Foil-Mediated Mutual-Impedance in Microwave Cavities with Enhanced Phase Response

Abstract

We formulate and validate an equivalent circuit model describing mutual coupling between three microwave cavity resonators interconnected via thin metallic foils. Each cavity is represented as a lumped LCR circuit, while the foils act as interfaces that mediate energy exchange via mutual impedance. This coupling mechanism produces interference effects and a controllable antiresonance when the input resonators are amplitude- and phase-balanced, while enabling coupling across a continuous metallic interface without direct antenna or aperture coupling. All three resonators operated in the TM${}_{010}$ mode, where two input resonators each excited the third via a thin copper foil. Analytical expressions are derived for the mutual impedance and coupling coefficient of these foils in this geometry. Under balanced conditions, a sharp antiresonance emerges with a several-fold enhanced phase sensitivity at the resonant frequency of the output cavity, consistent with model predictions. The experimentally extracted normalised mutual coupling coefficients, ${\Delta }_{13}=(10.8\pm 0.3)\times 10^{-6}$ and ${\Delta }_{23}=(8.6\pm 0.2)\times 10^{-6}$, fall within the calculated range ${\Delta }_{n3}\approx (7\text{–}17)\times 10^{-6}$ derived from the foil's electromagnetic properties, where the spread is dominated by the estimated foil thickness uncertainty of $(8.9\pm 0.4)\mu \mathrm{m}$. These results demonstrate a foil-mediated physical implementation of weak mutual coupling across a metallic interface spanning multiple skin depths, providing a distinct route for engineering controlled interference in three-dimensional multi-resonator systems.