From Mie to Fresnel through effective medium approximation with multipole contributions

The Mie theory gives the exact solution to scattering from spherical particles while the Fresnel theory provides the solution to optical behavior of multilayer thin film structures. Often, the bridge between the two theories to explain the behavior of materials such as nanoparticles in a host dielectric matrix, is done by effective medium approximation (EMA) models which exclusively rely on the dipolar response of the scattering objects. Here, we present a way to capture multipole effects using EMA. The effective complex dielectric function of the composite is derived using the Clausius–Mossotti relation and the multipole coefficients of the approximate Mie theory. The optical density (OD) of the dielectric slab is then calculated using the Fresnel approach. We have applied the resulting equation to predict the particle size dependent dipole and quadrupole behavior for spherical Ag nanoparticles embedded in glass matrix. This dielectric function contains the relevant properties of EMA and at the same time predicts the multipole contributions present in the single particle Mie model.