## 1 Introduction  
Over recent years, a great number of substrate integrated circuits [1] have been developed. The characteristics of these circuits are midway between the characteristics of waveguide technologies and planar circuits. They have a higher quality factor and lower insertion losses than planar circuits, while also being low profile, low cost, and easily integrable in a printed circuit board. The substrate integrated waveguide (SIW) [2] and the substrate integrated coaxial line (SICL) [3] are two proposals of substrate integrated circuits that have been a great advance in this field. Several passive components developed in these technologies, including filters [4, 5], ultra-wideband filters [6, 7], antennas [8, 9], transitions and tapers [10, 11], baluns [12, 13], couplers [14, 15], power dividers [16], and new transmission lines [3, 17, 18, 19, 20], have been proposed. However, although their quality factor and losses are better than those of the planar circuits, these characteristics can be further improved(without losing their compactness and low-cost manufacturing) if the dielectric substrate is removed and the waves are allowed to propagate through the air. This is the purpose of some empty substrate integrated waveguides that have been recently proposed, such as the Modified SIW (MSIW) [21], the Empty SIW (ESIW) [22], the Air-filled SIW (AFSIW) [23], the Hollow SIW (HSIW) [24], or the Dielectricless SIW (DSIW) [25]. All these new transmission lines are integrated in a substrate and they are low profile, low cost, and have higher quality factor and less losses than the SIW, thanks to the complete or partial removal of the dielectric in the path of the electromagnetic (EM) fields. However, these are one conductor structures, which means that the fundamental mode is dispersive, with a cutoff frequency greater than zero, which limits the usable bandwidth of microwave devices implemented with such novel technologies. These limitations can be overcome with the use of an empty substrate integrated coaxial line (ESICL). The structure of an ESICL was first used in Reference [26] for feeding an antenna, and definitively integrated in a planar circuit in Reference [27], where a transition to a classical planar line (a grounded coplanar waveguide) was proposed for the first time. The ESICL is a two-conductor transmission line, whose cross section is a rectangular coaxial built with the superposition of five layers of printed circuit board, as shown in Figure 1. This novel transmission line has been recently proposed, and until now, it has only been applied to the implementation of wide band [27] and narrow band [28] microwave filters with a non-dispersive transversal electric and magnetic (TEM) propagating mode. There are, consequently, many possible applications that remain to be explored for this novel transmission line. One of the possible applications is its implementation in continuous profile filters. Conversely to conventional microwave filter topologies, where the filtering structure is based on sharp changes in the cross section that produce sharp impedance steps, continuous profile filters use a progressive and smooth variation of the cross section. This continuous variation has the advantage, at the cost of increasing the length of the filter, of providing a structure with much more resilience to manufacturing tolerances. Besides, the continuous profile filters can be easily and accurately designed with well-known techniques such as the ones described in Reference [29; 30]. All these techniques require the phase constant to be kept constant (for the same frequency) along the propagation direction. This can be achieved with a structure that propagates a TEM mode (as is the case of the ESICL), or with a structure that propagates a non-TEM mode and that varies the cross section along the propagation direction in such a way that the cutoff frequency of this mode does not change. This can  
Figure 1: Simplest construction of an empty substrate integrated coaxial line (ESICL) (source Reference [27]).