Designing plasmonic systems using optical coupling between nanoparticles

A theory of the coupling of evanescent optical fields between metallic nanoparticles is developed to provide a basis for designing plasmonic systems. The interaction between metallic nanoparticles is investigated using an electrostatic approximation that describes the localized surface-plasmon resonances for particles much smaller than the wavelength of the exciting radiation. A coupling coefficient is derived relating the surface charge on one particle to the induced surface dipoles on another. The effect of dipole radiation damping is included to model the radiation broadening of the plasmon resonances. The theory enables an analysis of the key factors that control the coupling and that lead to resonances of the particle system. It is shown that the coupling between a simple pair of particles, in the form of stripes, leads to frequency splitting and the formation of a dark mode. The dark mode is associated with little scattering of light but large evanescent electric fields. It is shown that the dark mode has low radiation damping compared to an associated bright mode. The theory can be applied to an arbitrary number of interacting particles, allowing them to be configured to achieve the desired properties of the plasmonic system.

Figure 1

(Color online) This shows the triangle tessellation of the two numerical stripes used to model the coupling. The color (online version) represents the surface dipole distribution associated with the first resonance of each individual stripe, where blue is positive and red is negative.

Figure 2

(Color online) The permittivity associated with the resonance of the two-particle (stripes) system, based on Eq. (45) as a function of the separation of the stripes. The permittivities calculated from the eigenvalues of the coupled system are shown for the low-frequency (circles) and high-frequency (triangles) modes.

Figure 3

(Color online) The scattering cross section (arbitrary units) for the two stripe system for four different separations. The stripes were taken to be gold embedded in PMMA.

Figure 4

(Color online) (a) The real and imaginary parts of the component of the electric field parallel to the long axes of the stripes. The field is calculated at a point 10 units from the long axis of the longer stripe. The imaginary component indicates the wavelength where the coupled stripes resonate. Note the significant electric fields at 1010 nm even though the scattering cross section at this wavelength is small. The stripes were taken to be gold embedded in PMMA. (b) The surface dipole distributions for two modes when the stripes are far apart (400) and close (100): the arrow indicates the point where the electric field is calculated.

Figure 5

(Color online) The effect of radiation damping on the scattering cross section of a pair of stripes (gold embedded in PMMA) and on the electric field component 10 units beyond the long axis of the longer stripe. The scales refer to the length of the longer stripe. The radiation damping varies with the size scale whereas the unscaled stripe has no radiation damping.

Figure 6

(Color online) The asymmetry in coupling coefficients associated with identical asymmetric particles. (a) the triangular prisms showing the surface charge $\sigma$ and surface dipole $\tau$ eigenmodes, the coupling coefficients, and the “direction” of the coupling. (b) a graph of the coupling ratio vs separation between the prisms, demonstrating the asymmetry.