NATIONAL RADIO ASTRONOMY OBSERVATORY GREEN BANK, WEST VIRGINIA ENGINEERING DIVISION INTERNAL REPORT No1 112 ARECIBO THREE-MIRROR SYSTEMS: OPTIMIZING THE OPTICS SEBASTIAN VON HOERNER FEBRUARY 1983 NUMBER OF COPIES: 100 I. INTRODUCTION AND SUMMARY The spherical Arecibo telescope needs some focal correction device, and line feeds have been used up to now. Since they are narrow-band, a whole set must be specially designed, one feed for each frequency; and since they illuminate symmetrically, a large illuminated aperture will give vignetting from spillover for large pointing angles, for example, beyond 11° zenith 0 distance for a 700 ft aperture, while no vignetting at 20 pointing would allow only 450 ft aperture. Another correction device would be a secondary mirror, since exactly-focussing secondaries can be designed for almost any odd-shaped primary (von Hoerner 1976), but this would lead to undesirable aperture illuminations; for example, strong negative tapers for Gregorians. Thus, Frank Drake suggested (1980) to add a tertiary mirror, where proper "shaping" of both mirrors would give another degree of freedom, to be used for obtaining any wanted aperture illumination; for example, uniform illumination for maximum gain. Furthermore, an asymmetric illumination can now be chosen, avoiding any vignetting; for example, no vignetting for 700 ft aperture at 200 pointing demands an offset of 125 ft between the aperture center and the primary caustic axis. Drake investigated this case and found it feasible, but maybe the large secondary will pick up too much wind force and add too much weight. Then a smaller offset of 50 ft may be chosen as a compromise, with some vignetting but a smaller secondary. Recently, Tor Hagfors asked for a closer investigation of possible three-mirror systems, regarding feasibility, performance and cost. This must be approached from two sides. The limitations set by the present focal structure and its cable support must be determined, regarding maximum allowed weight and wind loads, both loads for survival limits, and maximum wind loads also for pointing errors. Also, possible ways of inexpen- sive stiffening should be looked for. This approach will be done later, after collecting all structural data. At present, only the optical design is approached: given the 125 or the 50 ft offset, what is the optimum configura- tion of secondary and tertiary mirrors? There are mainly two objectives for the optimization: minimizing the vertically projected area (wind force), and/or maximizing the over-all compactness through small sizes and distances (weight and cost). Constraints are: first, feed and surfaces must not be higher than 9 ft above the paraxial focus (carriage house at 10 ft); second, mirrors and feed should not cast large shadows in the aperture (gain loss, side lobes); third, the illumination pattern of the unshaped secondary must not be too odd (doubtful shaping convergence). In the following, only unshaped Gregorian secondaries are considered, because Cassegrains would be much larger. Unshaped means that the secondary is numerically calculated to give an exact secondary focus. The tertiary then is an exact ellipse if Gregorian, or an exact hyperbola if Cassegrain. Geometrical optics is used throughout. If a satisfactory three-mirror system is found that way, further detailed work is necessary: a shaping procedure must be applied to the upper two mirrors for obtaining the wanted aperture illumination for a given feed pattern. At the same time, following each geometrical ray through its three reflections, the polarization properties must be found; any resulting cross-polarization may hopefully be eliminated by changes of the feed location. Finally, the diffraction properties must be calculated for longer wavelengths. The shaping of asymmetric surfaces was proven impossible (Kinber 1962), but very good numerical solutions were found by an iterative relaxation method developed for this purpose (von Hoerner 1978); they may be considered exact solutions for all practical purposes. This method was written for a two- mirror system, where the iterations started out from an unshaped system of a parabolic primary with a hyperbolic secondary; the method would need some change of its equations for the present case. Convergence was fast and accurate with illumination changes up to 25 db. The shaping changed the two surfaces very little, mostly less than 0.003 of the aperture diameter. Thus the unshaped configurations of the present investigation may well be used for feasibility and cost studies, and later on they may serve as a start for the shaping iterations. Peter Napier (unpublished) investigated most of my shaped asymmetric systems for polarization. He obtained good results under two conditions: beam axis, offset, and feed should be located in one plane (which then is the single remaining plane of symmetry), and the feed axis should point parallel to the beam axis. This may either be a law of nature or chance. Assuming the former, all following systems have been designed to fulfill both these con- ditions. Whether this assumption is good or not will be found later by calculating the polarization properties of the shaped systems. The following examples, where optimization was tried, cover three types of asymmetry: (a) no offset, mainly serving for comparison as the limiting case; (b) 50 feet offset, as a compromise with some vignetting but less wind and weight loads; (c) 125 feet offset, as the desired one with no vignetting for 700 feet aperture diameter. For each of the latter two offsets, I select- ed three possible tertiary locations: to the right of the caustic, close to the caustical axis, and to the left of the caustic. For the last location, both Gregorian and Cassegrain tertiaries are treated. The present paper treats the optics only, giving the shape of secondary and tertiary mirror, and only in the plane of symmetry (containing the caustic axis and the secondary focus). All calculations were done with a "TI Program- mable 59" and its printer. The following results are obtained. From a total of over 50 calculated systems, a selection of 11 optimized systems is presented in this paper: 2 systems without offset, 5 with 50 feet offset, and 4 with 125 feet offset. The final evaluation must wait for estimates of loads and structural con- straints. So far, the best systems seem to be the ones of Table 1: Table 1. Summary of optimized systems Deepest point Offset (ft) Length of (ft, below paraxial focus) secondary (ft) secondary tertiary 0 67.5 - 9.2 -51 50 64.4 -33.8 -26 125 88.4 -70.3 -28 II. GENERAL CONSIDERATIONS 1. Calculation of Secondary Mirror Fig. 1 shows the geometry of primary and secondary mirror, in the plane of symmetry. For obtaining an exact focus at F 2 , the pathlength L of all incoming rays must be equal. For the axial ray, and starting to count at an aperture in the plane z=0, the pathlength is L = r + v + p2 (1) The numerical calculation of the secondary (point 5) is based on the following very easy geometrical construction. The incoming ray is reflected at point 1 (with z = r/2 1/( r 2 - a 2 ) by the angle 2a (with sin a = a/r). We mark 1 point 2 at the end of the pathlength ( L 12 = Lpz 1 ). The secondary mirror, point 5, then must be selected such that L 52 = L53 . Thus, we divide L in half, point 4; we erect the perpendicular bisector, and its 23 intersection with L yields point 5 of the secondary. 12 We start this procedure at the outermost left ray of the aperture (for example at a = -475 ft, for 700 ft aperture with 125 ft offset), and we 1 move in steps of Aa along the diameter to the outermost right ray (to a 2 = +225 ft in this example). We also calculate the illumination angle (P at F with tan (f) = (x - 2' 5 x3 )/(z 5z ). We neglect the second dimension and call i = A(P/Aa the 3 aperture illumination with regard to an omnidirectional feed at F . While 2 moving from a to a we note both extremes of i and call 1 2' illumination ratio I i /i •. (2) max min 2. Shadow Lines Fig. 2 shows a number of incoming equidistant rays, across the aperture, from the leftmost ray L o to the rightmost ray R o ; and it shows the once-reflected rays, from L 1 to R 1 . The secondary, as a Gregorian, must start at L 1 and go counterclockwise around the paraxial focus F 1 until it ends at R 1 . It casts its shadow S 2 on the aperture. In order not to cast shadow on itself, the secondary must start, for this large offset, to the right of point A (intersection of L 1 and R ). But 1 for offsets less than 435 feet, it must start to the right of point B (inter- section of L and caustic). 1 The plane of Fig. 2 is divided into several fields, and the numbers 0, 1, 2, 3 indicate how many independent shadows the tertiary would cast if located in one of these fields. The size of the shadow is proportional to the density of rays at this location. We see four fields with zero shadow, but the uppermost one is actually too small for any use. This leaves three fields without shadow: left-hand, right-hand, and lower center. We shall try to place the tertiary in these fields. But also outside of the two upper fields, in the region of only incoming rays, the ray density is so low that a tertiary may well be placed there because its shadow will be only small. Most other locations, however, should be avoided if possible.