Forward and Backward Extinction Measurements on a Single Supported Nanoparticle: Implications of the Generalized Optical Theorem

In most conventional optical experiments on substrate-supported nanoparticles, detectors are located in the far-field space whereas their intrinsic properties (scattering, absorption cross sections) are commonly computed in the near-field by using numerical approaches. In this respect, a connection between experiment and theory can be usefully made on the basis of the generalized optical theorem (GOT) that gives access, from the far-field expansion of the scattered wave, to the total extinction cross section as the sum of “forward” and “backward” contributions related to the power changes of the waves respectively transmitted and reflected by the substrate. The purpose of this work is to address this issue as clearly as possible by performing quantitative measurements on individual objects and on both type of waves, as far a comprehensive description of the global extinction process is intended. We show that the spatial modulation spectroscopy technique applied to a single nanoparticle is especially well suited in this context. This is illustrated by the model case of gold nanocubes supported either on a thin dielectric film or on a thick glass slide. Using the GOT in the particular case of a plane wave excitation at normal incidence, we were able to access the total extinction cross section of the supported scatterer. Moreover, we give evidence for the invariance of the extinction cross section relative to the transmitted wave, whether the nanoparticle is located in the front or behind the substrate with respect to the incoming beam. This effect can be seen as a manifestation of the optical reciprocity theorem with regard to this specific problem. We also discuss in which framework the experimental measurements can be pertinently reproduced by theoretical calculations, considering the setup geometry and the intrinsic optical response of the nanoparticle.