The non-radiative dielectric (NRD) waveguide was introduced by Yoneyama in 1981. In Fig. 1 the crosses shown: it consists of a dielectric rectangular slab of height (a) and width (b), which is placed between two metallic parallel plates of a suitable width. The structure is practically the same as the H waveguide, proposed by Tischer in 1953. Due to the dielectric slab, the electromagnetic field is confined in the vicinity of the dielectric region, whereas in the outside region for suitable frequencies, the electromagnetic field decays exponentially. Therefore, if the metallic plates are sufficiently extended, the field is practically negligible at the end of the plates and therefore the situation does not greatly differ from the ideal case in which the plates are infinitely extended. The polarization of the electric field in the required mode is mainly parallel to the conductive walls. As it is known, if the electric field is parallel to the walls, the conduction losses decrease in the metallic walls at the increasing frequency, whereas, if the field is perpendicular to the walls, losses increase at the increasing frequency. Since the NRD waveguide has been devised for its implementation at millimeter waves, the selected polarization minimizes the ohmic losses in the metallic walls. The essential difference between the H waveguide and the NRD guide is that in the latter the spacing between the metallic plates is less than half the wavelength in a vacuum, whereas in the H waveguide the spacing is greater. The conduction losses in the metallic plates decrease at the increasing spacing. Therefore, this spacing is larger in the H waveguide, used as a transmission medium for long distances; instead, the NRD waveguide is used for millimeter wave integrated circuit applications in which very short distances are typical. Thus an increase in losses is not of great importance. The choice of a little spacing between the metallic plates has a fundamental consequence that the required mode results below the cut-off in the outside air regions. In this way, any discontinuity, such as a bend or a junction, is purely reactive. This permits radiation and interference to be minimized (hence the name of the non-radiative guide); this fact is of vital importance in integrated circuit applications. Instead, in the case of the H waveguide, the above-mentioned discontinuities cause radiation and interference phenomena, as the desired mode, being above cutoff, can propagate towards the outside. In any case, it is important to notice that, if these discontinuities modify the symmetry of the structure with reference to the median horizontal plane, there is anyway radiation in the form of TEM mode in the parallel metallic plate guide and this mode results above cutoff, the distance between the plates may be no matter short. This aspect must always be considered in the design of the various components and junctions, and at the same time much attention has to be paid to the adherence of the dielectric slab to the metallic walls because the above-mentioned phenomena of losses are generated. This occurs when in general any asymmetry in the cross section confined mode into a "leaky" mode.
The dispersion relation in the NRD waveguide
The dispersion relation, the equation yielding the longitudinal propagation constant k z = β − j α {\displaystyle k_{z}=\beta -j\alpha } , is a function of the frequency and the geometric parameters for the various modes of the structure. In this case, however, this relation cannot be expressed explicitly, as it is verified in the most elementary case of the rectangular waveguide, but it is implicitly given by a transcendental equation.
The transverse resonance method
In order to obtain the dispersion relation it is possible to proceed in two different ways. The first one, which is simpler from the analytic point of view, consists of applying the transverse resonance method to obtain a transverse equivalent network. According to this method, the resonance condition along a transverse direction will be applied. This condition brings to a transcendental equation that, numerically solved, gives possible values for the transverse wavenumbers. Exploiting the well-known relation of separability which links the wavenumbers in the various directions and the frequency, it is possible to obtain the values of the longitudinal propagation constant kz for the various modes. It is supposed that the radiation losses, because actually the metallic plates have a finite width, are negligible. In fact, supposing that the field evanescent in the outside air-regions is negligible at the aperture, one can assume that the situation substantially coincides with the ideal case of the metallic plates having infinite width. Thus, one can assume the transverse equivalent network shown in Fig. 2. In it kxε and kx0 are the wavenumbers in the x transverse direction, in the dielectric and in the air, respectively; Yε and Y0 are the associated characteristic admittances of the equivalent transmission line. The presence of the metallic plates, considered perfectly conductive, imposes the possible values for the wavenumber in the y vertical direction: k y = m π a {\displaystyle k_{y}={\frac {m\pi }{a}}} , with m = 0, 1, 2, ... These values are the same in the air as in the dielectric regions. As above mentioned, the wavenumbers must satisfy the separability relations. In the air region, assimilated to a vacuum, one obtains
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