Closed form expressions for the modal dispersion equations and for the characteristic impedance of a metamaterial based gap waveguide

In a recent sequence of papers, a parallel-plate ridge gap waveguide has been introduced, that consists of a metal ridge in a metamaterial magnetic conductor surface, covered by a metallic plate at a small height above it. The gap waveguide is relatively simple to manufacture, especially at millimetre and sub-millimetre wave frequencies when compared with other solutions. The metamaterial surface is designed to provide a frequency band where parallel-plate modes are in cut-off, thereby allowing for a confined gap wave to propagate along the ridge. In a previous work, the authors have presented an approximate analytical solution for the confined quasi-TEM dominant mode of the ridge gap waveguide, when the surrounding metamaterial surface is in the form of a bed of nails. In this study, the authors investigation continues by providing an analytical expression of the modal dispersion equation of the first higher order ridge mode and of the characteristic impedance of the dominant mode. As in the previous paper, the field problem is divided in three regions, the central region above the ridge and the two surrounding side regions above the nails. Transverse mode-matching applied to a few modes representation in each region, results in a closed form expression of the dispersion equation of the first higher order mode. After summarising the formulation for the dominant quasi-TEM mode, the dispersion equation of the first higher order mode is derived, in order to give a criterion to maximise the unimodal bandwidth. Furthermore, three different closed form expressions of the dominant mode characteristic impedance are derived and compared with approximate expressions already used in literature.