Dr David Powell

Nonlinear Physics Centre

The Australian National University

- In metamaterials and nano-photonics, the elements often have a resonant response
- Dynamics should be described by modes
- In periodic structures Bloch modes or homogenisation approaches are appropriate
- Many structures of interest are not simple periodic, so modes of individual scatterers are preferred
- Radiative losses can be very high - not a perturbation
- The system open resonator, which is a non-Hermitian system with complex resonant frequencies
- Standard electromagnetic eigenvalue solvers cannot solve such problems

Fields at Resonance

Alonso-Gonzalez et al., Nano Lett., 11 3922- From numerics or experiment
- Need to separate overlapping modes

Dipole models

Sersic et al, Phys. Rev. B, 83 245102- Works well in far-field
- Need many multipoles in near-field

Equivalent circuit

Bilotti et al.,IEEE Trans. MTT 55 2865- Construct manually for each structure
- Coupling to radiation not explicit

In quantum mechanics, several approaches exist for open systems:

- Modes of the universe
- Solve the whole system, including the resonator
- Yields continuous eigenvalues instead of discrete modes

- System and bath
- Arbitrarily divide system into internal and external parts
- Solve separately and introduce coupling terms
- Requires a sharp boundary between the near-field and far-field regions

- Quasi-normal modes
- Find the field solutions which satisfy homogeneous conditions
- These field solutions diverge as $x\rightarrow\infty$, making them very inconvenient

These limitations can be overcome by expressing Maxwell's equations for the structure in integral equation form, and finding the operator's singularities.

Singularities of these operators were first used for solving radar scattering problems

Marin, Electromagnetics, 1 361 (1981)

Very recently they were considered for plasmonic structures

Mäkitalo, Phys Rev B., 89 165429 (2014)

In this work, it will be shown how to reliably find such singularities, and use them to construct **simple
yet highly accurate oscillator models** describing the full dynamics of the particles

- To construct a compact model which directly considers excitation on the scatterer, the electric field integral equation (EFIE) is used $$\mathbf{E}_{s}\left(\mathbf{r},s\right)=\iiint_{\Gamma}\overline{\overline{G}}_{0} (\mathbf{r}-\mathbf{r}',s)\cdot\mathbf{j}(\mathbf{r}',s)\mathrm{d^{3}\mathbf{r}}$$
- Current $\mathbf{j}$ can include both conduction current and dielectric polarization $\frac{\mathrm{d}P}{\mathrm{d}t}$
- $s=j\omega+\Omega$ - complex frequency (Laplace transform variable)
- $\overline{\overline{G}}_{0}$ - free space dyadic Green's function, $$\overline{\overline{G}}_{0}\left(\mathbf{r}\right)=\left[-s\mu\overline{\overline{I}}+\frac{1}{s\epsilon}\nabla\nabla\right]\frac{\exp(-\gamma|\mathbf{r}|)}{4\pi|\mathbf{r}|}$$
- By using a different Green's function, layered media or periodic systems can be modelled

- Surface currents on perfect conductors are considered here

(generalization to dielectric and plasmonic materials is also possible) - Current is expanded into basis functions $\mathrm{\mathbf{j}}\left(\mathbf{r}\right)=\sum_{n=1}^{N}I_{n}\mathbf{f}_{n}\left(\mathbf{r}\right)$.
- Incident field is weighted by the same functions $V_{n}=\iint_{T_{n}}\mathbf{f}_{n}\left(\mathbf{r}\right)\cdot\mathbf{E}_{i}\left(\mathbf{r}\right)\mathrm{d^{2}}\mathbf{r}$
- Result is a matrix equation relating the expansion coefficients $\mathrm{V}(s)=\mathrm{Z}(s)\cdot\mathrm{I}(s)$
- Impedance matrix $\mathrm{Z}$ is a discrete version of the EFIE $$\small Z_{mn}=\iint_{T_{m}}\iint_{T_{n}}\left(s\mu\mathbf{f}_{m}\!\left(\mathbf{r}\right)\!\cdot\!\mathbf{f}_{n}\!\left(\mathbf{r}'\right)+\frac{1}{s\varepsilon}\left[\nabla\!\cdot\!\mathbf{f}_{m}\!\left(\mathbf{r}\right)\right]\left[\nabla'\!\cdot\!\mathbf{f}_{n}\!\left(\mathbf{r}'\right)\right]\right)\frac{e^{-\gamma\left|\mathbf{r}-\mathbf{r'}\right|}}{4\pi\left|\mathbf{r}-\mathbf{r'}\right|}\mathrm{d^{2}}\mathbf{r}'\mathrm{d}^{2}\mathbf{r}$$
- Impedance is not just for circuits - it can be related to local density of states

(Greffet et al., Phys. Rev. Lett. 105 117701) - From this fully numerical description, the simple model is extracted

- Technique to reliably find modes of open radiating structures (meta-atoms, antennas, oligomers)
- Modes form the basis of simple models, with broadband accuracy
- Applicable to almost any particle shape
- Results published in
- Phys. Rev. B 90 075108
- arXiv:1405.3759

- Method is implemented in an open-source code OpenModes, see http://www.pythonhosted.org/OpenModes/