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.

中文翻译：

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一种替代透视磁感应波导色散图的间接方法

磁感应波导 (MIW) 的色散方程传统上直接求解传播常数 ( $\伽马$ ) 以获得衰减图 ( $\阿尔法$ ) 和相位 ( $\beta $ ）常数与角频率（ $\欧米茄$ ）。在这里，我们分两步介绍求解方程的间接方法。通过这样做，在中间步骤中提供了额外的信息和见解，这是传统直接方法无法获得的。报告方法的第一步分裂 $\阿尔法$进入传播通带中的衰减常数（ $\阿尔法'$ ) 和倏逝阻带 ( $\beta ''$ ); 和 $\beta $成传播通带中的相位常数（ $\beta '$ ) 和倏逝阻带 ( $\alpha ''$ ）。然后可以在第二步中适当地组合得到的四个图表以获得 $\阿尔法$和 $\beta $显示与直接方法极好的一致性的图表。获得的附加信息本质上是基本信息，可用于帮助理解和设计 MIW。例如，它可以帮助制定定量标准，以精确地将传播通带与倏逝阻带分开，这进一步为传播操作模式提供了理论界限。报告的间接方法是通用的，可用于任何类型的 MIW 结构和任何耦合顺序。