From green and Sommerfeld to Takahasi and Mori: Recent developments in Computational Electromagnetics

Since their introduction in the seventies, microstrip antennas have become one of the most popular and successful type of antenna. Nowadays, the original printed microstrip patch antenna has evolved into a very general class of planar antennas, used in frequencies from 100 MHz to 100 GHz, where thin metallizations (patches, ground planes, slots) occupy the interfaces of a multilayered dielectric substrate, with eventual metallic probes and/or via-holes possibly connecting the different interfaces. The most successful computational model for these antennas is provided by an integral equation formulation, where the associated Green´s functions are originally expressed as Sommerfeld integrals (SIs). Currently, some of the most popular commercial Computational ElectroMagnetics software codes are directly based on this strategy. Hence, the robust and accurate evaluation of Sommerfeld integrals is of paramount relevance, since any improvement in their calculation will directly result in faster and more accurate software tools. This paper reviews two of the most successful algorithms that have recently been applied to the numerical evaluation of these Sommerfeld integrals: the Weighted Averages (WA) algorithm and the Double Exponential (DE) quadrature. Although their use by the Antennas&Propagation community is quite recent, both algorithms are solidly rooted on well established numerical mathematics. Weighted Averages can be considered as an adaptation of the classic Euler transformation, while the Double Exponential quadrature was introduced in 1974 by two Japanese mathematicians, Takahasi and Mori. Together, WA and DE combine to provide a numerical evaluation of Sommerfeld integrals of unprecedented quality. This paper will provide a historical perspective of both algorithms together with numerical results showing their practical interest, which clearly go far beyond the Sommerfeld integrals´ realm.