Enhanced implementation of the complex images method to study bound and leaky regimes in layered planar printed lines

The discrete complex image technique, originally developed to deal with three-dimensional planar structures, has been recently applied to two-dimensional planar and nonplanar transmission lines embedded in layered substrates. In spite of successive refinements, some important practical issues concerning the sampling path, treatment of spectral Greenʹs functions poles, and extension to the leaky regime are still open and deserve deeper examination. It is shown that a suitable choice of the sampling path is key to reducing the number of images and to avoid the necessity of extracting spectral poles when the bound regime is considered. The situation arising from the analysis of the leaky regime becomes more complex and the adequate choice of the sampling path, as well as an appropriate pole extraction, turn out to be essential. In particular, a new pole-extraction strategy able to deal with poles associated with both proper and improper modes of the background waveguide is proposed. The advantages of using the theory in this paper are illustrated by diverse numerical results for the bound/leaky regime of various planar lines. Finally, the suitability of the method for covered planar lines is demonstrated.