  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, 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.
