e radiation characteristics of aperture antennas. Simple aperture distributions have analytic Fourier transforms, while more complex distributions can be solved numerically on a computer. The simplest aperture is a one-dimensional line source distribution of length ta serves to illustrate many of the features of aperture antennas. If the field in the aperture is constant, the radiated field is given by equation 9.3 as in equations 9.5 and 9.6. (9.5) (9.6) Chapter 9 Antennas J1(x)/x sin(x)/x Fig. 9.5 Dipole and radiation pattern. This distribution occurs widely in antenna theory. It is plotted in Figure 9.5. The beamwidth is inversely proportional to the aperture width and is 0.88 A/a. The first sidelobe level is at - -13.2 dB, which is one disadvantage of a uniform aperture distribution. The level can be reduced considerably by a tapered aperture distribution, where the field is greatest at the center of the aperture and tapers to a lower level at the edge. For example, if equation 9.7 holds, then the first sidelobe level is at -23 dB. Ea(x)=cos() (9.7) The energy that was in the sidelobes moves to the main beam with the result that the beamwidth broadens to 1.2 A/A. In practice almost all antennas have natural tapers across the aperture that result from boundary conditions and waveguide modes. Rectangular apertures are formed from two line source distributions in orthogonal planes. Circular apertures form the largest single class of aperture antennas. The parabolic reflector is widely used in communications and is often fed by a conical horn. Both the reflector and the horn are circular apertures. For an aperture distribution, which is indepen- dent of azimuthal angle, the simplest case is uniform illumination, which gives a radiated field as in equation 9.8, where J1(x) is a Bessel function of zero order. (9.8) This can be compared to sin(x)/x and is also plotted in Figure 9.5. The first sidelobe level is at - -17.6 dB. Table 9.2 lists a number of circular aperture distributions and corresponding radiation patt