Magnetism and homogenization of microresonators

Arrays of cylindrical metal microresonators embedded in a dielectric matrix were proposed by Pendry et al. [IEEE Trans. Microw. Theory Tech., 47 (1999), pp. 2075-2084] as a means of creating a microscopic structure that exhibits strong bulk magnetic behavior at frequencies not realized in nature. This behavior arises for H-polarized fields in the quasi-static regime, in which the scale of the microstructure is much smaller than the free-space wavelength of the fields. We carry out both formal and rigorous two-scale homogenization analyses, paying special attention to the appropriate method of averaging, which does not involve the usual cell averages. We show that the effective magnetic and dielectric coefficients obtained by means of such averaging characterize a bulk medium that, to leading order, produces the same scattering data as the microstructured composite.