Figure 1

Split ring radiation patterns corresponding to the polarizability tensor eigenvectors. (a) (Sketch) A single split ring resonator can have an electric dipole moment $\mathit{p}$ along the $x$ axis due to charging of the split. Circulating current $j$ in the ring gives rise to a magnetic dipole moment $\mathit{m}$ in the $z$ direction. (b),(c) Radiation patterns of the two eigenmodes of an SRR in the case of no off-diagonal magnetoelectric coupling (${\eta }_E=0.7,{\eta }_H=0.3,{\eta }_C=0$). The electric dipole moment oriented along the $x$ axis radiates most of its amplitude in the $\mathit{ky},\mathit{kz}$ plane, while the magnetic dipole oriented along the $z$ axis radiates mostly into the $\mathit{kx},\mathit{ky}$ plane. (d),(e) Radiation patterns of the eigenvectors with magnetoelectric cross coupling (${\eta }_C=0.4$). (f) Indication of the polarization of the light radiated by the eigenvector with a largest eigenvalue [(c)]. Light is linearly polarized for wave vectors along the Cartesian axes, but elliptically polarized in general. The direction of strongest circular dichroism in extinction and scattering is in the $\mathit{xz}$ plane.Reuse & Permissions

Figure 2

We calculate the extinction cross section ${\sigma }_{\mathrm{ext}}$ as a function of illumination angle and polarization. In panel (a), we sweep the incident wave vector over a total angular range of ${180}^{°}$, over a trajectory [see panel (a), top] starting from the $z$ axis (labeled $0^{°}$) to the $-y$ axis (labeled ${90}^{°}$), and then through the $\mathit{xy}$ plane to the $x$ axis (full ${180}^{°}$). The red dashed line represents ${\sigma }_{\mathrm{ext}}$ for linearly polarized incident illumination, where in (a) the polarization is kept in the $\mathit{xy}$ plane as shown (i.e., polarization is along $x$ for the first ${90}^{°}$ of the trajectory, rotating continuously to polarization vector along $y$ in the second part of the trajectory). The blue solid curve represents extinction for right-handed circularly polarized illumination, which for the incidence conditions in (a) is half the extinction seen for linearly polarized light. For the trajectory of incident wave vectors in panel (a), ${\sigma }_{\mathrm{ext}}$ at normal incidence to the split ring is a measure for only ${\alpha }_{\mathit{EE}}$ as $E_x$ is the only driving field. Increasing the angle to ${90}^{°}$, both polarizations $E_x$ and $H_z$ excite the dipoles in the SRR, so ${\sigma }_{\mathrm{ext}}$ is a measure for the sum of the terms on the diagonal of the polarizability tensor $({\alpha }_{\mathit{EE}}+{\alpha }_{\mathit{HH}})$. Changing the incidence condition to 180${}^{ˆ}$ removes the $E_x$ component of the field, leaving only $H_z$ of the incident light to drive the SRR. Hence ${\sigma }_{\mathrm{ext}}$ at 180${}^{ˆ}$ in panel (a) is a measure for purely ${\alpha }_{\mathit{HH}}$. In panel (b) we sweep the incident wave vector from the $-x$ axis (labeled $-{90}^{°}$) via the $z$ axis (again labeled 0${}^{ˆ}$) to the $+x$ axis (labeled 90${}^{ˆ}$). The color- and linetype coding is as in panel (a). However, now we constrain the linear polarization to have no $y$ component. Panel (b) ${\sigma }_{\mathrm{ext}}$ shows that as a function of the incident angle in the $\mathit{xz}$ plane for right-handed circular polarization, minima and maxima in ${\sigma }_{\mathrm{ext}}$ occur as a function of angle, which are a measure for the eigenpolarizabilities ${\alpha }_{-}$ and ${\alpha }_{+}$, respectively. Both sets of measurements in panels (a) and (b) together provide information on all the components of the polarizability tensor, ${\alpha }_{\mathit{EE}}$, ${\alpha }_{\mathit{HH}}$, and ${\alpha }_{\mathit{EH}}$.Reuse & Permissions

Figure 3

Extinction cross sections ${\sigma }_{\mathrm{ext}}$ vs frequency and twist angle for an SRR stereodimer structure. Panel (a) shows the geometry (top view and side view) in which two SRR’s are vertically stacked. The upper SRR is rotated around the $z$ axis by the twist angle $\psi$. We calculate extinction for light impinging from the $z$ direction with polarization along $x$, i.e., along the base of the lower SRR in (b), (c), (e), and (f). In (d) and (g), we use ${45}^{°}$ incidence in the $\mathit{xz}$ plane, so that the $\mathit{H}$ field of the excitation light directly couples also to the magnetic polarizability. Panels (b), (c), and (d) show extinction assuming no cross-coupling term (${\eta }_C=0$), while (e), (f), and (g) show extinction assuming strong magnetoelectric coupling (${\eta }_C=0.4$). Panels (b) and (e) assume the damping rate of gold $\gamma =1.25\times {10}^{14}$ ${\mathrm{s}}^{-1}$. To more clearly bring out the four mode structure, we reduce the damping tenfold for the calculations in (c), (d), (f), and (g). There are four modes present in the system. White lines in (b) and (e) indicate the frequencies of the modes, as taken from the resonances in the low-damping case, i.e., the resonances in panels (d) and (g).Reuse & Permissions