Description of Dipole Moment Coupling Depending on Time Using Catastrophe Theory in Metallic Nanoparticles

Abstract

We describe the non-harmonic dipole moment resulting of the interaction between a set of nanoparticles randomly distributed with a linear polarized plane wave. The study is performed by considering its structural parameters as function of time establishing a mechanical analogy with a spring less system. The simplest case occurs for two nanoparticles and the Mathieu differential equation describes the dipole interaction, from which interesting resonance effects are predicted. We show that an effective refractive index depending of time is obtained having the possibility to take negative values which offers interesting applications in the development of metamaterials. Finally associating a catastrophe function for the phase function, during the emission process we are able to describe the width of the frequency spectrum. The theoretical are well agree with experimental results previously reported in the optical literature.