Metamaterials are recent artificial media with very unusual electromagnetic properties. Waveguides containing metamaterials are very likely to be used in the design of future devices and components for the microwave regime and for millimeter-wave circuits. Several analytical methods, suitable for the characterization of the guided-wave propagation in metamaterial waveguides, are addressed in this paper. Approximate methods, like the effective index method, and more rigorous methods, like the transverse resonance method or the least squares boundary residual method, are independently applied and finally compared.
These analytical methods, which are well-known when applied to conventional waveguides, exhibit new features and require appropriate adaptation when dedicated to metamaterial waveguides. In most cases, the complete 3D waveguide analysis requires, in a first step, the characterization of two-dimensional metamaterial planar structures, according to a building block approach. Generally, these methods usually require a full-wave analysis of the lateral step discontinuities of the waveguide, through the application of the mode matching technique, and the discretization the continuous spectrum [2], [4]. On the other hand, approximate methods treat separately the field dependence on the transverse coordinates.
The numerical comparison between the values of the longitudinal wavenumber obtained by using the approximate effective index method and those computed through a more accurate mode-matching technique, shows that the approximate effective index method yields numerical results which can be sufficiently accurate. However, significantly differences may show, depending on the mode under analysis and how far from the mode cutoff is the frequency of operation. Moreover, the effective index method cannot predict leaky modes and certain new phenomena like the existence of super-slow modes, which are very typical in most types of metamaterial waveguides.