Equivalent Circuit Modeling of Mutually Resistively Coupled Microwave Cavities with Enhanced Phase Sensitivity Using Thin Metallic Foils

Abstract

We formulate and validate an equivalent circuit model describing mutual resistive 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 a dissipative interface that mediates energy exchange via mutual resistance. This coupling mechanism produces interference effects and a controllable anti-resonance when the input resonators are amplitude- and phase-balanced, a behavior not achievable with standard microwave antenna probes. 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 resistance and coupling coefficient of these foils in this geometry. Under balanced conditions, a sharp anti-resonance emerges with a near order-of-magnitude enhanced phase sensitivity at the resonant frequency of the output cavity, consistent with model predictions. The experimentally extracted mutual coupling coefficients, $Δ_{13}=(5.00\pm0.01)\times10^{-6}$ and $Δ_{23}=(4.10\pm0.01)\times10^{-6}$, fall within the calculated range $Δ_{n3}\approx(1\text{--}48)\times10^{-6}$ derived from the foil's electromagnetic properties, where the spread is dominated by the estimated foil thickness uncertainty of $(9\pm1)\,μ\mathrm{m}$. These results confirm that resistive coupling can occur across a number of skin depths of a metallic interface, providing a new means of engineering controlled interference in multi-resonator systems. The approach offers potential applications in precision microwave experiments, phase-sensitive detection, and tests of fundamental electromagnetic interactions.

Figures (5)

• Figure 1: (a) The experimental configuration used to excite a sharp anti-resonance. The vector network analyser (VNA) signal entered from Port 1 and is split into two signals via a 3dB power splitter, labelled i). One signal passes through ii) a variable attenuator, acquiring a relative attenuation factor $\alpha$. The other passes through iii) a microwave phase shifter, introducing a relative phase shift $\phi$ between the two signals. The two signals excite iv) the three-cavity system, where the TM$_{010}$ mode of resonators 1 and 2 was excited, and each resonator coupled to resonator 3 through a < 10µ m thick copper foil. The coupling between resonators 1--3 and 2--3 is represented by the mutual impedances $Z_{13}$ and $Z_{23}$, respectively, which can be approximated as mutual resistances $R_{13}$ and $R_{23}$ near resonance. Port 2 has three series output amplifiers (two Low Noise Factory $\approx37$ dB and Minicircuits $17$ dB) totaling $89.7$ dB of gain in the circuit. (b) Photograph of the fabricated three-cavity apparatus with shielding. The interferometric input and the remaining two output amplifiers are located outside the chamber.
• Figure 2: The equivalent circuit diagram of the three-cavity resonator system coupled through thin metal foils. The three resonators are modeled by standard LCR circuits with capacitance $C_{n}$, inductance $L_{n}$, resistance $R_{n}$, and probe resistances $R_{0n}$ for resonator $n$. The mutual resistance terms $R_{13}$ and $R_{23}$, are shown coupling the resonators. The currents entering resonators 1 and 2, are modelled by current sources $I_{1}$ and $I_{2}$, with $I_{3}$ the current entering resonator 3 due to mutual contributions. The output current $I_{out}$ passes through the third resonator's probe.
• Figure 3: Magnitude and phase versus frequency, with the best-fit transfer function from Eq. (\ref{['transfunction']}) plotted alongside the data. The corresponding fit parameters are listed in Table \ref{['fitparams']}. Panels (a)-(c) illustrate the incremental approach towards resonance as the fitted relative phase $\phi$ is varied from $\pi$ to near $2 \pi$ radians, respectively. The input power at port 1 was 20 dBm and the measured $S_{21}$ spectra are referenced to the input of the amplifier chain.
• Figure 4: Comparison between the simulated transfer function (using the parameters in Table \ref{['fitparams']}) for a near-$\pi$ span of $\phi$ (left) and the interpolated 98 measured spectra (right). The absolute phase reference is arbitrary; therefore a constant phase offset was applied for visual alignment between model and data. Both plots are shown for the experimentally determined $\phi$ range.
• Figure 5: Measured phase response near the balanced interference condition showing enhanced phase sensitivity. The local phase slope increases by nearly an order of magnitude relative to the near-resonant operating point shown in Fig. \ref{['manysharp']}, consistent with the behaviour expected near the antiresonance condition.