ized. This second, TM11, mode, is generated by a step change in the conical horn diameter, and the length of the horn is determined by the need to have the modes in the correct phase relationship the aperture. The dual-mode horn gives low cross-polarization over a narrow band of fre- quencies. They are simple to make and of low weight. Chapter 9 Antennas The concept of adding higher-order modes in a horn can be extended for other purposes. In tracking feeds a higher-order mode is used to provide tracking information. The inherent cross-polarization that occurs in offset reflectors can be canceled by the appropriate addition of higher-order modes (Love, Rudge, & Olver, 1986). Finally, the main beam can be shaped to provide higher efficiency in prime-focus reflectors, although only over a narrow fre- quency band. Conical Corrugated Horns Conical corrugated horns are the leading choice for a feed for dual-reflector and medium- size offset reflectors (see Figure 9.13(b)). They have excellent radiation pattern symmetry and radiate very low cross-polarization over a broad range of frequencies. A corrugated horn propagates a mixture of TE11 and TM11 modes, called a hybrid HE11 mode. The corrugations are approximately a quarter of a wavelength deep SO that the electric short circuit at the base of the slot is transformed into a magnetic short circuit at the top of the slot. The result is that the azimuthal magnetic field is forced to zero at the corrugations and the azimuthal electric field is zero because of the ridges. Consequently, the boundary conditions of the TE and TM modes are identical, and the mutual propagating modes are linear combinations of the two parts. The design procedure for corrugated horns is well understood (Clarricoats & Olver, 1984), and it is possible to accurately predict the radiation characteristics. Array Feeds Array feeds are used to form multiple-beam and shaped-beam reflector antennas used on satellites (see Figure 9.13(e)). The individual elements of the array can be any type of horn, alt