found at rrr/2 = 6.286, or r ~ 4. At that point, the intensity is just under 2xl0'5 of its peak value. Thus, this point can be considered to be the radius of the main lobe. By comparing the numerical integral over the main lobe with the analytic integral [8] over all two-dimensional space, Fideal = 0.9851 is obtained. Thus, an untapered beam puts 83.8% of the energy into the main lobe, while the various tapered beams under consideration place between 98 and nearly 100% of the energy into the main lobe. By using Equation 6a, with Fn(u) = 0.838 and solving for u, the radius of the portion of the diffraction pattern that contains 83.8% of the energy can be found for the tapered beams (see Table 1). For tapers with n > 3, the 83.8% capture area is larger than it is in the untapered case. In addition, although the edge of the main lobe is further from the center for the "ideal" case than it is for n = 2, if the "ideal" intensity distribution is integrated out to the distance of the first minimum of the n = 2 case, and it is seen that 98.4% of the energy is contained therein. Since integrating out to the first minimum of the "ideal" case increases this figure only slightly (to 98.5%), the "ideal" and n = 2 cases have nearly identical main lobes (see also Figure 2).
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