Resonant dynamics of arbitrarily shaped meta-atoms

Meta-atoms, nanoantennas, plasmonic particles, and other small scatterers are commonly modeled in terms of their modes. However these modal solutions are seldom determined explicitly, due to the conceptual and numerical difficulties in solving eigenvalue problems for open systems with strong radiative losses. Here these modes are directly calculated from Maxwell's equations expressed in integral operator form, by finding the complex frequencies which yield a homogeneous solution. This gives a clear physical interpretation of the modes, and enables their conduction or polarization current distribution to be calculated numerically for particles of arbitrary shape. By combining the modal current distribution with a scalar impedance function, simple yet accurate models of scatterers are constructed which describe their response to an arbitrary incident field over a broad bandwidth. These models generalize both equivalent-dipole and and equivalent-circuit models to finite-sized structures with multiple modes. They are applied here to explain the frequency-splitting for a pair of coupled split rings, and the accompanying change in radiative losses. The approach presented in this paper is made available in an open-source code.

Figure 1

Meta-atoms each showing charge (colors) and current (arrows) distributions for one of their modes. (a) Canonical spiral, (b) V antenna, (c) split-ring resonator, (d) sphere, (e) horseshoe, and (f) twisted crosses.

Figure 2

(a) Eigenvalue for the first mode of an SRR, having dimensions of impedance, and (b) its inverse which has dimensions of admittance, with a Lorentzian-like shape.

Figure 3

(a) Real and (b) imaginary parts of the first eigenvalue of an SRR, plotted as a function of complex frequency. The solid lines show where each part is zero. Their intersection gives the complex eigenfrequency.

Figure 4

(a) The complex singularities of a single split-ring resonator which give the resonant frequencies, and (b) the corresponding charge and current distributions.

Figure 5

Scalar admittance functions which describe the dynamics of the first four modes of a split-ring resonator.

Figure 6

The contribution of each mode to the (a) real and (b) imaginary parts of the complex extinction efficiency. The inset gives the polarization of the incident wave.

Figure 7

Total extinction efficiency calculated from the model and compared with the direct calculation, showing the (a) real and (b) imaginary parts.

Figure 8

The (a) capacitive and (b) inductive contributions to the mutual coupling between a pair of identical SRRs in broadside-coupled configuration.

Figure 9

The (a) real and (b) imaginary parts of the complex extinction spectrum for a broadside-coupled pair of SRRs, comparing the model with the directly calculated results. For reference, the extinction of a single SRR is also shown. The inset gives the polarization of the incident wave.

Figure 10

Improved accuracy of the model of a broadside-coupled SRR when including an additional mode in the calculation. Comparing the (a) real part and (b) imaginary part of the models against the direct calculation.