The Complex Permeability of Split-Ring Resonator Arrays Measured at Microwave Frequencies

Abstract: We have measured the relative permeability of split-ring resonator (SRR) arrays used in metamaterials designed to have $\mu^\prime< 0$ over a narrow range of microwave frequencies. The SRR arrays were loaded into the bore of a loop-gap resonator (LGR) and reflection coefficient measurements were used to determine both the real and imaginary parts of the array's effective permeability. Data were collected as a function of array size and SRR spacing. The results were compared to those obtained from continuous extended split-ring resonators (ESRRs). The arrays of planar SRRs exhibited enhanced damping and a narrower range of frequencies with $\mu^\prime<0$ when compared to the ESRRs. The observed differences in damping, however, were diminished considerably when the array size was expanded from a one-dimensional array of $N$ SRRs to a $2\times 2\times N$ array. Our method can also be used to experimentally determine the effective permeability of other metamaterial designs.