tion 17.3 as in Equations 17.5 and 17.6. 17.3.1 Radiation from apertures E, sin tru ) (17.5) The radiation from apertures illustrates most of the significant properties of pencil beam antennas. The radiation characteristics can be determined by simple mathematical relationships. If the " = " sin G (17.6) electric fields across an aperture, Figure 17.4, is Ea(x,y) then the radiated fields Ep( 0,0) is given by Equation 17.3, where S(0,0) This distribution occurs widely in antenna theory. It is plotted in is given by Equation 17.4. (Oliver, 1986; Milligan, 1985). Figure 17.5. The beamwidth is inversely proportional to the aperture 0.88A width The first sidelobe level is at -13.2dB which is a (17.3) disadvantage of a uniform aperture distribution. The level can be reduced considerably by a tapered aperture distribution where the (17.4) field is greatest at the centre of the aperture and tapers to a lower level at the edge of the aperture. For example if Equation 17.7 holds, then the first sidelobe level is at -23dB. For high or medium gain antennas the pencil beam radiation is (17.7) largely focused to a small range of angles around 0 = 0) In this case it can be seen from Equation 17.3 that the distant radiated fields, and the aperture fields are the Fourier transformation of each other. The energy which was in the sidelobes moves to the main beam Fourier transforms have been widely studied and their properties with the result that the beamwidth broadens to In practice can be used to understand the radiation characteristics of aperture antennas. Simple aperture distributions have analytic Fourier trans- almost all antennas have natural tapers across the aperture which forms, whilst more complex distributions can be solved numerically result from boundary conditions and waveguide modes. Rectangu- on a computer. lar apertures are formed from two line source distributions in ortho- The simplest aperture is a one dimensional line source distribution gonal planes. Circular apertures form the largest single class of apertu