Collective modes of interacting dielectric spheres

We present the theory of the collective modes of arrays of dielectric spheres, regarded here as each endowed with a frequency-dependent dielectric constant epsilon(omega). Our interest resides in arrays of small spheres, wherein the length scales are such that an electrostatic treatment suffices to describe the response of the individual spheres and also the interactions between them. The treatment thus applies to collections of nanoparticles. The formulation is in real space, so its general structure does not require the array to be periodic. We illustrate the nature of the collective modes with application to linear arrays of metallic spheres, similar to those grown recently on stepped surfaces. Particularly striking is a mode whose frequency softens greatly at long wavelength, in the limit the spheres just touch. Also, we discuss their response to applied electric fields whose wavelength is long compared to the length scales of the linear array.