Sommerfeld Integrals and Their Relation to the Development of Planar Microwave Devices

This paper deals with the mathematical expressions called Sommerfeld integrals. Introduced by A. Sommerfeld in 1909, they are mathematically equivalent to inverse Hankel transforms. The original historical goal of these integrals was to provide an accurate mathematical description of the electromagnetic phenomena involved in long-distance wireless radio and telegraphy. However, their scope was quickly enlarged thanks to the so-called spectral-domain stratified theory, and now they are ubiquitous in the mathematical models associated to many electromagnetic technologies, ranging from EMC lightning modelization and ground penetrating radar to optical and plasmonic integrated devices and going through the familiar microwave and millimeter-wave planar structures using printed circuit technology. In all these areas, Sommerfeld integrals can provide direct evaluations of the involved electromagnetic fields or they can be used as Green's functions in the frame of integral equation formulations. Other disciplines involving stratified media, like seismology and geological prospection, also benefit from these integrals. After discussing the most canonical Sommerfeld integral, appearing in the so-called Sommerfeld identity, this paper reviews three classical structures, namely, the original Sommerfeld problem involving two semi-infinite media, and the microstrip and stripline geometries. It is shown that Sommerfeld integrals provide a unifying treatment of these three problems and that their mathematical features have a direct translation in terms of the physical properties exhibited by the electromagnetic fields that can exist in them.