Application of the transverse resonance method for efficient extraction of the dispersion relation of arbitrary layers in silicon interposers

In this article, the dispersion relation for horizontal wave propagation in silicon interposers consisting of arbitrary numbers of silicon and silicon dioxide layers between metal layers is investigated. Results are obtained by the transverse resonance method (TRM). The method is verified by comparison to results of finite element based full-wave simulations. The results of the TRM show good correspondence with those obtained by full-wave simulations and can be obtained within significantly shorter calculation times. Results for a typical layout of a silicon interposer and examples for thin silicon dioxide layers and a large number of layers are also presented. Further, an approximate solution for the fundamental mode of typical structures is given and examples show the applicability and limitations of the approximate solution.