The transition between reactive and radiative regimes for leaky modes in planar waveguides based on homogenized partially reflecting surfaces

The gradual transition from a reactive to a radiative regime is studied for leaky modes supported by multilayered planar open waveguides. The so-called leaky cutoff condition, i.e., the frequency for which the leaky phase constant $eta $ equals the leaky attenuation constant $alpha $ , originally introduced for microstrip lines and other printed structures, is investigated here with the aim of providing detailed information on the relative amount of reactive and radiative attenuation for leaky modes excited by finite sources and propagating as cylindrical waves along general planar waveguides. Analytical results are derived on the basis of a lossy parallel-plate-waveguide model and are validated through full-wave numerical simulations of 2-D leaky-wave structures based on grounded slabs covered with lossless or lossy partially reflecting surfaces (including, e.g., graphene layers) that can be treated as homogenized sheets. An analysis of the complex wave impedance of the considered leaky modes is also provided, in order to assess the frequency ranges where a good input matching can be expected for practical sources. In this regard, an ad hoc impedance matching network is designed and full-wave validated for a specific case to show that is indeed possible to achieve a good impedance matching below the cutoff in practical designs