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Metamaterials consisting of long, circular, cylinders are very popular. It is a fundamental challenge to characterize the effective electromagnetic response of such composites. In this framework, the radius of cylinder is assumed to be considerably smaller than the external wave length, thus the dominant scattered EM fields can be approximately replaced by dipole fields. Previous works dealt mainly with two dimensional (2D) scenarios, i.e., characterizing the effective electromagnetic response for light propagation perpendicular to the cylinder axis. In this work, we generalize this treatment to three dimensions (3D), i.e., we characterize the effective electromagnetic response for light propagation at any angle, and find that the resulting electromagnetic response is non-local, i.e., it depends on the wavevector component parallel to the cylinder axis. We retrieve analytically, the full polarizability tensor and show that it has different contributions for different polarized incoming EM waves (transverse electric and transverse magnetic with respect to the cylindrical axis). It is also diagonal, i.e., it contains no magneto-electric coupling, showing that claims in previous studies were incorrect. Having closed form expressions for polarizability allows us to use effective medium approximation methods, and tailor the spectral response for both electric and magnetic dipolar contributions. It is important to emphasize that for the first time, this gives a fully systematic way to characterize the magnetism. Our analysis holds for additional structures based on cylindrical geometry, such as hole arrays, all-dielectric metamaterials, and multi-layer cylinders. It can be used to explain the electromagnetic response of wire media attributed with a negative refractive index, effective magnetism and hyperbolic dispersion relations. In addition, this approach can be applied to more complex unit cells e.g., consisting of clusters of parallel cylinders.
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