Periodic arrays of long circular cylinders, whether metallic or dielectric, are key components of a multitude of metamaterial designs. These include all-dielectric optical metamaterials, hyperbolic media, super-resolution endoscopes, drawn metamaterial fibers, and single-molecule bio-sensors.

Characterizing the response of individual cylinders is a fundamental step towards describing the effective electromagnetic response. Despite their simplicity, individual cylinders offer rich design potential, by exploiting the interplay between electric and magnetic Mie resonances. For optically thin cylinders, scattering is adequately described by only dipolar fields, considerably simplifying the subsequent treatment by effective medium theories for arrays of cylinders. Analytical results are available, enabling a systematic design process.

This proceeds by analogy to the polarizability tensor α of a molecule, which quantifies the distortion of an electron cloud under the action of impinging fields, ultimately yielding the macroscopic constitutive relations ε and μ. When the tensor is diagonal, only electric fields induce electric dipoles, but a general metamaterial element may have non-zero off-diagonal blocks, representing a magneto-electric coupling whereby magnetic fields induce electric dipole moments. 

Due to its fundamental importance, a number of recent papers discuss the retrieval of the dipolar polarizability of infinitely long cylinders under normal incidence. Yet the literature on retrieving the full polarizability tensor under oblique incidence is sparse. Only one publication to our knowledge considers this case, but surprisingly a magneto-electric coupling was reported [1], which usually only arises in non-centrosymmetric scatterers such as split ring resonators.

We delve into the physical origins of this coupling, establishing insight which is critical to the design of metamaterials exhibiting artificial magnetism. We develop two independent arguments, serving two different purposes. Firstly, we provide formal symmetry arguments based on group theory to reconcile the unexpected magnetoelectric coupling with existing symmetry theorems. Our second argument is heavily motivated by physical insight, deriving a reformulation of the polarizability tensor which restores its expected diagonal response. This is achieved via a simple decomposition of the tensor into its TM and TE components through a transformation which we also derive. An advantage of this reformulation is that it enables insight into the physical origins and behavior of the magnetoelectric coupling based on the basic geometry of planes waves, thus eliminating the need to invoke abstract symmetry arguments.

Our key insight is that the off-diagonal magnetoelectric coupling terms account for the difference between TM and TE responses, allowing two dissimilar diagonal responses to be combined onto a single tensor. The magnetoelectric coupling of the infinite cylinder behaves differently from the more familiar magnetoelectric coupling of say split ring resonators, due to their different physical origins.  Our TM/TE reformulation provides ready explanations for the unusual magnetoelectric coupling of infinite cylinders, which for example changes sign when the angle of incidence is inverted, and disappears when counterpropagating waves are impinging.

Another advantage of the TM/TE reformulation is that it also allows insight into the quantitative properties of magnetoelectric coupling. For dielectric cylinders coupling strength has a weak k⁴ scaling at long wavelengths, but scales as k² for perfect electric conductors. We retrace these weak scaling laws to the difference between TM and TE responses, which are identical to leading order. Magnetoelectric coupling does become prominent though at Mie resonances of the cylinder, since these often have either a TM or TE character. However, our analysis reveals that unlike structures without inversion symmetry, the strength of the magnetoelectric resonance cannot differ from that of both the electric and magnetic resonances.

[1] D. Strickland, A. Ayón and A. Alù. Dynamic polarizability tensor for circular cylinders. Phys. Rev. B 91 (8), 085104 (2015)