Probing split-ring resonator permeabilities with loop-gap resonators

A method is proposed to experimentally determine the effective complex permeability of split-ring resonator (SRR) arrays used in the design of metamaterials at microwave frequencies. We analyze the microwave response of a loop-gap resonator (LGR) whose bore has been partially loaded with one or more SRRs. Our analysis reveals that the resonance frequency, magnetic plasma frequency, and damping constant of the effective permeability of the SRR array can be extracted from fits to the reflection coefficient (S11) of an inductively-coupled LGR. We propose LGR designs that would allow both a one-dimensional array of SRRs and small three-dimensional arrays of SRRs to be characterized. Finally, we demonstrate the method using a toroidal LGR loaded with a single extended SRR of length z.