Modes and resonances of plasmonic scatterers

We present a rigorous full-wave electromagnetic approach to analyze the modes and resonances of dielectric and plasmonic nanoparticles of practically any geometry. Using boundary integral operators, we identify the resonances as inherent properties of the particles and propose a modal expansion for their optical response. We show that the resonance frequencies are isolated points on the complex plane. The approach allows the particles to be analyzed without specifying an incident field, which can be separately tailored for the desired interaction with the modes. We also connect the general theory to the Mie theory in spherical geometry and provide a connection to the quasistatic theory. In comparison to earlier work on modes and resonances of scatterers, our approach has the benefit that modes are defined entirely over a compact boundary surface of the scatterer. Furthermore, the boundary integral operator is of second-kind Fredholm type, enabling the rigorous characterization of the resonances.

Figure 1

Prototype geometry and parameters of the scattering problem.

Figure 2

Illustration of the singularities of inverse boundary integral operator for Drude-type dispersion.

Figure 3

Plot of the function ${log}_{10}|\mathrm{I}+\mathrm{A}\left(\omega \right)|$ at complex frequencies for a gold sphere with radius 200 nm. The first three TM modes with $l=1,2,3$ are considered.

Figure 4

Extinction cross section (solid) and field enhancement (dashed) of a gold sphere with radius 200 nm for a linearly polarized incident plane wave. TM contributions are plotted separately from the full-wave case.

Figure 5

Eigenvalues of $\mathrm{I}+\mathrm{A}$ at frequency 1.7 PHz for a gold sphere of radius 200 nm. Dots are calculated by method of moments and circles by analytic theory.

Figure 6

Resonance frequencies in the complex frequency plane. Numbers and colors indicate the varied parameter of particular structure. Dotted lines illustrate the trends in resonance shifts. The degenerate resonance of the disk clearly splits into branches for the bar and dimer. Circles correspond to a down-scaled disk that approaches the quasistatic limit, whence the resonances approach the branch cut. The encircled marks correspond to higher-order resonances.

Figure 7

Mode charge densities and radiation patterns at fixed real frequency 3.14 PHz for the gold nanodisk of diameter 120 nm. The charge densities correspond to a time instant of harmonic oscillation and are normalized to unity.

Figure 8

Mode charge densities and radiation patterns at fixed real frequency 3.14 PHz for the gold nanodisk dimer of gap length 20 nm. The color scale is that of Fig. 7.

Figure 9

Mode charge densities and radiation patterns at fixed real frequency 2.38 PHz for the gold nanobar dimer of length 200 nm. The color scale is that of Fig. 7.

Figure 10

Diagram of the strategy towards the definition of modes and resonances in scattering problems by boundary integral operators.