Abstract
Accurate and efficient modeling of discontinuous, randomly distributed entities is a computationally challenging task, especially in the presence of large and inhomogeneous electric near-fields of plasmons. Simultaneously, the anisotropy of sensed entities and their overlap with inhomogeneous fields means that typical effective medium approaches may fail at describing their optical properties. Here, we extend the Maxwell Garnett mixing formula to overcome this limitation by introducing a gradient within the effective medium description of inhomogeneous nanoparticle layers. The effective medium layer is divided into slices with a varying volume fraction of the inclusions and, consequently, a spatially varying effective permittivity. This preserves the interplay between an anisotropic particle distribution and an inhomogeneous electric field and enables more accurate predictions than with a single effective layer. We demonstrate the usefulness of the gradient effective medium in FDTD modeling of indirect plasmonic sensing of nanoparticle sintering. First of all, it yields accurate results significantly faster than with explicitly modeled nanoparticles. Moreover, by employing the gradient effective medium approach, we prove that the detected signal is proportional to not only the nanoparticle size but also its size dispersion and potentially shape. This implies that the simple volume fraction parameter is insufficient to properly homogenize these types of nanoparticle layers and that in order to quantify optically the state of the layer more than one independent measurement should be carried out. These findings extend beyond nanoparticle sintering and could be useful in analysis of average signals in both plasmonic and dielectric systems to unveil dynamic changes in exosomes or polymer brushes, phase changes of nanoparticles, or quantifying light absorption in plasmon assisted catalysis.