For n = 1 (untapered beam), this becomes: which was seen in Reference 2. For the general case, Equation Ila shows that r, depends on Is to the power of -l/(2n + l) and frequency to the power of (l-2n)/ (l+2n). This is summarized in Table 2. Two additional advantages of beam tapering are thus illustrated. As the taper is increased, changing the safety threshold has less effect on the exclusion zone radius, making strict thresholds more feasible. An increased taper also allows for a more rapid contraction of exclusion zone radius with increasing frequency. Since main lobe radius varies as the inverse of frequency, an increased taper will allow exclusion zone radius to vary in a similar manner. Thus, both land and rectenna cost can be minimized. (The possibility that higher frequencies will involve rectennas with a higher cost per unit area has not been considered.) The above estimates of exclusion zones apply if the boundary is far from the main lobe. This occurs when the safety threshold is very small compared with the peak beam intensity, i.e., for small n, high frequency, and/or low threshold. For thresholds that are somewhat larger, the exclusion zone boundary is near, or perhaps even within, the main lobe. In addition, the "ideal" case does not lend itself to as simple an exclusion zone estimate. lire exclusion zone radii were therefore computed for direct numerical comparison of Equations 2 and 3 with various safety thresholds. A total power of 5 GW was considered, as in the NASA/DoE reference design (4], For simplicity, power beamed from a 1 km2 circular antenna to a rectenna at the equator was considered. Standards for microwave exposure range from 0.01 mW/cm2 (Eastern Europe [13], [14]) to 1 mW/cm2 (Canada [11]) to 10 mW/cm2 (US and Western Europe [11]). The US and Western European standard is based mainly on thermal biological effects [11], while the Eastern European standard was established to avoid neurological effects [14]. Since there is not yet a general agreement on the existence of, and safety threshold for, nonthermal effects, more research must be done in this area. The cases considered here will therefore span this entire range of safety thresholds. The US and Western European standard should be considered a "best case" scenario in that exclusion zones will have to be at least as large as this case requires, and may have to be much larger. Standards much stricter than that of
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