Modes in open metamaterial and nanophotonic systems

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

Modelling meta-atoms and nano-particles

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

The physics of open resonators

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

Electric Field Integral Equation

  • 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

Impedance Matrix

  • 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

Eigenvalue decomposition

  • Eigenproblem for impedance matrix $$\mathrm{Z}(s)\cdot\mathrm{I}^{(\alpha)}\left(s\right)=z^{(\alpha)}\left(s\right)\mathrm{G}\cdot\mathrm{I}^{(\alpha)}\left(s\right)$$
  • $\mathrm{I}^{(\alpha)}\left(s\right)$ - eigenvector, currents of mode
  • $z^{(\alpha)}(s)$ - eigenvalue, scalar impedance
  • Gram matrix $\mathrm{G}$ ensures physically correct scaling
  • After normalising $\mathrm{I}^{(\alpha)}\left(s\right)$, decompose $\mathrm{Z}$: $$\mathrm{Z}(s)=\sum_{\alpha=1}^{N}z^{(\alpha)}\left(s\right)\mathrm{I^{(\alpha)}}\left(s\right)\otimes\mathrm{I}^{(\alpha)}\left(s\right)$$
  • Required modes $N$ can be small (or 1)
  • Eigenvalues (impedance) of first mode of single ring SRR
  • The inverse quantity (admittance) clearly shows the resonance
  • Also demonstrated for plasmonic dolmen in Zheng, et al., Optics Express, 21 31105

Analytic Extension

  • Eigenvalue expansion gives intuitive model
  • Need to calculate and factor $\mathrm{Z}$ at each frequency - computationally expensive like full numerics
  • More efficient procedure is possible
  • Consider eigenvalues $z^{(\alpha)}$ for complex freq $s$
  • At resonant frequency $z^{(\alpha)}\left(s^{\left(\alpha\right)}\right)=0$, current sustained with no incident field
  • Finding singularities is a nonlinear eigenvalue problem since $\mathrm{Z}$ is function of $s$
  • Automated initial guess allows iterative search to find $s^{\left(\alpha\right)}$ to accuracy $10^{-8}$ within 10 iterations


  • Increased decay rate for higher modes - higher radiation losses
  • Impedance is a real function in the time-domain, so singularities come in complex-conjugate pairs
  • For each mode real part of current and charge shown
  • Increasing oscillation with mode order just like closed cavity
  • Imaginary charge and current much smaller

Fitted Models

  • Evaluate $\mathrm{d}z^{(\alpha)}/\mathrm{d}s$ at singularity to fit model to scalar impedance $z^{(\alpha)}$ $$\small z^{(\alpha)}\left(s\right)=\cssId{capacitance_term}{\frac{z_{-1}^{(\alpha)}}{s}}+z_{0}^{(\alpha)}+z_{1}^{(\alpha)}s+z_{2}^{(\alpha)}s^{2}$$
  • Interpret terms as capacitance, ohmic resistance, inductance and radiation resistance
  • Inverse $1/z^{(\alpha)}\left(s\right)$ is admittance with Lorentzian line shape
  • Broader peaks for higher modes, corresponding to damping rate $\Omega^{\left(\alpha\right)}$


  • Extinction characterises total work done by incident field $$Q_{ext}\propto\iint_{\partial\Gamma}\mathbf{\overline{E}}_{i}\left(\mathbf{r}\right)\cdot\mathbf{j}\left(\mathbf{r}\right)\mathrm{d^{2}}\mathbf{r}$$
  • Analogous to complex power in circuit theory
  • Real part is power delivered to scattering + absorption (zero for PEC)
  • Imaginary part is reactive energy stored in near field. Important when energy is to be extracted from an emitter
  • Proportional to eigenvalue (impedance of mode), scaled by projection of $\mathbf{E}_{i}$ onto mode
  • Fourth mode has quadrupolar distribution, not excited at normal incidence

Comparison with direct calculation

  • Compare between direct calculation and sum of mode contributions
  • Very accurate over broad bandwidth - well beyond homogenisation or quasi-static limit
  • Once model is extracted, calculations for different frequencies or incident fields have trivial computational cost
  • Other quantities can be derived from the model
    • near-field excitation by local source (density of states, Purcell factor etc)
    • far-field radiation pattern and antenna gain
    • absorption cross section if losses are included
    • coupling between meta-atoms

Coupling open resonators

  • Lagrangian model of coupling between plasmonic or metamaterial elements works well in the quasi-static limit, but fails when retardation is significant
  • The impedance matrix from the EFIE naturally describes all these effects. The modal approach is extended to account for mutual impedance of modes of different rings
  • This gives the frequency-dependent coupling coefficients for arbitrary separation
  • The extinction of a single SRR is shown
  • The model shows the splitting of modes due to coupling
  • The change in line-width due to enhancement/suppression of radiation is automatically given
  • Extra modes of each ring can easily be included for increased accuracy


  • 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