gated by the moderate power levels (compared to the 5 GW SPS), and antenna thermal restrictions assume greater importance because of the relatively compact sizes of short range antennas and rectennas. Taper options for a satellite microwave system range from a thermally advantageous uniform taper to an optimal high efficiency taper (Fig. 9). The pattern produced at the field rectenna is found theoretically by summing the electromagnetic wavefronts emitted from the antenna. Each subarray projects a rectangular planar wavefront which combines with the distinctly phased wavefronts from adjacent subarrays to form an irregular, highly varigated beam in the antenna's near field. The result at far field ranges (ranges > antenna diameter2/wavelength) is a narrow, coherent beam with the radial symmetry of the overall antenna. Assuming far-field propagation conditions, the electromagnetic pattern at the rectenna is mathematically given by the Fourier transform integral of the antenna aperture distribution (the Fourier-Bessel transform is used for circular antennas) (8). Thus, the rectenna beam pattern depends directly on the antenna taper selected. The beam characteristic of greatest interest for far-field power transmission is the power collection efficiency at the rectenna. As defined previously, this efficiency is equal to the received power density integrated across the rectenna divided by the total power transmitted. For generalized polynominal antenna tapers, the rectenna power density solutions and efficiency integrals involve complicated Bessel function series. Resulting efficiencies are usually determined by numerical integration. One recent paper however has found closed form matrix solutions to give the highest efficiency taper for any known set of beam transmission conditions (9). The collection efficiency of a uniform taper antenna can be expressed more simply in closed form, and is equally indicative of the physical and geometrical constrains upon high efficiency microwave beam transmission:
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