An Indirect Method for an Alternate Perspective to Dispersion Diagrams of Magnetoinductive Waveguides

The dispersion equation of magnetoinductive waveguides (MIWs) is traditionally solved directly for the propagation constant (
$\gamma $
) to obtain diagrams for the attenuation (
$\alpha $
) and phase (
$\beta $
) constants vs. angular frequency (
$\omega $
). Here, we introduce an indirect method of solving the equation, in two steps. By doing so, additional information and insights are provided in the intermediate step, not available via the traditional direct method. The first step of the reported approach splits
$\alpha $
into attenuation constant in the propagating passband (
$\alpha '$
) and the evanescent stopband (
$\beta ''$
); and
$\beta $
into phase constant in the propagating passband (
$\beta '$
) and the evanescent stopband (
$\alpha ''$
). The resulting four diagrams can then be suitably combined in the second step to obtain
$\alpha $
and
$\beta $
diagrams that show excellent congruence with the direct method. The additional information obtained is fundamental in nature and can be utilized to aid in the understanding and design of MIWs. For instance, it can help formulate quantitative criteria to precisely segregate the propagating passband from the evanescent stopband, which further enables a theoretical bound for the propagating mode of operation. The reported indirect method is generic and can be used for any type of MIW structure and any order of coupling.