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mif/

j jm2

m1

Fig. 3.76. Dual-purpose Newton telescope due to Loveday (1981)

Fig. 3.76. Dual-purpose Newton telescope due to Loveday (1981)

200 mm aperture working at f/24. As a tertiary in this Loveday geometry, the primary has a radius nearly three times that of a Paul-Baker telescope.

Recently (2000), a complete and masterly analysis of all potential 3-mirror telescopes was initiated by Rumsey and carried out by Rakich [3.162] [3.163]. The basis of his approach is the Burch "plate diagram" or "see-saw diagram" (see § 3.4). He also uses a theorem enunciated in 1900 by Aldis [3.164], stating that four spherical surfaces are sufficient to produce an anastigmatic image, and extended by Burch in 1942 [3.28] to show that 2 spherical surfaces and one aspheric surface or 2 aspheric surfaces alone are sufficient for anastigmatism. (This general theorem is not applicable to concentric systems such as the Schmidt telescope or more complex forms of concentric mirrors). Thus Rakich deals with the general case of anastigmatic solutions containing 3 powered mirrors of which 2 are spherical, the object conjugate being infinite. The theorems of Aldis and Burch, as used by Rakich, can also be derived as limit cases of the generalised Schwarzschild theorem given in § 3.6.5.3.

The extension of the Burch method is developed into an automatic program which, for the predefined input parameters t1 (separation of primary and secondary mirrors - d1 in our notation) and c2 (curvature of the secondary mirror), generates and solves a cubic equation. The solutions of this cubic equation define anastigmats. Different cubics are generated depending on which mirror, primary, secondary or tertiary, is aspherized. This enables a survey of the plane in parameter space defined by t1 and c2. The use in three programs of the three separate solutions gives rise to 3 families of solutions for each type of cubic equation. Each family is divided into sub-families, separate regions in the (t1,c2) plane between which no real anastigmats exist.

In this way, a total of 15 sub-families are generated. A further division defines sub-regions containing solutions with positive or negative Petzval sum, flat field solutions lying on curves where these regions touch.

"Filters" are built in as conditions which reject impractical systems because of too extreme parameters. As a result, many solutions are rejected and 5 potentially useful families remain. Rakich presents, as normal uniaxial or folded systems, 9 novel anastigmats, some of which are closely related to known systems and others completely new.

Finally, some very interesting "Schiefspiegler" systems are presented which are discussed in § 3.7.

By covering the whole effective parameter space of his 3-mirror solutions, using Rumsey's basic ideas, Rakich has presented a most elegant analysis which may be seen as the definitive triumph of classical third order theory as a tool for setting up telescope solutions.

Rakich [3.165] has subsequently attempted to extend his approach to 4-mirror (all spherical) telescopes. He points out that the dimensionality of the solution space goes from one dimension for 2 aspheric mirrors, through two dimensions for 3 mirrors with 1 aspheric to three dimensions for 4 spherical mirrors. Most solutions thrown up are blocked by his filters and none were attractive because of high central obstruction, etc. The solutions discussed in § 3.6.5.3 do indeed suggest that, at this level of parametric complication, intuition and experience may well be more fruitful than a general analytical solution. However, according to latest information from Rakich, he is still hoping to set up useful Schiefspiegler solutions with 4 spherical mirrors. The spherical mirror form makes the manufacture of such excentric systems much easier than that of excentric aspherics.

3.6.5.2 Recursion formulae for calculating aberration coefficients for the general case of n mirror surfaces. In § 3.2.4.3 analytical expressions were developed for the general case of 2-mirror telescopes. As was pointed out in connection with Tables 3.2 and 3.3, calculations based on the direct tracing of the paraxial aperture and paraxial principal rays are highly desirable as a check on systems set up from analytical formulae. Ray-trace calculations must follow the systematic sign convention given in § 2.2.3.

The analytical formulae given by Korsch in § 3.6.5.1 for the general solution of a 3-mirror system satisfying all four aberration conditions (Sj = Sjj = Sjjj = Sjv = 0) show the complexity of a general analytical solution even for 3 mirrors. Furthermore, these formulae apply the simplifying conditions that s1 = œ and spr1 = 0, i.e. object at infinity and entrance pupil at the primary.

Rather than extend such general analytical solutions to even more complex forms for n > 3, we will now give recursion formulae which enable the basic parameters of Eqs. (3.20) to be determined rapidly and simply without tracing paraxial rays. These formulae are derived, in fact, by tracing a paraxial ray in the general case and expressing the result only in basic constructional parameters or natural parameters of the system. These parameters are similar to those used by Korsch in Eqs. (3.317)-(3.323). Such recursion formulae represent a special case for reflecting surfaces of a general formulation known as Gaussian brackets [3.7(a)] [3.80] [3.81]. According to Herzberger [3.7(a)], Gauss devised this algorithm for the solution of a linear diophantine equation [3.81]. In spite of its apparently evident applicability to the linear paraxial equations that Gauss himself had formalised, he failed, according to Zimmer [3.80(a)], to recognize the utility of his algorithm in this

The general starting parameters (see Eqs. (2.36)) for a paraxial ray (either aperture or principal) incident on a telescope system are si (or spri) and u (or upri). Otherwise, all quantities must be expressed in terms of ri,r2,...,rn and di,d2,...,dn , the constructional parameters. The refractive indices n1, n2 ..., nn are not included because our purpose is to establish formulae for a system of n-mirrors in air, giving only sign inversions with n = ±1. This brings a considerable simplification, but for n > 3, the formulae still become complex in terms of these parameters. They can be simplified to elegant, recursive forms by introducing, similarly to Korsch, the parameters

in which expresses the inverse magnification at the v-surface of the aperture ray and the inverse magnification at the v-surface of the principal ray (inverse pupil magnification). Then

These are therefore the reciprocals of the quantities mv and pv in Eqs. (3.317). Our sign convention also strictly follows the ray-trace rules: because n'/n = -1 at the reflections, dv is negative if v is odd, positive if even. Radii are negative if concave to the incident light direction (from left to right). The signs of Mv and Mprv are given by the ratios defined in (3.332).

The general recursion formulae for the parameters of Eqs. (3.20) are, then, for mirror surface v with v > 1:

[-si + Modo + (MoMi)di + (MoMiM2)d2 + ... + (Mo-.Mv )dv ] —i [(-1)v 2 (MoMiM2 ... M(v-i)) (Mv - 1) —i [(-1)v(MoMiM2 ... M(v-i))(Mv + 1)] —i 