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Electrodynamics of a planar Archimedean spiral resonator

We present a theoretical and experimental study of electrodynamics of a planar spiral superconducting resonator of a finite length. The resonator is made in the form of a monofilar Archimedean spiral. By making use of a general model of inhomogeneous alternating current flowing along the resonator and specific boundary conditions on the surface of the strip, we obtain analytically the frequencies $f_n$ of resonances which can be excited in such system. We also calculate corresponding inhomogeneous RF current distributions $ψ_n (r)$, where $r$ is the coordinate across a spiral. We show that the resonant frequencies and current distributions are well described by simple relationships $f_n=f_1 n$, and $ψ_n(r)\simeq \sin[πn (r/R_e)^2]$, where $n=1,2...$, and $R_e$ is the external radius of the spiral. Our analysis of electrodynamic properties of spiral resonators' is in good accord with direct numerical simulations and measurements made using specifically designed magnetic probe and laser scanning microscope.