A simple tutorial on designing coupled plasmonic systems.

To describe the coupling between metal particles supporting localized surface plasmon (LSP) resonances we need to know a few things. Although the math looks messy it is really quite straightforward. Firstly, the LSPs represent the set of resonant electrical modes, or standing waves, on the surface of the metal particle. Not all modes can be excited by a given light field and some can be excited more strongly than others. We approximate the amplitude of the excitation of a mode by eq. (1). Since there can be many modes and many particles, we use a superscript k to represent the k-th mode and a subscript m for the m-th particle. The excitation amplitude depends on: the dipole moment p of the mode; the vector amplitude of the electric field of the incident light E; the resonant frequency of the mode; and the applied frequency (omega). Strictly we should use complex numbers and include a loss term (Gamma). If you want, you can substitute this in at the end by replacing the resonance frequency as shown in eq. (1). A is a constant I usually ignore.

When two or more particles come close to one another, the electric fields from the LSP interact and induce new surface charges. This changes the resonances, both in frequency and phase. The coupling is predominantly electrical in nature and can be described by Coulomb's law, if we know the surface charge distributions associated with the LSPs. For most LSPs the dominant mode has a strong dipole character so that the electric field is the same as that from a simple dipole. This gives rise to eq. (2) for the "geometric coupling".This depends on the dipole moments of the modes on particles n and m as well as the unit vector d pointing from m to n (or vice versa).

Two coupled particles showing the electric fields that change the excitations of the LSPs.

The coupling of particle m to particle n depends on how close the applied frequency is to the resonance of particle n. Combining the geometric coupling with the frequency dependence gives the total coupling C, shown in eq. (3).

And finally, we need to know how the excitation amplitudes are affected by the coupling. It turns out that this is given by a matrix equation, eq. (4). This represents the amplitude of a particle in the coupled system (represented by a tilde) in terms of the amplitudes of the particles when they are isolated. This is very useful, because the amplitudes of the isolated particles are given by the simple formula eq. (1). Although the equation looks complicated, it is relatively simple. The subscripts (n and m) represent the different particles, and the superscripts (k and j) represent the different resonant modes. Often only one mode dominates so we can drop the mode indices.

To see how all this is put together to model a coupled plasmonic system, go to the next page ...

For a more in-depth tutorial, have a look at my book chapter "Evanescent Coupling between Resonant Plasmonic Nanoparticles and the Design of Nanoparticle Systems" In: Plasmons: Theory and Applications ISBN: 978-1-61761-306-7 Editor: Kristina N. Helsey, pp.111-141 © 2011 Nova Science Publishers, Inc. This chapter is free to download.

Copyright Tim Davis 2012