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.

3. Compactness

In general, the secondary has three free parameters, see Fig. 1. The height v should be large for good illumination, but it is limited by the

carriage house, and v = 8 ft will mostly be used. This leaves two parameters,

T and p, describing the location of the secondary focus F2.

In order to reduce weight and cost, the secondary should be small or

compact. This means mostly that it should start as far inward as possible without casting shadow on itself, at point A of Fig. 2 for large offsets and

at point B for small ones (A and B coincide at about 435 ft offset, for 700 ft

aperture). The resulting location of F 2 is shown in Fig. 3. Locations

above the curve T(p) are excluded.

Regarding wind loads, the secondary should not reach down too far. This needs a compromise, however, because making the secondary less deep means making it very long. 4. Size of Tertiary and Feed

Drakes paper mentions A = 30 cm as the longest wavelength. Since this would demand extremely wide and long feeds for most cases, I have tentatively adopted A = 21 cm. The tertiary must be larger than the 10 A sometimes adopted for limited diffraction, for three reasons: (a) Avoiding spillover beyond the tertiary needs a narrow feed pattern, and the half-power width of that pattern must at least be 10 A. (b) The tertiary must be able to concen- trate a large number of feed rays onto a small central region of the secon- dary; the odder the unshaped illumination (2), the larger a tertiary will be needed. (c) What matters is not the actual size d of the tertiary, but 3 only its projection d onto a plane perpendicular to the feed axis. After 3p some tries and estimates, I have adopted a projected diameter d 3p of the tertiary, for all cases, of

d = 18 feet. (3) 3p

Regarding the width of the feed, d f , I demand that the half-power width of the feed pattern is 1/2 of d 3p . Calling fi 3 the angle under which the feed sees the tertiary, the feed size then must be, for A = 21 cm,

d = (91°/13 3 ) feet. (4) f

Both (3) and (4) are used in all calculated examples. But they are open for discussion, and may be changed in the future. III. VARIOUS SELECTED SYSTEMS

1. No Offset

It is always good to know the limiting case, for comparison, and as a kind of measuring stick. First, we ask for the smallest possible secondary mirror illuminating an aperture of 700 ft. Smallest means symmetric, without offset. And if nothing else matters, the most compact secondary would go from caustic to caustic (Point B of Fig. 2), covering all rays from L 1 to R1.

This is shown in Fig. 4, and entered as #1 in Table 2. The size is small but not extremely so, with d 2 = 41.7 ft as its largest diameter, and with z 2 =

-31.2 ft as its deepest point. It is not a realistic case, because the interior is much too crowded with rays to allow a tertiary of proper size.

Only a simple dipole at F would be possible. 2 Second, we ask for the smallest realistic case, still symmetric but with a tertiary mirror, see Fig. 5, and #2 in Table 2. The shadow zones show that the tertiary must be placed deep down, below point A of Fig. 2. For optimum compactness, F 2 would be right at point A, with a Gregorian tertiary extend- ing somewhat into the one-shadow zones on both sides.

This would be a good realistic possibility if no offset were demanded.

Although the length has now increased to d 2 = 67.5 ft, the deepest secondary point is only at z 2 = -9.2 ft (but the tertiary goes down to z 3 = -51 ft).

An awkward feature would be the large feed, 5.1 ft diameter for X = 21 cm, casting an aperture shadow of 185 ft. But this could be avoided by a feed location close to the tertiary, similar to Fig. 7. 2. Offset of 50 feet First, we place the tertiary to the right of the caustic into the no-shadow zone. For compactness and low depth of the secondary, a Gregorian is here better than a Cassegrain, and a high location would be best, which then must also go to the right because of ray 14 1 . But then the illumination ratio I of equation (2) would become extremely large; we make it as small as possible by letting the tertiary start right at point B. This is shown in

Fig. 6, and as #3 in Table 2. Good is the small depth of the secondary, z 2

= -16.2 ft, but worst is the large feed, with d f = 4.3 ft for X = 21 cm.

Second, we put the tertiary at the center, Fig. 7 and #4. Just as in

Fig. 2, it must go far down for avoiding excessive shadow. If we left the

feed high up at the secondary, it would need an extremely large size of 11 ft,

casting about 300 ft shadow. We thus place the feed close to the tertiary, which then must be Cassegrain lest the feed blocks too many rays. Now we have

a good feed size, but at an inconvenient location. The secondary is somewhat

long, d 2 = 76.2 ft, and the tertiary is deep down at z 3 = -63 ft. Not an

attractive system.

Third, we have a tertiary in the left-hand zone of now shadow. It could be either type, and Fig. 8 shows the Gregorian, #5 in Table 2. This gives the most compact secondary with d 2 = 64.4 ft, even smaller than the symmetrical

case of #2 (although reaching deeper down to z 2 = -33.8 ft). All data look acceptable. Fourth, we choose a Cassegrain instead, Fig. 9 and #6. It is a bit

longer but less deep. All looks ok except maybe the large feed of d f = 3.4

ft for X = 21 cm. 10

Fifth, we thus lift the secondary just as high as possible, for minimiz- ing the feed, see Fig. 10 and #7. All data look good except maybe the fact that the secondary reaches a bit deep down into the wind, to z 2 = -40.3 ft.

For a comparing evaluation, and for selecting the best system for 50 ft offset, we would first need estimates about wind forces and weights, in relation to the structural constraints, which shall be done later. From the present data, I would like to exclude #3 because of its feed size, and #4 because of sizes, inconvenience and shadow. The remaining three look accept- able, and #5 is most compact and seems best so far.

3. Offset of 125 Feet

First, we start again with the tertiary mirror to the right of the caustic, in the no-shadow zone. We found that this location will always yield very long secondaries and extremely large illumination ratios. We thus minimize both by placing the tertiary as far left as possible, at point A. We obtain Fig. 11 and #8 of Table 2. We still get unpleasant values, a length of d = 110.9 ft, and an illumination ratio of I = 58. In addition, a feed as 2 drawn, high up at the secondary, would be much too large, d f = 6.5 ft. It should be brought down as in Fig. 7, to a very inconvenient place. The only advantage of #8 is its low depth, z 2 = -44.5 ft.

Second, we try a tertiary at the center, again a Gregorian for compact- ness, Fig. 12 and #9. This system has about all the same disadvantages as #8, though less extreme. If it were considered seriously, the feed certainly must be brought down as in Fig. 7.

Third, a tertiary Cassegrain is placed to the left of the caustic. For optimum compactness, we start the secondary at point A, and the tertiary at the caustic. For minimum feed size, we bring the feed down as far as 11

possible, which means to ray R 2 . The result is Fig. 13 and #10. The tertiary cannot be placed in the no-shadow zone because of the depth T required for F 2 from Fig. 3. But the density of rays in Fig. 2 is only low at the place of the tertiary, thus its shadow is only small, 18 ft diameter

(its projected length d3p). All data look acceptable, except that the depth z = -70.3 ft may pick up too much wind force. 2 Fourth, we replace the Cassegrain by a Gregorian and obtain Fig. 14 and #11. The depth is unchanged, but the length d has somewhat decreased, the 2 tertiary is less deep, and the feed can be made very small by lowering F 3 as much as wanted. In this preliminary comparison, before further investigation of loads and constraints, it seems that system #11 is the best one for 125 ft offset. A complete view of it, with the primary sphere, is shown in Fig. 15. Table 2. Secondary and tertiary mirrors, and feeds.

Parameters, see Fig. 1; d i = largest diameter or

length; z i = z-coordinate of deepest point; a i = angular size of mirror as seen from its focus;

I = illumination ratio, equation (2); S i = length of (additional) shadow in aperture. All lengths in feet, angles in degrees.

Parameters Secondary Tertiary Feed Shadows # Fig. V T p d I Type d 3 z 3 d 5 3 , 2 z 2 2 3 f S f

No Offset 1 Ii ! 1 4 +2.8 41.7 -31.2 328 4 25 2 5 9 -40.5 67.5 - 9.2 94 15 G 18 -51 18 5.11 67 82 185 / 9 50 ft Offset 110 32 G 18 -47 21 4.3 74 0 21 3 6 8 -37 +25 j 74.4 -16.5 4 7 8 -77 + 2 76.2 -19.7 58 23 C 18 -63 115 0.8 74 96 22 5 8 8 -19 -11 64.4 -33.8 145 14 G 21 -26 42 2.2 54 0 17 6 9 8 -34 -29 71.8 -31.8 88 16 C 20 -33 27 3.4j 61 0 17 7 10 7 -10 -23 67.1 -40.3 145 24 C 23 -21 43 2.1 49 0 11 125 ft Offset 8 11 8 -65 +43 110.9 -44.5 105 58 G 18 -74 14 6.51100 0 23 9 12 9 -79 +12 1105.0 -50.3 82 37 G 18 -90 11 8.3 k 89 0 80 10 13 8 -27 -35 92.8 -70.3 94 17 C 26 -40 38 2.4 47 18 0 i i 11 14,15 /11 8 -23 - 9 88.4 -70.3 146 15 G 30 -28 83 1.11. 47 14 0 1 12

References

Drake, F. 1980: "Gregorian Feed Concepts for the Arecibo Telescope"; NAIC 147, Cornell University, Nov. 1980.

Kinber, B. Y. 1962: "On Two-Reflector Antenna"; Radio Eng. Electron. Phys.

7, 914. von Hoerner, S. 1976: "The Design of Correcting Secondary Reflectors"; IEEE

Trans. AP-24, 336. von Hoerner, S. 1978: "Minimum-Noise Maximum-Gain Telescopes and Relaxation

Method for Shaped Asymmetric Surfaces"; IEEE Trans. AP-26, 464. Fig. 1. , Geometry and definitions, for calculation of secondary mirror. The direction of the incoming rays defines the caustic and its axis, where a = distance of a ray from this axis. The sedondary mirror is defined by the choice of three parameters: V m height of secondary above paraxial focus, Ts y se coordinates of secondary focus. Fig. 2 0 Shadow zones.

Equidistant incoming rays (L0 to R0 ) are shown, and their first reflections

(L to R 1 ) which are all tangent to the caustic. The numbers 0, i 2, 3 1 t tell how many independent shadows a tertiary would cast if placed at these locations, and the density of rays tells the size of a shadow. Fig. 3. , Location T(Y ) of secondary focus F 2 , if secondary mirror starts as far inward as possible, for optimum compactness.

If F is placed below the curve drawn, the secondary will start 2 more outward, to the right-hand side in Fig. 2, But F2 can never be placed above the curve (or the secondary would cast shadow on itself). v -70 t2 /0 t 0 Yt)

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Fig. 40 Smallest secondary, illuminating 700 feet symmetrically. Limiting extreme, for comparison only; no tertiary

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Fig. 5. Secondary and tertiary mirrors, for symmetric aperture of 700 feet. Optimum compactness without double or large shadow. V yo -; /0 ST) 7b;

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Fig. /5. Same system as Fig. /4; complete view including primary.