Electromagnetic modes in conical transmission lines with application to the linearly tapered slot antenna

A transmission line analysis of the bow-tie antenna and the linearly tapered slot antenna (LTSA) is presented. These structures belong to the class of conical transmission lines defined here in terms of conical coordinates. A complete set of solutions of the Helmholtz equation is obtained exhibiting TE and TM modes. Modal fields are expressed by Lame (1837) and Bessel-Schelkunoff functions. TE and TM eigenmode analysis is particularized to the bow-tie structure. Bow-tie antenna and LTSA are shown to be dual conical transmission lines by the image method and Babinet's principle. The modes of LTSA are calculated on the basis of the results obtained for the bow-tie structure. The radiation pattern of the LTSA is computed as the integral of a closed-form expression of the dyadic Green's function weighted by the modal electric field distribution over the slot aperture. The obtained dominant mode radiation patterns are validated by measurements from the literature. The radiation patterns of the first two-order modes are calculated and compared.