Dynamic polarizability tensor for circular cylinders

Based on Mie scattering theory, we derive the complete dynamic polarizability tensor for circular, azimuthally symmetric cylinders excited by an arbitrary field distribution, and provide compact expressions for all of its elements. Our results comprise fully dynamic cylinder polarizabilities, improving existing approximate models that use averaged electric or magnetic current lines to describe the scattering response of moderately thin cylinders. We show that the derived polarizability tensor satisfies reciprocity and passivity relations, and analyze its response under different conditions, varying the excitation angle, material properties, and cylinder radius. Interestingly, magnetoelectric effects are shown to arise at oblique incidence, even in the case of centrosymmetric achiral thin cylinders, associated with a weak form of spatial dispersion. This finding is particularly relevant for the proper modeling of individual cylinders and arrays of them, as in the case of metamaterials. We expect this work to find applications in antenna and metamaterial design, and to improve the physical understanding of the wave interaction and spatial dispersion in artificial materials composed of elongated inclusions such as wire media.

Figure 1

The geometry of the problem and its reference system.

Figure 2

Schematics of linear dipoles' current distribution. Longitudinal and transverse dipole line sources, respectively: (a) ${\mathit{p}}_{\mathit{z}}$ and (b) ${\mathit{p}}_{\mathit{x}}$, with phased propagation $e^{i\beta z}$ along the line (snapshot in time).

Figure 3

A comparison with existing longitudinal electric polarizability models for homogeneous cylinders. We vary here the electrical size ${\mathit{k}}_0\mathit{a}$, the relative permittivity ${\mathit{\varepsilon }}_{\mathit{c}}$, and the incidence angle ψ, where normal incidence is π/2. In each panel, the black curves labeled ${\alpha }^{ee}$ refer to our proposed model, and the blue and green curves represent the dynamic axial electric polarizability and its long-wavelength approximation as defined in Ref. [8]: ${\alpha }^C$ and ${\alpha }^{Ca}$, respectively. All models converge in the long-wavelength limit, as expected. The plots show the real parts of the polarizabilities for loss free cylinders.

Figure 4

A comparison with existing transverse electric polarizability models for homogeneous cylinders. We vary here the electrical size ${\mathit{k}}_0\mathit{a}$, the relative permittivity ${\mathit{\varepsilon }}_{\mathit{c}}$, and the incidence angle ψ, where normal incidence is π/2. The polarizabilities include a static [8] as well as a quasistatic model derived for perpendicularly incident excitation [24] ${\alpha }^{Ca}$. In each panel, the black curves labeled ${\alpha }^{ee}$ refer to our proposed model, the blue lines to a preexisting static model, and the green lines to a quasistatic approximation. As expected, the quasistatic and proposed models converge approaching normal incidence, and all models meet in the long-wavelength limit. The plots show the real parts of the polarizabilities for loss free cylinders.

Figure 5

Dipole polarizability terms of dielectric cylinders with relative permittivity ${\mathit{\varepsilon }}_{\mathit{c}}=40$. The longitudinal polarizabilities ${\alpha }_z^{.}$ are compared in the left column, and the transverse ones ${\alpha }_t^{.}$ on the right. The magnetic components are shown in (a) and (b) at different angles of incidence. In a comparison of tensor elements in (c) and (d), it can be seen that all are important near resonances, although only the electric ones are significant for long wavelengths. Generally, the magnetic and magnetoelectric responses are stronger at oblique angles. The plots show the real parts of the polarizabilities for loss free cylinders.

Figure 6

Dipole polarizability for plasmonic cylinders. A Drude model was used to fit the real permittivity of silver between $9.4\mathrm{and}750\mathrm{TH}{\mathrm{z}}^{36}$ for all dipole polarizabilities. Even for plasmonic structures, small magnetic and magnetoelectric polarizabilities can exist. All curves show the real parts of the polarizabilities for loss free cylinders.

Figure 7

Effect of losses on the magnetoelectric polarizability response of a dielectric cylinder, where $tan\delta =\mathrm{Im}\left({\varepsilon }_c\right)/\mathrm{Re}\left({\varepsilon }_c\right)$, $\mathrm{Re}\left({\varepsilon }_c\right)=40,$ and $\mathit{\psi }=\pi /12$. The plot shows the real parts of the polarizability.