Electrodynamics of a ring-shaped spiral resonator

We present analytical, numerical, and experimental investigations of electromagnetic resonant modes of a compact monofilar Archimedean spiral resonator shaped in a ring, with no central part. Planar spiral resonators are interesting as components of metamaterials for their compact deep-subwavelength size. Such resonators couple primarily to the magnetic field component of the incident electromagnetic wave, offering properties suitable for magnetic meta-atoms. Surprisingly, the relative frequencies of the resonant modes follow the sequence of the odd numbers as f1:f2:f3:f4… = 1:3:5:7…, despite the nearly identical boundary conditions for electromagnetic fields at the extremities of the resonator. In order to explain the observed spectrum of resonant modes, we show that the current distribution inside the spiral satisfies a particular Carleman type singular integral equation. By solving this equation, we obtain a set of resonant frequencies. The analytically calculated resonance frequencies and the current distributions are in good agreement with experimental data and the results of numerical simulations. By using low-temperature laser scanning microscopy of a superconducting spiral resonator, we compare the experimentally visualized ac current distributions over the spiral with the calculated ones. Theory and experiment agree well with each other. Our analytical model allows for calculation of a detailed three-dimensional magnetic field structure of the resonators.