In this letter, a new formulation for calculating an induction logging tool response in a multilayered transversely isotropic (also called uniaxial anisotropic) dipping formation is proposed. Using transmission line theory, only five independent integrals are necessary to form all the components of the dyadic Green's function in order to calculate the tool response, resulting in a more succinct and efficient formulation. The fast Hankel transform (FHT) is then employed to calculate these spectral integrals efficiently. In special cases, such as horizontal and vertical wells where the direct FHT fails, alternative approaches are proposed. A singularity extraction is implemented to handle the divergent integrals for a horizontal well, whereas the double-exponential integration rule is implemented for a vertical well. In order to speed up the computation, the spectral transmission line currents and voltages are precalculated and then stored in memory, and parallel computing using OpenMP is employed. Numerical examples demonstrate a speedup factor of more than two orders of magnitude compared with other reported results based solely on the FHT.

This review paper summarizes various aspects of directive beaming and explains these aspects in terms of leaky waves. Directive beaming occurs in antenna design where a narrow beam is obtainable by using fairly simple planar structures excited by a single source. These structures include Fabry-Pérot cavity structures as well as metamaterial structures made from artificial low-permittivity media. Directive beaming also occurs in the optical area where it has been observed that highly directive beams can be produced from small apertures in a metal film when an appropriate periodic patterning is placed on the film. One aspect that these phenomena all have in common is that they are due to the excitation of one or more weakly attenuated leaky waves, the radiation from which forms the directive beam. This is established in each case by examining the role of the leaky waves in determining the near-field on the aperture of the structure and the far-field radiation pattern of the structure.