1=F/m=ƒ1/D, the DK primary focal ratio number). It puts the ƒ/15 DK @3x secondary magnification at the level of an ƒ/6.6 and ƒ/9, while the one @5x secondary magnification at the level of an ƒ/4.2 and ƒ/6.4 paraboloid, respectively.

The above approximations imply that the increase in secondary magnification at a given primary focal ratio doesn't result in larger system coma error. Hence the angular coma-free field doesn't change, while the linear field increases by the m'/m ratio, m' being the higher secondary magnification. The effective focal ratio number corresponding to the diffraction-limited linear field changes in proportion to (m'/m)1/3. Thus, if we'd increase secondary magnification from 5 to 8 in the above system with ƒ/3 primary, the resulting ƒ/24 DK system would keep angular coma-free field size about identical to that at ƒ/15, while the linear diffraction-limited field would be larger by a factor of ~(8/5)=1.6. The effective focal ratio number of corresponding parabola to this field size would increase by a factor of (8/5)1/3, to F'~7.5 (also from F'~1.26F1m'1/3).

The practical significance of the above consideration is that maintaining secondary magnification of Dall-Kirkham systems low - thus accepting relatively large central obstruction - doesn't result in better quality field coma-wise for given primary mirror. In fact, higher secondary magnification expands quality linear field in proportion to the magnification.

Since the quality field in the DK is considerably more limited than in either classical or aplanatic two-mirror telescopes, a sub-diameter lens corrector is desirable already with moderately fast systems (~f/10), especially if it is relatively simple to make and results in field quality superior to that in the two alternative two-mirror configurations.

Unusual two-mirror systems

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. 61). 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 aplant 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.

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

◄ 8.2.1. Classical and aplanatic   ▐    8.2.3. Miscollimation, close focusing ►

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