is below the main lobe by 38 dB plus the relative gain of the subarray patterns in that angular region. If there is also distortion in the subarray, then the problem becomes somewhat less obvious. The subarray levels probably become worse, although there might be a situation in which they would get better by about 12 dB (100,000 modules in contrast to 7000 subarrays). This effect should be studied. Even if the subarray is rigid, however, manufacturing and installation tolerances on the modules within the subarray will introduce a priori unknown position errors which will affect the average far-out sidelobe level. Thus, it is possible that power which is lost to the sidelobes may be somewhat higher than predicted. This effect should also be given some consideration. Beam Taper To achieve a high beam efficiency an array amplitude taper of 8-10 dB is indicated. This should not be approximated in a stair-step fashion by subarrays of constant amplitude. Instead each subarray should have its own amplitude taper in both coordinates, to obtain the smoothest approximation to the desired array excitation. Of course, with square subarrays a circularly symmetric array excitation cannot be exactly realized, but the stair-step can be bettered as it produces higher sidelobes than necessary. In choosing the overall excitation function, distributions which are easy to integrate should be avoided. Instead of easy mathematics the design should be based on good physics. The antenna art now designs distributions by placing zeroes at the proper places for shaping the sidelobe envelope. There are, in fact, rotationally symmetric distributions with low Q, adjustable sidelobe level, and high efficiency. A more advanced design process would construct a distribution that optimizes beam efficiency subject to constraints on distribution edge pedestal and on sidelobe envelope. Such an optimization can be formulated as the ratio of two quadratic Hermitian forms; exact solutions can be calculated. For a program of the importance (and size) of the SPS, crude handbook array distributions are not adequate. At least a distribution with proper zeroes should be used, and eventually a constrained synthesis should be employed to optimize beam efficiency. An improvement of 1 or 2 percent can easily pay for good antenna design. Lattice Design Regarding the spacetenna array element lattice, examination of the effective area of an element in an array or an examination of the element active gain which is equivalent, shows that for small arrays the lattice should be square with half-wave spacing in order to maximize gain. As the array gets larger the element spacing may be increased without serious gain loss. In the limit of an infinitely large array the element spacing can be just under one wavelength in a square lattice or 15% larger in a hexagonal lattice without any gain decrease. That is to say, the effective area of half-wave dipoles or slots is sufficient to allow these larger spacings in an infinite array. This was shown rigorously in Chapter 3, Volume 2 of Microwave Scanning Antennas (R. C. Hansen, ed. Academic Press, 1966). Thus, since the spacecraft antenna is very large in wavelengths, the slot spacing can, in principle, be made nearly one wavelength except for the elements near the edge. The practical implications are that the waveguides can be operated closer to cutoff to increase slot spacing and that the waveguides need not be contiguous but can be separated. This reduces the weight and complexity of the antenna.
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