◄ 8.2. Two-mirror telescopes   ▐    8.2.2. Dall-Kirkham, Loveday ►
 

8.2.1. Classical and aplanatic two-mirror telescopes

Classical two-mirror telescopes

 While spherical aberration, according to Eq. 80, can be corrected for any appropriate combination of the primary and secondary mirror conic, the aberration coefficients for coma and astigmatism for two-mirror system show that coma and astigmatism do vary, potentially significantly, with the choice of conics. The coma aberration coefficient (Eq. 82) indicates that a system with spherical primary will have stronger coma than one with paraboloidal primary. A particular value of K1 resulting in the zero sum in the brackets, would result in corrected lower-order coma, with the needed value for the secondary conic for corrected spherical aberration obtained from Eq. 80.

In the early days of telescopes, limitations in both, calculation and mirror-making and testing skills would not allow to determine precisely the coma-free conic combination. However, it was determined that paraboloidal primary significantly reduces coma, compared to spherical (Eq. 82.1 shows that astigmatism with paraboloidal primary actually increases, but insignificantly in comparison with the reduction in coma). Thus, the choice for primary conic was, for quite some time, paraboloid and, for that reason, this arrangement is known as a classical two-mirror telescope.

With paraboloidal primary (K1=-1), which is corrected for spherical aberration, zero system spherical aberration requires zero secondary spherical aberration, for which the needed secondary mirror conic is, from Eq. 80:

                                                            K2= -[(m+1)/(m-1)]2                                        (83)

which makes it a hyperboloid.

With spherical aberration corrected, remaining lower-order coma and astigmatism, as peak-to-valley wavefront error at diffraction focus for object at infinity are given by:

                                                                   Wc=αD/48F2                                             (84.1)   and

                                                       Wa=-(m2+h)Dα2/8(1+h)mF                                (84.2)

respectively, with α being the field angle in radians, D the aperture diameter and F the system F-number. Evidently, coma is identical to that of a paraboloid of the same F-number, while the astigmatism exceeds that of a paraboloid by a factor (m2+η)/(1+η)m. Sign of astigmatism in the Gregorian is opposite to that in the Cassegrain and Newtonian. Therefore, related geometric (ray) aberration can be determined from those given for a paraboloid in 2.2. Coma and 2.3 Astigmatism.

Petzval field curvature of any two-mirror system is given by 1/Rp=2[(1/R2)-(1/R1)], R1 and R2 being the radius of curvature of primary and secondary mirror, respectively. However, due to the presence of astigmatism, best image surface curvature varies. With the primary mirror astigmatism independent of the conic (for the stop at the surface), it is the secondary mirror conic and shape (convex/concave) that induces variations in the system astigmatism. For classical two-mirror systems, median field curvature is given by:

                                           1/Rm= 2[(m2-2)(m-h)+m(m+1)]/(1+h)R1m2                        (85)

The relation shows that, for given primary ƒ-ratio and secondary magnification, best (median) image surface is somewhat less curved in the Gregorian.
 

Aplanatic two-mirror telescopes

 It wasn't until 1910, when Ritchey in the U.S. and Chrétien in France have calculated the needed conics for a coma-free two-mirror (Cassegrain) system, that a "classical" two-mirror telescope finally was upgraded to its optimal version. It took another 17 years before the very first such telescope was successfully made (Ritchey, 1927).

Correction of coma in a two-mirror system requires an additional, relatively small modification of both optical surfaces. In the Cassegrain configuration, secondary needs to be more strongly aspherized in order to correct for the coma. So does the primary, in order to compensate for additional spherical under-correction induced by the more strongly aspherized, aplanatic secondary. In the Gregorian, however, both mirrors need to be less strongly aspherized than in the classical arrangement. The needed conics for an aplanatic two-mirror telescope are:

                                                                 K1= -1-2(1+h)/(m-h)m2                                 (86)

                                                 K2= -[(m+1)/(m-1)]2- 2m(m+1)/(m-h)(m-1)3             (87)

Alternately, the conics can be expressed in terms of the primary focal length ƒ1, system focal length ƒ, and mirror separation s as K1=-1-2ƒ12(ƒ1-s)/sƒ2 and K2=-[s(ƒ-ƒ1)(ƒ+ƒ1)2+2ƒƒ13]/s(ƒ-ƒ1)3. The relations somewhat simplify if the parameters are expressed in units of the primary's focal length ƒ1, in which case ƒ→m, and the relations become K1=-1-2(1-ṡ)/ṡm2, K2=-[ṡ(m-1)(m+1)2+2m]/ṡ(m-1)3, ṡ being the mirror separation in units of the primary's focal length.

With lower-order spherical aberration and coma corrected, the remaining aberrations are astigmatism, field curvature and distortion. The P-V wavefront error of lower-order astigmatism is given by:

                                                      Wa= -[m(2m+1)+h]Dα2/16(1+h)mF                         (88)

This gives astigmatism in the aplanatic Cassegrain - also known as Ritchey-Chrétien - greater, and in the aplantic Gregorian smaller by a factor of (2m2+m+η)/(2m2+η) compared to the classical types. Neglecting distortion, the remaining aberration is best, or median field curvature, given as:

                                                     1/Rm= 2(m+1)[m2-(m-1)h]/(1+h)R1m2                     (89)

Again, for an equal set of parameters, aplanatic Cassegrain has somewhat stronger best field curvature than aplanatic Gregorian.
 

◄ 8.2. Two-mirror telescopes   ▐    8.2.2. Dall-Kirkham, Loveday ►

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